http://www.cfd-online.com/W/index.php?title=Special:Contributions/Discoganya&feed=atom&limit=50&target=Discoganya&year=&month=CFD-Wiki - User contributions [en]2016-08-27T00:52:04ZFrom CFD-WikiMediaWiki 1.16.5http://www.cfd-online.com/Wiki/Adams_bashforthAdams bashforth2005-12-10T00:02:09Z<p>Discoganya: </p>
<hr />
<div>#REDIRECT[[Adams methods]]</div>Discoganyahttp://www.cfd-online.com/Wiki/Adams_methodsAdams methods2005-12-10T00:01:27Z<p>Discoganya: </p>
<hr />
<div>Adams methods are a subset of the general family of multistep methods used for the numerical integration of initial value problems based on odes. Multistep methods benefit from the fact that the computation has been going on for a while and use previously computed values of the solution (BDF methods) or the right hand side (Adams methods) to approximate the solution at the next step.<br />
<br />
Adams methods begin by the integral approach,<br />
<br />
:<math><br />
y^\prime = f(t,y)<br />
</math><br />
<br />
<br />
:<math><br />
y(t_{N+1}) = y(t_{n}) + \int_{t_n}^{t_{n+1}} y^\prime (t) dt = \int_{t_n}^{t_{n+1}} f(t,y(t)) dt<br />
</math><br />
<br />
Since <math>f</math> is unknown in the interval <math>t_n</math> to <math>t_{n+1}</math> it is approximated by an interpolating [[polynomial]] <math>p(t)</math> using the previously computed steps <math>t_{n},t_{n-1},t_{n-2} ...</math> and the current step at <math>t_{n+1}</math> if an implicit method is desired.</div>Discoganyahttp://www.cfd-online.com/Wiki/Adams_methodsAdams methods2005-12-10T00:00:18Z<p>Discoganya: </p>
<hr />
<div>Adams methods are a subset of the general family of multistep methods used for the numerical integration of initial value problems based on odes. Multistep methods benefit from the fact that the computation has been going on for a while and use previously computed values of the solution (BDF methods) or the right hand side (Adams methods) to approximate the solution at the next step.<br />
<br />
Adams methods begin by the integral approach,<br />
<br />
:<math><br />
y^\prime = f(t,y)<br />
</math><br />
<br />
<br />
:<math><br />
y(t_{N+1}) = y(t_{n}) + \int_{t_n}^{t_{n+1}} y^\prime (t) dt = \int_{t_n}^{t_{n+1}} f(t,y(t)) dt<br />
</math><br />
<br />
Since <math>f</math> is unknown in the interval <math>t_n</math> to <math>t_{n+1}</math> it is approximated by an interpolating [[polynomial]] <math>p(t)</math> using the previously computed steps <math>t_{n},t_{n-1},t_{n-2} ...</math></div>Discoganyahttp://www.cfd-online.com/Wiki/Adams_methodsAdams methods2005-12-09T23:58:11Z<p>Discoganya: </p>
<hr />
<div>Adams methods are a subset of the general family of multistep methods used for the numerical integration of initial value problems based on odes. Multistep methods benefit from the fact that the computation has been going on for a while and use previously computed values of the solution (BDF methods) or the right hand side (Adams methods) to approximate the solution at the next step.<br />
<br />
Adams methods begin by the integral approach,<br />
<br />
:<math><br />
y^\prime = f(t,y)<br />
</math><br />
<br />
<br />
:<math><br />
y(t_{N+1}) = y(t_{n}) + \int_{t_n}^{t_{n+1}} y^\prime (t) dt = \int_{t_n}^{t_{n+1}} f(t,y(t)) dt<br />
</math><br />
<br />
Since f is unknown in the interval t_n to t_{n+1}</div>Discoganyahttp://www.cfd-online.com/Wiki/Adams_methodsAdams methods2005-12-09T23:57:26Z<p>Discoganya: </p>
<hr />
<div>Adams methods are a subset of the general family of multistep methods used for the numerical integration of initial value problems based on odes. Multistep methods benefit from the fact that the computation has been going on for a while and use previously computed values of the solution (BDF methods) or the right hand side (Adams methods) to approximate the solution at the next step.<br />
<br />
Adams methods begin by the integral approach,<br />
<br />
:<math><br />
y^\prime = f(t,y)<br />
</math><br />
<br />
<br />
:<math><br />
y(t_{N+1}) = y(t_{n}) + \int_{t_n}^{t_{n+1}} y^\prime (t) dt = \int_{t_n}^{t_{n+1}} f(t,y(t)) dt<br />
</math></div>Discoganyahttp://www.cfd-online.com/Wiki/Adams_methodsAdams methods2005-12-09T23:57:02Z<p>Discoganya: </p>
<hr />
<div>Adams methods are a subset of the general family of multistep methods used for the numerical integration of initial value problems based on odes. Multistep methods benefit from the fact that the computation has been going on for a while and use previously computed values of the solution (BDF methods) or the right hand side (Adams methods) to approximate the solution at the next step.<br />
<br />
Adams methods begin by the integral approach,<br />
<br />
:<math><br />
y^\prime = f(t,y)<br />
</math><br />
<br />
<br />
:<math><br />
y(t_{N+1}) = y(t_{n}) + \int_{t_n}^{t_{n+1}} y^\prime (s) ds = \int_{t_n}^{t_{n+1}} f(s,y(s)) ds<br />
</math></div>Discoganyahttp://www.cfd-online.com/Wiki/Runge_kuttaRunge kutta2005-12-09T23:50:32Z<p>Discoganya: </p>
<hr />
<div>#REDIRECT [[Runge Kutta methods]]</div>Discoganyahttp://www.cfd-online.com/Wiki/Runge_Kutta_methodsRunge Kutta methods2005-11-23T18:29:02Z<p>Discoganya: </p>
<hr />
<div>Runge Kutta (RK) methods are an important class of methods for integrating initial value problems formed by [[ODE]]s. Runge Kutta methods encompass a wide selection of numerical methods and some commonly used methods such as Explicit or Implicit [[Euler's Method]], the implicit midpoint rule and the trapezoidal rule are actually simplified versions of a general RK method.<br />
<br />
For the ODE,<br />
<br />
:<math><br />
y^\prime = f(t,y)<br />
</math><br />
<br />
the basic idea is to build a series of "stages", <math>k_i</math> that approximate the solution <math>y</math> at various points using samples of <math>f</math> from other stages. Finally, the numerical solution <math>u_{n+1}</math> is constructed from a linear combination of <math>u_n</math> and all the precomputed stages.<br />
<br />
Since the computation of one stage may involve other stages <math>k_i</math> the right hand side <math>f</math> is evaluated in a complicated nonlinear way. The most famous classical RK scheme is described below.<br />
<br />
= Fourth order Runge-Kutta method =<br />
<br />
The fourth order Runge-Kutta method could be summarized as:<br />
<br />
==Algorithm==<br />
::<math>y^\prime = f\left( {t,y} \right) </math><br />
::<math>k_1 = hf\left( {t_n ,y_n } \right) </math><br />
