From equation (5), the Reynolds stress is:

\[ -r = \frac{n\left( s - 1\right)}{s\left( n - 1\right)}\left(\left( 1-\eta\right)^{2n-1} -\left( 1-\eta\right)\right) \tag{7} \]

It is easy to show that \( r \) attains a maximum at \( \eta^* \) given by

\[ \eta^* = 1 - \left(\frac{1}{2n-1}\right)^{\frac{1}{2\left( n - 1\right)}} \tag{8} \]

As in the previous page, choose a value of \( R_m \) and get the corresponding values of \( R_\tau , s , n\). The Reynolds stress \( r \left(\eta\right) \) will be plotted on the right.

Also note the values of \( \eta^* \) and \( y^+ \). For example, if \( R_m = 12300 \), then \( \eta^* = 0.1082 \) and \( y^+ = 58.28 \), i.e. in the log-wall region. Notice that \( \left( 1 - \eta \right) \) is an asymptote (in green) and that for larger values of \( R_m \) the boundary layer near \( \eta = 0 \) becomes thinner.