https://www.cfd-online.com/W/index.php?title=Special:Contributions/Malkavian_GT&feed=atom&limit=50&target=Malkavian_GT&year=&month=CFD-Wiki - User contributions [en]2017-11-18T14:16:11ZFrom CFD-WikiMediaWiki 1.16.5https://www.cfd-online.com/Wiki/Introduction_to_turbulence/Reynolds_averaged_equationsIntroduction to turbulence/Reynolds averaged equations2011-04-15T14:11:01Z<p>Malkavian GT: /* Equations for the average velocity */</p>
<hr />
<div>{{Introduction to turbulence menu}}<br />
<br />
== Equations governing instantaneous fluid motion ==<br />
<br />
All fluid motions, whether turbulent or not, are governed by the dynamical equations for a fluid. These can be written using Cartesian tensor notation as:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\rho\left[\frac{\partial \tilde{u}_i}{\partial t}+\tilde{u}_j\frac{\partial \tilde{u}_i}{\partial x_j}\right] = -\frac{\partial \tilde{p}}{\partial x_i}+\frac{\partial \tilde{T}_{ij}^{(v)}}{\partial x_j}</math><br />
</td><td width="5%">(1)</td></tr></table><br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\left[\frac{\partial \tilde{\rho}}{\partial t}+\tilde{u}_j\frac{\partial \tilde{\rho}}{\partial x_j}\right]+ \tilde{\rho}\frac{\partial \tilde{u}_j}{\partial x_j}= 0 </math><br />
</td><td width="5%">(2)</td></tr></table><br />
<br />
where <math>\tilde{u_i}(\vec{x},t)</math> represents the i-the component of the fluid velocity at a point in space,<math>[\vec{x}]_i=x_i</math>, and time,t. Also <br />
<math>\tilde{p}(\vec{x},t)</math> represents the static pressure, <math>\tilde{T}_{ij}^{(v)}(\vec{x},t)</math>, the viscous(or deviatoric) stresses, and <math>\tilde\rho</math> the fluid density. The tilde over the symbol indicates that an instantaneous quantity is being considered. Also the [[einstein summation convention]] has been employed.<br />
<br />
In equation 1, the subscript <math>i</math> is a free index which can take on the values 1,2 and 3. Thus equation 1 is in reality three separate equations. These three equations are just Newton's second law written for a continuum in a spatial(or Eulerian) reference frame. Together they relate the rate of change of momentum per unit mass <math>(\rho{u_i})</math>,a vector quantity, to the contact and body forces.<br />
<br />
Equation 2 is the equation for mass conservation in the absence of sources(or sinks) of mass. Almost all flows considered in this material will be incompressible, which implies that derivative of the density following the fluid material[the term in brackets] is zero. Thus for incompressible flows, the mass conservation equation reduces to:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\frac{D \tilde{\rho}}{Dt}=\frac{\partial \tilde{\rho}}{\partial t}+\tilde{u}_j\frac{\partial \tilde{\rho}}{\partial x_j}= 0</math><br />
</td><td width="5%">(3)</td></tr></table><br />
<br />
From equation 2 it follows that for incompressible flows,<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\frac{\partial \tilde{u}_j}{\partial x_j}= 0</math><br />
</td><td width="5%">(4)</td></tr></table><br />
<br />
The viscous stresses(the stress minus the mean normal stress) are represented by the tensor<math>\tilde{T}_{ij}^{(v)}</math>. From its definition,<math>\tilde{T}_{kk}^{(v)}</math>=0. In many flows of interest, the fluid behaves as a Newtonian fluid in which the viscous stress can be related to the fluid motion by a constitutive relation of the form.<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math>\tilde{T}_{ij}^{(v)}= 2\mu[\tilde{s}_{ij}-\frac{1}{3}\tilde{s}_{kk}\delta_{ij}] </math><br />
</td><td width="5%">(5)</td></tr></table><br />
<br />
The viscosity, <math>\mu</math>, is a property of the fluid that can be measured in an independent experiment. <math>\tilde s_{ij}</math> is the instantaneous strain rate tensor defined by<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math>\tilde{s}_{ij}= \frac{1}{2}\left[\frac{\partial \tilde u_i}{\partial x_j}+\frac{\partial \tilde u_j}{\partial x_i}\right] </math><br />
</td><td width="5%">(6)</td></tr></table><br />
<br />
From its definition, <math>\tilde s_{kk}=\frac{\partial \tilde u_k}{\partial x_k}</math>. If the flow is incompressible, <math>\tilde s_{kk}=0</math> and the Newtonian constitutive equation reduces to<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math>\tilde{T}_{ij}^{(v)}= 2\mu\tilde{s}_{ij}</math><br />
</td><td width="5%">(7)</td></tr></table><br />
<br />
Throughout this material, unless explicitly stated otherwise, the density <math>\tilde\rho=\rho</math> and the viscosity <math>\mu</math> will be assumed constant. With these assumptions, the instantaneous momentum equations for a Newtonian Fluid reduce to:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\left[\frac{\partial \tilde{u}_i}{\partial t}+\tilde{u}_j\frac{\partial \tilde{u}_i}{\partial x_j}\right] = -\frac {1}{\tilde\rho}\frac{\partial \tilde{p}}{\partial x_i}+\nu\frac{\partial^2 {\tilde{u}_i}}{\partial x_j^2}</math><br />
</td><td width="5%">(8)</td></tr></table><br />
<br />
where the kinematic viscosity, <math>\nu</math>, has been defined as:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math>\nu\equiv\frac{\mu}{\rho}</math><br />
</td><td width="5%">(9)</td></tr></table><br />
<br />
Note that since the density is assumed constant, the tilde is no longer necessary.<br />
<br />
Sometimes it will be more instructive and convenient to not explicitly include incompressibilty in the stress term, but to refer to the incompressible momentum equation in the following form:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\rho\left[\frac{\partial \tilde{u}_i}{\partial t}+\tilde{u}_j\frac{\partial \tilde{u}_i}{\partial x_j}\right] = -\frac{\partial \tilde{p}}{\partial x_i}+\frac{\partial \tilde{T}_{ij}^{(v)}}{\partial x_j}</math><br />
</td><td width="5%">(10)</td></tr></table><br />
<br />
This form has the advantage that it is easier to keep track of the exact role of the viscous stresses.<br />
<br />
== Equations for the average velocity ==<br />
<br />
Although laminar solutions to the equations often exist that are consistent with the boundary conditions, perturbations to these solutions(sometimes even infinitesimal) can cause them to become turbulent. To see how this can happen, it is convenient to analyze the flow in two parts, a mean(or average) component and a fluctuating component. Thus the instantaneous velocity and stresses can be written as:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\tilde {u}_i=U_i+u_i<br />
</math><br />
:<math><br />
\tilde p=P+p<br />
</math><br />
:<math><br />
\tilde T_{ij}^{(v)}=T_{ij}^{(v)}+\tau_{ij}^{(v)}<br />
</math><br />
</td><td width="5%">(11)</td></tr><br />
</table><br />
<br />
Where <math>U_i</math>, <math>P</math> and <math>T_{ij}^{(v)}</math> represent the mean motion, and <math>u_i</math>, <math>p</math> and <math>\tau_{ij}^{(v)}</math> the fluctuating motions. This technique for decomposing the instantaneous motion is referred to as the '''''Reynolds decomposition.''''' Note that if the averages are defined as ensemble means, they are, in general, time-dependent. For the remainder of this material unless other wise stated, the density will be assumed constant so<math>\tilde{\rho}\equiv\rho</math>,and its fluctuation is zero.<br />
<br />
Substitution of equations 11 into equations 10 yields<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\rho\left[\frac{\partial (U_i+u_i)}{\partial t}+(U_j+u_j)\frac{\partial (U_i+u_i)}{\partial x_j}\right] = -\frac{\partial (P+p)}{\partial x_i}+\frac{\partial (T_{ij}^{(v)}+\tau_{ij}^{(v)})}{\partial x_j}</math><br />
</td><td width="5%">(12)</td></tr></table><br />
<br />
This equation can now be averaged to yield an equation expressing momentum conservation for the averaged motion. Note that the operations of averaging and differentiation commute; i.e., the average of a derivative is the same as the derivative of the average. Also the average of a fluctuating quantity is zero. Thus the equation for the averaged motion reduces to:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\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_{ij}^{(v)}}{\partial x_j}-\rho\left \langle u_j\frac{\partial u_i }{\partial x_j} \right \rangle</math><br />
</td><td width="5%">(13)</td></tr></table><br />
<br />
where the remaining fluctuating product term has been moved to the right hand side of the equation. Whether or not the last term is zero like the other fluctuating term depends on the correlation of the terms in the product. In general, these correlations are not zero.<br />
<br />
The mass conservation equation can be similarly decomposed. In incompressible form, substitution of equations 11 into equation 4 yields:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\frac{\partial (U_j+u_j)}{\partial x_j}=0</math><br />
</td><td width="5%">(14)</td></tr></table><br />
<br />
of which average is <br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\frac{\partial U_j}{\partial x_j}=0</math><br />
</td><td width="5%">(15)</td></tr></table><br />
<br />
It is clear from equation 15 that the averaged motion satisfies the same form of the mass conservation equation as does the instantaneous motion at least for incompressible flows. How much simpler the turbulence problem would be if the same were true for the momentum! Unfortunately, as is easily seen from equation 13, such is not the case.<br />
<br />
Equation 15 can be subtracted from equation 14 to yield an equation for instantaneous motion alone; i.e,<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\frac{\partial u_j}{\partial x_j}=0</math><br />
</td><td width="5%">(16)</td></tr></table><br />
<br />
Again, like the mean, the form of the original instantaneous equation is seen to be preserved. The reason, of course, is obvious: the continuity equation is linear. The momentum equation , on the other hand, is not; hence the difference.<br />
<br />
Equation 16 can be used to rewrite the last term in equation 13 for the mean momentum. Multiplying equation 16 by <math>u_i</math> yields:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\left \langle u_i\frac{\partial u_j}{\partial x_j}\right \rangle=0</math><br />
</td><td width="5%">(17)</td></tr></table><br />
<br />
Now add:<math>\left \langle u_j\frac{\partial u_i}{\partial x_j}\right \rangle</math> to both sides of eqaution 17, then using the product rule you obtain:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\left \langle u_j\frac{\partial u_i}{\partial x_j}\right \rangle +0=\left \langle u_j\frac{\partial u_i}{\partial x_j}\right \rangle+ \left \langle u_i\frac{\partial u_j}{\partial x_j}\right \rangle =\frac{ \partial}{\partial x_j}{\left \langle u_iu_j\right \rangle} </math><br />
</td><td width="5%">(18)</td></tr></table><br />
<br />
Where again the fact that arithmetic and averaging operations commute has been used.<br />
<br />
The equation for the averaged momentum, equation 13 can now be rewritten as:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\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_{ij}^{(v)}}{\partial x_j}-\frac{ \partial}{\partial x_j}{(\rho\left \langle u_iu_j\right \rangle)}</math><br />
</td><td width="5%">(19)</td></tr></table><br />
<br />
The last two terms on the right hand side are both divergence terms and can be combined; the result is:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\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 }{\partial x_j}[T_{ij}^{(v)}-{\rho\left \langle u_iu_j\right \rangle}]</math><br />
</td><td width="5%">(20)</td></tr></table><br />
<br />
Now the terms in square brackets on the right have the dimensions of stress. The first term is, in fact , the viscous stress. The second term, on the other hand, is not a stress at all, but simply a re-worked version of the fluctuating contribution to the non-linear acceleration terms. The fact that it can be written this way, however, indicates that at least as far as the motion is concerned, it acts as though it were a stress- hence its name, the '''Reynolds stress'''. In the succeeding sections the consequences of this difference will be examined.<br />
<br />
== The turbulence problem ==<br />
<br />
It is the appearance of the Reynolds stress which makes the turbulence problem so difficult - at least from the engineers perspective. Even though we can pretend it is a stress, the physics which give rise to it are very different from the viscous stress. The viscous stress can be related directly to the other flow properties by constitutive equations, which in turn depend only on the properties of the fluid (as in equation 5 for a Newtonian fluid). The reason this works is that when we make such closure approximations for a fluid, we are averaging over characteristic length and time scales much smaller than those of the flows we are interested in. Yet at the same time, these scales are much larger than the molecular length and time scales which characterize the molecular interactions that are actually causing the momentum transfer. (This is what the continuum approximation is all about).<br />
<br />
The '''''Reynolds stress''''', on the other hand, arises from the flow itself! Worse, the scales of the fluctuating motion which give rise to it are the scales we are interested in. This means that the closure ideas which worked so well for the viscous stress, should not be expected to work too well for the Reynolds stress. And as we shall see, they do not.<br />
<br />
This leaves us in a terrible position. Physics and engineering are all about writing equations(and boundary conditions) so we can solve them to make predictions. We don't want to have a build prototype airplanes first to see if they will they fall out of the sky. Instead we want to be able to analyze our designs before building, to save the cost in money and lives if our ideas are wrong. The same is true for dams and bridges and tunnels and automobiles. If we had confidence in our turbulence models, we could even build huge one-offs and expect them to work the first time. Unfortunately, even though turbulence models have improved to the point where we can use them in design, we still cannot trust them enough to eliminate expensive wind tunnel and model studies. And recent history is full of examples to prove this.<br />
<br />
The turbulence problem (from the engineers perspective) is then three-fold:<br />
<br />
* '''The averaged equations are not closed.''' Count the number of unknowns in equation 20 above. Then count the number of equations. Even with the continuity equation we have atleast six equations too few.<br />
<br />
* '''The simple ideas to provide the extra equations usually do not work.''' And even when we can fix them up for a particular class of flows (like the flow in a pipe, for example), they will most likely not be able to predict what happens in even a slightly different environment (like a bend).<br />
<br />
