= Solution
The constant <Reynolds stress> and the logarithmic shear imply the <eddy viscosity> $\nu_T=u_*\kappa y$. On the other hand the <K-epsilon model> gives $\nu_T=C_\mu k^2/\epsilon$. Insert $\epsilon=u_*^3/(\kappa y)$:
$$
u_*\kappa y=\frac{C_\mu k^2\kappa y}{u_*^3}.
$$
Canceling the positive factors yields $C_\mu k^2=u_*^4$. Since energy is nonnegative, the <log-layer solution of the k-epsilon model> has
$$
\boxed{k=\frac{u_*^2}{\sqrt{C_\mu}},}
$$
independent of $y$ to this leading overlap-layer accuracy. In particular $k_y=0$, and the $k$ diffusion term vanishes, confirming the local production–dissipation balance in the energy equation.
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