Mean Strain rate

rembe · January 10, 2013, 07:53

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

Rami · January 14, 2013, 03:11

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.

January 10, 2013, 07:53	Mean Strain rate	#1
rembe New Member Michael Rembe Join Date: Oct 2009 Location: Germany Posts: 4 Rep Power: 16	Hi, I'm not an expert in fluid mechanics. In the SMAGORINSKY-model I need the square root of SijSij. 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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January 14, 2013, 03:11		#2
Rami Senior Member Rami Ben-Zvi Join Date: Mar 2009 Posts: 155 Rep Power: 17	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 =SijSij 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.