*'''Even the last resort of compiling engineering tables for design handbooks carries substantial risk.''' This is the last resort for the engineer who lacks equations or cannot trust them. Even when based on a wealth of experience, they require expensive model testing to see if they can be extrapolated to a particular situation. Often they cannot, so infinitely clever is Mother Nature in creating turbulence that is unique to a particular set of boundary conditions.<br />
<br />
'''Turbulent flows are indeed flows!'''. And that is the problem.<br />
<br />
== Origins of turbulence==<br />
<br />
Turbulent flows can often be observed to arise from laminar flows as the Reynolds number, (or someother relevant parameter) is increased. This happens because small disturbances to the flow are no longer damped by the flow, but begin to grow by taking energy from the original laminar flow. This natural process is easily visualized by watching the simple stream of water from a faucet (or even a pitcher). Turn the flow on very slow (or pour) so the stream is very smooth initially, at least near the outlet. Now slowly open the faucet (or pour faster) abd observe what happens, first far away, then closer to the spout. The surface begins to exhibit waves or ripples which appear to grow downstream . In fact, they are growing by extracting energy from the primary flow. Eventually they grow enough that the flow breaks into drops. These are capillary instabilities arisiing from surface tension, but regardless of the type of instability, the idea is the same -small (or infinitesimal ) disturbances have grown to disrupt the serenity (and simplicity) of laminar flow.<br />
<br />
The manner in which the instabilities grow naturally in a flow can be examined using the equations we have already developed above. We derived them by decomposing the motion into a mean and fluctuating part. But suppose instead we had decomposed the motion into a base flow part (the initial laminar part) and into a disturbance which represents a fluctuating part superimposed on the base flow. The result of substituting such a decomposition into the full Navier-Stokes equations and averaging is precisely that given by equations (13) and (15). But the very important difference is the additional restriction that what was previously identified as the mean (or averaged ) motion is now also the base or laminar flow.<br />
<br />
Now if the base flow is really laminar flow (which it must be by our original hypothesis), then our averaged equations governing the base flow must yield the same mean flow as the original laminar flow on which the disturbances was superimposed. But this can happen only if these new averaged equations reduce to '''exactly''' the same lamiane flow equations without any evidence of a disturbance. Clearly from equations 13 and 15, this can happen ''only if all the Reynolds stress terms vanish identically!'' Obviously this requires that the disturbances be infintesimal so the extra terms can be neglected - hence our interest in infinitesimal disturbances.<br />
<br />
So we hypothesized a base flow which was laminar and showed that it is unchanged even with the imposition of infintesimal disturbances on it - ''but only as long as the disturbances'' '''remain''' ''infinitesimal!'' What happens if the disturbance starts to grow? Obviously before we conclude that all laminar flows are laminar forever we better investigate whether or not these infinitesimal disturbances can grow to ''finite'' size. To do this we need an equation for the fluctuation itself.<br />
<br />
An equation for the fluctuation (which might be an imposed disturbance) can be obtained by subtracting the equation for the mean (or base) flow from that for the instantaneous motion. We already did this for the continuity equation. Now we will do it for the momentum equation. Subtracting equation 13 from equation 11 yields an equation for the fluctuation as:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\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\tau^{(v)}_{ij}}{\partial x_{j}} - \rho \left[ u_{j}\frac{\partial U_{i}}{\partial x_{j}} \right] - \left\{ u_{j} \frac{\partial u_{i}}{ \partial x_{j}} - \rho \left\langle u_{j} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right\}<br />
</math><br />
</td><td width="5%">(21)</td></tr></table><br />
<br />
It is very important to note the type and character of the terms in this equation. First note that the left-hand side is the derivative of the ''fluctuating'' velocity following the ''mean'' motion. This is exactly like the term which appears on the left-hand side of the equation for the mean velocity, equation 13. The first two terms on the right-hand side are also like those in the mean motion, and represent the fluctuating pressure gradient and the fluctuating viscous stresses. The third term on the right-hand is is new, and will be seen later to represent the primary means by which fluctuations (and turbulence as well!) extract energy from the mean flow, the so-called ''production terms''. The last term is quadratic in the fluctuating velocity, unlike all the otherwhich are linear. Note that all of the terms vanish identically if the equation is averaged, the last because its mean is subtructed from it.<br />
<br />
Now we want to examine what happens if the disturbance is small. In the limit as the amplitude of the disturbance (or fluctuation) is ''infinitesmal'', the bracketed term in the equationfor the fluctuation vanishes (since it involves productsof infinitesimals) , and the remaining equation is ''linear in the disturbance''. The study of whether or not such infinitesmal disturbances can grow is called '''Linear Fluid Dynamic Stability Theory'''. These linearized equations are very different from those govering turbulence. Unlike the equations for disturbances of ''finite'' amplitude, the linearized equations are well-posed (or closed) since the Reynolds stress terms are gone.<br />
<br />
The absence of the non-linear terms, however, constrains the validity of the linear analysis to only the initial stage of disturbance growth. This is because as soon as the fluctuations begin to grow, their amplitudes can no longer be assumed infinitesmal and the Reynolds stress (or more properly, the non-linear fluctuating terms) become important. As a result the base flow equations begin to be modified so that the solution to them can no longer be identical to the laminar flow (or base flow) from which it arose. Thus while linear stability theory can predict ''when'' many flows become ''unstable'', it can say very little about ''transition to turbulence'' since this progress is highly non-linear.<br />
<br />
== Importance of non-linearity ==<br />
<br />
We saw in the preceding section that non-linearity was one of essential features of turbulence. When small disturbances grow large enough to interact ''with each other'', we enter a whole new world of complex behavior. Most of the rules we learned for linear system do not apply. Since most of your mathematical training has been for linear equations, most of your mathematical intuition therefore will not apply either. On the other hand, you may surprise yourself by discovering how much your ''non-mathematical'' intuition already recognizes non-linear behavior and accounts for it.<br />
<br />
Considering the following simple example. Take a long stick with one person holding each end and stand at the corner of a building. Now place the middle of against the building and let each person apply pressure in the same direction so as to bend the stick. If the applied force is small, the stick deflects (or bends) a small amount. Double the force, and the deflection is approximately doubled. Quadruple the force and the deflection is quadrupled. Now you don't need a Ph.D. in Engineering to know what is going to happen if you continue this process. '''The stick is going to break!'''<br />
<br />
But where in the equations for the deflection of the stick is there anything that predicts this can happen? Now if you are only like engineer, you are probably thinking: he's asking a stupid question. Of course you can't continue to increase the force because you will exceed first the yield stress, then the breaking limit, and of course the stick will break.<br />
<br />
But pretend I am the company president with nothing more than MBA. I don't know much about these things, but you have told me in the past that your computers have equations to predict everything. So I repeat: Where in the equations for the deflection of this stick does it tell me this going to happen?<br />
<br />
The answer is very simple: '''There is''' ''nothing'' '''in the equations that will predict this.''' And the reason is also quite simple: You lost the ability to predict catasrophes like breaking when you linearized the fundamental equations - which started out as Newton's Law too. In fact, before linearization, they were exactly the same as those for a fluid, only the constitutive equation was different.<br />
<br />
If we had NOT linearized these equations and had constituve equations that were more general, then we possibly could apply these equation right to and past the limit. The point of fracture would be a bifurcation point for the solution.<br />
<br />
Now the good news is that for things like reasonable deflections of beams linearization work woderfully since we hope most things we build don't deflect too much. Unfortunately, as we noted above, for fluids the disturbances tend to quickly become dominated by non-linear terms. This, of course, means our linear analytical techniques are pretty useless for fluid mechanics, and especially turbulence.<br />
<br />
But all is not lost. Just as we learned to train ourselves to anticipate when sticks break, we have to train ourselves to anticipate how non-linear fluid phenomena behave. Toward that end we will consider two simple examples: one from algebra - the logistic map, and one from fluid mechanics - simple vortex streching.<br />
<br />
'''Example 1:''' An experiment with the logistic map. <br />
<br />
Consider the behavior of the simple equation:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
y_{n+1}= r y_{n} \left(1- y_{n} \right)<br />
</math><br />
</td><td width="5%">(22)</td></tr></table><br />
<br />
where <math> n= 1,2..., 0<y < 1 </math> and <math> r > 0 </math>. The idea is that you pick any value for <math> y_{1}</math>, use the equation to find <math> y_{2}</math>, then insert that value on the right-hand side to find <math> y_{3}</math>, and just continue the process as long as you like. Make sure you note any dependece of the final result on the initial value for <math> y </math>.<br />
<br />
*First notice what happens if you linearize this equation by disregarding the term in parentheses; i.e., comsider the simpler equation <math> y_{1+1} = r y_{n}</math>. My guess is that you won't find this too eciting - unless, of course, you are one of those rare individuals who likes watching grass grow.<br />
<br />
* Now consider the full equation and note what happens for <math> r<3 </math> , and especially what happens for very small values of <math> r </math>. Run as many iterations as necessary to make sure your answer has converged. Do NOT try to take short-cuts by programming all the steps at once. Do them one at time so you can see what is happening. It will be much easier this way in the long run.<br />
<br />
* Now research carefully what happens when <math> r=3.1, 3.5, </math> and <math> 3.8 </math>. Can you recognize any patterns.<br />
<br />
* Vary <math> r </math> between 3 and 4 to see if you can find the boundaries for what you are observing. <br />
<br />
* Now try values of <math> r>4 </math>. How do you explain this<br />
<br />
'''Example 2:''' Stretching of a simple vortex.<br />
<br />
Imagine a simple vortex filament that looks about like a strand of spaghetti. Now suppose it is in otherwise steady inviscid incompressible flow. Use the vorticity equation to examine the following: <br />
<br />
* Examine first what happens to it in two-dimensional velocity field. Note particularly whether any new vorticity can be produced; i.e., can the material derivative of the vorticity ever be greater than zero? (Hint: look at the <math> \omega_{j} \partial u_{i}/ \partial x_{j} <br />
</math> - term)<br />
<br />
* Now consider the same vortex filament in a three-dimensional flow. Note particularly the various ways new vorticity can be produced - if you have some to start with! Does all this have anything to do with non-linearities?<br />
<br />
== Turbulence closure problem and eddy viscosity ==<br />
<br />
From the point of view of the averaged motion, at least, the problem with the non-linearity of the instaneous equations is that they introduce new unknowns, the Reynolds stress into the averaged equations. There are six individual stress components we must deal with to be exact: <math> \left\langle u^{2}_{1} \right\rangle </math> , <math> \left\langle u^{2}_{2} \right\rangle </math> , <math> \left\langle u^{2}_{3} \right\rangle </math> , <math> \left\langle u_{1}u_{2} \right\rangle </math> , <math> \left\langle u_{1}u_{3} \right\rangle </math> , and <math> \left\langle u_{2}u_{3} \right\rangle </math>. These have to be related to the mean motion itself before the equations can be solved, since the number of unknows and number of equations must be equal. The absence of these additional equations is often reffered to as '''the Turbulence Closure Problem'''.<br />
<br />
A similar problem arose when the instantaneous equations were written (equations 1 2), since relations had to be introduced to relate the stresses (in particular, the viscous stresses) to the motion itself. These relations (or constitutive equations) ''depended only on the properties of the fluid material, and not on the flow itself''. Because of this fact, it is possible to carry out independent experiments, called viscometric experiments, in which these fluid properties can be determinded once and for all. Equation 5 provides an example just such a constitutive relation, the viscosity, <math>\mu </math>, depending only in the choice of the material. For example, once the viscosity of water at given temperature is determined, this value can be used in all flows at that temperature, not just the one in which the evaluation was made.<br />
<br />
It is tempting to try such an approach for the turbulence Reynolds stresses (even though we know the underlying requirements of scale separation are not satisfied). For example, a Newtonian type closure for the Reynolds stresses, often referred to as an "eddy" or "turbulent" viscosity model, looks like:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
- \rho \left\langle u_{i} u_{j} \right\rangle = 2 \mu_{t} \left[ S_{ij} - \frac{1}{3} S_{kk} \delta_{ij} \right]<br />
</math><br />
</td><td width="5%">(23)</td></tr></table><br />
<br />
where <math>\mu_{t} </math> is the turbulence "viscosity" (also called the eddy viscosity) and <math>S_{ij} </math> is the ''mean'' strain rate defined by:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
S_{ij}= \frac{1}{2} \left[ \frac{\partial U_{i}}{\partial x_{j}} + \frac{\partial U_{j}}{\partial x_{i}} \right]<br />