::<math>k_2 = hf\left( {t_n + {h \over 2},y_n + {{k_1 } \over 2}} \right) </math><br />
::<math>k_3 = hf\left( {t_n + {h \over 2},y_n + {{k_2 } \over 2}} \right) </math><br />
::<math>k_4 = hf\left( {t_n + h,y_n + k_3 } \right) </math><br />
::<math>y_{n + 1} = y_n + {{k_1 } \over 6} + {{k_2 } \over 3} + {{k_3 } \over 3} + {{k_4 } \over 6} </math><br />
<br />
<br />
<br />
{{stub}} <br />
----<br />
<i> Return to [[Numerical methods | Numerical Methods]] </i></div>Discoganyahttp://www.cfd-online.com/Wiki/Runge_Kutta_methodsRunge Kutta methods2005-11-23T18:27:50Z<p>Discoganya: </p>
<hr />
<div>Runge Kutta (RK) methods are an important class of methods for integrating initial value problems formed by [[ODE]]s. Runge Kutta methods encompass a wide selection of numerical methods and some commonly used methods such as Explicit or Implicit [[Euler's Method]], the implicit midpoint rule and the trapezoidal rule are actually simplified versions of a general RK method.<br />
<br />
For the ODE,<br />
<br />
:<math><br />
y^\prime = f(t,y)<br />
</math><br />
<br />
the basic idea is to build a series of "stages", <math>k_i</math> that approximate the solution <math>y</math> at various points using samples of <math>f</math> from other stages. Finally, the numerical solution <math>u_{n+1}</math> is constructed from a linear combination of <math>u_n</math> and all the precomputed stages.<br />
<br />
Since the computation of one stage may involve other stages <math>k_i</math> the right hand side <math>f</math> is evaluated in a complicated nonlinear way. The most famous classical RK scheme is described below.<br />
<br />
= Fourth order Runge-Kutta method =<br />
<br />
The fourth order Runge-Kutta method could be summarized as:<br />
<br />
==Algorithm==<br />
::<math>\dot y = f\left( {x,y} \right) </math><br />
::<math>k_1 = hf\left( {x_n ,y_n } \right) </math><br />
::<math>k_2 = hf\left( {x_n + {h \over 2},y_n + {{k_1 } \over 2}} \right) </math><br />
::<math>k_3 = hf\left( {x_n + {h \over 2},y_n + {{k_2 } \over 2}} \right) </math><br />
::<math>k_4 = hf\left( {x_n + h,y_n + k_3 } \right) </math><br />
::<math>y_{n + 1} = y_n + {{k_1 } \over 6} + {{k_2 } \over 3} + {{k_3 } \over 3} + {{k_4 } \over 6} </math><br />
<br />
<br />
<br />
{{stub}} <br />
----<br />
<i> Return to [[Numerical methods | Numerical Methods]] </i></div>Discoganyahttp://www.cfd-online.com/Wiki/CFD-ACE_FAQCFD-ACE FAQ2005-11-23T18:08:35Z<p>Discoganya: </p>
<hr />
<div>== CFD-ACE ==<br />
<br />
=== Question 1 ===<br />
<br />
Answer 1<br />
<br />
=== Question 2 ===<br />
<br />
Answer 2<br />
<br />
== CFD-GEOM ==<br />
<br />
[[Category: FAQ's]]<br />
<br />
{{Stub}}</div>Discoganyahttp://www.cfd-online.com/Wiki/FAQ%27sFAQ's2005-11-23T18:07:24Z<p>Discoganya: </p>
<hr />
<div>;[[General CFD FAQ]]<br />
:An FAQ for general CFD related questions.<br />
<br />
;[[CFD-ACE FAQ]]<br />
:An FAQ for the software sold by ESI Software - [[CFD-ACE]], [[CFD-GEOM]] ...<br />
<br />
;[[Ansys FAQ]]<br />
:An FAQ for the software products sold by Ansys Inc - CFX, ICEM CFD, ...<br />
<br />
;[[CD-adapco FAQ]]<br />
:An FAQ for the software products sold by CD-adapco - STAR-CD, STAR-CCM+, ...<br />
<br />
;[[Fluent FAQ]]<br />
:An FAQ for the software products sold by Fluent Inc - FLUENT, Gambit, Tgrid, POLYFLOW, FloWizard, IcePak, AirPak, ...<br />
<br />
;[[Numeca FAQ]]<br />
:An FAQ for the software sold by Numeca - Fine, HEXPRESS, ...<br />
<br />
;[[CHAM FAQ]]<br />
:An FAQ for the software sold by CHAM - Phoenics, ...</div>Discoganyahttp://www.cfd-online.com/Wiki/LES_filtersLES filters2005-09-22T20:52:04Z<p>Discoganya: </p>
<hr />
<div>In [[Large eddy simulation]] (LES) only the large scale motions of the flow are solved for by filtering out the small and [[universal]] eddies. In practical applications of some [[SGS models]], implicit filtering is done by the grid itself and programmers need not worry about the filtering operation. The values of velocity on the grid are the filtered values of velocity. However, for some SGS models, such as the [[Dynamic subgrid-scale model]] an explicit filtering step is required to compute the [[SGS stress]] tensor. Additionally, in the theoretical analysis of LES, filtering a function is defined as convoluting the function with a filtering kernel, just as is typically done in electrical engineering.<br />
<br />
Some of the commonly used filters are defined below. In all cases, <math> /Delta </math> is the filter width, <math> G(x) </math> is the filtering kernel in physical space and <math>\widehat{G(k)}</math> is the filtering kernel in [[Fourier transform|Fourier]]-[[wavenumber]] space.<br />
<br />
== Box filter ==<br />
<br />
The Box filter is the same as the "grid filter" whereby the filter cuts off the values of the function beyond a half filter width away.<br />
<br><br />
:<math><br />
G(x) = \frac{1}{\Delta} H\left( \frac{1}{2}\Delta -|x| \right)<br />
</math><br />
where H is the [[Heaviside function]],<br />
<br><br />
:<math><br />
\widehat{G(k)} = \frac{\sin \left (\frac{1}{2} k \Delta \right)}{ \frac{1}{2} k \Delta}<br />
</math><br />
<br />
== Gaussian filter ==<br />
<br />
The Gaussian filter is a normalized [[Gaussian function]]. The Fourier transform of a Gaussian function is also a Gaussian, hence the G(x) and \widehat{G(k)} have very similar forms,<br />
<br><br />
:<math><br />
G(x) = \left( \frac{6}{\pi \Delta^2} \right)^{1/2} e^{ \frac{-6 x^2}{\Delta^2}}<br />
</math><br />
<br><br />
:<math><br />
\widehat{G(k)} = e^{ \frac{-k^2 \Delta^2}{24} }<br />
</math></div>Discoganyahttp://www.cfd-online.com/Wiki/LES_filtersLES filters2005-09-22T20:37:35Z<p>Discoganya: </p>
<hr />
<div>In [[Large eddy simulation]] (LES) only the large scale motions of the flow are solved for by filtering out the small and [[universal]] eddies. In practical applications of some [[SGS models]], implicit filtering is done by the grid itself and programmers need not worry about the filtering operation. The values of velocity on the grid are the filtered values of velocity. However, for some SGS models, such as the [[Dynamic subgrid-scale model]] an explicit filtering step is required to compute the [[SGS stress]] tensor. Additionally, in the theoretical analysis of LES, filtering a function is defined as convoluting the function with a filtering kernel, just as is typically done in electrical engineering.<br />
<br />
