CFD-Wiki - User contributions [en]

Automotive

2012-08-06T15:31:03Z

Jola: Removed link to deleted page

==Introduction==
An Automobile is a combination of several industrial technologies in a miniaturised form. CFD can be applied in different areas and would then have to be approached as per the characteristics of those physics.

The fundamental concepts of the Aerodynamics of an automobile can be found in several excellent books.

A few are listed here :

1. Aerodynamics of Road Vehicles -Hucho

2. Aerodynamics of Race Cars - J.Katz

CFD can be applied in situations pertaining to the external car aerodynamics, climate control, engine cooling, combustion in engines, exhaust flow and design

FEM or FDM technique can be used to find out the response of a temperature sensor mounted on exhaust gas manifold.

FEM techniques are widely used to check whether a sensor mounted would withstand the engine vibration and its life

==Sections Based on Application : ==

[[Exhaust System Analysis]]

[[IC Engine applications]]

==External Links==

*Autosim - http://www.autosim.org - a collaborative project, sponsored by NAFEMS, to look at the use of simulation in the automotive industry, including CFD.

{{stub}}

CFD-Wiki:Donated texts

2012-08-06T11:43:57Z

Jola: Removed link to deleted file

Please list all donated documents on this page. For more information on how to donate a text see [[CFD-Wiki:Donate texts|this page]].

===Donated texts:===

''Add more donated documents here''
*[[Media:Turbulence_Modeling_For_Beginners.pdf‎ | Turbulence Modeling For Beginners]]
:Author: Tony Saad
:Date written: Apr 2001
:Date Reviewed: Dec 2005
:Date donated: Dec 2005
:Summary: An introduction to turbulence modeling.

*[[Media:Old_fluent_faq_by_burley.zip | Old Fluent FAQ]]
:Author: Burley Wang
:Date written: Dec 2000
:Date donated: Nov 2005
:Summary: This is an old Fluent FAQ written by Burley Wang for CFD Online back in 2000. The FAQ was never finished and it was never published either. It is based on questions and answers found on the Fluent forum. Use it to cut-and-paste relevant things from into the [[Fluent FAQ]].

*[[Media:PhD_Thesis_Jonas_Larsson.tar.gz | Numerical Simulation of Turbulent Flows for Turbine Blade Heat Transfer Applications]]
:Author: Jonas Larsson
:Date written: Dec 1998
:Date donated: Nov 2005
:Summary: The first part of Jonas Larsson's PhD thesis from Chalmers University, Gothenburg. The thesis excludes papers which are copyrighted by ASME etc. It covers some basics on governing equations, turbine blade heat transfer and descriptions of several low-Re two-equation models (Chien model, Launder-Sharma, Nagano-Tagawa, Shih, ...) The report also includes various model improvements like the Kato-Launder modification, the Yap correction etc.

*[[Media:DEA_P6_09_2000.zip | Validation of an RSM closure on airfoil geometries (in French)]]
:Author: Ioannis Nousis
:Date written: Sep 2000
:Date donated: Jan 2006
:Summary: The MSc thesis of Ioannis (Yannis) Nousis at the Université Pierre et Marie Curie. The work consists in validating a Reynolds stress turbulence closure on airfoil geometries for transonic compressible viscous flow regimes. Prediction of airfoil aerodynamic performance for several test-cases (NACA 0012 - RAE 2822). Validation of the turbulence model against wind tunnel experiments. A digitized experimental data base is included in the Annexes. Key words: transonic viscous flows, numerical simulation, external aerodynamics, shoc wave-boundary layer interaction, biharmonic structured grid.

*[[Media:Nina_Shokina_phd_thesis.pdf | Numerical Modelling of Multi-dimensional Steady Ideal Gas and Fluid Flows ]]
:Author: Nina Shokina
:Date written: May 2000
:Date donated: Mar 2006
:Summary: The PhD thesis of Nina Shokina (Technical University of Darmstadt, Germany). The work is devoted to the numerical modelling of multi-dimensional steady ideal gas and fluid flows using adaptive grids.

*[[Media:Rui_Thesis.pdf‎ | Numerical Simulation of the Filling and Curing Stages in Reaction Injection Moulding, using CFX ]]
:Author: Rui Igreja
:Date written: June 2007
:Date donated: July 2007
:Summary: Rui Igreja's master thesis at the University of Aveiro, Portugal.<br> Commonly used methods for injection moulding simulation involve a considerable number of simplifications, leading to a significant reduction of the computational effort but, in some cases also to limitations. In this work, Reaction Injection Moulding (RIM) simulations are performed with a minimum of simplifications, by using the general purpose CFD software package CFX, designed for numerical simulation of fluid flow and heat and mass transfer.<br>The CFX’s homogeneous multiphase flow model, which is generally considered to be the appropriate choice for modelling free surface flows where the phases are completely stratified and the interface is well defined, is shown to be unable to model the filling process correctly. This problem is overcome through the implementation of the inhomogeneous model in combination with the free-slip boundary condition for the air phase.<br>The cure reaction is implemented in the code as a transport equation for an additional scalar variable, with a source term. Various transient and advection schemes are tested to determine which ones produce the most accurate results.<br>Finally, the mass conservation, momentum, cure and energy equations are implemented all together to simulate the simultaneous filling and curing processes present in the RIM process. The obtained numerical results show a good global accuracy when compared with other available numerical and experimental results, though considerably long computation times are required to perform the simulations.<br>( <i>A better quality version (6.1MB .pdf) is available from:</i><b> http://sites.google.com/site/ruiigreja2/MyMasterThesis</b> )

User talk:Jola

2010-05-03T15:50:24Z

Jola:

Hello Dear Jonas!

Glad to see You again at CFD-Wiki!

I went out from the hospital, solved some of my life problems and now I would like to continue my work in CFD. I'll continue to fill the turbulence section and would like to fit my SIMPLE-algorithm code (for solving lid-driven cavity flow test) into the Wiki as I did in approximation schemes section. I think that it will be good for all newcomers.

My congratulations with Your appointment in Chalmers! May I ask You a favor? If You remember, I applied to Chalmers in 2005. But my application was rejected because Lars Davidson said that I already knew enough. May be he is right, but I wish to get MSc degree in CFD. You know me, and may I ask You to speak to Lars Davidson about myself? I mean to persuade him to accept me on CFD MSc course in Chalmers. Not now, I still can't afford to study abroad (besides I still have difficulties with my juridical status in EU) but I hope that it occurs in the nearest future. I'll be very grateful to You for Your help.

Best wishes, sincerely Yours, Michail

Hi Michail,

Nice to see you back on CFD Wiki again. Many users have benefited from your previous additions to CFD Wiki.

Cheers'

--[[User:Jola|Jola]] 15:50, 3 May 2010 (UTC)

Best practice guidelines for turbomachinery CFD

2010-02-19T19:48:56Z

Jola: Reverted edits by EvrSIW (Talk) to last version by DavidF

This article contains a summary of the most important knowledge and experience a CFD engineer needs in order to perform CFD simulations of turbomachinery components. The guide is mainly aimed at axial turbomachinery. The goal is to give a CFD engineer, who has just started working with turbomachinery simulations, a head start and avoid some of the most difficult pit-falls. Experienced turbomachinery CFD engineers can also use the guidelines in order to learn what other experts consider best practice. The intended audience is expected to know basic CFD terminology and have a basic turbomachinery knowledge, but no detailed knowledge about CFD for turbomachinery is needed.

== Deciding what type of simulation to do ==

Before starting a new turbomachinery simulation it is wise to think carefully of what it is that should be predicted and what physical phenomena that affect the results. This chapter contains a brief overview of the various types of simulations and some hints of what can be predicted with them.

=== 2D, Quasi-3D or 3D ===

2D simulations are often used in the early design phase in order to obtain a typical 2D section of a blade. For cases with many long blades or vanes, like low-pressure turbines, a 2D simulation can also provide reasonable results. If the area of the flow-path changes significantly in the axial direction it might be necessary to instead make a quasi-3D simulation. A quasi-3D simulation is a 2D simulation in which extra source terms are used to account for the acceleration/deceleration caused by a changing channel height or growing end-wall boundary layers. Codes focused on turbomachinery applications often have the possibility to perform quasi-3D simulations, but most general purpose CFD codes can not do this type of simulations, or require user coding to implement the correct source terms in the equations. Please contact your software distributor if your code does not have the quasi-3d possibility and you require it. Many codes require special routines or hidden commands to enable this feature.

Full 3D simulations are necessary if a true 3D geometry is needed to obtain correct secondary flows and/or shock locations. For low-aspect-ratio cases with only a few short blades, like for example structurally loaded turbine outlet guide vanes, the secondary flow development is important and a 3D simulation is often necessary in order to obtain reasonable results. For applications where the end-wall boundary layers grow very quickly and interact with a large part of the flow-field it is necessary to perform a full 3D simulation. This is often the case in compressors and fans, where the negative pressure gradients make the boundary layers grow much quicker than what they do in for example turbines. For cases where the shock location is very critical, like in transonic compressors, it is also often necessary to perform a 3D simulation in order to obtain reasonable shock locations.

=== Inviscid or viscid ===

For attached flows close to the design point and without any large separations it is often sufficient with an inviscid Euler simulation in order to obtain reasonable blade loadings and pressure distributions. Note that inviscid Euler simulations should only be used if the boundary layers are judged to not have a significant effect on the global flow-field. A viscid Navier-Stokes simulation is necessary in order to predict losses, secondary flows and separations. As soon as separations are of interest it is of course also necessary to do a viscid simulation. Note that with todays computers it is often not time and resources that make users run inviscid Euler simulations. Running viscid Navier-Stokes simulations is now so quick that it is not a time problem any more. Euler simulations are still interesting though, since with an inviscid Euler simulation you don't have to worry about wall resolutions, y+ values, turbulence modeling errors etc.

=== Transient or Stationary ===

Most turbomachinery simulations are performed as stationary simulations. Transient simulations are done when some kind of transient flow behaviour has a strong influence on the global flow field. Examples of transient simulations are detailed simulations of rotor-stator interaction effects, simulations of large unsteady separations etc. Sometimes when you perform a steady stationary simulation you can see tendencies of unsteady behaviour like for example periodic vortex shedding behind blunt trailing edges. This is often first seen as periodical variations of the residuals. If the unsteady tendencies are judged to not affect the overall simulation results it might be necessary to coarsen the mesh close to the vortex shedding or run a different turbulence model in order to make the simulation converge. Sometimes you are still forced to run a transient simulation and average the results if you don't obtain a converged steady solution.

== Meshing ==

In turbomachinery applications structured multi-block hexahedral meshes are most often used for flow-path simulations. In most solvers a structured grid requires less memory, provides superior accuracy and allows a better boundary-layer resolution than an unstructured grid. By having cells with a large aspect ratio around sharp leading and trailing edges a structured grid also provides a better resolution of these areas. Many companies have automatic meshing tools that automatically mesh blade sections with a structured mesh without much user intervention.

Unstructured meshes are used for more complex and odd geometries where a structured mesh is difficult to create. Typical examples where unstructured meshes are often used are blade tip regions, areas involving leakage flows and secondary air systems, film cooling ducts etc.

When meshing avoid to create large jumps in cell sizes. Typically the cell size should not change with more than a factor of 1.25 between neighbouring cells. For structured meshes also try to create fairly continous mesh lines and avoid discontinuities where the cell directions suddenly change. For multi-block structured meshes avoid placing the singular points where blocks meet in regions with strong flow gradients since most schemes have a lower accuracy in these singular points.

=== Mesh size guidelines ===

It is difficult to define, a priori, the mesh size. The required mesh size depends on the purpose of the simulation.

If the main goal is to obtain static pressure forces a coarse mesh is often able to obtain a good solution, especially when an accurate resolution of the boundary layers is not required. For 2D inviscid simulations of one blade a mesh with say 3,000 cells is most often sufficient. For 3D inviscid blade simulations a mesh size of about 40,000 cells is usually sufficient. On inviscid Euler simulations the cells should be fairly equal in size and no boundary layer resolution should be present. Avoid having too skewed cells.

For loss predictions and cases where boundary layer development and separation is important the mesh needs to have a boundary layer resolution. The boundary layer resolution can either be coarse and suitable for a [[Wall functions|wall function]] simulation or very fine and suitable for a [[Low-Re resolved boundary layers|low-Re]] simulation. For further information about selecting the near-wall turbulence model please see the [[#Turbulence modeling|turbulence modeling]] section. In 3D single-blade simulations a decent wall-function mesh typically has around 100,000 cells. This type of mesh size is suitable for quick design iterations where it is not essential to resolve all secondary flows and vortices. A good 3D wall-function mesh of a blade section intended to resolve secondary flows well should have at least 400,000 cells . A good low-Re mesh with resolved boundary layers typically has around 1,000,000 cells.

In 2D blade simulations a good wall-function mesh has around 20,000 cells and a good low-Re mesh with resolved boundary layers has around 50,000 cells.

Along the suction and pressure surfaces it is a good use about 100 cells in the streamwise direction. In the radial direction a good first approach is to use something like 30 cells for a wall-function mesh and 100 cells for a low-Re mesh.

It is important to resolve leading and trailing edges well. Typically at least 10 cells, preferably 20 should be used around the leading and trailing edges. For very blunt and large leading edges, like those commonly found on HP turbine blades, 30 or more cells can be necessary.

Cases which are difficult to converge with a steady simulation and which show tendencies of periodic vortex shedding from the trailing edge, can sometimes be "tamed" by using a coarse mesh around the trailing edge. This, of course, reduces the accuracy but can be a trick to obtain a converged solution if time and computer resources does not allow a transient simulation to be performed.

=== Boundary layer mesh ===

For design iteration type of simulations where a [[Wall functions|wall function]] approach is sufficient [[Dimensionless wall distance (y plus) | y+]] for the first cell should be somewhere between 20 and 200. The outer limit is dependent on the actual Re number of the simulations. For cases with fairly low Re numbers make sure to keep the maximum y+ as low as possible. For more accurate simulations with resolved boundary layers the mesh should have a y+ for the first cell which is below 1. Some new codes are now using a hybrid wall treatment that allows a smooth transition from a coarse wall-function mesh to a resolved low-Re mesh. Use some extra care when using this type of hybrid technique since it is still fairly new and unproven.

Outside of the first cell at a wall a good rule of thumb is to use a growth ratio normal to the wall in the boundary layer of maximum 1.25. For a low-Re mesh this usually gives around 40 cells in the boundary layer whereas a wall-function mesh does not require more than 10 cells in the boundary layer.

If you are uncertain of which wall distance to mesh with you can use a [http://www.cfd-online.com/Links/tools.html#yplus y+ estimation tool] to estitmate the distance needed to obtain the desired y+. These estimation tools are ''very handy'' if you have not done any previous similar simulations.

As a rule of thumb a wall-function mesh typically requires areound 5 to 10 cells in the boundary layer whereas a resolved low-Re mesh requires about 40 cells in the boundary layer.

== Boundary conditions ==

Describe different types of boundary conditions and when they should be used:

* Total pressure in, static pressure out
* Absorbing boundary conditions
* ...

There are different types of boundary conditions you can use:
*Mass flow inlet, static pressure outlet.
To put this BC you must allocate inlet surface enough far away from the turbomachine impeller. The velocity distribution is not
constant over all the surface. If you put this "lie" away from your problem it works well.

=== Turbulence inlet conditions ===

Prescribing realistic turbulence inlet conditions is important and usually difficult. For two-equation turbulence models two different turbulence values need to be specified on the inlet. The most common way of specifying the inlet turbulence variables is to give an inlet [[Turbulence intensity|turbulence level]] and [[Turbulence length scale|length-scale]]. For simulations where the incoming boundary layers are estimated to not have any significant effect on the secondary flows and global flow-field downstream it is usually sufficient to specify a constant incoming turbulence level and length-scale.

The incoming [[Turbulence intensity|turbulence level]] is dependent on which component is analyzed. Best is of course to have measurments from a similar application in order to be able to prescribe a fairly realistic value. Without measurements it is necessary to make an educated guess. Simulations with very complex and turbulence-generating components upstream will have a very high incoming turbulence level. For example, a high-pressure turbine just downstream of a turbulence generating combustor might have incoming turbulence levels up to 20%. Fans and low-pressure compressors with not many components upstream might have as low incoming turbulence level as 1%. For components between these two extreme examples, like high-pressure compressors, low-pressure turbines etc. a turbulence level of around 5% might be realistic. If you do not have any measurements and are unsure about which incoming turbulence level to prescribe it is always a good idea to run a few different simulations with say one half and one double incoming turbulence level in order to estimate how important it is for the simulation results.

The incoming [[Turbulence length scale|turbulence length-scale]] is often even more difficult to guess than the incoming turbulence level. Measurements might sometimes include the incoming turbulence level, but very seldom also include the incoming turbulent length-scale. The best way of guessing a realistic incoming length-scale is to use the geometrical properties of the upstream components. The incoming turbulence length-scale can be estimated as say the thickness of upstream blades or somewhere between 2% and 20% of the incoming channel heigth. Fortunately the incoming turbulence length-scale is usually not that important for the end results. However, some k-omega models can have an unrelisticly strong influence on the incoming turbulence length-scale. Hence, be extra carefull about the length-scale when using k-omega models. Note that the [[SST k-omega model]] does not have this unrealistic sensitivity to the inlet turbulence length-scale.

Instead of specifying an incoming turbulence length-scale it is sometimes more convenient to specify an incoming [[Eddy viscosity ratio|eddy viscosity ratio]]. For low-turbulence cases with no well-specified turbulence generating components upstream it can be difficult to estimate a turbulence length-scale. Specifying an eddy viscosity ratio instead gives a more direct control over how large the effect of the incoming turbulence is. The eddy viscosity ratio describes how large the turbulence viscosity is related to the molecular viscosity. In turbomachinery the eddy viscosity ratio can vary from say 10 in low-turbulence components like fans up to say 1,000 or in rare cases even 10,000 in high-turbulence components like high-pressure turbines. In components between these extremes a typical eddy viscosity ration might be about 200. Even if specifying an inlet turbulence length scale it can still be good to compute what eddy viscosity ratio this corresponds to in order to estimate if it sounds reasonable.

== Turbulence modeling ==

Selecting a suitable turbulence model for turbomachinery simulations can be a challenging task. There is no single model which is suitable for all types of simulations. Which turbulence model CFD engineers use has as much to do with beliefs and traditions as with knowledge and facts. There are many different schools. However, below follows some advices that most CFD engineers in the turbomachinery field tend to agree upon.

For attached flows close to the design point a simple algebraic model like the [[Baldwin-Lomax model]] can be used. Another common choice for design-iteration type of simulations is the one-equation model by [[Spalart-Allmaras model | Spalart-Allmaras]]. This model has become more popular in the last years due to the many inherent problems in more refined two-equation models. The big advantage with both the [[Baldwin-Lomax model]] and the [[Spalart-Allmaras model]] over more advanced models is that they are very robust to use and rarely produce completely unphysical results.

In order to accurately predict more difficult cases, like separating flows, rotating flows, flows strongly affected by secondary flows etc. it is often necessary to use a more refined turbulence model. Common choices are two-equation models like the <math>k-\epsilon</math> model.

Two-equation models are based on the [[Boussinesq eddy viscosity assumption]] and this often leads to an over-production of turbulent energy in regions with strong acceleration or deceleration, like in the leading edge region, regions around shocks and in the suction peak on the suction side of a blade. To reduce this problems it is common to use a special model variant using, for example, [[Durbin's realizability constraint]] or the [[Kato-Launder modification]]. Note that different two-equation models behave differently in these problematic stagnation and acceleration regions. Worst is probably the standard <math>k-\epsilon</math> model. <math>k-\omega</math> model are slightly better but still do not behave well. More modern variants like Menter's [[SST k-omega model]] also has problems, wheras the [[V2-f models|v2f model ]] by Durbin behaves better.

=== Near-wall treatment ===

For on-design simulations without any large separated regions it is often sufficient to use a [[wall-function model]] close to the wall, preferably with some form of non-equilibrium wall-function that is sensitised to streamwise pressure gradients.

For off-design simulation, or simulations involving complex secondary flows and separations, it is often necessary to use a [[low-Re model]]. There exist many low-Re models that have been used with success in turbomachinery simulations. A robust and often good choice is to use a one-equation model, like for example the [[Wolfstein model]], in the inner parts of the boundary layer. There are also several Low-Re <math>k-\epsilon</math> models that work well. Just make sure they don't suffer from the problem with overproduction of turbulent energy in regions with strong acceleration or deceleration. In the last few years Menter's low-Re <math>SST k-\omega</math> model has gained increased popularity.

=== Transition prediction ===

Transition refers to the process when a laminar boundary layer becomes unstable and transitions to a turbulent boundary layer. There are two types of transition - natural transition, where inherent instabilities in the boundary layer cause the transition and by-pass transition, where convection and diffusion of turbulence from the free-stream into the boundary layer cause the transition. Most transitions in turbomachinery are by-pass transitions caused by free-stream turbulence and other external disturbances like wakes, vortices and surface defects.

Simulating transition in a CFD code accurately is very difficult. Often a separate transition model needs to be solved in order to specify the transition location and length. Predicting natural transition in a pure CFD code is not possible. Predicting by-pass transition in a pure CFD code is almost impossible, although there are people who claim to be able to predict by-pass transition with low-Re two-equation models. However, this is usually on special test cases and with simulations that have been tuned for these special cases, see for example [Saville 2002]. In reality transition is a very complex and sensitive process where disturbances like incoming wakes and vortices from previous stages, surface roughness effects and small steps or gaps in the surfaces play a significant role.

The turbomachinery codes that have transition prediction models often use old ad-hoc models like the Abu-Ghannam and Shaw model [Abu-Ghannam 1980] or the Mayle model [Mayle 1991]. These models can be quite reliable if they have been validated and tuned for a similar application. Do not trust your transition predictions without having some form of experimental validation. Menter has also recently developed a new form of transition model that might work fairly well, but it is still too new and untested.

For some turbomachinery applications, like modern high-lift low-pressure turbines, transition is critical. For these applications a CFD code with a transition model that has been tuned for this type of applications should be used.

== Numerical considerations ==

Use at least a second order accurate scheme for the flow variables. Some codes require a first order scheme for the turbulent variables (<math>k</math> and <math>\epsilon</math>) in order to converge well. It might be sufficient with a first order scheme only on the turbulence variables, but a second order scheme is of course preferable.

=== Convergence criteria ===

To know when a solution is converged is not always that easy. You need some prior experience of your CFD code and your application to judge when a simulation is converged. For normal pure aero simulations without resolved walls, i.e. with wall functions or inviscid Euler simulations, convergence can most often be estimated just by looking at the residuals. Exacty what the residuals should be is not possible to say, it all depends on how your particular code computes and scales the residuals. Hence, make sure to read the manuals and plot the convergence of a few global parameters before you decide what the residuals should be for a solution to converege. Note also that many manuals for genereal purpose CFD codes list overly agressive convergence critera that often produce unconverged results.
For simulations with resolved walls it is good to look at the convergence of some global properties, like total pressure losses from the inlet to the outlet. For heat transfer simulations it is even more tricky since the aerodynamic field can look almost converged although the thermal field is not converged at all. If doing heat transfer simulations make sure to plot the heat-transfer, run for some time, and plot it again to make sure that it doesn't change anymore. With very well resolved walls and heat transfer it can sometimes take 10 times longer for the thermal field to converge.

=== Single or double precision ===

With todays computers and cheap memory prices it often does not cost much extra to run in double precision. Before using single precision you should first investigate how your software and hardware works with double precision. If the extra time and memory needed for double precision is negligible you should of course always run in double precision. With double precision you never have to worry about round-off errors. Always using double precision is one way of avoiding one type of pit-falls in the complex world of CFD simulations. Use double precision when you have resolved boundary layers (Y+ around 1) and when you use advanced physical models like combustion, free-surface simulations, spray and transient simulations with quick mesh motions.

== Multi-stage analysis ==

Multi-stage analysis can be done in different ways:

* Steady mixing-plane simulations
* Frozen rotor simulations
* Unsteady sliding-mesh stator-rotor simulations
* Other '''advanced''' multi-stage methods

=== Steady mixing-plane simulations ===

Since the mixing-plane method was first introduced in 1979 [Denton & Singh 1979] is has become the industry standard type of rotor-stator simulations. A mixing-plane simulation is steady and only requires one rotor blade and one stator blade per stage. Between the rotating blade passage and the steady vane passage the flow properties are circumferentially averaged in a so-called mixing-plane interface. This will of course remove all transient rotor-stator interactions, but it still gives fairly representative results. In some commercial codes (CFX for example) mixing-plane interfaces are also called stage-interfaces.

=== Frozen rotor simulations ===

In a frozen rotor simulation the rotating and the stationary parts have a fixed relative position. A frame transformation is done to include the rotating effect on the rotating sections. This will give a steady flow and no transient effects are included. With a frozen-rotor simulation rotating wakes, secondary flows, leading edge pressure increases etc. will always stay in exactly the same positions. This makes a frozen rotor simulation very dependent on exactly how the rotors and the stators are positioned. Most often a mixing-plane simulation gives better results. Frozen rotor simulations are mainly performed to obtain a good starting flow-field before doing a transient sliding-mesh simulation.

=== Unsteady sliding-mesh stator-rotor simulations ===

This is the most complete type of stator-rotor simulation. In most engines the number of stators and rotors do not have a common denominator (to avoid instabilities caused by resonance between different rings). Hence, to make a full unsteady sliding-mesh computation it is necessary to have a mesh which includes the full wheel with all stators vanes and all rotor blades. This is often not possible, instead it is necessary to reduce the number of vanes and blades by finding a denominator that is '''almost''' common and then scale the geometry slightly circumferentially. Here is an example:

*Real engine: 36 stator vanes, 41 rotor blades
*Approximated engine: 41 stator vanes, 41 rotor blades, making it possible to simulate only 1 stator vane and 1 rotor blade
*Scaling of stator: All stator vanes are scaled by 36/41 = 0.8780 circumferentially.

=== Other '''advanced''' multi-stage methods ===

Time-inclinded, Adamszyk stresses ...

== Heat transfer predictions ==

Besides listing the general heat transfer mechanisms involved
(namely conduction, convection, radiation)
heat transfer prediction in CFD may be seen as or split into two cases.

'''Mesh consists of fluid domain(s)''':
you want to know how heat inserted into the fluid domain (e.g. via the flowing medium) changes the temperature field along the flow path. Be it a steady or transient run, a fluid which enters with a given temperature will usually experience temperature variations, for instance, caused by convective boundary conditions, prescribed temperature profiles at flow obstacles etc.

'''Mesh consists of fluid and solid domain(s)''':
additionally to the above, you want info too wrt the (spatial, temporal or even spectral) temperature field distribution in surrounding, confining or immersed solids like channel walls or heat exchanger tubes. This is also called conjugate heattransfer CHT in the CFD context. CHT requires a good boundary layer resolution, usually the wall mesh needs to be rather refined, to obtain realistic heatflux results at the fluid/solid interface. Flow and heat transfer convergence require different time step settings, to properly capture changes in flow and heat quantities respectively.

In either case, verify (strict necessity depends on CFD code used, CFX for instance checks and assists in regard) that model dimensions, boundary conditions and properties are in consistent units, hold appropriate values. Check temperature-dependence of properties and other numbers before the run.

In heat transfer predictions (depending on the CFD code in use)
besides the flow solver, you may have to activate the thermal solver too,
as a job specification.

== Acoustics and noise ==

A whole separate research subject, difficult.

Tone noise possible. Often run with linearzised solvers in the frequency domain.

Jet noise possible. Often run with LES or DES simulations that either also resolve the sound waves or couples to a separate acoustic solver.

Turbomachinery broadband noise not possible yet, or at least a great challenge.

== Errors and uncertainties ==

CFD is still a tool which requires that a user has a good understanding of uncertainties and errors that might spoil a CFD similation. There exists no error control in CFD and any CFD simulation must be interpreted by an expirenced user to have some credibility. Without some knowledge about possible errors and how they can be handled a CFD simulation can not be trusted. Errors can occur at different places:

* Definition of the problem - What needs to be analyzed?
* Selection of the solution strategy - What physical models and what numerical tools should be used?
* Development of the computational model - How should the geometry and the numerical tools be set up?
* Analysis and interpretation of the results - How should the model be analyzed and the results be interpreted?

There exists many different definitions on errors. In this guide the errors are classified into four types of errors: problem definition errors, model errors, numerical errors and user and code errors. The chapters below describe these errors and give some guidelines on how to avoid them.

=== Problem definition errors ===

Problem definition errors are the most basic form of errors. In order to obtain usefull results a CFD simulation must of course analyze the correct problem, have suitable boundary contions and be based on a relevant geometry.

==== Wrong type of simulation ====

It is essential to have an overview of the physics involved and how the problem can best be analyzed. Running a 2D simulation in order to understand secondary flows or running a steady simulation in order to understand transient behaviour is of course no use. When assessing a CFD simulation the first thing to consider is what physical phenomena are important for the results and if the selected type of simulation is suitable to resolve this type of phenomena. For further information about selecting the most suitable type of simulation please see the previous chapter on [[#Deciding what type of simulation to do|deciding what type of simulation to perform]].

==== Incorrect or uncertain boundary conditions ====

A common source of errors is that incorrect boundary conditions are used. The boundary conditions must be specified in enough detail in order to resolve all the important physical features in the problem. For further information about specifying boundary conditions please see the previous chapter on [[#Boundary conditions|boundary conditions]].

==== Geometrical errors ====

It is almost always necessary to simplify the geometry in some form. When assessing a CFD simulation one should consider how the geometrical simplifications can affect the interesting physical phenomena. Typical geometrical errors are:

*Simplifications
:Small geometrical features like fillets, small steps or gaps etc. can often be disregarded. When disregarding this type of features one should consider if they might affect the important physics. For example, a very large fillet on the suction-side of a vane might affect corner separations near the end-walls. A large tip-leakage might affect the flow physics significantly in the upper part of a compressor.
*Tolerances and manufacturing discrepancies
:If the geometry has very large tolerances or is manufactured in a way which might produce a non-ideal shape or position it might be necessary to perform additional CFD simulations in order to cover the whole span of possible real geometries.
*Surface conditions - roughness, welds, steps, gaps etc.
:Often CFD simulations assume a perfectly smooth surface. A non-smooth surface which might have welds, steps or even gaps will of course produce different results. If the physical phenomena of interest might depend on the surface conditions these should of course be condidered. Typical phenomena that might be dependent on this type of errors are transition prediction, leakage flows etc.

=== Model errors ===

Errors related to the computational model.

==== Wrong physical models ====

Once the type of simulation has been selected the next step is to select what type of physical models the simulation should use. The following points should be considered:

* Gas data (incompressible/compressible, perfect gas/real gas, ...)
* Turbulence modeling (type of model, type of near-wall treatment, ...)
* Other models (combustion, sprays, ...)

When assessing model related errors it is important to know the features of the selected model and think carefully how these features and possible short comings might affect the predicted physical behaviour. Using the wrong turbulence model or combustion model can completely destroy the results of a CFD simulation.

=== Numerical errors ===

Errors related to the numerical solution of the developed model. Typical examples of numerical errors are discretization errors, convergence errors and round-off errors.

==== Discretization errors ====

Discretization errors can either be spatial errors in space or temporal errors in time.

Spatial discretization errors are what people normally call discretization errors. These errors are due to the difference between the exact solution and the numerical representation of the solution in space. Describing exactly what discretization different codes use and what errors this might lead to is not possible here. Instead some general rules to avoid these errors can be summarized as:
* Use at least a 2nd order accurate scheme, preferably a 3rd order accurate scheme. Some general purpose codes have a 1st order upwind scheme as default, this is a very diffusives schemes that often gives too smooth results.
* For new applications always run a simulation with a finer mesh to see how grid independent your solution is.
* Be aware of checker-board errors. Checker-board errors occur close to strong shocks and other large discontinuities and can be seen as a wavy pattern with a wavelength of two cells. Some schemes, especially those who behave like central differencing schemes, are more prone to checker-board errors. Upwind schemes are a bit better and schemes like TVD or chock-capturing schemes are even better.

Temporal discretization errors mainly effect transient simulations. However, some codes use a time-marching method also for steady simulations and then a temporal discretization error might affect the final steady solution slightly. The discretization in time can be done with 1st or 2nd order schemes or a Runge-Kutta method, which is more accurate and saves memory. Some codes can adapt the time-step, but often it is necessary to prescribe a time-step in advance. Think of the time-step as your grid in time and make sure that the grid-resolution in time is fine enough to resolve the highest frequencies. To avoid problems with temporal discretization errors the following should be considered:
* Use at least a 2nd order scheme in time.
* Do a physical estimation of the typical frequencies in time of the phenomena that you are interested in and select a time-step which is fine enough to resolve these frequencies well. After the simulation also look at the frequencies captured and make sure that they are well resolved by the chosen time-step.
* For new applications try a finer time-step to ensure that your solution in time is fairly grid independent in time.

==== Convergence errors ====

To judge when a CFD simulation is converged is not always that easy. Different codes and different applications behave very differently. For a pure aero-simulation on a fairly coarse grid convergence is easy to judge, but for more complex simulations involving resolved boundary layers, heat transfer, combustion etc. convergence can be very tricky. Aside from looking at residuals one should always also look at how global parameters like static pressure distributions, total pressure losses, skin friction, heat transfer etc. change in time.
For more information about how to avoid convergence errors see the previous chapter about [[#Convergence criteria]].

==== Round-off errors ====

When using single precision care needs to be taken to avoid round-off errors. Inviscid Euler simulations and simulations using wall-function meshes can most often be performed in single precision. For well resolved boundary layers with Y plus close to 1 it is often necessary to use double precision. If using double precision for very fine mesh resolutions make sure that you also create the mesh in double precision and not just run the solver in double precision. Sometimes a single precision solver converges slower than a double precision solver due to numerical errors caused by round-off errors. When using advanced physical models like combustion, free-surface simulations, spray and transient simulations with quick mesh motions it is also often necessary to use double precision.

=== User and code errors ===

Errors related to bugs in the code used or mistakes made by the CFD engineer.

=== What to trust and what not to trust ===

CFD is generally quite good at predicting surface static pressure distributions. With care CFD can also be used to predict performance, total-pressure losses and blade turning.

Predicting separation, stall and off-design performance can be a challenge and results with non-attached flows should be interpreted with care.

Heat transfer is often very difficult to predict accurately and it is common to obtain heat-transfer coefficients that are 100% wrong or more. Validation data is critical in order to be able to trust heat transfer simulations.

Transition is almost impossible to predict accurately in general. However, there exist models that have been tuned to predict transition and these tend to give acceptable results for cases close to the ones they were tuned for.

== References ==

* {{reference-paper|author=Abu-Ghannam, B.J. and Shaw, R.|year=1980|title=Natural Transition of Boundary Layers - the Effects of Turbulence, Pressure Gradient, and Flow History|rest=Journal of Mech. Eng. Science, vol. 22, no. 5, pp. 213–228}}

* {{reference-paper|author=Adamczyk, J. J.|year=1985|title=Model Equation for Simulating Flows in Multistage Turbomachinery|rest=ASME Paper 85-GT-226, also NASA TM-86869}}

* {{reference-paper|author=Adamczyk, J. J.|year=1999|title=Aerodynamic Analysis of Multistage Turbomachinery Flows in Support of Aerodynamic Design|rest=ASME Paper 99-GT-80.}}

* {{reference-paper|author=Baldwin, B. S. and Lomax, H.|year=1978|title=Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows|rest=AIAA Paper 78-257}}

* {{reference-paper|author=Chen, J. P. and Barter, J. W.|year=1998|title=Comparison of Time-Accurate Calculations for the Unsteady Interaction in Turbomachinery Stage|rest=AIAA Paper 98-3292}}

* {{reference-paper|author=Denton, J. and Singh, U.|year=1979|title=Time Marching Methods for Turbomachinery Flow Calculations|rest=VKI Lecture Series 1979-7, von Karman Institute}}

* {{reference-paper|author=Erdos, J., Alzner, E. and McNally, W.|year=1977|title=Numerical Solution of Periodic Transonic Flow Through a Transonic Fan Stage|rest=AIAA Journal, vol. 15, no. 11, pp. 1559-1568}}

* {{reference-paper|author=Gerolymos, G., Vinteler, D., Haugeard, R., Tsange, G. and Vallet, I.|year=1996|title=On the Computation of Unsteady Turbomachinery Flows – Part II – Rotor/Stator Interaction using Euler Equations|rest=AGARD Report CP-571}}

* {{reference-paper|author=Giles, M. B.|year=1990|title=Stator/Rotor Interaction in a Transonic Turbine|rest=AIAA Journal of Propulsion and Power, vol. 6, no. 5}}

* {{reference-paper|author=He, L, and Ning, W.|year=1998|title=Efficient Approach for Analysis of Unsteady Viscous Flows in Turbomachines|rest=AIAA Journal, vol. 36, no. 11, pp. 2005-2012}}

* {{reference-paper|author=Mayle, R. E.|year=1991|title=The Role of Laminar-Turbulent Transition in Gas Turbine Engines|rest=ASME Journal of Turbomachinery, vol. 113, pp. 509-537}}

* {{reference-paper|author=Menter, F.R. |year=1994|title=Two-equation eddy-viscosity turbulence models for engineering applications|rest=AIAA Journal, vol. 32, pp. 269-289}}

* {{reference-paper|author=Savill, M., Dick, E., Hanjalić, K. and Voke, P.|year=2002|title=Synthesis Report of the ERCOFTAC Transition Modeling – TRANSPRETURB Thematic Network Activities 1998-2002|rest=Ecoftac Bulletin 54, September 2002, pp. 5-16}}

* {{reference-paper|author=Wang, X. and Chen J.|year=2004|title=A Post-Processor to Render Turbomachinery Flows Using Phase-Lag Simulations|rest=AIAA Paper 04-615}}

== External links ==

* [http://www.qnet-cfd.net/newsletter/8th/n8_40-46.pdf QNET-CFD Best Practise Advice for Turbomachinery Internal Flows]
* [http://www.qnet-cfd.net/newsletter/7th/n7_05.pdf State of the art in Industrial CFD for Turbomachinery Flows, QNET-CFD Network Newsletter, Volume 2, No. 3 – December 2003]
* [https://pronet.wsatkins.co.uk/marnet/guidelines/guide.html MARNET-CFD Best Practice Guidelines for Marine Applications of CFD]
* [http://www.stanford.edu/group/ctr/ResBriefs/temp05/weide.pdf On Large Scale Turbomachinery Computations, by E. van der Weide et. al., Center for Turbulence Research, Annual Research Briefs 2005]

User:Jola

2010-01-30T13:21:29Z

Jola: updated information

== Jonas Larsson ==

I am 40 years old and live in Sweden with my wife and our five year old son.

Since 1998 I work for Volvo Aero, where I am a CFD and aero-design specialist, focusing on turbomachinery applications.

I have a PhD from Chalmers University, where I worked on heat-transfer simulations in turbines. I left Chalmers in 1998. In 2009 I was also appointed [http://en.wikipedia.org/wiki/Docent#Northern_Europe Docent] in Thermo and Fluid Dynamics at Chalmers University.

You can email me at jola@cfd-online.com.

NOGRID

2009-06-25T18:56:34Z

Jola: Removed marketing like text

NOGRID FPM is a commercial CFD code. It is a product of NOGRID GmbH.

Current version is 4.0.0 (at June. 1, 2009).

FPM can be applied in the case of all problems, where grid-based methods reach their limits. Examples are fluid dynamical problems with free surfaces, moving parts, multiphase flows, fluid-structure interactions with a strong change of the computing domain or mechanical problems with substantial structure changes.

Nogrid's Finite Pointset Method (FPM) is software for simulation tasks in flow and continuum mechanical problems and is based on a method, which use a local defined, non-stationary point cloud distribution for discretization of the Navier-Stokes equations. This point cloud is generated automatically by the software depending on users settings. Thus there is no need to generate a mesh as required in classical CFD methods. FPM points are automatically filled, moved, refilled and cleaned depending on user specifications. The point cloud can be defined variously, e.g. constant in the whole flow domain, changing with time or increasing/decreasing locally depending on the flow/geometry situation.

== External Links ==

* [http://www.nogrid.com www.nogrid.com]

User:Jola

2009-04-26T11:11:09Z

Jola:

== Jonas Larsson ==

I am 39 years old and live in Sweden with my wife and our four year old son.

Since 1998 I work for Volvo Aero, where I am a CFD and aero specialist, mainly focusing on turbomachinery applications.

I have a PhD from Chalmers University, where I worked on heat-transfer simulations in turbines (lot's of turbulence modeling problems mainly). I left Chalmers in 1998.

You can email me at jola@cfd-online.com.

User:Jola

2009-04-26T08:11:38Z

Jola:

== Jonas Larsson ==

I am 39 years old and live in Sweden with my wife and our three year old son.

Since 1998 I work for Volvo Aero, where I am a CFD and aero specialist, mainly focusing on turbomachinery applications.

I have a PhD from Chalmers University, where I worked on heat-transfer simulations in turbines (lot's of turbulence modeling problems mainly). I left Chalmers in 1998.

You can email me at jola@cfd-online.com.

Talk:Metacomp FAQ

2009-04-26T08:11:09Z

Jola: New page: I changed the sub-sections to "book" chapters in the Metacomp FAQ --~~~~

I changed the sub-sections to "book" chapters in the Metacomp FAQ --[[User:Jola|Jola]] 08:11, 26 April 2009 (UTC)

Metacomp FAQ

2009-04-26T08:09:29Z

Jola:

== Introduction to CFD++ ==

=== What is CFD++ ===

Explain here

== Efficiency Strategies ==

;[[Metacomp FAQ/Solution Re-interpolator|Solution Re-interpolator]]

;[[Metacomp FAQ/Equation Set Convert|Equation Set Convert]]

;[[Metacomp FAQ/CFL Ramping|CFL Ramping]]

;[[Metacomp FAQ/Initialisation|Initialisation]]

Initialisation

2009-04-26T08:07:10Z

Jola: Initialisation moved to Metacomp FAQ/Initialisation

#REDIRECT [[Metacomp FAQ/Initialisation]]

Metacomp FAQ/Initialisation

2009-04-26T08:07:10Z

Jola: Initialisation moved to Metacomp FAQ/Initialisation

This is especially important if you’ve got separated regions. Using tools such as reintson and reintsom to interpolate existing solutions to different grids can save you massive amounts of simulation time. These can also be used to extrude a 2D solution into 3D. I had problems with this option for axi-symmetric geometries (the Cartesian “nearest neighbour” approach doesn’t work very well for such things) so used the “cdeps1asn” tool to create a restart file from an ascii file that I generated myself that contained the x, y, z, P, T, U, V, W data.

If you don’t have an existing solution then initiating the domain using the XYZ boxes option to prescribe, for instance, zero velocity at the leading edge and a pseudo-recirculation bubble etc can also be a big help.

Another useful tool is the equation set convert option in the physics menu. You can use this to convert a laminar run to a turbulent run etc and then perform a restart.

CFL Ramping

2009-04-26T08:06:40Z

Jola: CFL Ramping moved to Metacomp FAQ/CFL Ramping

#REDIRECT [[Metacomp FAQ/CFL Ramping]]

Metacomp FAQ/CFL Ramping

2009-04-26T08:06:40Z

Jola: CFL Ramping moved to Metacomp FAQ/CFL Ramping

=== Introduction ===

CFD++ has an automatic CFL adjustment procedure. This detects problems before they arise and adjusts the CFL to stop the code going unstable. This can kick in early on in the run in areas such as the stagnation point at the leading edge as the flow adjusts from the prescribed initial (usually freestream) conditions. By default the CFL ramping is adjusted so that, provided there are no more CFL adjustments, it will reach 95% of the specified value. Clearly if there are a considerable number of adjustments then you can end up with a very low CFL number indeed. This should be avoided at all costs!

=== Methods to ensure a high CFL number===
*Change the default adjustment factor from 0.95 (perhaps to 0.995, 0.999 or even 1.0) in the time integration panel. If you’ve specified too high a CFL number, or a ramping schedule that is too aggressive, then you will end up with adjustments happening all the time. If you look at the residual plot you’ll see a rapid increase whenever this adjustment occurs, so it is in your interest to specify a level at which it can happily remain. Exercise some discretion and try different ramping schedules – I usually ramp from 0.001 to 15 over maybe 100-200 iterations.

*Make sure that the CFL ramping is complete within the 1st order portion of the run. In the spatial discretisation panel there is a blending from 1st order to 2nd order option. Make sure that you use this! I sometimes delay the onset of the ramping by a few hundred iterations and increase the number of iterations that the ramping is performed. Without a high CFL number in the 1st order portion then this option is more or less meaningless – you need to give it big time steps to get all of the crap out of the domain before the 2nd order simulation begins.

*Ensure that your meshes are as smooth as possible. Skewness increases the chances of the adjustment procedure kicking in, in addition to giving crappy solutions.

Solution Re-interpolator

2009-04-26T08:05:12Z

Jola: Solution Re-interpolator moved to Metacomp FAQ/Solution Re-interpolator: More correct title

#REDIRECT [[Metacomp FAQ/Solution Re-interpolator]]

Metacomp FAQ/Solution Re-interpolator

2009-04-26T08:05:12Z

Jola: Solution Re-interpolator moved to Metacomp FAQ/Solution Re-interpolator: More correct title

== Introduction ==

=== What is it? ===

This tool is a solution re-interpolator, which is intended to take the solution on an existing mesh and interpolate that solution to a new mesh. Cell centroids in the new mesh which lie outside the domain of the old grid are initialized by marching data from interior cells that were successfully interpolated.

=== When should I use it? ===

This tool can be used when you wish to transfer the existing solution from one mesh (old) to a another mesh (new). The old and new meshes must have the same geometry in general but can have a different number of distribution of grid points. This tool is especially handy if you want to perform a grid convergence study on three meshes, say a course, medium and fine mesh. The solution from course mesh could be interpolated onto the medium mesh then from the medium mesh to the fine mesh in order to significantly reduce the convergence time.

== Running from the GUI ==

=== Preparation Steps ===

In order to make things clear we will establish a naming convention. We will call the existing solution, the one we are interpolating FROM, the OLD solution. We therefore call the solution we are interpolating TO, the NEW solution.

1. First rename the following files in the NEW solution folder:

''cellsin.bin'' -------> ''new_cellsin.bin''
''nodesin.bin'' -------> ''new_nodesin.bin''

2. Copy theses two files from the NEW solution folder to the OLD solution folder

=== Running the Tool ===

To initiate the tool from the GUI go to Tools -> CFD++ Solution Tools -> Solution Re-interpolator. The following window will appear:

[[Image:reintsom.png]]

3. Leave the default values and select the "Run Tool" button to start the run.

=== Post Run Steps ===

4. Copy the ''new_cdepsout.bin'' file from the OLD solution folder to the NEW solution folder.

5. Remove the ''new_'' appendage from each of the following files:

''new_cellsin.bin''
''new_nodesin.bin''
''new_cdepsout.bin''

== Running from the Command Line ==

=== Preparations Steps ===

The preparation steps are identical to steps 1-2 above.

=== Running the Tool ===

Use the following command

''reintsom new_cellsin.bin new_nodesin.bin new_cdepsout.bin cdepsout.bin''

=== Post Run Steps ===

The post run steps are identical to steps 4-5 above.

Solution of Navier-Stokes equations

2009-04-23T10:01:48Z

Jola: Reverted edits by Diyaseb (Talk) to last version by Peter

==Introduction==
Most traditional CFD algorithms require the solution of the Navier-Stokes (N-S) equations. There are two important issues that arise in the solution process:
# The N-S equations are nonlinear.
# The pressure-velocity coupling often requires special consideration.

The details of the solution process depend upon the details of the flow to be solved. The solution process for a incompressible flow can be very different than that for a compressible flow.

