Closing on wall functions - part 7: starting from a profile
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^+ T^+ = Pr \left(\frac{{s_T^{-1}}^+}{y^+}\right) y^+](/Forums/vbLatex/img/bf4eadbf488ef695500ba80212ef37d4-1.gif)
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}} \frac{d}{dy^+}\left({s_T^{-1}}^+\right) = \frac{1}{1+\frac{Pr}{Pr_t}\frac{\mu_t}{\mu}}](/Forums/vbLatex/img/7df47c4a3c580b62929934b1f42cff58-1.gif)
from which, it follows that:
![\frac{Pr}{Pr_t}\frac{\mu_t}{\mu} = \frac{1}{\frac{d}{dy^+}\left({s_T^{-1}}^+\right)}-1 \frac{Pr}{Pr_t}\frac{\mu_t}{\mu} = \frac{1}{\frac{d}{dy^+}\left({s_T^{-1}}^+\right)}-1](/Forums/vbLatex/img/0ef92176ba88c949ad8bdd353f7dd497-1.gif)
which is all that is needed to compute (numerically if not doable analytically) the remaining integrals for non equilibrium and/or TKE production terms.
For the non equilibrium terms this leads to the following:
![{s_T^i}^+ = \int_0^{y+}{\frac{z^{i+1}}{\left[1+\left(\frac{Pr}{Pr_t}\right)\left(\frac{\mu_t}{\mu}\right)\right]}dz} = \int_0^{y+}z^{i+1}\frac{d}{dz}\left({s_T^{-1}}^+\right)dz {s_T^i}^+ = \int_0^{y+}{\frac{z^{i+1}}{\left[1+\left(\frac{Pr}{Pr_t}\right)\left(\frac{\mu_t}{\mu}\right)\right]}dz} = \int_0^{y+}z^{i+1}\frac{d}{dz}\left({s_T^{-1}}^+\right)dz](/Forums/vbLatex/img/24c695ea6dfc3137676d388d28b45f8c-1.gif)
Hence, integration by parts finally leads to:
![\left(\frac{{s_T^i}^+}{{y^+}^{i+2}}\right) = \left(\frac{{s_T^{-1}}^+}{y^+}\right)-\frac{\left(i+1\right)}{{y^+}^{i+2}} \int_0^{y+}z^is_T^{-1}dz \left(\frac{{s_T^i}^+}{{y^+}^{i+2}}\right) = \left(\frac{{s_T^{-1}}^+}{y^+}\right)-\frac{\left(i+1\right)}{{y^+}^{i+2}} \int_0^{y+}z^is_T^{-1}dz](/Forums/vbLatex/img/22e5873b1280b6612357fd42796d270c-1.gif)
Formally, this is the generalized trick that I used here to extend the Reichardt wall law to constant only non equilibrium cases (i.e., i=0).
![T^+ = Pr \left(\frac{{s_T^{-1}}^+}{y^+}\right) y^+ T^+ = Pr \left(\frac{{s_T^{-1}}^+}{y^+}\right) y^+](/Forums/vbLatex/img/bf4eadbf488ef695500ba80212ef37d4-1.gif)
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}} \frac{d}{dy^+}\left({s_T^{-1}}^+\right) = \frac{1}{1+\frac{Pr}{Pr_t}\frac{\mu_t}{\mu}}](/Forums/vbLatex/img/7df47c4a3c580b62929934b1f42cff58-1.gif)
from which, it follows that:
![\frac{Pr}{Pr_t}\frac{\mu_t}{\mu} = \frac{1}{\frac{d}{dy^+}\left({s_T^{-1}}^+\right)}-1 \frac{Pr}{Pr_t}\frac{\mu_t}{\mu} = \frac{1}{\frac{d}{dy^+}\left({s_T^{-1}}^+\right)}-1](/Forums/vbLatex/img/0ef92176ba88c949ad8bdd353f7dd497-1.gif)
which is all that is needed to compute (numerically if not doable analytically) the remaining integrals for non equilibrium and/or TKE production terms.
For the non equilibrium terms this leads to the following:
![{s_T^i}^+ = \int_0^{y+}{\frac{z^{i+1}}{\left[1+\left(\frac{Pr}{Pr_t}\right)\left(\frac{\mu_t}{\mu}\right)\right]}dz} = \int_0^{y+}z^{i+1}\frac{d}{dz}\left({s_T^{-1}}^+\right)dz {s_T^i}^+ = \int_0^{y+}{\frac{z^{i+1}}{\left[1+\left(\frac{Pr}{Pr_t}\right)\left(\frac{\mu_t}{\mu}\right)\right]}dz} = \int_0^{y+}z^{i+1}\frac{d}{dz}\left({s_T^{-1}}^+\right)dz](/Forums/vbLatex/img/24c695ea6dfc3137676d388d28b45f8c-1.gif)
Hence, integration by parts finally leads to:
![\left(\frac{{s_T^i}^+}{{y^+}^{i+2}}\right) = \left(\frac{{s_T^{-1}}^+}{y^+}\right)-\frac{\left(i+1\right)}{{y^+}^{i+2}} \int_0^{y+}z^is_T^{-1}dz \left(\frac{{s_T^i}^+}{{y^+}^{i+2}}\right) = \left(\frac{{s_T^{-1}}^+}{y^+}\right)-\frac{\left(i+1\right)}{{y^+}^{i+2}} \int_0^{y+}z^is_T^{-1}dz](/Forums/vbLatex/img/22e5873b1280b6612357fd42796d270c-1.gif)
Formally, this is the generalized trick that I used here to extend the Reichardt wall law to constant only non equilibrium cases (i.e., i=0).
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