OpenFOAM's Energy Equation 'h' Form: Replacing 'Fourie' with Gradient of 'hk/cp'

ztdep · January 27, 2024, 20:01

In the "h" form of the energy equation, OpenFOAM has replaced the "Fourier Law" with the gradient of "h" multiplied with "k/cp". I want to know what kind of assumptions should be satisfied to validate this. I think Cp must be a constant value, but in reality , most fluids don't satisfy this condition.

fvm::laplacian(turbulence->alphaEff(), he)

$\frac{\partial h}{\partial x} = \frac{\partial h}{\partial T} \frac{\partial T}{\partial x} = C_p \frac{\partial T}{\partial x}$

Tobermory · January 29, 2024, 09:09

No - I don't believe so. You have written out the chain rule which demonstrates that grad T can be replaced with grad h divided by Cp ... there is no need for Cp to be constant. However, note that:

$C_p \equiv (\frac{\partial h}{\partial T})_p$

and this provides you your assumption ... the equivalence holds for constant pressure, which is generally acceptable for many flows but may prove problematic for strongly compressible flows.

January 27, 2024, 20:01	OpenFOAM's Energy Equation 'h' Form: Replacing 'Fourie' with Gradient of 'hk/cp'	#1
ztdep Senior Member p ding Join Date: Mar 2009 Posts: 427 Rep Power: 19	In the "h" form of the energy equation, OpenFOAM has replaced the "Fourier Law" with the gradient of "h" multiplied with "k/cp". I want to know what kind of assumptions should be satisfied to validate this. I think Cp must be a constant value, but in reality , most fluids don't satisfy this condition. fvm::laplacian(turbulence->alphaEff(), he) $\frac{\partial h}{\partial x} = \frac{\partial h}{\partial T} \frac{\partial T}{\partial x} = C_p \frac{\partial T}{\partial x}$

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January 29, 2024, 09:09		#2
Tobermory Senior Member Join Date: Apr 2020 Location: UK Posts: 668 Rep Power: 14	No - I don't believe so. You have written out the chain rule which demonstrates that grad T can be replaced with grad h divided by Cp ... there is no need for Cp to be constant. However, note that: $C_p \equiv (\frac{\partial h}{\partial T})_p$ and this provides you your assumption ... the equivalence holds for constant pressure, which is generally acceptable for many flows but may prove problematic for strongly compressible flows.