==Incompressible flow==
In situations in which the density is approximately constant, the flow may be termed [[Incompressible flow |incompressible]]. The Navier-Stokes equation may then be written as a continuity equation

:<math>
\nabla \cdot \vec U = 0 </math>

and as a momentum equation

:<math> {{\partial \vec U} \over {\partial t}} + \nabla \cdot \left( {\vec U\vec U} \right) = - \nabla p + \nabla \cdot\left( {\nu \nabla \vec U} \right). </math>

Here, we have written the nonlinear convective term (the second term on the right-hand side) in the so-called conservative form. If a turbulence model is to be employed, then the equations will change. If an eddy viscosity approach is to be used, then there are three likely modifications:

*the flow variables will represent average (or filtered) quantities,
*the viscosity <math>\nu</math> will actually be the sum of the fluid property and the calculated eddy viscosity (and will then be nonconstant, justifying keeping it inside the divergence operator), and
*the pressure will be modified to include normal-stress-like terms arising from the eddy viscosity assumption.

Thus, the equation as written will be valid for both turbulent or laminar flows, with some modification of the actual meaning of individual terms.

It is somewhat difficult to categorize the solution methods for the incompressible N-S equations, as over the years many have been developed that are very similar. However, the manner in which the difficulties associated with the pressure solution are handled (a process made difficult by the fact that pressure does not appear in the continuity equation) can be used to broadly categorize methods.

==Compressible Flow==

Codes

2009-04-21T17:19:00Z

Jola:

An overview of both free and commercial CFD software. Here you will find short descriptions of codes along with links to resources.

'''Note to contributers:''' Please try to keep descriptions short and to the point (approximately 200 words) and avoid long lists of features or capabilities. Also keep in mind that all contributions are considered to be released under the GNU Free Documentation License 1.2 (see [[Project:Copyrights]] for details). Also note that all information should be verifiable and objective truths that also competitors to the code in question will agree upon. This is especially important if you are an employee of the company selling the code. See the [[CFD-Wiki:Policy]] for further information.

== Free codes ==

This section lists codes that are in the public domain, and codes that are available under GPL, BSD or similar licenses.

=== Solvers ===
* ADFC -- [http://adfc.sourceforge.net/index.html ADFC homepage]
* Applied Computational Fluid Dynamics -- [http://www.partenovcfd.com Solver homepage]
* CFD2k -- [http://www.cfd2k.eu/ CFD2k: a 2D-solver for compressible ideal gases - homepage]
* Channelflow -- [http://www.cns.gatech.edu/channelflow/ Channelflow: a spectral Navier-Stokes simulator in C++ homepage]
* Code_Saturne -- [http://rd.edf.com/code_saturne Code_Saturne homepage]
* COOLFluiD -- [http://coolfluidsrv.vki.ac.be/coolfluid COOLFluiD homepage]
* Diagonalized Upwind Navier Stokes -- [http://duns.sourceforge.net DUNS homepage]
* [[Dolfyn]] -- [http://www.dolfyn.net/dolfyn/index_en.html dolfyn a 3D unstructured general purpose solver - homepage]
*[[Edge]] -- [http://www.foi.se/edge Edge homepage: 2D & 3D compressible RANS / Euler flow solver on unstructured and hybrid grids]
*[[ELMER]] -- [http://www.csc.fi/elmer/ ELMER homepage]
* [[FDS]] -- [http://www.fire.nist.gov/fds/ FDS homepage]
* Featflow -- [http://www.featflow.de Featflow homepage]
* Femwater -- [http://www.epa.gov/ceampubl/gwater/femwater/index.htm Femwater code]
* FreeFEM -- [http://www.freefem.org FreeFEM homepage]
*[[Gerris Flow Solver]] -- [http://gfs.sourceforge.net/ Gerris Flow Solver homepage]
* IMTEK Mathematica Supplement (IMS) -- [http://www.imtek.uni-freiburg.de/simulation/mathematica/IMSweb/ IMTEK Mathematica Supplement (IMS) homepage]
* iNavier -- [http://inavier.sourceforge.net/ iNavier Solver Home Page]
* ISAAC -- [http://isaac-cfd.sourceforge.net ISAAC Home Page]
* MFIX -- [http://www.mfix.org Computational multiphase flow homepage]
*[[NaSt2D-2.0]] -- [http://home.arcor.de/drklaus.bauerfeind/nast/eNaSt2D.html NaSt2D-2.0 homepage]
*[[NEK5000]] -- [http://nek5000.mcs.anl.gov NEK5000 homepage]
*[[NSC2KE]] -- [http://www-rocq1.inria.fr/gamma/cdrom/www/nsc2ke/eng.htm NSC2KE homepage]
* NUWTUN -- [http://nuwtun.berlios.de NUWTUN Home Page]
*[[OpenFlower]] -- [http://sourceforge.net/projects/openflower/ OpenFlower homepage]
*[[OpenFOAM]] -- [http://www.openfoam.org/ OpenFOAM homepage]
* OpenFVM -- [http://openfvm.sourceforge.net/ OpenFVM homepage]
* PETSc-FEM -- [http://www.cimec.org.ar/petscfem PETSc-FEM homepage]
* PP3D -- [http://www.featflow.de/ parpp3d++ homepage]
* SLFCFD -- [http://slfcfd.sourceforge.net SLFCFD homepage]
*[[SSIIM]] -- [http://folk.ntnu.no/nilsol/cfd/ CFD at NTNU]
*[[Tochnog]] -- [http://tochnog.sourceforge.net Tochnog homepage]
* Typhon solver -- [http://typhon.sf.net Typhon solver homepage]

=== Grid generation ===
*[[Delaundo]] -- [http://www.cerfacs.fr/~muller/delaundo.html Delaundo homepage]
* GMSH -- [http://www.geuz.org/gmsh/ GMSH hompage]
* NETGEN -- [http://www.hpfem.jku.at/netgen/ NETGEN homepage]
* SALOME -- [http://www.salome-platform.org SALOME homepage]
* TETGEN -- [http://tetgen.berlios.de/ TETGEN hompage]
* CartGen -- [http://mehr.sharif.ir/~tav/cartgen.htm CartGen homepage]
*[[Triangle]] -- [http://www.cs.cmu.edu/~quake/triangle.html Triangle homepage]
* gridgen -- [http://www.marine.csiro.au/~sakov Pavel Sakov's homepage]

=== Visualization ===
*[[DISLIN]] -- [http://www.mps.mpg.de/dislin/server.html DISLIN homepage]
* GMV -- [http://www-xdiv.lanl.gov/XCM/gmv/ GMV homepage]
*[[Gnuplot]] -- [http://www.gnuplot.info/ gnuplot homepage]
* GRI -- [http://gri.sourceforge.net/ GRI homepage]
*[[Mayavi]] -- [http://mayavi.sourceforge.net/ MayaVi homepage]
*[[OpenDX]] -- [http://www.opendx.org OpenDX homepage]
*[[ParaView]] -- [http://www.paraview.org/HTML/Index.html ParaView homepage]
*[[Tioga]] -- [http://www.kitp.ucsb.edu/~paxton/tioga.html Tioga homepage]
*[[Vigie]] -- [http://www-sop.inria.fr/sinus/Softs/vigie.html Vigie homepage]
*[[Visit]] -- [http://www.llnl.gov/visit Visit homepage]
*[[vtk]] -- [http://www.vtk.org vtk homepage]
*[[vtk.Net]] -- [http://vtkdotnet.sourceforge.net/ vtk.Net homepage]

=== Miscellaneous ===

*[[Engauge Digitizer]] -- [http://digitizer.sourceforge.net Engauge Digitizer homepage]
*[[Ftnchek]] -- [http://www.dsm.fordham.edu/~ftnchek/ ftnchek homepage]
*[[g3data]] -- [http://www.frantz.fi/software/g3data.php g3data homepage]
* GIFMerge -- [http://www.the-labs.com/GIFMerge/ GIFMerge homepage]
*[[Gifsicle]] -- [http://www.lcdf.org/~eddietwo/gifsicle/ Gifsicle homepage]
*[[ImageMagick]] -- [http://www.imagemagick.org ImageMagick homepage]
* nnbathy (natural neighbor interpolation) -- [http://www.marine.csiro.au/~sakov Pavel Sakov's home page]

== Commercial codes ==

=== Solvers ===
* EasyCFD -- [http://www.easycfd.net EasyCFD homepage]
* Applied Computational Fluid Dynamics -- [http://www.partenovcfd.com Solver homepage]
* AcuSolve -- [http://www.acusim.com/ ACUSIM Software's homepage]
* ADINA-F -- [http://www.adina.com/index.html ADINA's homepage]
* ADINA-FSI -- [http://www.adina.com/index.html ADINA's homepage]
* ANSWER -- [http://www.acricfd.com/ ACRi's homepage]
*[http://www.cfd-online.com/W/index.php?title=CFD%2B%2B CFD++] -- [http://www.metacomptech.com Metacomp Techonlogies' homepage]
* CFD2000 -- [http://www.adaptive-research.com/ Adaptive Research's homepage]
*[[CFD-FASTRAN]] -- [http://www.esi-group.com/SimulationSoftware/advanced.html ESI Group's homepage]
* CFD-ACE -- [http://www.esi-group.com/SimulationSoftware/advanced.html ESI Group's homepage]
* CFdesign -- [http://www.cfdesign.com CFdesign's homepage]
* CFX -- [http://www.ansys.com/ ANSYS homepage]
*[[FENSAP-ICE]] -- [http://www.newmerical.com/ NTI' homepage]
* FINE -- [http://www.numeca.be/ Numeca's homepage]
* FIRE -- [http://www.avl.com/ AVL's homepage]
*[[FLACS]] -- [http://www.gexcon.com/index.php?src=flacs/overview.html GexCon's homepage]
* COMSOL -- [http://www.comsol.com/ COMSOL's homepage]
* FloEFD -- [http://www.mentor.com/products/mechanical/products/floefd Mentor's FloEFD homepage]
* FloTHERM-- [http://www.mentor.com/products/mechanical/products/flotherm Mentor's FloTHERM homepage]
* FloVENT-- [http://www.mentor.com/products/mechanical/products/flovent Mentor's FloVENT homepage]
* FLOW-3D -- [http://www.flow3d.com/ Flow Science's homepage]
* FLOWVISION -- [http://www.fv-tech.com FlowVision's homepage]
*[[FLUENT]] -- [http://www.fluent.com Fluent's homepage]
* [[FLUIDYN]] -- [http://www.fluidyn.com Fluidyn's homepage]
* FluSol -- [http://www.cfd-rocket.com FluSol's hompage]
* Flowz--[http://www.zeusnumerix.com Zeus Numerix's homepage ]
*[[J-FLO]] -- [http://www.newmerical.com NTI's homepage]
* Kameleon FireEx - KFX -- [http://www.computit.com ComputIT's homepage]
* KINetics Reactive Flows -- [http://www.ReactionDesign.com Reaction Design's homepage]
* KIVA--[http://www.lanl.gov/orgs/t/t3/codes/kiva.shtml Los Alamos homepage]
*[[NOGRID FPM]] -- [http://www.no-grid.com NOGRIDS's homepage]
* NX Electronic Systems Cooling -- [http://www.mayahtt.com/index.php?option=com_content&task=view&id=69&Itemid=237 MAYA's NX ESC page]
* NX Advanced Flow -- [http://www.mayahtt.com/index.php?option=com_content&task=view&id=1&Itemid=115 MAYA HTT's NX Adv. Flow page]
* NX Flow -- [http://www.mayahtt.com/index.php?option=com_content&task=view&id=2&Itemid=116 MAYA HTT's NX Flow page]
*[[PHOENICS]] -- [http://www.cham.co.uk CHAM's homepage]
* PowerFLOW -- [http://www.exa.com/pages/pflow/pflow_main.html Exa PowerFLOW homepage]
* PumpLinx -- [http://www.simerics.com Simerics' homepage]
* [[Siemens PLM Software CFD]] -- [http://www.plm.automation.siemens.com/en_us/products/nx/simulation/advanced/index.shtml Siemens PLM Software NX CAE page]
* [[Solution of Boltzmann Equation]] -- [http://www.elegant-mathematics.com/ Elegant Mathematics homepage]
*[[SPLASH]] -- [http://www.panix.com/~brosen SPLASH's homepage]
*[[STAR-CD]] -- [http://www.cd-adapco.com CD-adapco's homepage]
*[[STAR-CCM+]] -- [http://www.cd-adapco.com CD-adapco's homepage]
*[[Tdyn]] -- [http://www.compassis.com CompassIS' homepage]
* TMG-Flow -- [http://www.mayahtt.com/index.php?option=com_content&task=view&id=82&Itemid=283 MAYA HTT's CFD page]
* Turb'Flow -- [http://www.fluorem.com Fluorem's hompage]
* TURBOcfd -- [http://adtechnology.co.uk/products/turbocfd/ TURBOcfd's hompage]

=== Grid generation ===

* ADINA-AUI -- [http://www.adina.com/index.html ADINA's homepage]
* Centaur -- [http://www.centaursoft.com CentaurSoft homepage]
*[[CFD-GEOM]] -- [http://www.esi-group.com/ ESI's homepage]
*[[CFD-VISCART]] -- [http://www.esi-group.com/ ESI's homepage]
* CFDExpert-GridZ --[http://www.zeusnumerix.com/ Zeus Numerix's homepage]
*[[Gridgen]] -- [http://www.pointwise.com/ Pointwise's homepage]
*[[ GridPro]] -- [http://www.gridpro.com/ PDC's homepage]
* Harpoon -- [http://www.ensight.com/ CEI's homepage]
* ICEM CFD -- [http://www.ansys.com/ ANSYS' homepage]
* +ScanFE -- [http://www.simpleware.com/ Simpleware's homepage]
* ANSA -- [http://www.beta-cae.gr/ BETA-CAE's homepage]

=== Visualization ===

* ADINA-AUI -- [http://www.adina.com/index.html ADINA's homepage]
*[[CFD-VIEW]] -- [http://www.esi-group.com/ ESI's homepage]
* CFX-Post -- [http://www.ansys.com/ ANSYS' homepage]
* COVISE -- [http://www.visenso.de/ Visenso's homepage]
* EnSight -- [http://www.ensight.com/ CEI's homepage]
* Fieldview -- [http://www.ilight.com/ Intelligent Light's homepage]
*[[Tecplot]] -- [http://www.tecplot.com/ Tecplot's homepage]
*ViewZ -- [http://www.zeusnumerix.com/ Zeus Numerix's homepage]

=== Systems ===

* ADINA -- [http://www.adina.com/index.html ADINA's homepage]
* Flowmaster -- [http://www.flowmaster.com/index.html Flowmaster's homepage]
* Flownex -- [http://www.flownex.com/ Flownex's homepage]

== Online tools and services ==

*[[CFDNet]] -- [http://www.cfdnet.com/ CFDNet homepage]

Codes

2009-04-21T17:15:25Z

Jola: Corrected links to Mentor

An overview of both free and commercial CFD software. Here you will find short descriptions of codes along with links to resources.

'''Note to contributers:''' Please try to keep descriptions short and to the point (approximately 200 words) and avoid long lists of features or capabilities. Also keep in mind that all contributions are considered to be released under the GNU Free Documentation License 1.2 (see [[Project:Copyrights]] for details). Also note that all information should be verifiable and objective truths that also competitors to the code in question will agree upon. This is especially important if you are an employee of the company selling the code. See the [[CFD-Wiki:Policy]] for further information.

== Free codes ==

This section lists codes that are in the public domain, and codes that are available under GPL, BSD or similar licenses.

=== Solvers ===
* ADFC -- [http://adfc.sourceforge.net/index.html ADFC homepage]
* Applied Computational Fluid Dynamics -- [http://www.partenovcfd.com Solver homepage]
* CFD2k -- [http://www.cfd2k.eu/ CFD2k: a 2D-solver for compressible ideal gases - homepage]
* Channelflow -- [http://www.cns.gatech.edu/channelflow/ Channelflow: a spectral Navier-Stokes simulator in C++ homepage]
* Code_Saturne -- [http://rd.edf.com/code_saturne Code_Saturne homepage]
* COOLFluiD -- [http://coolfluidsrv.vki.ac.be/coolfluid COOLFluiD homepage]
* Diagonalized Upwind Navier Stokes -- [http://duns.sourceforge.net DUNS homepage]
* [[Dolfyn]] -- [http://www.dolfyn.net/dolfyn/index_en.html dolfyn a 3D unstructured general purpose solver - homepage]
*[[Edge]] -- [http://www.foi.se/edge Edge homepage: 2D & 3D compressible RANS / Euler flow solver on unstructured and hybrid grids]
*[[ELMER]] -- [http://www.csc.fi/elmer/ ELMER homepage]
* [[FDS]] -- [http://www.fire.nist.gov/fds/ FDS homepage]
* Featflow -- [http://www.featflow.de Featflow homepage]
* Femwater -- [http://www.epa.gov/ceampubl/gwater/femwater/index.htm Femwater code]
* FreeFEM -- [http://www.freefem.org FreeFEM homepage]
*[[Gerris Flow Solver]] -- [http://gfs.sourceforge.net/ Gerris Flow Solver homepage]
* IMTEK Mathematica Supplement (IMS) -- [http://www.imtek.uni-freiburg.de/simulation/mathematica/IMSweb/ IMTEK Mathematica Supplement (IMS) homepage]
* iNavier -- [http://inavier.sourceforge.net/ iNavier Solver Home Page]
* ISAAC -- [http://isaac-cfd.sourceforge.net ISAAC Home Page]
* MFIX -- [http://www.mfix.org Computational multiphase flow homepage]
*[[NaSt2D-2.0]] -- [http://home.arcor.de/drklaus.bauerfeind/nast/eNaSt2D.html NaSt2D-2.0 homepage]
*[[NEK5000]] -- [http://nek5000.mcs.anl.gov NEK5000 homepage]
*[[NSC2KE]] -- [http://www-rocq1.inria.fr/gamma/cdrom/www/nsc2ke/eng.htm NSC2KE homepage]
* NUWTUN -- [http://nuwtun.berlios.de NUWTUN Home Page]
*[[OpenFlower]] -- [http://sourceforge.net/projects/openflower/ OpenFlower homepage]
*[[OpenFOAM]] -- [http://www.openfoam.org/ OpenFOAM homepage]
* OpenFVM -- [http://openfvm.sourceforge.net/ OpenFVM homepage]
* PETSc-FEM -- [http://www.cimec.org.ar/petscfem PETSc-FEM homepage]
* PP3D -- [http://www.featflow.de/ parpp3d++ homepage]
* SLFCFD -- [http://slfcfd.sourceforge.net SLFCFD homepage]
*[[SSIIM]] -- [http://folk.ntnu.no/nilsol/cfd/ CFD at NTNU]
*[[Tochnog]] -- [http://tochnog.sourceforge.net Tochnog homepage]
* Typhon solver -- [http://typhon.sf.net Typhon solver homepage]

=== Grid generation ===
*[[Delaundo]] -- [http://www.cerfacs.fr/~muller/delaundo.html Delaundo homepage]
* GMSH -- [http://www.geuz.org/gmsh/ GMSH hompage]
* NETGEN -- [http://www.hpfem.jku.at/netgen/ NETGEN homepage]
* SALOME -- [http://www.salome-platform.org SALOME homepage]
* TETGEN -- [http://tetgen.berlios.de/ TETGEN hompage]
* CartGen -- [http://mehr.sharif.ir/~tav/cartgen.htm CartGen homepage]
*[[Triangle]] -- [http://www.cs.cmu.edu/~quake/triangle.html Triangle homepage]
* gridgen -- [http://www.marine.csiro.au/~sakov Pavel Sakov's homepage]

=== Visualization ===
*[[DISLIN]] -- [http://www.mps.mpg.de/dislin/server.html DISLIN homepage]
* GMV -- [http://www-xdiv.lanl.gov/XCM/gmv/ GMV homepage]
*[[Gnuplot]] -- [http://www.gnuplot.info/ gnuplot homepage]
* GRI -- [http://gri.sourceforge.net/ GRI homepage]
*[[Mayavi]] -- [http://mayavi.sourceforge.net/ MayaVi homepage]
*[[OpenDX]] -- [http://www.opendx.org OpenDX homepage]
*[[ParaView]] -- [http://www.paraview.org/HTML/Index.html ParaView homepage]
*[[Tioga]] -- [http://www.kitp.ucsb.edu/~paxton/tioga.html Tioga homepage]
*[[Vigie]] -- [http://www-sop.inria.fr/sinus/Softs/vigie.html Vigie homepage]
*[[Visit]] -- [http://www.llnl.gov/visit Visit homepage]
*[[vtk]] -- [http://www.vtk.org vtk homepage]
*[[vtk.Net]] -- [http://vtkdotnet.sourceforge.net/ vtk.Net homepage]

=== Miscellaneous ===

*[[Engauge Digitizer]] -- [http://digitizer.sourceforge.net Engauge Digitizer homepage]
*[[Ftnchek]] -- [http://www.dsm.fordham.edu/~ftnchek/ ftnchek homepage]
*[[g3data]] -- [http://www.frantz.fi/software/g3data.php g3data homepage]
* GIFMerge -- [http://www.the-labs.com/GIFMerge/ GIFMerge homepage]
*[[Gifsicle]] -- [http://www.lcdf.org/~eddietwo/gifsicle/ Gifsicle homepage]
*[[ImageMagick]] -- [http://www.imagemagick.org ImageMagick homepage]
* nnbathy (natural neighbor interpolation) -- [http://www.marine.csiro.au/~sakov Pavel Sakov's home page]

== Commercial codes ==

=== Solvers ===
* EasyCFD -- [http://www.easycfd.net EasyCFD homepage]
* Applied Computational Fluid Dynamics -- [http://www.partenovcfd.com Solver homepage]
* AcuSolve -- [http://www.acusim.com/ ACUSIM Software's homepage]
* ADINA-F -- [http://www.adina.com/index.html ADINA's homepage]
* ADINA-FSI -- [http://www.adina.com/index.html ADINA's homepage]
* ANSWER -- [http://www.acricfd.com/ ACRi's homepage]
*[http://www.cfd-online.com/W/index.php?title=CFD%2B%2B CFD++] -- [http://www.metacomptech.com Metacomp Techonlogies' homepage]
* CFD2000 -- [http://www.adaptive-research.com/ Adaptive Research's homepage]
*[[CFD-FASTRAN]] -- [http://www.esi-group.com/SimulationSoftware/advanced.html ESI Group's homepage]
* CFD-ACE -- [http://www.esi-group.com/SimulationSoftware/advanced.html ESI Group's homepage]
* CFdesign -- [http://www.cfdesign.com CFdesign's homepage]
* CFX -- [http://www.ansys.com/ ANSYS homepage]
*[[FENSAP-ICE]] -- [http://www.newmerical.com/ NTI' homepage]
* FINE -- [http://www.numeca.be/ Numeca's homepage]
* FIRE -- [http://www.avl.com/ AVL's homepage]
*[[FLACS]] -- [http://www.gexcon.com/index.php?src=flacs/overview.html GexCon's homepage]
* COMSOL -- [http://www.comsol.com/ COMSOL's homepage]
* FloEFD -- [http://www.mentor.com/ Mentor's homepage]
* FloTHERM-- [http://www.mentor.com/ Mentor's homepage]
* FloVENT-- [http://www.mentor.com/ Mentor's homepage]
* FLOW-3D -- [http://www.flow3d.com/ Flow Science's homepage]
* FLOWVISION -- [http://www.fv-tech.com FlowVision's homepage]
*[[FLUENT]] -- [http://www.fluent.com Fluent's homepage]
* [[FLUIDYN]] -- [http://www.fluidyn.com Fluidyn's homepage]
* FluSol -- [http://www.cfd-rocket.com FluSol's hompage]
* Flowz--[http://www.zeusnumerix.com Zeus Numerix's homepage ]
*[[J-FLO]] -- [http://www.newmerical.com NTI's homepage]
* Kameleon FireEx - KFX -- [http://www.computit.com ComputIT's homepage]
* KINetics Reactive Flows -- [http://www.ReactionDesign.com Reaction Design's homepage]
* KIVA--[http://www.lanl.gov/orgs/t/t3/codes/kiva.shtml Los Alamos homepage]
*[[NOGRID FPM]] -- [http://www.no-grid.com NOGRIDS's homepage]
* NX Electronic Systems Cooling -- [http://www.mayahtt.com/index.php?option=com_content&task=view&id=69&Itemid=237 MAYA's NX ESC page]
* NX Advanced Flow -- [http://www.mayahtt.com/index.php?option=com_content&task=view&id=1&Itemid=115 MAYA HTT's NX Adv. Flow page]
* NX Flow -- [http://www.mayahtt.com/index.php?option=com_content&task=view&id=2&Itemid=116 MAYA HTT's NX Flow page]
*[[PHOENICS]] -- [http://www.cham.co.uk CHAM's homepage]
* PowerFLOW -- [http://www.exa.com/pages/pflow/pflow_main.html Exa PowerFLOW homepage]
* PumpLinx -- [http://www.simerics.com Simerics' homepage]
* [[Siemens PLM Software CFD]] -- [http://www.plm.automation.siemens.com/en_us/products/nx/simulation/advanced/index.shtml Siemens PLM Software NX CAE page]
* [[Solution of Boltzmann Equation]] -- [http://www.elegant-mathematics.com/ Elegant Mathematics homepage]
*[[SPLASH]] -- [http://www.panix.com/~brosen SPLASH's homepage]
*[[STAR-CD]] -- [http://www.cd-adapco.com CD-adapco's homepage]
*[[STAR-CCM+]] -- [http://www.cd-adapco.com CD-adapco's homepage]
*[[Tdyn]] -- [http://www.compassis.com CompassIS' homepage]
* TMG-Flow -- [http://www.mayahtt.com/index.php?option=com_content&task=view&id=82&Itemid=283 MAYA HTT's CFD page]
* Turb'Flow -- [http://www.fluorem.com Fluorem's hompage]
* TURBOcfd -- [http://adtechnology.co.uk/products/turbocfd/ TURBOcfd's hompage]

=== Grid generation ===

* ADINA-AUI -- [http://www.adina.com/index.html ADINA's homepage]
* Centaur -- [http://www.centaursoft.com CentaurSoft homepage]
*[[CFD-GEOM]] -- [http://www.esi-group.com/ ESI's homepage]
*[[CFD-VISCART]] -- [http://www.esi-group.com/ ESI's homepage]
* CFDExpert-GridZ --[http://www.zeusnumerix.com/ Zeus Numerix's homepage]
*[[Gridgen]] -- [http://www.pointwise.com/ Pointwise's homepage]
*[[ GridPro]] -- [http://www.gridpro.com/ PDC's homepage]
* Harpoon -- [http://www.ensight.com/ CEI's homepage]
* ICEM CFD -- [http://www.ansys.com/ ANSYS' homepage]
* +ScanFE -- [http://www.simpleware.com/ Simpleware's homepage]
* ANSA -- [http://www.beta-cae.gr/ BETA-CAE's homepage]

=== Visualization ===

* ADINA-AUI -- [http://www.adina.com/index.html ADINA's homepage]
*[[CFD-VIEW]] -- [http://www.esi-group.com/ ESI's homepage]
* CFX-Post -- [http://www.ansys.com/ ANSYS' homepage]
* COVISE -- [http://www.visenso.de/ Visenso's homepage]
* EnSight -- [http://www.ensight.com/ CEI's homepage]
* Fieldview -- [http://www.ilight.com/ Intelligent Light's homepage]
*[[Tecplot]] -- [http://www.tecplot.com/ Tecplot's homepage]
*ViewZ -- [http://www.zeusnumerix.com/ Zeus Numerix's homepage]

=== Systems ===

* ADINA -- [http://www.adina.com/index.html ADINA's homepage]
* Flowmaster -- [http://www.flowmaster.com/index.html Flowmaster's homepage]
* Flownex -- [http://www.flownex.com/ Flownex's homepage]

== Online tools and services ==

*[[CFDNet]] -- [http://www.cfdnet.com/ CFDNet homepage]

CFD-Wiki:Sandbox

2008-08-24T12:07:07Z

Jola:

{{Sandbox table of content}}

<math>F(x, y) = z</math>

ö

[[Page name test]]

[[page name test]]

[[Page Name Test]]

[[Media:Foobar.tar.gz | test of uploaded tar.gz file]]

[[Schemes by Leonard - structured grids#SHARP | SHARP]]

<math>\mu_t</math>

[[foo]]

[[bar]]

test
[http://www.example.com link title][http://www.example.com link title]

CFD-Wiki:Sandbox

2008-08-24T12:02:05Z

Jola:

{{Sandbox table of content}}

<math>F(x, y, a, b) = z</math>

ö

[[Page name test]]

[[page name test]]

[[Page Name Test]]

[[Media:Foobar.tar.gz | test of uploaded tar.gz file]]

[[Schemes by Leonard - structured grids#SHARP | SHARP]]

<math>\mu_t</math>

[[foo]]

[[bar]]

test
[http://www.example.com link title][http://www.example.com link title]

Codes

2008-07-09T18:03:58Z

Jola: Removed advertising

An overview of both free and commercial CFD software. Here you will find short descriptions of codes along with links to resources.

'''Note to contributers:''' Please try to keep descriptions short and to the point (approximately 200 words) and avoid long lists of features or capabilities. Also keep in mind that all contributions are considered to be released under the GNU Free Documentation License 1.2 (see [[Project:Copyrights]] for details). Also note that all information should be verifiable and objective truths that also competitors to the code in question will agree upon. This is especially important if you are an employee of the company selling the code. See the [[CFD-Wiki:Policy]] for further information.

== Free codes ==

This section lists codes that are in the public domain, and codes that are available under GPL, BSD or similar licenses.

=== Solvers ===
* Applied Computational Fluid Dynamics -- [http://www.partenovcfd.com Solver homepage]
* CFD2k -- [http://www.cfd2k.eu/ CFD2k: a 2D-solver for compressible ideal gases - homepage]
* Channelflow -- [http://www.cns.gatech.edu/channelflow/ Channelflow: a spectral Navier-Stokes simulator in C++ homepage]
* Code_Saturne -- [http://rd.edf.com/code_saturne Code_Saturne homepage]
* COOLFluiD -- [http://coolfluidsrv.vki.ac.be/coolfluid COOLFluiD homepage]
* ADFC -- [http://adfc.sourceforge.net/index.html ADFC homepage]
* Diagonalized Upwind Navier Stokes -- [http://duns.sourceforge.net DUNS homepage]
* [[Dolfyn]] -- [http://www.dolfyn.net/dolfyn/index_en.html dolfyn a 3D unstructured general purpose solver - homepage]
*[[Edge]] -- [http://www.foi.se/edge Edge homepage: 2D & 3D compressible RANS / Euler flow solver on unstructured and hybrid grids]
*[[ELMER]] -- [http://www.csc.fi/elmer/ ELMER homepage]
* [[FDS]] -- [http://www.fire.nist.gov/fds/ FDS homepage]
* Featflow -- [http://www.featflow.de Featflow homepage]
* Femwater -- [http://www.epa.gov/ceampubl/gwater/femwater/index.htm Femwater code]
* FreeFEM -- [http://www.freefem.org FreeFEM homepage]
*[[Gerris Flow Solver]] -- [http://gfs.sourceforge.net/ Gerris Flow Solver homepage]
* IMTEK Mathematica Supplement (IMS) -- [http://www.imtek.uni-freiburg.de/simulation/mathematica/IMSweb/ IMTEK Mathematica Supplement (IMS) homepage]
* iNavier -- [http://inavier.sourceforge.net/ iNavier Solver Home Page]
* ISAAC -- [http://isaac-cfd.sourceforge.net ISAAC Home Page]
* MFIX -- [http://www.mfix.org Computational multiphase flow homepage]
*[[NaSt2D-2.0]] -- [http://home.arcor.de/drklaus.bauerfeind/nast/eNaSt2D.html NaSt2D-2.0 homepage]
*[[NSC2KE]] -- [http://www-rocq1.inria.fr/gamma/cdrom/www/nsc2ke/eng.htm NSC2KE homepage]
* NUWTUN -- [http://nuwtun.berlios.de NUWTUN Home Page]
*[[OpenFlower]] -- [http://sourceforge.net/projects/openflower/ OpenFlower homepage]
*[[OpenFOAM]] -- [http://www.openfoam.org/ OpenFOAM homepage]
* OpenFVM -- [http://openfvm.sourceforge.net/ OpenFVM homepage]
* PETSc-FEM -- [http://www.cimec.org.ar/petscfem PETSc-FEM homepage]
* PP3D -- [http://www.featflow.de/ parpp3d++ homepage]
* SLFCFD -- [http://slfcfd.sourceforge.net SLFCFD homepage]
*[[SSIIM]] -- [http://folk.ntnu.no/nilsol/cfd/ CFD at NTNU]
*[[Tochnog]] -- [http://tochnog.sourceforge.net Tochnog homepage]
* Typhon solver -- [http://typhon.sf.net Typhon solver homepage]

=== Grid generation ===
*[[Delaundo]] -- [http://www.cerfacs.fr/~muller/delaundo.html Delaundo homepage]
* GMSH -- [http://www.geuz.org/gmsh/ GMSH hompage]
* NETGEN -- [http://www.hpfem.jku.at/netgen/ NETGEN homepage]
* SALOME -- [http://www.salome-platform.org SALOME homepage]
* TETGEN -- [http://tetgen.berlios.de/ TETGEN hompage]
* CartGen -- [http://mehr.sharif.ir/~tav/cartgen.htm CartGen homepage]
*[[Triangle]] -- [http://www.cs.cmu.edu/~quake/triangle.html Triangle homepage]
* gridgen -- [http://www.marine.csiro.au/~sakov Pavel Sakov's homepage]

=== Visualization ===
*[[DISLIN]] -- [http://www.mps.mpg.de/dislin/server.html DISLIN homepage]
* GMV -- [http://www-xdiv.lanl.gov/XCM/gmv/ GMV homepage]
*[[Gnuplot]] -- [http://www.gnuplot.info/ gnuplot homepage]
* GRI -- [http://gri.sourceforge.net/ GRI homepage]
*[[Mayavi]] -- [http://mayavi.sourceforge.net/ MayaVi homepage]
*[[OpenDX]] -- [http://www.opendx.org OpenDX homepage]
*[[ParaView]] -- [http://www.paraview.org/HTML/Index.html ParaView homepage]
*[[Tioga]] -- [http://www.kitp.ucsb.edu/~paxton/tioga.html Tioga homepage]
*[[Vigie]] -- [http://www-sop.inria.fr/sinus/Softs/vigie.html Vigie homepage]
*[[Visit]] -- [http://www.llnl.gov/visit Visit homepage]
*[[vtk]] -- [http://www.vtk.org vtk homepage]
*[[vtk.Net]] -- [http://vtkdotnet.sourceforge.net/ vtk.Net homepage]

=== Miscellaneous ===

*[[Engauge Digitizer]] -- [http://digitizer.sourceforge.net Engauge Digitizer homepage]
*[[Ftnchek]] -- [http://www.dsm.fordham.edu/~ftnchek/ ftnchek homepage]
*[[g3data]] -- [http://www.frantz.fi/software/g3data.php g3data homepage]
* GIFMerge -- [http://www.the-labs.com/GIFMerge/ GIFMerge homepage]
*[[Gifsicle]] -- [http://www.lcdf.org/~eddietwo/gifsicle/ Gifsicle homepage]
*[[ImageMagick]] -- [http://www.imagemagick.org ImageMagick homepage]
* nnbathy (natural neighbor interpolation) -- [http://www.marine.csiro.au/~sakov Pavel Sakov's home page]

== Commercial codes ==

=== Solvers ===
* Applied Computational Fluid Dynamics -- [http://www.partenovcfd.com Solver homepage]
* AcuSolve -- [http://www.acusim.com/ ACUSIM Software's homepage]
* ADINA-F -- [http://www.adina.com/index.html ADINA's homepage]
* ADINA-FSI -- [http://www.adina.com/index.html ADINA's homepage]
* ANSWER -- [http://www.acricfd.com/ ACRi's homepage]
*[http://www.cfd-online.com/W/index.php?title=CFD%2B%2B CFD++] -- [http://www.metacomptech.com Metacomp Techonlogies' homepage]
* CFD2000 -- [http://www.adaptive-research.com/ Adaptive Research's homepage]
*[[CFD-FASTRAN]] -- [http://www.esi-group.com/SimulationSoftware/advanced.html ESI Group's homepage]
* CFD-ACE -- [http://www.esi-group.com/SimulationSoftware/advanced.html ESI Group's homepage]
* CFdesign -- [http://www.cfdesign.com CFdesign's homepage]
* CFX -- [http://www.ansys.com/ ANSYS homepage]
* COMSOL -- [http://www.comsol.com/ COMSOL's homepage]
* EFD -- [http://www.nika.biz/ Flomerics/NIKA homepage]
*[[FENSAP-ICE]] -- [http://www.newmerical.com/ NTI' homepage]
* FINE -- [http://www.numeca.be/ Numeca's homepage]
* FIRE -- [http://www.avl.com/ AVL's homepage]
*[[FLACS]] -- [http://www.gexcon.com/index.php?src=flacs/overview.html GexCon's homepage]
* FLOW-3D -- [http://www.flow3d.com/ Flow Science's homepage]
* FLOTHERM-- [http://www.flomerics.com Flomerics' homepage]
* FLOVENT-- [http://www.flomerics.com Flomerics' homepage]
* FLOWVISION -- [http://www.fv-tech.com FlowVision's homepage]
*[[FLUENT]] -- [http://www.fluent.com Fluent's homepage]
* FluSol -- [http://www.cfd-rocket.com FluSol's hompage]
* Flowz--[http://www.zeusnumerix.com Zeus Numerix's homepage ]
*[[J-FLO]] -- [http://www.newmerical.com NTI's homepage]
* KINetics Reactive Flows -- [http://www.ReactionDesign.com Reaction Design's homepage]
* KIVA--[http://www.lanl.gov/orgs/t/t3/codes/kiva.shtml Los Alamos homepage]
*[[NOGRID FPM]] -- [http://www.no-grid.com NOGRIDS's homepage]
* NX Electronic Systems Cooling -- [http://www.mayahtt.com/index.php?option=com_content&task=view&id=69&Itemid=237 MAYA's NX ESC page]
* NX Advanced Flow -- [http://www.mayahtt.com/index.php?option=com_content&task=view&id=1&Itemid=115 MAYA HTT's NX Adv. Flow page]
* NX Flow -- [http://www.mayahtt.com/index.php?option=com_content&task=view&id=2&Itemid=116 MAYA HTT's NX Flow page]
*[[PHOENICS]] -- [http://www.cham.co.uk CHAM's homepage]
* PumpLinx -- [http://www.simerics.com Simerics' homepage]
* [[Siemens PLM Software CFD]] -- [http://www.plm.automation.siemens.com/en_us/products/nx/simulation/advanced/index.shtml Siemens PLM Software NX CAE page]
*[[SPLASH]] -- [http://www.panix.com/~brosen SPLASH's homepage]
*[[STAR-CD]] -- [http://www.cd-adapco.com CD-adapco's homepage]
*[[STAR-CCM+]] -- [http://www.cd-adapco.com CD-adapco's homepage]
*[[Tdyn]] -- [http://www.compassis.com CompassIS' homepage]
* TMG-Flow -- [http://www.mayahtt.com/index.php?option=com_content&task=view&id=82&Itemid=283 MAYA HTT's CFD page]
* Turb'Flow -- [http://www.fluorem.com Fluorem's hompage]

=== Grid generation ===

* ADINA-AUI -- [http://www.adina.com/index.html ADINA's homepage]
* Centaur -- [http://www.centaursoft.com CentaurSoft homepage]
*[[CFD-GEOM]] -- [http://www.esi-group.com/ ESI's homepage]
*[[CFD-VISCART]] -- [http://www.esi-group.com/ ESI's homepage]
* CFDExpert-GridZ --[http://www.zeusnumerix.com/ Zeus Numerix's homepage]
*[[Gridgen]] -- [http://www.pointwise.com/ Pointwise's homepage]
*[[ GridPro]] -- [http://www.gridpro.com/ PDC's homepage]
* Harpoon -- [http://www.ensight.com/ CEI's homepage]
* ICEM CFD -- [http://www.ansys.com/ ANSYS' homepage]
* +ScanFE -- [http://www.simpleware.com/ Simpleware's homepage]

=== Visualization ===

* ADINA-AUI -- [http://www.adina.com/index.html ADINA's homepage]
*[[CFD-VIEW]] -- [http://www.esi-group.com/ ESI's homepage]
* CFX-Post -- [http://www.ansys.com/ ANSYS' homepage]
* COVISE -- [http://www.visenso.de/ Visenso's homepage]
* EnSight -- [http://www.ensight.com/ CEI's homepage]
* Fieldview -- [http://www.ilight.com/ Intelligent Light's homepage]
*[[Tecplot]] -- [http://www.tecplot.com/ Tecplot's homepage]
*ViewZ -- [http://www.zeusnumerix.com/ Zeus Numerix's homepage]

=== Systems ===

* ADINA -- [http://www.adina.com/index.html ADINA's homepage]
* Flowmaster -- [http://www.flowmaster.com/index.html Flowmaster's homepage]
* Flownex -- [http://www.flownex.com/ Flownex's homepage]

== Online tools and services ==

*[[CFDNet]] -- [http://www.cfdnet.com/ CFDNet homepage]

Schemes by Leonard - structured grids

2008-07-01T20:23:20Z

Jola: Reverted edits by Rebel (Talk); changed back to last version by Michail

=== SHARP - Simple High Accuracy Resolution Program ===

B. P. Leonard. Simple high-accuracy resolution program for convective modelling of discontinuities.

International Journal for Numerical Methods in Fluids, 8:1291–1318, 1988.

=== ULTIMATE - Universal Limiter for Transport Interpolation Modelling of the Advective Transport Equation ===

B. P. Leonard. Universal limiter for transient interpolation modelling of the advective transport
equations. Technical Memorandum TM-100916 ICOMP-88-11, NASA, 1988.

=== ULTIMATE-QUICKEST ===

B. P. Leonard. The ULTIMATE conservative difference scheme applied to unsteady one–dimensional advection. Computer Methods in Applied Mechanics and Engineering, 88:17–74,
June 1991.

=== ULTRA-SHARP : Universal Limiter for Thight Resolution and Accuracy in combination with the Simple High-Accuracy Resolution Program (also ULTRA-QUICK) ===

B. P. Leonard and S. Mokhtari.
Beyond first-order upwinding: the ULTRA-SHARP alternative for non-oscillatory steady state simulation of convection. International Journal of Numerical
Methods in Engineering, 30:729–766, 1990.

B. P. Leonard and S. Mokhtari.
ULTRA-SHARP nonoscillatory convection schemes for highspeed steady multidimensional flow. Technical Memorandum TM-102568 ICOMP-90-12,
NASA, April 1990.

=== UTOPIA - Uniformly Third Order Polynomial Interpolation Algorithm ===

B. P. Leonard, M. K. MacVean, and A. P. Lock.

Positivity-preserving numerical schemes for multidimensional advection. Technical Memorandum TM-106055 ICOMP-93-05, NASA, March 1993.

=== NIRVANA - Non-oscilatory Integrally Reconstructed Volume-Avaraged Numerical Advection scheme ===

B. P. Leonard, A. P. Lock, and M. K. MacVean.
The NIRVANA scheme applied to one–dimensional advection. International Journal of Numerical Methods in Heat and Fluid Flow,
5:341–377, 1995.

=== ENIGMATIC - Extended Numerical Integration for Genuinely Multidimensional Advective Transport Insuring Conservation ===

B. P. Leonard, A. P. Lock, and M. K. MacVean.

Extended numerical integration for genuinely multidimensional advective transport insuring conservation.

In C. Taylor and P. Durbetaki, editors, Numerical Methods in Laminar and Turbulent Flow, volume 9, pages 1–12. Pineridge
Press, 1995.

=== MACHO : Multidimensional Advective - Conservative Hybrid Operator ===

B. P. Leonard, A. P. Lock, and M. K. MacVean.

Conservative explicit unrestricted-timestep multidimensional constancy-preserving advection schemes. Monthly Weather Review,
124:2588–2606, November 1996.

=== COSMIC : Conservative Operator Splitting for Multidimensions with Internal Constancy ===

B. P. Leonard, A. P. Lock, and M. K. MacVean. Conservative explicit unrestricted-timestep multidimensional constancy-preserving advection schemes. Monthly Weather Review, 124:2588–2606, November 1996.

=== QUICKEST - Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms ===

B. P. Leonard. Elliptic systems: Finite-difference method IV. In W. J. Minkowycz, E. M. Sparrow, G. E. Schneider, and R. H. Pletcher, editors, Handbook of Numerical Heat Transfer, pages 347–378. Wiley, New York, 1988.

=== AQUATIC - Adjusted Quadratic Upstream Algorithm for Transient Incompressible Convection ===

B. P. Leonard
A survey of finite differences with upwinding for numerical modelling of the incompressible convective diffusion equation.

In C. Taylor and K. Morgan, editors, Computational
Methods in Transient and Turbulent Flow, pages 1–35. Pineridge Press, Swansea, 1981.

=== EXQUISITE - Exponential or Quadratic Upstream Interpolation for Solution of the Incompressible Transport Equation ===

B. P. Leonard.

A survey of finite differences with upwinding for numerical modelling of the incompressible convective diffusion equation.

In C. Taylor and K. Morgan, editors, Computational Methods in Transient and Turbulent Flow, pages 1–35. Pineridge Press, Swansea, 1981.

=== EULER-QUICK ===

B.P.Leonard

Locally modified QUICK scheme for highly convective 2D and 3D flows,

in: Proc. 5th International Conf. on Numerical Methods in Laminar and Turbulent Flow, Montreal (1987) 35-47.

---------------------------------------------------------------------
----
<i> Return to [[Numerical methods | Numerical Methods]] </i>

<i> Return to [[Approximation Schemes for convective term - structured grids]] </i>

FAQ's

2008-04-12T19:04:01Z

Jola:

;[[General CFD FAQ]]
:An FAQ for general CFD related questions.

;[[CFD-Wiki:FAQ|CFD-Wiki FAQ]]
:FAQ's about CFD-Wiki.

==Commercial codes==

;[[Ansys FAQ]]
:An FAQ for the software products sold by Ansys Inc - CFX, ICEM CFD, ...

;[[CD-adapco FAQ]]
:An FAQ for the software products sold by CD-adapco - STAR-CD, STAR-CCM+, ...

;[[CFD-ACE FAQ]]
:An FAQ for the software sold by ESI Software - CFD-ACE, [[CFD-GEOM]] ...

;[[CHAM FAQ]]
:An FAQ for the software sold by CHAM - Phoenics, ...

;[[EFD and FloWorks FAQ]]
:An FAQ for the software sold by Flomerics - EFD, FloWorks, ...

;[[Flomerics FAQ]]
:An FAQ for the software products sold by Flomerics - Flotherm, EFD.Pro, EFD.Lab, EFD.V5, FloPCB, FloPACK....

;[[Fluent FAQ]]
:An FAQ for the software products sold by Fluent Inc - FLUENT, Gambit, Tgrid, POLYFLOW, FloWizard, IcePak, AirPak, ...

;[[Numeca FAQ]]
:An FAQ for the software sold by Numeca - Fine, HEXPRESS, ...

;[[Siemens PLM Software CFD FAQ]]
:An FAQ for the software products sold by Siemens PLM Software - NX Flow, NX Electronic Systems Cooling, NX Thermal, ...

