Standard $k\!-\!\varepsilon$ Model
As stated by \(\boxed{\nu_t
= C_1\,\sqrt{k}\,l
= C_2\,\frac{k^{2}}{\varepsilon}
= C_3\,k\left(\frac{\overline{\omega_i\omega_i}}{2}\right)^{1/2}
= \cdots}\), in the $k\!-\!\varepsilon$ model the eddy viscosity is written as
$$
\nu_t=C_\mu\,\frac{k^{2}}{\varepsilon}
$$
that is, the velocity scale for $\nu_t$ is taken as $k^{1/2}$ and the length scale as $k^{3/2}/\varepsilon$. Here $C_\mu$ is a constant
\[
\overline{u_i u_j}
=\frac{2}{3}\,k\,\delta_{ij}
-\nu_t\!\left(\frac{\partial U_i}{\partial x_j}
+\frac{\partial U_j}{\partial x_i}\right)
\Rightarrow -\overline{u_i u_j}
=2\,C_\mu\,\frac{k^{2}}{\varepsilon}\,S_{ij}
-\frac{2}{3}\,k\,\delta_{ij}
\] where $S_{ij}=(\partial U_i/\partial x_j+\partial U_j/\partial x_i)/2$ is the mean strain rate tensor
Define the turbulent Reynolds number as
$$
\mathrm{Re}_t=\frac{k^{2}}{\varepsilon\,\nu}
$$
In turbulent flows, if $\mathrm{Re}_t\gg1$ (or $\nu_t\gg\nu$) in a region of space, the flow in that region is called high-Reynolds-number turbulence; otherwise it is called low-Reynolds-number turbulence
In general, fully developed free turbulence is of high Reynolds number, whereas flows in the laminar-turbulent transition region or turbulence very near solid walls are typically of low Reynolds number.
A standard turbulence model refers to one intended for high-Reynolds-number flows without strong complicating effects (e.g.,
pronounced three-dimensionality, near-wall effects, strong curvature effects, or nonequilibrium flow). For flows where such effects are important, turbulence models are usually obtained by adding appropriate corrections to the corresponding baseline model.
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