Defined below are some of the commonly used filters:</div>Discoganyahttp://www.cfd-online.com/Wiki/Rossby_numberRossby number2005-09-22T20:29:13Z<p>Discoganya: </p>
<hr />
<div>[[Category:Dimensionless parameters]]<br />
<br />
In [[rotating flows]] (e.g. [[geophysical flows]]), the Rossby number is defined as the ratio of the advective acceleration to the [[Corioilis force|Coriolis acceleration]]. Alternately, it may be thought of as the ratio of the [[inertial force]] to the [[Coriolis force]].<br />
<br />
:<math><br />
Ro = \frac{\left( \frac{U^2}{L} \right) }{ \left( \Omega U \right) } = \frac{U}{\Omega L}<br />
</math><br />
<br />
where <math>\Omega</math> is the angular speed of rotation, U is the velocity scale and L is the lenth scale.</div>Discoganyahttp://www.cfd-online.com/Wiki/Rossby_numberRossby number2005-09-22T20:28:50Z<p>Discoganya: </p>
<hr />
<div>In [[rotating flows]] (e.g. [[geophysical flows]]), the Rossby number is defined as the ratio of the advective acceleration to the [[Corioilis force|Coriolis acceleration]]. Alternately, it may be thought of as the ratio of the [[inertial force]] to the [[Coriolis force]].<br />
<br />
:<math><br />
Ro = \frac{\left( \frac{U^2}{L} \right) }{ \left( \Omega U \right) } = \frac{U}{\Omega L}<br />
</math><br />
<br />
where <math>\Omega</math> is the angular speed of rotation, U is the velocity scale and L is the lenth scale.</div>Discoganyahttp://www.cfd-online.com/Wiki/Knudsen_numberKnudsen number2005-09-22T20:23:47Z<p>Discoganya: </p>
<hr />
<div>[[Category: Dimensionless parameters]]<br />
<br />
The Knudsen number is defined as <br />
:<math> Kn\equiv \frac{\lambda}{L} </math><br />
<br />
where <br />
*<math> \lambda </math> is the mean free path of the molecules<br />
*<math> L </math> is the characteristic dimension<br />
<br />
Knudsen number is a measure of the rarefaction of the flow. The various [[Flow regimes ]] can be classifed based on Knudsen number.<br />
<br />
The Knudsen number is related to the [[Mach number]] and the [[Reynolds number]] by the following relation:<br />
<br />
:<math><br />
Kn = \frac{Ma}{Re} \; \sqrt{ \frac{\pi \gamma}{2}}<br />
</math><br />
<br />
where <math>\gamma</math> is the [[ratio of specific heats]].</div>Discoganyahttp://www.cfd-online.com/Wiki/Capillary_numberCapillary number2005-09-22T20:20:13Z<p>Discoganya: </p>
<hr />
<div>[[Category:Dimensionless parameters]]<br />
In small scales flows where the effect of surface tension is important, the Capillary number is defined as the ratio of the viscous force to the surface tension force.<br />
<br />
:<math><br />
Ca = \frac{\mu U}{\sigma}<br />
</math><br />
<br />
where <math>\mu</math> is the [[dynamic viscosity]], <math>U</math> is the velocity scale and <math>\sigma</math> is the surface tension.</div>Discoganyahttp://www.cfd-online.com/Wiki/Smagorinsky-Lilly_modelSmagorinsky-Lilly model2005-09-21T13:51:50Z<p>Discoganya: latex stuff</p>
<hr />
<div>The Smagorinsky model could be summerised as:<br />
:<math><br />
\tau _{ij} - \frac{1}{3}\tau _{kk} \delta _{ij} = - 2\left( {C_s \bar \Delta } \right)^2 \left| {\bar S} \right|S_{ij} <br />
</math> <br><br />
<br />
In the Smagorinsky-Lilly model, the eddy-viscosity is modeled by <br><br />
<br />
<br />
:<math><br />
\mu _{sgs} = \rho \left( {C_s \bar \Delta } \right)^2 \left| {\bar S} \right|<br />
</math> <br />
<br><br />
<br />
Where the filter width is usually taken to be<br />
:<math><br />
\bar \Delta = \left( \mbox{Volume} \right)^{\frac{1}{3}} <br />
</math> <br />
<br><br />
and <br />
:<math><br />
\bar S = \sqrt {2S_{ij} S_{ij} } <br />
</math><br />
<br />
The effective viscosity is calculated from <br><br />
:<math><br />
\mu _{eff} = \mu _{mol} + \mu _{sgs} <br />
</math><br />
The Smagorinsky constant usually has the value: <br />
:<math><br />
C_s = 0.1 - 0.2<br />
</math></div>Discoganyahttp://www.cfd-online.com/Wiki/Talk:Smagorinsky-Lilly_modelTalk:Smagorinsky-Lilly model2005-09-20T23:19:22Z<p>Discoganya: </p>
<hr />
<div>The filter width is the nabla operator to the one-third power? I understand that sometimes the filter width is the cube root of the cell volume...</div>Discoganyahttp://www.cfd-online.com/Wiki/Richardson_numberRichardson number2005-09-20T18:07:53Z<p>Discoganya: </p>
<hr />
<div>[[Category:Dimensionless parameters]]<br />
In the stability of continuously stratified parallel shear flows the ratio of (the squares of) the buoyancy frequency to the background velocity gradient is known as the (gradient) Richardson number.<br />
<br><br />
:<math><br />
Ri = \frac{N^2}{U_z^2} <br />
</math><br />
<br><br />
:<math><br />
N = \mbox{Buoyancy frequncy} = -\frac{g}{\rho_0} \frac{\partial \bar{\rho}}{\partial z} <br />
</math><br />
<br />
Here <math> \rho_0 </math> is the reference density and <math> \bar{\rho} </math> is the background density field.</div>Discoganyahttp://www.cfd-online.com/Wiki/Taylor_numberTaylor number2005-09-20T18:00:16Z<p>Discoganya: </p>
<hr />
<div>[[Category:Dimensionless parameters]]<br />
For the flow between two concentric cylinders with radii <math>R_1</math> and <math>R_2</math>, rotating at <math>\Omega_1</math> and <math>\Omega_2</math>, the Taylor number is defined as, <br />
:<math><br />
Ta = \frac{\mbox{Centrifugal Force}}{\mbox{Viscous Force}} = 4 \left( \frac{\Omega_1 R_1^2 - \Omega_2 R_2^2}{R_2^2 - R_1^2}\right) \; \frac{\Omega_1 d^4}{\nu}<br />
</math><br />
<br />
where <math> d =</math> gap width between the cylinders and <math> \nu =</math> the [[dynamic viscosity]].</div>Discoganyahttp://www.cfd-online.com/Wiki/Ekman_numberEkman number2005-09-20T17:52:14Z<p>Discoganya: </p>
<hr />
<div>[[Category:Dimensionless parameters]]<br />
:<math><br />
Ek = \frac{\nu}{\Omega R^2}<br />
</math></div>Discoganyahttp://www.cfd-online.com/Wiki/Ekman_numberEkman number2005-09-20T17:52:00Z<p>Discoganya: </p>
<hr />
<div>[[Catergory:Dimensionless parameters]]<br />
:<math><br />
Ek = \frac{\nu}{\Omega R^2}<br />
</math></div>Discoganyahttp://www.cfd-online.com/Wiki/Rayleigh_numberRayleigh number2005-09-20T17:50:28Z<p>Discoganya: </p>
<hr />
<div>[[Category:Dimensionless parameters]]<br />
The Rayleigh number is the ratio of the buoyancy force to the viscous force in a medium.<br />
<br><br />
:<math><br />
Ra = \frac{g \alpha \Gamma d^4}{\nu \kappa}<br />
</math><br />
<br />
where <math>g =</math> acceleration due to [[gravity]]<br />
<math>\alpha =</math> [[coefficient of thermal expansion]]<br />
<math>\Gamma =</math> background [[temperature]] [[gradient]]<br />
