  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.
