 Solution

Here $k$ is turbulent kinetic energy per unit mass, $k=\frac{1}{2}\overline{u^{\prime 2}+v^{\prime 2}+w^{\prime 2}}$, rather than the permeability of a porous medium used in the porous-flow questions. The K-epsilon turbulence model represents the high-Reynolds-number transport of this energy and its turbulent kinetic energy dissipation rate $ϵ$.

In the $k$ equation, $d[({\nu }_T/{\sigma }_k)k_y]/dy$ is the divergence of modeled turbulent diffusion, $\mathcal{P}$ is mean-shear production, and $-ϵ$ is molecular conversion of fluctuation energy into heat. Statistical stationarity and fully developed flow have removed storage and streamwise advection from this reduction.

In the $ϵ$ equation, $d[({\nu }_T/{\sigma }_{ϵ}){ϵ}_y]/dy$ transports dissipation intensity; $C_{ϵ1}\mathcal{P}ϵ/k$ models its generation associated with energy production; and $-C_{ϵ2}{ϵ}^2/k$ models its destruction. The time scale $k/ϵ$ supplies the source-rate dimension. The $ϵ$ equation is a closure, not another energy-conservation equation. Finally, ${\nu }_T=C_{\mu }{k}^2/ϵ$ has dynamic viscosity dimensions and parametrizes Reynolds stress transport; ${\sigma }_k,{\sigma }_{ϵ}$ set the relative diffusion strengths, and the remaining constants calibrate the closure.