<math>d =</math> length scale of flow<br />
<math>\nu =</math> [[kinematic viscosity]]<br />
<math>\kappa =</math> [[thermal diffusivity]]</div>Discoganyahttp://www.cfd-online.com/Wiki/Froude_numberFroude number2005-09-20T17:43:37Z<p>Discoganya: </p>
<hr />
<div>[[Category:Dimensionless parameters]]<br />
<br />
The Froude number is the ratio of the flow speed to the speed of infinitesimal (incompressible) gravity waves in the same medium. It has application in phenomena such as [[Hydraulic Jump]].<br />
<br><br />
:<math><br />
Fr = \frac{U}{\sqrt{gH}}<br />
</math></div>Discoganyahttp://www.cfd-online.com/Wiki/Froude_numberFroude number2005-09-20T17:42:40Z<p>Discoganya: </p>
<hr />
<div>[[Category:Dimensionless Parameters]]<br />
<br />
The Froude number is the ratio of the flow speed to the speed of infinitesimal (incompressible) gravity waves in the same medium. It has application in phenomena such as [[Hydraulic Jump]].<br />
<br><br />
:<math><br />
Fr = \frac{U}{\sqrt{gH}}<br />
</math></div>Discoganyahttp://www.cfd-online.com/Wiki/User:DiscoganyaUser:Discoganya2005-09-20T16:26:25Z<p>Discoganya: </p>
<hr />
<div>I'm a grad student at the [http://www.berkeley.edu University of California, Berkeley].</div>Discoganyahttp://www.cfd-online.com/Wiki/CFD-Wiki:Community_portalCFD-Wiki:Community portal2005-09-20T16:25:53Z<p>Discoganya: /* Wiki editors - Who we are */</p>
<hr />
<div>This section is intended for people who work on adding content to the Wiki. If you still haven't contributed to the Wiki please do so! We need your help and everyone is welcome to join our team of Wiki editors. For technical details and guidelines on how to contribute material to the Wiki read the [[Help:Contents]] page. We also have a [http://www.cfd-online.com/Forum/wiki.cgi discussion forum] for us Wiki editors. These can also be reached via the links in the navigation section to the left.<br />
<br />
The Wiki has gotten off to a flying start. We have a basic structure and more and more people are joining us. The plan is to launch the Wiki publicly by the end of October (which year depends on us ;-)<br />
<br />
== What's in the works ==<br />
<br />
You who do significant additions to the Wiki, please add some information about your work, plans and progress here so that others can see what you are working on and perhaps help, monitor, come with suggestions and most importantly, be inspired by.<br />
<br />
* We now have a [http://www.cfd-online.com/Forum/wiki.cgi dicussion forum] where we can discuss our work on the Wiki. We should use the forum to coordinate our efforts and agree on common standards and styles. --[[User:Jola|Jola]] 10:44, 18 September 2005 (MDT)<br />
<br />
* I'm working hard on recruiting more Wiki editors. We need all the help we can get with this ambitious project. I'm in email contact with several experienced CFD guys. In addition, I have posted several invitations to join us on the discussion forums. I also posted a job-ad in the jobs database here at CFD Online. --[[User:Jola|Jola]] 10:44, 18 September 2005 (MDT)<br />
<br />
* The [[Numerical methods|numerics section]] is groving very quickly now and has a very ambitious table of content. Michail and zxaar are working hard on it. It is a very large area though so if you can help please do so. --[[User:Jola|Jola]] 10:44, 18 September 2005 (MDT)<br />
<br />
* We have been allowed to base the [[Turbulence|turbulence section]] on an excellent book on turbulence written by Professor William K. George. Pavitran is responsible for this work. Expect to see significant additions in this section. --[[User:Jola|Jola]] 10:44, 18 September 2005 (MDT)<br />
<br />
* The [[Validation and test cases|validation and test-case section]] has gotten off to a flying start. Both Praveen and Jasond have already added several cases. --[[User:Jola|Jola]] 10:44, 18 September 2005 (MDT)<br />
<br />
* We have a first [[Special topics|special topics section]] on [[Combustion|combustion]]. ForMat has created a basic structure and are adding content to it. --[[User:Jola|Jola]] 10:44, 18 September 2005 (MDT)<br />
<br />
* I've started work on creating a first best-practise guide. We need some sort of first example for how one of these should look. I chose to start with a guide for [[Best practise guidelines for turbomachinery CFD | turbomachinery CFD]]. --[[User:Jola|Jola]] 10:44, 18 September 2005 (MDT)<br />
<br />
== What needs to be done ==<br />
<br />
* We have many turbulence models listed in the [[turbulence modeling]] section which still lack any description. Feel free to pick a model that you are familiar with and write a description of it. --[[User:Jola|Jola]] 01:50, 13 September 2005 (MDT)<br />
<br />
* The [[FAQ's | FAQ]] section is still very thin. If you are familiar with one of the larger CFD codes please consider adding a few questions and answers to the FAQ. --[[User:Jola|Jola]] 08:28, 13 September 2005 (MDT)<br />
<br />
* If you are an experienced CFD engineer and an expert in a special application area you are very welcome to start a [[Best practise guidelines|best practise guideline]] for your speciality. --[[User:Jola|Jola]] 10:44, 18 September 2005 (MDT)<br />
<br />
* ''... add your suggestions on what should be done here''<br />
<br />
== Wiki editors - Who we are ==<br />
<br />
Add your name here if you make contributions to the wiki. The order is alphabetical based on the last name.<br />
<br />
* [[User:praveen]] - Praveen. C<br />
* [[User:jasond]] - Jason D.<br />
* [[User:Pavitran]] - Pavitran. D<br />
* [[User:ForMat]] - Matej Forman<br />
* [[User:harish]] - Harish Gopalan<br />
* [[User:Michail]] - Michail Kirichkov<br />
* [[User:Discoganya]] - Sujit Kirpekar<br />
* [[User:jola]] - Jonas Larsson<br />
* [[User:anurag]] - Anurag Sharma<br />
* [[User:zxaar]] - Arjun Yadav</div>Discoganyahttp://www.cfd-online.com/Wiki/Strouhal_numberStrouhal number2005-09-20T00:18:47Z<p>Discoganya: </p>
<hr />
<div>The Strouhal Number is a dimensionless parameter describing the oscillating frequency of a flow. Usually it is defined as:<br />
<br><br />
:<math><br />
St = \frac{fL}{U}<br />
</math><br />
where <math>f</math> is the frequency of the flow (e.g. vortex shedding frequency), <math>L</math> is a characteristic length scale of the flow and <math>U</math> is a characteristic velocity.<br />
<p><br />
