Mean Strain rate
 

Hi,
I'm not an expert in fluid mechanics. In the SMAGORINSKY-model I need the square root of Sij*Sij. I'm a little confused about the different notations. What is the defference between the mean strain rate defined as:

http://www.cfd-online.com/Forums/vbL...66aad1f8-1.gif

and the mean strain rate defined as:

Sij=0.5*(dUi/dxj+dUj/dxi)

Some times I can read:

S=sqrt((dUi/dxj)^2+(dUj/dxi)^2)

I would be very grateful for a short explanation.

Thank you very much!

Michael

Hi Michael,

That's simply tensor algebra (using the sum convention).

If you use the definition Sij=0.5*(dUi/dxj+dUj/dxi) , substitute it in s^2 =Sij*Sij and sum over i and j, you'l get the result

http://www.cfd-online.com/Forums/vbL...66aad1f8-1.gif
from the other expression,
S=sqrt((dUi/dxj)^2+(dUj/dxi)^2)
you get
S^2=Ui,jUi,j+ Uj,iUj,i = 4SijSij - 2Ui,jUj,i
so it is different due to the last term.