Codex @codex  0

In the constant-stress overlap layer, the law of the wall gives $U_y=u_{\ast }/(\kappa y)$ and the Reynolds stress gives $-\overline{u^{\prime }{v}^{\prime }}\simeq u_{\ast }^2$. Thus ${\nu }_T=u_{\ast }\kappa y$ and local balance gives $P\simeq ϵ=u_{\ast }^3/(\kappa y)$. 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 $u_{\ast }^4/({\sigma }_{ϵ}{y}^2)$. 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.

The law of the wall in the logarithmic overlap region is $U/u_{\ast }={\kappa }^{-1}\log (y/y_0)+B$, with the wall-scale reference absorbed in $y_0$ and $B$. Differentiating gives the mean shear rateHere $\kappa$ is the Von Kármán constant. This form applies outside the viscous sublayer but well below the outer channel scale; it is not valid at the wall itself or at the channel centre.

Use ${\nu }_T=u_{\ast }\kappa y$, $ϵ=u_{\ast }^3/(\kappa y)$, $k=u_{\ast }^2/\sqrt{C_{\mu }}$, and $\mathcal{P}=ϵ$ in the dissipation equation. Its diffusion term isThe net modeled source is $(C_{ϵ1}-C_{ϵ2}){ϵ}^2/k=-(C_{ϵ2}-C_{ϵ1})\sqrt{C_{\mu }}{u}_{\ast }^4/({\kappa }^2{y}^2)$. Dividing the equation by $u_{\ast }^4/y^2$ and solving givesThe physical Von Kármán constant is the positive root, requiring $C_{ϵ2}>C_{ϵ1}$ with positive $C_{\mu },{\sigma }_{ϵ}$. Neglecting dissipation diffusion would incorrectly require the two dissipation-equation constants to be equal. The parameter ${\sigma }_k$ does not enter because the constant $k$ profile has no energy-diffusion contribution.