Leray-alpha simulations of wall-bounded turbulent flows

The Leray-$\alpha$ model reduces the range of active scales of the Navier-Stokes equations by smoothing the advective transport. Here we assess the potential of the Leray-$\alpha$ model in its standard formulation to simulate wall-bounded flows. Three flow cases are considered: plane channel flow at $\Re_\tau=590$, Rayleigh-B\'{e}nard convection at $\Ra=10^7$ and $\Pr=1$, and a side-heated vertical channel at $\Ra=5 \times 10^6$ and $\Pr=0.7$. The simulations are compared to results from a well resolved and coarse DNS. It is found that for all three flow cases, the variance in the velocity field increases as the filter width parameter $a$ is increased, where $a$ is connected to the filter width as $\alpha_i = a \Delta x_i$, with $\Delta x_i$ the local grid size. Furthermore, the viscous and diffusive wall regions tend to thicken relative to the coarse DNS results as a function of $a$. In the cases where coarse DNS overpredicts wall gradients (as for Rayleigh-B\'{e}nard convection and the side-heated vertical channel), the thickening is beneficial. However, for the plane channel flow, coarse DNS underpredicts the wall-shear velocity, and increasing $a$ only degrades the results. It is shown that buoyancy effects need to be included with care, because of the close relation of turbulent heat flux and the production of turbulent kinetic energy.