  We perform direct numerical simulations of wall sheared Rayleigh-B\'enard
(RB) convection for Rayleigh numbers up to $Ra=10^8$, Prandtl number unity, and
wall shear Reynolds numbers up to $Re_w=10000$. Using the Monin-Obukhov length
$L_{MO}$ we identify three different flow states, a buoyancy dominated regime
($L_{MO} \lesssim \lambda_{\theta}$; with $\lambda_{\theta}$ the thermal
boundary layer thickness), a transitional regime ($0.5H \gtrsim L_{MO} \gtrsim
\lambda_{\theta}$; with $H$ the height of the domain), and a shear dominated
regime ($L_{MO} \gtrsim 0.5H$). In the buoyancy dominated regime the flow
dynamics are similar to that of turbulent thermal convection. The transitional
regime is characterized by rolls that are increasingly elongated with
increasing shear. The flow in the shear dominated regime consists of very
large-scale meandering rolls, similar to the ones found in conventional Couette
flow. As a consequence of these different flow regimes, for fixed $Ra$ and with
increasing shear, the heat transfer first decreases, due to the breakup of the
thermal rolls, and then increases at the beginning of the shear dominated
regime. For $L_{MO} \gtrsim 0.5H$ the Nusselt number $Nu$ effectively scales as
$Nu \sim Ra^{\alpha}$, with $\alpha \ll 1/3$ while we find $\alpha \simeq 0.31$
in the buoyancy dominated regime. In the transitional regime the effective
scaling exponent is $\alpha > 1/3$, but the temperature and velocity profiles
in this regime are not logarithmic yet, thus indicating transient dynamics and
not the ultimate regime of thermal convection.
