[aspect-devel] Thermodynamic consistency of Aspect's temperature and momentum equations

[Timo Heister]

Hey everyone,

Ian approached me about this and I asked him to write it down here.
Does anyone have any feedback about this, especially (assuming this is
correct), what to do in the compressible case?

On Wed, Feb 6, 2013 at 6:33 PM, Ian Rose <ian.rose at berkeley.edu> wrote:
> Hi Aspect folks,
>
> I was working through some tests with Aspect and came across what I believe
> is an inconsistency in the governing equations.
>
> For incompressible Boussinesq flow, the global viscous dissipation should
> exactly cancel the global adiabatic heating.  This can be seen by
> multiplying the momentum equation by velocity and integrating over the
> domain.
>
> As it stands in assembly.cc, the formula used for calculating adiabatic
> heating is different from that you would get by integrating the momentum
> equation.  I wrote a simple postprocessor that compares the two integrated
> quantities which I am attaching.  The difference is quite a lot for the
> current formula.
>
> Put another way, this is the formula that is currently used:
>
>    Q_a = ( velocity * gravity ) * alpha * density * temperature
>
> The density at this point however, has already been adjusted for
> temperature, so we are in effect double counting the thermal expansion.
> Instead, I believe it should be
>
>   Q_a = ( velocity * gravity ) * ( density - reference_density )
>
>
> The compressible case, too, should require some thought, though I have not
> gone through the paces there.
>
> Thoughts?
>
> Best,
> Ian
>
> PS, for some details on the derivations, I refer you to Leng and Zhong
> (2008)
>
>
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-- 
Timo Heister
http://www.math.tamu.edu/~heister/