</math><br />
</td><td width="5%">(24)</td></tr></table><br />
<br />
The second term of course, vanishes identically for incompressible flow. For the simple case of a two-dimensional shear flow, equation 3.23 for the Reynolds shear stress reduces to<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
- \rho \left\langle u_{1} u_{2} \right\rangle = 2 \mu_{t} \frac{\partial U_{1}}{\partial x_{2}} <br />
</math><br />
</td><td width="5%">(25)</td></tr></table><br />
<br />
That such a simple model can adequately describe the mean motion in at least one flow is illustrated by the axisymmetric buoyant plume sketched in <font color="orange">Figure 3.1. Figures 3.2 and 3.3</font> show the calculation of the mean velocity and temperature profiles respectively. Obviously the mean velocity and temperature are reasonably accurately computed, as are the Reynolds shear stress and lateral turbulent heat flux shown in <font color ="orange">Figures 3.4 and 3.5</font>.<br />
<br />
The succes of the eddy viscosity in the preceding example is more apparent than real, however, since the value of the eddy viscosity and eddy diffusivity (for turbulent heat flux) have been chosen to give the best possible agreement with the data. This, in itself, would not be a problem if that chosen values could have been obtained in advance of the computation, or even if they could be used to successfully predict other flows. In fact, the values used work only for this flow, thus the computation is ''not a prediction at all, but a'' '''postdiction''' ''or'' '''hindcast''' from which no extrapolation to the future can be made. In other words, our turbulence "model" is about as useful as having a program to predict yesterday's weather. Thus the closure problem still very much remains.<br />
<br />
<font color="orange" size="3">Figure 3.1 not uploaded yet</font><br />
<br />
<font color="orange" size="3">Figure 3.2 not uploaded yet</font><br />
<br />
<font color="orange" size="3">Figure 3.3 not uploaded yet</font><br />
<br />
<font color="orange" size="3">Figure 3.4 not uploaded yet</font><br />
<br />
<font color="orange" size="3">Figure 3.5 not uploaded yet</font><br />
<br />
<font color="orange" size="3">Figure 3.6 not uploaded yet</font><br />
<br />
Another problem with the eddy viscosity in the example above is that it fails to calculate the vertical components of the Reynolds stress and turbulent heat flux. An attempt at such a computation is shown in <font color ="orange">Figure 3.6</font> where the vertical turbulent heat flux is shown to be severely underestimated. Clearly the value of the in the vertical direction must be different than in the radial direction. In other words, the turbulence for which a constituve equation is being written is ''not an isotropic "medium" ''. In fact, in this specific example the problem is that the vertical component of the heat flux is produced more by the interaction of buoyancy and the turbulence, than it is by the working of turbulence against mean gradient in the flow. We will discuss this in more detail in the next chapter when we consider the turbulence energy balances, but note for now that simple gradient closure models never work unless gradient production dominates. This rules out many flows involving buoyancy, and also many involving recirculations or separation where the local turbulence is convected in from somewhere else. <br />
<br />
A more general form of constitutive equation which would allow for the nonisotropic nature of the "medium" (in this case the turbulence itself) would be<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
- \rho \left\langle u_{i} u_{j} \right\rangle = \mu_{ijkl} \left[ S_{kl} - \frac{1}{3} S_{mm} \delta_{kl} \right]<br />
</math><br />
</td><td width="5%">(26)</td></tr></table><br />
<br />
This closure relation allows each component of the Reynolds stress to have its own unique value of the eddy viscosity. It is easy to see that it is unlikely this will solve the closure problem since the original six unknowns the <math>\left\langle u_{i} u_{j} \right\rangle </math> have been traded for eighty-one new ones, <math> \mu_{ijkl} </math>. Even if some can be removed by symmetries, the remaining number is still formidable. More important than the number of unknowns, however, is that there is no independent or general means for selecting them without considering a particular flow. This is because ''turbulence is indeed a property of the flow, not of the fluid''.<br />
<br />
== Reynolds stress equations ==<br />
<br />
It is clear from the preceding section that the simple idea of an eddy viscosity might not be the best way to approach the problem of relating the Reynolds stress to the mean motion. An alternative approach is to try to derive dynamical equations for the Reynolds stresses from the equations governing the fluctuations themselves. Such an approach recognizes that the Reynolds stress is really a functional of the velocity; that is, the stress at a point depends on the velocity everywhere and for all past times, not just at the point in question and at a particular instant in time.<br />
<br />
The analysis begins with the equation for the instantaneous fluctuating velocity, equation 21. This can be rewritten for a Newtonian fluid with constant viscosity as: <br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\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 \tau ^{ ( v ) }_{ij}}{ \partial x_{j}} - \rho \left[ u_{j} \frac{\partial U_{i}}{ \partial x_{j} } \right] - \rho \left\{ u_{j} \frac{\partial u_{i}}{ \partial x_{j} } - \left\langle u_{j} \frac{\partial u_{i} }{\partial x_{j} } \right\rangle \right\}<br />
</math><br />
</td><td width="5%">(27)</td></tr></table><br />
<br />
Note that the free index in this equation is <math> i </math>. Also, since we are talking about turbulence again, the capital letters represent mean or averaged quantities. <br />
<br />
Multiplying equation 27 by <math> u_{k} </math> and averaging yields:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\rho \left[ \left\langle u_{k} \frac{\partial u_{i} }{\partial t} \right\rangle + U_{j} \left\langle u_{k} \frac{\partial u_{i} }{\partial x_{j}} \right\rangle \right] = & - & \left\langle u_{k} \frac{\partial p }{\partial x_{i} } \right\rangle + \left\langle u_{k} \frac{\partial \tau^{(v)}_{ij} }{\partial x_{j}} \right\rangle \\<br />
& - & \rho \left[ \left\langle u_{k}u_{j} \right\rangle \frac{\partial U_{i} }{\partial x_{j} } \right] - \rho \left\{ \left\langle u_{k}u_{j} \frac{\partial u_{i} }{\partial x_{j} } \right\rangle \right\} \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(28)</td></tr></table><br />
<br />
Now since both <math> i </math> and <math> k </math> are free indices they can be interchanged to yield a second equation given by (alternatively equation 21 can be rewritten with free index <math> k </math>, then multiplied by <math> u_{i} </math> and averaged):<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\rho \left[ \left\langle u_{i} \frac{\partial u_{k} }{\partial t} \right\rangle + U_{j} \left\langle u_{i} \frac{\partial u_{k} }{\partial x_{j}} \right\rangle \right] = & - & \left\langle u_{i} \frac{\partial p }{\partial x_{k} } \right\rangle + \left\langle u_{i} \frac{\partial \tau^{(v)}_{kj} }{\partial x_{j}} \right\rangle \\<br />
& - & \rho \left[ \left\langle u_{i}u_{j} \right\rangle \frac{\partial U_{k} }{\partial x_{j} } \right] - \rho \left\{ \left\langle u_{i}u_{j} \frac{\partial u_{k} }{\partial x_{j} } \right\rangle \right\} \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(29)</td></tr></table><br />
<br />
Equations 28 and 29 can be added together to yield an equation for the Reynolds stress<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\frac{\partial\left\langle u_{i}u_{k} \right\rangle}{\partial t} + U_{j}\frac{\partial\left\langle u_{i}u_{k} \right\rangle}{\partial x_{j}} <br />
& = & - \frac{1}{\rho} \left[ \left\langle u_{i} \frac{\partial p}{\partial x_{k}} \right\rangle + \left\langle u_{k} \frac{\partial p}{\partial x_{i}} \right\rangle \right] \\<br />
& - & \left[ \left\langle u_{i}u_{j} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle u_{k}u_{j} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right] \\<br />
& + & \frac{1}{\rho} \left[ \left\langle u_{i} \frac{\partial \tau^{(v)}_{kj} }{\partial x_{j}} \right\rangle + \left\langle u_{k} \frac{\partial \tau^{(v)}_{ij} }{\partial x_{j}} \right\rangle \right] \\<br />
& - & \left[ \left\langle u_{i} u_{j} \right\rangle \frac{\partial U_{k}}{\partial x_{j}} + \left\langle u_{k} u_{j} \right\rangle \frac{\partial U_{i}}{\partial x_{j}} \right] \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(30)</td></tr></table><br />
<br />
It is customary to rearrange the first term on the right hand side in the following way:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\left[ \left\langle u_{i} \frac{\partial p}{\partial x_{k}} \right\rangle + \left\langle u_{k} \frac{\partial p}{\partial x_{i}} \right\rangle \right] = & & \left\langle p \left[ \frac{\partial u_{i}}{\partial x_{k}} + \frac{\partial u_{k}}{\partial x_{i}} \right] \right\rangle \\<br />
& - & \frac{\partial}{\partial x_{j}} \left[ \left\langle pu_{i} \right\rangle \delta_{kj} + \left\langle pu_{k} \right\rangle \delta_{ij} \right] \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(31)</td></tr></table><br />
<br />
The first term on the right is generally referred to as the ''pressure strain-rate'' term. The second term is written as a divergence term, and generally referred to as the ''pressure diffusion'' term. We shal see later that divergence terms can never create nor destroy anything; they can simple move it around from one place to another. <br />
<br />
The third term on the right-hand side of equation 30 can similarly be rewritten as:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\left[ \left\langle u_{i} \frac{\partial \tau^{(v)}_{kj}}{\partial x_{j}} \right\rangle + \left\langle u_{k} \frac{\partial \tau^{(v)}_{ij}}{\partial x_{j}} \right\rangle \right] = & - & \left[ \left\langle \tau^{(v)}_{ij} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle \tau^{(v)}_{kj} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right] \\<br />
& + & \frac{\partial}{\partial x_{j}} \left[ \left\langle u_{i} \tau^{(v)}_{kj} \right\rangle + \left\langle u_{k} \tau^{(v)}_{ij} \right\rangle \right] \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(32)</td></tr></table><br />
<br />
The first of these is also a divergence term. For a Newtonian fluid, the last is the so-called "dissipation of Reynolds stress" by the turbulence viscous stresses. This is easily seen by substituting the Newtonian constitutive relation to obtain:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left[ \left\langle \tau^{(v)}_{ij} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle \tau^{(v)}_{kj} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right] = 2 \nu \left[ \left\langle s_{ij} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle s_{kj} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right]<br />
</math><br />
</td><td width="5%">(33)</td></tr></table><br />
<br />
It is not at all obvious what this has to do with dissipation, but it will become clear later on when we consider the trace of the Reynolds stress equation, which is the ''kinetic energy'' equation for the turbulence.<br />
<br />
Now if we use the same trick from before using the continuity equation, we can rewrite the third term in equation 30 to obtain:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left[ \left\langle u_{i}u_{j} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle u_{k}u_{j} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right] = \frac{\partial}{\partial x_{j}} \left\langle u_{i} u_{k} u_{j} \right\rangle<br />
</math><br />
</td><td width="5%">(34)</td></tr></table><br />
<br />
This is also a divergence term.<br />
<br />
We can use all of the pieces we have developed above to rewrite equation 30 as <br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\frac{\partial}{\partial t} \left\langle u_{i}u_{k} \right\rangle & + & U_{j} \frac{\partial}{\partial x_{j}} \left\langle u_{i}u_{k} \right\rangle = - \left\langle \frac{p}{\rho} \left[ \frac{\partial u_{i}}{\partial x_{k}} + \frac{\partial u_{i}}{\partial x_{k}} \right] \right\rangle \\<br />
& + & \frac{\partial}{\partial x_{j}} \left\{ - \left[ \left\langle pu_{k} \right\rangle \delta_{ij} + \left\langle pu_{i} \right\rangle \delta_{kj} \right] - \left\langle u_{i}u_{k}u_{j} \right\rangle + 2\nu \left[ \left\langle s_{ij}u_{k} \right\rangle + \left\langle s_{ij}u_{k} \right\rangle \right] \right\}\\<br />
& - & \left[ \left\langle u_{i} u_{j} \right\rangle \frac{\partial U_{k}}{\partial x_{j}} + \left\langle u_{k} u_{j} \right\rangle \frac{\partial U_{i}}{\partial x_{j}} \right] \\<br />
& - & 2\nu \left[ \left\langle s_{ij} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle s_{kj} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right] \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(35)</td></tr></table><br />
<br />
This is the so-called '''Reynolds Stress Equation''' which has been the primary vehicle for much of the turbulence modelling efforts of the past few decades.<br />
<br />
The left hand side of the Reynolds Stress Equation can easily be recognized as the rate of change of the Reynolds stress following the mean motion. It seems to provide exactly what we need: nine new equations for the nine unknowns we can not account for. The problems are all on the right-hand side. These terms are referred to respectively as<br />
<br />
* the pressure-strain rate term<br />
* the turbulence transport (or divergence) term<br />
* the "production" term, and<br />
* the "dissipation" term.<br />
<br />
Obviously these equations do not involve only <math> U_{i} </math> and <math> \left\langle u_{i} u_{j} \right\rangle </math>, but depend on many more new unknowns.<br />
<br />
It is clear that, contrary to our hopes, we have not derived a single equation relating the Reynolds stress to the mean motion. Instead, our Reynolds stress transport transport equation is exceedingly complex. Whereas the process of averaging the equation for the mean motion introduced only six new, independent unknowns, the Reynolds stress, <math> \left\langle u_{i} u_{j} \right\rangle </math>, the search for a transport equation which will relate these to the mean motion has produced many more unknowns. They are<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left\langle p u_{i} \right\rangle - 3<br />
</math><br />
</td><td width="5%">(36)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left\langle u_{i} s_{jk} \right\rangle - 27<br />
</math><br />
</td><td width="5%">(37)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left\langle s_{ij}s_{jk} \right\rangle - 9<br />
</math><br />
</td><td width="5%">(38)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left\langle u_{i}u_{k}u_{j} \right\rangle - 27<br />
</math><br />
