A scaling theory for the passive scalar transport in Couette flow, i.e. the flow between two parallel plates moving with different velocities, is proposed. This flow is determined by the bulk Reynolds number $Re_b$ and the Prandtl number $Pr$ . In the turbulent regime, for moderate shear Reynolds number $Re_{\tau }$ and moderate $Pr$ , we derive that the passive scalar transport characterised by the Nusselt number $Nu$ scales as $Nu \sim Pr^{1/2}Re_{\tau }^{2}Re_b^{-1}$ . We then use the well-established scaling for the friction coefficient $C_f \sim Re_b^{-1/4}$ (corresponding to a shear Reynolds number $Re_{\tau } \sim Re_b^{7/8}$ ) which holds reasonably well within the range $3\times 10^{3} \leqslant Re_b \leqslant 10^{5}$ , to obtain $Nu \sim Pr^{1/2}Re_b^{3/4}$ for the Nusselt number scaling. The theoretical results are tested against direct numerical simulations of Couette flows for the parameter ranges $81 \leqslant Re_b \leqslant 22361$ and $0.1 \leqslant Pr \leqslant 10$ , finding good agreement. Analyses of the numerically obtained turbulent flow fields confirm logarithmic mean wall-parallel profiles of the velocity and the passive scalar in the inertial sublayer.

We investigate the counter-intuitive initial decrease and subsequent increase in the Nusselt number $Nu$ with increasing wall Reynolds number $Re_w$ in the sheared Rayleigh–Bénard (RB) system by studying the energy spectra of convective flux and turbulent kinetic energy for Rayleigh number $Ra = 10^{7}$ , Prandtl number $Pr=1.0$ and inverse Richardson numbers $0 \leq 1/Ri \leq 10$ . These energy spectra show two distinct high-energy regions corresponding to the large-scale superstructures in the bulk and small-scale structures in the boundary layer (BL) regions. A greater separation between these scales at the thermal BL height correlates to a higher $Nu$ and indicates that the BLs are more turbulent. The minimum $Nu$ , which occurs at $1/Ri=1.0$ , is accompanied by the smallest separation between the large- and small-scale structures at the thermal BL height. At $1/Ri=1.0$ , we also observe the lowest value of turbulent kinetic energy normalized with the square of friction velocity within the thermal BL. Additionally, we find that the domain size has a limited effect on the heat and momentum transfer in the sheared RB system as long as the domain can accommodate the small-scale convective structures at the thermal BL height, signifying that capturing the large-scale superstructures is not essential to obtain converged values of $Nu$ and shear Reynolds number $Re_{\tau }$ . When the domain is smaller than these small-scale convective structures, the overall heat and momentum transfer reduces drastically.