Passive scalar transport in Couette flow

Abstract

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.