Finally, the production of turbulent kinetic energy, expressed as \( v \left(\frac{du}{dy}\right)^2 \) is \[ \left( 1 - \eta \right)^2\left( 1 - \left( 1 - \eta\right)^{2n-2}\right)\left(\alpha + n\beta\left( 1 - \eta\right)^{2n - 2}\right) \tag{11} \] where the multiplying factor has been scaled out and \( \beta = 1 - \alpha \).

As before, choose a value of \( R_m \) and thus get the corresponding values of \( R_\tau, s, n \) and plot turbulent kinetic energy on the right. This plot resembles the classical distribution of Klebanoff [6] obtained for zero pressure gradient boundary layers. AS the maximum turbulent kinetic energy is attained at the point at which the maximum energy production is attained [7], this point may be obtained from \[ n\beta\left( 2n-1 \right)\chi^2 - n\left( n\beta - \alpha\right)\chi - \alpha = 0 \] where \( \chi = \left( 1 - \eta\right)^{2n-2} \).

For the value of \( R_m \) we get the corresponding value of \( \eta_{maxprod} \) in inner wall variable \( y^+_{maxprod} \). For \( R_m = 12300 \) this yields \( y^+_{maxprod} = 11.47 \) confirming that the maximum turbulent kinetic energy occurs at the edge of the viscous sublayer (cf. the previous page).