</td><td width="5%">(39)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left\langle p\frac{\partial u_{i}}{\partial x_{j}} \right\rangle - 9<br />
</math><br />
</td><td width="5%">(40)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
TOTAL - 75<br />
</math><br />
</td><td width="5%">(41)</td></tr></table><br />
<br />
Not at all of these are independent, since some can be derived from others. Even so, our goal of reducing the number of unknowns has clearly not been met.<br />
<br />
Equations governing each of these new quantities can be derived from the original dynamical equations, just as we did for the Reynolds stress. Unfortunately new quantities continue to be introduced with each new equation, and at a faster rate than the icrease in the number of equations. Now the full implications of the closure problem introduced by the Reynolds decomposition and averaging has become apparent. No matter how many new equations are derived, the number of new unknown quantities introduced will always increase more rapidly.<br />
<br />
Our attempt to solve the turbulence problem by considering averages illustrates a general principle. Any time we try to full Mother Nature by averaging out her details, she gets her revenge by leaving us with closure problem - more equations than unknowns. In thermodynamics, we tried to simplify the consideration of molecules by averaging over them, and were left with the need for an equation of state. In heat transfer, we tried to simplify considerations by which molecules transfer their kinetic energy, and found we were lacking a relation between the heat flux and the temperature field. And in fluid mechanics, we tried to simplify consideration of the "mean" motionof molecules and ended up with viscous stress. In all of these cases we were able to make simple physical models which worked at least some of the time; e.g., ideal gas, Fourier-Newtonian fluid. And these models all worked because we were able to make assumptions about the underlying molecular processes and assume them to be independent of the macroscopic flows of interest. Unfortunately such assumptions are rarely satisfied in turbulence. <br />
<br />
It should be obvious by now that the turbulence closure problem will not be solved by the straight-forward derivation of new equations, nor by direct analogy with viscous stresses. Rather, ''closure attempts will have to depend on an intimate knowledge of the dynamics of the turbulence itself''. Only by understanding how the turbulence behaves can one hope to ''guess'' an appropriate set of constitutive equations '''AND''' ''understand the limits of them''. This is, of course, another consequence of the fact that the ''turbulence is a property of the flow itself, and not of the fluid''!<br />
<br />
{{Turbulence credit wkgeorge}}<br />
<br />
{{Chapter navigation|Statistical analysis|Turbulence kinetic energy}}</div>Malkavian GThttps://www.cfd-online.com/Wiki/Introduction_to_turbulence/Reynolds_averaged_equationsIntroduction to turbulence/Reynolds averaged equations2011-04-15T14:10:02Z<p>Malkavian GT: /* Equations for the average velocity */</p>
<hr />
<div>{{Introduction to turbulence menu}}<br />
<br />
== Equations governing instantaneous fluid motion ==<br />
<br />
All fluid motions, whether turbulent or not, are governed by the dynamical equations for a fluid. These can be written using Cartesian tensor notation as:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\rho\left[\frac{\partial \tilde{u}_i}{\partial t}+\tilde{u}_j\frac{\partial \tilde{u}_i}{\partial x_j}\right] = -\frac{\partial \tilde{p}}{\partial x_i}+\frac{\partial \tilde{T}_{ij}^{(v)}}{\partial x_j}</math><br />
</td><td width="5%">(1)</td></tr></table><br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\left[\frac{\partial \tilde{\rho}}{\partial t}+\tilde{u}_j\frac{\partial \tilde{\rho}}{\partial x_j}\right]+ \tilde{\rho}\frac{\partial \tilde{u}_j}{\partial x_j}= 0 </math><br />
</td><td width="5%">(2)</td></tr></table><br />
<br />
where <math>\tilde{u_i}(\vec{x},t)</math> represents the i-the component of the fluid velocity at a point in space,<math>[\vec{x}]_i=x_i</math>, and time,t. Also <br />
<math>\tilde{p}(\vec{x},t)</math> represents the static pressure, <math>\tilde{T}_{ij}^{(v)}(\vec{x},t)</math>, the viscous(or deviatoric) stresses, and <math>\tilde\rho</math> the fluid density. The tilde over the symbol indicates that an instantaneous quantity is being considered. Also the [[einstein summation convention]] has been employed.<br />
<br />
In equation 1, the subscript <math>i</math> is a free index which can take on the values 1,2 and 3. Thus equation 1 is in reality three separate equations. These three equations are just Newton's second law written for a continuum in a spatial(or Eulerian) reference frame. Together they relate the rate of change of momentum per unit mass <math>(\rho{u_i})</math>,a vector quantity, to the contact and body forces.<br />
<br />
Equation 2 is the equation for mass conservation in the absence of sources(or sinks) of mass. Almost all flows considered in this material will be incompressible, which implies that derivative of the density following the fluid material[the term in brackets] is zero. Thus for incompressible flows, the mass conservation equation reduces to:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\frac{D \tilde{\rho}}{Dt}=\frac{\partial \tilde{\rho}}{\partial t}+\tilde{u}_j\frac{\partial \tilde{\rho}}{\partial x_j}= 0</math><br />
</td><td width="5%">(3)</td></tr></table><br />
<br />
From equation 2 it follows that for incompressible flows,<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\frac{\partial \tilde{u}_j}{\partial x_j}= 0</math><br />
</td><td width="5%">(4)</td></tr></table><br />
<br />
The viscous stresses(the stress minus the mean normal stress) are represented by the tensor<math>\tilde{T}_{ij}^{(v)}</math>. From its definition,<math>\tilde{T}_{kk}^{(v)}</math>=0. In many flows of interest, the fluid behaves as a Newtonian fluid in which the viscous stress can be related to the fluid motion by a constitutive relation of the form.<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math>\tilde{T}_{ij}^{(v)}= 2\mu[\tilde{s}_{ij}-\frac{1}{3}\tilde{s}_{kk}\delta_{ij}] </math><br />
</td><td width="5%">(5)</td></tr></table><br />
<br />
The viscosity, <math>\mu</math>, is a property of the fluid that can be measured in an independent experiment. <math>\tilde s_{ij}</math> is the instantaneous strain rate tensor defined by<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math>\tilde{s}_{ij}= \frac{1}{2}\left[\frac{\partial \tilde u_i}{\partial x_j}+\frac{\partial \tilde u_j}{\partial x_i}\right] </math><br />
</td><td width="5%">(6)</td></tr></table><br />
<br />
From its definition, <math>\tilde s_{kk}=\frac{\partial \tilde u_k}{\partial x_k}</math>. If the flow is incompressible, <math>\tilde s_{kk}=0</math> and the Newtonian constitutive equation reduces to<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math>\tilde{T}_{ij}^{(v)}= 2\mu\tilde{s}_{ij}</math><br />
</td><td width="5%">(7)</td></tr></table><br />
<br />
Throughout this material, unless explicitly stated otherwise, the density <math>\tilde\rho=\rho</math> and the viscosity <math>\mu</math> will be assumed constant. With these assumptions, the instantaneous momentum equations for a Newtonian Fluid reduce to:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\left[\frac{\partial \tilde{u}_i}{\partial t}+\tilde{u}_j\frac{\partial \tilde{u}_i}{\partial x_j}\right] = -\frac {1}{\tilde\rho}\frac{\partial \tilde{p}}{\partial x_i}+\nu\frac{\partial^2 {\tilde{u}_i}}{\partial x_j^2}</math><br />
</td><td width="5%">(8)</td></tr></table><br />
<br />
where the kinematic viscosity, <math>\nu</math>, has been defined as:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math>\nu\equiv\frac{\mu}{\rho}</math><br />
</td><td width="5%">(9)</td></tr></table><br />
<br />
Note that since the density is assumed constant, the tilde is no longer necessary.<br />
<br />
Sometimes it will be more instructive and convenient to not explicitly include incompressibilty in the stress term, but to refer to the incompressible momentum equation in the following form:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\rho\left[\frac{\partial \tilde{u}_i}{\partial t}+\tilde{u}_j\frac{\partial \tilde{u}_i}{\partial x_j}\right] = -\frac{\partial \tilde{p}}{\partial x_i}+\frac{\partial \tilde{T}_{ij}^{(v)}}{\partial x_j}</math><br />
</td><td width="5%">(10)</td></tr></table><br />
<br />
This form has the advantage that it is easier to keep track of the exact role of the viscous stresses.<br />
<br />
== Equations for the average velocity ==<br />
<br />
Although laminar solutions to the equations often exist that are consistent with the boundary conditions, perturbations to these solutions(sometimes even infinitesimal) can cause them to become turbulent. To see how this can happen, it is convenient to analyze the flow in two parts, a mean(or average) component and a fluctuating component. Thus the instantaneous velocity and stresses can be written as:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\tilde {u}_i=U_i+u_i<br />
</math><br />
:<math><br />
\tilde p=P+p<br />
</math><br />
:<math><br />
\tilde T_{ij}^{(v)}=T_{ij}^{(v)}+\tau_{ij}^{(v)}<br />
</math><br />
</td><td width="5%">(11)</td></tr><br />
</table><br />
<br />
Where <math>U_i</math>, <math>P</math> and <math>T_{ij}^{(v)}</math> represent the mean motion, and <math>u_i</math>, <math>p</math> and <math>\tau_{ij}^{(v)}</math> the fluctuating motions. This technique for decomposing the instantaneous motion is referred to as the '''''Reynolds decomposition.''''' Note that if the averages are defined as ensemble means, they are, in general, time-dependent. For the remainder of this material unless other wise stated, the density will be assumed constant so<math>\tilde{\rho}\equiv\rho</math>,and its fluctuation is zero.<br />
<br />
Substitution of equations 11 into equations 10 yields<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\rho\left[\frac{\partial (U_i+u_i)}{\partial t}+(U_j+u_j)\frac{\partial (U_i+u_i)}{\partial x_j}\right] = -\frac{\partial (P+p)}{\partial x_i}+\frac{\partial (T_{ij}^{(v)}+\tau_{ij}^{(v)})}{\partial x_j}</math><br />
</td><td width="5%">(12)</td></tr></table><br />
<br />
This equation can now be averaged to yield an equation expressing momentum conservation for the averaged motion. Note that the operations of averaging and differentiation commute; i.e., the average of a derivative is the same as the derivative of the average. Also the average of a fluctuating quantity is zero. Thus the equation for the averaged motion reduces to:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\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_{ij}^{(v)}}{\partial x_j}-\rho\left \langle u_j\frac{\partial u_i }{\partial x_j} \right \rangle</math><br />
</td><td width="5%">(13)</td></tr></table><br />
<br />
where the remaining fluctuating product term has been moved to the right hand side of the equation. Whether or not the last term is zero like the other fluctuating term depends on the correlation of the terms in the product. In general, these correlations are not zero.<br />
<br />
The mass conservation equation can be similarly decomposed. In incompressible form, substitution of equations 11 into equation 4 yields:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\frac{\partial (U_j+u_j)}{\partial x_j}=0</math><br />
</td><td width="5%">(14)</td></tr></table><br />
<br />
of which average is <br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\frac{\partial U_j}{\partial x_j}=0</math><br />
</td><td width="5%">(15)</td></tr></table><br />
<br />
It is clear from equation 15 that the averaged motion satisfies the same form of the mass conservation equation as does the instantaneous motion at least for incompressible flows. How much simpler the turbulence problem would be if the same were true for the momentum! Unfortunately, as is easily seen from equation 13, such is not the case.<br />
<br />
Equation 15 can be subtracted from equation 14 to yield an equation for instantaneous motion alone; i.e,<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\frac{\partial u_j}{\partial x_j}=0</math><br />
</td><td width="5%">(16)</td></tr></table><br />
<br />
Again, like the mean, the form of the original instantaneous equation is seen to be preserved. The reason, of course, is obvious: the continuity equation is linear. The momentum equation , on the other hand, is not; hence the difference.<br />
<br />
Equation 16 can be used to rewrite the last term in equation 13 for the mean momentum. Multiplying equation 16 by <math>u_i</math> yields:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\left \langle u_i\frac{\partial u_j}{\partial x_j}\right \rangle=0</math><br />
</td><td width="5%">(17)</td></tr></table><br />
<br />
Now add:<math>\left \langle u_j\frac{\partial u_i}{\partial x_j}\right \rangle</math> to both sides of eqaution 18, then using the product rule you obtain:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\left \langle u_j\frac{\partial u_i}{\partial x_j}\right \rangle +0=\left \langle u_j\frac{\partial u_i}{\partial x_j}\right \rangle+ \left \langle u_i\frac{\partial u_j}{\partial x_j}\right \rangle =\frac{ \partial}{\partial x_j}{\left \langle u_iu_j\right \rangle} </math><br />
</td><td width="5%">(18)</td></tr></table><br />
<br />
Where again the fact that arithmetic and averaging operations commute has been used.<br />
<br />
The equation for the averaged momentum, equation 13 can now be rewritten as:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\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_{ij}^{(v)}}{\partial x_j}-\frac{ \partial}{\partial x_j}{(\rho\left \langle u_iu_j\right \rangle)}</math><br />
</td><td width="5%">(19)</td></tr></table><br />
<br />
The last two terms on the right hand side are both divergence terms and can be combined; the result is:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\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 }{\partial x_j}[T_{ij}^{(v)}-{\rho\left \langle u_iu_j\right \rangle}]</math><br />
</td><td width="5%">(20)</td></tr></table><br />
<br />
Now the terms in square brackets on the right have the dimensions of stress. The first term is, in fact , the viscous stress. The second term, on the other hand, is not a stress at all, but simply a re-worked version of the fluctuating contribution to the non-linear acceleration terms. The fact that it can be written this way, however, indicates that at least as far as the motion is concerned, it acts as though it were a stress- hence its name, the '''Reynolds stress'''. In the succeeding sections the consequences of this difference will be examined.<br />
<br />
== The turbulence problem ==<br />
<br />