The Strouhal Number is a function of the [[Reynolds Number]] typically between 20 < Re < 20,000.</div>Discoganyahttp://www.cfd-online.com/Wiki/Large_Eddy_SimulationLarge Eddy Simulation2005-09-20T00:13:44Z<p>Discoganya: </p>
<hr />
<div>#REDIRECT[[Large eddy simulation (LES)]]</div>Discoganyahttp://www.cfd-online.com/Wiki/Large_eddy_simulationLarge eddy simulation2005-09-20T00:12:40Z<p>Discoganya: </p>
<hr />
<div>#REDIRECT[[Large eddy simulation (LES)]]</div>Discoganyahttp://www.cfd-online.com/Wiki/LesLes2005-09-20T00:12:06Z<p>Discoganya: </p>
<hr />
<div>#REDIRECT [[Large eddy simulation (LES)]]</div>Discoganyahttp://www.cfd-online.com/Wiki/Flow_across_a_square_cylinderFlow across a square cylinder2005-09-20T00:08:30Z<p>Discoganya: </p>
<hr />
<div>The flow across a square cylinder is an important test case for the validation of separated flows in the turbulent regime. In the past, it has been regularly used to validate [[Large Eddy Simulation]] (LES) models.<br />
<br />
Experiments studying the flow across a square cylinder have been available since about the time of Vickery (1966), while modern simulations usually tend to reproduce the results of Lyn and Rodi (1994) whose data set for [[Reynolds Number|Re]] = 22,000 is publicly available.<br />
<br />
The flow involves separation and coherent vortex shedding. In trying to benchmark a code with the experimental data, the following items are usually compared:<br />
<br />
* [[Strouhal Number]] which is the easiest to reproduce.<br />
* Mean recirculation length.<br />
* Global drag and lift statistics, e.g. mean lift coefficient, rms lift coefficient, mean drag coefficient and rms drag coefficient.<br />
* Spatial distributions of time-averaged and phase-averaged velocities, pressure and Reynolds stresses.<br />
<br />
= References =<br />
<br />
== Experiments ==<br />
<br />
*<b>B. J. Vickery.</b> Fluctuating lift and drag on a long cylinder of square cross section in a smooth and in a turbulent stream. Journal of Fluid Mechanics, 25: 481-494, 1966.<br />
*<b>B. E. Lee.</b> The effect of turbulence on the surface pressure field of a square prism. Journal of Fluid Mechanics, 69(2): 263-292, 1975.<br />
*<b>P. W. Bearman and E. D. Obasaju.</b> An experimental study of pressure fluctuations on fixed and oscillating square-section cylinders. 119: 297-321, 1982.<br />
*<b>I. McLean and I. Gartshore.</b> Spanwise correlations of pressure on a rigid square section cylinder. 41-44:779-808, 1992.<br />
*<b>C. Norberg.</b> Flow around rectangular cylinders: Pressure forces and wake frequencies. Journal of Wind Engineering and Industrial Aerodynamics, 49: 187-196, 1993.<br />
*<b>D. A. Lyn and W. Rodi.</b> The flapping shear layer formed by flow separation from the forward corner of a square cylinder. Journal of Fluid Mechanics, 267: 353-376, 1994.<br />
*<b>D. A. Lyn, S. Einav, W. Rodi, and J.-H. Park.</b> A laser-doppler velocimetry study of ensemble-averaged characteristics of the turbulent near wake of a square cylinder. Journal of Fluid Mechanics, 304: 285-319, 1995.<br />
<br />
== Computations ==<br />
<br />
*<b>P. R. Voke.</b> Direct and Large Eddy Simulation II, pages 355-422. Kulwer Acadamic Publishers, 1997.<br />
*<b>W. Rodi, J. H. Ferziger, M. Breuer, and M. Pourquie.</b> Status of large eddy simulation: Results of a workshop. ASME Journal of Fluids Engineering, 119: 248-261, 1997.<br />
*<b>A. Sohankar, L. Davidson, and C. Norberg.</b> Large eddy simulation of flow past a square cylinder: Comparison of different subgrid scale models. ASME Journal of Fluids Engineering, 122: 39-47, 2000.</div>Discoganyahttp://www.cfd-online.com/Wiki/Flow_across_a_square_cylinderFlow across a square cylinder2005-09-20T00:07:40Z<p>Discoganya: </p>
<hr />
<div>The flow across a square cylinder is an important test case for the validation of separated flows in the turbulent regime. In the past, it has been regularly used to validate [[Large Eddy Simulation]] (LES) models.<br />
<br />
Experiments studying the flow across a square cylinder have been available since about the time of Vickery (1966), while modern simulations usually tend to reproduce the results of Lyn and Rodi (1994) whose data set for [[Reynolds Number|Re]] = 22,000 is publicly available.<br />
<br />
The flow involves separation and coherent vortex shedding. In trying to benchmark a code with the experimental data, the following items are usually compared:<br />
<br />
* [[Strouhal Number]] which is the easiest to reproduce.<br />
* Mean recirculation length.<br />
* Global drag and lift statistics, e.g. mean lift coefficient, rms lift coefficient, mean drag coefficient and rms drag coefficient.<br />
* Spatial distributions of time-averaged and phase-averaged velocities, pressure and Reynolds stresses.<br />
<br />
= References =<br />
<br />
== Experiments ==<br />
<br />
*<b>B. J. Vickery.</b> Fluctuating lift and drag on a long cylinder of square cross section in a smooth and in a turbulent stream. Journal of Fluid Mechanics, 25: 481-494, 1966.<br />
*<b>B. E. Lee.</b> The effect of turbulence on the surface pressure field of a square prism. Journal of Fluid Mechanics, 69(2): 263-292, 1975.<br />
*<b>P. W. Bearman and E. D. Obasaju.</b> An experimental study of pressure fluctuations on fixed and oscillating square-section cylinders. 119: 297-321, 1982.<br />
*<b>I. McLean and I. Gartshore.</b> Spanwise correlations of pressure on a rigid square section cylinder. 41-44:779-808, 1992.<br />
*<b>C. Norberg. Flow around rectangular cylinders: Pressure forces and wake frequencies. Journal of Wind Engineering and Industrial Aerodynamics, 49: 187-196, 1993.<br />
*<b>D. A. Lyn and W. Rodi.</b> The flapping shear layer formed by flow separation from the forward corner of a square cylinder. Journal of Fluid Mechanics, 267: 353-376, 1994.<br />
*<b>D. A. Lyn, S. Einav, W. Rodi, and J.-H. Park.</b> A laser-doppler velocimetry study of ensemble-averaged characteristics of the turbulent near wake of a square cylinder. Journal of Fluid Mechanics, 304: 285-319, 1995.<br />
<br />
== Computations ==<br />
<br />
*<b>P. R. Voke.</b> Direct and Large Eddy Simulation II, pages 355-422. Kulwer Acadamic Publishers, 1997.<br />
*<b>W. Rodi, J. H. Ferziger, M. Breuer, and M. Pourquie.</b> Status of large eddy simulation: Results of a workshop. ASME Journal of Fluids Engineering, 119: 248-261, 1997.<br />
