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Closing on wall functions - part 8: coupled/thin wall boundary conditions

Posted April 27, 2022 at 08:37 by sbaffini (NuTBox)
Updated May 16, 2022 at 11:51 by sbaffini

It might happen that wall functions for temperature or scalars are needed when the assigned boundary condition is not simply the assigned flux or temperature/scalar, but rather a more complex one. One example is when the wall is a coupled one, with either a fluid or a solid on the other side, or it has some thickness, possibly with source terms in it and, say, radiation or convective boundary conditions assigned on the other side, or maybe some other combination of the above.

In all...

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Closing on wall functions - part 7: starting from a profile

Posted April 26, 2022 at 12:12 by sbaffini (NuTBox)
Updated April 27, 2022 at 18:57 by sbaffini

It might happens that one doesn't have a turbulent viscosity profile but actually has just an equilibrium profile for velocity or temperature. More specifically:

$T^+ = Pr \left(\frac{{s_T^{-1}}^+}{y^+}\right) y^+$

with its obvious extension to the velocity case. In order to go back to the framework presented here one should notice that:

$\frac{d}{dy^+}\left({s_T^{-1}}^+\right) = \frac{1}{1+\frac{Pr}{Pr_t}\frac{\mu_t}{\mu}}$ ...

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Closing on wall functions - part 6: viscous dissipation

Posted April 24, 2022 at 10:20 by sbaffini (NuTBox)
Updated June 1, 2022 at 12:28 by sbaffini

One thing which is missing in the previous derivations is the viscous dissipation term in the temperature equation. Let's reconsider the initial temperature equation when it is present:

$\frac{d}{dy}\left[C_p\left(\frac{\mu}{Pr}+\frac{\mu_t}{Pr_t}\right)\frac{dT}{dy}\right]=F_T - \frac{d}{dy}\left[\left(\mu+\mu_t\right)U\frac{dU}{dy}\right]$

A first integration leads to:

$\left(\frac{C_p \mu}{Pr}\right)\left(1+\frac{Pr}{Pr_t}\frac{\mu_t}{\mu}\right)\frac{dT}{dy}=q_w+$ ...

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Closing on wall functions - part 5: testing scripts

Posted April 23, 2022 at 21:26 by sbaffini (NuTBox)
Updated June 3, 2022 at 08:02 by sbaffini

I provide here a set of MATLAB scripts to test all the claims made in the first 4 parts.

The first group of scripts is actually made of functions, that you are not supposed to directly call or modify:

muskersp.m: returns $\left(\frac{{s_{U,T}^i}^+}{{y^+}^{i+2}}\right)$ , $\left(\frac{{p^i}^+}{{y^+}^{i+2}}\right)$ and ${q^i}^+$ as shown here. It only works for N up to 0 (constant non equilibrium terms) EDIT: There is an apparently innocuous mistake in the limiting behavior of s,

...

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wallfunction.zip (14.2 KB, 523 views)

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Closing on wall functions - part 4: a Musker-Monkewitz wall function

Posted April 23, 2022 at 17:53 by sbaffini (NuTBox)
Updated April 26, 2022 at 16:49 by sbaffini

In the Musker-Monkewitz wall function the following assumption is made on the turbulent viscosity profile:

$\frac{\mu_t}{\mu} = \frac{\left(\kappa y^+\right)^3}{\left( \kappa y^+\right)^2+\left(\kappa a \right)^3-\left(\kappa a \right)^2}$

where $\kappa$ is the von Karman constant and