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J. Roued March 10, 1999 11:43

LES and the Strain rate factor

I have a question about why the evaluatin of the strain rate factor are different when used in LES.

|S| = 1/2 * ( S_ij S_ij)^{0.5}


|S| = 2 * ( S_ij S_ij)^{0.5}

Where S_ij = 1/2 * ( du_i/dx_j + du_j/dx_i )

are there any one that can explain the differences.

Thanks in advance.

Mayank Tyagi March 10, 1999 20:36

Re: LES and the Strain rate factor

Could you provide the expression for dynamic coefficient and eddy viscosity where these relations are used?

My guess is final expression must be same... should not differ by the constant scale either...

one more point ... Look at Lilly's paper or for that matter any other paper... the relation is |S| = (2*S_ij S_ij)^{0.5} I'm not sure where did you find your relations.. but so long as the final expression for eddy viscosty is same ... you don't have to worry about it.


Gary Dantinne March 11, 1999 03:42

Re: LES and the Strain rate factor

Usually as you said the strain rate is defined by |S| = (Sij Sij)^1/2 in LES conventionnaly we use for the filtered strain rate |S| = (2 Sij Sij)^1/2. I think the only reason of this is historical. It may be interesting to look at the Smagorinsky's paper in 1968 and the way he wrote his model.

Wu Chun-Kai March 11, 1999 19:21

Re: LES and the Strain rate factor
Hi! J. Roued,

I'm a PhD student. I am working and calculating with a single airfoil in free stream by using LES. I use the definition for incompressible flow

Mut = (Cs)^2 * (filter width)^2 * |S|

with |S| = ( S_ij S_ij)^{0.5} ,

S_ij = 1/2 * ( du_i/dx_j + du_j/dx_i )

as in the "Computational Methods for Fluid Dynamics"--by Joel H.Ferziger and Milovan Peric.

I also adopt that, for a box filter,

(filter width)=grid size=h

h = (h1*h1+h2*h2)^(0.5) for my 2-D problem.

I found that the LES works well with a Smagorinsky constant Cs=0.1 as comparing with the experimental results.

The computational case with experimental data is got from the ERCOFTAC database for a DSMA671 airfoil, Re=1.2*10^6 and Alfa=4.

But be note that the choice of Cs value might be sensitive to the used mesh.


Tony Chen March 11, 1999 23:16

Re: LES and the Strain rate factor
Gary is right, It is a matter of convention. You can check that one of the definitions of |S| can be directly related to the second invariant of Sij, which is mathematically convenient.


Mayank Tyagi March 12, 1999 07:16

Re: LES and the Strain rate factor
Hi Wowakai...

The Value of Cs is mesh dependent because you are using a pure smagorinsky model... In case of dynamic model the Cs can only be function of the filter width ratio at test and grid level.. and hence independent of the mesh size... In fact dynamic procedure is the only way to make Cs independent of the mesh size as far as i know...

responding for the convention used for |s| ... it's totally upto you so long as final expression for SGS model cantains the right value for the constant...


Ravi Krishnamurthy March 19, 1999 17:25

Re: LES and the Strain rate factor
The stress tensor on the right side of the N.S. due to LES is

q_ij = -2 Nu_t Sij

where Nu_t = C_s delta^2 |Sij|

where Sij = 0.5 * (u_i/x_j + u_j/x_i) and

|Sij| = SQRT( 2 Sij.Sij)

In general Nu_t ~ l^2 |Sij| by definition where |Sij| is the magnitude of the Strain tensor.Any tensor for that matter. Hence the definition.

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