The $k-\varepsilon$ Model

Abstract

Turbulent flows are considered as random motions, the velocity v and the pressure p being random variables. After having defined a clear probabilistic framework, we look for equations satisfied by the expectations $\overline{\mathbf{v}}$ and $\overline{p}$ of v and p, which yield the Reynolds stress $\boldsymbol{\sigma }^{(R)} = \overline{\mathbf{v}^{\prime} \otimes \mathbf{v}^{\prime}}$. In keeping with the Boussinesq assumption, we write $\boldsymbol{\sigma }^{(R)}$ in terms of the mean deformation tensor $D\overline{\mathbf{v}}$ and the eddy viscosity ν _t , which must be modeled. The modeling process leads to the assumption that ν _t depends on the turbulent kinetic energy $k = (1/2)\overline{\vert \mathbf{v}^{\prime}\vert ^{2}}$ and the turbulent diffusion $\mathcal{E} = 2\nu \overline{\vert D\mathbf{v}^{\prime}\vert ^{2}}$, addressing the issue of finding equations for k and $\mathcal{E}$ to compute ν _t , and therefore $(\overline{\mathbf{v}},\overline{p})$. To realize this, hypotheses about turbulence are necessary, such as local homogeneity, expressed by the local invariance of the correlation tensors under translations. The concept of mild homogeneity is introduced, which is the minimal hypothesis about correlations that allows the derivation of the $k -\mathcal{E}$ model carried out in this chapter, using additional standard closure assumptions.