*<b>A. Sohankar, L. Davidson, and C. Norberg.</b> Large eddy simulation of flow past a square cylinder: Comparison of different subgrid scale models. ASME Journal of Fluids Engineering, 122: 39-47, 2000.</div>Discoganyahttp://www.cfd-online.com/Wiki/Flow_across_a_square_cylinderFlow across a square cylinder2005-09-20T00:07:09Z<p>Discoganya: </p>
<hr />
<div>The flow across a square cylinder is an important test case for the validation of separated flows in the turbulent regime. In the past, it has been regularly used to validate [[Large Eddy Simulation]] (LES) models.<br />
<br />
Experiments studying the flow across a square cylinder have been available since about the time of Vickery (1966), while modern simulations usually tend to reproduce the results of Lyn and Rodi (1994) whose data set for [[Re|Reynolds Number]] = 22,000 is publicly available.<br />
<br />
The flow involves separation and coherent vortex shedding. In trying to benchmark a code with the experimental data, the following items are usually compared:<br />
<br />
* [[Strouhal Number]] which is the easiest to reproduce.<br />
* Mean recirculation length.<br />
* Global drag and lift statistics, e.g. mean lift coefficient, rms lift coefficient, mean drag coefficient and rms drag coefficient.<br />
* Spatial distributions of time-averaged and phase-averaged velocities, pressure and Reynolds stresses.<br />
<br />
= References =<br />
<br />
== Experiments ==<br />
<br />
*<b>B. J. Vickery.</b> Fluctuating lift and drag on a long cylinder of square cross section in a smooth and in a turbulent stream. Journal of Fluid Mechanics, 25: 481-494, 1966.<br />
*<b>B. E. Lee.</b> The effect of turbulence on the surface pressure field of a square prism. Journal of Fluid Mechanics, 69(2): 263-292, 1975.<br />
*<b>P. W. Bearman and E. D. Obasaju.</b> An experimental study of pressure fluctuations on fixed and oscillating square-section cylinders. 119: 297-321, 1982.<br />
*<b>I. McLean and I. Gartshore.</b> Spanwise correlations of pressure on a rigid square section cylinder. 41-44:779-808, 1992.<br />
*<b>C. Norberg. Flow around rectangular cylinders: Pressure forces and wake frequencies. Journal of Wind Engineering and Industrial Aerodynamics, 49: 187-196, 1993.<br />
*<b>D. A. Lyn and W. Rodi.</b> The flapping shear layer formed by flow separation from the forward corner of a square cylinder. Journal of Fluid Mechanics, 267: 353-376, 1994.<br />
*<b>D. A. Lyn, S. Einav, W. Rodi, and J.-H. Park.</b> A laser-doppler velocimetry study of ensemble-averaged characteristics of the turbulent near wake of a square cylinder. Journal of Fluid Mechanics, 304: 285-319, 1995.<br />
<br />
== Computations ==<br />
<br />
*<b>P. R. Voke.</b> Direct and Large Eddy Simulation II, pages 355-422. Kulwer Acadamic Publishers, 1997.<br />
*<b>W. Rodi, J. H. Ferziger, M. Breuer, and M. Pourquie.</b> Status of large eddy simulation: Results of a workshop. ASME Journal of Fluids Engineering, 119: 248-261, 1997.<br />
*<b>A. Sohankar, L. Davidson, and C. Norberg.</b> Large eddy simulation of flow past a square cylinder: Comparison of different subgrid scale models. ASME Journal of Fluids Engineering, 122: 39-47, 2000.</div>Discoganyahttp://www.cfd-online.com/Wiki/Flow_across_a_square_cylinderFlow across a square cylinder2005-09-20T00:01:51Z<p>Discoganya: </p>
<hr />
<div>The flow across a square cylinder is an important test case for the validation of separated flows in the turbulent regime. In the past, it has been regularly used to validate [[Large Eddy Simulation]] (LES) models.<br />
<br />
Experiments studying the flow across a square cylinder have been available since about the time of Vickery (1966), while modern simulations usually tend to reproduce the results of Lyn and Rodi (1994) whose data set is publicly available.<br />
<br />
<br />
= References =<br />
<br />
== Experiments ==<br />
<br />
*<b>B. J. Vickery.</b> Fluctuating lift and drag on a long cylinder of square cross section in a smooth and in a turbulent stream. Journal of Fluid Mechanics, 25: 481-494, 1966.<br />
*<b>B. E. Lee.</b> The effect of turbulence on the surface pressure field of a square prism. Journal of Fluid Mechanics, 69(2): 263-292, 1975.<br />
*<b>P. W. Bearman and E. D. Obasaju.</b> An experimental study of pressure fluctuations on fixed and oscillating square-section cylinders. 119: 297-321, 1982.<br />
*<b>I. McLean and I. Gartshore.</b> Spanwise correlations of pressure on a rigid square section cylinder. 41-44:779-808, 1992.<br />
*<b>C. Norberg. Flow around rectangular cylinders: Pressure forces and wake frequencies. Journal of Wind Engineering and Industrial Aerodynamics, 49: 187-196, 1993.<br />
*<b>D. A. Lyn and W. Rodi.</b> The flapping shear layer formed by flow separation from the forward corner of a square cylinder. Journal of Fluid Mechanics, 267: 353-376, 1994.<br />
*<b>D. A. Lyn, S. Einav, W. Rodi, and J.-H. Park.</b> A laser-doppler velocimetry study of ensemble-averaged characteristics of the turbulent near wake of a square cylinder. Journal of Fluid Mechanics, 304: 285-319, 1995.<br />
<br />
== Computations ==<br />
<br />
*<b>P. R. Voke.</b> Direct and Large Eddy Simulation II, pages 355-422. Kulwer Acadamic Publishers, 1997.<br />
*<b>W. Rodi, J. H. Ferziger, M. Breuer, and M. Pourquie.</b> Status of large eddy simulation: Results of a workshop. ASME Journal of Fluids Engineering, 119: 248-261, 1997.<br />
*<b>A. Sohankar, L. Davidson, and C. Norberg.</b> Large eddy simulation of flow past a square cylinder: Comparison of different subgrid scale models. ASME Journal of Fluids Engineering, 122: 39-47, 2000.</div>Discoganyahttp://www.cfd-online.com/Wiki/Large_eddy_simulation_(LES)Large eddy simulation (LES)2005-09-18T19:14:34Z<p>Discoganya: /* Sub-grid Scale models */</p>
<hr />
<div>Large eddy simulation (LES) is a popular technique for simulating turbulent flows. A common deduction of [[Kolmogorov]]'s (1941) theory of self similarity is that the large eddies of the flow are dependant on the geometry while the smaller scales more [[universal]]. This feature allows one to explicitly solve for the large eddies in a calculation and implicitly account for the small eddies by using a [[sub-grid scale model]] (SGS model).<br />
<br />