It is the appearance of the Reynolds stress which makes the turbulence problem so difficult - at least from the engineers perspective. Even though we can pretend it is a stress, the physics which give rise to it are very different from the viscous stress. The viscous stress can be related directly to the other flow properties by constitutive equations, which in turn depend only on the properties of the fluid (as in equation 5 for a Newtonian fluid). The reason this works is that when we make such closure approximations for a fluid, we are averaging over characteristic length and time scales much smaller than those of the flows we are interested in. Yet at the same time, these scales are much larger than the molecular length and time scales which characterize the molecular interactions that are actually causing the momentum transfer. (This is what the continuum approximation is all about).<br />
<br />
The '''''Reynolds stress''''', on the other hand, arises from the flow itself! Worse, the scales of the fluctuating motion which give rise to it are the scales we are interested in. This means that the closure ideas which worked so well for the viscous stress, should not be expected to work too well for the Reynolds stress. And as we shall see, they do not.<br />
<br />
This leaves us in a terrible position. Physics and engineering are all about writing equations(and boundary conditions) so we can solve them to make predictions. We don't want to have a build prototype airplanes first to see if they will they fall out of the sky. Instead we want to be able to analyze our designs before building, to save the cost in money and lives if our ideas are wrong. The same is true for dams and bridges and tunnels and automobiles. If we had confidence in our turbulence models, we could even build huge one-offs and expect them to work the first time. Unfortunately, even though turbulence models have improved to the point where we can use them in design, we still cannot trust them enough to eliminate expensive wind tunnel and model studies. And recent history is full of examples to prove this.<br />
<br />
The turbulence problem (from the engineers perspective) is then three-fold:<br />
<br />
* '''The averaged equations are not closed.''' Count the number of unknowns in equation 20 above. Then count the number of equations. Even with the continuity equation we have atleast six equations too few.<br />
<br />
* '''The simple ideas to provide the extra equations usually do not work.''' And even when we can fix them up for a particular class of flows (like the flow in a pipe, for example), they will most likely not be able to predict what happens in even a slightly different environment (like a bend).<br />
<br />
*'''Even the last resort of compiling engineering tables for design handbooks carries substantial risk.''' This is the last resort for the engineer who lacks equations or cannot trust them. Even when based on a wealth of experience, they require expensive model testing to see if they can be extrapolated to a particular situation. Often they cannot, so infinitely clever is Mother Nature in creating turbulence that is unique to a particular set of boundary conditions.<br />
<br />
'''Turbulent flows are indeed flows!'''. And that is the problem.<br />
<br />
== Origins of turbulence==<br />
<br />
Turbulent flows can often be observed to arise from laminar flows as the Reynolds number, (or someother relevant parameter) is increased. This happens because small disturbances to the flow are no longer damped by the flow, but begin to grow by taking energy from the original laminar flow. This natural process is easily visualized by watching the simple stream of water from a faucet (or even a pitcher). Turn the flow on very slow (or pour) so the stream is very smooth initially, at least near the outlet. Now slowly open the faucet (or pour faster) abd observe what happens, first far away, then closer to the spout. The surface begins to exhibit waves or ripples which appear to grow downstream . In fact, they are growing by extracting energy from the primary flow. Eventually they grow enough that the flow breaks into drops. These are capillary instabilities arisiing from surface tension, but regardless of the type of instability, the idea is the same -small (or infinitesimal ) disturbances have grown to disrupt the serenity (and simplicity) of laminar flow.<br />
<br />
The manner in which the instabilities grow naturally in a flow can be examined using the equations we have already developed above. We derived them by decomposing the motion into a mean and fluctuating part. But suppose instead we had decomposed the motion into a base flow part (the initial laminar part) and into a disturbance which represents a fluctuating part superimposed on the base flow. The result of substituting such a decomposition into the full Navier-Stokes equations and averaging is precisely that given by equations (13) and (15). But the very important difference is the additional restriction that what was previously identified as the mean (or averaged ) motion is now also the base or laminar flow.<br />
<br />
Now if the base flow is really laminar flow (which it must be by our original hypothesis), then our averaged equations governing the base flow must yield the same mean flow as the original laminar flow on which the disturbances was superimposed. But this can happen only if these new averaged equations reduce to '''exactly''' the same lamiane flow equations without any evidence of a disturbance. Clearly from equations 13 and 15, this can happen ''only if all the Reynolds stress terms vanish identically!'' Obviously this requires that the disturbances be infintesimal so the extra terms can be neglected - hence our interest in infinitesimal disturbances.<br />
<br />
So we hypothesized a base flow which was laminar and showed that it is unchanged even with the imposition of infintesimal disturbances on it - ''but only as long as the disturbances'' '''remain''' ''infinitesimal!'' What happens if the disturbance starts to grow? Obviously before we conclude that all laminar flows are laminar forever we better investigate whether or not these infinitesimal disturbances can grow to ''finite'' size. To do this we need an equation for the fluctuation itself.<br />
<br />
An equation for the fluctuation (which might be an imposed disturbance) can be obtained by subtracting the equation for the mean (or base) flow from that for the instantaneous motion. We already did this for the continuity equation. Now we will do it for the momentum equation. Subtracting equation 13 from equation 11 yields an equation for the fluctuation as:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\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\tau^{(v)}_{ij}}{\partial x_{j}} - \rho \left[ u_{j}\frac{\partial U_{i}}{\partial x_{j}} \right] - \left\{ u_{j} \frac{\partial u_{i}}{ \partial x_{j}} - \rho \left\langle u_{j} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right\}<br />
</math><br />
</td><td width="5%">(21)</td></tr></table><br />
<br />
It is very important to note the type and character of the terms in this equation. First note that the left-hand side is the derivative of the ''fluctuating'' velocity following the ''mean'' motion. This is exactly like the term which appears on the left-hand side of the equation for the mean velocity, equation 13. The first two terms on the right-hand side are also like those in the mean motion, and represent the fluctuating pressure gradient and the fluctuating viscous stresses. The third term on the right-hand is is new, and will be seen later to represent the primary means by which fluctuations (and turbulence as well!) extract energy from the mean flow, the so-called ''production terms''. The last term is quadratic in the fluctuating velocity, unlike all the otherwhich are linear. Note that all of the terms vanish identically if the equation is averaged, the last because its mean is subtructed from it.<br />
<br />
Now we want to examine what happens if the disturbance is small. In the limit as the amplitude of the disturbance (or fluctuation) is ''infinitesmal'', the bracketed term in the equationfor the fluctuation vanishes (since it involves productsof infinitesimals) , and the remaining equation is ''linear in the disturbance''. The study of whether or not such infinitesmal disturbances can grow is called '''Linear Fluid Dynamic Stability Theory'''. These linearized equations are very different from those govering turbulence. Unlike the equations for disturbances of ''finite'' amplitude, the linearized equations are well-posed (or closed) since the Reynolds stress terms are gone.<br />
<br />
The absence of the non-linear terms, however, constrains the validity of the linear analysis to only the initial stage of disturbance growth. This is because as soon as the fluctuations begin to grow, their amplitudes can no longer be assumed infinitesmal and the Reynolds stress (or more properly, the non-linear fluctuating terms) become important. As a result the base flow equations begin to be modified so that the solution to them can no longer be identical to the laminar flow (or base flow) from which it arose. Thus while linear stability theory can predict ''when'' many flows become ''unstable'', it can say very little about ''transition to turbulence'' since this progress is highly non-linear.<br />
<br />
== Importance of non-linearity ==<br />
<br />
We saw in the preceding section that non-linearity was one of essential features of turbulence. When small disturbances grow large enough to interact ''with each other'', we enter a whole new world of complex behavior. Most of the rules we learned for linear system do not apply. Since most of your mathematical training has been for linear equations, most of your mathematical intuition therefore will not apply either. On the other hand, you may surprise yourself by discovering how much your ''non-mathematical'' intuition already recognizes non-linear behavior and accounts for it.<br />
<br />
Considering the following simple example. Take a long stick with one person holding each end and stand at the corner of a building. Now place the middle of against the building and let each person apply pressure in the same direction so as to bend the stick. If the applied force is small, the stick deflects (or bends) a small amount. Double the force, and the deflection is approximately doubled. Quadruple the force and the deflection is quadrupled. Now you don't need a Ph.D. in Engineering to know what is going to happen if you continue this process. '''The stick is going to break!'''<br />
<br />
But where in the equations for the deflection of the stick is there anything that predicts this can happen? Now if you are only like engineer, you are probably thinking: he's asking a stupid question. Of course you can't continue to increase the force because you will exceed first the yield stress, then the breaking limit, and of course the stick will break.<br />
<br />
But pretend I am the company president with nothing more than MBA. I don't know much about these things, but you have told me in the past that your computers have equations to predict everything. So I repeat: Where in the equations for the deflection of this stick does it tell me this going to happen?<br />
<br />
The answer is very simple: '''There is''' ''nothing'' '''in the equations that will predict this.''' And the reason is also quite simple: You lost the ability to predict catasrophes like breaking when you linearized the fundamental equations - which started out as Newton's Law too. In fact, before linearization, they were exactly the same as those for a fluid, only the constitutive equation was different.<br />
<br />
If we had NOT linearized these equations and had constituve equations that were more general, then we possibly could apply these equation right to and past the limit. The point of fracture would be a bifurcation point for the solution.<br />
<br />
Now the good news is that for things like reasonable deflections of beams linearization work woderfully since we hope most things we build don't deflect too much. Unfortunately, as we noted above, for fluids the disturbances tend to quickly become dominated by non-linear terms. This, of course, means our linear analytical techniques are pretty useless for fluid mechanics, and especially turbulence.<br />
<br />
But all is not lost. Just as we learned to train ourselves to anticipate when sticks break, we have to train ourselves to anticipate how non-linear fluid phenomena behave. Toward that end we will consider two simple examples: one from algebra - the logistic map, and one from fluid mechanics - simple vortex streching.<br />
<br />
'''Example 1:''' An experiment with the logistic map. <br />
<br />
Consider the behavior of the simple equation:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
y_{n+1}= r y_{n} \left(1- y_{n} \right)<br />
</math><br />
</td><td width="5%">(22)</td></tr></table><br />
<br />
where <math> n= 1,2..., 0<y < 1 </math> and <math> r > 0 </math>. The idea is that you pick any value for <math> y_{1}</math>, use the equation to find <math> y_{2}</math>, then insert that value on the right-hand side to find <math> y_{3}</math>, and just continue the process as long as you like. Make sure you note any dependece of the final result on the initial value for <math> y </math>.<br />
<br />
*First notice what happens if you linearize this equation by disregarding the term in parentheses; i.e., comsider the simpler equation <math> y_{1+1} = r y_{n}</math>. My guess is that you won't find this too eciting - unless, of course, you are one of those rare individuals who likes watching grass grow.<br />
<br />
* Now consider the full equation and note what happens for <math> r<3 </math> , and especially what happens for very small values of <math> r </math>. Run as many iterations as necessary to make sure your answer has converged. Do NOT try to take short-cuts by programming all the steps at once. Do them one at time so you can see what is happening. It will be much easier this way in the long run.<br />
<br />
* Now research carefully what happens when <math> r=3.1, 3.5, </math> and <math> 3.8 </math>. Can you recognize any patterns.<br />
<br />
* Vary <math> r </math> between 3 and 4 to see if you can find the boundaries for what you are observing. <br />
<br />
* Now try values of <math> r>4 </math>. How do you explain this<br />
<br />
'''Example 2:''' Stretching of a simple vortex.<br />
<br />
Imagine a simple vortex filament that looks about like a strand of spaghetti. Now suppose it is in otherwise steady inviscid incompressible flow. Use the vorticity equation to examine the following: <br />
<br />
* Examine first what happens to it in two-dimensional velocity field. Note particularly whether any new vorticity can be produced; i.e., can the material derivative of the vorticity ever be greater than zero? (Hint: look at the <math> \omega_{j} \partial u_{i}/ \partial x_{j} <br />
</math> - term)<br />
<br />
* Now consider the same vortex filament in a three-dimensional flow. Note particularly the various ways new vorticity can be produced - if you have some to start with! Does all this have anything to do with non-linearities?<br />
