A starting point is the Boussinesq approximation [9]
\[ -\rho\overline{u^\prime v^\prime} = \mu_\tau \frac{d\bar{u}}{dy} \tag{2} \]
which introduces the turbulent viscosity \( \mu_\tau \) int erms of known quantities such as \( \bar{u} \) or turbulent kinetic energy
\( k = \left( \overline{u^n} + \overline{v^n} + \overline{w^n}\right) / 2 \) depending on the choice of the turbulence model
(e.g. the Prandtl mixing length hypothesis or the \( k - \varepsilon \) turbulence model still used in major commercial computational
fluid dynamics).
The Boussinesq approximation assumes that the flow is isotropic when in wall bounded flows the flow is in fact anisotropic.
Integrating equation (1) with respect to why yields
\[ p + \rho\overline{v^n} = P_w\left( x\right) \tag{3} \]
where \( P_w\left( x\right) \) is the pressure on either wall of the channel. Let us define the total shear stress as
\( \tau := \mu\frac{du}{dy} - \rho\overline{u^\prime v^\prime} \). The integrating equation (2) yields
\[ \tau = -\left( H - y\right)\frac{dP_w}{dx} \tag{4} \]
where \( H \) is the half-width of the channel. The wall shear stress at \( y = 0 \) given by \( \tau_w \) is \( -H\frac{dP_w}{dx} \).
It is useful to define a friction velocity \( u_\tau^2 = \frac{\tau_w}{\rho} \). Then if
\[ r = \frac{\overline{u^\prime v^\prime}}{u_\tau^2} , V = \frac{u}{u_\tau} , \eta = \frac{y}{H} , R_\tau = \frac{u_\tau H}{v} \]
equation (4) reduces to
\[ -r = \frac{1}{R_\tau} \frac{dV}{d\eta} - \left( 1 - \eta \right) \tag{5} \]
