Standard $k\!-\!\varepsilon$ Model
For flows that satisfy the Boussinesq approximation, the transport equations for the turbulent kinetic energy \(k\) and its dissipation rate \(\varepsilon\) \(\Longrightarrow \frac{\partial \varepsilon}{\partial t} + U_k \frac{\partial \varepsilon}{\partial x_k} = -\frac{\partial}{\partial x_j} \left( \nu u_j\overline{\frac{\partial u_i}{\partial x_k},\frac{\partial u_i}{\partial x_k}} -\nu\frac{\partial \varepsilon}{\partial x_k} \right) - 2\nu\frac{\partial^2 U_i}{\partial x_j\partial x_k} \overline{u_j \frac{\partial u_i}{\partial x_k}} - 2\nu \overline{\frac{\partial u_i}{\partial x_j} \frac{\partial u_i}{\partial x_k} \frac{\partial u_j}{\partial x_k}} - 2\nu^{2} \overline{\frac{\partial^{2} u_i}{\partial x_j\partial x_k} \frac{\partial^{2} u_i}{\partial x_j\partial x_k}}\)
Term Order of Magnitude Normalized Order of Magnitude
Convection \[ \frac{U \varepsilon}{L} \] \[ 1 \]
Turbulent Diffusion \[ \frac{k^{1/2}\varepsilon}{L} \, Re_t^{-1/2} \] \[ \frac{k^{1/2}}{U} \, Re_t^{-1/2} \]
Molecular Diffusion \[ \frac{\nu \varepsilon}{L^2} \] \[ \frac{l}{L} \frac{k^{1/2}}{U} Re_t^{-1} \]
Small-Eddy Stretching Production \[ \varepsilon \left( \frac{\varepsilon}{\nu} \right)^{1/2} \] \[ \frac{l}{L} \frac{k^{1/2}}{U} Re_t^{1/2} \]
Mean Vorticity Gradient Production \[ \nu \frac{U}{L^{2}}\frac{k}{l} \] \[ \frac{l}{L} Re_t^{-1} \]
Viscous Dissipation \[ \varepsilon^{3/2}\,\nu^{-1/2} \] \[ \frac{L}{l} \frac{k^{1/2}}{U} Re_t^{1/2} \]

If \(Re_t \gg 1\), the small-eddy stretching production term and the viscous dissipation term are much larger than the other terms in the equation. For the equation to be satisfied, these two dominant terms must have opposite signs: the dissipation term acts as a negative source term, while the production term acts as a positive source term. In other words, the stretching of small eddies induced by strain-rate fluctuations is the primary factor responsible for the increase of \(\varepsilon\) along the mean flow streamlines, and this effect is almost entirely balanced by viscous dissipation. Because the rate of change of \(\varepsilon\) depends on the difference between these two dominant terms, the order of magnitude of this difference is at least \(Re_t^{1/2}\) times smaller than either term individually. Moreover, both terms represent correlation quantities of small-scale velocity fluctuations, and it is very difficult to evaluate them quantitatively either theoretically or experimentally. Therefore, direct modeling is challenging. A reasonable approach is to treat the difference of these two dominant terms as a single combined quantity for modeling purposes.

1Song Fu & Liang Wang (2023). Theory of Turbulence Modelling. ISBN 978-7-03-074639-9.

◀ ▶