can energy equations be written in conservative form?

hunt_mat · May 22, 2024, 12:36

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 · May 22, 2024, 13:29

Quote:

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 · May 22, 2024, 20:55

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

FMDenaro · May 23, 2024, 07:20

Quote:

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 · May 23, 2024, 09:46

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.

FMDenaro · May 23, 2024, 10:40

Quote:

Originally Posted by hunt_mat

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.

hunt_mat · June 3, 2024, 05:44

Quote:

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 · June 3, 2024, 06:37

Quote:

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 · June 3, 2024, 08:26

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 · June 3, 2024, 20:53

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.

May 22, 2024, 12:36	can energy equations be written in conservative form?	#1
hunt_mat Senior Member Matthew Join Date: Mar 2022 Location: United Kingdom Posts: 178 Rep Power: 4	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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May 22, 2024, 20:55		#3
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,683 Rep Power: 66	In case there is any doubt.... see Noether's theorem

May 23, 2024, 09:46		#5
hunt_mat Senior Member Matthew Join Date: Mar 2022 Location: United Kingdom Posts: 178 Rep Power: 4	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.

June 3, 2024, 08:26		#9
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,163 Blog Entries: 29 Rep Power: 39	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.

June 3, 2024, 20:53		#10
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,683 Rep Power: 66	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.