# A generalized thermal/dynamic wall function: Part 4

Posted February 7, 2019 at 11:29 by sbaffini

In previous posts of this series I presented an elaboration of the Musker-Monkewitz analytical wall function that allowed extensions to non equilibrium cases and to thermal (scalar) cases with, in theory, arbitrary Pr/Pr_t (Sc/Sc_t) ratios.

In the meanwhile, I worked on a rationalization and generalization of the framework, derivation of averaged production term for the TKE equation, etc.

While the new material is presented in a substantially different manner and will require a dedicated post (probably a simple link to the material posted elsewhere), few details emerged that are still worth mentioning in this posts series.

In particular, what is worth discussing here is the fit of the presented wall function (and, for that matter, of wall functions in general) to a particular turbulence model. Indeed, while wall functions are typically presented in stand alone fashion, without particular reference to the turbulence model in use (and, indeed, as done here too in the previous posts), it is important that their analytical profile actually fits, as close as possible, the one expected from the tubulence model in use. This becomes of paramount importance when using an y+ insensitive formulation (as the presented one is intended to be), or such insensitivity is not really achieved.

Thus, for example, using the Reichardt or the Spalding profile (which are well known y+ insensitive formulations) with a turbulence model that, when resolved to the wall, provides a different velocity profile, is not optimal and is not going to produce the expected results of insensitivity.

Things get particularly troublesome when going to the thermal (scalar) case with high Pr/Pr_t (Sc/Sc_t) ratios. Indeed, as this ratio ideally (or practically, depending from the specific formulation) multiplies the viscosity ratio underlying the wall function, even minute differences, not typically relevant for Pr/Pr_t (Sc/Sc_t) < 1 (e.g., the velocity case), are instead amplified.

Thus, for example, using for the temperature the well known Kader formulation with the Jayatilleke term for the log part is not typically going to match the results of a given turbulence model at all the Pr/Pr_t ratios. The same just happens also for the presented Musker-Monkewitz wall function, that has its own peculiar dependence from the ratio Pr/Pr_t.

With this post I just want to present a fitting of the basic profile constant a in the Musker-Monkewitz wall function that can be used to match, approximately, the Spalart-Allmaras temperature/scalar profile. I have similar fittings also for other all y+ models, but SA is relevant because the viscosity ratio profile is simple and can be integrated with a very simple routine (and is thus included for comparison in the attached one).

Just change, as usual, the file extensions from .txt in .m and launch comparewf (having musker.m in the same folder). The adjusted Munker wall function gets compared with the numerically integrated SA profile and the reference Kader wall function.

You can play, in comparewf (don't touch musker.m), with the Pr/Pr_t ratio and the non-equilibrium source term FT (but then note that the Kader profile does not include its effects) and see how the fit works relatively better than the Kader profile for SA.

In particular, the present fit for the constant a is:

where the constant values can be found in the attached file. Of course, this fit is just an attempt and should not be taken as etched in stone. In particular it is based on the SA profile using the von Karman constant vk = 0.4187.

Note also that, with respect to the previous posts, where I made a mistake, the suggested default value for the profile constant is the one of the original authors (of course), a0 = 10.306.

In the meanwhile, I worked on a rationalization and generalization of the framework, derivation of averaged production term for the TKE equation, etc.

While the new material is presented in a substantially different manner and will require a dedicated post (probably a simple link to the material posted elsewhere), few details emerged that are still worth mentioning in this posts series.

In particular, what is worth discussing here is the fit of the presented wall function (and, for that matter, of wall functions in general) to a particular turbulence model. Indeed, while wall functions are typically presented in stand alone fashion, without particular reference to the turbulence model in use (and, indeed, as done here too in the previous posts), it is important that their analytical profile actually fits, as close as possible, the one expected from the tubulence model in use. This becomes of paramount importance when using an y+ insensitive formulation (as the presented one is intended to be), or such insensitivity is not really achieved.

Thus, for example, using the Reichardt or the Spalding profile (which are well known y+ insensitive formulations) with a turbulence model that, when resolved to the wall, provides a different velocity profile, is not optimal and is not going to produce the expected results of insensitivity.

Things get particularly troublesome when going to the thermal (scalar) case with high Pr/Pr_t (Sc/Sc_t) ratios. Indeed, as this ratio ideally (or practically, depending from the specific formulation) multiplies the viscosity ratio underlying the wall function, even minute differences, not typically relevant for Pr/Pr_t (Sc/Sc_t) < 1 (e.g., the velocity case), are instead amplified.

Thus, for example, using for the temperature the well known Kader formulation with the Jayatilleke term for the log part is not typically going to match the results of a given turbulence model at all the Pr/Pr_t ratios. The same just happens also for the presented Musker-Monkewitz wall function, that has its own peculiar dependence from the ratio Pr/Pr_t.

With this post I just want to present a fitting of the basic profile constant a in the Musker-Monkewitz wall function that can be used to match, approximately, the Spalart-Allmaras temperature/scalar profile. I have similar fittings also for other all y+ models, but SA is relevant because the viscosity ratio profile is simple and can be integrated with a very simple routine (and is thus included for comparison in the attached one).

Just change, as usual, the file extensions from .txt in .m and launch comparewf (having musker.m in the same folder). The adjusted Munker wall function gets compared with the numerically integrated SA profile and the reference Kader wall function.

You can play, in comparewf (don't touch musker.m), with the Pr/Pr_t ratio and the non-equilibrium source term FT (but then note that the Kader profile does not include its effects) and see how the fit works relatively better than the Kader profile for SA.

In particular, the present fit for the constant a is:

where the constant values can be found in the attached file. Of course, this fit is just an attempt and should not be taken as etched in stone. In particular it is based on the SA profile using the von Karman constant vk = 0.4187.

Note also that, with respect to the previous posts, where I made a mistake, the suggested default value for the profile constant is the one of the original authors (of course), a0 = 10.306.

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