[mmkr825]

Dear all,
I would like to share your opinion on my doubt.

- The rate of strain tensor is written as(if i have not understand wrongly):

E=symm(grad(U))=1/2[grad(U)+grad(U)^T].......................(1)

- According to linear algebra the magnitude of any tensor S is:

|S|=sqrt(2S:S) .........................(2)

- According to Eq. (2) the magnitude of rate of strain tensor is:

|E| = sqrt(2E:E). ..............................(3)

= sqrt(2*(E_ij*E_ij))...................... (4)

- In OpenFOAM notation the magnitude of rate of strain tensor can be written as:

|E|=sqrt(2*symm(grad(U))).......................(5)

But for the testing purpose i defined a Tensor Q as [1 0 0 0 0.8 0 0 0 0.5]. According to Eq. (2), it's magnitude is 1.944222. But the OpenFOAM is giving 1.3747727.
So the definition of the magnitude of tensor according to OpenFOAM is:

|S| = sqrt(S:S)=sqrt(S_ij*S_ij)...........................(6)

Which contradicts Eq. (2).

Can any one clarify my doubt and how to write rate of strain tensor in OpenFOAM notation. Thanks in advance.

yours
Mallikarjuna Reddy

[fumiya]

Hi,

As you wrote, the magnitude of a tensor is defined in OpenFOAM as eq.(6).
This is also described in the programmer's guide p-16.

Fumiya

[mmkr825]

Hi Fumiya
Thanks for quick response. But in most of the books the shear rate is written as

|E| = sqrt(2E:E)

where rate of strain tensor is: E=symm(grad(U))=1/2[grad(U)+grad(U)^T]

So according to OpenFOAM notation, the shear rate will be

|E| = sqrt(E:E) = sqrt(magSqr(symm(fvc::grad(U))))

Could you please clarify whether this notation is correct or not.

Thanks
Mallikarjuna Reddy

[anishtain4]

I don't understand why the magnitude of tensor should be twice its double contraction?
As a simple oppose example I can name vectors which are first order tensor and its magnitude would be sqrt(Ux^2+Uy^2+Uz^2). Still you may have shear stress right as:
|E| = sqrt(E:E) = sqrt(2*magSqr(symm(fvc::grad(U))))