[mihirmakwana6]

How can I use FVM to discretize the term

sigma : D


in the internal energy equation

http://i.imgur.com/3RaYm5p.jpg

where

sigma and D are given by

http://i.imgur.com/6K65Lih.jpg


I am not able to expand the term sigma : D

Please help.

Thanks in Advance

- Mihir

[FMDenaro]

If you want using a FVM, you have to recast the equation in the divergence form and integrate over a finite volume each term.
Using the expression of sigma, the term D:D you get is a scalar function that appears as volume source

[mihirmakwana6]

Originally Posted by FMDenaro

If you want using a FVM, you have to recast the equation in the divergence form and integrate over a finite volume each term.
Using the expression of sigma, the term D:D you get is a scalar function that appears as volume source

sir, if i use tensors then sigma : D gives

http://i.imgur.com/UmzDCrI.jpg

How do i convert this equation to divergence form ?

[mihirmakwana6]

sir say i split D

then one of the term is

http://i.imgur.com/zkbee0D.jpg


here Tau is sigma

so the 1st term on r.h.s is in divergence form but the second is not

[FMDenaro]

no, you do not have all terms in te internal energy equation in divergence form...that respects the physical fact that such energy form does not have a conservative formulation. The term D: D is a pointwise source term, you have to integrate over the local volume in a FVM and discretize the volume integral.
If you want to work with a fully divergence form then you have to adopt the total energy equation

[mihirmakwana6]

Originally Posted by FMDenaro

no, you do not have all terms in te internal energy equation in divergence form...that respects the physical fact that such energy form does not have a conservative formulation. The term D: D is a pointwise source term, you have to integrate over the local volume in a FVM and discretize the volume integral.
If you want to work with a fully divergence form then you have to adopt the total energy equation

ok.

1) As i am using FVM, i cannot integrate the second term in RHS of

http://i.imgur.com/zkbee0D.jpg

Right ??

or is there a way I can integrate it over the C.V

2) what is D: D ?

[FMDenaro]

you can always integrate each term of the equation over a finite volume...when the term is in divergence form you can apply Gauss and get the surface integral of the fluxes, conversely the integral to discretize remains the volume integral (see for example the book of Peric & Ferziger).

D: D is nothing else that the double dot product between the symmetric velocity gradient

[mihirmakwana6]

Originally Posted by FMDenaro

you can always integrate each term of the equation over a finite volume...when the term is in divergence form you can apply Gauss and get the surface integral of the fluxes, conversely the integral to discretize remains the volume integral (see for example the book of Peric & Ferziger).

D: D is nothing else that the double dot product between the symmetric velocity gradient

1) ok.

2) but i need to find sigma : D and not D : D

[FMDenaro]

sigma: D is nothing else that a product by an isotropic tensor (I: D) added with D: D

for brevity I disregarded the coefficients