Turbulent heat flux

Eicosane · April 14, 2023, 15:10

Greetings, everyone!

I work on the modeling of compressible turbulent flows, and as a part of simulation i should somehow model turbulent diffusive heat flux, which appears in filtered equation of total energy. There is an ambiguity in the papers on how to do it.
The two main approaches are
$q_{t} = -\frac{\mu_{t}}{Pr_{t}}\cdot \nabla \tilde e$ and $q_{t} = -\lambda_{t}\cdot \nabla \bar T$ , where gradients of internal energy per mass and temperature are used. The second approach, in turn, again splits in two ways by the definition of turbulent heat conductivity:
$\lambda_{t} = -\frac{\bar C_{p} \cdot \mu_{t}}{Pr_{t}}$ ,
$\lambda_{t} = -\frac{\bar C_{V} \cdot \mu_{t}}{Pr_{t}}$ .
Where $\bar C_{p}$ --- isobaric heat capacity and $\bar C_{V}$ --- isochoric heat capacity.
Authors of papers do not provide arguments on the choice of the diffusive flow representation, and so i struggle to choose, which variant i should implement in my code?

LuckyTran · April 17, 2023, 04:00

It depends on whether your energy equation is written in internal energy or enthalpy and how the total energy is tracked (whether it includes pressure work or not). In other words, you have to write out the entire energy equation, you cannot get your answer by only looking at this one term. It is indeed ambiguous when you leave out the entire equation.

Eicosane · April 24, 2023, 13:07

Quote:

Originally Posted by LuckyTran

It depends on whether your energy equation is written in internal energy or enthalpy and how the total energy is tracked...

Thank you for your answer.

So it depends on the form of the energy equation solved, i.e. for
$\frac{\partial \rho E}{\partial t} + \nabla \cdot (\rho E \cdot \vec{\upsilon}) + \nabla \cdot (P\cdot \vec{\upsilon}) = ...$
$\bar{C}_{V}$ must be used, and for
$\frac{\partial \rho E}{\partial t} + \nabla \cdot (\rho h \cdot \vec{\upsilon}) = ...$
or
$\frac{\partial \rho h}{\partial t} + \nabla \cdot (\rho h \cdot \vec{\upsilon}) - \frac{\partial p}{\partial t}= ...$
it must be $\bar{C}_{P}$ ?

April 14, 2023, 15:10	Turbulent heat flux	#1
Eicosane New Member Maksim Join Date: Jul 2020 Posts: 10 Rep Power: 5	Greetings, everyone! I work on the modeling of compressible turbulent flows, and as a part of simulation i should somehow model turbulent diffusive heat flux, which appears in filtered equation of total energy. There is an ambiguity in the papers on how to do it. The two main approaches are $q_{t} = -\frac{\mu_{t}}{Pr_{t}}\cdot \nabla \tilde e$ and $q_{t} = -\lambda_{t}\cdot \nabla \bar T$ , where gradients of internal energy per mass and temperature are used. The second approach, in turn, again splits in two ways by the definition of turbulent heat conductivity: $\lambda_{t} = -\frac{\bar C_{p} \cdot \mu_{t}}{Pr_{t}}$ , $\lambda_{t} = -\frac{\bar C_{V} \cdot \mu_{t}}{Pr_{t}}$ . Where $\bar C_{p}$ --- isobaric heat capacity and $\bar C_{V}$ --- isochoric heat capacity. Authors of papers do not provide arguments on the choice of the diffusive flow representation, and so i struggle to choose, which variant i should implement in my code?

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April 17, 2023, 04:00		#2
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,704 Rep Power: 66	It depends on whether your energy equation is written in internal energy or enthalpy and how the total energy is tracked (whether it includes pressure work or not). In other words, you have to write out the entire energy equation, you cannot get your answer by only looking at this one term. It is indeed ambiguous when you leave out the entire equation.