[hunt_mat]

The mass and momentum equations can usually be written in conservative form which is useful for the finite volume method. My question is: Can the conservation of energy be written in this form? I'm thinking of an application of solid mechanics where you have movement.

[FMDenaro]

Originally Posted by hunt_mat

The mass and momentum equations can usually be written in conservative form which is useful for the finite volume method. My question is: Can the conservation of energy be written in this form? I'm thinking of an application of solid mechanics where you have movement.

The total energy has indeed a conservation form! You can find that in any fluid mechanics textbook

[LuckyTran]

In case there is any doubt.... see Noether's theorem

[FMDenaro]

Originally Posted by hunt_mat

The mass and momentum equations can usually be written in conservative form which is useful for the finite volume method. My question is: Can the conservation of energy be written in this form? I'm thinking of an application of solid mechanics where you have movement.

Here is the form

[hunt_mat]

It's a little more difficult than that, the equation that I have been working with is:


I don't know what the ``energy'' is in this case.

[FMDenaro]

Originally Posted by hunt_mat

It's a little more difficult than that, the equation that I have been working with is:


I don't know what the ``energy'' is in this case.

Actually, this is the balance equation for the temperature. It is the quasi-linear form of the internal energy (rho c T). No one of such variables has a conservation property, only the total energy does.
Even if you write the temperature equation in integral form, that is not a conservation equation, you have a production term.

[hunt_mat]

Originally Posted by FMDenaro

Actually, this is the balance equation for the temperature. It is the quasi-linear form of the internal energy (rho c T). No one of such variables has a conservation property, only the total energy does.
Even if you write the temperature equation in integral form, that is not a conservation equation, you have a production term.

I'm aware that is temperature, what I am really interested in is if I can write it in conservative form, so I can use the finite volume method.

[FMDenaro]

Originally Posted by hunt_mat

I'm aware that is temperature, what I am really interested in is if I can write it in conservative form, so I can use the finite volume method.

You can write an integral equation for the internal energy, however it has a production term that remains in the form of volume integral, you cannot use Gauss to convert in the surface integral of the fluxes.
This is the form you can use for a FV discretization

[sbaffini]

In summary, not surprisingly, you can write in fully conservative form only equations for fully conserved quantities. Total energy is fully conserved, not internal or kinetic separately.

[LuckyTran]

Just to give a 3rd redundant answer...

Any transport equation works with FVM. FVM doesn't breakdown simply because you choose temperature (a non-conservative property) as your transport variable. You just integrate the production term (sources/sinks) over the volume and FVM still works.