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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.

The K-epsilon turbulence model evolves turbulent kinetic energy $k$ and its turbulent kinetic energy dissipation rate $ϵ$, rather than resolving turbulent fluctuations. It closes the Reynolds stress with an isotropic eddy viscosity, ${\nu }_T=C_{\mu }{k}^2/ϵ$, and models transport of $k$ and $ϵ$ by gradient diffusion with coefficients ${\nu }_T/{\sigma }_k$ and ${\nu }_T/{\sigma }_{ϵ}$. Production is tied to mean strain; dissipation-equation production and destruction are modeled terms with empirical constants. The characteristic velocity and length are $\sqrt{k}$ and $k^{3/2}/ϵ$. These assumptions are primarily for high Reynolds number, and strongly anisotropic flow, sharp interfaces and molecular-viscosity-dominated regions need additional treatment. The dissipation equation is a closure, not a second exact energy-conservation law.

The first similarity hypothesis of Kolmogorov 1941 theory states that at sufficiently large Reynolds number, the small-scale statistics in the equilibrium range are locally universal and determined by the mean turbulent kinetic energy dissipation rate $ϵ$ and kinematic viscosity $\nu$, together with the separation or wavenumber being measured. Large-scale forcing details are excluded only at scales well below the integral scale of turbulence.

These parameters define the Kolmogorov microscalesDimensional analysis gives universal functions $F$ and $G$ withThe dimensional factors follow because $S_2$ has units of velocity squared, whereas $E$ has units of velocity squared times length. The universal equilibrium forms include viscous effects; they do not impose a power law through the dissipation range.

The second similarity hypothesis asserts independence of kinematic viscosity at scales large compared with $\eta$ but still small compared with the integral scale. In this inertial range, $S_2$ can depend only on $ϵ$ and $r$. Balancing time and length dimensions forces powers $2/3$ and $2/3$. The same calculation for $E$, using $k$ instead of $r$, forces powers $2/3$ and $-5/3$. HenceThe constants $C_2$ and $C_K$ are universal within the hypotheses and the stated normalization. They are related by the exact structure-function spectral filter, rather than by assigning all energy to one sharp scale.

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.

Equating the two expressions for eddy viscosity gives directlyFrom the preceding part, $u_{\ast }^2=\sqrt{C_{\mu }}k$. Replace one factor of $k$ using that identity, and then replace $u_{\ast }$ itself by $C_{\mu }^{1/4}{k}^{1/2}$. This proves all three equivalent dissipation expressions:Each is the same turbulent kinetic energy dissipation rate, with dimensions of velocity cubed divided by length. Its equivalent friction-velocity form is $u_{\ast }^3/(\kappa y)$.

The fluctuation speed $q$ and the integral scale of turbulence $L_u$ give a turnover time $L_u/q$. Dividing the velocity-variance scale $q^2$ by that time gives a turbulent kinetic energy dissipation rate proportional to $q^3/L_u$. Likewise mass density variance $\overline{{\rho }^{\prime 2}}$ removed over scalar mixing time $L_{\rho }/q$ gives the density-variance dissipation rate proportional to $\overline{{\rho }^{\prime 2}}q/L_{\rho }$. Absorbing dimensionless proportionality coefficients into the specified length scales givesThese are integral-scale closure estimates, not exact consequences of dimensional analysis alone. In particular $q^2=2k$ for the turbulent-energy convention in Question 3. The normalization of $\chi$ is the one fixed by the printed scalar balance; if the budget is written for the full variance rather than one half of it, both production and dissipation terms acquire a factor two.