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can energy equations be written in conservative form?

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Old   May 22, 2024, 12:36
Default can energy equations be written in conservative form?
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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.
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Old   May 22, 2024, 13:29
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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
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Old   May 22, 2024, 20:55
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In case there is any doubt.... see Noether's theorem
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Old   May 23, 2024, 07:20
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Quote:
Originally Posted by hunt_mat View Post
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
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Old   May 23, 2024, 09:46
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It's a little more difficult than that, the equation that I have been working with is:
\rho c\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}=\kappa\frac{\partial^{2}T}{\partial x^{2}}+\alpha\rho T\frac{\partial u}{\partial x}-K\rho\frac{\partial u}{\partial x}

I don't know what the ``energy'' is in this case.
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Old   May 23, 2024, 10:40
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Quote:
Originally Posted by hunt_mat View Post
It's a little more difficult than that, the equation that I have been working with is:
\rho c\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}=\kappa\frac{\partial^{2}T}{\partial x^{2}}+\alpha\rho T\frac{\partial u}{\partial x}-K\rho\frac{\partial u}{\partial x}

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.
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Old   June 3, 2024, 05:44
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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.
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Old   June 3, 2024, 06:37
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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
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Old   June 3, 2024, 08:26
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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.
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Old   June 3, 2024, 20:53
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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.
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