[[Category: FAQ's]]

EFD and FloWorks FAQ

2008-04-12T19:02:52Z

Jola: just an empty stub

== EFD ==

=== Question 1 ===

Answer 1

=== Question 2 ===

Answer 2

== FloWorks ==

[[Category: FAQ's]]

{{Stub}}

CFD-ACE FAQ

2008-04-12T19:01:40Z

Jola: Reverted edits by Sivaramakrishnaiah (Talk); changed back to last version by Jola

== CFD-ACE ==

=== Question 1 ===

Answer 1

=== Question 2 ===

Answer 2

== CFD-GEOM ==

[[Category: FAQ's]]

{{Stub}}

Application areas

2008-03-20T15:57:22Z

Jola: Changed order to albhabetic

This section is not very well developed yet. Most application areas just contain a short introduction and some links. Please feel free to expand an application area. Also make sure to name sub-pages so that their names clearly show which application area the page belongs to. The following application areas have so far been listed here:

*[[Aerospace]]
*[[Architecture]]
*[[Automotive]]
*[[Civil engineering]]
*[[Movies and computer graphics]]
*[[Process industry]]
*[[Semiconductor industry]]
*[[Steel industry]]
*[[Turbomachinery]]

{{stub}}

User talk:Combustion

2008-03-12T19:41:35Z

Jola: New page: I saw your attempts to edit the Combustion article. It seems as if your browser for some reason damages the content formatting, removes new-lines etc. I do not know why and I reverted your...

I saw your attempts to edit the Combustion article. It seems as if your browser for some reason damages the content formatting, removes new-lines etc. I do not know why and I reverted your edits to restore the original formatting. You might try to use another browser to see if that solves the problem. Which browser were you using? --[[User:Jola|Jola]] 13:41, 12 March 2008 (MDT)

Combustion

2008-03-12T19:38:53Z

Jola: Reverted edits by Combustion (Talk); changed back to last version by Joan

== What is combustion -- physics versus modelling ==

Combustion phenomena consist of many physical and chemical processes which exhibit a
broad range of time and length scales. A mathematical description of combustion is not
always trivial, although some analytical solutions exist for simple situations of
laminar flame. Such analytical models are usually restricted to problems in zero or one-dimensional space.

= Fundamental Aspects =

== Main Specificities of Combustion Chemistry ==

Combustion can be split into two processes interacting with each other: thermal, and chemical.

The chemistry is highly exothermal (this is the reason of its use) but also highly temperature dependent, thus highly self-accelerating. In a simplified form, combustion can be represented by a single irreversible reaction involving 'a' fuel and 'an' oxidizer:
:<math> \frac{\nu_F}{\bar M_F}Y_F + \frac{\nu_O}{\bar M_O} Y_O \rightarrow Product + Heat </math>
Althgough very simplified compared to real chemistry involving hundreds of species (and their individual transport properties) and elemental reactions, this rudimentary chemistry has been the cornerstone of combustion analysis and modelling.

The most widely used form for the rate of the above reaction is the Arrhénius law:
:<math> \dot\omega = \rho A \left ( \frac{Y_F}{\bar M_F} \right )^{n_F} \left (\frac{Y_O}{\bar M_O}\right )^{n_O} \exp^{-T_a/T} </math>
<math> T_a </math> is the activation temperature, high in combustion, consistently with the temperature dependence.
This is where the high non-linearity in temperature is modelled. ''A'' is the pre-exponential constant. One of the interpretation of the Arrhénius law comes from gas kinetic theory: the number of molecules whose kinetic energy is larger than the minimum value allowing a collision to be energetic enough to trig a reaction is proportional to the exponential term introduced above divided by the square root of the temperature. This interpretation allows one to think that the temperature dependence of ''A'' is very weak compared to the exponential term. ''A'' is eventually considered as constant.
The reaction rate is also naturally proportional to the molecular density of each of the reactant. Nonetheless, the orders of reaction <math> n_i</math> are different from the stoichiometric coefficients as the single-step reaction is global, not governed by collision for it represents hundreds of elementary reactions.
If one goes into the details, combustion chemistry is based on chain reactions, decomposed into three main steps: (i) generation (where radicals are created from the fresh mixture), (ii) branching (where products and new radicals appear from interaction of radicals with reactants), and (iii) termination (where radicals collide and turn into products). The branching step tends to accelerate the production of active radicals (autocatalytic). The impact is nevertheless small compared to the high non-linearity in temperature. This explains why single-step chemistry has been sufficient for most of the combustion modelling work up to now.

The fact that a flame is a very thin reaction zone separating, and making the transition between, a frozen mixture and an equilibrium is explained by the high temperature
dependence of the reaction term, modelled by a large activation temperature, and a large heat release (the ratio of the burned and fresh gas temperatures is about 7 for typical hydrocarbon flames) leading to a sharp self-acceleration in a very narrow area. To evaluate the order of magnitude of the quantities, the terms in the exponential argument are normalized:
:<math> \beta=\alpha\frac{T_a}{T_s} \qquad \alpha=\frac{T_s-T_f}{T_s} </math>
<math>\beta</math> is named the Zeldovitch number and <math>\alpha</math> the heat release factor.
Here, <math> T_s</math> has been used instead of <math> T_b</math>, the conventional notation for burned gas temperature (at final equilibrium). <math> T_s</math> is actually <math> T_b</math>
for a mixture at stoichiometry and when the flame is adiabatic, i.e. this is the reference highest temperature that can be
obtained in the system. That said, typical value for <math>\beta</math> and <math>\alpha</math> are 10 and 0.9, giving
a good taste of the level of non-linearity of the combustion process with respect to temperature.
Actually, the reaction rate is rewritten as:
:<math> \dot\omega = \rho A \left ( \frac{Y_F}{\bar M_F} \right )^{n_F} \left (\frac{Y_O}{\bar M_O}\right )^{n_O}
\exp^{-\frac{\beta}{\alpha}}\exp^{-\beta\frac{1-\theta}{1-\alpha(1-\theta)}} </math>
where the non-dimensionalized temperature is:
:<math>\theta=\frac{T-T_f}{T_s-T_f}</math>
The non-linearity of the reaction rate is seen from the exponential term:
:* <math> {\mathcal O}(\exp^{-\beta}) </math> for <math>\theta</math> far from unity (in the fresh gas)
:* <math> {\mathcal O}(1) </math> for <math>\theta</math> close to unity (in the reaction zone close to the burned gas whose temperature must be close to the adiabatic one <math> T_s </math>), more exactly <math> 1-\theta \sim {\mathcal O}(\beta^{-1})</math>

[[Image:NonLinearite.jpg|thumb|Temperature Non-Linearity of the Source Term: the Temperature-Dependent Factor of the Reaction Term for Some Values of the Zeldovtich and Heat Release Parameters]]Note that for an infinitely high activation energy, the reaction rate is piloted by a <math>\delta(\theta)</math> function. The figure, beside, illustrates how common values of <math>\beta</math> around 10 tend to make the reaction rate singular around <math>\theta</math> of unity. Two set of values are presented: <math>
\beta = 10</math> and <math>\beta = 8</math>. The first magnitude is the representative value while the second one is a smoother one usually used to ease numerical simulations. In the same way, two values for the heat release <math>\alpha</math> 0.9 and 0.75 are explored. The heat release is seen to have a minor impact on the temperature non-linearity.

== Transport Equations ==

Additionally to the Navier-Stokes equations, at least with variable density, the transport equations for a reacting flow are the energy and species transport equations. In usual notations, the specie ''i'' transport equation is written as:
:<math>\frac{D \rho Y_i}{D t} = \nabla\cdot \rho D_i\vec\nabla Y_i - \nu_i\bar M_i\dot\omega</math>
and the temperature transport equation:
:<math>\frac{D\rho C_p T}{Dt} = \nabla\cdot \lambda\vec\nabla T + Q\nu_F \bar M_F \dot\omega </math>
The diffusion is modelled thanks to Fick's law that is a (usually good) approximation to the rigorous diffusion velocity calculation. Regarding the temperature transport equation, it is derived from the energy transport equation under the assumption of a low-Mach number flow (compressibility and viscous heating neglected). The low-Mach number approximation is suitable for the deflagration regime (as it will be demonstrated [[#Premixed|below]]), which is the main focus of combustion modelling. Hence, the transport equation for temperature, as a simplified version of the energy transport equation, is usually retained for the study of combustion and its modelling.

=== Low-Mach Number Equations ===
In compressible flows, when the motion of the fluid is not negligible compared to the speed of sound (which is the speed at which the molecules can reorganize themselves), the heap of molecules results in a local increase of pressure and temperature moving as an acoustic wave. It means that, in such a system, a proper reference velocity is the speed of sound and a proper pressure reference is the kinetic pressure. A contrario, in low-Mach number flows, the reference speed is the natural representative speed of the flow and the reference pressure is the thermodynamic pressure. Hence, the set of reference quantities to characterize a low-Mach number flow is given in the table below:

Density <math>\rho_o</math> A reference density (upstream, average, etc.)

Velocity <math>U_o</math> A reference velocity (inlet average, etc.)

Temperature <math>T_o</math> A reference temperature (upstream, average, etc.)

Pressure (static) <math>P_o=\rho_o \bar r T_o</math> From Boyle-Mariotte

Length <math>L_o</math> A reference length (representative of the domain)

Time <math>L_o/U_o</math>

Energy <math>C_p T_o</math> Internal energy at constant reference pressure

The equations for fluid mechanics properly adimensionalized can be written:

Mass conservation:
:<math> \frac{D\rho}{Dt} =0 </math>

Momentum:
:<math>\frac{D\rho\vec U}{Dt}=-\frac{1}{\gamma M}\vec\nabla P+\nabla\cdot\frac{1}{Re}\bar\bar\Sigma</math>

Total energy:
:<math>\frac{D\rho e_T}{Dt}-\frac{1}{RePr}\nabla\cdot\lambda\vec\nabla T=-\frac{\gamma-1}{\gamma}\nabla\cdot P\vec U
+\frac{1}{Re}M^2(\gamma-1)\nabla\cdot \vec U\bar\bar\Sigma + \frac{\rho}{C_pT_o(U_o/L_o)}Q\nu_F \bar M_F\dot\omega</math>

Specie:
:<math>\frac{D\rho Y}{Dt}-\frac{1}{ScRe}\nabla\cdot\rho D\vec\nabla Y=-\frac{\rho}{U_o/L_o}\nu\bar M\dot\omega</math>

State law:
:<math> P=\rho T</math>

The low-Mach number equations are obtained considering that <math> M^2 </math> is small. 0.1 is usually taken as the limit, which recovers the value of a Mach number of 0.3 to characterize the incompressible regime.

Considering the energy equation, in addition to the terms with <math> M^2 </math> in factor in the equation, the total energy reduces to internal energy as: <math> e_T = T/\gamma+M^2(\gamma-1)\vec U^2/2 </math>. Moreover, the work of pressure is considered as negligible because the gradient of pressure is negligible (low-Mach number approximation is indeed also named ''isobaric'' approximation) and the flow is assumed close to a divergence-free state.
For the same reason, volumic energy and enthalpy variations are assumed equal as they only differ through the addition of pressure. Hence, redimensionalized, the low-Mach number energy equation leads to the temperature equation as used in combustion analysis:
:<math>\frac{D\rho C_p T}{Dt} = \nabla\cdot \lambda\vec\nabla T + Q\nu_F \bar M_F \dot\omega </math>

______________________________

'''Note:''' The species and temperature equations are not closed as the fields of velocity and density also need to be computed. Through intense heat release in a very small area (the jump in temperature in typical hydrocarbon flames is about seven and so is the drop in density in this isobaric process, and the thickness of a flame is of the order of the millimetre), combustion influences the flow field. Nevertheless, the vast majority of combustion modelling has been developed based on the species and temperature equations, assuming simple flow fields.

=== The Damk&oumlaut;hler Number ===

A flame is a reaction zone. From this simple point of view, two aspects have to be considered: (i) the rate at which it is fed by reactants, let call <math> \tau_d </math> the characteristic time, and
the strength of the chemistry to consume them, let call the characteristic chemical time <math> \tau_c </math>. In combustion, the Damk&oumlaut;hler number, ''Da'', compares these both time scales and, for that
reason, it is one of the most integral non-dimensional groups:
:<math>Da=\frac{\tau_d}{\tau_c}</math>.
If ''Da'' is large, it means that the chemistry has always the time to fully consume the fresh mixture and turn it into equilibrium. Real flames are usually close to this state. The characteristic reaction time, <math> (Ae^{-T_a/T_s})^{-1} </math>, is
estimated of the order of the tenth of a ms. When ''Da'' is low, the fresh mixture cannot be converted by a too weak chemistry. The flow remains frozen. This situation happens with ignition or misfire, for instance.

The picture of a deflagration lends itself to a description based on the Damk&oumlaut;hler number. A reacting wave progresses towards the fresh mixture through preheating of the upstream closest layer. The elevation of the temperature strengthens the chemistry and reduces its characteristic time such that the mixture changes from a low-''Da'' region (far upstream, frozen) to a high-''Da'' region in the flame (intense reaction to equilibrium).

== Conservation Laws ==

The processus of combustion transforms the chemical enthalpy into sensible enthalpy (i.e. rise the temperature of the gases thanks to the heat released). Simple relations can be drawn between species and temperature by studying the source terms appearing in the above equations:
:<math> \frac{Y_F}{\nu_F\bar M_F} - \frac{Y_O}{\nu_O\bar M_O} = \frac{Y_{F,u}}{\nu_F\bar M_F} - \frac{Y_{O,u}}{\nu_O\bar M_O} </math>
:<math> Y_F + \frac{Cp T}{Q} = Y_{F,u} + \frac{CpT_u}{Q} </math>
Hence <math> T_b = T_u + \frac{Q Y_{F,u}}{Cp} </math>, <math> Y_{O,b} = Y_{O,u} - \frac{\nu_O \bar M_O}{\nu_F \bar M_F} Y_{F,u} </math> and <math> Y_{F,b} = 0 </math>. Here, the example has been taken for a lean case.

As mentioned in [[#Main Specificities of Combustion Chemistry|Sec. Main Specificities]], the stoichiometric state is used to non-dimensionalize the conservation equations:
:<math> Y_i^* = Y_{i,u}^* - \theta \qquad ; \qquad Y_i^*=Y_i/Y_{i,s}</math>.

A comprehensive form of the reaction rate can be reconstituted to understand the difficulty of numerically resolving the reaction zone:
:<math> \dot\omega = B \prod_{i=O,F}(Y_{i,u}^*-\theta)^{n_i} \exp{\left ( -\beta\frac{1-\theta}{1-\alpha(1-\theta)}\right)} </math>
where <math> B </math> stands for all the constant terms present in this reaction rate, plus density.

[[Image:NonLineariteii.jpg|thumb|Source Term versus Temperature]]
For the stoichiometric case and a global order of two, the reaction rate is graphed versus the reduced temperature for different values of the heat release and Zeldovitch parameter. A high value of <math>\beta</math> makes the reaction rate very sharp, versus temperature. It means that reaction is significant beyond a temperature level (sometimes called ignition temperature) that is close to one (the exponential term above is non-negligible for <math>1-\theta \sim \beta^{-1}</math>). The heat release has qualitatively the same impact but not so strong. Transposed to the case of a flame sheet, it effectively shows that the reaction exists only in a fraction of the thermal thickness of the flame (the region close to the flame that the latter preheats, hence, where the reduced temperature rises from 0 to 1 here) where the temperature deviates few from the maximal one (density can be assumed as constant and equal to its burned-gas value). Numerically capturing such a sharp reaction zone can be costly and the lower values of
<math>\beta</math> and <math>\alpha</math> as presented here are usually preferred whenever possible.

Most problems in combustion involve turbulent flows, gas and liquid
fuels, and pollution transport issues (products of combustion as well as for example noise
pollution). These problems require not only extensive experimental
work, but also numerical modelling. All combustion models must be validated
against the experiments as each one has its own drawbacks and limits. In this article, we will address the modeling fundamentals only.

In addition to the flow parameters used in fluid mechanics,
new dimensionless parameters are introduced, the most important of which are the [[Karlovitz number]] and the [[Damkholer number]] which represent ratios of chemical and flow time scales, and
the [[Lewis number]] which compares the diffusion speeds of species.
The combustion models are often classified on their capability to deal with the different combustion regimes. <br>
[[Image:SUMMARY_OF_COMBUSTION_MODELS.jpg]]

= Three Combustion Regimes =

Depending on how fuel and oxidizer are brought into contact in the combustion system, different combustion modes or regimes are identified. Traditionally, two regimes have been recognized: the ''premixed'' regime and the ''non-premixed'' regime. Over the last two decades, a third regime, sometime considered as a hybrid of the two former ones to a certain extend, has risen. It has been named ''partially-premixed'' regime.

== The Non-Premixed Regime ==
[[Image:DiffusionFlame.jpg|thumb|Sketch of a diffusion flame]]
This regime is certainly the easiest to understand. Everybody has already seen a lighter, candle or gas-powered stove. Basically, the fuel issues from a nozzle or a simple duct into the atmosphere. The combustion reaction is the oxidization of the fuel. Because fuel and oxidizer are in contact only in a limited region but are separated elsewhere (especially in the feeding system) this configuration is the safest. The non-premixed flame has some other advantages. By controlling the flows of both reactants, it is (theoretically) possible to locate the stoichiometric interface, and thus, the location of the flame sheet. Moreover, the strength of the flame can also be controlled through the same process. Depending on the width of the transition region from the oxidizer to the fuel side, the species (fuel and oxidizer) feed the flame at different rates.
This is because the diffusion of the species is directly dependent on the unbalance (gradient) of their distribution. A sharp transition from fuel to oxidizer creates intense diffusion of those species towards the flame, increasing its burning rate. This burning rate control through the diffusion process is certainly one of the reasons of the alternate name of such a flame and combustion mode: ''diffusion'' flame and ''diffusion'' regime.

Because a diffusion flame is fully determined by the inter-penetration of the fuel and oxidizer streams, it has been convenient to introduce a tracer of the state of the mixture. This is the role of the ''mixture fraction'', usually called ''Z'' or ''f''. Z is usually taken as unity in the fuel stream and is null in the oxidizer stream. It varies linearly between this two bounds such that at any point of a frozen flow the fuel mass fraction is given by <math> Y_F=ZY_{F,o} </math> and the oxidizer mass fraction by <math> Y_O = (1-Z)Y_{O,o} </math>. <math> Y_{F,o} </math> and <math> Y_{O,o} </math> are the fuel and oxidizer mass fractions in the fuel and oxidizer streams, respectively.
The mixture fraction posses a transport equation that is expected to not have any source term as a tracer of a mixture must be a conserved scalar. First, the fuel and oxidizer mass fraction transport equations are written in usual notations:
:<math>\frac{D \rho Y_F}{D t} = \nabla\cdot \rho D\vec\nabla Y_F - \nu_F\bar M_F\dot\omega</math>
:<math>\frac{D \rho Y_O}{D t} = \nabla\cdot \rho D\vec\nabla Y_O - \nu_O\bar M_O\dot\omega</math>
The two above equations are linearly combined in a single one in a manner that the source term disappears:
:<math>\frac{D \rho (\nu_O\bar M_O Y_F-\nu_F\bar M_F Y_O)}{D t} = \nabla\cdot \rho D\vec\nabla (\nu_O\bar M_O Y_F-\nu_F\bar M_F Y_O)</math>
The quantity <math>(\nu_O\bar M_O Y_F-\nu_F\bar M_F Y_O)</math> is thus a conserved scalar. The last step is to normalize it such that it equals unity in the pure fuel stream (<math> Y_F=Y_{F,o}</math> and <math> Y_O=0 </math>) and is null in the pure oxidizer stream
(<math> Y_F=0</math> and <math> Y_O=Y_{O,o} </math>). The resulting normalized passive scalar is the mixture fraction:
:<math>Z=\frac{\Phi(Y_F/Y_{F,o})-(Y_O/Y_{O,o})+1}{\Phi+1}\qquad \Phi=\frac{\nu_O\bar M_O Y_{F,o}}{\nu_F\bar M_F Y_{O,o}}</math>
governed by the transport equation
:<math>\frac{D \rho Z}{D t} = \nabla\cdot \rho D\vec\nabla Z </math>
The stoichiometric interface location (and thus the approximate location of the flame if the flow is reacting) is where <math>\Phi(Y_F/Y_{F,o})-(Y_O/Y_{O,o}) </math> vanishes (or <math>Y_F</math> and <math> Y_O </math> are both null in the reacting case). This leads to a stoichiometry definition:
:<math>Z_s=\frac{1}{1+\Phi}</math>

As the mixture fraction qualifies the degree of inter-penetration of fuel and oxidizer, the elements originally present in these molecules are conserved and can be directly traced back to the mixture fraction. This has led to an alternate defintion of the mixture fraction, based on ''element conservation''.
First, the elemental mass fraction <math> X_{j} </math> of element ''j'' is linked to the species mass fraction <math> Y_i </math>:
:<math> X_j = \sum_{i=1}^{n} \frac{a_{i,j} \bar M_j}{\bar M_i} Y_i </math>
where <math> a_{i,j} </math> is a matrix counting the number of element ''j'' atoms in specie molecule named ''i'' and ''n'' is the number of species in the mixture.
The group pictured by the summation above is a linear combination of <math> Y_i </math>. Because the transport equations of species mass fraction, few lines earlier, are also linear, a transport equation for the elemental mass fraction can be written:
:<math>\frac{D \rho X_j}{D t} = \nabla\cdot \sum_{i=1}^n\frac{a_{i,j}\bar M_j}{\bar M_i}\rho D_i\vec\nabla Y_i</math>
For mass is conserved, the linear combination of the source terms vanishes. Furthermore, by taking the same diffusion coefficient <math> D_i </math> for all the species, the elemental mass fraction transport equation has exactly the same form as the specie transport equation (except the source term). Notice that the assumption of equal diffusion coefficient was also made in the previous definition of the mixture fraction and is justified in turbulent combustion modelling by the turbulence diffusivity flattening the diffusion process in high Reynolds number flows. Hence, the elemental mass fraction transport equation has the same structure as the mixture fraction transport equation seen above. Properly renormalized to reach unity in the fuel stream and zero in the oxidizer stream, the elemental mass fraction is a convenient way of determining the mixture fraction field in a flow. Indeed, it is widely used in practice for this purpose.

==== Dissipation Rate ====
A very important quantity, derived from the mixture fraction concept, is the ''scalar dissipation rate'', usually noted: <math> \chi </math>. In the above introduction to non-premixed combustion, it has been said that a diffusion flame is fully controlled through: (i) the position of the stoichiometric line, dictating where the flame sheet lies; (ii) the gradients of fuel on one side and oxidizer on the other side, dictating the feeding rate of the reaction zone through diffusion and thus the strength of combustion. According the the mixture fraction definition, the location of the stoichiometric line is naturally tracked through the <math>Z_s</math> iso-line and it is seen here how the mixture fraction is a convenient tracer to locate the flame.
In the same manner, the mixture fraction field should also be able to give information on the strength of the chemistry as the gradients of reactants are directly linked to the mixture fraction distribution. The feeding rate of the reaction zone is characterized through the inverse of a time. Because it is done through diffusion, it must be obtained through a combination of mixture fraction gradient and diffusion coefficient (dimensional analysis):
:<math> \chi_s = \frac{(\rho D)_s}{\rho_s} ||\vec \nabla Z||^2_s \qquad (\rho D)_s = \left ( \frac{\lambda}{C_p} \right )_s</math>
where the subscript ''s'' refers to quantities taken effectively where the reacting sheet is supposed to be, close to stoichiometry. This deduction of the scalar dissipation rate, scaling the feeding rate of the flame, is obtained here through physical arguments and is to be derived from equations below in a more mathematical manner. Note that the transport coefficient for the mixture fraction is identified to the one for temperature. This notation is usually used in the literature to emphasize that the rate of temperature diffusion (that is commensurable to the rate of species diffusion) is the retained parameter (as introduced in the following approach highlighting the role of the scalar dissipation rate).

Because combustion is highly temperature-dependent, ''T'' is certainly the scalar to which attention must be paid for. The temperature equation in the Low-Mach Number regime ([[#Low-Mach Number Equations|Sec. Low-Mach Number Equations]]) is written below in steady-state:
:<math>\rho\vec U \cdot \vec\nabla T = \nabla\cdot \frac{\lambda}{C_p}\vec\nabla T + \frac{Q}{C_p}\nu_F \bar M_F \dot\omega </math>
In order to make this equation easily tractable, the Howarth-Dorodnitzyn transform and the Chapman approximation are applied.
In the Chapman approximation, the thermal dependence of <math>\lambda/C_p</math> is approximated as <math>\rho^{-1}</math>.
The Howarth-Dorodnitzyn transform introduces <math>\rho</math> in the space coordinate system: <math> \vec\nabla \rightarrow \rho\vec\nabla \quad ; \quad \nabla\cdot\rightarrow \rho\nabla\cdot</math>. The effect of these both mathematical operations is to `digest' the thermal variation of quantities such as density or transport coefficient. Hence, the temperature equation comes in a simpler mathematical shape:
:<math>\frac{\rho}{\rho_s}\vec U \cdot \vec\nabla T = \left ( \frac{\lambda}{C_p}\right )_s \nabla\cdot \vec\nabla T + \frac{Q}{\rho\rho_s C_p}\nu_F \bar M_F \dot\omega </math>
Here the references are taken in the flame, i.e. close to the stoichiometric line (''s'' subscript).

In a non-premixed system, strictly speaking, <math> \vec U </math>, is not really relevant as the flame is fully controlled by the diffusion process. Notwithstanding, in practice, non-premixed flames must be stabilized by creating a strain in the direction of diffusion. This is the reason why the velocity is left in the equation. Because a diffusion flame is fully described by the mixture fraction field, a change in coordinate can be applied:
:<math> \left (
\begin{array}{c}
x \\
y
\end{array}\right ) \longrightarrow
\left (
\begin{array}{c}
x \\
Z
\end{array}\right )
</math>
where ''x'' is the coordinate tangential to the iso-<math>Z_s</math> (hence to the flame, in a first approximation) and ''y'' is perpendicular.
The Jacobian of the transform is given as:
:<math> \left [
\begin{array}{cc}
\frac{\partial x}{\partial x} & \frac{\partial Z}{\partial x} \\
\frac{\partial x}{\partial y} & \frac{\partial Z}{\partial y}
\end{array}\right ] =
\left [
\begin{array}{cc}
1 & 0 \\
0 & l_d^{-1}
\end{array}\right ]</math>
Note that the diffusive layer of thickness <math>l_d</math> is defined as the region of transition between fuel and oxidizer and is thus given by the gradient of ''Z'' along the ''y'' direction.
This transform is applied to the vectorial operators:
:<math> \nabla\cdot = \nabla_x\cdot+\nabla_y\cdot=\nabla_x\cdot+\nabla_y\cdot Z\nabla_Z\cdot</math>
:<math> \vec\nabla = \vec\nabla_x+\vec\nabla_y=\vec\nabla_x+\vec\nabla_y Z\nabla_Z\cdot</math>

With this transform, the above temperature equation looks like:
:<math>\frac{\rho}{\rho_s}\vec U \cdot (\vec\nabla_x T + \vec\nabla Z \nabla_Z\cdot T) = \left (\frac{\lambda}{C_p}\right )_s \left ( \nabla_x\cdot \vec\nabla_x T + \nabla_y\cdot Z \nabla_Z\cdot \vec\nabla_y Z \nabla_Z\cdot T + \nabla_y\cdot Z \nabla_Z\cdot \vec\nabla_x T + \nabla_x\cdot \vec\nabla_y Z \nabla_Z\cdot T \right ) + \frac{Q}{\rho\rho_s C_p}\nu_F \bar M_F \dot\omega </math>
As mentioned above, the velocity and the variation along the tangential direction to the main flame structure ''x'' are not supposed to play a major role. By emphasizing the role of the gradient of ''Z'' along ''y'' as a key parameter defining the configuration the following equation is obtained:
:<math>0 = \left (\frac{\lambda}{C_p}\right )_s ||\vec\nabla_y Z||^2 \Delta_Z T + \frac{Q}{\rho\rho_s C_p}\nu_F \bar M_F \dot\omega </math>
This equation (sometimes named the ''flamelet equation'') serves as the basic framework to study the structure of diffusion flames. It highlights the role of the dissipation rate with respect to the strength of the source term and shows that the dissipation rate calibrates the combustion intensity.

To describe the structure of the diffusion flame, the ''reduced mixture fraction'' is set:
:<math> \xi = \frac{Z-Z_s}{Z_s(1-Z_s)\varepsilon} </math>
The utility of the reduced mixture fraction is to focus on the reaction zone. This reaction zone is supposed located on the stoichiometric line (this is why the reduced mixture fraction is centred on <math>Z_s</math>) and to be very thin (reason of the introduction of the magnifying factor <math>\varepsilon</math>).

== The Premixed Regime ==
[[Image:Premixed.jpg|thumb|Sketch of a premixed flame]]
In contrast to the non-premixed regime above, the reactants are here well mixed before entering the combustion chamber. Chemical reaction can occur everywhere and this flame can propagate upstream into the feeding system as a subsonic (deflagration regime) chemical wave. This presents lots of safety issues. Some situations prevent them: (i) the mixture is made too rich (lot of fuel compared to oxidizer) or too lean (too much oxidizer) such that the flame is close to its flammability limits (it cannot easily propagate); (ii) the feeding system and regions where the flame is not wanted are designed such that they impose strong heat loss to the flame in order to quench it. For a given thermodynamical state of the mixture (composition, temperature, pressure), the flame has its own dynamics (speed, heat release, etc) on which there is few control: the wave exchanges mass and energy through diffusion process in the fresh gases. On the other hand, those well defined quantities are convenient to describe the flame characteristics. The mechanism of spontaneous propagation
towards fresh gas through the thermal transfer from the combustion zone to the immediate slice of fresh gas
such that the ignition temperature is eventually reached for this latter was highlighted as early as by the end of the 19th century by Mallard and LeChatelier.
The reason the chemical wave is contained in a narrow region of reaction propagating upstream is the consequence of the discussion on the non-linearity of the combustion with temperature in the
[[#Fundamental Aspects|Sec. Fundamental Aspects]]. It is of interest to compare the orders of magnitude of the temperature dependent term <math> \exp{(-\beta(1-\theta)/(1-\alpha(1-\theta)))}</math> of the reaction source upstream in the fresh gas (<math>\theta\rightarrow 0</math>) and in the reaction zone close to equilibrium temperature (<math>\theta\rightarrow 1 </math>) for the set of representative values: <math> \beta = 10 </math> and
<math>\alpha=0.9</math>. It is found that the reaction is about <math>10^{43}</math> times slower in the fresh
gas than close to the burned gas. It is known that the chemical time scale is about 0.1 ms in the reaction zone of a typical flame, then the typical reaction time in the fresh gas in normal conditions is about
<math> 10^{39} s </math>. To be compared with the order of magnitude of the estimated Universe age: <math> 1 0^{17} s </math>. Non-negligible chemistry is only confined in a thin reaction zone stuck to the hot burned gas at equilibrium temperature. In this zone, the [[#Damk&oumlaut;hler]] number is high, in contrast to in the fresh mixture. It is natural and convenient to consider that the reaction rate is strictly zero everywhere except in this small reaction zone (one recovers the Dirac-like shape of the reaction profile,
provided that one can see the upstream flow as a region of increasing temperature towards the combustion zone and the downstream flow as in fully equilibrium).

As the premixed flame is a reaction wave propagating from burned to fresh gases, the basic parameter is known to be the ''progress variable''. In the fresh gas, the progress variable is conventionally put to zero. In the burned gas, it equals unity. Across the flame, the intermediate values describe the progress of the reaction to turn into burned gas the fresh gas penetrating the flame sheet. A progress variable can be set with the help of any quantity like temperature, reactant mass fraction, provided it is bounded by a single value in the burned gas and another one in the fresh gas. The progress variable is usually named ''c'', in usual notations:
:<math>c=\frac{T-T_f}{T_b-T_f}</math>
It is seen that ''c'' is a normalization of a scalar quantity. As mentioned above, the scalar transport equations are assumed linear such that the transport equation for ''c'' can be obtained directly.
Actually, the transport equation for ''T'' ([[#Transport Equations|Sec. Transport Equations]]) is linear if constant heat capacity is further assumed (combustion of hydrocarbon in air implies a large excess of nitrogen whose heat capacity is only slightly varying) and the progress variable equation is directly
obtained (here for a default of fuel - lean combustion):
:<math>\frac{D\rho c}{Dt}=\nabla\cdot\rho D\vec\nabla c + \frac{\nu_F\bar M_F}{Y_{F,u}}\dot\omega \qquad \rho D = \frac{\lambda}{C_p} </math>

The fact that the default or excess of fuel has been discussed above leads to the introduction of another quantity: the ''equivalence ratio''. The equivalence ratio, usually noted <math>\Phi</math>, is the ratio of two ratios. The first one is the ratio of the mass of fuel with the mass of oxidizer in the mixture. The second one is the same ratio for a mixture at stoichiometry. Hence, when the equivalence ratio equals unity, the mixture is at stoichiometry. If it is greater than unity, the mixture is named ''rich'' as there is an excess of fuel. In contrast, when it is smaller than unity the mixture is named ''lean''. The equivalence ratio presented here for premixed flames has little connection with the equivalence ratio introduced earlier regarding the non-premixed regime. Basically, the equivalence ratio as defined for non-premixed flames gives the equivalence ratio of a premixed mixture with the same mass of fuel and oxidizer. Moreover, the equivalence ratio as defined for a premixed mixture can be obtained based on the mixture fraction (it is thus the local equivalence ratio at a point in the non-homogeneous mixture described by the mixture fraction). From the definitions given above:
:<math> \Phi=\frac{Z}{1-Z}\frac{1-Z_s}{Z_s}</math>

==== Premixed Flame Péclet Number ====

Earlier in this section, it has been said that a premixed flame posses its own dynamics, as a free propagating surface, and has thus characteristic quantities. For this reason, a Péclet number may be defined, based on these quantities. The Péclet number has the same structure as the Reynolds number but the dynamical viscosity is replaced by the ratio of the thermal conductivity and the heat capacity of the mixture. The thickness <math>\delta_L </math> of a premixed flame is essentially thermal. It means that it corresponds to the distance of the temperature rise between fresh and burned gases. This thickness is below the millimetre for conventional flames. The width of the reaction zone inside this flame is even smaller, by about one order of magnitude. This reaction zone is stuck to the hot side of the flame due to the high thermal dependency of the combustion reactions, as seen above. Hence, the flame region is essentially governed by a convection-diffusion process, the source term being negligible in most of it.

It is convenient to write the progress variable transport equation in a steady-state framework. The quantities at flame temperature (<math>_f</math>) are used to non-dimensionalize the equation:
:<math> \overbrace{(\rho S_L)}^{\dot M}\frac{\vec S_L}{S_L}\vec\nabla c = (\rho D)_f \nabla\cdot (\rho D)^* \vec\nabla c</math>
Note that the source term is neglected, consistently with what has been said above.
This convection-diffusion equation makes appear a first approximation of a flame Péclet number:
:<math> Pe_f = \frac{\dot M \delta_L}{(\rho D)_f} \approx 1 </math>

From the Péclet number, it is possible to obtain an expression for the flame velocity (remembering that <math> \delta_L/S_{L,f} \approx \tau_c</math>, vid. inf. [[#Three Turbulent-Flame Interaction Regimes| Sec. Three Turbulent-Flame Interaction Regimes]]):
:<math> S_{L,f}^2\approx \frac{(\rho D)_f}{\rho_f \tau_c}</math>
For typical hydrocarbon flames, the speed is some tens of centimetres per second and the diffusivity is some
<math> 10^{-5} </math> square metres per second. The chemical time in the reaction zone of about one tenth of a millisecond is recovered.

==== Details of the Premixed Unstrained Planar Flame ====

A plane combustion wave propagating in a homogeneous fresh mixture is the reference case to describe the premixed regime. At constant speed, it is convenient to see the flame at rest with a flowing upstream mixture. This is actually the way propagating flames are usually stabilized. In the frame of description, the
physics is 1-D, steady with a uniform (the flame is said unstrained) mass flowing across the system. Two types of equations are thus sufficient to describe the problem, the temperature transport equation and the species transport equations, as in [[#Transport Equations|Sec. Transport Equations]]. The transport coefficients will be chosen as equal: <math> \rho D_i = \lambda / C_p </math> (unity Lewis numbers). Suppose the 1-D domain is described thanks to a conventional (''Ox'') axis with a flame propagating towards negative ''x'' (this is the conventional usage), the boundary conditions are:
:* in the frozen mixture:
:** <math>Y_i \rightarrow Y_{i,u} \qquad ; \qquad x\rightarrow-\infty </math>
:** <math>T \rightarrow T_u \qquad ; \qquad x\rightarrow-\infty </math>
:* in the burned gas region supposed at equilibrium:
:** <math>Y_i \rightarrow Y_{i,b} \qquad ; \qquad x\rightarrow+\infty </math>
:** <math>T \rightarrow T_b \qquad ; \qquad x\rightarrow+\infty </math>

:<math>Y_{i,b}</math> and <math>T_b</math> are obtained from [[#Conservation Laws|Sec. Conservation Laws]].

The quantities that have been mentioned just above (scalar and temperature profiles, mass flow rate through the system) are the solution to be sought.

According to the discussions above, the temperature transport equation in its full normalized form may be written as (lean /stoichiometric case):
:<math> \dot M \frac{\partial \theta}{\partial x} = \frac{\partial \ }{\partial x}\frac{\lambda}{Cp} \frac{\partial \theta}{\partial x} + \frac{\nu_F \bar M_F B}{Y_{F,s}} \prod_{i=O,F}(Y_{i,u}^*-\theta)^{n_i} \exp{-\beta\frac{1-\theta}{1-\alpha(1-\theta)}} </math>
This equation is further simplified by the variable change <math>d\xi=\dot M/(\lambda/Cp)dx</math>:
:<math>\frac{\partial \theta}{\partial \xi}=\frac{\partial^2 \theta}{\partial \xi^2} + \overbrace{\frac{\lambda/Cp}{\dot M^2}\frac{\nu_F \bar M_F B}{Y_{F,s}}}^{\Lambda} \prod_{i=O,F}(Y_{i,u}^*-\theta)^{n_i} \exp{-\beta\frac{1-\theta}{1-\alpha(1-\theta)}}</math>

== The Partially-Premixed Regime ==
[[Image:ppf.jpg|thumb| Ideal sketch of a partially-premixed flame]]
This regime is somewhat less academics and has been recognized two decades ago. It is acknowledged as a hybrid of the premixed and the non-premixed regimes but the degree of interaction of these two modes of combustion to accurately describe a partially-premixed flame is still not well understood. It can be simply pictured by a lifted diffusion flame. Let us consider fuel issuing from a nozzle into the air. If the exit velocity is large enough, for some fuels, the flame lifts off the rim of the nozzle. It means that below the flame base, fuel and oxidizer have room to premix. Hence, the flame base propagates into a premixed mixture. However, it cannot be reduced to a premixed flame (although it is often simplified as this): (i) the mixing is not perfect and the different parts of the flame front constituting the flame base burn in mixtures of different thermodynamical states. This provides those parts with different deflagration capabilities such that the flame base has a complex shape. Indeed, it is convex, naturally leaded by the part burning at stoichiometry, unless `exotic' feeding temperatures are used. (ii) Because the mixture is not homogeneous, transfer of species and temperature driven by diffusion occurs in a direction perpendicular to the propagation of the flame base. Because the flame front is not flat, those transfers act as a connection vehicle across the different parts of the leading front. (iii) The unburned left downstream by the sections of the leading front not burning at stoichiometry diffuse towards each other to form a diffusion flame as described above. The connection of the leading front with the trailing diffusion flame has been evidenced as complicated and the siege of transfers of species and temperature. These two last items are the state-of-the-art difficulties in understanding those flames and do not appear in the models although it has been demonstrated they have a major impact and are certainly a fundamental characteristic of partially-premixed flames.

The partially-premixed flame is usually described using ''c'' and ''Z'' as introduced earlier. Because the framework is essentially non-premixed, the mixture fraction is primarily used to describe the flame. Regarding the head of the flame where partial-premixing has an impact, each part of the front is described with a local progress variable. The need of defining a local progress variable is that each section of the partially-premixed front has a different equivalence ratio leading to a different definition of ''c'':
:<math> c=\frac{T-T_u}{T_b(Z)-T_u}</math>

= Three Turbulent-Flame Interaction Regimes =
It may appear odd to try to describe here what is the reason of combustion modelling research: the interaction of the turbulence with chemistry. However, one of the first steps in building knowledge in turbulent combustion was the qualitative exploration of what might be the dynamics of a flame in a turbulent environment. This led to what is now known as ''combustion diagrams''. As explained above, the premixed regime lends itself the easiest to such an approach as it exhibits natural intrinsic quantities which are not as objectively identifiable in the other combustion regimes. Note that these quantities may depend
on the geometry of the flame: for instance turbulence can bend a flame sheet, leading to a change in its
dynamics compared to the flat flame propagating in a medium at rest. In this section, the turbulence-flame interaction modes will be described for a premixed flame. Only remarks will be added regarding the non-premixed and partially-premixed regimes.

An integral quantity to assess the interaction between a premixed flame sheet and the turbulence
is the Karlovitz number ''Ka''. It compares the characteristic time of flame displacement with the characteristic time of the smallest structures (that are also the fastest) of the turbulence.
:<math> Ka= \frac{\tau_c}{\tau_k}</math>

<math>\tau_c</math> is the chemical time of the flame. To estimate it, it is necessary to come back to the above progress variable transport equation in a steady-state framework.
:<math> \overbrace{(\rho S_L)}^{\dot M}\frac{\vec S_L}{S_L}\vec\nabla c = (\rho D)_f\nabla\cdot (\rho D)^* \vec\nabla c + \dot\omega_c</math>
The premixed wave propagates at a speed <math>S_L</math> because it is fed by reactants diffusing inside the combustion zone and which are preheated because temperature diffuses in the reverse direction. The speed at which the flame progresses is thus related to the rate of species diffusion into the reaction zone which are then consumed. As the premixed flame is a free propagating wave whose speed of propagation is only limited by the chemical strength, the characteristic chemical time is based only on the diffusion and the mass flow rate experienced by the flame:
:<math> \tau_c = \frac{\rho (\rho D)_f}{\dot M^2}</math>

The smallest eddies are the ones being dissipated by the viscous forces. Their characteristic time is estimated thanks to a combination of the viscosity and the flux of turbulent energy to be dissipated (also called turbulent dissipation <math> \varepsilon=u'^3/l_t </math>):
:<math>\tau_k = \sqrt{\frac{l_t\nu}{u'^3}}</math>

Thanks to those definitions of chemical and small structure times, it is possible to give another definition of the Karlovitz number:
:<math> Ka=\left (\frac{\delta_L}{l_k} \right)^2</math>
which is the square of the ratio between the premixed flame thickness and the small structure scale: ''Ka'' actually compares scales. To arrive to this latter result, the three following assumptions must be used: (i) the flame thickness is obtained thanks to the premixed flame Péclet number (''vid. sup.''); (ii) the turbulence small structure (''Kolmogorov eddies'') scale is given by: <math> l_k=(\nu^3/\varepsilon)^{1/4} </math> following the same dimensional argument as for the estimation of its time; and (iii) scalar diffusion scales with viscosity.

==== Remark Regarding the Diffusion Flame ====
From what has been presented above, a diffusion flame does not have characteristic scales. Setting a turbulence combustion regime classification for non-premixed flames has still not been answered by research. Some laws of behaviour will only be drawn around the scalar dissipation rate which is the parameter of integral importance for a diffusion flame.

Indeed, the dynamics of a diffusion flame is determined by the strain rate imposed by the turbulence. As for the premixed flames, the shortest eddies (Kolmogorov) are the ones having the largest impact. The diffusive layer is thus given by the size of the Kolmogrov eddies: <math> l_d\approx l_k</math> and the typical diffusion time scale (feeding rate of the reaction zone) is given by the characteristic time of the Kolmogorov eddies: <math> \tau_k^{-1}\approx \chi_s</math> as the Reynolds number of the Kolmogorov structures is unity. Here, <math>\chi_s</math> is the sample-averaging of <math>\chi</math> based on (conditioned) stoichiometric conditions, where the flame is expected to be.

== The Wrinkled Regime ==
[[Image:wrinkled.jpg|thumb| Wrinkled flamelet regime]]
This regime is also called the ''flamelet regime''. Basically, it assumes that the flame structure is not affected by turbulence. The flame sheet is convoluted and wrinkled by eddies but none of them is small enough to enter it.
Locally magnifying, the laminar flame structure is maintained.

This regime exists for a Karlovitz number below unity (vid. sup.), i.e. chemical time smaller than the small structure time or flame thickness smaller than small structure scale. Notwithstanding, the laminar flame dynamics can be disrupted for <math> u'>S_L</math>. In that case, although the flame structure is not altered by the small structures, it can be convected by large structures such that areas of different locations in the front interact. It shows that, even with a small Karlovitz number, the turbulence effect is not always weak.

== The Corrugated Regime ==
[[Image:Corrugated.jpg|thumb|Corrugated flamelet regime]]
The formal definition of a flame is the region of temperature rise. However, the volume where the reaction takes place is about one order of magnitude smaller, embedded inside the temperature rise region and close to its high temperature end. Hence, there exist some levels of turbulence creating eddies able to enter the flame zone but still large enough to not affect the internal reaction sheet. In other words, the flame thermal region is thickened by turbulence but the reaction zone is still in the wrinkled regime.
This situation is called the ''Corrugated Regime''.

Due to the structure of the Karlovitz number, once written in terms of length scales (vid. sup.), this situation arises for an
increase of the Karlovitz number by two orders of magnitude compared to the value for the wrinkled regime. Hence, in the range
<math> 1 < Ka < 100 </math>, the laminar structure of the reaction zone is still preserved but not the one of the preheat zone.

== The Thickened Regime ==
[[Image:thickened.jpg|thumb|Thickened regime]]
In this last case, turbulence is intense enough to generate eddies able to affect the structure of the reaction zone as well. In practice, it is expected that those eddies are in the tail of the energy spectrum such that their lifetime is very short. Their impact on the reaction zone is thus limited.

Obviously, ''Ka > 100''. A topological description is of little relevance here and a ''well-stirred reactor model'' fits better.

== Reaction mechanisms ==

Combustion is mainly a chemical process. Although we can, to some extent,
describe a flame without any chemistry information, modelling of the flame
propagation requires the knowledge of speeds of reactions, product concentrations,
temperature, and other parameters.
Therefore fundamental information about reaction kinetics is
essential for any combustion model.
A fuel-oxidizer mixture will generally combust if the reaction is fast enough to
prevail until all of the mixture is burned into products. If the reaction
is too slow, the flame will extinguish. If too fast, explosion or even
detonation will occur. The reaction rate of a typical combustion reaction
is influenced mainly by the concentration of the reactants, temperature, and pressure.

A stoichiometric equation for an arbitrary reaction can be written as:

<table width="100%">
<tr><td>
:<math>
\sum_{j=1}^{n}\nu' (M_j) = \sum_{j=1}^{n}\nu'' (M_j),
</math></td></tr></table>

where <math> \nu </math> denotes the stoichiometric coefficient, and <math>M_j</math> stands for an arbitrary species. A one prime holds for the reactants while a double prime holds for the products of the reaction.

The reaction rate, expressing the rate of disappearance of reactant <b>i</b>, is defined as:

<table width="100%">
<tr><td>
:<math>
RR_i = k \, \prod_{j=1}^{n}(M_j)^{\nu'},
</math></td></tr></table>

in which <b>k</b> is the specific reaction rate constant. Arrhenius found that this constant is a function of temperature only and is defined as:

<table width="100%">
<tr><td>
:<math>
k= A T^{\beta} \, exp \left( \frac{-E}{RT}\right)
</math></td></table>

where <b>A</b> is pre-exponential factor, <b>E</b> is the activation energy, and <math>\beta</math> is a temperature exponent. The constants vary from one reaction to another and can be found in the literature.

