= Log-layer solution of the k-epsilon model
{title2=$k=u_*^2/\sqrt{C_\mu},\quad\kappa^2=\sigma_\epsilon\sqrt{C_\mu}(C_{\epsilon2}-C_{\epsilon1})$}
In the constant-stress overlap layer, the <law of the wall> gives $U_y=u_*/(\kappa y)$ and the <Reynolds stress> gives $-\overline{u'v'}\simeq u_*^2$. Thus $\nu_T=u_*\kappa y$ and local balance gives $P\simeq\epsilon=u_*^3/(\kappa y)$. Combining with the <K-epsilon model> viscosity formula gives the displayed constant $k$. The energy diffusion term then vanishes. The dissipation diffusion term does not: it is $u_*^4/(\sigma_\epsilon y^2)$. Balancing it against the dissipation-equation source terms yields the displayed <Von Kármán constant> relation. Neglecting dissipation diffusion would incorrectly require equal production and destruction constants.
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