<br />
== Turbulence closure problem and eddy viscosity ==<br />
<br />
From the point of view of the averaged motion, at least, the problem with the non-linearity of the instaneous equations is that they introduce new unknowns, the Reynolds stress into the averaged equations. There are six individual stress components we must deal with to be exact: <math> \left\langle u^{2}_{1} \right\rangle </math> , <math> \left\langle u^{2}_{2} \right\rangle </math> , <math> \left\langle u^{2}_{3} \right\rangle </math> , <math> \left\langle u_{1}u_{2} \right\rangle </math> , <math> \left\langle u_{1}u_{3} \right\rangle </math> , and <math> \left\langle u_{2}u_{3} \right\rangle </math>. These have to be related to the mean motion itself before the equations can be solved, since the number of unknows and number of equations must be equal. The absence of these additional equations is often reffered to as '''the Turbulence Closure Problem'''.<br />
<br />
A similar problem arose when the instantaneous equations were written (equations 1 2), since relations had to be introduced to relate the stresses (in particular, the viscous stresses) to the motion itself. These relations (or constitutive equations) ''depended only on the properties of the fluid material, and not on the flow itself''. Because of this fact, it is possible to carry out independent experiments, called viscometric experiments, in which these fluid properties can be determinded once and for all. Equation 5 provides an example just such a constitutive relation, the viscosity, <math>\mu </math>, depending only in the choice of the material. For example, once the viscosity of water at given temperature is determined, this value can be used in all flows at that temperature, not just the one in which the evaluation was made.<br />
<br />
It is tempting to try such an approach for the turbulence Reynolds stresses (even though we know the underlying requirements of scale separation are not satisfied). For example, a Newtonian type closure for the Reynolds stresses, often referred to as an "eddy" or "turbulent" viscosity model, looks like:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
- \rho \left\langle u_{i} u_{j} \right\rangle = 2 \mu_{t} \left[ S_{ij} - \frac{1}{3} S_{kk} \delta_{ij} \right]<br />
</math><br />
</td><td width="5%">(23)</td></tr></table><br />
<br />
where <math>\mu_{t} </math> is the turbulence "viscosity" (also called the eddy viscosity) and <math>S_{ij} </math> is the ''mean'' strain rate defined by:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
S_{ij}= \frac{1}{2} \left[ \frac{\partial U_{i}}{\partial x_{j}} + \frac{\partial U_{j}}{\partial x_{i}} \right]<br />
</math><br />
</td><td width="5%">(24)</td></tr></table><br />
<br />
The second term of course, vanishes identically for incompressible flow. For the simple case of a two-dimensional shear flow, equation 3.23 for the Reynolds shear stress reduces to<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
- \rho \left\langle u_{1} u_{2} \right\rangle = 2 \mu_{t} \frac{\partial U_{1}}{\partial x_{2}} <br />
</math><br />
</td><td width="5%">(25)</td></tr></table><br />
<br />
That such a simple model can adequately describe the mean motion in at least one flow is illustrated by the axisymmetric buoyant plume sketched in <font color="orange">Figure 3.1. Figures 3.2 and 3.3</font> show the calculation of the mean velocity and temperature profiles respectively. Obviously the mean velocity and temperature are reasonably accurately computed, as are the Reynolds shear stress and lateral turbulent heat flux shown in <font color ="orange">Figures 3.4 and 3.5</font>.<br />
<br />
The succes of the eddy viscosity in the preceding example is more apparent than real, however, since the value of the eddy viscosity and eddy diffusivity (for turbulent heat flux) have been chosen to give the best possible agreement with the data. This, in itself, would not be a problem if that chosen values could have been obtained in advance of the computation, or even if they could be used to successfully predict other flows. In fact, the values used work only for this flow, thus the computation is ''not a prediction at all, but a'' '''postdiction''' ''or'' '''hindcast''' from which no extrapolation to the future can be made. In other words, our turbulence "model" is about as useful as having a program to predict yesterday's weather. Thus the closure problem still very much remains.<br />
<br />
<font color="orange" size="3">Figure 3.1 not uploaded yet</font><br />
<br />
<font color="orange" size="3">Figure 3.2 not uploaded yet</font><br />
<br />
<font color="orange" size="3">Figure 3.3 not uploaded yet</font><br />
<br />
<font color="orange" size="3">Figure 3.4 not uploaded yet</font><br />
<br />
<font color="orange" size="3">Figure 3.5 not uploaded yet</font><br />
<br />
<font color="orange" size="3">Figure 3.6 not uploaded yet</font><br />
<br />
Another problem with the eddy viscosity in the example above is that it fails to calculate the vertical components of the Reynolds stress and turbulent heat flux. An attempt at such a computation is shown in <font color ="orange">Figure 3.6</font> where the vertical turbulent heat flux is shown to be severely underestimated. Clearly the value of the in the vertical direction must be different than in the radial direction. In other words, the turbulence for which a constituve equation is being written is ''not an isotropic "medium" ''. In fact, in this specific example the problem is that the vertical component of the heat flux is produced more by the interaction of buoyancy and the turbulence, than it is by the working of turbulence against mean gradient in the flow. We will discuss this in more detail in the next chapter when we consider the turbulence energy balances, but note for now that simple gradient closure models never work unless gradient production dominates. This rules out many flows involving buoyancy, and also many involving recirculations or separation where the local turbulence is convected in from somewhere else. <br />
<br />
A more general form of constitutive equation which would allow for the nonisotropic nature of the "medium" (in this case the turbulence itself) would be<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
- \rho \left\langle u_{i} u_{j} \right\rangle = \mu_{ijkl} \left[ S_{kl} - \frac{1}{3} S_{mm} \delta_{kl} \right]<br />
</math><br />
</td><td width="5%">(26)</td></tr></table><br />
<br />
This closure relation allows each component of the Reynolds stress to have its own unique value of the eddy viscosity. It is easy to see that it is unlikely this will solve the closure problem since the original six unknowns the <math>\left\langle u_{i} u_{j} \right\rangle </math> have been traded for eighty-one new ones, <math> \mu_{ijkl} </math>. Even if some can be removed by symmetries, the remaining number is still formidable. More important than the number of unknowns, however, is that there is no independent or general means for selecting them without considering a particular flow. This is because ''turbulence is indeed a property of the flow, not of the fluid''.<br />
<br />
== Reynolds stress equations ==<br />
<br />
It is clear from the preceding section that the simple idea of an eddy viscosity might not be the best way to approach the problem of relating the Reynolds stress to the mean motion. An alternative approach is to try to derive dynamical equations for the Reynolds stresses from the equations governing the fluctuations themselves. Such an approach recognizes that the Reynolds stress is really a functional of the velocity; that is, the stress at a point depends on the velocity everywhere and for all past times, not just at the point in question and at a particular instant in time.<br />
<br />
The analysis begins with the equation for the instantaneous fluctuating velocity, equation 21. This can be rewritten for a Newtonian fluid with constant viscosity as: <br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\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 \tau ^{ ( v ) }_{ij}}{ \partial x_{j}} - \rho \left[ u_{j} \frac{\partial U_{i}}{ \partial x_{j} } \right] - \rho \left\{ u_{j} \frac{\partial u_{i}}{ \partial x_{j} } - \left\langle u_{j} \frac{\partial u_{i} }{\partial x_{j} } \right\rangle \right\}<br />
</math><br />
</td><td width="5%">(27)</td></tr></table><br />
<br />
Note that the free index in this equation is <math> i </math>. Also, since we are talking about turbulence again, the capital letters represent mean or averaged quantities. <br />
<br />
Multiplying equation 27 by <math> u_{k} </math> and averaging yields:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\rho \left[ \left\langle u_{k} \frac{\partial u_{i} }{\partial t} \right\rangle + U_{j} \left\langle u_{k} \frac{\partial u_{i} }{\partial x_{j}} \right\rangle \right] = & - & \left\langle u_{k} \frac{\partial p }{\partial x_{i} } \right\rangle + \left\langle u_{k} \frac{\partial \tau^{(v)}_{ij} }{\partial x_{j}} \right\rangle \\<br />
& - & \rho \left[ \left\langle u_{k}u_{j} \right\rangle \frac{\partial U_{i} }{\partial x_{j} } \right] - \rho \left\{ \left\langle u_{k}u_{j} \frac{\partial u_{i} }{\partial x_{j} } \right\rangle \right\} \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(28)</td></tr></table><br />
<br />
Now since both <math> i </math> and <math> k </math> are free indices they can be interchanged to yield a second equation given by (alternatively equation 21 can be rewritten with free index <math> k </math>, then multiplied by <math> u_{i} </math> and averaged):<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\rho \left[ \left\langle u_{i} \frac{\partial u_{k} }{\partial t} \right\rangle + U_{j} \left\langle u_{i} \frac{\partial u_{k} }{\partial x_{j}} \right\rangle \right] = & - & \left\langle u_{i} \frac{\partial p }{\partial x_{k} } \right\rangle + \left\langle u_{i} \frac{\partial \tau^{(v)}_{kj} }{\partial x_{j}} \right\rangle \\<br />
& - & \rho \left[ \left\langle u_{i}u_{j} \right\rangle \frac{\partial U_{k} }{\partial x_{j} } \right] - \rho \left\{ \left\langle u_{i}u_{j} \frac{\partial u_{k} }{\partial x_{j} } \right\rangle \right\} \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(29)</td></tr></table><br />
<br />
Equations 28 and 29 can be added together to yield an equation for the Reynolds stress<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\frac{\partial\left\langle u_{i}u_{k} \right\rangle}{\partial t} + U_{j}\frac{\partial\left\langle u_{i}u_{k} \right\rangle}{\partial x_{j}} <br />
& = & - \frac{1}{\rho} \left[ \left\langle u_{i} \frac{\partial p}{\partial x_{k}} \right\rangle + \left\langle u_{k} \frac{\partial p}{\partial x_{i}} \right\rangle \right] \\<br />
& - & \left[ \left\langle u_{i}u_{j} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle u_{k}u_{j} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right] \\<br />
& + & \frac{1}{\rho} \left[ \left\langle u_{i} \frac{\partial \tau^{(v)}_{kj} }{\partial x_{j}} \right\rangle + \left\langle u_{k} \frac{\partial \tau^{(v)}_{ij} }{\partial x_{j}} \right\rangle \right] \\<br />
& - & \left[ \left\langle u_{i} u_{j} \right\rangle \frac{\partial U_{k}}{\partial x_{j}} + \left\langle u_{k} u_{j} \right\rangle \frac{\partial U_{i}}{\partial x_{j}} \right] \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(30)</td></tr></table><br />
<br />
It is customary to rearrange the first term on the right hand side in the following way:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\left[ \left\langle u_{i} \frac{\partial p}{\partial x_{k}} \right\rangle + \left\langle u_{k} \frac{\partial p}{\partial x_{i}} \right\rangle \right] = & & \left\langle p \left[ \frac{\partial u_{i}}{\partial x_{k}} + \frac{\partial u_{k}}{\partial x_{i}} \right] \right\rangle \\<br />
& - & \frac{\partial}{\partial x_{j}} \left[ \left\langle pu_{i} \right\rangle \delta_{kj} + \left\langle pu_{k} \right\rangle \delta_{ij} \right] \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(31)</td></tr></table><br />
<br />
The first term on the right is generally referred to as the ''pressure strain-rate'' term. The second term is written as a divergence term, and generally referred to as the ''pressure diffusion'' term. We shal see later that divergence terms can never create nor destroy anything; they can simple move it around from one place to another. <br />
<br />
The third term on the right-hand side of equation 30 can similarly be rewritten as:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\left[ \left\langle u_{i} \frac{\partial \tau^{(v)}_{kj}}{\partial x_{j}} \right\rangle + \left\langle u_{k} \frac{\partial \tau^{(v)}_{ij}}{\partial x_{j}} \right\rangle \right] = & - & \left[ \left\langle \tau^{(v)}_{ij} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle \tau^{(v)}_{kj} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right] \\<br />
& + & \frac{\partial}{\partial x_{j}} \left[ \left\langle u_{i} \tau^{(v)}_{kj} \right\rangle + \left\langle u_{k} \tau^{(v)}_{ij} \right\rangle \right] \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(32)</td></tr></table><br />
<br />
The first of these is also a divergence term. For a Newtonian fluid, the last is the so-called "dissipation of Reynolds stress" by the turbulence viscous stresses. This is easily seen by substituting the Newtonian constitutive relation to obtain:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left[ \left\langle \tau^{(v)}_{ij} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle \tau^{(v)}_{kj} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right] = 2 \nu \left[ \left\langle s_{ij} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle s_{kj} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right]<br />
</math><br />
</td><td width="5%">(33)</td></tr></table><br />
<br />
It is not at all obvious what this has to do with dissipation, but it will become clear later on when we consider the trace of the Reynolds stress equation, which is the ''kinetic energy'' equation for the turbulence.<br />
<br />
Now if we use the same trick from before using the continuity equation, we can rewrite the third term in equation 30 to obtain:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left[ \left\langle u_{i}u_{j} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle u_{k}u_{j} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right] = \frac{\partial}{\partial x_{j}} \left\langle u_{i} u_{k} u_{j} \right\rangle<br />
</math><br />
</td><td width="5%">(34)</td></tr></table><br />
<br />
This is also a divergence term.<br />
<br />
We can use all of the pieces we have developed above to rewrite equation 30 as <br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\frac{\partial}{\partial t} \left\langle u_{i}u_{k} \right\rangle & + & U_{j} \frac{\partial}{\partial x_{j}} \left\langle u_{i}u_{k} \right\rangle = - \left\langle \frac{p}{\rho} \left[ \frac{\partial u_{i}}{\partial x_{k}} + \frac{\partial u_{i}}{\partial x_{k}} \right] \right\rangle \\<br />