Reaction mechanisms can be deduced from experiments (for every resolved reaction), they can also be constructed numerically by the '''automatic generation method''' (see [Griffiths (1994)] for a review on reaction mechanisms).
For simple hydrocarbons, tens to hundreds of reactions are involved.
By analysis and systematic reduction of reaction mechanisms, global reactions
(from one to five step reactions) can be found (see [Westbrook (1984)]).

== Governing equations for chemically reacting flows ==

Together with the usual Navier-Stokes equations for compresible flow (See [[Governing equations]]), additional equations are needed in reacting flows.

The transport equation for the mass fraction <math> Y_k </math> of <i>k-th</i> species is

:<math>
\frac{\partial}{\partial t} \left( \rho Y_k \right) +
\frac{\partial}{\partial x_j} \left( \rho u_j Y_k\right) =
\frac{\partial}{\partial x_j} \left( \rho D_k \frac{\partial Y_k}{\partial x_j}\right)+ \dot \omega_k
</math>

where Ficks' law is assumed for scalar diffusion with <math> D_k </math> denoting the species difussion coefficient, and <math> \dot \omega_k </math> denoting the species reaction rate.

A non-reactive (passive) scalar (like the mixture fraction <math> Z </math>) is goverened by the following transport equation

:<math>
\frac{\partial}{\partial t} \left( \rho Z \right) +
\frac{\partial}{\partial x_j} \left( \rho u_j Z \right) =
\frac{\partial}{\partial x_j} \left( \rho D \frac{\partial Z}{\partial x_j}\right)
</math>

where <math> D </math> is the diffusion coefficient of the passive scalar.

=== RANS equations ===

In turbulent flows, [[Favre averaging]] is often used to reduce the scales (see [[Reynolds averaging]]) and the
mass fraction transport equation is transformed to

:<math>
\frac{\partial \overline{\rho} \widetilde{Y}_k }{\partial t} +
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Y}_k}{\partial x_j}=
\frac{\partial} {\partial x_j} \left( \overline{\rho D_k \frac{\partial Y_k} {\partial x_j} } -
\overline{\rho} \widetilde{u''_i Y''_k } \right)
+ \overline{\dot \omega_k}
</math>

where the turbulent fluxes <math> \widetilde{u''_i Y''_k} </math> and reaction terms
<math> \overline{\dot \omega_k} </math> need to be closed.

The passive scalar turbulent transport equation is

:<math>
\frac{\partial \overline{\rho} \widetilde{Z} }{\partial t} +
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Z} }{\partial x_j}=
\frac{\partial} {\partial x_j} \left( \overline{\rho D \frac{\partial Z} {\partial x_j} } -
\overline{\rho} \widetilde{u''_i Z'' } \right)
</math>

where <math> \widetilde{u''_i Z''} </math> needs modelling. A common practice is to model the turbulent fluxes using the
gradient diffusion hypothesis. For example, in the equation above the flux <math> \widetilde{u''_i Z''} </math> is modelled as

:<math>
\widetilde{u''_i Z''} = -D_t \frac{\partial \tilde Z}{\partial x_i}
</math>

where <math> D_t </math> is the turbulent diffusivity. Since <math> D_t >> D </math>, the first term inside the parentheses on the right hand of the mixture fraction transport equation is often neglected (Peters (2000)). This assumption is also used below.

In addition to the mean passive scalar equation,
an equation for the Favre variance <math> \widetilde{Z''^2}</math> is often employed

:<math>
\frac{\partial \overline{\rho} \widetilde{Z''^2} }{\partial t} +
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Z''^2} }{\partial x_j}=
\frac{\partial}{\partial x_j}
\left( \overline{\rho} \widetilde{u''_i Z''^2} \right) -
2 \overline{\rho} \widetilde{u''_i Z'' }
- \overline{\rho} \widetilde{\chi}
</math>

where <math> \widetilde{\chi} </math> is the mean [[Scalar dissipation]] rate
defined as <math> \widetilde{\chi} =
2 D \widetilde{\left| \frac{\partial Z''}{\partial x_j} \right|^2 } </math>
This term and the variance diffusion fluxes needs to be modelled.

=== LES equations ===
The [[Large eddy simulation (LES)]] approach for reactive flows introduces equations for the filtered species mass fractions within the compressible flow field.
Similar to the [[#RANS equations]], but using Favre filtering instead
of [[Favre averaging]], the filtered mass fraction transport equation is

:<math>
\frac{\partial \overline{\rho} \widetilde{Y}_k }{\partial t} +
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Y}_k}{\partial x_j}=
\frac{\partial} {\partial x_j} \left( \overline{\rho D_k \frac{\partial Y_k} {\partial x_j} } -
J_j \right)
+ \overline{\dot \omega_k}
</math>

where <math> J_j </math> is the transport of subgrid fluctuations of mass fraction
:<math>
J_j = \widetilde{u_jY_k} - \widetilde{u}_j \widetilde{Y}_k
</math>
and has to be modelled.

Fluctuations of diffusion coefficients are often ignored and their contributions
are assumed to be much smaller than the apparent turbulent diffusion due to transport of subgrid fluctuations.
The first term on the right hand side is then
:<math>
\frac{\partial} {\partial x_j}
\left(
\overline{ \rho D_k \frac{\partial Y_k} {\partial x_j} }
\right)
\approx
\frac{\partial} {\partial x_j}
\left(
\overline{\rho} D_k \frac{\partial \widetilde{Y}_k} {\partial x_j}
\right)
</math>

[[Category:Equations]]

== Infinitely fast chemistry ==
All combustion models can be divided into two main groups according to the
assumptions on the reaction kinetics.
We can either assume the reactions to be infinitely fast - compared to
e.g. mixing of the species, or comparable to the time scale of the mixing
process. The simple approach is to assume infinitely fast chemistry. Historically, mixing of the species is the older approach, and it is still in wide use today. It is therefore simpler to solve for [[#Finite rate chemistry]] models, at the overshoot of introducing errors to the solution.

=== Premixed combustion ===
Premixed flames occur when the fuel and oxidiser are homogeneously mixed prior to ignition. These flames are not
limited only to gas fuels, but also to the pre-vaporised fuels.
Typical examples of premixed laminar flames is bunsen burner, where
the air enters the fuel stream and the mixture burns in the wake of the
riser tube walls forming nice stable flame. Another example of a premixed system is the solid rocket motor where oxidizer and fuel and properly mixed in a gum-like matrix that is uniformly distributed on the periphery of the chamber.
Premixed flames have many advantages in terms of control of temperature and
products and pollution concentration. However, they introduce some dangers like the autoignition (in the supply system).

[[Image:Premixed.jpg]]
==== Turbulent flame speed model ====

==== Eddy Break-Up model ====

The Eddy Break-Up model is the typical example of '''mixed-is-burnt''' combustion model.
It is based on the work of Magnussen, Hjertager, and Spalding and can be found in all commercial CFD packages.
The model assumes that the reactions are completed at the moment of mixing, so that the reaction rate is completely controlled by turbulent mixing. Combustion is then described by a single step global chemical reaction

<table width="100%">
<tr><td>
:<math>
F + \nu_s O \rightarrow (1+\nu_s) P
</math></td><td width="5%"></td></tr></table>

in which <b>F</b> stands for fuel, <b>O</b> for oxidiser, and <b>P</b> for products of the reaction. Alternativelly we can have a multistep scheme, where each reaction has its own mean reaction rate.
The mean reaction rate is given by

<table width="100%">
<tr><td>
:<math>
\bar{\dot\omega}_F=A_{EB} \frac{\epsilon}{k}
min\left[\bar{C}_F,\frac{\bar{C}_O}{\nu},
B_{EB}\frac{\bar{C}_P}{(1+\nu)}\right]
</math></td><td width="5%"></td></tr></table>

where <math>\bar{C}</math> denotes the mean concentrations of fuel, oxidiser, and products
respectively. <b>A</b> and <b>B</b> are model constants with typical values of 0.5
and 4.0 respectively. The values of these constants are fitted according
to experimental results and they are suitable for most cases of general interest. It is important to note that these constants are only based on experimental fitting and they need not be suitable for <b>all</b> the situations. Care must be taken especially in highly strained regions, where the ratio of <math>k</math> to <math>\epsilon</math> is large (flame-holder wakes, walls ...). In these, regions a positive reaction rate occurs and an artificial flame can be observed.
CFD codes usually have some remedies to overcome this problem.

This model largely over-predicts temperatures and concentrations of species like <i>CO</i> and other species. However, the Eddy Break-Up model enjoys a popularity for its simplicity, steady convergence, and implementation.

==== Bray-Moss-Libby model ====

=== Non-premixed combustion ===

Non premixed combustion is a special class of combustion where fuel and oxidizer
enter separately into the combustion chamber. The diffusion and mixing of the two streams
must bring the reactants together for the reaction to occur.
Mixing becomes the key characteristic for diffusion flames.
Diffusion burners are easier and safer to operate than premixed burners.
However their efficiency is reduced compared to premixed burners.
One of the major theoretical tools in non-premixed combustion
is the passive scalar [[mixture fraction]] <math> Z </math> which is the
backbone on most of the numerical methods in non-premixed combustion.

==== Conserved scalar equilibrium models ====
The reactive problem is split into two parts. First, the problem of <i> mixing </i>, which consists of the location of the flame surface which is a non-reactive problem concerning the propagation of a passive scalar; And second, the <i> flame structure </i> problem, which deals with the distribution of the reactive species inside the flamelet.

To obtain the distribution inside the flame front we assume it is locally one-dimensional and
depends only on time and the scalar coodinate.

We first make use of the following chain rules
:<math>
\frac{\partial Y_k}{\partial t} = \frac{\partial Z}{\partial t}\frac{\partial Y_k}{\partial Z}
</math>
<br>
:<math>
\frac{\partial Y_k}{\partial x_j} = \frac{\partial Z}{\partial x_j}\frac{\partial Y_k}{\partial Z}
</math>
and transformation
:<math>
\frac{\partial }{\partial t}= \frac{\partial }{\partial t} + \frac{\partial Z}{\partial t} \frac{\partial }{\partial Z}
</math>

upon substitution into the species transport equation (see [[#Governing Equations for Reacting Flows]]), we obtain

:<math>
\rho \frac{\partial Y_k}{\partial t} + Y_k \left[
\frac{\partial \rho}{\partial t} + \frac{\partial \rho u_j}{\partial x_j}
\right]
+ \frac{\partial Y_k}{\partial Z} \left[
\rho \frac{\partial Z}{\partial t} + \rho u_j \frac{\partial Z}{\partial x_j} -
\frac{\partial}{\partial x_j}\left( \rho D \frac{\partial Z}{\partial x_j} \right)
\right]
=
\rho D \left( \frac{\partial Z}{\partial x_j} \frac{\partial Z}{\partial x_j} \right)
\frac{\partial^2 Y_k}{\partial Z^2} + \dot \omega_k

</math>

The second and third terms in the LHS cancel out due to continuity and mixture fraction transport.
At the outset, the equation boils down to

:<math>
\frac{\partial Y_k}{\partial t} = \frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot \omega_k
</math>

where <math> \chi = 2 D \left( \frac{\partial Z}{\partial x_j} \right)^2 </math> is called the <b>scalar dissipation</b>
which controls the mixing, providing the interaction between the flow and the chemistry.

If the flame dependence on time is dropped, even though the field <math> Z </math> still depends on it.

:<math>
\dot \omega_k= -\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2}
</math>

If the reaction is assumed to be infinetly fast, the resultant flame distribution is in equilibrium.
and <math> \dot \omega_k= 0</math>. When the flame is in equilibrium, the flame configuration <math> Y_k(Z) </math> is independent of strain.

===== Burke-Schumann flame structure =====
The Burke-Schuman solution is valid for irreversible infinitely fast chemistry.
With a reaction in the form of
<table width="100%">
<tr><td>
:<math>
F + \nu_s O \rightarrow (1+\nu_s) P
</math></td><td width="5%"></td></tr></table>

If the flame is in equilibrium and therefore the reaction term is 0.
Two possible solution exists, one with pure mixing (no reaction)
and a linear dependence of the species mass fraction with <math> Z </math>.
Fuel mass fraction
:<math>
Y_F=Y_F^0 Z
</math>
Oxidizer mass fraction
:<math>
Y_O=Y_O^0(1-Z)
</math>
Where <math> Y_F^0 </math> and <math> Y_O^0 </math> are fuel and oxidizer mass fractions
in the pure fuel and oxidizer streams respectively.

The other solution is given by a discontinuous slope at stoichiometric mixture fraction
<math> Z_{st}</math> and two linear profiles (in the rich and lean side) at either
side of the stoichiometric mixture fraction.
Both concentrations must be 0 at stoichiometric, the reactants become products infinitely fast.

:<math>
Y_F=Y_F^0 \frac{Z-Z_{st}}{1-Z_{st}}
</math>
and oxidizer mass fraction
:<math>
Y_O=Y_O^0 \frac{Z-Z_{st}}{Z_{st}}
</math>

== Finite rate chemistry ==

=== Premixed combustion ===

==== Coherent flame model ====

==== Flamelets based on G-equation ====
==== Flame surface density model ====

=== Non-premixed combustion ===

==== Flamelets based on conserved scalar ====

Peters (2000) define Flamelets as "thin diffusion layers embedded in a turbulent non-reactive flow field".
If the chemistry is fast enough, the chemistry is active within a thin region
where the chemistry conditions are in (or close to) stoichiometric conditions, the "flame" surface.
This thin region is assumed to be smaller than Kolmogorov length scale and therefore the
region is locally laminar. The flame surface is defined as an iso-surface of a certain scalar <math> Z </math>,
mixture fraction in [[#Non premixed combustion]].

The same equation used in [[#Conserved scalar models]] for equilibrium chemistry
is used here but with chemical source term different from 0

:<math>
\frac{\partial Y_k}{\partial t} =
\frac{\chi}{2} \frac{\partial ^2 Y_k}{\partial Z^2} + \dot \omega_k.
</math>

This approach is called the Stationary Laminar Flamelet Model (SLFM) and has the advantage that
flamelet profiles <math> Y_k=f(Z,\chi)</math>
can be pre-computed and stored in a dtaset or file which is called a "flamelet library" with all the required complex chemistry.
For the generation of such libraries ready to use software is avalable such as Softpredict's Combustion Simulation Laboratory COSILAB [http://www.softpredict.com] with its relevant solver RUN1DL, which can be used for a variety of relevant geometries; see various [http://www.softpredict.com/?page=989 publications that are available for download]. Other software tools are available such as CHEMKIN [http://www.reactiondesign.com] and
CANTERA [http://www.cantera.org].

===== Flamelets in turbulent combustion =====

In turbulent flames the interest is <math> \widetilde{Y}_k </math>.
In flamelets, the flame thickness is assumed to be much smaller than Kolmogorov scale
and obviously is much smaller than the grid size.
It is therefore needed a distribution of the passive scalar within the cell.
<math> \widetilde{Y}_k </math> cannot be obtained directly from the flamelets library
<math> \widetilde{Y}_k \neq Y_F(Z,\chi) </math>, where <math> Y_F(Z,\chi) </math> corresponds
to the value obtained from the flamelets libraries.
A generic solution can be expressed as
:<math>
\widetilde{Y}_k= \int Y_F( \widetilde{Z},\widetilde{\chi}) P(Z,\chi) dZ d\chi
</math>
where <math> P(Z,\chi) </math> is the joint [[Probability density function | Probability Density Function]] (PDF) of the mixture fraction
and scalar dissipation which account for the scalar distribution inside the cell and "a priori"
depends on time and space.

The most simple assumption is to use a constant distribution of the scalar dissipation within the cell and
the above equation reduces to
:<math>
\widetilde{Y}_k= \int Y_F(\widetilde{Z},\widetilde{\chi}) P(Z) dZ
</math>

<math> P(Z) </math> is the PDF of the mixture fraction scalar and simple models (such as Gaussian or a [[beta PDF]])
can be build depending only on two moments of the scalar
mean and variance,<math> \widetilde{Z},Z''</math>.

If the mixture fraction and scalar dissipation are consider independent variables,<math> P(Z,\chi) </math>
can be written as <math> P(Z) P(\chi)</math>. The PDF of the scalar dissipation is assumed to be log-normal with
variance unity.
:<math>
\widetilde{Y}_k= \int Y_F(\widetilde{Z},\widetilde{\chi}) P(Z) P(\chi) dZ d\chi
</math>
In [[Large eddy simulation (LES)]] context (see [[#LES equations]] for reacting flow),
the [[probability density function]] is replaced by a [[subgrid PDF]] <math> \widetilde{P}</math>.
The same equation hold by replacing averaged values with filtered values.
:<math>
\widetilde{Y}_k= \int Y_F(\widetilde{Z},\widetilde{\chi}) \widetilde{P}(Z) \widetilde{P}(\chi) dZ d\chi
</math>
The assumptions made regarding the shapes of the PDFs are still justified. In LES combustion the
[[subgrid variance]] is smaller than RANS counterpart (part of the large-scale fluctuations are solved)
and therefore the modelled PDFs are thinner.

====== Unsteady flamelets ======
====Intrinsic Low Dimensional Manifolds (ILDM)====

Detailed mechanisms describing ignition, flame propagation and pollutant formation typically involve several hundred species and elementary reactions, prohibiting their use in practical three-dimensional engine simulations. Conventionally reduced mechanisms often fail to predict minor radicals and pollutant precursors. The ILDM-method is an automatic reduction of a detailed mechanism, which assumes local equilibrium with respect to the fastest time scales identified by a local eigenvector analysis. In the reactive flow calculation, the species compositions are constrained to these manifolds. Conservation equations are solved for only a small number of reaction progress variables, thereby retaining the accuracy of detailed chemical mechanisms. This gives an effective way of coupling the details of complex chemistry with the time variations due to turbulence.

The intrinsic low-dimensional manifold (ILDM) method {Maas:1992,Maas:1993} is a method for ''in-situ'' reduction of a detailed chemical mechanism based on a local time scale analysis. This method is based on the fact that different trajectories in state space start from a high-dimensional point and quickly relax to lower-dimensional manifolds due to the fast reactions. The movement along these lower-dimensional manifolds, however, is governed by the slow reactions. It exploits the variety of time scales to systematically reduce the detailed mechanism. For a detailed chemical mechanism with N species, N different time scales govern the process. An assumption that all the time scales are relaxed results in assuming complete equilibrium, where the only variables required to describe the system are the mixture fraction, the temperature and the pressure. This results in a zero-dimensional manifold. An assumption that all but the slowest 'n' time scales are relaxed results in a 'n' dimensional manifold, which requires the additional specification of 'n' parameters (called progress variables). In the ILDM method, the fast chemical reactions do not need to be identified a priori. An eigenvalue analysis of the detailed chemical mechanism is carried out which identifies the fast processes in dynamic equilibrium with the slow processes. The computation of ILDM points can be expensive, and hence an in-situ tabulation procedure is used, which enables the calculation of only those points that are needed during the CFD calculation.

--[[User:Fredgauss|Fredgauss]] 07:37, 25 August 2006 (MDT)

U. Maas, S.B. Pope. Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space. Comb. Flame 88,
239, 1992.

Ulrich Maas. Automatische Reduktion von Reaktionsmechanismen zur Simulation reaktiver Str¨omungen. Habilitationsschrift, Universit
¨at Stuttgart, 1993.

==== Conditional Moment Closure (CMC)====
In Conditional Moment Closure (CMC) methods we assume that the species mass fractions are all correlated with
the mixture fraction (in non premixed combustion).

From [[Probability density function]] we have
:<math>
\overline{Y_k}= \int <Y_k|\eta> P(\eta) d\eta
</math>
where <math> \eta </math> is the sample space for <math> Z </math>.

CMC consists of providing a set of transport equations for the conditional moments which define the
flame structure.

Experimentally, it has been observed that temperature and chemical radicals are
strong non-linear functions of mixture fraction. For a given species mass fraction we can decomposed
it into a mean and a fluctuation:
:<math>
Y_k= \overline{Y_k} + Y'_k
</math>
The fluctuations <math> Y_k' </math> are usually very strong in time and space which makes the closure
of <math> \overline{\omega_k} </math> very difficult.
However, the alternative decomposition
:<math>
Y_k= <Y_k|\eta> + y'_k
</math>
where <math> y'_k </math> is the fluctuation around the conditional mean or the "conditional fluctuation".
Experimentally, it is observed that <math> y'_k<< Y'_k </math>, which forms the basic assumption of the CMC method.
Closures. Due to this property better closure methods can be used reducing the non-linearity
of the mass fraction equations.

The [[Derivation of the CMC equations]] produces the following CMC transport equation
where <math> Q \equiv <Y_k|\eta> </math> for simplicity.
:<math>
\frac{ \partial Q}{\partial t} + <u_j|\eta> \frac{\partial Q}{\partial x_j} =
\frac{<\chi|\eta> }{2} \frac{\partial ^2 Q}{\partial \eta^2} +
\frac{ < \dot \omega_k|\eta> }{ <\rho| \eta >}
</math>

In this equation, high order terms in Reynolds number have been neglected.
(See [[Derivation of the CMC equations]] for the complete series of terms).

It is well known that closure of the unconditional source term
<math> \overline {\dot \omega_k} </math> as a function of the
mean temperature and species (<math> \overline{Y}, \overline{T}</math>) will give rise to large errors.
However, in CMC the conditional averaged mass fractions contain more information and fluctuations around the mean are much smaller.
The first order closure
<math>
< \dot \omega_k|\eta> \approx \dot \omega_k \left( Q, <T|\eta> \right)
</math>
is a good approximation in zones which are not close to extinction.

===== Second order closure =====

A second order closure can be obtained if conditional fluctuations are taken into account.
For a chemical source term in the form <math> \dot \omega_k = k Y_A Y_B </math> with the rate constant in Arrhenius form
<math> k=A_0 T^\beta exp [-Ta/T] </math>
the second order closure is (Klimenko and Bilger 1999)
:<math>
< \dot \omega_k|\eta> \approx < \dot \omega_k|\eta >^{FO}

\left[1+ \frac{< Y''_A Y''_B |\eta>}{Q_A Q_B}+ \left( \beta + T_a/Q_T \right)
\left(
\frac{< Y''_A T'' |\eta>}{Q_AQ_T} + \frac{< Y''_B T'' |\eta>}{Q_BQ_T}
\right) + ...

\right]

</math>

where <math> < \dot \omega_k|\eta >^{FO} </math> is the first order CMC closure and
<math> Q_T \equiv <T|\eta> </math>.
When the temperature exponent <math> \beta </math> or <math> T_a/Q_T </math>
are large the error of taking the first order approximation increases.
Improvement of small pollutant predictions can be obtained using the above reaction
rate for selected species like CO and NO.

===== Double conditioning =====

Close to extinction and reignition. The conditional fluctuations can be very large
and the primary closure of CMC of "small" fluctuations is not longer valid.
A second variable <math> h </math> can be chosen to define a double conditioned mass fraction
:<math>
Q(x,t;\eta,\psi) \equiv <Y_i(x,t) |Z=\eta,h=\psi >
</math>
Due to the strong dependence on chemical reactions to temperature, <math> h </math>
is advised to be a temperature related variable (Kronenburg 2004).
Scalar dissipation is not a good choice, due to its log-normal behaviour
(smaller scales give highest dissipation). A must better choice is the sensible enthalpy
or a progress variable.
Double conditional variables have much smaller conditional fluctuations and allow
the existence of points with the same chemical composition which can be fully burning
(high temperature) or just mixing (low temperature).
The range of applicability is greatly increased and allows non-premixed and premixed problems
to be treated without ad-hoc distinctions.
The main problem is the closure of the new terms involving cross scalar transport.

The double conditional CMC equation is obtained in a similar manner than the conventional
CMC equations

===== LES modelling =====
In a LES context a [[conditional filtering]] operator can be defined
and <math> Q </math> therefore represents a conditionally filtered reactive scalar.

=== Linear Eddy Model ===
The Linear Eddy Model (LEM) was first developed by Kerstein(1988).
It is an one-dimensional model for representing the flame structure in turbulent flows.

In every computational cell a molecular, diffusion and chemical model is defined as

:<math>
\frac{\partial}{\partial t} \left( \rho Y_k \right) =
\frac{\partial}{\partial \eta} \left( \rho D_k \frac{\partial Y_k}{\partial \eta }\right)+ \dot \omega_k
</math>

where <math> \eta </math>is a spatial coordinate. The scalar distribution obtained can be seen as a
one-dimensional reference field between Kolmogorov scale and grid scales.

In a second stage a series of re-arranging stochastic event take place.
These events represent the effects
of a certain turbulent structure of size <math> l </math>, smaller than the grid size at a location <math> \eta_0 </math>
within the one-dimensional domain. This vortex distort the <math> \eta </math> field obtain by the one-dimensional equation,
creating new maxima and minima in the interval <math> (\eta_0, \eta + \eta_0) </math>.
The vortex size <math> l </math> is chosen randomly based on the inertial scale range while
<math> \eta_0 </math> is obtained from a uniform distribution in <math> \eta </math>.
The number of events is chosen to match the turbulent diffusivity of the flow.

=== PDF transport models ===

[[Probability Density Function]] (PDF) methods are not exclusive to combustion, although they are particularly attractive to them. They provided more information than moment closures and they are used to compute inhomegenous turbulent flows, see reviews in Dopazo (1993) and Pope (1994).

PDF methods are based on the transport equation of the joint-PDF of the scalars.
Denoting <math> P \equiv P(\underline{\psi}; x, t) </math> where
<math> \underline{\psi} = ( \psi_1,\psi_2 ... \psi_N) </math> is the phase space for the reactive scalars
<math> \underline{Y} = ( Y_1,Y_2 ... Y_N) </math>.
The transport equation of the joint PDF is:

:<math>
\frac{\partial <\rho | \underline{Y}=\underline{\psi}> P }{\partial t} + \frac{
\partial <\rho u_j | \underline{Y}=\underline{\psi}> P }{\partial x_j} =
\sum^N_\alpha \frac{\partial}{\partial \psi_\alpha}\left[ \rho \dot{\omega}_\alpha P \right]
- \sum^N_\alpha \sum^N_\beta \frac{\partial^2}{\partial \psi_\alpha \psi_\beta}
\left[ <D \frac{\partial Y_\alpha}{\partial x_i} \frac{\partial Y_\beta}{\partial x_i} | \underline{Y}=\underline{\psi}> \right] P
</math>

where the chemical source term is closed. Another term appeared on the right hand side which accounts for the effects of
the molecular mixing on the PDF, is the so called "micro-mixing " term.
Equal diffusivities are used for simplicity <math> D_k = D </math>

A more general approach is the velocity-composition joint-PDF
with <math> P \equiv P(\underline{V},\underline{\psi}; x, t) </math>, where
<math> \underline{V} </math> is the sample space of the velocity field
<math> u,v,w </math>. This approach has the advantage of avoiding gradient-diffusion
modelling. A similar equation to the above is obtained combining the momentum
and scalar transport equation.

The PDF transport equation can be solved in two ways: through a Lagrangian approach
using stochastic methods or in a Eulerian ways using stochastic fields.

==== Lagrangian ====

The main idea of Lagrangian methods is that the flow can be represented by an
ensemble of fluid particles. Central to this approach is the stochastic differential equations and in particular the [[Langevin equation]].

==== Eulerian ====

Instead of stochastic particles, smooth stochastic fields can be used
to represent the [[probability density function ]] (PDF) of a scalar (or joint PDF) involved in transport (convection), diffusion
and chemical reaction (Valino 1998). A similar formulation was proposed by Sabelnikov and Soulard 2005, which removes
part of the a-priori assumption of "smoothness" of the stochastic fields.
This approach is purely Eulerian and offers implementations advantages compared to
Lagrangian or semi-Eulerian methods.
Transport equations for scalars are often easy to programme and normal
CFD algorithms can be used (see [[Discretisation of convective term]]).
Although discretization errors are introduced by solving transport equations,
this is partially compesated by the error introduced in Lagrangian approaches due to the numerical evaluation of means.

A new set of <math> N_s </math> scalar variables
(the stochastic field <math> \xi </math>) is used to represent the
PDF
:<math>
P (\underline{\psi}; x,t) = \frac{1}{N} \sum^{N_s}_{j=1} \prod^{N}_{k=1} \delta \left[\psi_k -\xi_k^j(x,t) \right]
</math>

===Other combustion models===

==== MMC ====

The Multiple Mapping Conditioning (MMC) (Klimenko and Pope 2003) is an
extension of the [[#Conditional Moment Closure (CMC)]] approach combined with
[[probability density function]] methods. MMC
looks for the minimum set of variables that describes the particular turbulent combustion
system.

==== Fractals ====
Derived from the [[#Eddy Dissipation Concept (EDC)]].

== References ==

*{{reference-paper|author=Dopazo, C.|year=1993|title=Recent development in PDF methods|rest=Turbulent Reacting Flows, ed. P. A. Libby and F. A. Williams }}
*{{reference-book|author=Fox, R.O.|year=2003|title=Computational Models for Turbulent Reacting Flows|rest=ISBN 0-521-65049-6,Cambridge University Press}}
*{{reference-paper|author=Kerstein, A. R.|year=1988|title=A linear eddy model of turbulent scalar transport and mixing|rest=Comb. Science and Technology, Vol. 60,pp. 391}}
*{{reference-paper|author=Klimenko, A. Y., Bilger, R. W.|year=1999|title=Conditional moment closure for turbulent combustion|rest=Progress in Energy and Combustion Science, Vol. 25,pp. 595-687}}
*{{reference-paper|author=Klimenko, A. Y., Pope, S. B.|year=2003|title=The modeling of turbulent reactive flows based on multiple mapping conditioning|rest=Physics of Fluids, Vol. 15, Num. 7, pp. 1907-1925}}
*{{reference-paper|author=Kronenburg, A.,|year=2004|title=Double conditioning of reactive scalar transport equations in turbulent non-premixed flames|rest=Physics of Fluids, Vol. 16, Num. 7, pp. 2640-2648}}
*{{reference-paper|author=Griffiths, J. F.|year=1994|title=Reduced Kinetic Models and Their Application to Practical Combustion Systems |rest=Prog. in Energy and Combustion Science,Vol. 21, pp. 25-107}}
*{{reference-book|author=Peters, N.|year=2000|title=Turbulent Combustion|rest=ISBN 0-521-66082-3,Cambridge University Press}}
*{{reference-book|author=Poinsot, T.,Veynante, D.|year=2001|title=Theoretical and Numerical Combustion|rest=ISBN 1-930217-05-6, R. T Edwards}}
*{{reference-paper|author=Pope, S. B.|year=1994|title=Lagrangian PDF methods for turbulent flows|rest=Annu. Rev. Fluid Mech, Vol. 26, pp. 23-63}}
*{{reference-paper|author=Sabel'nikov, V.,Soulard, O.|year=2005|title=Rapidly decorrelating velocity-field model as a tool for solving one-point Fokker-Planck equations for probability density functions of turbulent reactive scalars|rest=Physical Review E, Vol. 72,pp. 016301-1-22}}
*{{reference-paper|author=Valino, L.,|year=1998|title=A field montecarlo formulation for calculating the probability density function of a single scalar in a turbulent flow|rest=Flow. Turb and Combustion, Vol. 60,pp. 157-172}}
*{{reference-paper|author=Westbrook, Ch. K., Dryer,F. L.,|year=1984|title=Chemical Kinetic Modeling of Hydrocarbon Combustion|rest=Prog. in Energy and Combustion Science,Vol. 10, pp. 1-57}}

== External links and sources ==

Talk:Combustion

2008-03-12T19:37:39Z

Jola:

I think it is worthy to somewhat modify the introduction in order to describe in length both the combustion and turbulent regimes (flamelet,etc) early in the article. I plan to do it in the coming days or weeks. Please, let know here whether somebody has any concern. --[[User:Joan|Joan]] 07:51, 5 September 2007 (MDT)

I would like to move the premixed flame picture provided by Salva earlier in the article as I think it fits well with the premixed regime introduction. Let know here if any concern.
--[[User:Joan|Joan]] 12:44, 12 September 2007 (MDT)

As combustion modelling is far from being mature, the users need to have a clear knowledge of what is behind the models and what is known about flames today. For that reason, the best way is to open another article in combustion dedicated to fundamentals. The role of this article will be twofold: to give a background, and to serve as a database to which results and equations coming with the models can be referred. Let know if any concern.--[[User:Joan|Joan]] 13:14, 29 October 2007 (MDT)

It is impossible to create a new article for a contributor. Hence, I am going to develop the ideas into the existing introduction. If an administrator thinks it fits, please, move the matter into a new article focusing on fundamentals, otherwise, simply revert it.--[[User:Joan|Joan]] 08:25, 30 October 2007 (MDT)

Talk:Valve industry

2008-03-02T08:48:16Z

Jola: New page: This entire page looks like advertising. That is not allowed in CFD-Wiki. There is no general information and it directly mentiones commercial companies. I will remove it soon unless peopl...

This entire page looks like advertising. That is not allowed in CFD-Wiki. There is no general information and it directly mentiones commercial companies. I will remove it soon unless people object here. --[[User:Jola|Jola]] 01:48, 2 March 2008 (MST)

Introduction to turbulence

2008-02-25T09:28:07Z

Jola:

{{Introduction to turbulence menu}}

__NOTOC__

'''This section is currently undergoing heavy reorganization and editing. Please excuse any errors or unfinished parts. --[[User:Jola|Jola]] 07:01, 21 June 2007 (MDT)'''

== [[Introduction to turbulence/Nature of turbulence|Nature of turbulence]] ==

* [[Introduction to turbulence/Nature of turbulence#The turbulent world around us|The turbulent world around us]]
* [[Introduction to turbulence/Nature of turbulence#What is turbulence?|What is turbulence?]]
* [[Introduction to turbulence/Nature of turbulence#Why study turbulence?|Why study turbulence?]]
* [[Introduction to turbulence/Nature of turbulence#The cost of our ignorance|The cost of our ignorance]]
* [[Introduction to turbulence/Nature of turbulence#What do we really know for sure?|What do we really know for sure?]]

== [[Introduction to turbulence/Statistical analysis|Statistical analysis]] ==

* [[Introduction to turbulence/Statistical analysis/Ensemble average|Ensemble average]]
** [[Introduction to turbulence/Statistical analysis/Ensemble average#Mean or ensemble average|Mean or ensemble average]]
** [[Introduction to turbulence/Statistical analysis/Ensemble average#Fluctuations about the mean|Fluctuations about the mean]]
** [[Introduction to turbulence/Statistical analysis/Ensemble average#Higher moments|Higher moments]]
* [[Introduction to turbulence/Statistical analysis/Probability|Probability]]
** [[Introduction to turbulence/Statistical analysis/Probability#Histogram and probability density function|Histogram and probability density function]]
** [[Introduction to turbulence/Statistical analysis/Probability#Probability distribution|Probability distribution]]
** [[Introduction to turbulence/Statistical analysis/Probability#Gaussian (or normal) distributions|Gaussian (or normal) distributions]]
** [[Introduction to turbulence/Statistical analysis/Probability#Skewness and kurtosis|Skewness and kurtosis]]
* [[Introduction to turbulence/Statistical analysis/Multivariate random variables|Multivariate random variables]]
** [[Introduction to turbulence/Statistical analysis/Multivariate random variables#Joint pdfs and joint moments|Joint pdfs and joint moments]]
** [[Introduction to turbulence/Statistical analysis/Multivariate random variables#The bi-variate normal (or Gaussian) distribution|The bi-variate normal (or Gaussian) distribution]]
** [[Introduction to turbulence/Statistical analysis/Multivariate random variables#Statistical independence and lack of correlation|Statistical independence and lack of correlation]]
* [[Introduction to turbulence/Statistical analysis/Estimation from a finite number of realizations|Estimation from a finite number of realizations]]
** [[Introduction to turbulence/Statistical analysis/Estimation from a finite number of realizations#Estimators for averaged quantities|Estimators for averaged quantities]]
** [[Introduction to turbulence/Statistical analysis/Estimation from a finite number of realizations#Bias and convergence of estimators|Bias and convergence of estimators]]
* [[Introduction to turbulence/Statistical analysis/Generalization to the estimator of any quantity|Generalization to the estimator of any quantity]]

== [[Introduction to turbulence/Reynolds averaged equations|Reynolds averaged equations and the turbulence closure problem]] ==

* [[Introduction to turbulence/Reynolds averaged equations#Equations governing instantaneous fluid motion|Equations governing instantaneous fluid motion]]
* [[Introduction to turbulence/Reynolds averaged equations#Equations for the average velocity|Equations for the average velocity]]
* [[Introduction to turbulence/Reynolds averaged equations#The turbulence problem|The turbulence problem]]
* [[Introduction to turbulence/Reynolds averaged equations#Origins of turbulence|Origins of turbulence]]
* [[Introduction to turbulence/Reynolds averaged equations#Importance of non-linearity|Importance of non-linearity]]
* [[Introduction to turbulence/Reynolds averaged equations#Turbulence closure problem and eddy viscosity|Turbulence closure problem and eddy viscosity]]
* [[Introduction to turbulence/Reynolds averaged equations#Reynolds stress equations|Reynolds stress equations]]

== [[Introduction to turbulence/Turbulence kinetic energy|Turbulence kinetic energy]] ==
* [[Introduction to turbulence/Turbulence kinetic energy#Fluctuating kinetic energy|Fluctuating kinetic energy]]
* [[Introduction to turbulence/Turbulence kinetic energy#Rate of dissipation of the turbulence kinetic energy|Rate of dissipation of the turbulence kinetic energy]]
* [[Introduction to turbulence/Turbulence kinetic energy#Kinetic energy of the mean motion and production of turbulence|Kinetic energy of the mean motion and production of turbulence]]
* [[Introduction to turbulence/Turbulence kinetic energy#Transport or divergence terms|Transport or divergence terms]]
* [[Introduction to turbulence/Turbulence kinetic energy#Intercomponent transfer of energy|Intercomponent transfer of energy]]

== [[Introduction to turbulence/Stationarity and homogeneity|Stationarity and homogeneity]] ==
* [[Introduction to turbulence/Stationarity and homogeneity#Processes statistically stationary in time|Processes statistically stationary in time]]
* [[Introduction to turbulence/Stationarity and homogeneity#Autocorrelation|Autocorrelation]]
* [[Introduction to turbulence/Stationarity and homogeneity#Autocorrelation coefficient|Autocorrelation coefficient]]
* [[Introduction to turbulence/Stationarity and homogeneity#Integral scale|Integral scale]]
* [[Introduction to turbulence/Stationarity and homogeneity#Temporal Taylor microscale|Temporal Taylor microscale]]
* [[Introduction to turbulence/Stationarity and homogeneity#Time averages of stationary processes|Time averages of stationary processes]]
* [[Introduction to turbulence/Stationarity and homogeneity#Bias and variability of time estimators|Bias and variability of time estimators]]
* [[Introduction to turbulence/Stationarity and homogeneity#Random fields of space and time|Random fields of space and time]]
* [[Introduction to turbulence/Stationarity and homogeneity#Multi-point statistics in homogeneous field|Multi-point statistics in homogeneous field]]
* [[Introduction to turbulence/Stationarity and homogeneity#Spatial integral and Taylor microscales|Spatial integral and Taylor microscales]]

== [[Introduction to turbulence/Homogeneous turbulence|Homogeneous turbulence]] ==

== [[Introduction to turbulence/Free turbulent shear flows|Free turbulent shear flows]] ==

== [[Introduction to turbulence/Wall bounded turbulent flows|Wall bounded turbulent flows]] ==

{{Turbulence credit wkgeorge}}

[[Category: Turbulence]]

Introduction to turbulence

2008-02-25T09:26:14Z

Jola:

{{Introduction to turbulence menu}}

__NOTOC__

'''This section is currently undergoing heavy reorganization and editing. Please excuse any errors or unfinished parts. --[[User:Jola|Jola]] 07:01, 21 June 2007 (MDT)'''

== [[Introduction to turbulence/Nature of turbulence|Nature of turbulence]] ==

* [[Introduction to turbulence/Nature of turbulence#The turbulent world around us|The turbulent world around us]]
* [[Introduction to turbulence/Nature of turbulence#What is turbulence?|What is turbulence?]]
* [[Introduction to turbulence/Nature of turbulence#Why study turbulence?|Why study turbulence?]]
* [[Introduction to turbulence/Nature of turbulence#The cost of our ignorance|The cost of our ignorance]]
* [[Introduction to turbulence/Nature of turbulence#What do we really know for sure?|What do we really know for sure?]]

== [[Introduction to turbulence/Statistical analysis|Statistical analysis]] ==

* [[Introduction to turbulence/Statistical analysis/Ensemble average|Ensemble average]]
** [[Introduction to turbulence/Statistical analysis/Ensemble average#Mean or ensemble average|Mean or ensemble average]]
** [[Introduction to turbulence/Statistical analysis/Ensemble average#Fluctuations about the mean|Fluctuations about the mean]]
** [[Introduction to turbulence/Statistical analysis/Ensemble average#Higher moments|Higher moments]]
* [[Introduction to turbulence/Statistical analysis/Probability|Probability]]
** [[Introduction to turbulence/Statistical analysis/Probability#Histogram and probability density function|Histogram and probability density function]]
** [[Introduction to turbulence/Statistical analysis/Probability#Probability distribution|Probability distribution]]
** [[Introduction to turbulence/Statistical analysis/Probability#Gaussian (or normal) distributions|Gaussian (or normal) distributions]]
** [[Introduction to turbulence/Statistical analysis/Probability#Skewness and kurtosis|Skewness and kurtosis]]
* [[Introduction to turbulence/Statistical analysis/Multivariate random variables|Multivariate random variables]]
** [[Introduction to turbulence/Statistical analysis/Multivariate random variables#Joint pdfs and joint moments|Joint pdfs and joint moments]]
** [[Introduction to turbulence/Statistical analysis/Multivariate random variables#The bi-variate normal (or Gaussian) distribution|The bi-variate normal (or Gaussian) distribution]]
** [[Introduction to turbulence/Statistical analysis/Multivariate random variables#Statistical independence and lack of correlation|Statistical independence and lack of correlation]]
* [[Introduction to turbulence/Statistical analysis/Estimation from a finite number of realizations|Estimation from a finite number of realizations]]
** [[Introduction to turbulence/Statistical analysis/Estimation from a finite number of realizations#Estimators for averaged quantities|Estimators for averaged quantities]]
** [[Introduction to turbulence/Statistical analysis/Estimation from a finite number of realizations#Bias and convergence of estimators|Bias and convergence of estimators]]
* [[Introduction to turbulence/Statistical analysis/Generalization to the estimator of any quantity|Generalization to the estimator of any quantity]]

== [[Introduction to turbulence/Reynolds averaged equations|Reynolds averaged equations and the turbulence closure problem]] ==

* [[Introduction to turbulence/Reynolds averaged equations#Equations governing instantaneous fluid motion|Equations governing instantaneous fluid motion]]
* [[Introduction to turbulence/Reynolds averaged equations#Equations for the average velocity|Equations for the average velocity]]
* [[Introduction to turbulence/Reynolds averaged equations#The turbulence problem|The turbulence problem]]
* [[Introduction to turbulence/Reynolds averaged equations#Origins of turbulence|Origins of turbulence]]
* [[Introduction to turbulence/Reynolds averaged equations#Importance of non-linearity|Importance of non-linearity]]
* [[Introduction to turbulence/Reynolds averaged equations#Turbulence closure problem and eddy viscosity|Turbulence closure problem and eddy viscosity]]
* [[Introduction to turbulence/Reynolds averaged equations#Reynolds stress equations|Reynolds stress equations]]

== [[Introduction to turbulence/Turbulence kinetic energy|Turbulence kinetic energy]] ==
* [[Introduction to turbulence/Turbulence kinetic energy#Fluctuating kinetic energy|Fluctuating kinetic energy]]
* [[Introduction to turbulence/Turbulence kinetic energy#Rate of dissipation of the turbulence kinetic energy|Rate of dissipation of the turbulence kinetic energy]]
* [[Introduction to turbulence/Turbulence kinetic energy#Kinetic energy of the mean motion and production of turbulence|Kinetic energy of the mean motion and production of turbulence]]
* [[Introduction to turbulence/Turbulence kinetic energy#Transport or divergence terms|Transport or divergence terms]]
* [[Introduction to turbulence/Turbulence kinetic energy#Intercomponent transfer of energy|Intercomponent transfer of energy]]

== [[Introduction to turbulence/Stationarity and homogeneity|Stationarity and homogeneity]] ==
* [[Introduction to turbulence/Processes statistically stationary in time|Processes statistically stationary in time]]
* [[Introduction to turbulence/Autocorrelation|Autocorrelation]]
* [[Introduction to turbulence/Autocorrelation coefficient|Autocorrelation coefficient]]
* [[Introduction to turbulence/Integral scale|Integral scale]]
* [[Introduction to turbulence/Temporal Taylor microscale|Temporal Taylor microscale]]
* [[Introduction to turbulence/Time averages of stationary processes|Time averages of stationary processes]]
* [[Introduction to turbulence/Bias and variability of time estimators|Bias and variability of time estimators]]
* [[Introduction to turbulence/Random fields of space and time|Random fields of space and time]]
* [[Introduction to turbulence/Multi-point statistics in homogeneous field|Multi-point statistics in homogeneous field]]
* [[Introduction to turbulence/Spatial integral and Taylor microscales|Spatial integral and Taylor microscales]]

== [[Introduction to turbulence/Homogeneous turbulence|Homogeneous turbulence]] ==

== [[Introduction to turbulence/Free turbulent shear flows|Free turbulent shear flows]] ==

== [[Introduction to turbulence/Wall bounded turbulent flows|Wall bounded turbulent flows]] ==

{{Turbulence credit wkgeorge}}

[[Category: Turbulence]]

Stationarity and homogeneity

2008-02-25T09:19:50Z

Jola: Stationarity and homogeneity moved to Introduction to turbulence/Stationarity and homogeneity: Correct book title

#REDIRECT [[Introduction to turbulence/Stationarity and homogeneity]]

Introduction to turbulence/Stationarity and homogeneity

2008-02-25T09:19:50Z

Jola: Stationarity and homogeneity moved to Introduction to turbulence/Stationarity and homogeneity: Correct book title

{{Introduction to turbulence menu}}

== Processes statistically stationary in time ==

Many random processes have the characteristic that their statistical properties do not appear to depend directly on time, even though the random variables themselves are time-dependent. For example, consider the signals shown in Figures 2.2 and 2.5

When the statistical properties of a random process are independent of time, the random process is said to be ''stationary''. For such a process all the moments are time-independent, e.g., <math> \left\langle \tilde{ u \left( t \right)} \right\rangle = U </math>, etc. In fact, the probability density itself is time-independent, as should be obvious from the fact that the moments are time independent.

An alternative way of looking at ''stationarity'' is to note that ''the statistics of the process are independent of the origin in time''. It is obvious from the above, for example, that if the statistics of a process are time independent, then <math> \left\langle u^{n} \left( t \right) \right\rangle = \left\langle u^{n} \left( t + T \right) \right\rangle </math> , etc., where <math> T </math> is some arbitrary translation of the origin in time. Less obvious, but equally true, is that the product <math> \left\langle u \left( t \right) u \left( t' \right) \right\rangle </math> depends only on time difference <math> t'-t </math> and not on <math> t </math> (or <math> t' </math> ) directly. This consequence of stationarity can be extended to any product moment. For example <math> \left\langle u \left( t \right) v \left( t' \right) \right\rangle </math> can depend only on the time difference <math> t'-t </math>. And <math> \left\langle u \left( t \right) v \left( t' \right) w \left( t'' \right)\right\rangle </math> can depend only on the two time differences <math> t'- t </math> and <math> t'' - t </math> (or <math> t'' - t' </math> ) and not <math> t </math> , <math> t' </math> or <math> t'' </math> directly.

