= Deviatoric eddy-viscosity closure
{title2=$-\rho\overline{u_i'u_j'}=2\mu_TS_{ij}-2\rho e_t\delta_{ij}/d$}
For an incompressible mean flow, an <eddy viscosity> can model the non-isotropic part of the <Reynolds stress> through the <strain-rate tensor> $S_{ij}=(U_{i,j}+U_{j,i})/2$. The isotropic part is fixed separately by the <turbulent kinetic energy> $e_t=\overline{u_k'u_k'}/2$ in $d$ fluctuating dimensions. Taking the <trace> proves that the isotropic term is necessary: $S_{ii}=0$, whereas $-\rho\overline{u_i'u_i'}=-2\rho e_t$. In the mean momentum equation the isotropic term can be absorbed into $P_{\mathrm{eff}}=P+2\rho e_t/d$.
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