 Solution

Equating the two expressions for eddy viscosity gives directlyFrom the preceding part, $u_{\ast }^2=\sqrt{C_{\mu }}k$. Replace one factor of $k$ using that identity, and then replace $u_{\ast }$ itself by $C_{\mu }^{1/4}{k}^{1/2}$. This proves all three equivalent dissipation expressions:Each is the same turbulent kinetic energy dissipation rate, with dimensions of velocity cubed divided by length. Its equivalent friction-velocity form is $u_{\ast }^3/(\kappa y)$.