== Autocorrelation ==

One of the most useful statistical moments in the study of stationary random processes (and turbulence, in particular) is the '''autocorrelation''' defined as the average of the product of the random variable evaluated at two times, i.e. <math> \left\langle u \left( t \right) u \left( t' \right)\right\rangle </math>. Since the process is assumed stationary, this product can depend only on the time difference <math> \tau = t' - t </math>. Therefore the autocorrelation can be written as:

<table width="70%"><tr><td>
:<math>
C \left( \tau \right) \equiv \left\langle u \left( t \right) u \left( t + \tau \right) \right\rangle
</math>
</td><td width="5%">(1)</td></tr></table>

The importance of the autocorrelation lies in the fact that it indicates the "memory" of the process; that is, ''the time over which is correlated with itself''. Contrast the two autocorrelation of deterministic sine wave is simply a cosine as can be easily proven. Note that there is no time beyond which it can be guaranteed to be arbitrarily small since it always "remembers" when it began, and thus always remains correlated with itself. By contrast, a stationary random process like the one illustrated in the figure will eventually lose all correlation and go to zero. In other words it has a "finite memory" and "forgets" how it was. Note that one must be careful to make sure that a correlation really both goes to zero and ''stays down'' before drawing conclusions, since even the sine wave was zero at some points. Stationary random process ''always'' have two-time correlation functions which eventually go to zero and stay there.

'''Example 1.'''

Consider the motion of an automobile responding to the movement of the wheels over a rough surface. In the usual case where the road roughness is randomly distributed, the motion of the car will be a weighted history of the road's roughness with the most recent bumps having the most influence and with distant bumps eventually forgotten. On the other hand, if the car is travelling down a railroad track, the periodic crossing of the railroad ties represents a determenistic input an the motion will remain correlated with itself indefinitely, a very bad thing if the tie crossing rate corresponds to a natural resonance of the suspension system of the vehicle.

Since a random process can never be more than perfectly correlated, it can never achieve a correlation greater than is value at the origin. Thus

<table width="70%"><tr><td>
:<math>
\left| C \left( \tau \right) \right| \leq C\left( 0 \right)
</math>
</td><td width="5%">(2)</td></tr></table>

An important consequence of stationarity is that the autocorrelation is symmetric in the time difference <math> \tau = t' - t </math>. To see this simply shift the origin in time backwards by an amount <math> \tau </math> and note that independence of origin implies:

<table width="70%"><tr><td>
:<math>
\left\langle u \left( t \right) u \left( t + \tau \right) \right\rangle = \left\langle u \left( t - \tau \right) u \left( t \right) \right\rangle
</math>
</td><td width="5%">(3)</td></tr></table>

Since the right hand side is simply <math> C \left( - \tau \right) </math>, it follows immediately that:

<table width="70%"><tr><td>
:<math>
C \left( \tau \right) = C \left( - \tau \right)
</math>
</td><td width="5%">(4)</td></tr></table>

== Autocorrelation coefficient ==

It is convenient to define the ''autocorrelation coefficient'' as:

<table width="70%"><tr><td>
:<math>
\rho \left( \tau \right) \equiv \frac{ C \left( \tau \right)}{ C \left( 0 \right)} = \frac{\left\langle u \left( t \right) u \left( t + \tau \right) \right\rangle}{ \left\langle u'^{2} \right\rangle }
</math>
</td><td width="5%">(5)</td></tr></table>

where

<table width="70%"><tr><td>
:<math>
\left\langle u^{2} \right\rangle = \left\langle u \left( t \right) u \left( t \right) \right\rangle = C \left( 0 \right) = var \left[ u \right]
</math>
</td><td width="5%">(6)</td></tr></table>

Since the autocorrelation is symmetric, so is its coefficient, i.e.,

<table width="70%"><tr><td>
:<math>
\rho \left( \tau \right) = \rho \left( - \tau \right)
</math>
</td><td width="5%">(7)</td></tr></table>

It is also obvious from the fact that the autocorrelation is maximal at the origin that the autocorrelation coefficient must also be maximal there. In fact from the definition it follows that

<table width="70%"><tr><td>
:<math>
\rho \left( 0 \right) = 1
</math>
</td><td width="5%">(8)</td></tr></table>

and

<table width="70%"><tr><td>
:<math>
\rho \left( \tau \right) \leq 1
</math>
</td><td width="5%">(9)</td></tr></table>

for all values of <math> \tau </math> .

== Integral scale ==

One of the most useful measures of the length of a time a process is correlated with itself is the integral scale defined by

<table width="70%"><tr><td>
:<math>
T_{int} \equiv \int^{\infty}_{0} \rho \left( \tau \right) d \tau
</math>
</td><td width="5%">(10)</td></tr></table>

It is easy to see why this works by looking at Figure 5.2. In effect we have replaced the area under the correlation coefficient by a rectangle of height unity and width <math> T_{int} </math> .

== Temporal Taylor microscale ==

The autocorrelation can be expanded about the origin in a MacClaurin series; i.e.,

<table width="70%"><tr><td>
:<math>
C \left( \tau \right) = C \left( 0 \right) + \tau \frac{ d C }{ d t }|_{\tau = 0} + \frac{1}{2} \tau^{2} \frac{d^{2} C}{d t^{2} }|_{\tau = 0} + \frac{1}{3!} \tau^{3} \frac{d^{3} C}{d t^{3} }|_{\tau = 0}
</math>
</td><td width="5%">(11)</td></tr></table>

But we know the aoutocorrelation is symmetric in <math> \tau </math> , hence the odd terms in <math> \tau </math> must be identically to zero (i.e., <math> dC / dt |_{\tau = 0} = 0 </math> , <math> d^{3}C / dt^{3} |_{\tau = 0} = 0 </math>, etc.). Therefore the expansion of the autocorrelation near the origin reduces to:

<table width="70%"><tr><td>
:<math>
C \left( \tau \right) = C \left( 0 \right) + \frac{1}{2} \tau^{2} \frac{d^{2} C}{d t^{2} }|_{\tau = 0} + \cdots
</math>
</td><td width="5%">(12)</td></tr></table>

Similary, the autocorrelation coefficient near the origin can be expanded as:

<table width="70%"><tr><td>
:<math>
\rho \left( \tau \right) = 1 + \frac{1}{2}\frac{d^{2}\rho}{d t^{2}}|_{\tau = 0} \tau^{2}+ \cdots
</math>
</td><td width="5%">(13)</td></tr></table>

where we have used the fact that <math> \rho \left( 0 \right) = 1 </math> . If we define <math> ' = d / dt </math> we can write this compactly as:

<table width="70%"><tr><td>
:<math>
\rho \left( \tau \right) = 1 + \frac{1}{2} \rho '' \left( 0 \right) \tau^{2} + \cdots
</math>
</td><td width="5%">(14)</td></tr></table>

Since <math> \rho \left( \tau \right) </math> has its maximum at the origin, obviously <math> \rho'' \left( 0 \right) </math> must be negative.

We can use the correlation and its second derivative at the origin to ''define'' a special time scale, <math> \lambda_{\tau} </math> (called the Taylor microscale) by:

<table width="70%"><tr><td>
:<math>
\lambda^{2}_{\tau} \equiv - \frac{2}{\rho'' \left( 0 \right)}
</math>
</td><td width="5%">(15)</td></tr></table>

Using this in equation 14 yields the expansion for the correlation coefficient near the origin as:

<table width="70%"><tr><td>
:<math>
\rho \left( \tau \right) = 1 - \frac{\tau^{2}}{\lambda^{2}_{\tau}} + \cdots
</math>
</td><td width="5%">(16)</td></tr></table>

Thus very near the origin the correlation coefficient (and the autocorrelation as well) simply rolls off parabolically; i.e.,

<table width="70%"><tr><td>
:<math>
\rho \left( \tau \right) \approx 1 - \frac{\tau^{2}}{\lambda^{2}_{\tau}}
</math>
</td><td width="5%">(17)</td></tr></table>

This parabolic curve is shown in Figure 3 as the osculating (or 'kissing') parabola which approaches zero exactly as the autocorrelation coefficient does. The intercept of this osculating parabola with the <math> \tau </math> -axis is the Taylor microscale, <math> \lambda_{\tau} </math>.

The Taylor microscale is significant for a number of reasons. First, for many random processes (e.g., Gaussian), the Taylor microscale can be proven to be the average distance between zero-crossing of a random variable in time. This is approximately true for turbulence as well. Thus one can quickly estimate the Taylor microscale by simply observing the zero-crossings using an oscilloscope trace.

The Taylor microscale also has a special relationship to the mean square time derivative of the signal, <math> \left\langle \left[ d u / d t \right]^{2} \right\rangle </math>. This is easiest to derive if we consider two stationary random signals at two different times say <math> u = u \left( t \right) </math> and <math> u' = u' \left( t' \right) </math>. The derivative of the first signal is <math> d u / d t </math> and the second <math> d u' / d t' </math>. Now lets multiply these together and rewrite them as:

<table width="70%"><tr><td>
:<math>
\frac{du'}{dt'} \frac{du}{dt} = \frac{d^{2}}{dtdt'} u \left( t \right) u' \left( t' \right)
</math>
</td><td width="5%">(18)</td></tr></table>

where the right-hand side follows from our assumption that <math> u </math> is not a function of <math> t' </math> nor <math> u' </math> a function of <math> t </math>.

Now if we average and interchenge the operations of differentiation and averaging we obtain:

<table width="70%"><tr><td>
:<math>
\left\langle \frac{du'}{dt'} \frac{du}{dt} \right\rangle = \frac{d^{2}}{dtdt'} \left\langle u \left( t \right) u' \left( t' \right) \right\rangle
</math>
</td><td width="5%">(19)</td></tr></table>

Here comes the first trick: we simply take <math> u' </math> to be exactly <math> u </math> but evaluated at time <math> t' </math>. So <math> u \left( t \right) u' \left( t' \right) </math> simply becomes <math> u \left( t \right) u \left( t' \right) </math> and its average is just the autocorrelation, <math> C \left( \tau \right) </math>. Thus we are left with:

<table width="70%"><tr><td>
:<math>
\left\langle \frac{du'}{dt'} \frac{du}{dt} \right\rangle = \frac{d^{2}}{dtdt'} C \left( t' - t \right)
</math>
</td><td width="5%">(20)</td></tr></table>

Now we simply need to use the chain-rule. We have already defined <math> \tau = t' - t </math>. Let's also define <math> \xi = t' + t </math> and transform the derivatives involving <math> t </math> and <math> t' </math> to derivatives involving <math> \tau </math> and <math> \xi </math>. The result is:

<table width="70%"><tr><td>
:<math>
\frac{d^{2}}{dtdt'} = \frac{d^{2}}{d \xi^{2}} - \frac{d^{2}}{d \tau^{2}}
</math>
</td><td width="5%">(21)</td></tr></table>

So equation 20 becomes

<table width="70%"><tr><td>
:<math>
\left\langle \frac{du'}{dt'} \frac{du}{dt} \right\rangle = \frac{d^{2}}{d \xi^{2}}C \left( \tau \right) - \frac{d^{2}}{d \tau^{2}} C \left( \tau \right)
</math>
</td><td width="5%">(22)</td></tr></table>

But since <math> C </math> is a function only of <math> \tau </math>, the derivative of it with respect to <math> \xi </math> is identically zero. Thus we are left with:

<table width="70%"><tr><td>
:<math>
\left\langle \frac{du'}{dt'} \frac{du}{dt} \right\rangle = - \frac{d^{2}}{d \tau^{2}} C \left( \tau \right)
</math>
</td><td width="5%">(23)</td></tr></table>

And finally we need the second trick. Let's evaluate both sides at <math> t = t' </math> (or <math> \tau = 0 </math> ) to obtain the ''mean square derivative'' as:

<table width="70%"><tr><td>
:<math>
\left\langle \left( \frac{du}{dt} \right)^{2} \right\rangle = - \frac{d^{2}}{d \tau^{2}} C \left( \tau \right)|_{ \tau = 0}
</math>
</td><td width="5%">(24)</td></tr></table>

But from our definition of the Taylor microscale and the facts that <math> C \left( 0 \right) = \left\langle u^{2} \right\rangle </math> and <math> C \left( \tau \right) = \left\langle u^{2} \right\rangle \rho \left( \tau \right) </math>, this is exactly the same as:

<table width="70%"><tr><td>
:<math>
\left\langle \left( \frac{du}{dt} \right)^{2} \right\rangle = 2 \frac{ \left\langle u^{2} \right\rangle}{\lambda^{2}_{\tau}}
</math>
</td><td width="5%">(25)</td></tr></table>

This amasingly simple result is very important in the study of turbulence, especially after we extend it to spatial derivatives.

== Time averages of stationary processes ==

It is common practice in many scientific disciplines to define a time average by integrating the random variable over a fixed time interval, i.e. ,

<table width="70%"><tr><td>
:<math>
U_{T} \equiv \frac{1}{T} \int^{T_{2}}_{T_{1}} u \left( t \right) dt
</math>
</td><td width="5%">(26)</td></tr></table>

For the stationary random processes we are considering here, we can define <math> T_{1} </math> to be the origin in time and simply write:

<table width="70%"><tr><td>
:<math>
U_{T} \equiv \frac{1}{T} \int^{T}_{0} u \left( t \right) dt
</math>
</td><td width="5%">(27)</td></tr></table>

where <math> T = T_{2} - T_{1} </math> is the integration time.

Figure 5.4. shows a portion of a stationary random signal over which such an integration might be performed. The ime integral of <math> u \left( t \right) </math> over the integral <math> \left( O, T \right) </math> corresponds to the shaded area under the curve. Now since <math> u \left( t \right) </math> is random and since it formsthe upper boundary of the shadd area, it is clear that the time average, <math> U_{T} </math> is a lot like the estimator for the mean based on a finite number of independent realization, <math> X_{N} </math> we encountered earlier in section ''Estimation from a finite number of realizations'' (see ''Elements of statistical analysis'')

It will be shown in the analysis presented below that ''if the signal is stationary'', the time average defined by equation 27 is an unbiased estimator of the true average <math> U </math>. Moreover, the estimator converges to <math> U </math> as the time becomes infinite; i.e., for stationary random processes

<table width="70%"><tr><td>
:<math>
U = \lim_{T \rightarrow \infty} \frac{1}{T} \int^{T}_{0} u \left( t \right) dt
</math>
</td><td width="5%">(28)</td></tr></table>

Thus the time and ensemble averages are equivalent in the limit as <math> T \rightarrow \infty </math>, ''but only for a stationary random process''.

== Bias and variability of time estimators ==

It is easy to show that the estimator, <math> U_{T} </math>, is unbiased by taking its ensemble average; i.e.,

<table width="70%"><tr><td>
:<math>
\left\langle U_{T} \right\rangle = \left\langle \frac{1}{T} \int^{T}_{0} u \left( t \right) dt \right\rangle = \frac{1}{T} \int^{T}_{0} \left\langle u \left( t \right) \right\rangle dt
</math>
</td><td width="5%">(29)</td></tr></table>

Since the process has been assumed stationary, <math> \left\langle u \left( t \right) \right\rangle </math> is independent of time. It follows that:

<table width="70%"><tr><td>
:<math>
\left\langle U_{T} \right\rangle = \frac{1}{T} \left\langle u \left( t \right) \right\rangle T = U
</math>
</td><td width="5%">(30)</td></tr></table>

To see whether the etimate improves as <math> T </math> increases, the variability of <math> U_{T} </math> must be examined, exactly as we did for <math> X_{N} </math> earlier in section Bias and convergence of estimators (see chapter The elements of statistical analysis). To do this we need the variance of <math> U_{T} </math> given by:

<table width="70%"><tr><td>
:<math>
\begin{matrix}
var \left[ U_{T} \right] & = & \left\langle \left[ U_{T} - \left\langle U_{T} \right\rangle \right]^{2} \right\rangle = \left\langle \left[ U_{T} - U \right]^{2} \right\rangle \\
& = & \frac{1}{T^{2}} \left\langle \left\{ \int^{T}_{0} \left[ u \left( t \right) - U \right] \right\}^{2} \right\rangle \\
& = & \frac{1}{T^{2}} \left\langle \int^{T}_{0} \int^{T}_{0} \left[ u \left( t \right) - U \right] \left[ u \left( t' \right) - U \right] dtdt' \right\rangle \\
& = & \frac{1}{T^{2}} \int^{T}_{0} \int^{T}_{0} \left\langle u'\left( t \right) u'\left( t' \right) \right\rangle dtdt' \\
\end{matrix}
</math>
</td><td width="5%">(31)</td></tr></table>

But since the process is assumed stationary <math> \left\langle u' \left( t \right) u' \left( t' \right) \right\rangle = C \left( t' - t \right) </math> where <math> C \left( t' - t \right) = \left\langle u^{2} \right\rangle \rho \left( t'-t \right) </math> is the correlation coefficient. Therefore the integral can be rewritten as:

<table width="70%"><tr><td>
:<math>
\begin{matrix}
var \left[ U_{T} \right] & = & \frac{1}{T^{2}} \int^{T}_{0} \int^{T}_{0} C \left( t' - t \right) dtdt' \\
& = & \frac{ \left\langle u^{2} \right\rangle }{ T^{2} } \int^{T}_{0} \int^{T}_{0} \rho \left( t' - t \right) dtdt' \\
\end{matrix}
</math>
</td><td width="5%">(33)</td></tr></table>

Now we need to apply some fancy calculus. If new variables <math> \tau= t'-t </math> and <math> \xi= t'+t </math> are defined, the double integral can be transformed to (see Figure 5.5):

<table width="70%"><tr><td>
:<math>
var \left[ U_{T} \right] = \frac{var \left[ u \right]}{2 T^{2}} \left[ \int^{T}_{0} d \tau \int^{T-\tau}_{\tau} d \xi \rho \left( \tau \right) + \int^{0}_{-T} d \tau \int^{T+\tau}_{-\tau} d \xi \rho \left( \tau \right) \right]
</math>
</td><td width="5%">(35)</td></tr></table>

where the factor of <math> 1/2 </math> arises from the Jacobian of the transformation. The integrals over <math> d \xi </math> can be evaluated directly to yield:

<table width="70%"><tr><td>
:<math>
var \left[ U_{T} \right] = \frac{var \left[ u \right]}{2 T^{2}} \left\{ \int^{T}_{0} \rho \left( \tau \right) \left[ T - \tau \right] d \tau + \int^{0}_{-T} \rho \left( \tau \right) \left[ T + \tau \right] \right\}
</math>
</td><td width="5%">(36)</td></tr></table>

By noting that the autocorrelation is symmetric, the second integral can be transformed and added to the first to yield at last the result we seek as:

<table width="70%"><tr><td>
:<math>
var \left[ U_{T} \right] = \frac{var \left[ u \right]}{T} \int^{T}_{-T} \rho \left( \tau \right) \left[ 1 - \frac{ \left| \tau \right| }{T} \right] d \tau
</math>
</td><td width="5%">(37)</td></tr></table>

Now if our averaging time, <math> T </math>, is chosen so large that <math> \left| \tau \right| / T << 1 </math> over the range for which <math> \rho \left( \tau \right) </math> is non-zero, the integral reduces:

<table width="70%"><tr><td>
:<math>
\begin{matrix}
var \left[ U_{T} \right] & \approx & \frac{2 var \left[ u \right]}{T} \int^{T}_{0} \rho \left( \tau \right) d \tau \\
& = & \frac{2 T_{int}}{T} var \left[ u \right] \\
\end{matrix}
</math>
</td><td width="5%">(38)</td></tr></table>

where <math> T_{int} </math> is the integral scale defined by equation 10. Thus the ''variability'' of our estimator is given by:

<table width="70%"><tr><td>
:<math>
\epsilon^{2}_{U_{T}} = \frac{2T_{int}}{T}
</math>
</td><td width="5%">(39)</td></tr></table>

Therefore the estimator does, in fact, converge (in mean square) to the correct result as the averaging time, <math> T </math> increases relative to the integral scale, <math> T_{int} </math>.

There is a direct relationship between equation 39 and equation 52 in chapter The elements of statistical analysis ( section Bias and convergence of estimators) which gave the mean square variability for the ensemble estimate from a finite number of statistically independent realizations, <math> X_{N} </math>. Obviously the effective number of independent realizations for the finite time estimator is:

<table width="70%"><tr><td>
:<math>
N_{eff} = \frac{2T_{int}}{T}
</math>
</td><td width="5%">(40)</td></tr></table>

so that the two expressions are equivalent. Thus, in effect, ''portions of the record separated by two integral scales behave as though they were statistically independent, at least as far as convergence of finite time estimators is concerned''.

Thus what is required for convergence is again, many ''independent'' pieces of information. This is illustrated in Figure 5.6. That the length of the recordn should be measured in terms of the integral scale should really be no surprise since it is a measure of the rate at which a process forgets its past.

'''Example'''

It is desired to mesure the mean velocity in a turbulent flow to within an rms error of 1% (i.e. <math> \epsilon = 0.01 </math> ). The expected fluctuation level of the signal is 25% and integral scale is estimated as 100 ms. What is the required averaging time?

From equation 39

<table width="70%"><tr><td>
:<math>
\begin{matrix}
T & = & \frac{2T_{int}}{\epsilon^{2}} \frac{var \left[ u \right]}{U^{2}} \\
& = & 2 \times 0.1 \times (0.25)^{2} / (0.01)^{2} = 125 sec \\
\end{matrix}
</math>
</td><td width="5%">(41)</td></tr></table>

Similar considerations apply to any other finite time estimator and equation 55 from chapter Statistical analysis can be applied directly as long as equation 40 is used for the number of independent samples.

It is common common experimental practice to not actually carry out an analog integration. Rather the signal is sampled at fixed intervals in time by digital means and the averages are computed as for an esemble with a finite number of realizations. Regardless of the manner in which the signal is processed, only a finite portion of a stationary time series can be analyzed and the preceding considerations always apply.

It is important to note that data sampled more rapidly than once every two integral scales do '''not''' contribute to the convergence of the estimator since they can not be considered independent. If <math> N </math> is the actual number of samples acquired and <math> \Delta t </math> is the time between samples, then the effective number of independent realizations is

<table width="70%"><tr><td>
:<math>
N_{eff} = \left\{
\begin{array}{lll}
N \Delta t /T_{int} & if & \Delta t < 2T_{int} \\
N & if & \Delta t \geq 2T_{int} \\
\end{array}
\right.
</math>
</td><td width="5%">(42)</td></tr></table>

It should be clear that if you sample faster than <math> \Delta t = 2T_{int} </math> you are processing unnecessary data which does not help your statistics converge.

You may wonder why one would ever take data faster than absolutely necessary, since it simply it simply fills up your computer memory with lots of statistically redundant data. When we talk about measuring spectra you will learn that for spectral measurements it is necessary to sample much faster to avoid spactral aliasing. Many wrongly infer that they must sample at these higher rates even when measuring just moments. Obviously this is not the case if you are not measuring spectra.

== Random fields of space and time ==

To this point only temporally varying random fields have been discussed. For turbulence however, random fields can be functions of both space and time. For example, the temperature <math> \theta </math> could be a random scalar function of time <math> t </math> and position <math> \stackrel{\rightarrow}{x} </math>, i.e.,

<table width="70%"><tr><td>
:<math>
\theta = \theta \left( \stackrel{\rightarrow}{x} , t \right)
</math>
</td><td width="5%">(43)</td></tr></table>

The velocity is another example of a random vector function of position and time, i.e.,

<table width="70%"><tr><td>
:<math>
\stackrel{\rightarrow}{u} = \stackrel{\rightarrow}{u} \left( \stackrel{\rightarrow}{x},t \right)
</math>
</td><td width="5%">(44)</td></tr></table>

or in tensor notation,

<table width="70%"><tr><td>
:<math>
u_{i} = u_{i} \left( \stackrel{\rightarrow}{x},t \right)
</math>
</td><td width="5%">(45)</td></tr></table>

In the general case, the ensemble averages of these quantities are functions of both positon and time; i.e.,

<table width="70%"><tr><td>
:<math>
\left\langle \theta \left( \stackrel{\rightarrow}{x},t \right) \right\rangle \equiv \Theta \left( \stackrel{\rightarrow}{x},t \right)
</math>
</td><td width="5%">(46)</td></tr></table>

<table width="70%"><tr><td>
:<math>
\left\langle u_{i} \left( \stackrel{\rightarrow}{x},t \right) \right\rangle \equiv U_{i} \left( \stackrel{\rightarrow}{x},t \right)
</math>
</td><td width="5%">(47)</td></tr></table>

If only ''stationary'' random processes are considered, then the averages do not depend on time and are functions of <math> \stackrel{\rightarrow}{x} </math> only; i.e.,

<table width="70%"><tr><td>
:<math>
\left\langle \theta \left( \stackrel{\rightarrow}{x},t \right) \right\rangle \equiv \Theta \left( \stackrel{\rightarrow}{x} \right)
</math>
</td><td width="5%">(48)</td></tr></table>

<table width="70%"><tr><td>
:<math>
\left\langle u_{i} \left( \stackrel{\rightarrow}{x},t \right) \right\rangle \equiv U_{i} \left( \stackrel{\rightarrow}{x}\right)
</math>
</td><td width="5%">(49)</td></tr></table>

Now the averages may not be position dependent either. For example, if the averages are ''independent of the origin in position'', then the field is said to be '''homogeneous'''. '''Homogenity''' (the noun corresponding to the adjective homogeneous) is exactly analogous to stationarity except that position is now the variable, and not time.

It is, of course, possible (at least in concept) to have homogeneous fields which are either stationary or non stationary. Since position, unlike time, is a vector quantity it is also possible to have only partial homogeneity. For example, a field can be homogeneous in the <math> x_{1}- </math> and <math> x_{3}- </math> directions, but not in the <math> x_{2}- </math> direction so that <math> U_{i}=U_{i}(X_{2}) </math> only. In fact, it appears to be dynamically impossible to have flows which are honogeneous in all variables and stationary as well, but the concept is useful, nonetheless.

Homogeneity will be seen to have powerful consequences for the equations govering the averaged motion, since the spatial derivative of any averaged quantity must be identically zero. Thus even homogeneity in only one direction can considerably simplify the problem. For example, in the Reynolds stress transport equation, the entire turbulence transport is exactly zero if the field is homogeneous.

== Multi-point statistics in homogeneous field ==

The concept of homogeneity can also be extended to multi-point statistics. Consider for example, the correlation between the velocity at one point and that at another as illustrated in Figure 5.7. If the time dependence is suppressed and the field is assumed statistically ''homogeneous'', this correlation is a function only of the separation of the two points, i.e.,

<table width="70%"><tr><td>
:<math>
\left\langle u_{i} \left( \stackrel{\rightarrow}{x} , t \right) u_{j} \left( \stackrel{\rightarrow}{x'} , t \right) \right\rangle \equiv B_{i,j} \left( \stackrel{\rightarrow}{r} \right)
</math>
</td><td width="5%">(50)</td></tr></table>

where <math> \stackrel{\rightarrow}{r} </math> is the separation vector defined by

<table width="70%"><tr><td>
:<math>
\stackrel{\rightarrow}{r} = \stackrel{\rightarrow}{x'} - \stackrel{\rightarrow}{x}
</math>
</td><td width="5%">(51)</td></tr></table>

or

<table width="70%"><tr><td>
:<math>
r_{i} = x'_{i} - x_{i}
</math>
</td><td width="5%">(52)</td></tr></table>

Note that the convention we shall follow for vector quantities is that the first subscript on <math> B_{i,j} </math> is the component of velocity at the first position, <math> \stackrel{\rightarrow}{x} </math> , and the second subscript is the component of velocity at the second, <math> \stackrel{\rightarrow}{x'} </math>. For scalar quantities we shall simply put a simbol for the quantity to hold the place. For example, we would write the two-point temperature correlation in a homogeneous field by:

<table width="70%"><tr><td>
:<math>
\left\langle \theta \left( \stackrel{\rightarrow}{x},t \right) \theta \left( \stackrel{\rightarrow}{x'},t \right) \right\rangle \equiv B_{\theta , \theta} \left( \stackrel{\rightarrow}{r} \right)
</math>
</td><td width="5%">(53)</td></tr></table>

A mixed vector/scalar correlation like the two-point temperature velocity correlation would be written as:

<table width="70%"><tr><td>
:<math>
\left\langle u_{i} \left( \stackrel{\rightarrow}{x} , t \right) \theta \left( \stackrel{\rightarrow}{x'},t \right) \right\rangle \equiv B_{i,\theta } \left( \stackrel{\rightarrow}{r} \right)
</math>
</td><td width="5%">(54)</td></tr></table>

On the other hand, if we meant for the temperature to be evaluated at <math> \stackrel{\rightarrow}{x} </math> and the velocity at <math> \stackrel{\rightarrow}{x'} </math> we would have to write:

<table width="70%"><tr><td>
:<math>
\left\langle \theta \left( \stackrel{\rightarrow}{x},t \right) u_{i} \left( \stackrel{\rightarrow}{x'},t \right) \right\rangle \equiv B_{ \theta, i } \left( \stackrel{\rightarrow}{r} \right)
</math>
</td><td width="5%">(55)</td></tr></table>

Now most books don't bother with the subscript notation, and simply give each new correlation a new symbol. At first this seems much simpler; and it is as long as you are only dealing with one or two different correlations. But introduce a few more, then read about a half-dozen pages, and you will find you completely forget what they are or how they were put together. It is usually very important to know exactly what you are talking about, so we will use this comma system to help us remember.

It is easy to see that the consideration of vector quantities raises special considerations. For example, the correlation between a scalar function of position at two points is symmetrical in <math> \stackrel{\rightarrow}{r} </math> , i.e.,

<table width="70%"><tr><td>
:<math>
B_{\theta,\theta} \left( \stackrel{\rightarrow}{r} \right) = B_{\theta,\theta} \left( - \stackrel{\rightarrow}{r} \right)
</math>
</td><td width="5%">(56)</td></tr></table>

This is easy to show from the definition of <math> B_{\theta,\theta} </math> and the fact that the field is homogeneous. Simply shift each of the position vectors by the same amount <math> - \stackrel{\rightarrow}{r} </math> as shown in Figure 5.8 to obtain:

<table width="70%"><tr><td>
:<math>
\begin{matrix}
B_{\theta,\theta}\left( \stackrel{\rightarrow}{r},t \right) & \equiv & \left\langle \theta\left( \stackrel{\rightarrow}{x}, t \right) \theta\left( \stackrel{\rightarrow}{x'}, t \right) \right\rangle \\
& = & \left\langle \theta \left( \stackrel{\rightarrow}{x} - \stackrel{\rightarrow}{r} , t \right) \theta \left( \stackrel{\rightarrow}{x'} - \stackrel{\rightarrow}{r} , t \right) \right\rangle \\
& = & B_{\theta,\theta}\left( - \stackrel{\rightarrow}{r},t \right) \\
\end{matrix}
</math>
</td><td width="5%">(57)</td></tr></table>

since <math> \stackrel{\rightarrow}{x'} - \stackrel{\rightarrow}{r} = \stackrel{\rightarrow}{x} </math> ; i.e., the points are reversed and the separation vector is pointing the opposite way.

Such is not the case, in general, for ''vector'' functions of position. For example, see if you can prove to yourself the following:

<table width="70%"><tr><td>
:<math>
B_{\theta,i} \left( \stackrel{\rightarrow}{r} \right) = B_{i,\theta} \left( - \stackrel{\rightarrow}{r} \right)
</math>
</td><td width="5%">(58)</td></tr></table>

and

<table width="70%"><tr><td>
:<math>
B_{i,j} \left( \stackrel{\rightarrow}{r} \right) = B_{j,i} \left( - \stackrel{\rightarrow}{r} \right)
</math>
</td><td width="5%">(59)</td></tr></table>

Clearly the latter is symmetrical in the variable <math> \stackrel{\rightarrow}{r} </math> only when <math> i = j </math> .

These properties of the two-point correlation function will be seen to play an important role in determining the interrelations among the different two-point statistical quantities. They will be especially important when we talk about spectral quantities.

== Spatial integral and Taylor microscales ==

Just as for a stationary random process, correlations between spatially varying, but ''statistically homogeneous'', random quantities ultimately go to zero;, i.e., they become uncorrelated as their locations become widely separated. Because position (o relative position) is a vector quantity, however, the correlation the carrelation may die off at different rates in different directions. Thus direction must be an important part of the definitions of the integral scales and microscales.

Consider for example the one-dimensional spatial correlation which is obtained by measuring the correlation between the temperature at two points along a line in the x-direction, say,

<table width="70%"><tr><td>
:<math>
B^{(1)}_{\theta,\theta} \left( r \right) \equiv \left\langle \theta \left( x_{1} + r , x_{2} , x_{3} , t \right) \theta \left( x_{1} , x_{2} , x_{3} , t \right) \right\rangle
</math>
</td><td width="5%">(60)</td></tr></table>

The superscript "(1)" denotes "the coordinate direction in which the separation occurs". This distinguishes it from the vector separation of <math> B_{\theta,\theta} </math> above. Also, note that the correlation at zero separationis just the variance; i.e.,

<table width="70%"><tr><td>
:<math>
B^{(1)}_{\theta,\theta} \left( 0 \right) = \left\langle \theta^{2} \right\rangle
</math>
</td><td width="5%">(61)</td></tr></table>

The integral scale in the <math> x </math>-direction can be defined as:

<table width="70%"><tr><td>
:<math>
L^{(1)}_{\theta} \equiv \frac{1}{ \left\langle \theta^{2} \right\rangle} \int^{\infty}_{0} \left\langle \theta \left( x + r, y,z,t \right) \theta \left( x,y,z,t \right) \right\rangle dr
</math>
</td><td width="5%">(62)</td></tr></table>

It is clear that there are at least two more integral scales which could be defined by considering separations in the y and z directions. Thus

<table width="70%"><tr><td>
:<math>
L^{(2)}_{\theta} \equiv \frac{1}{ \left\langle \theta^{2} \right\rangle} \int^{\infty}_{0} \left\langle \theta \left( x,y + r,z,t \right) \theta \left( x,y,z,t \right) \right\rangle dr
</math>
</td><td width="5%">(63)</td></tr></table>

and

<table width="70%"><tr><td>
:<math>
L^{(3)}_{\theta} \equiv \frac{1}{ \left\langle \theta^{2} \right\rangle} \int^{\infty}_{0} \left\langle \theta \left( x,y,z + r,t \right) \theta \left( x,y,z,t \right) \right\rangle dr
</math>
</td><td width="5%">(64)</td></tr></table>

In fact, an integral scale could be defined for ''any'' direction simply by choosing the components of the separation vector <math> \stackrel{\rightarrow}{r} </math>. This situation is even more complicated when correlations of vector quantities are considered. For example, consider the correlation of the velocity vectors at two points, <math> B_{i,j} \left( \stackrel{\rightarrow}{r} \right) </math>. Clearly <math> B_{i,j} \left( \stackrel{\rightarrow}{r} \right) </math> is not a single correlation, but rather nine separate correlations: <math> B_{1,1} \left( \stackrel{\rightarrow}{r} \right) </math> , <math> B_{1,2} \left( \stackrel{\rightarrow}{r} \right) </math> , <math> B_{1,3} \left( \stackrel{\rightarrow}{r} \right) </math> , <math> B_{2,1} \left( \stackrel{\rightarrow}{r} \right) </math> , <math> B_{2,2} \left( \stackrel{\rightarrow}{r} \right) </math> , etc. For each of these an integral scale can be defined once a direction for the separation vector is chosen. For example, the integral scales associated with <math> B_{1,1} </math> for the principal directions are

<table width="70%"><tr><td>
:<math>
L^{(1)}_{1,1} \equiv \frac{1}{\left\langle u^{2}_{1} \right\rangle} \int^{\infty}_{0} B_{1,1} \left( r,0,0 \right) dr
</math>
</td><td width="5%">(65)</td></tr></table>

<table width="70%"><tr><td>
:<math>
L^{(2)}_{1,1} \equiv \frac{1}{\left\langle u^{2}_{1} \right\rangle} \int^{\infty}_{0} B_{1,1} \left( 0,r,0 \right) dr
</math>
</td><td width="5%">(66)</td></tr></table>

<table width="70%"><tr><td>
:<math>
L^{(3)}_{1,1} \equiv \frac{1}{\left\langle u^{2}_{1} \right\rangle} \int^{\infty}_{0} B_{1,1} \left( 0,0,r \right) dr
</math>
</td><td width="5%">(67)</td></tr></table>

Similar integral scales can be defined for the other componentsof the correlation tensor. Two of particular importance in the development of the turbulence theory are:

<table width="70%"><tr><td>
:<math>
L^{(2)}_{1,1} \equiv \frac{1}{\left\langle u^{2}_{1} \right\rangle} \int^{\infty}_{0} B_{1,1} \left( 0,r,0 \right) dr
</math>
</td><td width="5%">(68)</td></tr></table>

<table width="70%"><tr><td>
:<math>
L^{(1)}_{2,2} \equiv \frac{1}{\left\langle u^{2}_{2} \right\rangle} \int^{\infty}_{0} B_{2,2} \left( r,0,0 \right) dr
</math>
</td><td width="5%">(69)</td></tr></table>

In general, each of these integral scales will be different, unless restrictions beyond simple homogeneity are placed on the process (e.g., like ''isotropy'' discussed below). Thus, it is important to specify precisely which integral scale is being referred to; i.e., which components of the vector quantities are being used and in which direction the integration is being performed.

Similar considerations apply to the Taylor microscales, regardless of whether they are being determined from the correlations at small separations, or from the mean square fluctuating gradients. The two most commonly used Taylor microscales are often referred to as <math> \lambda_{f} </math> and <math> \lambda_{g} </math> and are defined by

<table width="70%"><tr><td>
:<math>
\lambda^{2}_{f} \equiv 2 \frac{ \left\langle u^{2}_{1} \right\rangle }{ \left\langle \left[ \partial u_{1} / \partial x_{1} \right]^{2} \right\rangle }
</math>
</td><td width="5%">(70)</td></tr></table>

and

<table width="70%"><tr><td>
:<math>
\lambda^{2}_{g} \equiv 2 \frac{ \left\langle u^{2}_{1} \right\rangle }{ \left\langle \left[ \partial u_{1} / \partial x_{2} \right]^{2} \right\rangle }
</math>
</td><td width="5%">(71)</td></tr></table>

The subscripts f and g refer to the autocorrelation coefficients defined by:

<table width="70%"><tr><td>
:<math>
f \left( r \right) \equiv \frac{\left\langle u_{1} \left( x_{1} + r,x_{2},x_{3} \right) u_{1} \left( x_{1},x_{2},x_{3} \right) \right\rangle}{ \left\langle u^{2}_{1} \right\rangle } = \frac{B_{1,1} \left( r,0,0 \right)}{ B_{1,1} \left( 0,0,0 \right) }
</math>
</td><td width="5%">(72)</td></tr></table>

and

<table width="70%"><tr><td>
:<math>
g \left( r \right) \equiv \frac{\left\langle u_{1} \left( x_{1},x_{2}+r,x_{3} \right) u_{1} \left( x_{1},x_{2},x_{3} \right) \right\rangle}{ \left\langle u^{2}_{1} \right\rangle } = \frac{B_{1,1} \left( 0,r,0 \right)}{ B_{1,1} \left( 0,0,0 \right) }
</math>
</td><td width="5%">(73)</td></tr></table>

It is straightforward to show from the definitions that <math> \lambda_{f} </math> and <math> \lambda_{g} </math> are related to the curvature of the <math> f </math> and <math> g </math> correlation functions at <math> r=0 </math>. Specifically,

<table width="70%"><tr><td>
:<math>
\lambda^{2}_{f}= \frac{2}{d^{2} f / dr^{2} |_{r=0} }
</math>
</td><td width="5%">(74)</td></tr></table>

and

<table width="70%"><tr><td>
:<math>
\lambda^{2}_{g}= \frac{2}{d^{2} g / dr^{2} |_{r=0} }
</math>
</td><td width="5%">(75)</td></tr></table>

Since both <math> f </math> and <math> g </math> are symmetrical functions of <math> r </math>, <math> df/dr </math> and <math> dg/dr </math> must be zero at <math> r=0 </math>. It follows immediately that the leading <math> r </math>-dependent term in the expansions about the origin of both autocorrelations are of parabolic form; i.e.,

<table width="70%"><tr><td>
:<math>
f \left( r \right) = 1 - \frac{r^{2}}{\lambda^{2}_{f}} + \cdots
</math>
</td><td width="5%">(76)</td></tr></table>

and

<table width="70%"><tr><td>
:<math>
g \left( r \right) = 1 - \frac{r^{2}}{\lambda^{2}_{g}} + \cdots
</math>
</td><td width="5%">(77)</td></tr></table>

This is illustrated in Figure 5.9 which shows that the Taylor microscales are the intersection with the <math> r </math>-axis of a parabola fitted to the appropriate correlation function at the origin. Fitting a parabola is a common way to determine the Taylor microscale, but to do so you must make sure you resolve accurately to scales much smaller than it (typically an order of magnitude smaller is required). Otherwise you are simply determining the spatial filtering of your probe or numerical algorithm.

{{Turbulence credit wkgeorge}}

{{Chapter navigation|Turbulence kinetic energy|Homogeneous turbulence}}

Introduction to turbulence/Turbulence kinetic energy

2008-02-25T09:19:02Z

Jola:

{{Introduction to turbulence menu}}

== Fluctuating kinetic energy ==

It is clear from the previous chapter that the straightforward application of ideas that worked well for viscous stresses do not work too well for turbulence Reynolds stresses. Moreover, even the attempt to directly derive equations for the Reynolds stresses using the Navier-Stokes equations as a starting point has left us with far more equations than unknowns. Unfortunately this means that the turbulence
problem for engineers is not going to have a simple solution: we simply cannot produce a set of reasonably universal equations. Obviously we are going to have to study the turbulence fluctuations in more detail and learn how they get their energy (usually from the mean flow somehow), and what they ultimately do with it. Our hope is that by understanding more about turbulence itself, we will gain insight into how we might make closure approximations that will work, at least
sometimes. Hopefully, we will also gain an understanding of when and why they will not work.

An equation for the fluctuating kinetic energy for constant density flow can be obtained directly from the Reynolds stress equation derived earlier (see equation 35 in the chapter on [[Introduction_to_turbulence/Reynolds_averaged_equations|Reynolds averaged equations]]) by contracting the free indices. The result is:

<table width="70%"><tr><td>
:<math>
\begin{matrix}
\left[ \frac{\partial}{\partial t} \left\langle u_{i} u_{i} \right\rangle + U_{j} \frac{\partial }{\partial x_{j} } \left\langle u_{i} u_{i} \right\rangle \right] \\
& = & \frac{\partial}{\partial x_{j}} \left\{ -\frac{2}{\rho} \left\langle p u_{i} \right\rangle \delta_{ij} - \left\langle q^{2} u_{j} \right\rangle + 4 \nu \left\langle s_{ij} u_{i} \right\rangle \right\} \\
& & - 2 \left\langle u_{i}u_{j} \right\rangle \frac{\partial U_{i}}{\partial x_{j}} - 4 \nu \left\langle s_{ij} \frac{\partial u_{i}}{\partial x_{j} } \right\rangle \\
\end{matrix}
</math>
</td><td width="5%">(1)</td></tr></table>

where the incompressibility condition ( <math> \partial u_{j} / \partial x_{j} = 0 </math> ) has been used to eliminate the pressure-strain rate term, and <math> q^{2} \equiv u_{i} u_{i}</math>.

The last term can be simplified by recalling that the velocity deformation rate tensor, <math> \partial u_{i} / \partial x_{j} </math>, can be decomposed into symmetric and anti-symmetric parts; i.e.,

<table width="70%"><tr><td>
:<math>
\frac{\partial u_{i}}{\partial x_{j}} = s_{ij} + \omega_{ij}
</math>
</td><td width="5%">(2)</td></tr></table>

where the symmetric part is the strain-rate tensor, <math> s_{ij} </math>, and the anti-symmetric part is the rotation-rate tensor <math> \omega_{ij} </math>, defined by:

<table width="70%"><tr><td>
:<math>
\omega_{ij} = \frac{1}{2} \left[ \frac{\partial u_{i}}{\partial x_{j}} - \frac{\partial u_{j}}{\partial x_{i}} \right]
</math>
</td><td width="5%">(3)</td></tr></table>

Since the double contraction of a symmetric tensor with an anti-symmetric tensor is identically zero, it follows immediately that:

<table width="70%"><tr><td>
:<math>
\begin{matrix}
\left\langle s_{ij} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle & = & \left\langle s_{ij} s_{ij} \right\rangle + \left\langle s_{ij} \omega_{ij} \right\rangle \\
& = & \left\langle s_{ij} s_{ij} \right\rangle \\
\end{matrix}
</math>
</td><td width="5%">(4)</td></tr></table>

Now it is customary to define a new variable k, the average fluctuating kinetic energy per unit mass, by:

<table width="70%"><tr><td>
:<math>
k \equiv \frac{1}{2} \left\langle u_{i}u_{i} \right\rangle = \frac{1}{2} \left\langle q^{2} \right\rangle = \frac{1}{2} \left[ \left\langle u^{2}_{1} \right\rangle + \left\langle u^{2}_{2} \right\rangle + \left\langle u^{2}_{3} \right\rangle \right]
</math>
</td><td width="5%">(5)</td></tr></table>

By dividing equation 1 by equation 2 and inserting this definition, the equation for the average kinetic energy per unit mass of the fluctuating motion can be re-written as:

<table width="70%"><tr><td>
:<math>
\begin{matrix}
\left[ \frac{\partial}{\partial t} + U_{j} \frac{\partial}{\partial x_{j}} \right] k & = & \frac{\partial}{\partial x_{j}} \left\{ - \frac{1}{\rho} \left\langle pu_{i} \right\rangle \delta_{ij} - \frac{1}{2} \left\langle q^{2} u_{j} \right\rangle + 2 \nu \left\langle s_{ij}u_{i} \right\rangle \right\} \\
& & - \left\langle u_{i}u_{j} \right\rangle \frac{\partial U_{i}}{\partial x_{j} } - 2 \nu \left\langle s_{ij} s_{ij} \right\rangle \\
\end{matrix}
</math>
</td><td width="5%">(6)</td></tr></table>

The role of each of these terms will be examined in detail later. First note that an alternative form of this equation can be derived by leaving the viscous stress in terms of the strain rate. We can obtain the appropriate form of the equation for the fluctuating momentum from equation 21 in the chapter on[[Introduction to turbulence/Reynolds averaged equations#Origins of turbulence|origins of turbulence]] by substituting the incompressible Newtonian constitutive equation into it to obtain:

<table width="70%"><tr><td>
:<math>
\left[ \frac{\partial }{\partial t } + U_{j} \frac{\partial }{\partial x_{j} } \right] u_{i} = - \frac{1}{\rho} \frac{\partial p}{\partial x_{i}} + \nu \frac{\partial^{2} u_{i}}{ \partial x^{2}_{j}} - \left[ u_{j} \frac{\partial U_{i}}{\partial x_{j} } \right] - \left\{ u_{j} \frac{\partial u_{i}}{ \partial x_{j}} - \left\langle u_{j} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right\}
</math>
</td><td width="5%">(7)</td></tr></table>

If we take the scalar product of this with the fluctuating velocity itself and average, it follows (after some rearrangement) that:

<table width="70%"><tr><td>
:<math>
\begin{matrix}
\left[ \frac{\partial}{\partial t} + U_{j} \frac{\partial}{\partial x_{j}} \right] k & = & \frac{\partial }{ \partial x_{j} } \left\{ - \frac{1}{\rho} \left\langle pu_{i} \right\rangle \delta_{ij} - \frac{1}{2} \left\langle q^{2} u_{j} \right\rangle + \nu \frac{\partial}{\partial x_{j} } k \right\} \\
& & - \left\langle u_{i} u_{j} \right\rangle \frac{\partial U_{i}}{\partial x_{j}} - \nu \left\langle \frac{\partial u_{i}}{\partial x_{j}} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle\\
\end{matrix}
</math>
</td><td width="5%">(8)</td></tr></table>

Both equations 6 and 8 play an important role in the study of turbulence. The first form given by equation 6 will provide the framework for understanding the dynamics of turbulent motion. The second form, equation 8 forms the basis for most of the second-order closure attempts at turbulence modelling; e.g., the socalled k-e models ( usually referred to as the “k-epsilon models”). This because it has fewer unknowns to be modelled, although this comes at the expense of some extra assumptions about the last term. It is only the last term in equation 6 that can be identified as the true rate of dissipation of turbulence kinetic energy, unlike the last term in equation 8 which is only the dissipation when the flow is ''homogeneous''. We will talk about homogeniety below, but suffice it to say now that it never occurs in nature. Nonetheless, many flows can be assumed to be homogeneous ''at the scales of turbulence which are important to this term'', so-called ''local homogeniety''.

