<div dir="ltr">he Timo,<div><br></div><div style>there is no such thing as adiabatic heating in the incompressible Boussinesq case Di (alpha*g/cp) is assumed zero .</div><div style>for extended Boussinesq there should also be no problem since there is no density in the net adiabatic heating term.</div>
<div style><br></div><div style>setting thermal diffusion, viscous dissipation and internal heating to zero (dS/dt=0) we end up with </div><div style>rhocp(dT/dt) - alphaTdP/dt=0 </div><div style>or <br></div><div style>
rho*cp*(dT/dt) - alpha*rho*g*u_r*T=0</div><div style><br></div><div style>this will give for an adiabatic temperature profile</div><div style>T(r) = T_0*exp(alpha*g*r/cp) </div><div style><br></div><div style>iow the density does not play a role since its devided out of the equation.</div>
<div style><br></div><div style>this also holds for the compressible case i would say.</div><div style><br></div><div style>cheers</div><div style>Thomas</div><div style><br></div></div><div class="gmail_extra"><br><br><div class="gmail_quote">
On Tue, Feb 12, 2013 at 5:57 AM, Timo Heister <span dir="ltr"><<a href="mailto:heister@math.tamu.edu" target="_blank">heister@math.tamu.edu</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
Hey everyone,<br>
<br>
Ian approached me about this and I asked him to write it down here.<br>
Does anyone have any feedback about this, especially (assuming this is<br>
correct), what to do in the compressible case?<br>
<div><div class="h5"><br>
On Wed, Feb 6, 2013 at 6:33 PM, Ian Rose <<a href="mailto:ian.rose@berkeley.edu">ian.rose@berkeley.edu</a>> wrote:<br>
> Hi Aspect folks,<br>
><br>
> I was working through some tests with Aspect and came across what I believe<br>
> is an inconsistency in the governing equations.<br>
><br>
> For incompressible Boussinesq flow, the global viscous dissipation should<br>
> exactly cancel the global adiabatic heating. This can be seen by<br>
> multiplying the momentum equation by velocity and integrating over the<br>
> domain.<br>
><br>
> As it stands in assembly.cc, the formula used for calculating adiabatic<br>
> heating is different from that you would get by integrating the momentum<br>
> equation. I wrote a simple postprocessor that compares the two integrated<br>
> quantities which I am attaching. The difference is quite a lot for the<br>
> current formula.<br>
><br>
> Put another way, this is the formula that is currently used:<br>
><br>
> Q_a = ( velocity * gravity ) * alpha * density * temperature<br>
><br>
> The density at this point however, has already been adjusted for<br>
> temperature, so we are in effect double counting the thermal expansion.<br>
> Instead, I believe it should be<br>
><br>
> Q_a = ( velocity * gravity ) * ( density - reference_density )<br>
><br>
><br>
> The compressible case, too, should require some thought, though I have not<br>
> gone through the paces there.<br>
><br>
> Thoughts?<br>
><br>
> Best,<br>
> Ian<br>
><br>
> PS, for some details on the derivations, I refer you to Leng and Zhong<br>
> (2008)<br>
><br>
><br>
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<span class="HOEnZb"><font color="#888888"><br>
<br>
<br>
--<br>
Timo Heister<br>
<a href="http://www.math.tamu.edu/~heister/" target="_blank">http://www.math.tamu.edu/~heister/</a><br>
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