Question about LES derivation

siw · May 18, 2014, 06:27

Hi,

I was reading the CFD Wiki page about LES (http://www.cfd-online.com/Wiki/Large...tion_%28LES%29). It shows the instantaneous incompressible momentum equation and says in it is substituted in the decomposition of the velocity and pressure, so that gives:

$\frac{\partial \left( \bar{u_{i}} + u_{i}' \right)}{\partial t} + \left( \bar{u_{j}} + u_{j}' \right) \frac{\partial \left( \bar{u_{i}} + u_{i}' \right)}{\partial x_{j}} = - \frac{1}{\rho} \frac{\partial \left( \bar{p} + p' \right)}{\partial x_{i}} + \nu \frac{\partial^{2} \left( \bar{u_{i}} + u_{i}' \right)}{\partial x_{j}^{2}}$

but what is the next step? The wiki page just says "filtering" but how does that look in the equation for ending up with:

$\frac{\partial \bar{u_{i}}}{\partial t} + \bar{u_{j}} \frac{\partial \bar{u_{i}}}{\partial x_{j}} = - \frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_{i}} + \nu \frac{\partial^{2} \bar{u_{i}}}{\partial x_{j}^{2}} + \frac{1}{\rho} \frac{\partial \tau_{ij}}{x_{j}}$

I cannot see the step-by-step method as the books just ignore such detail. Is it something similar (but not the same) as applying the Reynolds decomposition, time-averaging the terms and then eliminating those that are zero for RANS? A useful step-by-step guide was given for the non-dimensional NSEs at http://www.cfd-online.com/Forums/mai...er-stokes.html.

Thanks

FMDenaro · May 18, 2014, 07:32

First, I suggest to have a reading of some book illustrating LES (for example Sagaut).
Second, starting from the NS equation, a "filter" is formally applied on each term. If it does commute with derivatives you have

d u_f/dt + Div (uu)_f + grad p_f = Div (2 mu Grad u_f)

You can see that

Div (uu)_f = Div (u_f u_f) + [ Div (uu)_f -Div (u_f u_f)]

The last term in square brackets is the unresolved term that must be modelled

The wiki page addresses an equation decomposed by using the quasi-linear form of the momentum equation but is quite simple to obtain the same procedure

May 18, 2014, 06:27	Question about LES derivation	#1
siw Senior Member Stuart Join Date: Jul 2009 Location: Portsmouth, England Posts: 733 Rep Power: 25	Hi, I was reading the CFD Wiki page about LES (http://www.cfd-online.com/Wiki/Large...tion_%28LES%29). It shows the instantaneous incompressible momentum equation and says in it is substituted in the decomposition of the velocity and pressure, so that gives: $\frac{\partial \left( \bar{u_{i}} + u_{i}' \right)}{\partial t} + \left( \bar{u_{j}} + u_{j}' \right) \frac{\partial \left( \bar{u_{i}} + u_{i}' \right)}{\partial x_{j}} = - \frac{1}{\rho} \frac{\partial \left( \bar{p} + p' \right)}{\partial x_{i}} + \nu \frac{\partial^{2} \left( \bar{u_{i}} + u_{i}' \right)}{\partial x_{j}^{2}}$ but what is the next step? The wiki page just says "filtering" but how does that look in the equation for ending up with: $\frac{\partial \bar{u_{i}}}{\partial t} + \bar{u_{j}} \frac{\partial \bar{u_{i}}}{\partial x_{j}} = - \frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_{i}} + \nu \frac{\partial^{2} \bar{u_{i}}}{\partial x_{j}^{2}} + \frac{1}{\rho} \frac{\partial \tau_{ij}}{x_{j}}$ I cannot see the step-by-step method as the books just ignore such detail. Is it something similar (but not the same) as applying the Reynolds decomposition, time-averaging the terms and then eliminating those that are zero for RANS? A useful step-by-step guide was given for the non-dimensional NSEs at http://www.cfd-online.com/Forums/mai...er-stokes.html. Thanks

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
LES filtering question	martyn88	OpenFOAM Running, Solving & CFD	1	October 18, 2012 03:08
LES supersonic flow	ziooscuro	OpenFOAM Running, Solving & CFD	1	March 5, 2011 00:56
question regarding LES	usker	Siemens	1	June 29, 2008 18:33
LES beginner question	Shuo	Main CFD Forum	4	July 9, 2007 08:40
Basic LES question	toyCFD	Main CFD Forum	3	August 23, 2006 15:09

May 18, 2014, 07:32		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,768 Rep Power: 71	First, I suggest to have a reading of some book illustrating LES (for example Sagaut). Second, starting from the NS equation, a "filter" is formally applied on each term. If it does commute with derivatives you have d u_f/dt + Div (uu)_f + grad p_f = Div (2 mu Grad u_f) You can see that Div (uu)_f = Div (u_f u_f) + [ Div (uu)_f -Div (u_f u_f)] The last term in square brackets is the unresolved term that must be modelled The wiki page addresses an equation decomposed by using the quasi-linear form of the momentum equation but is quite simple to obtain the same procedure