Mathematically, one may think of separating the velocity field into a resolved and sub-grid part. The resolved part of the field represent the "large" eddies, while the sub-grid part of the velocity represent the "small scales" whose effect on the resolved field is included through the sub-grid scale model. Formally, one may think of filtering as the convolution of a function with a [[LES filters|filtering kernel]]. However, most practical (and commercial) implimentations of LES, use the grid itself as the filter, and perform no explicit filtering. More information about the theory and application of filters is found [[LES filters|here]].<br />
<br />
This page is mainly focused on LES of incompressible flows. For compressible flows, see [[Favre averaged Navier-Stokes equations]]. <br />
<br />
Typically, one would begin with the incompressible [[Navier-Stokes equations]] of motion, <br><br />
:<math><br />
\frac{\partial{u_i}}{\partial t} + u_j \frac{\partial u_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial p}{\partial x_i} + \nu \nabla^2 u_i <br />
</math><br />
<br />
and by the application of a filtering kernel, derive the equations of motion for the resolved field,<br />
<br />
:<math><br />
<br />
\frac{\partial{\bar{u_i}}}{\partial t} + \bar{u_j} \frac{\partial \bar{u_i}}{\partial x_j} = -\frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i} + \nu \nabla^2 \bar{u_i}<br />
+ \frac{\partial \tau_{ij}}{\partial x_j}<br />
</math><br />
<br />
Velocities and pressures with an overbar denote the resolved field after the application of the filtering operation. Similar equations can be derived for the sub-grid scale field (i.e. the residual field). An extra term <math> \frac{\partial \tau_{ij}}{\partial x_j} </math> arises from the non-linear advection terms, due to the fact that<br />
:<math><br />
\overline{ u_j \frac{\partial u_i}{\partial x_j} } \ne<br />
\bar{u_j} \frac{\partial \bar{u_i}}{\partial x_j}<br />
</math><br />
<br />
and hence <br />
:<math><br />
\tau_{ij} = \bar{u_i} \bar{u_j} - \overline{u_i u_j}<br />
</math><br />
<br />
Subgrid-scale turbulence models usually employ the [[Boussinesq hypothesis]], and seek to calculate (the deviatoric part of) the SGS stress using: <br><br />
:<math><br />
\tau _{ij} - \frac{1}{3}\tau _{kk} \delta _{ij} = - 2\mu _\tau \bar S_{ij} <br />
</math><br />
<br />
where <math><br />
\bar S_{ij} <br />
</math> is the rate-of-strain tensor for the resolved scale defined by <br><br />
<br />
:<math><br />
\bar S_{ij} = \frac{1}{2}\left( {\frac{{\partial \bar u_i }}{{\partial x_j }} + \frac{{\partial \bar u_j }}{{\partial x_i }}} \right)<br />
</math><br />
<br><br />
and <math> \mu _\tau </math> is the subgrid-scale turbulent viscosity.<br />
<br />
== Sub-grid Scale models ==<br />
<br />
*[[Smagorinsky-Lilly model|Smagorinsky's model]] (Smagorinsky, 1963)<br />
*[[Dynamic subgrid-scale model|Algebraic Dynamic model]] (Germano, et. al., 1991)<br />
*[[Kinetic energy subgrid-scale model|Localized Dynamic model]] (Kim & Menon, 1993)<br />
*[[Wall-adapting local eddy-viscosity (WALE) model|WALE (Wall-Adapting Local Eddy-viscosity) model]] (Nicoud and Ducros, 1999)<br />
*[[RNG-LES model]]<br />
<br />
== References ==<br />
<br />
*<b>J. Smagorinsky.</b> General circulation experiments with the primitive equations, i. the basic experiment. Monthly Weather Review, 91: 99-164, 1963.<br />
*<b>M. Germano, U. Piomelli, P. Moin, and W. H. Cabot.</b> A dynamic sub-grid scale eddy viscosity model. Physics of Fluids, A(3): 1760-1765, 1991.<br />
*<b>W. Kim and S. Menon.</b> A new dynamic one-equation subgrid-scale model for large eddy simulation. In 33rd Aerospace Sciences Meeting and Exhibit, Reno, NV, 1995.<br />
*<b>F. Nicoud and F. Ducros.</b> Subgrid-scale modelling based on the square of the velocity gradient tensor. Flow, Turbulence and Combustion, 62: 183-200, 1999.</div>Discoganyahttp://www.cfd-online.com/Wiki/Large_eddy_simulation_(LES)Large eddy simulation (LES)2005-09-18T17:49:22Z<p>Discoganya: </p>
<hr />
<div>Large eddy simulation (LES) is a popular technique for simulating turbulent flows. A common deduction of [[Kolmogorov]]'s (1941) theory of self similarity is that the large eddies of the flow are dependant on the geometry while the smaller scales more [[universal]]. This feature allows one to explicitly solve for the large eddies in a calculation and implicitly account for the small eddies by using a [[sub-grid scale model]] (SGS model).<br />
<br />
Mathematically, one may think of separating the velocity field into a resolved and sub-grid part. The resolved part of the field represent the "large" eddies, while the sub-grid part of the velocity represent the "small scales" whose effect on the resolved field is included through the sub-grid scale model. Formally, one may think of filtering as the convolution of a function with a [[LES filters|filtering kernel]]. However, most practical (and commercial) implimentations of LES, use the grid itself as the filter, and perform no explicit filtering. More information about the theory and application of filters is found [[LES filters|here]].<br />
<br />
This page is mainly focused on LES of incompressible flows. For compressible flows, see [[Favre averaged Navier-Stokes equations]]. <br />
<br />
Typically, one would begin with the incompressible [[Navier-Stokes equations]] of motion, <br><br />
:<math><br />
\frac{\partial{u_i}}{\partial t} + u_j \frac{\partial u_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial p}{\partial x_i} + \nu \nabla^2 u_i <br />
</math><br />
<br />
and by the application of a filtering kernel, derive the equations of motion for the resolved field,<br />
<br />
:<math><br />
<br />
\frac{\partial{\bar{u_i}}}{\partial t} + \bar{u_j} \frac{\partial \bar{u_i}}{\partial x_j} = -\frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i} + \nu \nabla^2 \bar{u_i}<br />
+ \frac{\partial \tau_{ij}}{\partial x_j}<br />
</math><br />
<br />
Velocities and pressures with an overbar denote the resolved field after the application of the filtering operation. Similar equations can be derived for the sub-grid scale field (i.e. the residual field). An extra term <math> \frac{\partial \tau_{ij}}{\partial x_j} </math> arises from the non-linear advection terms, due to the fact that<br />
:<math><br />
\overline{ u_j \frac{\partial u_i}{\partial x_j} } \ne<br />
\bar{u_j} \frac{\partial \bar{u_i}}{\partial x_j}<br />
</math><br />
<br />
and hence <br />
:<math><br />
\tau_{ij} = \bar{u_i} \bar{u_j} - \overline{u_i u_j}<br />
</math><br />
<br />
Subgrid-scale turbulence models usually employ the [[Boussinesq hypothesis]], and seek to calculate (the deviatoric part of) the SGS stress using: <br><br />
:<math><br />
