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Old   July 31, 2012, 10:06
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Hi all,

I have been reading about non-Newtonian fluids recently and I have a question about the scalar shear strain rate. In the Ansys CFX theory manual, the scalar shear strain rate is defined as follows:

gamma = [2*(delUi/delXj : Dij)]^0.5

Here, the tensor Dij denotes the strain rate (or rate of deformation) tensor. In the Ansys Fluent theory manual, the scalar shear strain rate is then defined as follows:

gamma = [0.5*(Dij : Dij)]^0.5

Finally, in two journal publications that I have found, the scalar shear strain rate is alternately defined as follows:

gamma = [(Dij : Dij)]^0.5
gamma = [2*(Dij : Dij)]^0.5

Why are there so many different definitions for the scalar shear strain rate? Any insight would be greatly appreciated.

Thanks,
Dave
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Old   August 1, 2012, 07:29
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So I think I may have figured this out.

The rate of deformation (or strain rate) tensor is typically defined as follows:

Dij=0.5*(delUi/delXj+delUj/delXi)

In the Ansys Fluent theory manual the 0.5 is neglected in this definition which leads to the following definition of the scalar shear strain rate:

gamma = [0.5*(Dij : Dij)]^0.5

Including the 0.5 in the definition of the rate of deformation tensor leads to the following definition of the scalar shear strain rate:

gamma = [2*(Dij : Dij)]^0.5

Finally, the definition of the scalar shear strain rate given in the Ansys CFX theory which reads as follows:

gamma = [2*(delUi/delXj : Dij)]^0.5

leads to the same equation as the above definitions. So thats three out of four definitions accounted for. Still not sure of the last one.

Cant understand Why the 0.5 would be left out of the definition for the rate of deformation tensor though.

Any ideas?
Dave
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Old   October 24, 2013, 09:08
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Hi, Dave.
Using tensor, you can easily prove that:
gamma = [2*(delUi/delXj : Dij)]^0.5

is SAME as
gamma = [2*(Dij : Dij)]^0.5

Therefore, for CFX
We can write:
sstrnr=gamma = [2*(Dij : Dij)]^0.5

To Prove:
write Dij in Matrix form as 3 by 3 Matrix. Multiply Dij by Dij using Matrix multiplication.
Sum up the diagonal elements of the Result Matrix and Multiply by 2.

After Manipulation, you will get the same result for sstrnr as ANSYS CFX Manual.
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Old   October 24, 2013, 10:20
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Hi cfd_begin,

I understand that [2*(delUi/delXj : Dij)]^0.5 = [2*(Dij : Dij)]^0.5.

My problem was the definitions of the scalar shear strain rate in ANSYS CFX and ANSYS Fluent are slightly different:

In CFX Theory Manual: gamma = [2*(delUi/delXj : Dij)]^0.5 = [2*(Dij : Dij)]^0.5

In Fluent User Manual: gamma = [0.5*(Dij : Dij)]^0.5

I believe this is because:

In CFX Theory Manual: Dij = 0.5*(delUi/delXj+delUj/delXi)

In Fluent User Manual: Dij = (delUi/delXj+delUj/delXi)

As I said before, I'm not sure why the 0.5 would is left out of the rate of deformation tensor.

Kind Regards,
Dave
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Old   October 24, 2013, 22:19
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Hi Dave,

I think that the deformation tensor Dij is defined for compatibility with the definition of gamma.

In CFX Theory Manual: gamma = [2*(delUi/delXj : Dij)]^0.5 = [2*(Dij : Dij)]^0.5

In Fluent User Manual: gamma = [0.5*(Dij : Dij)]^0.5
Its just my opinion !
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Old   October 25, 2013, 04:08
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Hi cfd_begin,

In all of the continuum mechanics books that I have read, the rate of deformation tensor is defined as 0.5*(delUi/delXj+delUj/delXi).

I'm sure that there is probably a very good reason for dropping the 0.5 in ANSYS Fluent. I just never figured it out.

I can't imagine that a different definition of Dij would be used just to satisfy a different definition of gamma? Why not use conventional definitions?

Kind Regards,
Dave
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