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Old   January 10, 2013, 08:53
Default Mean Strain rate
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Michael Rembe
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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:



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
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Old   January 14, 2013, 04:11
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Rami Ben-Zvi
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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


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.
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