& + & \frac{\partial}{\partial x_{j}} \left\{ - \left[ \left\langle pu_{k} \right\rangle \delta_{ij} + \left\langle pu_{i} \right\rangle \delta_{kj} \right] - \left\langle u_{i}u_{k}u_{j} \right\rangle + 2\nu \left[ \left\langle s_{ij}u_{k} \right\rangle + \left\langle s_{ij}u_{k} \right\rangle \right] \right\}\\<br />
& - & \left[ \left\langle u_{i} u_{j} \right\rangle \frac{\partial U_{k}}{\partial x_{j}} + \left\langle u_{k} u_{j} \right\rangle \frac{\partial U_{i}}{\partial x_{j}} \right] \\<br />
& - & 2\nu \left[ \left\langle s_{ij} \frac{\partial u_{k}}{\partial x_{j}} \right\rangle + \left\langle s_{kj} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right] \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(35)</td></tr></table><br />
<br />
This is the so-called '''Reynolds Stress Equation''' which has been the primary vehicle for much of the turbulence modelling efforts of the past few decades.<br />
<br />
The left hand side of the Reynolds Stress Equation can easily be recognized as the rate of change of the Reynolds stress following the mean motion. It seems to provide exactly what we need: nine new equations for the nine unknowns we can not account for. The problems are all on the right-hand side. These terms are referred to respectively as<br />
<br />
* the pressure-strain rate term<br />
* the turbulence transport (or divergence) term<br />
* the "production" term, and<br />
* the "dissipation" term.<br />
<br />
Obviously these equations do not involve only <math> U_{i} </math> and <math> \left\langle u_{i} u_{j} \right\rangle </math>, but depend on many more new unknowns.<br />
<br />
It is clear that, contrary to our hopes, we have not derived a single equation relating the Reynolds stress to the mean motion. Instead, our Reynolds stress transport transport equation is exceedingly complex. Whereas the process of averaging the equation for the mean motion introduced only six new, independent unknowns, the Reynolds stress, <math> \left\langle u_{i} u_{j} \right\rangle </math>, the search for a transport equation which will relate these to the mean motion has produced many more unknowns. They are<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left\langle p u_{i} \right\rangle - 3<br />
</math><br />
</td><td width="5%">(36)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left\langle u_{i} s_{jk} \right\rangle - 27<br />
</math><br />
</td><td width="5%">(37)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left\langle s_{ij}s_{jk} \right\rangle - 9<br />
</math><br />
</td><td width="5%">(38)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left\langle u_{i}u_{k}u_{j} \right\rangle - 27<br />
</math><br />
</td><td width="5%">(39)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left\langle p\frac{\partial u_{i}}{\partial x_{j}} \right\rangle - 9<br />
</math><br />
</td><td width="5%">(40)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
TOTAL - 75<br />
</math><br />
</td><td width="5%">(41)</td></tr></table><br />
<br />
Not at all of these are independent, since some can be derived from others. Even so, our goal of reducing the number of unknowns has clearly not been met.<br />
<br />
Equations governing each of these new quantities can be derived from the original dynamical equations, just as we did for the Reynolds stress. Unfortunately new quantities continue to be introduced with each new equation, and at a faster rate than the icrease in the number of equations. Now the full implications of the closure problem introduced by the Reynolds decomposition and averaging has become apparent. No matter how many new equations are derived, the number of new unknown quantities introduced will always increase more rapidly.<br />
<br />
Our attempt to solve the turbulence problem by considering averages illustrates a general principle. Any time we try to full Mother Nature by averaging out her details, she gets her revenge by leaving us with closure problem - more equations than unknowns. In thermodynamics, we tried to simplify the consideration of molecules by averaging over them, and were left with the need for an equation of state. In heat transfer, we tried to simplify considerations by which molecules transfer their kinetic energy, and found we were lacking a relation between the heat flux and the temperature field. And in fluid mechanics, we tried to simplify consideration of the "mean" motionof molecules and ended up with viscous stress. In all of these cases we were able to make simple physical models which worked at least some of the time; e.g., ideal gas, Fourier-Newtonian fluid. And these models all worked because we were able to make assumptions about the underlying molecular processes and assume them to be independent of the macroscopic flows of interest. Unfortunately such assumptions are rarely satisfied in turbulence. <br />
<br />
It should be obvious by now that the turbulence closure problem will not be solved by the straight-forward derivation of new equations, nor by direct analogy with viscous stresses. Rather, ''closure attempts will have to depend on an intimate knowledge of the dynamics of the turbulence itself''. Only by understanding how the turbulence behaves can one hope to ''guess'' an appropriate set of constitutive equations '''AND''' ''understand the limits of them''. This is, of course, another consequence of the fact that the ''turbulence is a property of the flow itself, and not of the fluid''!<br />
<br />
{{Turbulence credit wkgeorge}}<br />
<br />
{{Chapter navigation|Statistical analysis|Turbulence kinetic energy}}</div>Malkavian GThttps://www.cfd-online.com/Wiki/Introduction_to_turbulence/Turbulence_kinetic_energyIntroduction to turbulence/Turbulence kinetic energy2011-03-20T18:46:51Z<p>Malkavian GT: /* Rate of dissipation of the turbulence kinetic energy */</p>
<hr />
<div>{{Introduction to turbulence menu}}<br />
<br />
== Fluctuating kinetic energy ==<br />
<br />
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<br />
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<br />
sometimes. Hopefully, we will also gain an understanding of when and why they will not work.<br />
<br />
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:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\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] \\<br />
& = & \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\} \\<br />
& & - 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 \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(1)</td></tr></table><br />
<br />
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>. <br />
<br />
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., <br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\frac{\partial u_{i}}{\partial x_{j}} = s_{ij} + \omega_{ij}<br />
</math><br />
</td><td width="5%">(2)</td></tr></table><br />
<br />
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:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\omega_{ij} = \frac{1}{2} \left[ \frac{\partial u_{i}}{\partial x_{j}} - \frac{\partial u_{j}}{\partial x_{i}} \right]<br />
</math><br />
</td><td width="5%">(3)</td></tr></table><br />
<br />
Since the double contraction of a symmetric tensor with an anti-symmetric tensor is identically zero, it follows immediately that:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\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 \\<br />
& = & \left\langle s_{ij} s_{ij} \right\rangle \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(4)</td></tr></table><br />
<br />
Now it is customary to define a new variable k, the average fluctuating kinetic energy per unit mass, by:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
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]<br />
</math><br />
</td><td width="5%">(5)</td></tr></table><br />
<br />
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:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\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\} \\<br />
& & - \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 \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(6)</td></tr></table><br />
<br />
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:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\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\}<br />
</math><br />
</td><td width="5%">(7)</td></tr></table><br />
<br />
If we take the scalar product of this with the fluctuating velocity itself and average, it follows (after some rearrangement) that:<br />
<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\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\} \\<br />
& & - \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\\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(8)</td></tr></table><br />
<br />
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''.<br />
<br />
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:<br />
<br />
* Rate of change of kinetic energy per unit mass due to non-stationarity; i.e., time dependence of the mean:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\frac{\partial k}{\partial t}<br />
</math><br />
</td><td width="5%">(9)</td></tr></table><br />
<br />
* Rate of change of kinetic energy per unit mass due to convection (or advection) by the mean flow through an inhomogenous field :<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
U_{j} \frac{\partial k}{\partial x_{j}}<br />
</math><br />
</td><td width="5%">(10)</td></tr></table><br />
<br />
* Transport of kinetic energy in an inhomogeneous field due respectively to the pressure fluctuations, the turbulence itself, and the viscous stresses:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\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\}<br />
</math><br />
</td><td width="5%">(11)</td></tr></table><br />
<br />
* Rate of production of turbulence kinetic energy from the mean flow(gradient):<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
- \left\langle u_{i}u_{j} \right\rangle \frac{\partial U_{i}}{\partial x_{j}}<br />
</math><br />
</td><td width="5%">(12)</td></tr></table><br />
<br />
* Rate of dissipation of turbulence kinetic energy per unit mass due to viscous stresses:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\epsilon \equiv 2\nu \left\langle s_{ij}s_{ij} \right\rangle<br />
</math><br />
</td><td width="5%">(12)</td></tr></table><br />
<br />
These terms will be discussed in detail in the succeeding sections, and the role of each examined carefully.<br />
<br />
== Rate of dissipation of the turbulence kinetic energy ==<br />
<br />
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.,<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\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_{j} }{\partial x_{i} } \right\rangle \left\langle \frac{\partial u_{i} }{\partial x_{j} } + \frac{\partial u_{j} }{\partial x_{i} } \right\rangle \right\}<br />
</math><br />
</td><td width="5%">(14)</td></tr></table><br />
<br />
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''. <br />
<br />
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. <br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\epsilon = \left\langle \tau^{(v)}_{ij} s_{ij} \right\rangle<br />
</math><br />
</td><td width="5%">(15)</td></tr></table><br />
<br />
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.<br />
<br />
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:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\eta_{K} \equiv \left(\frac{\nu^{3}}{\epsilon} \right)^{1/4}<br />
</math><br />
</td><td width="5%">(16)</td></tr></table><br />
<br />
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.<br />
<br />
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:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\epsilon \propto \frac{u^{3}}{L}<br />
</math><br />
</td><td width="5%">(17)</td></tr></table><br />
<br />
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>.<br />
<br />
Sometimes it is convenient to just ''define'' the "length scale of the energy containing eddies" (or the ''pseudo-integral scale'') as:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
l \equiv \frac{u^{3}}{\epsilon} <br />
</math><br />
</td><td width="5%">(18)</td></tr></table><br />
<br />
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.<br />
<br />
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<br />
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<br />
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<br />
of like the weather man describing the weather. Using equation 18, the Reynolds number dependence of the ratio of the<br />
Kolmorgorov microscale, <math>K</math>, to the pseudo-integral scale, <math>l</math>, can be obtained as:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\frac{\eta_k}{l} = R_l^{-3/4} <br />
</math><br />
</td><td width="5%">(19)</td></tr></table><br />
<br />
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>. <br />
<br />
Where the turbulence Reynolds number, <math>R_l</math>, is defined by:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
R_l \equiv \frac{u l}{\nu} = \frac{u^4}{\nu \epsilon}<br />
</math><br />
</td><td width="5%">(20)</td></tr></table><br />
<br />
'''Example:''' Estimate the Kolmogorov microscale for <math>u = 1 m/s</math> and <math>L = 0.1 m</math> for air and water.<br />
<br />
:'''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>.<br />
<br />
:'''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>.<br />
<br />
'''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<br />
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<br />
reduction in the number of unknowns, particularly those determined primarily by the dissipative scales of motion.<br />
<br />
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<br />
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<br />
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.<br />
<br />
'''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?<br />
<br />
== Kinetic energy of the mean motion and production of turbulence ==<br />
<br />
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:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\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)<br />
</math><br />
</td><td width="5%">(21)</td></tr></table><br />
<br />
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:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
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} } <br />
</math><br />
</td><td width="5%">(22)</td></tr></table><br />
<br />
Unlike the fluctuating equations, there is no need to average here, since all the terms are already averages.<br />
<br />
In exactly the same manner that we rearranged the terms in the equation for the kinetic energy of the fluctuations, we can rearrange the equation for the kinetic energy of the mean flow to obtain:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
<br />
\left[ \frac{\partial}{\partial t} + U_{j} \frac{\partial}{\partial x_{j}} \right] K = \\<br />
