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