Each term in the equation for the kinetic energy of the turbulence has a distinct role to play in the overall kinetic energy balance. Briefly these are:

* Rate of change of kinetic energy per unit mass due to non-stationarity; i.e., time dependence of the mean:

<table width="70%"><tr><td>
:<math>
\frac{\partial k}{\partial t}
</math>
</td><td width="5%">(9)</td></tr></table>

* Rate of change of kinetic energy per unit mass due to convection (or advection) by the mean flow through an inhomogenous field :

<table width="70%"><tr><td>
:<math>
U_{j} \frac{\partial k}{\partial x_{j}}
</math>
</td><td width="5%">(10)</td></tr></table>

* Transport of kinetic energy in an inhomogeneous field due respectively to the pressure fluctuations, the turbulence itself, and the viscous stresses:

<table width="70%"><tr><td>
:<math>
\frac{\partial}{\partial x_{j}} \left\{-\frac{1}{\rho} \left\langle pu_{i} \right\rangle \delta_{ij} - \frac{1}{2} \left\langle q^{2} u_{j} \right\rangle + 2\nu \left\langle s_{ij}u_{i} \right\rangle \right\}
</math>
</td><td width="5%">(11)</td></tr></table>

* Rate of production of turbulence kinetic energy from the mean flow(gradient):

<table width="70%"><tr><td>
:<math>
- \left\langle u_{i}u_{j} \right\rangle \frac{\partial U_{i}}{\partial x_{j}}
</math>
</td><td width="5%">(12)</td></tr></table>

* Rate of dissipation of turbulence kinetic energy per unit mass due to viscous stresses:

<table width="70%"><tr><td>
:<math>
\epsilon \equiv 2\nu \left\langle s_{ij}s_{ij} \right\rangle
</math>
</td><td width="5%">(12)</td></tr></table>

These terms will be discussed in detail in the succeeding sections, and the role of each examined carefully.

== Rate of dissipation of the turbulence kinetic energy ==

The last term in the equation for the kinetic energy of the turbulence has been identified as the rate of dissipation of the turbulence energy per unit mass; i.e.,

<table width="70%"><tr><td>
:<math>
\epsilon = 2\nu \left\langle s_{ij} s_{ij} \right\rangle = \nu \left\{ \left\langle \frac{\partial u_{i} }{\partial x_{j} } \frac{\partial u_{i} }{\partial x_{j} } \right\rangle + \left\langle \frac{\partial u_{i} }{\partial x_{j} } \frac{\partial u_{j} }{\partial x_{i} } \right\rangle \right\}
</math>
</td><td width="5%">(14)</td></tr></table>

It is easy to see that <math> \epsilon \geq 0 </math> always, since it is a sum of the average of squared quantities only (i.e. <math> \left\langle s_{ij} s_{ij} \right\rangle \geq 0 </math> ). Also, since it occurs on the right hand side of the kinetic energy equation for the fluctuating motions preceded by a minus sign, it is clear that it can act only to ''reduce'' the kinetic energy of the flow. Therefore it causes a ''negative'' rate of change of kinetic energy; hence the name ''dissipation''.

Physically, enegry is dissipated because of the work done by the fluctuating viscous stresses in resisting deformation of the fluid material by the fluctuating strain rates; i.e.

<table width="70%"><tr><td>
:<math>
\epsilon = \left\langle \tau^{(v)}_{ij} s_{ij} \right\rangle
</math>
</td><td width="5%">(15)</td></tr></table>

This reduces to equation 14 only for a Newtonian fluid. In non-Newtonian fluids, protions of this product may not be negative implying that it may not all represent an irrecoverable loss of fluctuating kinetic energy.

It will be shown in the following chapter on [[Introduction to turbulence/Stationarity and homogenity|stationarity and homogenity]] that the dissipation of turbulence energy mostly takes place at the smallest turbulence scales, and that those scales can be characterized by so-called Kolmogorov microscale defined by:

<table width="70%"><tr><td>
:<math>
\eta_{K} \equiv \left(\frac{\nu^{3}}{\epsilon} \right)^{1/4}
</math>
</td><td width="5%">(16)</td></tr></table>

In atmospheric motions where the length scale for those eddies having the most turbulence energy (and responsible for the Reynolds stress) can be measured in kilometers, typical values of the Kolmogorov microscale range from 0.1 - 10 ''millimeters''. In laboratory flows where the overall scale of the flow is greatly reduced, much smaller values of <math> \eta_{K} </math> are not uncommon. The small size of these dissipative scales greately complicates measurement of energy balances, since the largest measuring dimension must be about equal to twice the Kolmogorov microscale. And it is the range of scales, <math> L / \eta </math>, which makes direct numerical simulation of most interesting flows impossible, since the required number of computational cells is several orders of magnitude greater that <math> (L / \eta )^{3} </math>. This same limitation also affects experiments as well, which must often be quite large to be useful.

One of the consequences of this great separation of scales between those containing the bulk of the turbulence energy and those dissipating it is that ''the dissipation rate is primarily determined by the large scales and not the small''. This is because the viscous scales (which operate on a time scale of <math> t_{K} = ( \nu / \epsilon )^{1/2}</math> ) dissipate rapidly any energy sent down to them by non-linear processes of scale to scale energy transfer. Thus the overall rate of dissipation is controlled by the rate of energy transfer ''from'' the energetic scales, primarily by the non-linear scale-to-scale transfer. This will be discussed later when we consider the energy spactrum. But for now it is important only note that a consequence of this is that the dissipation rate is given approximately as:

<table width="70%"><tr><td>
:<math>
\epsilon \propto \frac{u^{3}}{L}
</math>
</td><td width="5%">(17)</td></tr></table>

where <math> u^{2} \equiv \left\langle q^{2} \right\rangle / 3 </math> and <math> L </math> is an integral length scale. It is easy to remember this relation if you note that the time scale of the energetic turbulent eddies can be estimated as <math> L/u </math>. Thus <math> d3u^{2} / dt </math> can estimated as <math> \left( 3u^{2} /2 \right) / \left( L / u \right) </math>.

Sometimes it is convenient to just ''define'' the "length scale of the energy containing eddies" (or the ''pseudo-integral scale'') as:

<table width="70%"><tr><td>
:<math>
l \equiv \frac{u^{3}}{\epsilon}
</math>
</td><td width="5%">(18)</td></tr></table>

Almost always <math> l \propto L </math>, but the relation is at most only exact theoretically in the limit of infinite Reynolds number since the constant of proportionality is Reynolds number dependent. The Reynolds number dependence of the ratio <math> L/l </math> for grid turbulence is illustrated in <font color=orange>Figure 4.1</font>. Many interpret this data to suggest that this ratioapproaches a constant and ignore the scatter. In fact some assume ratio to be constant and even refer to <math> l </math> though it were the real integral scale. Others argue that the scatter is because of the differing upstream conditions and that the ratio may not be constant at all. It is really hard to tell who is right in the absence of facilities or simulations in which the Reynolds number can vary very much for fixed initial conditions. This all may leave you feeling a bit confused, but that’s the way turbulence is right now. It’s a lot easier to teach if we just tell you one view, but that’s not very good preparation for the future.

Here is what we can say for sure. Only the integral scale, <math>L</math>, is a physical length scale, meaning that it can be directly observed in the flow by spectral or correlation measurements (as shown in the following chapters on [[Introduction to turbulence/Stationarity and homogenity|stationarity and homogenity]] and [[Introduction to turbulence/Homogenous turbulence|homogenous turbulence]]). The pseudo-integral scale, <math>l</math>, on the other hand is simply a definition; and it is only at infinite turbulence Reynolds number that it may have physical significance. But it is certainly a useful
approximation at large, but finite, Reynolds numbers. We will talk about these subtle but important distinctions later when we consider homogeneous flows, but it is especially important when considering similarity theories of turbulence. For
now simply file away in your memory a note of caution about using equation 17 too freely. And do not be fooled by the cute description this provides. It is just that, a description, and not really an explanation of why all this happens — sort
of like the weather man describing the weather. Using equation 18, the Reynolds number dependence of the ratio of the
Kolmorgorov microscale, <math>K</math>, to the pseudo-integral scale, <math>l</math>, can be obtained as:

<table width="70%"><tr><td>
:<math>
\frac{\eta_k}{l} = R_l^{-3/4}
</math>
</td><td width="5%">(19)</td></tr></table>

Figure 4.1: Ratio of physical integral length scale to pseudo-integral length scale in homogeneous turbulence as function of local Reynolds number, <math>R_\lambda</math>.

Where the turbulence Reynolds number, <math>R_l</math>, is defined by:

<table width="70%"><tr><td>
:<math>
R_l \equiv \frac{u l}{\nu} = \frac{u^4}{\nu \epsilon}
</math>
</td><td width="5%">(20)</td></tr></table>

'''Example:''' Estimate the Kolmogorov microscale for <math>u = 1 m/s</math> and <math>L = 0.1 m</math> for air and water.

:'''air''' For air, <math>R_l = 1 \cdot (0.1) / 15 \cdot 10^{-6} \approx 7 \cdot 10^3 </math>. Therefore <math>l/\eta_K \approx 8 \cdot 10^2</math>, so <math>\eta_K \approx 1.2 \cdot 10^{-4} m</math> or <math>0.12 mm</math>.

:'''water''' For water, <math>R_l = 1 \cdot (0.1) / 10^{-6} \approx 10^5 </math>. Therefore <math>l/\eta_K \approx 5 \cdot 10^3</math>, so <math>\eta_K \approx 2 \cdot 10^{-5} m</math> or <math>0.02 mm</math>.

'''Exercise:''' Find the dependence on <math>R_l</math> of the time-scale ration between the Kolmorogov microtime and the time scale of the energy-containing eddies. It will also be argued later that these small dissipative scales of motion at very
high Reynolds number tend to be statistically nearly isotropic; i.e., their statistical character is independent of direction. We will discuss some of the implications of isotropy and local isotropy later, but note for now that it makes possible a huge
reduction in the number of unknowns, particularly those determined primarily by the dissipative scales of motion.

Thus the dissipative scales are all much smaller than those characterizing the energy of the turbulent fluctuations, and their relative size decreases with increasing Reynolds number. Note that in spite of this, the Kolmogorov scales all increase
with increasing energy containing scales for fixed values of the Reynolds number. This fact is very important in designing laboratory experiments at high turbulence Reynolds number where the finite probe size limits spatial resolution. The
rather imposing size of some experiments is an attempt to cope with this problem by increasing the size of the smallest scales, thus making them larger than the resolution limits of the probes being used.

'''Exercise:''' Suppose the smallest probe you can build can only resolve <math>0.1 mm</math>. Also to do an experiment which is a reasonable model of a real engineering flow (like a hydropower plant), you need (for reason that will be clear later) a scale separation of at least <math>L/\eta_K = 10^4</math>. If your facility has to be at least a factor of ten larger than <math>L</math> (which you estimate as <math>l</math>), what is its smallest dimension?

== Kinetic energy of the mean motion and production of turbulence ==

An equation for the kinetic energy of the ''mean motion'' can be derived by a procedure exactly analogous to that applied to the fluctuating motion. The mean motion was shown in 19 in the chapter on [[Introduction_to_turbulence/Reynolds_averaged_equations|Reynolds averaged equations]] to be given by:

<table width="70%">
<tr><td>
:<math>
\rho \left[ \frac{\partial U_{i}}{\partial t} + U_{j}\frac{\partial U_{i}}{\partial x_{j}} \right] = -\frac{\partial P}{\partial x_{i}} + \frac{\partial T^{(v)}_{ij}}{\partial x_{j}}- \frac{\partial }{\partial x_{j}}\left(\rho \left\langle u_{i}u_{j} \right\rangle \right)
</math>
</td><td width="5%">(21)</td></tr></table>

By taking the scalar product of this equation with the mean velocity,<math> U_{i}</math>, we can obtain an equation for the kinetic energy of the ''mean'' motion as:

<table width="70%">
<tr><td>
:<math>
U_{i}\left[ \frac{\partial}{\partial t} + U_{j} \frac{\partial}{\partial x_{j}} \right] U_{i} = - \frac{U_{i}}{\rho} \frac{\partial P}{\partial x_{i}} + \frac{U_{i}}{\rho} \frac{\partial T^{(v)}_{ij} }{\partial x_{j}} - U_{i} \frac{\partial \left\langle u_{i}u_{j} \right\rangle}{\partial x_{j} }
</math>
</td><td width="5%">(22)</td></tr></table>

Unlike the fluctuating equations, there is no need to average here, since all the terms are already averages.

In exactly the same manner that we rearrannged the terms in the eqyation for the kinetic energy of the fluctuations, we can rearrange the equation for the kinetic energy of the mean flow to obtain:

<table width="70%"><tr><td>
:<math>
\begin{matrix}

\left[ \frac{\partial}{\partial t} + U_{j} \frac{\partial}{\partial x_{j}} \right] K = \\
& = & \frac{\partial}{\partial x_{j}} \left\{ - \frac{1}{\rho} \left\langle PU_{i} \right\rangle \delta_{ij} - \frac{1}{2} \left\langle u_{i}u_{j} \right\rangle U_{i} + 2 \nu \left\langle S_{ij} U_{i} \right\rangle \right\} +\\
& + & \left\langle u_{i} u_{j} \right\rangle \frac{\partial U_{i}}{\partial x{j}} - 2 \nu \left\langle S_{ij} S_{ij} \right\rangle \\

\end{matrix}
</math>
</td><td width="5%">(23)</td></tr></table>

where

<table width="70%">
<tr><td>
:<math>
K\equiv \frac{1}{2} Q^{2} = \frac{1}{2} U_{i}U_{i}
</math>
</td><td width="5%">(24)</td></tr></table>

The role of all of the terms can immediately be recognized since each term has its counterpart in the equation for the average fluctuating kinetic energy.

Comparison of equations 23 and 6 reveals that the term <math>-\left\langle u_{i}u_{j}\right\rangle \partial U_{i}/\partial x_{j}</math> appears in the equations for the kinetis energy of BOTH the mean and the fluctuations. There is, however, one VERY important difference. This "production" term has the opposite sign in the equationfor the mean kinetic energy than in that for the mean fluctuating kinetic energy! Therefore, ''whatever its effect on the kinetic energy of the mean, its effect on the kinetic energy of the fluctuations will be the opposite''. Thus kinetic energy can be interchanged between the mean and fluctuating motions. In fact, the only other term involving fluctuations in the equation for the kinetic energy of the mean motion is divergence term; therefore it can only move the kinetic energy of the mean flow from one place to another. Therefore this "production" term provides the ''only'' means by which energy can be interchanged between the mean flow and fluctuations.

Understanding the manner in which this energy exchange between mean and fluctuating motions is accomplished represents one of the most challenging problems in turbulence. The overall exchange can be understood by exploiting the analogy which treats <math>-\rho \left\langle u_{i}u_{j}\right\rangle </math> as a stress, the Reynolds stress. The term <math>-\rho \left\langle u_{i}u_{j}\right\rangle \partial U_{i}/\partial x_{j}</math> can be thought of as the working of the Reynolds stress against the mean velocity gradient of the flow, exactly as the viscous stresses resist deformation by the instantaneous velocity gradients. This energy expended against the Reynolds stress during deformation by the mean motion ends up in the fluctuating motions, however, while that expended against viscous stresses goes directly to internal energy. As we have already seen, the viscous deformation work from the fluctuating motions (or dissipation) will eventually send this fluctuating kinetic energy on to internal energy as well.

Now, just in case you are not all that clear exactly how the dissipation terms really accomplish this for the instantaneous motion, it might be useful to examine exactly how the above works. We begin by decomposing the mean deformation rate tensor <math>\partial U_{i}/\partial x_{j}</math> into its symmetric and antisymmetric parts, exactly as we did for the instantaneous deformation rate tensor in Chapter 3; i.e.,

<table width="70%">
<tr><td>
:<math>
\frac{\partial U_{i}}{\partial x_{j} } = S_{ij} + \Omega_{ij}
</math>
</td><td width="5%">(25)</td></tr></table>

where the mean strain rate <math>S_{ij}</math> is defined by

<table width="70%">
<tr><td>
:<math>
S_{ij}=\frac{1}{2}\left[ \frac{\partial U_{i}}{\partial x_{j}} + \frac{\partial U_{j}}{\partial x_{i}} \right]
</math>
</td><td width="5%">(26)</td></tr></table>

and the mean rotation rate is defined by
<table width="70%">
<tr><td>
:<math>
\Omega_{ij} = \frac{1}{2}\left[ \frac{\partial U_{i}}{\partial x_{j}} - \frac{\partial U_{j}}{\partial x_{i}} \right]
</math>
</td><td width="5%">(27)</td></tr></table>

Since <math>\Omega_{ij}</math> is antisymmetric and <math> -\left\langle u_{i}u_{j}\right\rangle </math> is symmetric, their contraction is zero so it follows that:

<table width="70%">
<tr><td>
:<math>
- \left\langle u_{i} u_{j} \right\rangle \frac{\partial U_{i}}{\partial x_{j}} = - \left\langle u_{i} u_{j} \right\rangle S_{ij}
</math>
</td><td width="5%">(28)</td></tr></table>

Equation 28 is an analog to the mean viscous dissipation term given for incompressible flow by:

<table width="70%">
<tr><td>
:<math>
T^{(v)}_{ij} \frac{\partial U_{i}}{\partial x_{j}} = T^{(v)}_{ij} S_{ij} = 2 \mu S_{ij}S_{ij}
</math>
</td><td width="5%">(29)</td></tr></table>

It is easy to show that this term transfers (or dissipates) the mean kinetic energy directly to internal energy, since exactly the same term appears with the opposite sing in the internal energy equations. Moreover, since <math>S_{ij}S_{ij}\geq 0</math> always, this is a one-way process and kinetic energy is decreased while internal energy is increased. Hence it can be referred to either as "dissipation" of kinetic energy, or as "production" of internal energy. As surprising as it may seem, this direct dissipation of energy by the mean flow is usually negligible compared to the energy lost to the turbulence through the Reynolds stress term.(Remember, there is a term exactly like this in the kinetic energy equation for the fluctuating motion, but involving only fluctuating quantities; namely <math> 2 \mu \left\langle s_{ij} s_{ij} \right\rangle </math> .) We shall show later that <math> \left\langle s_{ij}s_{ij} \right\rangle >> \left\langle S_{ij}S_{ij} \right\rangle </math>. What this means is that most of the energy dissipation is due to the turbulence.

There is a very important difference between equations 28 and 29. Whereas the effect of the viscous stress working against the deformation (in a Newtonian fluid) is ''always'' to remove energy from the flow (since <math>S_{ij}S_{ij}\geq 0</math> always), ''the effect of the Reynolds stress working against the mean gradient can be of either sign'', at least in principle. That is, it can either transfer energy ''from'' the mean motion ''to'' the fluctuating motion, or ''vice versa''.

Almost always (and especially in situations of engineering importance), <math>- \left\langle u_{i}u_{j}\right\rangle S_{ij} > 0 </math> almost always so kinetic energy is removed from the mean motion and added to the fluctuations. Since the term <math> P = - \left\langle u_{i}u_{j}\right\rangle \partial U_{i}/\partial x_{j}</math> usually acts to increase the turbulence kinetic energy, it is usually referred to as the "rate of turbulence energy production", or simply "production".

Now that we have identified how the averaged equations account for the ‘production’ of turbulence energy from the mean motion, it is tempting to think we have understood the problem. In fact, labelling phenomenon is not the same as understanding them. The manner in which the turbulence motions cause this exchange of kinetic energy between the mean and fluctuating motions varies from flow to flow, and is really very poorly understood. Saying that it is the Reynolds stress working against the mean velocity gradient is true, but like saying that
money comes from a bank. If we want to examine the energy transfer mechanism in detail we must look beyond the single point statistics, so this will have to be a story for another time.

== Transport or divergence terms ==

The overall role of the transport terms is best understood by considering a turbulent flow which is completely confined by rigid walls as in Figure 4.2. First consider only the turbulence transport term. If the volume within the confinement is denoted by <math> V_{o}</math> and its bounding surface is <math> S_{o}</math>, then first term on the right-hand side of equation 4.6 for the fluctuating kinetic energy can be integrated over the volume to yield:

<table width="70%"><tr><td>
:<math>
\begin{matrix}
& & \int \int \int_{V_{o}} \frac{\partial}{\partial x_{j}} \left[ - \frac{1}{\rho} \left\langle pu_{i} \right\rangle \delta_{ij} - \frac{1}{2} \left\langle q^{2} u_{j} \right\rangle + \nu \left\langle s_{ij} u_{i} \right\rangle \right] dV \\
& = & \int \int_{S_{o}} \left[ - \frac{1}{\rho} \left\langle pu_{i} \right\rangle \delta_{ij} - \frac{1}{2} \left\langle q^{2} u_{j} \right\rangle + \nu \left\langle s_{ij} u_{i} \right\rangle \right] n_{j} dS \\
\end{matrix}
</math>
</td><td width="5%">(30)</td></tr></table>

where we have used the divergence theorem - again!

We assumed our enclosure to have rigid walls; therefore the normal component of the mean velocity <math> ( u_{n}= u_{j}n_{j}) </math> must be zero on the surface since there can be no flow through it (the kinematic boundary condition). This immediately eliminates the contributions to the surface integral from the <math> \left\langle p u_{j} n_{j} \right\rangle </math> and <math>\left\langle q^{2} u_{j} n_{j} \right\rangle </math> terms. But the last term is zero on the surface also. This can be seen in two ways: either by invoking the no-slip condition which together with the kinematic boundary condition insures that <math> u_{i} </math> is zero on the boundary, or by noting from Cauchy's theorem that <math> \nu s_{ij} n_{j} </math> is the viscous contribution to the normal contact force per unit area on the surface (i.e., <math> t^{(v)}_{n} </math> ) whose scalar product with <math> u_{i} </math> must be identically zero since <math> u_{n} </math> is zero. Therefore the entire integral is identically zero and its net contribution to the rate of change of kinetic energy is zero.

Thus the only effect of the turbulence transport terms (in a fixed volume at least) can be to move energy from one place to another, neither creating nor destroying it in the process. This is, of course, why they are collectively called the ''transport terms''. This spatial transport of kinetic energy is accomplished by the acceleration of adjacent fluid due to pressure and viscous stresses (the first and last terms respectively), and by the physical transport of fluctuating kinetic energy by the turbulence itself (the middle term).

This role of these turbulence transport terms in moving kinetic energy around is often exploited by turbulence modellers. It is argued, that ''on the average'', these terms will only act to move energy from regions of higher kinetic energy to lower. Thus a ''plausible'' first-order hypothesis is that this "diffusion" of kinetic energy should be proportioned to gradients of the kinetic energy itself. That is,

<table width="70%"><tr><td>
:<math>
- \frac{1}{\rho}\left\langle pu_{j} \right\rangle - \frac{1}{2} \left\langle q^{2} u_{j} \right\rangle + \nu \left\langle s_{ij} u_{i} \right\rangle = \nu_{ke} \frac{\partial k}{\partial x_{j}}
</math>
</td><td width="5%">(31)</td></tr></table>

where <math> \nu_{ke} </math> is an effective diffusivity like the eddy viscosity discussed earlier. If we use the alternative form of the kinetic energy equation (equation 4.8), there is no need to model the viscous term (since it involves only <math> k </math> itself). Therefore our model might be:

<table width="70%"><tr><td>
:<math>
- \frac{1}{\rho} \left\langle pu_{j} \right\rangle - \frac{1}{2} \left\langle q^{2} u_{j} \right\rangle = \nu_{ke alt} \frac{\partial k}{\partial x_{j}}
</math>
</td><td width="5%">(32)</td></tr></table>

If You think about it, that such a simple closure is worth mentioning at all is pretty amazing. We took 9 unknowns, lumped them together, and replaced their net effect by simple gradient of something we did know (or at least were calculating), <math> k </math>. And surprisingly, this simple idea works pretty well in many flows, wspecially if the value of the turbulent viscosity is itself related to other quantities like <math> k </math> and <math> \epsilon </math>. In fact this simple gradient hypothesis for the turbulence transport terms is at the root of all engineering turbulence models.

There are a couple of things to note about such simple closures though, before getting too enthused about them. First such an assumption rules out a counter-gradient diffusion of kinetic energy which is known to exist in some flows. In such situations the energy appears to flow ''up'' the gradient. While this may seem unphysical, remember we only ''assumed'' it flowed ''down'' the gradient in the first place. This is the whole problem with a ''plausibility'' argument. Typically energy does tend to be transported from regions of high kinetic energy to low kinetic energy, but there is really no reason for it always to do so, especially if there are other mechanisms at work. And certainly there is no reason for it to always be true locally, and the gradient of anything is a local quantity.

Let me illustrate this by a simple example. Let's apply a gradient hypothesis to the economy - a plausibility hypothesis if you will By this simple model, money would always flow from the rich who have the most, to the poor who have the least. In fact, as history has shown, in the absence of other forces (like revolutions, beheadings, and taxes) this almost never happens. The rich will always get richer, and the poor poorer. And the reason is quite simple, the poor are usually borrowing, while the rich are loaning - with interest. Naturally there are indidual exceptions and great success stories among the poor. And there are wealthy people who give everything away. But mostly in a completely free economy, the money flows in a counter-gradient manner. So society (and the rich in particular) have a choice - risk beheading and revolution, or find a peaceful means to redistribute the wealth - like taxes. While the general need for the latter is recognized (especially among those who have the least), there is, of course, considerable disagreement of how much tax is reasonable to counter the natural gradient.

Just as the simple eddy viscosity closure for the mean flow can be more generally written as a tensor, so can it be here. In fact the more sophisticated models write it as second or fourth-order tensors. More importantly, they include other gradients in the model so that the gradient of one quantity can influence the gradient of another. Such models can sometimes even accont for counter-gradient behavior. If your study of turbulence takes you into the study of turbulence models watch for these subtle differences among them. And don't let yourself be annoyed or intimidated by their complexity. Instead marvel at the physics behind them, and try to appreciate the wonderful manner in which mathematics has been used to make them properly invariant so you don't have to worry about whether they work in any particular coordinate system. It is all these extra terms that give you reason to hope it might work at all.

== The Intercomponent Transfer of Energy ==

The objective of this section is to examine how kinetic energy produced in one velocity component of the turbulence can be transferred to the other velocity components of the fluctuating motion. This is very important since often energy is transferred from the mean flow to a only a single component of the fluctuating motion. Yet somehow all three components of the kinetic energy end up being about the same order of magnitude. The most common exception to this is very close to surfaces where the normal component is suppressed by the kinematic boundary condition. To understand what is going on, it is necessary to develop even a few more equations; in particular, equations for ''each component of the kinetic energy''. The procedure is almost identical to that used to derive the kinetic energy equation itself.

Consider first the equation for the 1-component of the fluctuating momentum. We can do this by simply setting <math> i=1 </math> and <math> k=1 </math> in the equation 35 in the chapter on [[Introduction_to_turbulence/Reynolds_averaged_equations|Reynolds averaged equations]] , or derive it from scratch by setting the free index in equation 27 in the chapter [[Introduction_to_turbulence/Reynolds_averaged_equations|Reynolds averaged equations]]
equal to unity (i.e. i=1); i.e.,

<table width="70%"><tr><td>
:<math>
\left[ \frac{ \partial u_{1}}{\partial t} + U_{j} \frac{ \partial u_{1} }{ \partial x_{j} } \right] = - \frac{1}{ \rho } \frac{ \partial p}{ \partial x_{1} } + \frac{1}{ \rho } \frac{\partial \tau^{(v)}_{1j}}{\partial x_{j}} - \left[ u_{j} \frac{ \partial U_{1} }{ \partial x_{j} } \right] - \left\{ u_{j} \frac{ \partial u_{1} }{ \partial x_{j} } - \left\langle u_{j} \frac{ \partial u_{1} }{ \partial x_{j} } \right\rangle \right\}
</math>
</td><td width="5%">(33)</td></tr></table>

Multiplying this equation by <math> u_{1} </math>, averaging, and rearranging the pressure-velocity gradient term using the chain rule for products yields:

'''1-component'''

<table width="70%"><tr><td>
:<math>
\begin{matrix}
\left[ \frac{ \partial }{ \partial t} + U_{j} \frac{ \partial }{ \partial x_{j} } \right] \frac{1}{2} \left\langle u^{2}_{1}\right\rangle = \\
& = & \left\langle p \frac{\partial u_{1}}{\partial x_{1} } \right\rangle + \\
& + & \frac{ \partial }{ \partial x_{j}} \left\{ -\frac{1}{\rho} \left\langle p u_{1} \right\rangle \delta_{1j} - \frac{1}{2} \left\langle u^{2}_{1} u_{j} \right\rangle + 2 \nu \left\langle s_{1j} u_{1} \right\rangle \right\} - \\
& - & \left\langle u_{1} u_{j} \right\rangle \frac{\partial U_{1}}{ \partial x_{j}} - 2 \nu \left\langle s_{1j} s_{1j} \right\rangle \\
\end{matrix}
</math>
</td><td width="5%">(34)</td></tr></table>

All of the terms except one look exactly like the their counterparts in equation 6 for the average of the total fluctuating kinetic energy. The single exception is the first term on the right-hand side which is the contribution from the pressure-strain rate. This will be seen to be exactly the term we are looking for to move energy among the three components.

Similar equations can be derived for the other fluctuating components with the result that

'''2-component'''

<table width="70%"><tr><td>
:<math>
\begin{matrix}
\left[ \frac{ \partial }{ \partial t} + U_{j} \frac{ \partial }{ \partial x_{j} } \right] \frac{1}{2} \left\langle u^{2}_{2}\right\rangle = \\
& = & \left\langle p \frac{\partial u_{2}}{\partial x_{2} } \right\rangle + \\
& + & \frac{ \partial }{ \partial x_{j}} \left\{ -\frac{1}{\rho} \left\langle p u_{2} \right\rangle \delta_{2j} - \frac{1}{2} \left\langle u^{2}_{2} u_{j} \right\rangle + 2 \nu \left\langle s_{2j} u_{2} \right\rangle \right\} - \\
& - & \left\langle u_{2} u_{j} \right\rangle \frac{\partial U_{2}}{ \partial x_{j}} - 2 \nu \left\langle s_{2j} s_{2j} \right\rangle \\
\end{matrix}
</math>
</td><td width="5%">(35)</td></tr></table>

and

'''3-component'''

<table width="70%"><tr><td>
:<math>
\begin{matrix}
\left[ \frac{ \partial }{ \partial t} + U_{j} \frac{ \partial }{ \partial x_{j} } \right] \frac{1}{2} \left\langle u^{2}_{3}\right\rangle = \\
& = & \left\langle p \frac{\partial u_{3}}{\partial x_{3} } \right\rangle + \\
& + & \frac{ \partial }{ \partial x_{j}} \left\{ -\frac{1}{\rho} \left\langle p u_{3} \right\rangle \delta_{3j} - \frac{1}{2} \left\langle u^{2}_{3} u_{j} \right\rangle + 2 \nu \left\langle s_{3j} u_{3} \right\rangle \right\} - \\
& - & \left\langle u_{3} u_{j} \right\rangle \frac{\partial U_{3}}{ \partial x_{j}} - 2 \nu \left\langle s_{3j} s_{3j} \right\rangle \\
\end{matrix}
</math>
</td><td width="5%">(36)</td></tr></table>

Note that in each equation a new term involving a pressure-strain rate has appeared as the first term on the right-hand side. It is straightforward to show that these three equations sum to the kinetic energy equation given by equation 6, the extra pressure terms vanishing for the incompressible flow assumed here. In fact, the vanishing of the pressure-strain rate terms when the three equations are added together gives a clue as to their role. Obviously they can neither create nor destroy kinetic energy, only move it from one component of the kinetic energy to another.

The precise role of the pressure terms can be seen by noting that incompressibility implies that:

<table width="70%"><tr><td>
:<math>
\left\langle p \frac{\partial u_{j} }{ \partial x_{j} } \right\rangle = 0
</math>
</td><td width="5%">(37)</td></tr></table>

It follows immediately that:

<table width="70%"><tr><td>
:<math>
\left\langle p \frac{\partial u_{1} }{ \partial x_{1} } \right\rangle = - \left[ \left\langle p \frac{\partial u_{2} }{ \partial x_{2} } \right\rangle + \left\langle p \frac{\partial u_{3} }{ \partial x_{3} } \right\rangle \right]
</math>
</td><td width="5%">(38)</td></tr></table>

Thus equation 34 can be rewritten as:

<table width="70%"><tr><td>
:<math>
\begin{matrix}
\left[ \frac{ \partial }{ \partial t} + U_{j} \frac{ \partial }{ \partial x_{j} }\right] \frac{1}{2} \left\langle u^{2}_{1} \right\rangle = \\
= & - & \left[ \left\langle p \frac{\partial u_{2} }{ \partial x_{2} } \right\rangle + \left\langle p \frac{\partial u_{3} }{ \partial x_{3} } \right\rangle \right] + \\
& + & \frac{ \partial }{ \partial x_{j}} \left\{ -\frac{1}{\rho} \left\langle p u_{1} \right\rangle \delta_{1j} - \frac{1}{2} \left\langle u^{2}_{1} u_{j} \right\rangle + 2 \nu \left\langle s_{1j} u_{1} \right\rangle \right\} - \\
& - & \left\langle u_{1} u_{j} \right\rangle \frac{\partial U_{1}}{ \partial x_{j}} - 2 \nu \left\langle s_{1j} s_{1j} \right\rangle \\
\end{matrix}
</math>
</td><td width="5%">(39)</td></tr></table>

Comparison of equation 39 with equations 35 and 36 make it immediately apparent that ''the pressure strain rate terms act to exchange energy between components of the turbulence''. If <math> \left\langle p \partial u_{2} / \partial x_{2} \right\rangle </math> and <math> \left\langle p \partial u_{3} / \partial x_{3} \right\rangle </math> are both positive, then energy is removed from the 1-equation and put into the 2- and 3-equations since the same terms occur with opposite sign. O vice versa.

The role of the pressure strain rate terms can best be illustrated by looking at simple example. Consider a simple homogeneous shear flow in which <math> U_{i} = U \left( x_{2} \right) \delta_{1i} </math> and in which the turbulence is homogeneous. For this flow, the assumption of homogeneity insures that all terms involving gradients of average quantities vanish (except for <math> d U_{1} / d x_{2} </math> ). This leaves only the pressure-strain rate, production and dissipation terms; therefore equations 35, 36, 39 reduce to:

'''1-component'''

<table width="70%"><tr><td>
:<math>
\frac{\partial \left\langle u^{2}_{1} \right\rangle}{ \partial t} = - \left[ \left\langle p \frac{ \partial u_{2} }{ \partial x_{2} } \right\rangle + \left\langle p \frac{ \partial u_{3} }{ \partial x_{3} } \right\rangle \right] - \left\langle u_{1} u_{2} \right\rangle \frac{ \partial U_{1} }{ \partial x_{2}} - \epsilon_{1}
</math>
</td><td width="5%">(40)</td></tr></table>

'''2-component'''

<table width="70%"><tr><td>
:<math>
\frac{\partial \left\langle u^{2}_{2} \right\rangle}{ \partial t} = \left\langle p \frac{ \partial u_{2}}{ \partial x_{2} } \right\rangle - \epsilon_{2}
</math>
</td><td width="5%">(41)</td></tr></table>

'''3-component'''

<table width="70%"><tr><td>
:<math>
\frac{\partial \left\langle u^{2}_{3} \right\rangle}{ \partial t} = \left\langle p \frac{ \partial u_{3}}{ \partial x_{3} } \right\rangle - \epsilon_{3}
</math>
</td><td width="5%">(42)</td></tr></table>

where

<table width="70%"><tr><td>
:<math>
\epsilon_{1} \equiv 2 \nu \left\langle s_{1j} s_{1j} \right\rangle
</math>
</td><td width="5%">(43)</td></tr></table>

<table width="70%"><tr><td>
:<math>
\epsilon_{2} \equiv 2 \nu \left\langle s_{2j} s_{2j} \right\rangle
</math>
</td><td width="5%">(44)</td></tr></table>

<table width="70%"><tr><td>
:<math>
\epsilon_{3} \equiv 2 \nu \left\langle s_{3j} s_{3j} \right\rangle
</math>
</td><td width="5%">(45)</td></tr></table>

It is immediately apparent that only <math> \left\langle u^{2}_{1} \right\rangle </math> can receive energy from the mean flow because only the first equation has a non-zero production term.

Now let's further assume that the smallest scales of the turbulece can be ''assumed'' to be ''locally isotropic''. While not always true, this is a pretty good approximation for high Reynolds number flows. (Note that it ''might'' be exactly true in many flows in the limit of infinite Reynolds number, at least away from walls.) Local isotropy implies that the component dissipation rates are equal; i.e., <math> \epsilon_{1}= \epsilon_{2}= \epsilon_{3} </math>. But where does the energy in the 2 and 3-components come from? Obviously the pressure-strain-rate terms must act to remove energyfrom the 1-component and redistribute it to the others.

As the preceding example makes clear, the role of the pressure-strain-rate terms is to attempt to distribute the energy ''among'' the various components of the turbulence. In the absence of other influences, they are so successful that the dissipation by each component is almost equal, at least at high turbulence Reynolds numbers. In fact, because of the energy re-distribution by the the pressure strain rate terms, it is uncommon to find a turbulent shear flow away from boundaries where the kinetic energy of the turbulence components differ by more than 30-40%, no matter which component gets the energy from the mean flow.

'''Example:''' In simple turbulent free shear flows like wakes or jets where the energy is primarily produced in a single component (as in the example above), typically <math> \left\langle u^{2}_{1} \right\rangle \approx \left\langle u^{2}_{2} \right\rangle + \left\langle u^{2}_{3} \right\rangle </math> where <math> \left\langle u^{2}_{1} \right\rangle </math> is the kinetic of the component produced directly by the action of Reynolds stresses against the mean velocity gradient. Moreover <math> \left\langle u^{2}_{2} \right\rangle \approx \left\langle u^{2}_{3} \right\rangle </math>. This, of course, makes some sense in light of the above, since both off-axis components get most of their energy from the pressure-strain rate terms.

It is possible to show that the pressure-strain rate terms vanish in isotropic turbulence. This suggests (at least to some) that the natural state for turbulence in the absence of other influences is the isotropic state. This has also been exploited by the turbulence modelers. One of the most common assumptions involves setting these pressure-strain rate terms (as they occur in the Reynolds shear equation) proportional to the anisotropy of the flow defined by:

<table width="70%"><tr><td>
:<math>
a_{ij} = \left\langle u_{i} u_{j} \right\rangle - \left\langle q^{2} \right\rangle \delta_{ij} / 3
</math>
</td><td width="5%">(46)</td></tr></table>

Models accounting for this are said to include a "''return-to-isotropy''" term. An additional term must also be included to account for the direct effect of the mean shear on the pressure-strain rate correlation, and this is reffered to as the "''rapid term''". The reasons for this latter term are not easy to see from single point equations, but fall out rather naturally from the two-point Reynolds stress equations we shall discuss later.

{{Turbulence credit wkgeorge}}

{{Chapter navigation|Reynolds averaged equations|Stationarity and homogeneity}}

Template:Introduction to turbulence menu

2008-02-25T09:18:09Z

Jola:

{| class="infobox bordered" style="vertical-align: top; text-align: left;"
|+ style="background: #ccf; font-size: larger;" align="center" | '''[[Introduction to turbulence|Introduction to turbulence]]'''
|-
| '''[[Introduction to turbulence/Nature of turbulence|Nature of turbulence]]'''
|-
| '''[[Introduction to turbulence/Statistical analysis|Statistical analysis]]'''
|-
|
* [[Introduction to turbulence/Statistical analysis/Ensemble average|Ensemble average]]
* [[Introduction to turbulence/Statistical analysis/Probability|Probability]]
* [[Introduction to turbulence/Statistical analysis/Multivariate random variables|Multivariate random var ...]]
* [[Introduction to turbulence/Statistical analysis/Estimation from a finite number of realizations|Estimation from a finite ...]]
* [[Introduction to turbulence/Statistical analysis/Generalization to the estimator of any quantity|Generalization to the esti ...]]
|-
| '''[[Introduction to turbulence/Reynolds averaged equations|Reynolds averaged eq ...]]'''
|-
| '''[[Introduction to turbulence/Turbulence kinetic energy|Turbulence kinetic energy]]'''
|-
| '''[[Introduction to turbulence/Stationarity and homogeneity|Stationarity and homogeneity]]'''
|-
| '''[[Introduction to turbulence/Study_questions|Study questions]]'''
... template not finished yet!
|}
<noinclude>[[Category: Navigation templates]]</noinclude>
[[Category: Introduction to turbulence]]

Introduction to turbulence/Stationarity and homogeneity

2008-02-25T09:15:50Z

Jola: Removed initial "the" in titles - not wiki standard.

Introduction to turbulence/Stationarity and homogeneity

2008-02-25T09:13:09Z

Jola:

{{Introduction to turbulence menu}}

== Processes statistically stationary in time ==

Many random processes have the characteristic that their statistical properties do not appear to depend directly on time, even though the random variables themselves are time-dependent. For example, consider the signals shown in Figures 2.2 and 2.5

When the statistical properties of a random process are independent of time, the random process is said to be ''stationary''. For such a process all the moments are time-independent, e.g., <math> \left\langle \tilde{ u \left( t \right)} \right\rangle = U </math>, etc. In fact, the probability density itself is time-independent, as should be obvious from the fact that the moments are time independent.

An alternative way of looking at ''stationarity'' is to note that ''the statistics of the process are independent of the origin in time''. It is obvious from the above, for example, that if the statistics of a process are time independent, then <math> \left\langle u^{n} \left( t \right) \right\rangle = \left\langle u^{n} \left( t + T \right) \right\rangle </math> , etc., where <math> T </math> is some arbitrary translation of the origin in time. Less obvious, but equally true, is that the product <math> \left\langle u \left( t \right) u \left( t' \right) \right\rangle </math> depends only on time difference <math> t'-t </math> and not on <math> t </math> (or <math> t' </math> ) directly. This consequence of stationarity can be extended to any product moment. For example <math> \left\langle u \left( t \right) v \left( t' \right) \right\rangle </math> can depend only on the time difference <math> t'-t </math>. And <math> \left\langle u \left( t \right) v \left( t' \right) w \left( t'' \right)\right\rangle </math> can depend only on the two time differences <math> t'- t </math> and <math> t'' - t </math> (or <math> t'' - t' </math> ) and not <math> t </math> , <math> t' </math> or <math> t'' </math> directly.

== The autocorrelation ==

One of the most useful statistical moments in the study of stationary random processes (and turbulence, in particular) is the '''autocorrelation''' defined as the average of the product of the random variable evaluated at two times, i.e. <math> \left\langle u \left( t \right) u \left( t' \right)\right\rangle </math>. Since the process is assumed stationary, this product can depend only on the time difference <math> \tau = t' - t </math>. Therefore the autocorrelation can be written as:

<table width="70%"><tr><td>
:<math>
C \left( \tau \right) \equiv \left\langle u \left( t \right) u \left( t + \tau \right) \right\rangle
</math>
</td><td width="5%">(1)</td></tr></table>

The importance of the autocorrelation lies in the fact that it indicates the "memory" of the process; that is, ''the time over which is correlated with itself''. Contrast the two autocorrelation of deterministic sine wave is simply a cosine as can be easily proven. Note that there is no time beyond which it can be guaranteed to be arbitrarily small since it always "remembers" when it began, and thus always remains correlated with itself. By contrast, a stationary random process like the one illustrated in the figure will eventually lose all correlation and go to zero. In other words it has a "finite memory" and "forgets" how it was. Note that one must be careful to make sure that a correlation really both goes to zero and ''stays down'' before drawing conclusions, since even the sine wave was zero at some points. Stationary random process ''always'' have two-time correlation functions which eventually go to zero and stay there.

'''Example 1.'''

Consider the motion of an automobile responding to the movement of the wheels over a rough surface. In the usual case where the road roughness is randomly distributed, the motion of the car will be a weighted history of the road's roughness with the most recent bumps having the most influence and with distant bumps eventually forgotten. On the other hand, if the car is travelling down a railroad track, the periodic crossing of the railroad ties represents a determenistic input an the motion will remain correlated with itself indefinitely, a very bad thing if the tie crossing rate corresponds to a natural resonance of the suspension system of the vehicle.

Since a random process can never be more than perfectly correlated, it can never achieve a correlation greater than is value at the origin. Thus

<table width="70%"><tr><td>
:<math>
\left| C \left( \tau \right) \right| \leq C\left( 0 \right)
</math>
</td><td width="5%">(2)</td></tr></table>

An important consequence of stationarity is that the autocorrelation is symmetric in the time difference <math> \tau = t' - t </math>. To see this simply shift the origin in time backwards by an amount <math> \tau </math> and note that independence of origin implies:

<table width="70%"><tr><td>
:<math>
\left\langle u \left( t \right) u \left( t + \tau \right) \right\rangle = \left\langle u \left( t - \tau \right) u \left( t \right) \right\rangle
</math>
</td><td width="5%">(3)</td></tr></table>

Since the right hand side is simply <math> C \left( - \tau \right) </math>, it follows immediately that:

<table width="70%"><tr><td>
:<math>
C \left( \tau \right) = C \left( - \tau \right)
</math>
</td><td width="5%">(4)</td></tr></table>

== The autocorrelation coefficient ==

It is convenient to define the ''autocorrelation coefficient'' as:

<table width="70%"><tr><td>
:<math>
\rho \left( \tau \right) \equiv \frac{ C \left( \tau \right)}{ C \left( 0 \right)} = \frac{\left\langle u \left( t \right) u \left( t + \tau \right) \right\rangle}{ \left\langle u'^{2} \right\rangle }
</math>
</td><td width="5%">(5)</td></tr></table>

where

<table width="70%"><tr><td>
:<math>
\left\langle u^{2} \right\rangle = \left\langle u \left( t \right) u \left( t \right) \right\rangle = C \left( 0 \right) = var \left[ u \right]
</math>
</td><td width="5%">(6)</td></tr></table>

Since the autocorrelation is symmetric, so is its coefficient, i.e.,

<table width="70%"><tr><td>
:<math>
\rho \left( \tau \right) = \rho \left( - \tau \right)
</math>
</td><td width="5%">(7)</td></tr></table>

It is also obvious from the fact that the autocorrelation is maximal at the origin that the autocorrelation coefficient must also be maximal there. In fact from the definition it follows that

<table width="70%"><tr><td>
:<math>
\rho \left( 0 \right) = 1
</math>
</td><td width="5%">(8)</td></tr></table>

and

<table width="70%"><tr><td>
:<math>
\rho \left( \tau \right) \leq 1
</math>
</td><td width="5%">(9)</td></tr></table>

for all values of <math> \tau </math> .

