[er99]

Hi,

I would like some help in clarifying the implementation of the energy equation in sonicFoam in OpenFOAM:

fvScalarMatrix EEqn
(
fvm::ddt(rho, e) + fvm::div(phi, e)
+ fvc::ddt(rho, K) + fvc::div(phi, K)
+ fvc::div(fvc::absolute(phi/fvc::interpolate(rho), U), p, "div(phiv,p)")
- fvm::laplacian(turbulence->alphaEff(), e)
==
fvOptions(rho, e)
);

Specifically, is alphaEff an effective thermal diffusivity which accounts for laminar and turbulent contributions? If so, how is this expressed in an equation?

i.e. alphaEff = alpha_lam + alpha_turb? Or another formulation?

thanks!

[LuckyTran]

To put it simply, yes alphaEff is alpha_laminar + alpha_turbulent.


Eventually thermo.alphaEff will be called which you can find in the appropriate thermophysical source file for your thermophysical model. E.g. /src/thermophysicalModels/basic/heThermo/heThermo.C

[er99]

Originally Posted by LuckyTran

To put it simply, yes alphaEff is alpha_laminar + alpha_turbulent.


Eventually thermo.alphaEff will be called which you can find in the appropriate thermophysical source file for your thermophysical model. E.g. /src/thermophysicalModels/basic/heThermo/heThermo.C

Thanks for your answer? What would the answer be if you didn't have to put it simply? Happy to hear the more complicated answer!

I am using sensibleInternalEnthalpy and just found this post that states alphaEff is actually:

alphaEff = cp/cv(alpha_lam + alpha_turb)

alphaEff

Is it right?

Thanks again for your help!

[LuckyTran]

There is a lot of juggling that happens under-the-hood.


Recall that you're looking at the heat diffusion/conduction term which is divergence of thermal conductivity (kappa in OpenFOAM) times the temperature gradient, which looks very different than laplacian of alphaEff*e.



There isn't a mistake in EEqn.h, but the reason for the Cp/Cv is elusive and I have to dig it up later. I'll probably edit this post when I get around to it.

[er99]

Originally Posted by LuckyTran

There is a lot of juggling that happens under-the-hood.


Recall that you're looking at the heat diffusion/conduction term which is divergence of thermal conductivity (kappa in OpenFOAM) times the temperature gradient, which looks very different than laplacian of alphaEff*e.



There isn't a mistake in EEqn.h, but the reason for the Cp/Cv is elusive and I have to dig it up later. I'll probably edit this post when I get around to it.

Thanks for your time! Could you reply to this message so I get a notification of when you get back to it? I'm just trying to express alphaEff in terms of the laminar and turb components, and right now this is what I've got for the energy equation in latex format:

\begin{equation}
\frac{\partial }{\partial t}\Big[\rho(e + k)\Big] = -\nabla \cdot \Big[\rho U(e + k)\Big] - \nabla \cdot ({\bf{U}} p) + \frac{c_p}{c_v}\Big[\alpha_{l} + \alpha_{t}\Big]\nabla(e)
\end{equation}

Does that look ok?

Thanks!

[LuckyTran]

Btw I hope you meant sensibleInternalEnergy and not sensibleInternalEnthalpy.

It's not cp/cv but actually a variable called cpBycpv. cpBycpv when not needed is set to 1 (i.e. when using sensibleEnthalpy) but set to cpBycpv = gamma when needed (i.e. in sensibleInternalEnergy).

It turned out to be much simpler than I thought. You have:

de = Cv*dT
and
alpha = kappa / Cp

Hence in order to get something that looks like kappa*T, you ordinarily want kappa * T= alpha*Cp * (e / Cv)

Conveniently when you call thermo.alpha in OpenFOAM, you actually get (not alpha but something else) thermo.alpha = cpBycpv*alpha which is alpha*gamma or alpha*Cp/Cv for sensibleInternalEnergy...

The reason for setting up thermo.alpha in this way is for versatility. Someone might decide to use a sensibleEnthalpy formulation instead of sensibleInternalEnergy formulation.

So finally, alphaEff is indeed alpha_laminar + alpha_turbulent, but the way it is written in OpenFOAM in many places, by shoving this cpBycpv in front of it makes it seem elusive. The trick is to not get fooled by this variable that is labeled alpha/alphaEff in OpenFOAM but doesn't actually return the property alpha/alphaEff.