& = & \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\} +\\<br />
& + & \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 \\<br />
<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(23)</td></tr></table><br />
<br />
where<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
K\equiv \frac{1}{2} Q^{2} = \frac{1}{2} U_{i}U_{i} <br />
</math><br />
</td><td width="5%">(24)</td></tr></table><br />
<br />
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. <br />
<br />
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 kinetic energy of BOTH the mean and the fluctuations. There is, however, one VERY important difference. This "production" term has the opposite sign in the equation for 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.<br />
<br />
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.<br />
<br />
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.,<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\frac{\partial U_{i}}{\partial x_{j} } = S_{ij} + \Omega_{ij}<br />
</math><br />
</td><td width="5%">(25)</td></tr></table><br />
<br />
where the mean strain rate <math>S_{ij}</math> is defined by<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
S_{ij}=\frac{1}{2}\left[ \frac{\partial U_{i}}{\partial x_{j}} + \frac{\partial U_{j}}{\partial x_{i}} \right]<br />
</math><br />
</td><td width="5%">(26)</td></tr></table><br />
<br />
and the mean rotation rate is defined by<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
\Omega_{ij} = \frac{1}{2}\left[ \frac{\partial U_{i}}{\partial x_{j}} - \frac{\partial U_{j}}{\partial x_{i}} \right]<br />
</math><br />
</td><td width="5%">(27)</td></tr></table><br />
<br />
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:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
- \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}<br />
</math><br />
</td><td width="5%">(28)</td></tr></table><br />
<br />
Equation 28 is an analog to the mean viscous dissipation term given for incompressible flow by:<br />
<br />
<table width="70%"><br />
<tr><td><br />
:<math><br />
T^{(v)}_{ij} \frac{\partial U_{i}}{\partial x_{j}} = T^{(v)}_{ij} S_{ij} = 2 \mu S_{ij}S_{ij}<br />
</math><br />
</td><td width="5%">(29)</td></tr></table><br />
<br />
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.<br />
<br />
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''. <br />
<br />
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".<br />
<br />
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<br />
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.<br />
<br />
== Transport or divergence terms ==<br />
<br />
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:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
& & \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 \\<br />
& = & \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 \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(30)</td></tr></table><br />
<br />
where we have used the divergence theorem - again!<br />
<br />
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.<br />
<br />
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).<br />
<br />
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, <br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
- \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}}<br />
</math><br />
</td><td width="5%">(31)</td></tr></table><br />
<br />
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:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
- \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}}<br />
</math><br />
</td><td width="5%">(32)</td></tr></table><br />
<br />
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.<br />
<br />
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.<br />
<br />
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. <br />
<br />
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.<br />
<br />
== The Intercomponent Transfer of Energy ==<br />
<br />
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.<br />
<br />
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]]<br />
equal to unity (i.e. i=1); i.e.,<br />
<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\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\}<br />
</math><br />
</td><td width="5%">(33)</td></tr></table><br />
<br />
Multiplying this equation by <math> u_{1} </math>, averaging, and rearranging the pressure-velocity gradient term using the chain rule for products yields:<br />
<br />
'''1-component'''<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\left[ \frac{ \partial }{ \partial t} + U_{j} \frac{ \partial }{ \partial x_{j} } \right] \frac{1}{2} \left\langle u^{2}_{1}\right\rangle = \\<br />
& = & \left\langle p \frac{\partial u_{1}}{\partial x_{1} } \right\rangle + \\ <br />
& + & \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\} - \\<br />
& - & \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 \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(34)</td></tr></table><br />
<br />
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. <br />
<br />
Similar equations can be derived for the other fluctuating components with the result that<br />
<br />
'''2-component'''<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\left[ \frac{ \partial }{ \partial t} + U_{j} \frac{ \partial }{ \partial x_{j} } \right] \frac{1}{2} \left\langle u^{2}_{2}\right\rangle = \\<br />
& = & \left\langle p \frac{\partial u_{2}}{\partial x_{2} } \right\rangle + \\ <br />
& + & \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\} - \\<br />
& - & \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 \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(35)</td></tr></table><br />
<br />
and <br />
<br />
'''3-component'''<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\left[ \frac{ \partial }{ \partial t} + U_{j} \frac{ \partial }{ \partial x_{j} } \right] \frac{1}{2} \left\langle u^{2}_{3}\right\rangle = \\<br />
& = & \left\langle p \frac{\partial u_{3}}{\partial x_{3} } \right\rangle + \\ <br />
& + & \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\} - \\<br />
& - & \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 \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(36)</td></tr></table><br />
<br />
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. <br />
<br />
The precise role of the pressure terms can be seen by noting that incompressibility implies that:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left\langle p \frac{\partial u_{j} }{ \partial x_{j} } \right\rangle = 0<br />
</math><br />
</td><td width="5%">(37)</td></tr></table><br />
<br />
It follows immediately that:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\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]<br />
</math><br />
</td><td width="5%">(38)</td></tr></table><br />
<br />
Thus equation 34 can be rewritten as:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
\left[ \frac{ \partial }{ \partial t} + U_{j} \frac{ \partial }{ \partial x_{j} }\right] \frac{1}{2} \left\langle u^{2}_{1} \right\rangle = \\<br />
= & - & \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] + \\<br />
& + & \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\} - \\<br />
& - & \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 \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(39)</td></tr></table><br />
<br />
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.<br />
<br />
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:<br />
<br />
'''1-component'''<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\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} <br />
</math><br />
</td><td width="5%">(40)</td></tr></table><br />
<br />
'''2-component'''<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\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}<br />
</math><br />
</td><td width="5%">(41)</td></tr></table><br />
<br />
'''3-component'''<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\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}<br />
</math><br />
</td><td width="5%">(42)</td></tr></table><br />
<br />
where<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\epsilon_{1} \equiv 2 \nu \left\langle s_{1j} s_{1j} \right\rangle<br />
</math><br />
</td><td width="5%">(43)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\epsilon_{2} \equiv 2 \nu \left\langle s_{2j} s_{2j} \right\rangle<br />
</math><br />
</td><td width="5%">(44)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\epsilon_{3} \equiv 2 \nu \left\langle s_{3j} s_{3j} \right\rangle<br />
</math><br />
</td><td width="5%">(45)</td></tr></table><br />
<br />
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.<br />
<br />
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.<br />
<br />
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.<br />
<br />
'''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.<br />
<br />
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:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
a_{ij} = \left\langle u_{i} u_{j} \right\rangle - \left\langle q^{2} \right\rangle \delta_{ij} / 3<br />
</math><br />
</td><td width="5%">(46)</td></tr></table><br />
<br />
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.<br />
<br />
<br />
{{Turbulence credit wkgeorge}}<br />
<br />
{{Chapter navigation|Reynolds averaged equations|Stationarity and homogeneity}}</div>Malkavian GThttps://www.cfd-online.com/Wiki/Wilcox%27s_k-omega_modelWilcox's k-omega model2011-03-08T16:52:52Z<p>Malkavian GT: /* References */</p>
<hr />
<div>{{Turbulence modeling}}<br />
==Kinematic Eddy Viscosity ==<br />
:<math><br />
\nu _T = {k \over \omega }<br />
</math><br />
<br />
== Turbulence Kinetic Energy ==<br />
:<math><br />
{{\partial k} \over {\partial t}} + U_j {{\partial k} \over {\partial x_j }} = \tau _{ij} {{\partial U_i } \over {\partial x_j }} - \beta ^* k\omega + {\partial \over {\partial x_j }}\left[ {\left( {\nu + \sigma ^* \nu _T } \right){{\partial k} \over {\partial x_j }}} \right]<br />
</math><br />
<br />
== Specific Dissipation Rate==<br />
:<math><br />
{{\partial \omega } \over {\partial t}} + U_j {{\partial \omega } \over {\partial x_j }} = \alpha {\omega \over k}\tau _{ij} {{\partial U_i } \over {\partial x_j }} - \beta \omega ^2 + {\partial \over {\partial x_j }}\left[ {\left( {\nu + \sigma \nu _T } \right){{\partial \omega } \over {\partial x_j }}} \right]<br />
</math><br />
<br />
==Closure Coefficients and Auxilary Relations==<br />
<br />
:<math><br />
\alpha = {{5} \over {9}} <br />
</math><br />
<br />
:<math><br />
\beta = {{3} \over {40}} <br />
</math><br />
<br />
:<math><br />
\beta^* = {9 \over {100}}<br />
</math><br />
<br />
:<math><br />
\sigma = {1 \over 2} <br />
</math><br />
<br />
:<math><br />
\sigma ^* = {1 \over 2} <br />
</math><br />
<br />
:<math><br />
\varepsilon = \beta ^* \omega k<br />
</math><br />
<br />
== References ==<br />
<br />
#{{reference-paper|author=Wilcox, D.C. |year=1988|title=Re-assessment of the scale-determining equation for advanced turbulence models|rest=AIAA Journal, vol. 26, no. 11, pp. 1299-1310}}<br />
<br />
[[Category:Turbulence models]]</div>Malkavian GThttps://www.cfd-online.com/Wiki/SST_k-omega_modelSST k-omega model2011-02-28T21:36:21Z<p>Malkavian GT: </p>
<hr />
<div>{{Turbulence modeling}}<br />
<br />
The SST k-ω turbulence model [Menter 1993] is a [[Two equation turbulence models|two-equation]] [[Eddy viscosity|eddy-viscosity]] model which has become very popular. The shear stress transport (SST) formulation combines the best of two worlds. The use of a k-ω formulation in the inner parts of the boundary layer makes the model directly usable all the way down to the wall through the viscous sub-layer, hence the SST k-ω model can be used as a [[Low-Re turbulence model]] without any extra damping functions. The SST formulation also switches to a k-ε behaviour in the free-stream and thereby avoids the common k-ω problem that the model is too sensitive to the [[Turbulence free-stream boundary conditions|inlet free-stream turbulence properties]]. Authors who use the SST k-ω model often merit it for its good behaviour in adverse pressure gradients and separating flow. The SST k-ω model does produce a bit too large turbulence levels in regions with large normal strain, like stagnation regions and regions with strong acceleration. This tendency is much less pronounced than with a normal k-ε model though.<br />
<br />
==Kinematic Eddy Viscosity ==<br />
:<math><br />
\nu _T = {a_1 k \over \mbox{max}(a_1 \omega, S F_2) }<br />
</math><br />
<br />
== Turbulence Kinetic Energy ==<br />
:<math><br />
{{\partial k} \over {\partial t}} + U_j {{\partial k} \over {\partial x_j }} = P_k - \beta ^* k\omega + {\partial \over {\partial x_j }}\left[ {\left( {\nu + \sigma_k \nu _T } \right){{\partial k} \over {\partial x_j }}} \right]<br />
</math><br />
<br />
== Specific Dissipation Rate==<br />
:<math><br />
{{\partial \omega } \over {\partial t}} + U_j {{\partial \omega } \over {\partial x_j }} = \alpha S^2 - \beta \omega ^2 + {\partial \over {\partial x_j }}\left[ {\left( {\nu + \sigma_{\omega} \nu _T } \right){{\partial \omega } \over {\partial x_j }}} \right] + 2( 1 - F_1 ) \sigma_{\omega 2} {1 \over \omega} {{\partial k } \over {\partial x_i}} {{\partial \omega } \over {\partial x_i}} <br />
</math><br />
<br />
==Closure Coefficients and Auxilary Relations==<br />
<br />
:<math><br />
F_2=\mbox{tanh} \left[ \left[ \mbox{max} \left( { 2 \sqrt{k} \over \beta^* \omega y } , { 500 \nu \over y^2 \omega } \right) \right]^2 \right]<br />
</math><br />
<br />
:<math><br />
P_k=\mbox{min} \left(\tau _{ij} {{\partial U_i } \over {\partial x_j }} , 10\beta^* k \omega \right)<br />
</math><br />
<br />
:<math><br />
F_1=\mbox{tanh} \left\{ \left\{ \mbox{min} \left[ \mbox{max} \left( {\sqrt{k} \over \beta ^* \omega y}, {500 \nu \over y^2 \omega} \right) , {4 \sigma_{\omega 2} k \over CD_{k\omega} y^2} \right] \right\} ^4 \right\}<br />
</math><br />
<br />
:<math><br />
CD_{k\omega}=\mbox{max} \left( 2\rho\sigma_{\omega 2} {1 \over \omega} {{\partial k} \over {\partial x_i}} {{\partial \omega} \over {\partial x_i}}, 10 ^{-10} \right )<br />
</math><br />
<br />
:<math><br />
\phi = \phi_1 F_1 + \phi_2 (1 - F_1)<br />
</math><br />
<br />
:<math><br />
\alpha_1 = {{5} \over {9}}, \alpha_2 = 0.44 <br />
</math><br />
<br />
:<math><br />
\beta_1 = {{3} \over {40}}, \beta_2 = 0.0828 <br />
</math><br />
<br />
:<math><br />
\beta^* = {9 \over {100}}<br />
</math><br />
<br />
:<math><br />
\sigma_{k1} = 0.85, \sigma_{k2} = 1<br />
</math><br />
<br />
:<math><br />
\sigma_{\omega 1} = 0.5, \sigma_{\omega 2} = 0.856<br />
</math><br />
<br />
== References ==<br />
<br />
#{{reference-paper|author=Menter, F. R.|year=1993|title=Zonal Two Equation k-ω Turbulence Models for Aerodynamic Flows|rest=AIAA Paper 93-2906}}<br />
#{{reference-paper|author=Menter, F. R. |year=1994|title=Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications|rest=AIAA Journal, vol. 32, no 8. pp. 1598-1605}}<br />
<br />
[[Category:Turbulence models]]</div>Malkavian GT