== The integral scale ==

One of the most useful measures of the length of a time a process is correlated with itself is the integral scale defined by

<table width="70%"><tr><td>
:<math>
T_{int} \equiv \int^{\infty}_{0} \rho \left( \tau \right) d \tau
</math>
</td><td width="5%">(10)</td></tr></table>

It is easy to see why this works by looking at Figure 5.2. In effect we have replaced the area under the correlation coefficient by a rectangle of height unity and width <math> T_{int} </math> .

== The temporal Taylor microscale ==

The autocorrelation can be expanded about the origin in a MacClaurin series; i.e.,

<table width="70%"><tr><td>
:<math>
C \left( \tau \right) = C \left( 0 \right) + \tau \frac{ d C }{ d t }|_{\tau = 0} + \frac{1}{2} \tau^{2} \frac{d^{2} C}{d t^{2} }|_{\tau = 0} + \frac{1}{3!} \tau^{3} \frac{d^{3} C}{d t^{3} }|_{\tau = 0}
</math>
</td><td width="5%">(11)</td></tr></table>

But we know the aoutocorrelation is symmetric in <math> \tau </math> , hence the odd terms in <math> \tau </math> must be identically to zero (i.e., <math> dC / dt |_{\tau = 0} = 0 </math> , <math> d^{3}C / dt^{3} |_{\tau = 0} = 0 </math>, etc.). Therefore the expansion of the autocorrelation near the origin reduces to:

<table width="70%"><tr><td>
:<math>
C \left( \tau \right) = C \left( 0 \right) + \frac{1}{2} \tau^{2} \frac{d^{2} C}{d t^{2} }|_{\tau = 0} + \cdots
</math>
</td><td width="5%">(12)</td></tr></table>

Similary, the autocorrelation coefficient near the origin can be expanded as:

<table width="70%"><tr><td>
:<math>
\rho \left( \tau \right) = 1 + \frac{1}{2}\frac{d^{2}\rho}{d t^{2}}|_{\tau = 0} \tau^{2}+ \cdots
</math>
</td><td width="5%">(13)</td></tr></table>

where we have used the fact that <math> \rho \left( 0 \right) = 1 </math> . If we define <math> ' = d / dt </math> we can write this compactly as:

<table width="70%"><tr><td>
:<math>
\rho \left( \tau \right) = 1 + \frac{1}{2} \rho '' \left( 0 \right) \tau^{2} + \cdots
</math>
</td><td width="5%">(14)</td></tr></table>

Since <math> \rho \left( \tau \right) </math> has its maximum at the origin, obviously <math> \rho'' \left( 0 \right) </math> must be negative.

We can use the correlation and its second derivative at the origin to ''define'' a special time scale, <math> \lambda_{\tau} </math> (called the Taylor microscale) by:

<table width="70%"><tr><td>
:<math>
\lambda^{2}_{\tau} \equiv - \frac{2}{\rho'' \left( 0 \right)}
</math>
</td><td width="5%">(15)</td></tr></table>

Using this in equation 14 yields the expansion for the correlation coefficient near the origin as:

<table width="70%"><tr><td>
:<math>
\rho \left( \tau \right) = 1 - \frac{\tau^{2}}{\lambda^{2}_{\tau}} + \cdots
</math>
</td><td width="5%">(16)</td></tr></table>

Thus very near the origin the correlation coefficient (and the autocorrelation as well) simply rolls off parabolically; i.e.,

<table width="70%"><tr><td>
:<math>
\rho \left( \tau \right) \approx 1 - \frac{\tau^{2}}{\lambda^{2}_{\tau}}
</math>
</td><td width="5%">(17)</td></tr></table>

This parabolic curve is shown in Figure 3 as the osculating (or 'kissing') parabola which approaches zero exactly as the autocorrelation coefficient does. The intercept of this osculating parabola with the <math> \tau </math> -axis is the Taylor microscale, <math> \lambda_{\tau} </math>.

The Taylor microscale is significant for a number of reasons. First, for many random processes (e.g., Gaussian), the Taylor microscale can be proven to be the average distance between zero-crossing of a random variable in time. This is approximately true for turbulence as well. Thus one can quickly estimate the Taylor microscale by simply observing the zero-crossings using an oscilloscope trace.

The Taylor microscale also has a special relationship to the mean square time derivative of the signal, <math> \left\langle \left[ d u / d t \right]^{2} \right\rangle </math>. This is easiest to derive if we consider two stationary random signals at two different times say <math> u = u \left( t \right) </math> and <math> u' = u' \left( t' \right) </math>. The derivative of the first signal is <math> d u / d t </math> and the second <math> d u' / d t' </math>. Now lets multiply these together and rewrite them as:

<table width="70%"><tr><td>
:<math>
\frac{du'}{dt'} \frac{du}{dt} = \frac{d^{2}}{dtdt'} u \left( t \right) u' \left( t' \right)
</math>
</td><td width="5%">(18)</td></tr></table>

where the right-hand side follows from our assumption that <math> u </math> is not a function of <math> t' </math> nor <math> u' </math> a function of <math> t </math>.

Now if we average and interchenge the operations of differentiation and averaging we obtain:

<table width="70%"><tr><td>
:<math>
\left\langle \frac{du'}{dt'} \frac{du}{dt} \right\rangle = \frac{d^{2}}{dtdt'} \left\langle u \left( t \right) u' \left( t' \right) \right\rangle
</math>
</td><td width="5%">(19)</td></tr></table>

Here comes the first trick: we simply take <math> u' </math> to be exactly <math> u </math> but evaluated at time <math> t' </math>. So <math> u \left( t \right) u' \left( t' \right) </math> simply becomes <math> u \left( t \right) u \left( t' \right) </math> and its average is just the autocorrelation, <math> C \left( \tau \right) </math>. Thus we are left with:

<table width="70%"><tr><td>
:<math>
\left\langle \frac{du'}{dt'} \frac{du}{dt} \right\rangle = \frac{d^{2}}{dtdt'} C \left( t' - t \right)
</math>
</td><td width="5%">(20)</td></tr></table>

Now we simply need to use the chain-rule. We have already defined <math> \tau = t' - t </math>. Let's also define <math> \xi = t' + t </math> and transform the derivatives involving <math> t </math> and <math> t' </math> to derivatives involving <math> \tau </math> and <math> \xi </math>. The result is:

<table width="70%"><tr><td>
:<math>
\frac{d^{2}}{dtdt'} = \frac{d^{2}}{d \xi^{2}} - \frac{d^{2}}{d \tau^{2}}
</math>
</td><td width="5%">(21)</td></tr></table>

So equation 20 becomes

<table width="70%"><tr><td>
:<math>
\left\langle \frac{du'}{dt'} \frac{du}{dt} \right\rangle = \frac{d^{2}}{d \xi^{2}}C \left( \tau \right) - \frac{d^{2}}{d \tau^{2}} C \left( \tau \right)
</math>
</td><td width="5%">(22)</td></tr></table>

But since <math> C </math> is a function only of <math> \tau </math>, the derivative of it with respect to <math> \xi </math> is identically zero. Thus we are left with:

<table width="70%"><tr><td>
:<math>
\left\langle \frac{du'}{dt'} \frac{du}{dt} \right\rangle = - \frac{d^{2}}{d \tau^{2}} C \left( \tau \right)
</math>
</td><td width="5%">(23)</td></tr></table>

And finally we need the second trick. Let's evaluate both sides at <math> t = t' </math> (or <math> \tau = 0 </math> ) to obtain the ''mean square derivative'' as:

<table width="70%"><tr><td>
:<math>
\left\langle \left( \frac{du}{dt} \right)^{2} \right\rangle = - \frac{d^{2}}{d \tau^{2}} C \left( \tau \right)|_{ \tau = 0}
</math>
</td><td width="5%">(24)</td></tr></table>

But from our definition of the Taylor microscale and the facts that <math> C \left( 0 \right) = \left\langle u^{2} \right\rangle </math> and <math> C \left( \tau \right) = \left\langle u^{2} \right\rangle \rho \left( \tau \right) </math>, this is exactly the same as:

<table width="70%"><tr><td>
:<math>
\left\langle \left( \frac{du}{dt} \right)^{2} \right\rangle = 2 \frac{ \left\langle u^{2} \right\rangle}{\lambda^{2}_{\tau}}
</math>
</td><td width="5%">(25)</td></tr></table>

This amasingly simple result is very important in the study of turbulence, especially after we extend it to spatial derivatives.

== Time averages of stationary processes ==

It is common practice in many scientific disciplines to define a time average by integrating the random variable over a fixed time interval, i.e. ,

<table width="70%"><tr><td>
:<math>
U_{T} \equiv \frac{1}{T} \int^{T_{2}}_{T_{1}} u \left( t \right) dt
</math>
</td><td width="5%">(26)</td></tr></table>

For the stationary random processes we are considering here, we can define <math> T_{1} </math> to be the origin in time and simply write:

<table width="70%"><tr><td>
:<math>
U_{T} \equiv \frac{1}{T} \int^{T}_{0} u \left( t \right) dt
</math>
</td><td width="5%">(27)</td></tr></table>

where <math> T = T_{2} - T_{1} </math> is the integration time.

Figure 5.4. shows a portion of a stationary random signal over which such an integration might be performed. The ime integral of <math> u \left( t \right) </math> over the integral <math> \left( O, T \right) </math> corresponds to the shaded area under the curve. Now since <math> u \left( t \right) </math> is random and since it formsthe upper boundary of the shadd area, it is clear that the time average, <math> U_{T} </math> is a lot like the estimator for the mean based on a finite number of independent realization, <math> X_{N} </math> we encountered earlier in section ''Estimation from a finite number of realizations'' (see ''Elements of statistical analysis'')

It will be shown in the analysis presented below that ''if the signal is stationary'', the time average defined by equation 27 is an unbiased estimator of the true average <math> U </math>. Moreover, the estimator converges to <math> U </math> as the time becomes infinite; i.e., for stationary random processes

<table width="70%"><tr><td>
:<math>
U = \lim_{T \rightarrow \infty} \frac{1}{T} \int^{T}_{0} u \left( t \right) dt
</math>
</td><td width="5%">(28)</td></tr></table>

Thus the time and ensemble averages are equivalent in the limit as <math> T \rightarrow \infty </math>, ''but only for a stationary random process''.

== Bias and variability of time estimators ==

It is easy to show that the estimator, <math> U_{T} </math>, is unbiased by taking its ensemble average; i.e.,

<table width="70%"><tr><td>
:<math>
\left\langle U_{T} \right\rangle = \left\langle \frac{1}{T} \int^{T}_{0} u \left( t \right) dt \right\rangle = \frac{1}{T} \int^{T}_{0} \left\langle u \left( t \right) \right\rangle dt
</math>
</td><td width="5%">(29)</td></tr></table>

Since the process has been assumed stationary, <math> \left\langle u \left( t \right) \right\rangle </math> is independent of time. It follows that:

<table width="70%"><tr><td>
:<math>
\left\langle U_{T} \right\rangle = \frac{1}{T} \left\langle u \left( t \right) \right\rangle T = U
</math>
</td><td width="5%">(30)</td></tr></table>

To see whether the etimate improves as <math> T </math> increases, the variability of <math> U_{T} </math> must be examined, exactly as we did for <math> X_{N} </math> earlier in section Bias and convergence of estimators (see chapter The elements of statistical analysis). To do this we need the variance of <math> U_{T} </math> given by:

<table width="70%"><tr><td>
:<math>
\begin{matrix}
var \left[ U_{T} \right] & = & \left\langle \left[ U_{T} - \left\langle U_{T} \right\rangle \right]^{2} \right\rangle = \left\langle \left[ U_{T} - U \right]^{2} \right\rangle \\
& = & \frac{1}{T^{2}} \left\langle \left\{ \int^{T}_{0} \left[ u \left( t \right) - U \right] \right\}^{2} \right\rangle \\
& = & \frac{1}{T^{2}} \left\langle \int^{T}_{0} \int^{T}_{0} \left[ u \left( t \right) - U \right] \left[ u \left( t' \right) - U \right] dtdt' \right\rangle \\
& = & \frac{1}{T^{2}} \int^{T}_{0} \int^{T}_{0} \left\langle u'\left( t \right) u'\left( t' \right) \right\rangle dtdt' \\
\end{matrix}
</math>
</td><td width="5%">(31)</td></tr></table>

But since the process is assumed stationary <math> \left\langle u' \left( t \right) u' \left( t' \right) \right\rangle = C \left( t' - t \right) </math> where <math> C \left( t' - t \right) = \left\langle u^{2} \right\rangle \rho \left( t'-t \right) </math> is the correlation coefficient. Therefore the integral can be rewritten as:

<table width="70%"><tr><td>
:<math>
\begin{matrix}
var \left[ U_{T} \right] & = & \frac{1}{T^{2}} \int^{T}_{0} \int^{T}_{0} C \left( t' - t \right) dtdt' \\
& = & \frac{ \left\langle u^{2} \right\rangle }{ T^{2} } \int^{T}_{0} \int^{T}_{0} \rho \left( t' - t \right) dtdt' \\
\end{matrix}
</math>
</td><td width="5%">(33)</td></tr></table>

Now we need to apply some fancy calculus. If new variables <math> \tau= t'-t </math> and <math> \xi= t'+t </math> are defined, the double integral can be transformed to (see Figure 5.5):

<table width="70%"><tr><td>
:<math>
var \left[ U_{T} \right] = \frac{var \left[ u \right]}{2 T^{2}} \left[ \int^{T}_{0} d \tau \int^{T-\tau}_{\tau} d \xi \rho \left( \tau \right) + \int^{0}_{-T} d \tau \int^{T+\tau}_{-\tau} d \xi \rho \left( \tau \right) \right]
</math>
</td><td width="5%">(35)</td></tr></table>

where the factor of <math> 1/2 </math> arises from the Jacobian of the transformation. The integrals over <math> d \xi </math> can be evaluated directly to yield:

<table width="70%"><tr><td>
:<math>
var \left[ U_{T} \right] = \frac{var \left[ u \right]}{2 T^{2}} \left\{ \int^{T}_{0} \rho \left( \tau \right) \left[ T - \tau \right] d \tau + \int^{0}_{-T} \rho \left( \tau \right) \left[ T + \tau \right] \right\}
</math>
</td><td width="5%">(36)</td></tr></table>

By noting that the autocorrelation is symmetric, the second integral can be transformed and added to the first to yield at last the result we seek as:

<table width="70%"><tr><td>
:<math>
var \left[ U_{T} \right] = \frac{var \left[ u \right]}{T} \int^{T}_{-T} \rho \left( \tau \right) \left[ 1 - \frac{ \left| \tau \right| }{T} \right] d \tau
</math>
</td><td width="5%">(37)</td></tr></table>

Now if our averaging time, <math> T </math>, is chosen so large that <math> \left| \tau \right| / T << 1 </math> over the range for which <math> \rho \left( \tau \right) </math> is non-zero, the integral reduces:

<table width="70%"><tr><td>
:<math>
\begin{matrix}
var \left[ U_{T} \right] & \approx & \frac{2 var \left[ u \right]}{T} \int^{T}_{0} \rho \left( \tau \right) d \tau \\
& = & \frac{2 T_{int}}{T} var \left[ u \right] \\
\end{matrix}
</math>
</td><td width="5%">(38)</td></tr></table>

where <math> T_{int} </math> is the integral scale defined by equation 10. Thus the ''variability'' of our estimator is given by:

<table width="70%"><tr><td>
:<math>
\epsilon^{2}_{U_{T}} = \frac{2T_{int}}{T}
</math>
</td><td width="5%">(39)</td></tr></table>

Therefore the estimator does, in fact, converge (in mean square) to the correct result as the averaging time, <math> T </math> increases relative to the integral scale, <math> T_{int} </math>.

There is a direct relationship between equation 39 and equation 52 in chapter The elements of statistical analysis ( section Bias and convergence of estimators) which gave the mean square variability for the ensemble estimate from a finite number of statistically independent realizations, <math> X_{N} </math>. Obviously the effective number of independent realizations for the finite time estimator is:

<table width="70%"><tr><td>
:<math>
N_{eff} = \frac{2T_{int}}{T}
</math>
</td><td width="5%">(40)</td></tr></table>

so that the two expressions are equivalent. Thus, in effect, ''portions of the record separated by two integral scales behave as though they were statistically independent, at least as far as convergence of finite time estimators is concerned''.

Thus what is required for convergence is again, many ''independent'' pieces of information. This is illustrated in Figure 5.6. That the length of the recordn should be measured in terms of the integral scale should really be no surprise since it is a measure of the rate at which a process forgets its past.

'''Example'''

It is desired to mesure the mean velocity in a turbulent flow to within an rms error of 1% (i.e. <math> \epsilon = 0.01 </math> ). The expected fluctuation level of the signal is 25% and integral scale is estimated as 100 ms. What is the required averaging time?

From equation 39

<table width="70%"><tr><td>
:<math>
\begin{matrix}
T & = & \frac{2T_{int}}{\epsilon^{2}} \frac{var \left[ u \right]}{U^{2}} \\
& = & 2 \times 0.1 \times (0.25)^{2} / (0.01)^{2} = 125 sec \\
\end{matrix}
</math>
</td><td width="5%">(41)</td></tr></table>

Similar considerations apply to any other finite time estimator and equation 55 from chapter Statistical analysis can be applied directly as long as equation 40 is used for the number of independent samples.

It is common common experimental practice to not actually carry out an analog integration. Rather the signal is sampled at fixed intervals in time by digital means and the averages are computed as for an esemble with a finite number of realizations. Regardless of the manner in which the signal is processed, only a finite portion of a stationary time series can be analyzed and the preceding considerations always apply.

It is important to note that data sampled more rapidly than once every two integral scales do '''not''' contribute to the convergence of the estimator since they can not be considered independent. If <math> N </math> is the actual number of samples acquired and <math> \Delta t </math> is the time between samples, then the effective number of independent realizations is

<table width="70%"><tr><td>
:<math>
N_{eff} = \left\{
\begin{array}{lll}
N \Delta t /T_{int} & if & \Delta t < 2T_{int} \\
N & if & \Delta t \geq 2T_{int} \\
\end{array}
\right.
</math>
</td><td width="5%">(42)</td></tr></table>

It should be clear that if you sample faster than <math> \Delta t = 2T_{int} </math> you are processing unnecessary data which does not help your statistics converge.

You may wonder why one would ever take data faster than absolutely necessary, since it simply it simply fills up your computer memory with lots of statistically redundant data. When we talk about measuring spectra you will learn that for spectral measurements it is necessary to sample much faster to avoid spactral aliasing. Many wrongly infer that they must sample at these higher rates even when measuring just moments. Obviously this is not the case if you are not measuring spectra.

== Random fields of space and time ==

To this point only temporally varying random fields have been discussed. For turbulence however, random fields can be functions of both space and time. For example, the temperature <math> \theta </math> could be a random scalar function of time <math> t </math> and position <math> \stackrel{\rightarrow}{x} </math>, i.e.,

<table width="70%"><tr><td>
:<math>
\theta = \theta \left( \stackrel{\rightarrow}{x} , t \right)
</math>
</td><td width="5%">(43)</td></tr></table>

The velocity is another example of a random vector function of position and time, i.e.,

<table width="70%"><tr><td>
:<math>
\stackrel{\rightarrow}{u} = \stackrel{\rightarrow}{u} \left( \stackrel{\rightarrow}{x},t \right)
</math>
</td><td width="5%">(44)</td></tr></table>

or in tensor notation,

<table width="70%"><tr><td>
:<math>
u_{i} = u_{i} \left( \stackrel{\rightarrow}{x},t \right)
</math>
</td><td width="5%">(45)</td></tr></table>

In the general case, the ensemble averages of these quantities are functions of both positon and time; i.e.,

<table width="70%"><tr><td>
:<math>
\left\langle \theta \left( \stackrel{\rightarrow}{x},t \right) \right\rangle \equiv \Theta \left( \stackrel{\rightarrow}{x},t \right)
</math>
</td><td width="5%">(46)</td></tr></table>

<table width="70%"><tr><td>
:<math>
\left\langle u_{i} \left( \stackrel{\rightarrow}{x},t \right) \right\rangle \equiv U_{i} \left( \stackrel{\rightarrow}{x},t \right)
</math>
</td><td width="5%">(47)</td></tr></table>

If only ''stationary'' random processes are considered, then the averages do not depend on time and are functions of <math> \stackrel{\rightarrow}{x} </math> only; i.e.,

<table width="70%"><tr><td>
:<math>
\left\langle \theta \left( \stackrel{\rightarrow}{x},t \right) \right\rangle \equiv \Theta \left( \stackrel{\rightarrow}{x} \right)
</math>
</td><td width="5%">(48)</td></tr></table>

<table width="70%"><tr><td>
:<math>
\left\langle u_{i} \left( \stackrel{\rightarrow}{x},t \right) \right\rangle \equiv U_{i} \left( \stackrel{\rightarrow}{x}\right)
</math>
</td><td width="5%">(49)</td></tr></table>

Now the averages may not be position dependent either. For example, if the averages are ''independent of the origin in position'', then the field is said to be '''homogeneous'''. '''Homogenity''' (the noun corresponding to the adjective homogeneous) is exactly analogous to stationarity except that position is now the variable, and not time.

It is, of course, possible (at least in concept) to have homogeneous fields which are either stationary or non stationary. Since position, unlike time, is a vector quantity it is also possible to have only partial homogeneity. For example, a field can be homogeneous in the <math> x_{1}- </math> and <math> x_{3}- </math> directions, but not in the <math> x_{2}- </math> direction so that <math> U_{i}=U_{i}(X_{2}) </math> only. In fact, it appears to be dynamically impossible to have flows which are honogeneous in all variables and stationary as well, but the concept is useful, nonetheless.

Homogeneity will be seen to have powerful consequences for the equations govering the averaged motion, since the spatial derivative of any averaged quantity must be identically zero. Thus even homogeneity in only one direction can considerably simplify the problem. For example, in the Reynolds stress transport equation, the entire turbulence transport is exactly zero if the field is homogeneous.

== Multi-point statistics in homogeneous field ==

The concept of homogeneity can also be extended to multi-point statistics. Consider for example, the correlation between the velocity at one point and that at another as illustrated in Figure 5.7. If the time dependence is suppressed and the field is assumed statistically ''homogeneous'', this correlation is a function only of the separation of the two points, i.e.,

<table width="70%"><tr><td>
:<math>
\left\langle u_{i} \left( \stackrel{\rightarrow}{x} , t \right) u_{j} \left( \stackrel{\rightarrow}{x'} , t \right) \right\rangle \equiv B_{i,j} \left( \stackrel{\rightarrow}{r} \right)
</math>
</td><td width="5%">(50)</td></tr></table>

where <math> \stackrel{\rightarrow}{r} </math> is the separation vector defined by

<table width="70%"><tr><td>
:<math>
\stackrel{\rightarrow}{r} = \stackrel{\rightarrow}{x'} - \stackrel{\rightarrow}{x}
</math>
</td><td width="5%">(51)</td></tr></table>

or

<table width="70%"><tr><td>
:<math>
r_{i} = x'_{i} - x_{i}
</math>
</td><td width="5%">(52)</td></tr></table>

Note that the convention we shall follow for vector quantities is that the first subscript on <math> B_{i,j} </math> is the component of velocity at the first position, <math> \stackrel{\rightarrow}{x} </math> , and the second subscript is the component of velocity at the second, <math> \stackrel{\rightarrow}{x'} </math>. For scalar quantities we shall simply put a simbol for the quantity to hold the place. For example, we would write the two-point temperature correlation in a homogeneous field by:

<table width="70%"><tr><td>
:<math>
\left\langle \theta \left( \stackrel{\rightarrow}{x},t \right) \theta \left( \stackrel{\rightarrow}{x'},t \right) \right\rangle \equiv B_{\theta , \theta} \left( \stackrel{\rightarrow}{r} \right)
</math>
</td><td width="5%">(53)</td></tr></table>

A mixed vector/scalar correlation like the two-point temperature velocity correlation would be written as:

<table width="70%"><tr><td>
:<math>
\left\langle u_{i} \left( \stackrel{\rightarrow}{x} , t \right) \theta \left( \stackrel{\rightarrow}{x'},t \right) \right\rangle \equiv B_{i,\theta } \left( \stackrel{\rightarrow}{r} \right)
</math>
</td><td width="5%">(54)</td></tr></table>

On the other hand, if we meant for the temperature to be evaluated at <math> \stackrel{\rightarrow}{x} </math> and the velocity at <math> \stackrel{\rightarrow}{x'} </math> we would have to write:

<table width="70%"><tr><td>
:<math>
\left\langle \theta \left( \stackrel{\rightarrow}{x},t \right) u_{i} \left( \stackrel{\rightarrow}{x'},t \right) \right\rangle \equiv B_{ \theta, i } \left( \stackrel{\rightarrow}{r} \right)
</math>
</td><td width="5%">(55)</td></tr></table>

Now most books don't bother with the subscript notation, and simply give each new correlation a new symbol. At first this seems much simpler; and it is as long as you are only dealing with one or two different correlations. But introduce a few more, then read about a half-dozen pages, and you will find you completely forget what they are or how they were put together. It is usually very important to know exactly what you are talking about, so we will use this comma system to help us remember.

It is easy to see that the consideration of vector quantities raises special considerations. For example, the correlation between a scalar function of position at two points is symmetrical in <math> \stackrel{\rightarrow}{r} </math> , i.e.,

<table width="70%"><tr><td>
:<math>
B_{\theta,\theta} \left( \stackrel{\rightarrow}{r} \right) = B_{\theta,\theta} \left( - \stackrel{\rightarrow}{r} \right)
</math>
</td><td width="5%">(56)</td></tr></table>

This is easy to show from the definition of <math> B_{\theta,\theta} </math> and the fact that the field is homogeneous. Simply shift each of the position vectors by the same amount <math> - \stackrel{\rightarrow}{r} </math> as shown in Figure 5.8 to obtain:

<table width="70%"><tr><td>
:<math>
\begin{matrix}
B_{\theta,\theta}\left( \stackrel{\rightarrow}{r},t \right) & \equiv & \left\langle \theta\left( \stackrel{\rightarrow}{x}, t \right) \theta\left( \stackrel{\rightarrow}{x'}, t \right) \right\rangle \\
& = & \left\langle \theta \left( \stackrel{\rightarrow}{x} - \stackrel{\rightarrow}{r} , t \right) \theta \left( \stackrel{\rightarrow}{x'} - \stackrel{\rightarrow}{r} , t \right) \right\rangle \\
& = & B_{\theta,\theta}\left( - \stackrel{\rightarrow}{r},t \right) \\
\end{matrix}
</math>
</td><td width="5%">(57)</td></tr></table>

since <math> \stackrel{\rightarrow}{x'} - \stackrel{\rightarrow}{r} = \stackrel{\rightarrow}{x} </math> ; i.e., the points are reversed and the separation vector is pointing the opposite way.

Such is not the case, in general, for ''vector'' functions of position. For example, see if you can prove to yourself the following:

<table width="70%"><tr><td>
:<math>
B_{\theta,i} \left( \stackrel{\rightarrow}{r} \right) = B_{i,\theta} \left( - \stackrel{\rightarrow}{r} \right)
</math>
</td><td width="5%">(58)</td></tr></table>

and

<table width="70%"><tr><td>
:<math>
B_{i,j} \left( \stackrel{\rightarrow}{r} \right) = B_{j,i} \left( - \stackrel{\rightarrow}{r} \right)
</math>
</td><td width="5%">(59)</td></tr></table>

Clearly the latter is symmetrical in the variable <math> \stackrel{\rightarrow}{r} </math> only when <math> i = j </math> .

These properties of the two-point correlation function will be seen to play an important role in determining the interrelations among the different two-point statistical quantities. They will be especially important when we talk about spectral quantities.

== Spatial integral and Taylor microscales ==

Just as for a stationary random process, correlations between spatially varying, but ''statistically homogeneous'', random quantities ultimately go to zero;, i.e., they become uncorrelated as their locations become widely separated. Because position (o relative position) is a vector quantity, however, the correlation the carrelation may die off at different rates in different directions. Thus direction must be an important part of the definitions of the integral scales and microscales.

Consider for example the one-dimensional spatial correlation which is obtained by measuring the correlation between the temperature at two points along a line in the x-direction, say,

<table width="70%"><tr><td>
:<math>
B^{(1)}_{\theta,\theta} \left( r \right) \equiv \left\langle \theta \left( x_{1} + r , x_{2} , x_{3} , t \right) \theta \left( x_{1} , x_{2} , x_{3} , t \right) \right\rangle
</math>
</td><td width="5%">(60)</td></tr></table>

The superscript "(1)" denotes "the coordinate direction in which the separation occurs". This distinguishes it from the vector separation of <math> B_{\theta,\theta} </math> above. Also, note that the correlation at zero separationis just the variance; i.e.,

<table width="70%"><tr><td>
:<math>
B^{(1)}_{\theta,\theta} \left( 0 \right) = \left\langle \theta^{2} \right\rangle
</math>
</td><td width="5%">(61)</td></tr></table>

The integral scale in the <math> x </math>-direction can be defined as:

<table width="70%"><tr><td>
:<math>
L^{(1)}_{\theta} \equiv \frac{1}{ \left\langle \theta^{2} \right\rangle} \int^{\infty}_{0} \left\langle \theta \left( x + r, y,z,t \right) \theta \left( x,y,z,t \right) \right\rangle dr
</math>
</td><td width="5%">(62)</td></tr></table>

It is clear that there are at least two more integral scales which could be defined by considering separations in the y and z directions. Thus

<table width="70%"><tr><td>
:<math>
L^{(2)}_{\theta} \equiv \frac{1}{ \left\langle \theta^{2} \right\rangle} \int^{\infty}_{0} \left\langle \theta \left( x,y + r,z,t \right) \theta \left( x,y,z,t \right) \right\rangle dr
</math>
</td><td width="5%">(63)</td></tr></table>

and

<table width="70%"><tr><td>
:<math>
L^{(3)}_{\theta} \equiv \frac{1}{ \left\langle \theta^{2} \right\rangle} \int^{\infty}_{0} \left\langle \theta \left( x,y,z + r,t \right) \theta \left( x,y,z,t \right) \right\rangle dr
</math>
</td><td width="5%">(64)</td></tr></table>

In fact, an integral scale could be defined for ''any'' direction simply by choosing the components of the separation vector <math> \stackrel{\rightarrow}{r} </math>. This situation is even more complicated when correlations of vector quantities are considered. For example, consider the correlation of the velocity vectors at two points, <math> B_{i,j} \left( \stackrel{\rightarrow}{r} \right) </math>. Clearly <math> B_{i,j} \left( \stackrel{\rightarrow}{r} \right) </math> is not a single correlation, but rather nine separate correlations: <math> B_{1,1} \left( \stackrel{\rightarrow}{r} \right) </math> , <math> B_{1,2} \left( \stackrel{\rightarrow}{r} \right) </math> , <math> B_{1,3} \left( \stackrel{\rightarrow}{r} \right) </math> , <math> B_{2,1} \left( \stackrel{\rightarrow}{r} \right) </math> , <math> B_{2,2} \left( \stackrel{\rightarrow}{r} \right) </math> , etc. For each of these an integral scale can be defined once a direction for the separation vector is chosen. For example, the integral scales associated with <math> B_{1,1} </math> for the principal directions are

<table width="70%"><tr><td>
:<math>
L^{(1)}_{1,1} \equiv \frac{1}{\left\langle u^{2}_{1} \right\rangle} \int^{\infty}_{0} B_{1,1} \left( r,0,0 \right) dr
</math>
</td><td width="5%">(65)</td></tr></table>

<table width="70%"><tr><td>
:<math>
L^{(2)}_{1,1} \equiv \frac{1}{\left\langle u^{2}_{1} \right\rangle} \int^{\infty}_{0} B_{1,1} \left( 0,r,0 \right) dr
</math>
</td><td width="5%">(66)</td></tr></table>

<table width="70%"><tr><td>
:<math>
L^{(3)}_{1,1} \equiv \frac{1}{\left\langle u^{2}_{1} \right\rangle} \int^{\infty}_{0} B_{1,1} \left( 0,0,r \right) dr
</math>
</td><td width="5%">(67)</td></tr></table>

Similar integral scales can be defined for the other componentsof the correlation tensor. Two of particular importance in the development of the turbulence theory are:

<table width="70%"><tr><td>
:<math>
L^{(2)}_{1,1} \equiv \frac{1}{\left\langle u^{2}_{1} \right\rangle} \int^{\infty}_{0} B_{1,1} \left( 0,r,0 \right) dr
</math>
</td><td width="5%">(68)</td></tr></table>

<table width="70%"><tr><td>
:<math>
L^{(1)}_{2,2} \equiv \frac{1}{\left\langle u^{2}_{2} \right\rangle} \int^{\infty}_{0} B_{2,2} \left( r,0,0 \right) dr
</math>
</td><td width="5%">(69)</td></tr></table>

In general, each of these integral scales will be different, unless restrictions beyond simple homogeneity are placed on the process (e.g., like ''isotropy'' discussed below). Thus, it is important to specify precisely which integral scale is being referred to; i.e., which components of the vector quantities are being used and in which direction the integration is being performed.

Similar considerations apply to the Taylor microscales, regardless of whether they are being determined from the correlations at small separations, or from the mean square fluctuating gradients. The two most commonly used Taylor microscales are often referred to as <math> \lambda_{f} </math> and <math> \lambda_{g} </math> and are defined by

<table width="70%"><tr><td>
:<math>
\lambda^{2}_{f} \equiv 2 \frac{ \left\langle u^{2}_{1} \right\rangle }{ \left\langle \left[ \partial u_{1} / \partial x_{1} \right]^{2} \right\rangle }
</math>
</td><td width="5%">(70)</td></tr></table>

and

<table width="70%"><tr><td>
:<math>
\lambda^{2}_{g} \equiv 2 \frac{ \left\langle u^{2}_{1} \right\rangle }{ \left\langle \left[ \partial u_{1} / \partial x_{2} \right]^{2} \right\rangle }
</math>
</td><td width="5%">(71)</td></tr></table>

The subscripts f and g refer to the autocorrelation coefficients defined by:

<table width="70%"><tr><td>
:<math>
f \left( r \right) \equiv \frac{\left\langle u_{1} \left( x_{1} + r,x_{2},x_{3} \right) u_{1} \left( x_{1},x_{2},x_{3} \right) \right\rangle}{ \left\langle u^{2}_{1} \right\rangle } = \frac{B_{1,1} \left( r,0,0 \right)}{ B_{1,1} \left( 0,0,0 \right) }
</math>
</td><td width="5%">(72)</td></tr></table>

and

<table width="70%"><tr><td>
:<math>
g \left( r \right) \equiv \frac{\left\langle u_{1} \left( x_{1},x_{2}+r,x_{3} \right) u_{1} \left( x_{1},x_{2},x_{3} \right) \right\rangle}{ \left\langle u^{2}_{1} \right\rangle } = \frac{B_{1,1} \left( 0,r,0 \right)}{ B_{1,1} \left( 0,0,0 \right) }
</math>
</td><td width="5%">(73)</td></tr></table>

It is straightforward to show from the definitions that <math> \lambda_{f} </math> and <math> \lambda_{g} </math> are related to the curvature of the <math> f </math> and <math> g </math> correlation functions at <math> r=0 </math>. Specifically,

<table width="70%"><tr><td>
:<math>
\lambda^{2}_{f}= \frac{2}{d^{2} f / dr^{2} |_{r=0} }
</math>
</td><td width="5%">(74)</td></tr></table>

and

<table width="70%"><tr><td>
:<math>
\lambda^{2}_{g}= \frac{2}{d^{2} g / dr^{2} |_{r=0} }
</math>
</td><td width="5%">(75)</td></tr></table>

Since both <math> f </math> and <math> g </math> are symmetrical functions of <math> r </math>, <math> df/dr </math> and <math> dg/dr </math> must be zero at <math> r=0 </math>. It follows immediately that the leading <math> r </math>-dependent term in the expansions about the origin of both autocorrelations are of parabolic form; i.e.,

<table width="70%"><tr><td>
:<math>
f \left( r \right) = 1 - \frac{r^{2}}{\lambda^{2}_{f}} + \cdots
</math>
</td><td width="5%">(76)</td></tr></table>

and

<table width="70%"><tr><td>
:<math>
g \left( r \right) = 1 - \frac{r^{2}}{\lambda^{2}_{g}} + \cdots
</math>
</td><td width="5%">(77)</td></tr></table>

This is illustrated in Figure 5.9 which shows that the Taylor microscales are the intersection with the <math> r </math>-axis of a parabola fitted to the appropriate correlation function at the origin. Fitting a parabola is a common way to determine the Taylor microscale, but to do so you must make sure you resolve accurately to scales much smaller than it (typically an order of magnitude smaller is required). Otherwise you are simply determining the spatial filtering of your probe or numerical algorithm.

{{Turbulence credit wkgeorge}}

{{Chapter navigation|Turbulence kinetic energy|Homogeneous turbulence}}

Introduction to turbulence/Stationarity and homogeneity

2008-02-25T09:10:42Z

Jola: Added book navigation features and credit

{{Introduction to turbulence menu}}

== Processes statistically stationary in time ==

Many random processes have the characteristic that their statistical properties do not appear to depend directly on time, even though the random variables themselves are time-dependent. For example, consider the signals shown in Figures 2.2 and 2.5

When the statistical properties of a random process are independent of time, the random process is said to be ''stationary''. For such a process all the moments are time-independent, e.g., <math> \left\langle \tilde{ u \left( t \right)} \right\rangle = U </math>, etc. In fact, the probability density itself is time-independent, as should be obvious from the fact that the moments are time independent.

An alternative way of looking at ''stationarity'' is to note that ''the statistics of the process are independent of the origin in time''. It is obvious from the above, for example, that if the statistics of a process are time independent, then <math> \left\langle u^{n} \left( t \right) \right\rangle = \left\langle u^{n} \left( t + T \right) \right\rangle </math> , etc., where <math> T </math> is some arbitrary translation of the origin in time. Less obvious, but equally true, is that the product <math> \left\langle u \left( t \right) u \left( t' \right) \right\rangle </math> depends only on time difference <math> t'-t </math> and not on <math> t </math> (or <math> t' </math> ) directly. This consequence of stationarity can be extended to any product moment. For example <math> \left\langle u \left( t \right) v \left( t' \right) \right\rangle </math> can depend only on the time difference <math> t'-t </math>. And <math> \left\langle u \left( t \right) v \left( t' \right) w \left( t'' \right)\right\rangle </math> can depend only on the two time differences <math> t'- t </math> and <math> t'' - t </math> (or <math> t'' - t' </math> ) and not <math> t </math> , <math> t' </math> or <math> t'' </math> directly.

== The autocorrelation ==

One of the most useful statistical moments in the study of stationary random processes (and turbulence, in particular) is the '''autocorrelation''' defined as the average of the product of the random variable evaluated at two times, i.e. <math> \left\langle u \left( t \right) u \left( t' \right)\right\rangle </math>. Since the process is assumed stationary, this product can depend only on the time difference <math> \tau = t' - t </math>. Therefore the autocorrelation can be written as:

<table width="70%"><tr><td>
:<math>
C \left( \tau \right) \equiv \left\langle u \left( t \right) u \left( t + \tau \right) \right\rangle
</math>
</td><td width="5%">(1)</td></tr></table>

The importance of the autocorrelation lies in the fact that it indicates the "memory" of the process; that is, ''the time over which is correlated with itself''. Contrast the two autocorrelation of deterministic sine wave is simply a cosine as can be easily proven. Note that there is no time beyond which it can be guaranteed to be arbitrarily small since it always "remembers" when it began, and thus always remains correlated with itself. By contrast, a stationary random process like the one illustrated in the figure will eventually lose all correlation and go to zero. In other words it has a "finite memory" and "forgets" how it was. Note that one must be careful to make sure that a correlation really both goes to zero and ''stays down'' before drawing conclusions, since even the sine wave was zero at some points. Stationary random process ''always'' have two-time correlation functions which eventually go to zero and stay there.

'''Example 1.'''

Consider the motion of an automobile responding to the movement of the wheels over a rough surface. In the usual case where the road roughness is randomly distributed, the motion of the car will be a weighted history of the road's roughness with the most recent bumps having the most influence and with distant bumps eventually forgotten. On the other hand, if the car is travelling down a railroad track, the periodic crossing of the railroad ties represents a determenistic input an the motion will remain correlated with itself indefinitely, a very bad thing if the tie crossing rate corresponds to a natural resonance of the suspension system of the vehicle.

Since a random process can never be more than perfectly correlated, it can never achieve a correlation greater than is value at the origin. Thus

<table width="70%"><tr><td>
:<math>
\left| C \left( \tau \right) \right| \leq C\left( 0 \right)
</math>
</td><td width="5%">(2)</td></tr></table>

An important consequence of stationarity is that the autocorrelation is symmetric in the time difference <math> \tau = t' - t </math>. To see this simply shift the origin in time backwards by an amount <math> \tau </math> and note that independence of origin implies:

<table width="70%"><tr><td>
:<math>
\left\langle u \left( t \right) u \left( t + \tau \right) \right\rangle = \left\langle u \left( t - \tau \right) u \left( t \right) \right\rangle
</math>
</td><td width="5%">(3)</td></tr></table>

Since the right hand side is simply <math> C \left( - \tau \right) </math>, it follows immediately that:

<table width="70%"><tr><td>
:<math>
C \left( \tau \right) = C \left( - \tau \right)
</math>
</td><td width="5%">(4)</td></tr></table>

== The autocorrelation coefficient ==

It is convenient to define the ''autocorrelation coefficient'' as:

<table width="70%"><tr><td>
:<math>
\rho \left( \tau \right) \equiv \frac{ C \left( \tau \right)}{ C \left( 0 \right)} = \frac{\left\langle u \left( t \right) u \left( t + \tau \right) \right\rangle}{ \left\langle u'^{2} \right\rangle }
</math>
</td><td width="5%">(5)</td></tr></table>

where

<table width="70%"><tr><td>
:<math>
\left\langle u^{2} \right\rangle = \left\langle u \left( t \right) u \left( t \right) \right\rangle = C \left( 0 \right) = var \left[ u \right]
</math>
</td><td width="5%">(6)</td></tr></table>

Since the autocorrelation is symmetric, so is its coefficient, i.e.,

<table width="70%"><tr><td>
:<math>
\rho \left( \tau \right) = \rho \left( - \tau \right)
</math>
</td><td width="5%">(7)</td></tr></table>

It is also obvious from the fact that the autocorrelation is maximal at the origin that the autocorrelation coefficient must also be maximal there. In fact from the definition it follows that

<table width="70%"><tr><td>
:<math>
\rho \left( 0 \right) = 1
</math>
</td><td width="5%">(8)</td></tr></table>

and

<table width="70%"><tr><td>
:<math>
\rho \left( \tau \right) \leq 1
</math>
</td><td width="5%">(9)</td></tr></table>

for all values of <math> \tau </math> .

== The integral scale ==

One of the most useful measures of the length of a time a process is correlated with itself is the integral scale defined by

<table width="70%"><tr><td>
:<math>
T_{int} \equiv \int^{\infty}_{0} \rho \left( \tau \right) d \tau
</math>
</td><td width="5%">(10)</td></tr></table>

It is easy to see why this works by looking at Figure 5.2. In effect we have replaced the area under the correlation coefficient by a rectangle of height unity and width <math> T_{int} </math> .

== The temporal Taylor microscale ==

The autocorrelation can be expanded about the origin in a MacClaurin series; i.e.,

<table width="70%"><tr><td>
:<math>
C \left( \tau \right) = C \left( 0 \right) + \tau \frac{ d C }{ d t }|_{\tau = 0} + \frac{1}{2} \tau^{2} \frac{d^{2} C}{d t^{2} }|_{\tau = 0} + \frac{1}{3!} \tau^{3} \frac{d^{3} C}{d t^{3} }|_{\tau = 0}
</math>
</td><td width="5%">(11)</td></tr></table>

But we know the aoutocorrelation is symmetric in <math> \tau </math> , hence the odd terms in <math> \tau </math> must be identically to zero (i.e., <math> dC / dt |_{\tau = 0} = 0 </math> , <math> d^{3}C / dt^{3} |_{\tau = 0} = 0 </math>, etc.). Therefore the expansion of the autocorrelation near the origin reduces to:

<table width="70%"><tr><td>
:<math>
C \left( \tau \right) = C \left( 0 \right) + \frac{1}{2} \tau^{2} \frac{d^{2} C}{d t^{2} }|_{\tau = 0} + \cdots
</math>
</td><td width="5%">(12)</td></tr></table>

Similary, the autocorrelation coefficient near the origin can be expanded as:

<table width="70%"><tr><td>
:<math>
\rho \left( \tau \right) = 1 + \frac{1}{2}\frac{d^{2}\rho}{d t^{2}}|_{\tau = 0} \tau^{2}+ \cdots
</math>
</td><td width="5%">(13)</td></tr></table>

where we have used the fact that <math> \rho \left( 0 \right) = 1 </math> . If we define <math> ' = d / dt </math> we can write this compactly as:

<table width="70%"><tr><td>
:<math>
\rho \left( \tau \right) = 1 + \frac{1}{2} \rho '' \left( 0 \right) \tau^{2} + \cdots
</math>
</td><td width="5%">(14)</td></tr></table>

Since <math> \rho \left( \tau \right) </math> has its maximum at the origin, obviously <math> \rho'' \left( 0 \right) </math> must be negative.

We can use the correlation and its second derivative at the origin to ''define'' a special time scale, <math> \lambda_{\tau} </math> (called the Taylor microscale) by:

<table width="70%"><tr><td>
:<math>
\lambda^{2}_{\tau} \equiv - \frac{2}{\rho'' \left( 0 \right)}
</math>
</td><td width="5%">(15)</td></tr></table>

Using this in equation 14 yields the expansion for the correlation coefficient near the origin as:

<table width="70%"><tr><td>
:<math>
\rho \left( \tau \right) = 1 - \frac{\tau^{2}}{\lambda^{2}_{\tau}} + \cdots
</math>
</td><td width="5%">(16)</td></tr></table>

Thus very near the origin the correlation coefficient (and the autocorrelation as well) simply rolls off parabolically; i.e.,

<table width="70%"><tr><td>
:<math>
\rho \left( \tau \right) \approx 1 - \frac{\tau^{2}}{\lambda^{2}_{\tau}}
</math>
</td><td width="5%">(17)</td></tr></table>

This parabolic curve is shown in Figure 3 as the osculating (or 'kissing') parabola which approaches zero exactly as the autocorrelation coefficient does. The intercept of this osculating parabola with the <math> \tau </math> -axis is the Taylor microscale, <math> \lambda_{\tau} </math>.

The Taylor microscale is significant for a number of reasons. First, for many random processes (e.g., Gaussian), the Taylor microscale can be proven to be the average distance between zero-crossing of a random variable in time. This is approximately true for turbulence as well. Thus one can quickly estimate the Taylor microscale by simply observing the zero-crossings using an oscilloscope trace.

The Taylor microscale also has a special relationship to the mean square time derivative of the signal, <math> \left\langle \left[ d u / d t \right]^{2} \right\rangle </math>. This is easiest to derive if we consider two stationary random signals at two different times say <math> u = u \left( t \right) </math> and <math> u' = u' \left( t' \right) </math>. The derivative of the first signal is <math> d u / d t </math> and the second <math> d u' / d t' </math>. Now lets multiply these together and rewrite them as:

<table width="70%"><tr><td>
:<math>
\frac{du'}{dt'} \frac{du}{dt} = \frac{d^{2}}{dtdt'} u \left( t \right) u' \left( t' \right)
</math>
</td><td width="5%">(18)</td></tr></table>

where the right-hand side follows from our assumption that <math> u </math> is not a function of <math> t' </math> nor <math> u' </math> a function of <math> t </math>.

