## Properly include the volume work term in the energy balance

We currently implement

`\frac{\partial}{\partial t}(\rho u) = - \nabla \cdot (\rho \mathbf{v}h) + \nabla \cdot (\lambda \nabla T)`

with `h = u + \frac{p}{\rho}`

However, the correct form of the energy balance equation is (see !2473 (closed), !2471 (merged))

`\frac{\partial}{\partial t}(\rho u) = - \nabla \cdot (\rho \mathbf{v}u) -p \nabla \cdot \mathbf{v} + \nabla \cdot (\lambda \nabla T)`

Replacing enthalpy with internal energy in the first divergence term is simple but the discretization of

`p \nabla \cdot \mathbf{v}`

is not straight-forward.

A possible simple solution would be:

Include in calculation of flux terms, i.e., evaluate `\nabla \cdot \mathbf{v}`

and assume a cell-constant `p`

which is just multiplied to the term. This gives a first-order scheme but should be ok since the term is often small. For incompressible flow the term also vanishes discretely since `\nabla \cdot \mathbf{v} = 0`

.

This seems to work for cell-centered schemes without much additional effort (!2494). For the box method it seems more difficult since the code assumes that fluxes on scvfs are symmetric. However the energy contribution (assembled on the scvfs) is different depending on the control volume (different control volume pressure).