Extension of Boussinesq turbulence constitutive relation for partially-averaged velocity fields

Boussinesq hypothesis is still widely used to model Reynolds stresses in turbulent flows. Recently, there have been many attempts to extend the Boussinesq constitutive relation for modeling the sub-grid scale stresses (SGS) for filtered or partially-averaged velocity fields, for example, limited numerical scales method (LNS), flow simulation methodology (FSM), unsteady Reynolds averaged Navier-Stokes (URANS), very large eddy simulation (VLES) and partially-averaged Navier-Stokes (PANS). The major proposals can be classified into two categories. In VLES, FSM, and LNS the effective SGS viscosity is given by: $\nu_T =fC_{\mu} \frac{k^2}{\varepsilon}$ where $f$ is determined by the filter size. In PANS, the SGS viscosity is given by $\nu_{Tu} =C_{\mu} \frac{k_u^2}{\varepsilon_u}$ where $k_{u}$ and \textit{$\varepsilon$}$_{u}$ are the unresolved turbulent kinetic energy and dissipation rate respectively. The validity of these two proposals is investigated in two turbulent flow fields: (i) flow past a circular cylinder; (ii) flow past a backward facing step. We investigate the degree of eddy viscosity reduction obtained \textit{a posteriori} in the computations and compare them to the prescribed value. The effect of the two viscosity reduction proposals on the unsteadiness of the computed velocity fields is also examined.