Now if we average and interchenge the operations of differentiation and averaging we obtain:

<table width="70%"><tr><td>
:<math>
\left\langle \frac{du'}{dt'} \frac{du}{dt} \right\rangle = \frac{d^{2}}{dtdt'} \left\langle u \left( t \right) u' \left( t' \right) \right\rangle
</math>
</td><td width="5%">(19)</td></tr></table>

Here comes the first trick: we simply take <math> u' </math> to be exactly <math> u </math> but evaluated at time <math> t' </math>. So <math> u \left( t \right) u' \left( t' \right) </math> simply becomes <math> u \left( t \right) u \left( t' \right) </math> and its average is just the autocorrelation, <math> C \left( \tau \right) </math>. Thus we are left with:

<table width="70%"><tr><td>
:<math>
\left\langle \frac{du'}{dt'} \frac{du}{dt} \right\rangle = \frac{d^{2}}{dtdt'} C \left( t' - t \right)
</math>
</td><td width="5%">(20)</td></tr></table>

Now we simply need to use the chain-rule. We have already defined <math> \tau = t' - t </math>. Let's also define <math> \xi = t' + t </math> and transform the derivatives involving <math> t </math> and <math> t' </math> to derivatives involving <math> \tau </math> and <math> \xi </math>. The result is:

<table width="70%"><tr><td>
:<math>
\frac{d^{2}}{dtdt'} = \frac{d^{2}}{d \xi^{2}} - \frac{d^{2}}{d \tau^{2}}
</math>
</td><td width="5%">(21)</td></tr></table>

So equation 20 becomes

<table width="70%"><tr><td>
:<math>
\left\langle \frac{du'}{dt'} \frac{du}{dt} \right\rangle = \frac{d^{2}}{d \xi^{2}}C \left( \tau \right) - \frac{d^{2}}{d \tau^{2}} C \left( \tau \right)
</math>
</td><td width="5%">(22)</td></tr></table>

But since <math> C </math> is a function only of <math> \tau </math>, the derivative of it with respect to <math> \xi </math> is identically zero. Thus we are left with:

<table width="70%"><tr><td>
:<math>
\left\langle \frac{du'}{dt'} \frac{du}{dt} \right\rangle = - \frac{d^{2}}{d \tau^{2}} C \left( \tau \right)
</math>
</td><td width="5%">(23)</td></tr></table>

And finally we need the second trick. Let's evaluate both sides at <math> t = t' </math> (or <math> \tau = 0 </math> ) to obtain the ''mean square derivative'' as:

<table width="70%"><tr><td>
:<math>
\left\langle \left( \frac{du}{dt} \right)^{2} \right\rangle = - \frac{d^{2}}{d \tau^{2}} C \left( \tau \right)|_{ \tau = 0}
</math>
</td><td width="5%">(24)</td></tr></table>

But from our definition of the Taylor microscale and the facts that <math> C \left( 0 \right) = \left\langle u^{2} \right\rangle </math> and <math> C \left( \tau \right) = \left\langle u^{2} \right\rangle \rho \left( \tau \right) </math>, this is exactly the same as:

<table width="70%"><tr><td>
:<math>
\left\langle \left( \frac{du}{dt} \right)^{2} \right\rangle = 2 \frac{ \left\langle u^{2} \right\rangle}{\lambda^{2}_{\tau}}
</math>
</td><td width="5%">(25)</td></tr></table>

This amasingly simple result is very important in the study of turbulence, especially after we extend it to spatial derivatives.

== Time averages of stationary processes ==

It is common practice in many scientific disciplines to define a time average by integrating the random variable over a fixed time interval, i.e. ,

<table width="70%"><tr><td>
:<math>
U_{T} \equiv \frac{1}{T} \int^{T_{2}}_{T_{1}} u \left( t \right) dt
</math>
</td><td width="5%">(26)</td></tr></table>

For the stationary random processes we are considering here, we can define <math> T_{1} </math> to be the origin in time and simply write:

<table width="70%"><tr><td>
:<math>
U_{T} \equiv \frac{1}{T} \int^{T}_{0} u \left( t \right) dt
</math>
</td><td width="5%">(27)</td></tr></table>

where <math> T = T_{2} - T_{1} </math> is the integration time.

Figure 5.4. shows a portion of a stationary random signal over which such an integration might be performed. The ime integral of <math> u \left( t \right) </math> over the integral <math> \left( O, T \right) </math> corresponds to the shaded area under the curve. Now since <math> u \left( t \right) </math> is random and since it formsthe upper boundary of the shadd area, it is clear that the time average, <math> U_{T} </math> is a lot like the estimator for the mean based on a finite number of independent realization, <math> X_{N} </math> we encountered earlier in section ''Estimation from a finite number of realizations'' (see ''Elements of statistical analysis'')

It will be shown in the analysis presented below that ''if the signal is stationary'', the time average defined by equation 27 is an unbiased estimator of the true average <math> U </math>. Moreover, the estimator converges to <math> U </math> as the time becomes infinite; i.e., for stationary random processes

<table width="70%"><tr><td>
:<math>
U = \lim_{T \rightarrow \infty} \frac{1}{T} \int^{T}_{0} u \left( t \right) dt
</math>
</td><td width="5%">(28)</td></tr></table>

Thus the time and ensemble averages are equivalent in the limit as <math> T \rightarrow \infty </math>, ''but only for a stationary random process''.

== Bias and variability of time estimators ==

It is easy to show that the estimator, <math> U_{T} </math>, is unbiased by taking its ensemble average; i.e.,

<table width="70%"><tr><td>
:<math>
\left\langle U_{T} \right\rangle = \left\langle \frac{1}{T} \int^{T}_{0} u \left( t \right) dt \right\rangle = \frac{1}{T} \int^{T}_{0} \left\langle u \left( t \right) \right\rangle dt
</math>
</td><td width="5%">(29)</td></tr></table>

Since the process has been assumed stationary, <math> \left\langle u \left( t \right) \right\rangle </math> is independent of time. It follows that:

<table width="70%"><tr><td>
:<math>
\left\langle U_{T} \right\rangle = \frac{1}{T} \left\langle u \left( t \right) \right\rangle T = U
</math>
</td><td width="5%">(30)</td></tr></table>

To see whether the etimate improves as <math> T </math> increases, the variability of <math> U_{T} </math> must be examined, exactly as we did for <math> X_{N} </math> earlier in section Bias and convergence of estimators (see chapter The elements of statistical analysis). To do this we need the variance of <math> U_{T} </math> given by:

<table width="70%"><tr><td>
:<math>
\begin{matrix}
var \left[ U_{T} \right] & = & \left\langle \left[ U_{T} - \left\langle U_{T} \right\rangle \right]^{2} \right\rangle = \left\langle \left[ U_{T} - U \right]^{2} \right\rangle \\
& = & \frac{1}{T^{2}} \left\langle \left\{ \int^{T}_{0} \left[ u \left( t \right) - U \right] \right\}^{2} \right\rangle \\
& = & \frac{1}{T^{2}} \left\langle \int^{T}_{0} \int^{T}_{0} \left[ u \left( t \right) - U \right] \left[ u \left( t' \right) - U \right] dtdt' \right\rangle \\
& = & \frac{1}{T^{2}} \int^{T}_{0} \int^{T}_{0} \left\langle u'\left( t \right) u'\left( t' \right) \right\rangle dtdt' \\
\end{matrix}
</math>
</td><td width="5%">(31)</td></tr></table>

But since the process is assumed stationary <math> \left\langle u' \left( t \right) u' \left( t' \right) \right\rangle = C \left( t' - t \right) </math> where <math> C \left( t' - t \right) = \left\langle u^{2} \right\rangle \rho \left( t'-t \right) </math> is the correlation coefficient. Therefore the integral can be rewritten as:

<table width="70%"><tr><td>
:<math>
\begin{matrix}
var \left[ U_{T} \right] & = & \frac{1}{T^{2}} \int^{T}_{0} \int^{T}_{0} C \left( t' - t \right) dtdt' \\
& = & \frac{ \left\langle u^{2} \right\rangle }{ T^{2} } \int^{T}_{0} \int^{T}_{0} \rho \left( t' - t \right) dtdt' \\
\end{matrix}
</math>
</td><td width="5%">(33)</td></tr></table>

Now we need to apply some fancy calculus. If new variables <math> \tau= t'-t </math> and <math> \xi= t'+t </math> are defined, the double integral can be transformed to (see Figure 5.5):

<table width="70%"><tr><td>
:<math>
var \left[ U_{T} \right] = \frac{var \left[ u \right]}{2 T^{2}} \left[ \int^{T}_{0} d \tau \int^{T-\tau}_{\tau} d \xi \rho \left( \tau \right) + \int^{0}_{-T} d \tau \int^{T+\tau}_{-\tau} d \xi \rho \left( \tau \right) \right]
</math>
</td><td width="5%">(35)</td></tr></table>

where the factor of <math> 1/2 </math> arises from the Jacobian of the transformation. The integrals over <math> d \xi </math> can be evaluated directly to yield:

<table width="70%"><tr><td>
:<math>
var \left[ U_{T} \right] = \frac{var \left[ u \right]}{2 T^{2}} \left\{ \int^{T}_{0} \rho \left( \tau \right) \left[ T - \tau \right] d \tau + \int^{0}_{-T} \rho \left( \tau \right) \left[ T + \tau \right] \right\}
</math>
</td><td width="5%">(36)</td></tr></table>

By noting that the autocorrelation is symmetric, the second integral can be transformed and added to the first to yield at last the result we seek as:

<table width="70%"><tr><td>
:<math>
var \left[ U_{T} \right] = \frac{var \left[ u \right]}{T} \int^{T}_{-T} \rho \left( \tau \right) \left[ 1 - \frac{ \left| \tau \right| }{T} \right] d \tau
</math>
</td><td width="5%">(37)</td></tr></table>

Now if our averaging time, <math> T </math>, is chosen so large that <math> \left| \tau \right| / T << 1 </math> over the range for which <math> \rho \left( \tau \right) </math> is non-zero, the integral reduces:

<table width="70%"><tr><td>
:<math>
\begin{matrix}
var \left[ U_{T} \right] & \approx & \frac{2 var \left[ u \right]}{T} \int^{T}_{0} \rho \left( \tau \right) d \tau \\
& = & \frac{2 T_{int}}{T} var \left[ u \right] \\
\end{matrix}
</math>
</td><td width="5%">(38)</td></tr></table>

where <math> T_{int} </math> is the integral scale defined by equation 10. Thus the ''variability'' of our estimator is given by:

<table width="70%"><tr><td>
:<math>
\epsilon^{2}_{U_{T}} = \frac{2T_{int}}{T}
</math>
</td><td width="5%">(39)</td></tr></table>

Therefore the estimator does, in fact, converge (in mean square) to the correct result as the averaging time, <math> T </math> increases relative to the integral scale, <math> T_{int} </math>.

There is a direct relationship between equation 39 and equation 52 in chapter The elements of statistical analysis ( section Bias and convergence of estimators) which gave the mean square variability for the ensemble estimate from a finite number of statistically independent realizations, <math> X_{N} </math>. Obviously the effective number of independent realizations for the finite time estimator is:

<table width="70%"><tr><td>
:<math>
N_{eff} = \frac{2T_{int}}{T}
</math>
</td><td width="5%">(40)</td></tr></table>

so that the two expressions are equivalent. Thus, in effect, ''portions of the record separated by two integral scales behave as though they were statistically independent, at least as far as convergence of finite time estimators is concerned''.

Thus what is required for convergence is again, many ''independent'' pieces of information. This is illustrated in Figure 5.6. That the length of the recordn should be measured in terms of the integral scale should really be no surprise since it is a measure of the rate at which a process forgets its past.

'''Example'''

It is desired to mesure the mean velocity in a turbulent flow to within an rms error of 1% (i.e. <math> \epsilon = 0.01 </math> ). The expected fluctuation level of the signal is 25% and integral scale is estimated as 100 ms. What is the required averaging time?

From equation 39

<table width="70%"><tr><td>
:<math>
\begin{matrix}
T & = & \frac{2T_{int}}{\epsilon^{2}} \frac{var \left[ u \right]}{U^{2}} \\
& = & 2 \times 0.1 \times (0.25)^{2} / (0.01)^{2} = 125 sec \\
\end{matrix}
</math>
</td><td width="5%">(41)</td></tr></table>

Similar considerations apply to any other finite time estimator and equation 55 from chapter Statistical analysis can be applied directly as long as equation 40 is used for the number of independent samples.

It is common common experimental practice to not actually carry out an analog integration. Rather the signal is sampled at fixed intervals in time by digital means and the averages are computed as for an esemble with a finite number of realizations. Regardless of the manner in which the signal is processed, only a finite portion of a stationary time series can be analyzed and the preceding considerations always apply.

It is important to note that data sampled more rapidly than once every two integral scales do '''not''' contribute to the convergence of the estimator since they can not be considered independent. If <math> N </math> is the actual number of samples acquired and <math> \Delta t </math> is the time between samples, then the effective number of independent realizations is

<table width="70%"><tr><td>
:<math>
N_{eff} = \left\{
\begin{array}{lll}
N \Delta t /T_{int} & if & \Delta t < 2T_{int} \\
N & if & \Delta t \geq 2T_{int} \\
\end{array}
\right.
</math>
</td><td width="5%">(42)</td></tr></table>

It should be clear that if you sample faster than <math> \Delta t = 2T_{int} </math> you are processing unnecessary data which does not help your statistics converge.

You may wonder why one would ever take data faster than absolutely necessary, since it simply it simply fills up your computer memory with lots of statistically redundant data. When we talk about measuring spectra you will learn that for spectral measurements it is necessary to sample much faster to avoid spactral aliasing. Many wrongly infer that they must sample at these higher rates even when measuring just moments. Obviously this is not the case if you are not measuring spectra.

== Random fields of space and time ==

To this point only temporally varying random fields have been discussed. For turbulence however, random fields can be functions of both space and time. For example, the temperature <math> \theta </math> could be a random scalar function of time <math> t </math> and position <math> \stackrel{\rightarrow}{x} </math>, i.e.,

<table width="70%"><tr><td>
:<math>
\theta = \theta \left( \stackrel{\rightarrow}{x} , t \right)
</math>
</td><td width="5%">(43)</td></tr></table>

The velocity is another example of a random vector function of position and time, i.e.,

<table width="70%"><tr><td>
:<math>
\stackrel{\rightarrow}{u} = \stackrel{\rightarrow}{u} \left( \stackrel{\rightarrow}{x},t \right)
</math>
</td><td width="5%">(44)</td></tr></table>

or in tensor notation,

<table width="70%"><tr><td>
:<math>
u_{i} = u_{i} \left( \stackrel{\rightarrow}{x},t \right)
</math>
</td><td width="5%">(45)</td></tr></table>

In the general case, the ensemble averages of these quantities are functions of both positon and time; i.e.,

<table width="70%"><tr><td>
:<math>
\left\langle \theta \left( \stackrel{\rightarrow}{x},t \right) \right\rangle \equiv \Theta \left( \stackrel{\rightarrow}{x},t \right)
</math>
</td><td width="5%">(46)</td></tr></table>

<table width="70%"><tr><td>
:<math>
\left\langle u_{i} \left( \stackrel{\rightarrow}{x},t \right) \right\rangle \equiv U_{i} \left( \stackrel{\rightarrow}{x},t \right)
</math>
</td><td width="5%">(47)</td></tr></table>

If only ''stationary'' random processes are considered, then the averages do not depend on time and are functions of <math> \stackrel{\rightarrow}{x} </math> only; i.e.,

<table width="70%"><tr><td>
:<math>
\left\langle \theta \left( \stackrel{\rightarrow}{x},t \right) \right\rangle \equiv \Theta \left( \stackrel{\rightarrow}{x} \right)
</math>
</td><td width="5%">(48)</td></tr></table>

<table width="70%"><tr><td>
:<math>
\left\langle u_{i} \left( \stackrel{\rightarrow}{x},t \right) \right\rangle \equiv U_{i} \left( \stackrel{\rightarrow}{x}\right)
</math>
</td><td width="5%">(49)</td></tr></table>

Now the averages may not be position dependent either. For example, if the averages are ''independent of the origin in position'', then the field is said to be '''homogeneous'''. '''Homogenity''' (the noun corresponding to the adjective homogeneous) is exactly analogous to stationarity except that position is now the variable, and not time.

It is, of course, possible (at least in concept) to have homogeneous fields which are either stationary or non stationary. Since position, unlike time, is a vector quantity it is also possible to have only partial homogeneity. For example, a field can be homogeneous in the <math> x_{1}- </math> and <math> x_{3}- </math> directions, but not in the <math> x_{2}- </math> direction so that <math> U_{i}=U_{i}(X_{2}) </math> only. In fact, it appears to be dynamically impossible to have flows which are honogeneous in all variables and stationary as well, but the concept is useful, nonetheless.

Homogeneity will be seen to have powerful consequences for the equations govering the averaged motion, since the spatial derivative of any averaged quantity must be identically zero. Thus even homogeneity in only one direction can considerably simplify the problem. For example, in the Reynolds stress transport equation, the entire turbulence transport is exactly zero if the field is homogeneous.

== Multi-point statistics in homogeneous field ==

The concept of homogeneity can also be extended to multi-point statistics. Consider for example, the correlation between the velocity at one point and that at another as illustrated in Figure 5.7. If the time dependence is suppressed and the field is assumed statistically ''homogeneous'', this correlation is a function only of the separation of the two points, i.e.,

<table width="70%"><tr><td>
:<math>
\left\langle u_{i} \left( \stackrel{\rightarrow}{x} , t \right) u_{j} \left( \stackrel{\rightarrow}{x'} , t \right) \right\rangle \equiv B_{i,j} \left( \stackrel{\rightarrow}{r} \right)
</math>
</td><td width="5%">(50)</td></tr></table>

where <math> \stackrel{\rightarrow}{r} </math> is the separation vector defined by

<table width="70%"><tr><td>
:<math>
\stackrel{\rightarrow}{r} = \stackrel{\rightarrow}{x'} - \stackrel{\rightarrow}{x}
</math>
</td><td width="5%">(51)</td></tr></table>

or

<table width="70%"><tr><td>
:<math>
r_{i} = x'_{i} - x_{i}
</math>
</td><td width="5%">(52)</td></tr></table>

Note that the convention we shall follow for vector quantities is that the first subscript on <math> B_{i,j} </math> is the component of velocity at the first position, <math> \stackrel{\rightarrow}{x} </math> , and the second subscript is the component of velocity at the second, <math> \stackrel{\rightarrow}{x'} </math>. For scalar quantities we shall simply put a simbol for the quantity to hold the place. For example, we would write the two-point temperature correlation in a homogeneous field by:

<table width="70%"><tr><td>
:<math>
\left\langle \theta \left( \stackrel{\rightarrow}{x},t \right) \theta \left( \stackrel{\rightarrow}{x'},t \right) \right\rangle \equiv B_{\theta , \theta} \left( \stackrel{\rightarrow}{r} \right)
</math>
</td><td width="5%">(53)</td></tr></table>

A mixed vector/scalar correlation like the two-point temperature velocity correlation would be written as:

<table width="70%"><tr><td>
:<math>
\left\langle u_{i} \left( \stackrel{\rightarrow}{x} , t \right) \theta \left( \stackrel{\rightarrow}{x'},t \right) \right\rangle \equiv B_{i,\theta } \left( \stackrel{\rightarrow}{r} \right)
</math>
</td><td width="5%">(54)</td></tr></table>

On the other hand, if we meant for the temperature to be evaluated at <math> \stackrel{\rightarrow}{x} </math> and the velocity at <math> \stackrel{\rightarrow}{x'} </math> we would have to write:

<table width="70%"><tr><td>
:<math>
\left\langle \theta \left( \stackrel{\rightarrow}{x},t \right) u_{i} \left( \stackrel{\rightarrow}{x'},t \right) \right\rangle \equiv B_{ \theta, i } \left( \stackrel{\rightarrow}{r} \right)
</math>
</td><td width="5%">(55)</td></tr></table>

Now most books don't bother with the subscript notation, and simply give each new correlation a new symbol. At first this seems much simpler; and it is as long as you are only dealing with one or two different correlations. But introduce a few more, then read about a half-dozen pages, and you will find you completely forget what they are or how they were put together. It is usually very important to know exactly what you are talking about, so we will use this comma system to help us remember.

It is easy to see that the consideration of vector quantities raises special considerations. For example, the correlation between a scalar function of position at two points is symmetrical in <math> \stackrel{\rightarrow}{r} </math> , i.e.,

<table width="70%"><tr><td>
:<math>
B_{\theta,\theta} \left( \stackrel{\rightarrow}{r} \right) = B_{\theta,\theta} \left( - \stackrel{\rightarrow}{r} \right)
</math>
</td><td width="5%">(56)</td></tr></table>

This is easy to show from the definition of <math> B_{\theta,\theta} </math> and the fact that the field is homogeneous. Simply shift each of the position vectors by the same amount <math> - \stackrel{\rightarrow}{r} </math> as shown in Figure 5.8 to obtain:

<table width="70%"><tr><td>
:<math>
\begin{matrix}
B_{\theta,\theta}\left( \stackrel{\rightarrow}{r},t \right) & \equiv & \left\langle \theta\left( \stackrel{\rightarrow}{x}, t \right) \theta\left( \stackrel{\rightarrow}{x'}, t \right) \right\rangle \\
& = & \left\langle \theta \left( \stackrel{\rightarrow}{x} - \stackrel{\rightarrow}{r} , t \right) \theta \left( \stackrel{\rightarrow}{x'} - \stackrel{\rightarrow}{r} , t \right) \right\rangle \\
& = & B_{\theta,\theta}\left( - \stackrel{\rightarrow}{r},t \right) \\
\end{matrix}
</math>
</td><td width="5%">(57)</td></tr></table>

since <math> \stackrel{\rightarrow}{x'} - \stackrel{\rightarrow}{r} = \stackrel{\rightarrow}{x} </math> ; i.e., the points are reversed and the separation vector is pointing the opposite way.

Such is not the case, in general, for ''vector'' functions of position. For example, see if you can prove to yourself the following:

<table width="70%"><tr><td>
:<math>
B_{\theta,i} \left( \stackrel{\rightarrow}{r} \right) = B_{i,\theta} \left( - \stackrel{\rightarrow}{r} \right)
</math>
</td><td width="5%">(58)</td></tr></table>

and

<table width="70%"><tr><td>
:<math>
B_{i,j} \left( \stackrel{\rightarrow}{r} \right) = B_{j,i} \left( - \stackrel{\rightarrow}{r} \right)
</math>
</td><td width="5%">(59)</td></tr></table>

Clearly the latter is symmetrical in the variable <math> \stackrel{\rightarrow}{r} </math> only when <math> i = j </math> .

These properties of the two-point correlation function will be seen to play an important role in determining the interrelations among the different two-point statistical quantities. They will be especially important when we talk about spectral quantities.

== Spatial integral and Taylor microscales ==

Just as for a stationary random process, correlations between spatially varying, but ''statistically homogeneous'', random quantities ultimately go to zero;, i.e., they become uncorrelated as their locations become widely separated. Because position (o relative position) is a vector quantity, however, the correlation the carrelation may die off at different rates in different directions. Thus direction must be an important part of the definitions of the integral scales and microscales.

Consider for example the one-dimensional spatial correlation which is obtained by measuring the correlation between the temperature at two points along a line in the x-direction, say,

<table width="70%"><tr><td>
:<math>
B^{(1)}_{\theta,\theta} \left( r \right) \equiv \left\langle \theta \left( x_{1} + r , x_{2} , x_{3} , t \right) \theta \left( x_{1} , x_{2} , x_{3} , t \right) \right\rangle
</math>
</td><td width="5%">(60)</td></tr></table>

The superscript "(1)" denotes "the coordinate direction in which the separation occurs". This distinguishes it from the vector separation of <math> B_{\theta,\theta} </math> above. Also, note that the correlation at zero separationis just the variance; i.e.,

<table width="70%"><tr><td>
:<math>
B^{(1)}_{\theta,\theta} \left( 0 \right) = \left\langle \theta^{2} \right\rangle
</math>
</td><td width="5%">(61)</td></tr></table>

The integral scale in the <math> x </math>-direction can be defined as:

<table width="70%"><tr><td>
:<math>
L^{(1)}_{\theta} \equiv \frac{1}{ \left\langle \theta^{2} \right\rangle} \int^{\infty}_{0} \left\langle \theta \left( x + r, y,z,t \right) \theta \left( x,y,z,t \right) \right\rangle dr
</math>
</td><td width="5%">(62)</td></tr></table>

It is clear that there are at least two more integral scales which could be defined by considering separations in the y and z directions. Thus

<table width="70%"><tr><td>
:<math>
L^{(2)}_{\theta} \equiv \frac{1}{ \left\langle \theta^{2} \right\rangle} \int^{\infty}_{0} \left\langle \theta \left( x,y + r,z,t \right) \theta \left( x,y,z,t \right) \right\rangle dr
</math>
</td><td width="5%">(63)</td></tr></table>

and

<table width="70%"><tr><td>
:<math>
L^{(3)}_{\theta} \equiv \frac{1}{ \left\langle \theta^{2} \right\rangle} \int^{\infty}_{0} \left\langle \theta \left( x,y,z + r,t \right) \theta \left( x,y,z,t \right) \right\rangle dr
</math>
</td><td width="5%">(64)</td></tr></table>

In fact, an integral scale could be defined for ''any'' direction simply by choosing the components of the separation vector <math> \stackrel{\rightarrow}{r} </math>. This situation is even more complicated when correlations of vector quantities are considered. For example, consider the correlation of the velocity vectors at two points, <math> B_{i,j} \left( \stackrel{\rightarrow}{r} \right) </math>. Clearly <math> B_{i,j} \left( \stackrel{\rightarrow}{r} \right) </math> is not a single correlation, but rather nine separate correlations: <math> B_{1,1} \left( \stackrel{\rightarrow}{r} \right) </math> , <math> B_{1,2} \left( \stackrel{\rightarrow}{r} \right) </math> , <math> B_{1,3} \left( \stackrel{\rightarrow}{r} \right) </math> , <math> B_{2,1} \left( \stackrel{\rightarrow}{r} \right) </math> , <math> B_{2,2} \left( \stackrel{\rightarrow}{r} \right) </math> , etc. For each of these an integral scale can be defined once a direction for the separation vector is chosen. For example, the integral scales associated with <math> B_{1,1} </math> for the principal directions are

<table width="70%"><tr><td>
:<math>
L^{(1)}_{1,1} \equiv \frac{1}{\left\langle u^{2}_{1} \right\rangle} \int^{\infty}_{0} B_{1,1} \left( r,0,0 \right) dr
</math>
</td><td width="5%">(65)</td></tr></table>

<table width="70%"><tr><td>
:<math>
L^{(2)}_{1,1} \equiv \frac{1}{\left\langle u^{2}_{1} \right\rangle} \int^{\infty}_{0} B_{1,1} \left( 0,r,0 \right) dr
</math>
</td><td width="5%">(66)</td></tr></table>

<table width="70%"><tr><td>
:<math>
L^{(3)}_{1,1} \equiv \frac{1}{\left\langle u^{2}_{1} \right\rangle} \int^{\infty}_{0} B_{1,1} \left( 0,0,r \right) dr
</math>
</td><td width="5%">(67)</td></tr></table>

Similar integral scales can be defined for the other componentsof the correlation tensor. Two of particular importance in the development of the turbulence theory are:

<table width="70%"><tr><td>
:<math>
L^{(2)}_{1,1} \equiv \frac{1}{\left\langle u^{2}_{1} \right\rangle} \int^{\infty}_{0} B_{1,1} \left( 0,r,0 \right) dr
</math>
</td><td width="5%">(68)</td></tr></table>

<table width="70%"><tr><td>
:<math>
L^{(1)}_{2,2} \equiv \frac{1}{\left\langle u^{2}_{2} \right\rangle} \int^{\infty}_{0} B_{2,2} \left( r,0,0 \right) dr
</math>
</td><td width="5%">(69)</td></tr></table>

In general, each of these integral scales will be different, unless restrictions beyond simple homogeneity are placed on the process (e.g., like ''isotropy'' discussed below). Thus, it is important to specify precisely which integral scale is being referred to; i.e., which components of the vector quantities are being used and in which direction the integration is being performed.

Similar considerations apply to the Taylor microscales, regardless of whether they are being determined from the correlations at small separations, or from the mean square fluctuating gradients. The two most commonly used Taylor microscales are often referred to as <math> \lambda_{f} </math> and <math> \lambda_{g} </math> and are defined by

<table width="70%"><tr><td>
:<math>
\lambda^{2}_{f} \equiv 2 \frac{ \left\langle u^{2}_{1} \right\rangle }{ \left\langle \left[ \partial u_{1} / \partial x_{1} \right]^{2} \right\rangle }
</math>
</td><td width="5%">(70)</td></tr></table>

and

<table width="70%"><tr><td>
:<math>
\lambda^{2}_{g} \equiv 2 \frac{ \left\langle u^{2}_{1} \right\rangle }{ \left\langle \left[ \partial u_{1} / \partial x_{2} \right]^{2} \right\rangle }
</math>
</td><td width="5%">(71)</td></tr></table>

The subscripts f and g refer to the autocorrelation coefficients defined by:

<table width="70%"><tr><td>
:<math>
f \left( r \right) \equiv \frac{\left\langle u_{1} \left( x_{1} + r,x_{2},x_{3} \right) u_{1} \left( x_{1},x_{2},x_{3} \right) \right\rangle}{ \left\langle u^{2}_{1} \right\rangle } = \frac{B_{1,1} \left( r,0,0 \right)}{ B_{1,1} \left( 0,0,0 \right) }
</math>
</td><td width="5%">(72)</td></tr></table>

and

<table width="70%"><tr><td>
:<math>
g \left( r \right) \equiv \frac{\left\langle u_{1} \left( x_{1},x_{2}+r,x_{3} \right) u_{1} \left( x_{1},x_{2},x_{3} \right) \right\rangle}{ \left\langle u^{2}_{1} \right\rangle } = \frac{B_{1,1} \left( 0,r,0 \right)}{ B_{1,1} \left( 0,0,0 \right) }
</math>
</td><td width="5%">(73)</td></tr></table>

It is straightforward to show from the definitions that <math> \lambda_{f} </math> and <math> \lambda_{g} </math> are related to the curvature of the <math> f </math> and <math> g </math> correlation functions at <math> r=0 </math>. Specifically,

<table width="70%"><tr><td>
:<math>
\lambda^{2}_{f}= \frac{2}{d^{2} f / dr^{2} |_{r=0} }
</math>
</td><td width="5%">(74)</td></tr></table>

and

<table width="70%"><tr><td>
:<math>
\lambda^{2}_{g}= \frac{2}{d^{2} g / dr^{2} |_{r=0} }
</math>
</td><td width="5%">(75)</td></tr></table>

Since both <math> f </math> and <math> g </math> are symmetrical functions of <math> r </math>, <math> df/dr </math> and <math> dg/dr </math> must be zero at <math> r=0 </math>. It follows immediately that the leading <math> r </math>-dependent term in the expansions about the origin of both autocorrelations are of parabolic form; i.e.,

<table width="70%"><tr><td>
:<math>
f \left( r \right) = 1 - \frac{r^{2}}{\lambda^{2}_{f}} + \cdots
</math>
</td><td width="5%">(76)</td></tr></table>

and

<table width="70%"><tr><td>
:<math>
g \left( r \right) = 1 - \frac{r^{2}}{\lambda^{2}_{g}} + \cdots
</math>
</td><td width="5%">(77)</td></tr></table>

This is illustrated in Figure 5.9 which shows that the Taylor microscales are the intersection with the <math> r </math>-axis of a parabola fitted to the appropriate correlation function at the origin. Fitting a parabola is a common way to determine the Taylor microscale, but to do so you must make sure you resolve accurately to scales much smaller than it (typically an order of magnitude smaller is required). Otherwise you are simply determining the spatial filtering of your probe or numerical algorithm.

{{Turbulence credit wkgeorge}}

{{Chapter navigation|Reynolds averaged equations|Stationarity and homogenity}}

Talk:Fluent FAQ

2008-02-21T10:55:35Z

Jola: Reverted edits by Srinivas (Talk); changed back to last version by Jasond

I'm not sure that the question about Forum posting is the most important FAQ, and I it think would be better placed as part of a Forum FAQ (which we don't currently have...). While I agree that text-message-style posts are not the greatest, we have little chance of changing the culture here. I have softened the wording accordingly (and made some other editorial changes). Personally, I will answer a post as long as I can understand it (SMS abbreviations or not). In my opinion, the wording on the ANSYS FAQ should also be changed. --[[User:Jasond|Jasond]] 09:11, 18 August 2007 (MDT)

Please don't use this FAQ as a kind of discussion forum where you can just write your questions. The FAQ is the place where already answered questions from the forums should be written. --[[User:Jola|Jola]] 02:35, 22 March 2007 (MDT)

In case you have missed it there is an old fluent FAQ available [[CFD-Wiki:Donated_texts| here]]. If you feel like helping out a bit please take some time to go through this FAQ and move the questions and answers that are still relevant and interesting into this CFD-Wiki Fluent FAQ. --[[User:Jola|Jola]] 13:18, 14 December 2005 (MST)

I have added some of the questions/answers from the old FAQ, and I left them in the form of the old FAQ (with Q's and A's to distinguish them from the new content). Once I'm through adding, I'll either remove the Q's and A's or add them to the rest of the questions. --[[User:Jasond|Jasond]] 10:40, 22 July 2006 (MDT)

Stratford's separation criterion

2008-02-14T18:41:15Z

Jola:

Stratford's separation criterion is an old classical analytical way to assess if a turbulent boundary layer is likely to separate or not. Stratford's criteria says that from the start of the pressure recovery where the max velocity and the minimum static pressure is obtained the boundary layer is on the verge of separation when:

<math>
C'_p \cdot \sqrt{x' \frac{dC'_p}{dx}} = k \cdot \left( \frac{Re}{10^6} \right) ^ {0.1}
</math>

This formula is only valid as long as <math>C'_p < \frac{4}{7}</math>.

*<math>C'_p</math> is the canonical pressure distribution defined by:

:<math>C'_p = 1 - \left( \frac{U}{U_{max}} \right) ^ 2</math>
:<math>U</math> is the local velocity and <math>U_{max}</math> is the maximum velocity at the start of the pressure recovery.

*<math>k</math> is a constant which Stratford used the following values for:
:<math>
k =
\begin{cases}
0.35 & \mbox{if } \frac{d^2p}{dx^2} \le 0 \mbox{ (concave recovery)} \\
0.39 & \mbox{if } \frac{d^2p}{dx^2} > 0 \mbox{ (convex recovery)}
\end{cases}
</math>
Other researcers have used other values. Cebeci-Smith for example used <math>k = 0.5</math>, which is a bit less conservative than Statford's original values.

*<math>x'</math> is the effective length of the boundary layer. Note that computing <math>x'</math> can be a bit tricky. If the boundary layer is first accelerated up to the start of the recovery and is assumed to be turbulent all the time a turbulent boundary layer can be assumed to have the followig effective length:
:<math>x' = \int_0^{x^{turb}_{rec}} \left( \frac{U}{U_{max}} \right) ^ 3 dx + (x - x^{turb}_{rec})</math>
Note that this is only valid if the approaching boundary layer can be assumed to be fully turbulent. If the boundary layer is laminar, or undergoes transition, a different approximations needs to be done.

*<math>Re = \frac{U_{max} \cdot x'}{\nu}</math>
The Reynolds number above is based on the effective length of the bounday layer <math>x'</math> and the maximum velocity <math>U_{max}</math> at the start of the recovery.

Stratford's separation criteria is known to be conservative. It will most likely predict a bit too early separation.

== References ==

*{{reference-paper|author=Stratford, B. S.|year=1959|title=The Prediction of Separation of the Turbulent Boundary Layer|rest=Journal of Fluid Mechanics, Vol. 5, pp. 1-16}}

*{{reference-paper|author=Cebeci, T., Mosinskis, G. J., and Smith A. M. O|year=1972|title=Calculation of Separation Points in Incompressible Turbulent Flows|rest=Journal of Aircraft, Vol. 9., Sept. 1972, p. 618-624}}

Stratford's separation criterion

2008-02-14T18:40:45Z

Jola: Added a couple of references and an alternative constant from Cebeci-Smith

Stratford's separation criterion is an old classical analytical way to assess if a turbulent boundary layer is likely to separate or not. Stratford's criteria says that from the start of the pressure recovery where the max velocity and the minimum static pressure is obtained the boundary layer is on the verge of separation when:

<math>
C'_p \cdot \sqrt{x' \frac{dC'_p}{dx}} = k \cdot \left( \frac{Re}{10^6} \right) ^ {0.1}
</math>

This formula is only valid as long as <math>C'_p < \frac{4}{7}</math>.

*<math>C'_p</math> is the canonical pressure distribution defined by:

:<math>C'_p = 1 - \left( \frac{U}{U_{max}} \right) ^ 2</math>
:<math>U</math> is the local velocity and <math>U_{max}</math> is the maximum velocity at the start of the pressure recovery.

*<math>k</math> is a constant which Stratford used the following values for:
:<math>
k =
\begin{cases}
0.35 & \mbox{if } \frac{d^2p}{dx^2} \le 0 \mbox{ (concave recovery)} \\
0.39 & \mbox{if } \frac{d^2p}{dx^2} > 0 \mbox{ (convex recovery)}
\end{cases}
</math>
Other researcers have used other values. Cebeci-Smith for example used <math>k = 0.5</math>, which is a bit less conservative than Statford's original values.

*<math>x'</math> is the effective length of the boundary layer. Note that computing <math>x'</math> can be a bit tricky. If the boundary layer is first accelerated up to the start of the recovery and is assumed to be turbulent all the time a turbulent boundary layer can be assumed to have the followig effective length:
:<math>x' = \int_0^{x^{turb}_{rec}} \left( \frac{U}{U_{max}} \right) ^ 3 dx + (x - x^{turb}_{rec})</math>
Note that this is only valid if the approaching boundary layer can be assumed to be fully turbulent. If the boundary layer is laminar, or undergoes transition, a different approximations needs to be done.

*<math>Re = \frac{U_{max} \cdot x'}{\nu}</math>
The Reynolds number above is based on the effective length of the bounday layer <math>x'</math> and the maximum velocity <math>U_{max}</math> at the start of the recovery.

Stratford's separation criteria is known to be conservative. It will most likely predict a bit too early separation.

== References ==

*{{reference-paper|author=Stratford, B. S.|year=1959|title=The Prediction of Separation of the Turbulent Boundary Layer|rest=Journal of Fluid Mechanics, Vol. 5, pp. 1-16.}}

*{{reference-paper|author=Cebeci, T., Mosinskis, G. J., and Smith A. M. O|year=1972|title=Calculation of Separation Points in Incompressible Turbulent Flows|rest=Journal of Aircraft, Vol. 9., Sept. 1972, p. 618-624.}}

Stratford's separation criterion

2008-02-14T18:15:41Z

Jola: First version finished

Stratford's separation criterion is an old classical analytical way to assess if a turbulent boundary layer is likely to separate or not. Stratford's criteria says that from the start of the pressure recovery where the max velocity and the minimum static pressure is obtained the boundary layer is on the verge of separation when:

<math>
C'_p \cdot \sqrt{x' \frac{dC'_p}{dx}} = k \cdot \left( \frac{Re}{10^6} \right) ^ {0.1}
</math>

This formula is only valid as long as <math>C'_p < \frac{4}{7}</math>.

*<math>C'_p</math> is the canonical pressure distribution defined by:

:<math>C'_p = 1 - \left( \frac{U}{U_{max}} \right) ^ 2</math>
:<math>U</math> is the local velocity and <math>U_{max}</math> is the maximum velocity at the start of the pressure recovery.

*<math>k</math> is a constant which Stratford used the following values for:
:<math>
k =
\begin{cases}
0.35 & \mbox{if } \frac{d^2p}{dx^2} \le 0 \mbox{ (concave recovery)} \\
0.39 & \mbox{if } \frac{d^2p}{dx^2} > 0 \mbox{ (convex recovery)}
\end{cases}
</math>

*<math>x'</math> is the effective length of the boundary layer. Note that computing <math>x'</math> can be a bit tricky. If the boundary layer is first accelerated up to the start of the recovery and is assumed to be turbulent all the time a turbulent boundary layer can be assumed to have the followig effective length:
:<math>x' = \int_0^{x^{turb}_{rec}} \left( \frac{U}{U_{max}} \right) ^ 3 dx + (x - x^{turb}_{rec})</math>
Note that this is only valid if the approaching boundary layer can be assumed to be fully turbulent. If the boundary layer is laminar, or undergoes transition, a different approximations needs to be done.

*<math>Re = \frac{U_{max} \cdot x'}{\nu}</math>
The Reynolds number above is based on the effective length of the bounday layer <math>x'</math> and the maximum velocity <math>U_{max}</math> at the start of the recovery.

Stratford's separation criteria is known to be conservative. It will most likely predict a bit too early separation.

Stratford's separation criterion

2008-02-14T17:54:23Z

Jola: still more to add

Stratford's separation criteria is an old classical analytical way to assess if a turbulent boundary layer is likely to separate or not. Stratford's criteria says that from the start of the pressure recovery where the max velocity and the minimum static pressure is obtained the boundary layer is on the verge of separation when:

<math>
C'_p \cdot \sqrt{x' \frac{dC'_p}{dx}} = k \cdot \left( \frac{Re}{10^6} \right) ^ {0.1}
</math>

This formula is only valid as long as <math>C'_p < \frac{4}{7}</math>.

*<math>C'_p</math> is the canonical pressure distribution defined by:

:<math>C'_p = 1 - \left( \frac{U}{U_{max}} \right) ^ 2</math>
:<math>U</math> is the local velocity and <math>U_{max}</math> is the maximum velocity at the start of the pressure recovery.

*<math>k</math> is a constant which Stratford used the following values for:
:<math>
k =
\begin{cases}
0.35 & \mbox{if } \frac{d^2p}{dx^2} \le 0 \mbox{ (concave recovery)} \\
0.39 & \mbox{if } \frac{d^2p}{dx^2} > 0 \mbox{ (convex recovery)}
\end{cases}
</math>

*<math>x'</math> is the effective length of the boundary layer. Note that computing <math>x'</math> can be a bit tricky. If the boundary layer is first accelerated up to the start of the recovery and is assumed to be turbulent all the time a turbulent boundary layer can be assumed to have the followig effective length:
:<math>x' = \int_0^{x^{turb}_{rec}} \left( \frac{U}{U_{max}} \right) ^ 3 dx + (x - x^{turb}_{rec})</math>
Note that this is only valid if the approaching boundary layer can be assumed to be fully turbulent. If the boundary layer is laminar, or undergoes transition, a different approximations needs to be done.

Stratford's separation criteria is known to be conservative. It will most likely predict a bit too early separation.

Stratford's separation criteria

2008-02-13T18:20:21Z

Jola: Stratford's separation criteria moved to Stratford's separation criterion: Incorrect title

#REDIRECT [[Stratford's separation criterion]]

Stratford's separation criterion

2008-02-13T18:20:21Z

Jola: Stratford's separation criteria moved to Stratford's separation criterion: Incorrect title

Stratford's separation criteria is an old classical analytical way to assess if a turbulent boundary layer is likely to separate or not. Stratford's criteria says that from the start of the pressure recovery where the max velocity and the minimum static pressure is obtained the boundary layer is on the verge of separation when:

<math>
C'_p \cdot \sqrt{x' \frac{dC'_p}{dx}} = k \cdot \left( \frac{Re}{10^6} \right) ^ {0.1}
</math>

This formula is only valid as long as <math>C'_p < \frac{4}{7}</math>.

*<math>C'_p</math> is the canonical pressure distribution defined by:

:<math>C'_p = 1 - \left( \frac{U}{U_{max}} \right) ^ 2</math>
:<math>U</math> is the local velocity and <math>U_{max}</math> is the maximum velocity at the start of the pressure recovery.

*<math>k</math> is a constant which Stratford used the following values for:
:<math>
k =
\begin{cases}
0.35 & \mbox{if } \frac{d^2p}{dx^2} \le 0 \mbox{ (concave recovery)} \\
0.39 & \mbox{if } \frac{d^2p}{dx^2} > 0 \mbox{ (convex recovery)}
\end{cases}
</math>

*<math>x'</math> is the effective length of the boundary layer. Note that computing <math>x'</math> can be a bit tricky. If the boundary layer is first accelerated up to the start of the recovery a turbulent boundary layer can be assumed to have the followig effective length:

Stratford's separation criteria is known to be conservative. It will most likely predict a bit too early separation.

Stratford's separation criterion

2008-02-13T12:15:43Z

Jola: still not finished

Stratford's separation criteria is an old classical analytical way to assess if a turbulent boundary layer is likely to separate or not. Stratford's criteria says that from the start of the pressure recovery where the max velocity and the minimum static pressure is obtained the boundary layer is on the verge of separation when:

<math>
C'_p \cdot \sqrt{x' \frac{dC'_p}{dx}} = k \cdot \left( \frac{Re}{10^6} \right) ^ {0.1}
</math>

This formula is only valid as long as <math>C'_p < \frac{4}{7}</math>.

*<math>C'_p</math> is the canonical pressure distribution defined by:

:<math>C'_p = 1 - \left( \frac{U}{U_{max}} \right) ^ 2</math>
:<math>U</math> is the local velocity and <math>U_{max}</math> is the maximum velocity at the start of the pressure recovery.

*<math>k</math> is a constant which Stratford used the following values for:
:<math>
k =
\begin{cases}
0.35 & \mbox{if } \frac{d^2p}{dx^2} \le 0 \mbox{ (concave recovery)} \\
0.39 & \mbox{if } \frac{d^2p}{dx^2} > 0 \mbox{ (convex recovery)}
\end{cases}
</math>

*<math>x'</math> is the effective length of the boundary layer. Note that computing <math>x'</math> can be a bit tricky. If the boundary layer is first accelerated up to the start of the recovery a turbulent boundary layer can be assumed to have the followig effective length:

Stratford's separation criteria is known to be conservative. It will most likely predict a bit too early separation.

Stratford's separation criterion

2008-02-13T08:42:28Z

Jola: not finished yet

Stratford's separation criteria is an old classical analytical way to assess if a boundary layer is likely to separate or not. Stratford's criteria says that from the start of the pressure recovery (max velocity and minimum static pressure) the boundary layer is on the verge of separation when:

<math>
C'_p \cdot \sqrt{x' \frac{dC'_p}{dx}} = k \cdot \left( \frac{Re}{10^6} \right) ^ {0.1}
</math>

Where

<math>C'_p</math> is the canonical pressure distribution defined by:

:<math>C'_p = 1 - \left( \frac{U}{U_{max}} \right) ^ 2</math>
:<math>U</math> is the local velocity and <math>U_{max}</math> is the maximum velocity at the start of the pressure recovery.

<math>x'</math> is the effective length of the boundary layer. Note that computing <math>x'</math> can be a bit tricky. If the boundary layer is first accelerated up to the start of the recovery a turbulent boundary layer can be assumed to have the followig effective length:

Main Page

2008-02-09T10:46:33Z

Jola: Reverted edits by Deltrix (Talk); changed back to last version by Aawasthi

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The Baldwin-Lomax model is a classical algebraic turbulence model which is suitable for high-speed flows with thin attached boundary-layers, typically present in aerospace and turbomachinery applications. The Baldwin-Lomax model is not suitable for cases with large separated regions and significant curvature/rotation effects...
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GridPro

2008-01-27T16:22:01Z

Jola: Removed some advertising and made a few small other changes

GridPro is a structured multiblock meshing tool.

== Supported mesh types ==

GridPro is currently capable of generating structured multiblock meshes. However, the mesh can be exported also as an unstructured hexahedral mesh.

== Geometry modeling and import ==

GridPro primarily imports geometries in faceted format. GridPro has tied up with [http://www.transcendata.com/products/cadfix/index.htm Transcendata] to import other CAD formats also.

== CFD solver interfaces ==
== Topology input language ==
== External links ==