\tau _{ij} - \frac{1}{3}\tau _{kk} \delta _{ij} = - 2\mu _\tau \bar S_{ij} <br />
</math><br />
<br />
where <math><br />
\bar S_{ij} <br />
</math> is the rate-of-strain tensor for the resolved scale defined by <br><br />
<br />
:<math><br />
\bar S_{ij} = \frac{1}{2}\left( {\frac{{\partial \bar u_i }}{{\partial x_j }} + \frac{{\partial \bar u_j }}{{\partial x_i }}} \right)<br />
</math><br />
<br><br />
and <math> \mu _\tau </math> is the subgrid-scale turbulent viscosity.<br />
<br />
== Sub-grid Scale models ==<br />
<br />
*[[Smagorinsky's model]] (Smagorinsky, 1963)<br />
*[[Algebraic Dynamic model]] (Germano, et. al., 1991)<br />
*[[Localized Dynamic model]] (Kim & Menon, 1993)<br />
*[[WALE (Wall-Adapting Local Eddy-viscosity) model]] (Nicoud and Ducros, 1999)<br />
<br />
== References ==<br />
<br />
*<b>J. Smagorinsky.</b> General circulation experiments with the primitive equations, i. the basic experiment. Monthly Weather Review, 91: 99-164, 1963.<br />
*<b>M. Germano, U. Piomelli, P. Moin, and W. H. Cabot.</b> A dynamic sub-grid scale eddy viscosity model. Physics of Fluids, A(3): 1760-1765, 1991.<br />
*<b>W. Kim and S. Menon.</b> A new dynamic one-equation subgrid-scale model for large eddy simulation. In 33rd Aerospace Sciences Meeting and Exhibit, Reno, NV, 1995.<br />
*<b>F. Nicoud and F. Ducros.</b> Subgrid-scale modelling based on the square of the velocity gradient tensor. Flow, Turbulence and Combustion, 62: 183-200, 1999.</div>Discoganyahttp://www.cfd-online.com/Wiki/Large_eddy_simulation_(LES)Large eddy simulation (LES)2005-09-18T17:38:18Z<p>Discoganya: Added some introduction and showed the derivation of the main equations</p>
<hr />
<div>Large eddy simulation (LES) is a popular technique for simulating turbulent flows. A common deduction of [[Kolmogorov]]'s (1941) theory of self similarity is that the large eddies of the flow are dependant on the geometry while the smaller scales more [[universal]]. This feature allows one to explicitly solve for the large eddies in a calculation and implicitly account for the small eddies by using a [[sub-grid scale model]] (SGS model).<br />
<br />
Mathematically, one may think of separating the velocity field into a resolved and sub-grid part. The resolved part of the field represent the "large" eddies, while the sub-grid part of the velocity represent the "small scales" whose effect on the resolved field is included through the sub-grid scale model. Formally, one may think of filtering as the convolution of a function with a [[LES filters|filtering kernel]]. However, most practical (and commercial) implimentations of LES, use the grid itself as the filter, and perform no explicit filtering. More information about the theory and application of filters is found [[LES filters|here]].<br />
<br />
This page is mainly focused on LES of incompressible flows. For compressible flows, see [[Favre averaged Navier-Stokes equations]]. <br />
<br />
Typically, one would begin with the incompressible [[Navier-Stokes equations]] of motion, <br><br />
:<math><br />
\frac{\partial{u_i}}{\partial t} + u_j \frac{\partial u_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial p}{\partial x_i} + \nu \nabla^2 u_i <br />
</math><br />
<br />
and by the application of a filtering kernel, derive the equations of motion for the resolved field,<br />
<br />
:<math><br />
\frac{\partial{\bar{u_i}}}{\partial t} + \bar{u_j} \frac{\partial \bar{u_i}}{\partial x_j} = -\frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i} + \nu \nabla^2 \bar{u_i}<br />
+ \frac{\partial \tau_{ij}}{\partial x_j}<br />
</math><br />
<br />
Velocities and pressures with an overbar denote the resolved field after the application of the filtering operation. Similar equations can be derived for the sub-grid scale field (i.e. the residual field). An extra term <math> \frac{\partial \tau_{ij}}{\partial x_j} </math> arises from the non-linear advection terms, due to the fact that<br />
:<math><br />
\overline{ u_j \frac{\partial u_i}{\partial x_j} } \ne<br />
\bar{u_j} \frac{\partial \bar{u_i}}{\partial x_j}<br />
</math><br />
<br />
and hence <br />
:<math><br />
\tau_{ij} = \bar{u_i} \bar{u_j} - \overline{u_i u_j}<br />
</math><br />
<br />
Subgrid-scale turbulence models usually employ the [[Boussinesq hypothesis]], and seek to calculate (the deviatoric part of) the SGS stress using: <br><br />
:<math><br />
\tau _{ij} - \frac{1}{3}\tau _{kk} \delta _{ij} = - 2\mu _\tau \bar S_{ij} <br />
</math><br />
<br />
where <math><br />
\bar S_{ij} <br />
</math> is the rate-of-strain tensor for the resolved scale defined by <br><br />
<br />
:<math><br />
\bar S_{ij} = \frac{1}{2}\left( {\frac{{\partial \bar u_i }}{{\partial x_j }} + \frac{{\partial \bar u_j }}{{\partial x_i }}} \right)<br />
</math><br />
<br><br />
and <math> \mu _\tau </math> is the subgrid-scale turbulent viscosity.</div>Discoganyahttp://www.cfd-online.com/Wiki/Validation_and_test_casesValidation and test cases2005-09-16T19:09:28Z<p>Discoganya: /* 3-D test cases */</p>
<hr />
<div>== 1-D test cases ==<br />
<br />
*[[Shock tube problem]]<br />
<br />
== 2-D test cases ==<br />
<br />
*[[Ringleb flow]]<br />
*[[2-D vortex in isentropic flow]]<br />
*[[Viscous diffusion of multiple vortex system]]<br />
*[[Williams airfoil]]<br />
*[[Suddhoo-Hall airfoil]]<br />
*[[RAE2822 airfoil]]<br />
*[[Turbulent flat-plate]]<br />
*[[Lid-driven cavity problem]]<br />
<br />
== 3-D test cases ==<br />
<br />
*[[Onera M6 wing]]<br />
*[[Hypersonic blunt body flow]]<br />
*[[DARPA SUBOFF model]]<br />
*[[Ahmed body]]<br />
*[[Flow across a square cylinder]]<br />
<br />
== External links ==<br />
<br />
*[http://www.cfd-online.com/Links/refs.html#validation CFD Online links to validation cases]<br />
*[http://www.grc.nasa.gov/WWW/wind/valid/tutorial/tutorial.html Tutorial on CFD verification and validation]<br />
*[http://journaltool.asme.org/Templates/JFENumAccuracy.pdf Statement on the content of numerical accuracy] of [http://scitation.aip.org/ASMEJournals/Fluids/ Journal of Fluids Engineering]<br />
*[http://cfl3d.larc.nasa.gov/Cfl3dv6/cfl3dv6_testcases.html CFL3D test cases]<br />
*[http://ad-www.larc.nasa.gov/tsab/usm3d/TESTCASES.html USM3D test cases]<br />
*[http://aaac.larc.nasa.gov/tsab/cfdlarc/aiaa-dpw AIAA CFD drag prediction workshop]<br />
*[http://www.grc.nasa.gov/WWW/wind/valid/validation.html NPARC Alliance CFD verification and validation web-site]</div>Discoganya