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March 28, 2019, 11:46 
Query regarding law of wall for temperature?

#1 
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Delhi
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Hello everyone!
I am slightly confused about the validity of the law of temperature. I tried to go over literature and I couldn't find a concrete answer. I was wondering if the law of wall for temperature shifts when we can change the wall temperature. I ran RANS simulations in an inhouse code for a flat plate case with different wall temperature (isothermal boundary condition). I selected Twall/Tfreestream as 1.03, 1.04, 1.05, 1.1. I observed that T+ vs y+ was varying for both SA and kw SST turbulence models. I wondering if anyone ran a similar simulation in any other code. I want to check if this behavior is expected or not. It would be great to have the opinion of the CFD community about it. Thank you very much 

March 28, 2019, 12:59 

#2 
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It would be great if someone could give me their opinion regarding it from their experience or relevant literature.


March 29, 2019, 01:41 

#3 
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Lane Carasik
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March 29, 2019, 04:55 

#4 
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Also, show us what you mean by varying (some plots would be useful), how did you define your T+ and y+, your grid parameters, what other bc you used for your case, the properties you used (are you sure the Pr number is the same?).
As it is not expected in general, the first thing I can think of is that your grid is in a gray area, and as you changed the temperature difference, the y+ range chenaged as well. You should be aware that these WF are not perfect, they typically have some variability if you change your y+. 

March 29, 2019, 08:53 

#5  
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Quote:
Keeping this unnecessary discussion aside. Can you please give any technical advice to my question? Thank you! 

March 29, 2019, 09:04 

#6  
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I defined my variables as: where, I take is the freestream value of viscosity. where and is the wall temperature, and u are freestream density and velocity respectively. is specific heat at constant pressure and is wall heat flux. I should mention that this dependence on wall temperature is not due to the absolute value of Twall but the difference with the freestream temperature (). For low temperature difference, this behavior is obtained which I find weird as most of the experimental data were obtained with low temperature difference so as to do a constant parametric (with respect to T) measurements. The results are shown as follows and can be found here: https://drive.google.com/file/d/1wma...ew?usp=sharing Note that here Twall = 1.03 in the legend is basically Twall = 1.03*T_freestream. Where T_freestream is 300 K for all the simulations. The prandtl number (=0.71) and c_p are constant with temperature. These are not wall function simulations, they are lowReynolds number turbulence model simulations with y+ < 0.1. I did mesh convergence study and the result is not varying. It is important to tell here that the U+ vs y+ profile is almost identical to the analytic one for all wall temperatures. I understand your point about grid variability, however I am not running wall functions. It's the turbulence model solved to the wall with y+ < 0.1. The BC I used for the wall is isothermal wall. The case is exactly the 2D flat plate case in NASA turbulence modeling resource. Unfortunately, they don't provide heat transfer data for validation. The velocity profile shows good agreement with CFL3D data. 

March 29, 2019, 09:47 

#7 
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Let us assume that you are computing a constant density flow, otherwise things get much more involved (or, in any case, let us assume that density variations are negligible).
Let us also assume that the viscosity variation with temperature is negligible, as this is the case under which classical law of the walls actually hold. In this scenario, your definition of seems wrong to me, as it requires the local and not freestream value of u at the denominator. Second, how do you compute ? In any case, my experience is with the channel flow where, using a temperature bc instead of a fixed flux one, I also experienced non collapsing profiles, but that was for different Re numbers, which might be a different scenario. 

March 29, 2019, 09:56 

#8 
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I computed as from the CFD solution where where is the temperature at the first cell and is the wall distance. I checked my Stanton number results and they match with the empirical solutions. So, I guess computation (even though it is first order), is not the reason. I also tried greengauss to get the value of temperature gradient, however, results are still the same. Hmm, I realised that I am using viscosity at the wall for computation. This seems correct with the physics! Last edited by aviral2000; March 29, 2019 at 12:39. 

March 29, 2019, 14:53 

#9 
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Everything seems legit to me. As I said, I found a similar behavior when changing the Re number, for fixed wall temperature, in a periodic channel flow. Maybe the two things are related.
Yet, the fact that it gets worst with decreasing temperature difference, which is at your T+ numerator, is suspicious... maybe there is a utau dependence from the temperature difference? 

March 29, 2019, 15:37 

#10  
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Hmm, I don't think that there is dependence as the velocity profile and skinfriction results matches perfectly. As of the current moment, my guess is the computation of is not accurate or probably for the specified conditions i.e. Mach number (0.2 in my case), this behavior is expected. I am testing other mach numbers to understand this behaviour. I expect the to change slightly with Mach number. Do you think this is expected? Thanks for suggestions. 

March 30, 2019, 14:04 

#11  
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where is the adiabatic wall temperature. The use of this definition makes my Stanton number distribution quite independent of Mach number, however, the temperature distribution in the boundary layer is still affected. At lower mach number (M = 0.1), the T+ vs y+ curve for different wall temperature were matching perfectly. I also tried removing the viscous dissipation term from the RANS solver and observed that even though my St number result is different now (expected behaviour), the temperature profile for different mach numbers are matching nicely. Most of the T+ vs y+ curves and studies are for incompressible solver which does not model viscous dissipation term. Also, for higher compressible flows, VanDriest or other similar transformations are used to convert to velocity and temperature profile to analytic ones. The case I was doing was in sort of a gray area as I was using a compressible solver for a lowspeed flow without using these transformations for the velocity profile. That's my opinion regarding the issue. It will be great to have an opinion from you and other members of CFD community regarding it. Feel free to point out any mistakes in my reasoning or suggestions for additional verification tests. Thank you! 

March 30, 2019, 14:19 

#12  
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Quote:
In more general terms, I would conclude that the main mistake was in applying an incompressible law to a compressible case. Thus, besides the viscous dissipation term, a varying density and, in case, viscosity, might invalidate it as well. 

Tags 
heat transfer, law of wall, temperature, turbulence modeling 
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