\[\frac{\partial\rho\overline{v}}{\partial t}+\nabla\cdot\rho\overline{vv}=-\nabla\overline{p}-\nabla\cdot\tau^{T}+\rho\vec{g}\]
We can use two models for the turbulent viscosity: Eddy Viscosity and
Prandtl’s Mixing Length Model.
Eddy Viscosity
Using Newton’s viscosity model
\[\tau_{xy}^{t}=-\mu^{t}\frac{\partial\overline{v_{x}}}{\partial y}\]
but unfortunately, \(\mu=\mu\left(\vec{x}\right)\) and non-linear,
so instead we use PML
Prandtl’s Mixing Length Model
Momentum transfer at discrete intervals only after the eddy mass
(\(\delta m\)) has moved an entire length (\(l\)).
\[\begin{split}\underbrace{\delta m\delta v_{x}}_{\substack{\text{momentum}\\
\text{gain in the x}\\
\text{direction}
}
}\;\;\Longrightarrow\;\;\underbrace{\frac{\delta m}{\delta t}\delta v_{x}}_{\substack{\text{force}}
}\;\;\Longrightarrow\;\;\tau^{t}=\frac{F}{A}=\underbrace{-\frac{1}{A}\frac{\delta m}{\delta t}\delta v_{x}}_{\substack{\text{shear stress}\\
\text{acting against}\\
\delta v_{x}\text{ over area }A
}
}\end{split}\]
By assuming that \(l\) is very small:
\[ \begin{align}\begin{aligned}\delta v_{x}\approx\frac{\delta\overline{v_{x}}}{\delta y}l\\and applying continuity (or matching units)\end{aligned}\end{align} \]
\[ \begin{align}\begin{aligned}\frac{1}{A}\frac{\delta m}{\delta t}=\rho\left|v_{y}^{'}\right|\\And therefore we have\end{aligned}\end{align} \]
\[ \begin{align}\begin{aligned}\frac{\tau^{t}}{\rho}=-l\left|v_{y}^{'}\right|\frac{d\overline{v_{x}}}{dy}=-\varepsilon_{M}\frac{d\overline{v_{x}}}{dy}\\Applying Prandtl's Assumptions:\end{aligned}\end{align} \]
perpendicular velocity is proportional to the flow velocity
\[\left|v_{y}^{'}\right|=k_{1}\left|v_{x}^{'}\right|\]
fluctuation in x direction velocity is propotional to the rate of
change of x direction velocity times the length moved
\[v_{x}^{'}=k_{2}\delta v_{x}=k_{2}l\frac{d\overline{v_{x}}}{dy}\]
and, minorly, \(l=ky\)
\[\frac{\tau^{t}}{\rho}=-\left(ky\right)\left|k_{1}\left|k_{2}\left(ky\right)\frac{d\overline{v_{x}}}{dy}\right|\right|\frac{d\overline{v_{x}}}{dy}=-k^{2}y^{2}\left(\frac{d\overline{v_{x}}}{dy}\right)^{2}\]
therefore
\[\frac{d\overline{v_{x}}}{dy}=-\frac{1}{ky}\sqrt{\frac{\tau^{t}}{\rho}}\]
and we can assume that all of the turbulent stress comes from the wall
(\(\tau^{t}=\tau^{w}\)).
By non dimensionalizing with
\[v^{*}=\frac{v_{x}}{\sqrt{\nicefrac{\tau^{t}}{\rho}}}\;\;\&\;\;y^{*}=\frac{y\sqrt{\nicefrac{\tau^{t}}{\rho}}}{v}\]
we get
\[\frac{dv^{*}}{dy^{*}}=\frac{1}{ky^{*}}\;\;\therefore\;\;v^{*}=\frac{1}{R}\ln\left(y^{*}\right)+C\]
and we can evaluate how much this is using Fanning’s friction factor
\[\tau^{w}=\frac{1}{2}f_{i}\rho u^{2}\;\;\therefore\;\;\sqrt{\frac{\tau^{w}}{\rho}}=\sqrt{\frac{1}{2}f_{i}}u\]
and
\[y^{*}=Re\frac{\sqrt{\nicefrac{\tau^{w}}{\rho}}}{\nu}\frac{y}{D}\approx Re\frac{1}{40}\frac{y}{R}\]
Bibliography
