K-epsilon turbulence model (${\nu }_T=C_{\mu }{k}^2/ϵ$)

A two-equation closure transporting turbulent kinetic energy $k$ and its turbulent kinetic energy dissipation rate $ϵ$. Its eddy viscosity is $C_{\mu }{k}^2/ϵ$. In a high-Reynolds-number wall-normal reduction, turbulent diffusion has coefficients ${\nu }_T/{\sigma }_k$ and ${\nu }_T/{\sigma }_{ϵ}$; the energy equation has shear production $P$ and destruction $ϵ$, while the dissipation equation has modeled production $C_{ϵ1}Pϵ/k$ and destruction $C_{ϵ2}{ϵ}^2/k$. The dissipation production/destruction terms are closure terms, not a second conservation of energy.

A stationary turbulence edge with no production has an advection–turbulent-diffusion leading balance. For $k\propto x^p$, $ϵ\propto x^q$ and ${\nu }_T=C_{\mu }{k}^2/ϵ$, this balance gives $2p-q=1$ and $q/p={\sigma }_{ϵ}/{\sigma }_k=\sigma$. The integral length $k^{3/2}/ϵ$ and turnover frequency $ϵ/k$ both tend to zero when $1<\sigma <3/2$. The front viscosity is linear, with ${\nu }_T^{\prime }(0^{+})=(2{\sigma }_k-{\sigma }_{ϵ})U_0$. This is formal high-Reynolds-number consistency, not a uniform justification for neglecting molecular viscosity at the exact edge.

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.