Standard $k\!-\!\varepsilon$ Model
Consider flows that satisfy the Boussinesq approximation (i.e., the fluid is approximately incompressible, and any influence of non-uniform temperature on the mean velocity field acts only through buoyancy). In this case, the exact transport equation for turbulent kinetic energy (the $k\!-\!\varepsilon$ equation) is
$$
\frac{Dk}{Dt}
=\frac{\partial}{\partial x_j}\!\left[
\nu\,\frac{\partial k}{\partial x_j}
-\overline{\,u_j\!\left(\frac{u_i u_i}{2}+\frac{p}{\rho}\right)}\right]
-\overline{u_i u_j}\,\frac{\partial U_i}{\partial x_j}
-\beta\,g_i\,\overline{u_i\vartheta}
-\varepsilon
$$
where $D/Dt=\partial/\partial t+U_j\,\partial/\partial x_j$.
$\overline{u_i u_j}$ is given by \(\boxed{-\overline{u_i u_j}
=2\,C_\mu\,\frac{k^{2}}{\varepsilon}\,S_{ij}
-\frac{2}{3}\,k\,\delta_{ij}}\), while $k$ and $\varepsilon$ are unknowns to be solved for. Only the turbulent diffusion term and the buoyancy production term on the right-hand side need to be modeled approximately
Referring to the laws of molecular diffusion and molecular heat conduction, the turbulent diffusivity of $k$ and the turbulent heat flux are approximated by the following gradient models
$$
-\overline{\,u_j\!\left(\frac{u_i u_i}{2}+\frac{p}{\rho}\right)}
=\frac{\nu_t}{\sigma_k}\,\frac{\partial k}{\partial x_j}
\quad,\quad
-\overline{u_i\vartheta}
=\frac{\nu_t}{\sigma_t}\,\frac{\partial \Theta}{\partial x_i}
$$
where $\Theta$ is the mean relative temperature; $\sigma_k$ and $\sigma_t$ are empirical constants ($\sigma_t$ is named the turbulent Prandtl number)
Substituting \(\boxed{-\overline{u_i u_j}
=2\,C_\mu\,\frac{k^{2}}{\varepsilon}\,S_{ij}
-\frac{2}{3}\,k\,\delta_{ij}}\), \(\boxed{-\overline{\,u_j\!\left(\frac{u_i u_i}{2}+\frac{p}{\rho}\right)}
=\frac{\nu_t}{\sigma_k}\,\frac{\partial k}{\partial x_j}}\) and \(\boxed{-\overline{u_i\vartheta}
=\frac{\nu_t}{\sigma_t}\,\frac{\partial \Theta}{\partial x_i}}\) into \(\boxed{\frac{Dk}{Dt}
=\frac{\partial}{\partial x_j}\!\left[
\nu\,\frac{\partial k}{\partial x_j}
-\overline{\,u_j\!\left(\frac{u_i u_i}{2}+\frac{p}{\rho}\right)}\right]
-\overline{u_i u_j}\,\frac{\partial U_i}{\partial x_j}
-\beta\,g_i\,\overline{u_i\vartheta}
-\varepsilon}\), and neglecting molecular diffusion under high-Reynolds-number conditions, yields the $k\!-\!\varepsilon$ turbulence model
\[
\frac{Dk}{Dt}
=\frac{\partial}{\partial x_j}\!\left(\frac{\nu_t}{\sigma_k}\,\frac{\partial k}{\partial x_j}\right)
+2C_\mu\,\frac{k^2}{\varepsilon}\,S_{ij}S_{ij}
+\beta\,g_i\,\frac{\nu_t}{\sigma_t}\,\frac{\partial \Theta}{\partial x_i}
-\varepsilon
\]
For high-Reynolds-number turbulence, the empirical constants are determined experimentally as
$$
C_\mu=0.09,\qquad
\sigma_k=1.0,\qquad
\sigma_t=0.9
$$
Taking $C_\mu=0.09$ is a consequence of the local equilibrium characteristics of turbulent boundary layers.
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