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Ravenn April 2, 2013 19:45

Turbulent Prandtl number

I'm trying to run a given case and it is said that the turbulent Prandtl number is set to 0.9 (no more information).
I've searched in Fluent and the definition of the turbulent Prandtl number is unclear to me, there are at least 5 different turbulent Prandtl numbers (with SST):
- TKE inner
- TKE outer
- SDR inner
- SDR outer
- Energy
- Wall

I don't think the first four are related to the given turbulent Prandtl number in my paper. But the Energy and Wall turbulent Prandtl number have the same value, close to 0.9. Should I change both of them?
Do you have any information about turbulent Prandtl number?
I've looked at the definition of turbulent Prandtl number (wiki, cfd-online) and at the definition in Fluent Manual, but I can't really figure out which one is concerned here.

The case is about hypersonic interactions and heat transfer.

Thanks for your help,

oj.bulmer April 3, 2013 10:57

Can you give more information about the paper you are referring to?


pete April 3, 2013 11:21

The turbulent Prandtl number is used to relate turbulent heat flux with turbulent momentum flux. The definition is given in for example this CFD-Wiki page:

By setting a constant turbulent Prandtl number you can compute the turbulent heat flux based on the turbulent eddy-viscosity that a turbulence model predicts.

Changing the turbulent Prandtl number is a way to tune heat transfer results. So always be skeptical about papers where it is not stated what turbulent Prandtl number has been used.

Ravenn April 3, 2013 11:24

I only have hard copies of different papers:
- AIAA paper 93-0779: numerical simulation of crossing shock/turbulent boundary layer interaction at Mach 8.3
"The baldwin-Lomax and the Rodi k-epsilon turbulence models are employed, the molecular viscosity was specified by Sutherland's law. The molecular Prandtl number is 0.73 (air) and the turbulent Prandtl number is 0.9."

- AIAA Journal Vol 39 No 6 June 2001 p: 985-995: Insights in Turbulence Modeling for Crossing-Shock-Wave/Boundary-Layer Interactions
"The Sutherland's law is used to calculate the laminar viscosity,and a constant laminar Prandtl number of 0.72 is assumed. Central differencing is used to evaluate the viscous terms.The steady solution is obtained by applying a time-marching method based on the hybrid approximate actorization/relaxation algorithm. The first solutions presented here are computed with the k–x turbulence model by Wilcox and a constant turbulent Prandtl number of 0.9."


That's exactly the problem, I've seen in many papers "constant turbulent Prandtl number of 0.9" without more information, I suppose it is linked to the hypersonic characteristic of the flow, but it dos not seem like a "cheat" or something to obtain better results. Or someone proposed in the 50's to use this value for this case and ever since everybody uses it.

Edit 2:
In the cfd-online wiki page, it is said after equation 26 that:
"Where Prt is a turbulent Prandtl number. Often a constant Prt = 0.9 is used."
In that case, which one of the fluent turbulent Prandtl number is concerned?

Is it possible that turbulent Prendtl numbers in fluent are not this turbulent Prandtl number Prt defined here, hence the Prt would be a results of the calculation and not a property defined before the calculation?

pete April 3, 2013 11:50

In laminar flow the viscosity is used to compute the heat-flux using Fourier's law and a laminar, well defined, Prandtl number Pr \equiv \frac{C_p \mu}{\lambda} as:

q_j = -\lambda \frac{\partial T}{\partial x_j}
    \equiv -C_p \frac{\mu}{Pr} \frac{\partial T}{\partial x_j}

Most turbulence models just give a turbulent eddy viscosity \mu_t. By setting a turbulent Prandtl number the turbulent heat flux can be estimated in the same way by just using the turbulent eddy viscosity that the turbulence mode predicts:

q_j^{turb} \equiv
C_p \overline{\rho u''_j T} \approx
- C_p \frac{\mu_t}{Pr_t} \frac{\partial \widetilde{T}}{\partial x_j}

Using a constant turbulent Prandtl number is a simplification and it is not fully correct. Experimentally a value of something close to 0.9 has been measured. You can find more information and further references here:

Ravenn April 3, 2013 17:18

So, I've post-processed 2 different calculations, one with the default values for energy and wall Prandtl number, and the other with energy and wall Prandtl number set to 0.9.

I've read carefully fluent user manual and here is what I understood:
- k_{eff} = k + k_T
- \mu_{eff} = \mu + \mu_T
- Pr_{eff} = C_p \frac{\mu_{eff}}{k_{eff}}

In Fluent, it is possible to obtain k_{eff}, k and \mu_T. With these values on a line normal to a surface, I've calculated k_{T} = k_{eff}-k and then Pr_{T} = Cp \frac{\mu_T}{k_T}.

It appears that for the first calculation, Pr_{T}=0.85 outside the laminar part of boundary layer, which is the value of energy and wall Prandtl numbers.
For the second calculation where I changed the value of energy and wall Prandtl numbers to 0.9, Pr_{T}=0.9 outside the laminar part of boundary layer.

In the laminar part of boundary layer, k_{T}=0, hence Pr_{T} is undefined, or very high.

I'm wondering now what is the value of Pr_{T} in the laminar part of boundary layer for the papers where Pr_{T} is said to be fixed to 0.9.
In my opinion it depends on which parameter is used to calculate the other:
- Pr_{T}, fixed to 0.9, and \mu_{T} are used to calculate k_{T} (papers)
- \mu_{T} and k_{T} are calculated by the turbulence model, and Pr_{T} can be deduced with the same procedure I used. (Fluent)

Shamoon Jamshed May 23, 2016 13:06

Dear Ravenn

I was facing the same issue. I used two turbulence models, kepsilon and k-omega SST. I was getting incorrect Nu from the komega. So I decreased very much the Prt and got my result close to experiment. But since I used very low value (0.35) I am still doubtful. Are you still working on that too? yuor post seems 3 years old

Ravenn May 24, 2016 15:57

Hello Shamoon,

I'm sorry but it's been a while since I last worked on this subject (almost 3 years as you see) and I don't really remember everything, I've rediscovered the subject with your question. It appears I still receive the alerts for this post^^

Good luck for your research!

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