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

Posted April 26, 2022 at 13:12 by sbaffini (NuTBox)
Updated April 27, 2022 at 19:57 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:

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

sbaffini

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Closing on wall functions - part 6: viscous dissipation

Posted April 24, 2022 at 11:20 by sbaffini (NuTBox)
Updated June 1, 2022 at 13:28 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 22:26 by sbaffini (NuTBox)
Updated June 3, 2022 at 09:02 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,

...

Attached Files

wallfunction.zip (14.2 KB, 667 views)

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

Posted April 23, 2022 at 18:53 by sbaffini (NuTBox)
Updated April 26, 2022 at 17:49 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

is a constant that specifies the

for which $\frac{\mu_t}{\mu} =1$ but, in practical terms has the same role of

in the standard wall function of the previous post....

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Closing on wall functions - part 3: a standard wall function

Posted April 23, 2022 at 06:41 by sbaffini (NuTBox)
Updated May 15, 2022 at 04:20 by sbaffini

We make the following assumption for the turbulent viscosity ratio:

$\frac{\mu_t}{\mu} = \left\{ \begin{array}{ll} 0 & \text{for } y^+ < y_v^+ \\ \kappa y^+ & \text{for } y^+ \geq y_v^+ \end{array} \right.$

where $\kappa$ is the von Karman constant and