Boussinesq's Turbulent Viscosity Hypothesis

Any two of the variables $\ell,\ \tau,\ k,\ \varepsilon,\ \frac{\overline{\omega_i\omega_i}}{2}$, or any two independent combinations formed from them, may be used as scale variables to specify $\nu_t$. For example $$ \nu_t = C_1\,\sqrt{k}\,l = C_2\,\frac{k^{2}}{\varepsilon} = C_3\,k\left(\frac{\overline{\omega_i\omega_i}}{2}\right)^{1/2} = \cdots $$ where $C_1,\ C_2,\ C_3,\ldots$ are constants

The turbulence models commonly used at present, such as the $k\!-\!\ell$ model, the $k\!-\!\varepsilon$ model, and the $k\!-\!\omega$ model, are precisely formulated using two such scale variables indicated in their names. In these models the first scale variable is always chosen to be $k$, because

$\sqrt{k}$ is the best velocity scale for the energy-containing motions in the inertial range of turbulence
In the transport equation for $k$, each term has a clear physical meaning and is relatively easy to model
The magnitude of $k$ is easy to measure experimentally

Note that in some situations the single-scale assumption is not reasonable. The clearest example is turbulent flow near a solid wall, in such flows, the scales of the energy-containing eddies in the inertial region and the scales of the dissipative eddies both play important roles in determining turbulent behavior. In general, the length and time scales of turbulence are continuously distributed with a very broad spectrum, and interactions among motions at different scales are related to the spectral distribution of turbulent kinetic energy. For turbulence close to spectral equilibrium, single-scale turbulence models have many successful computations; for non-equilibrium turbulence, single-scale models still need improvement. There are also multi-scale turbulence models.

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

◀ ▶