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# Turbulent Kinetic Energy

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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).

$$R_\tau =$$ ,
$$n =$$ ,
$$s =$$
$$y^+_{maxprod} =$$