Influence of hydrodynamic slip on convective transport in flow past a circular cylinder

Abstract

The presence of a finite tangential velocity on a hydrodynamically slipping surface is known to reduce vorticity production in bluff body flows substantially while at the same time enhancing its convection downstream and into the wake. Here, we investigate the effect of hydrodynamic slippage on the convective heat transfer (scalar transport) from a heated isothermal circular cylinder placed in a uniform cross-flow of an incompressible fluid through analytical and simulation techniques. At low Reynolds (${\textit{Re}}\ll 1$) and high Péclet (${\textit{Pe}}\gg 1$) numbers, our theoretical analysis based on Oseen and thermal boundary layer equations allows for an explicit determination of the dependence of the thermal transport on the non-dimensional slip length $l_s$. In this case, the surface-averaged Nusselt number, Nu transitions gradually between the asymptotic limits of $Nu \sim {\textit{Pe}}^{1/3}$ and $Nu \sim {\textit{Pe}}^{1/2}$ for no-slip ($l_s \rightarrow 0$) and shear-free ($l_s \rightarrow \infty$) boundaries, respectively. Boundary layer analysis also shows that the scaling $Nu \sim {\textit{Pe}}^{1/2}$ holds for a shear-free cylinder surface in the asymptotic limit of ${\textit{Re}}\gg 1$ so that the corresponding heat transfer rate becomes independent of the fluid viscosity. At finite ${\textit{Re}}$, results from our two-dimensional simulations confirm the scaling $Nu \sim {\textit{Pe}}^{1/2}$ for a shear-free boundary over the range $0.1 \le {\textit{Re}}\le 10^3$ and $0.1\le {\textit{Pr}}\le 10$. A gradual transition from the lower asymptotic limit corresponding to a no-slip surface, to the upper limit for a shear-free boundary, with $l_s$, is observed in both the maximum slip velocity and the Nu. The local time-averaged Nusselt number $Nu_{\theta }$ for a shear-free surface exceeds the one for a no-slip surface all along the cylinder boundary except over the downstream portion where unsteady separation and flow reversal lead to an appreciable rise in the local heat transfer rates, especially at high ${\textit{Re}}$ and Pr. At a Reynolds number of $10^3$, the formation of secondary recirculating eddy pairs results in appearance of additional local maxima in $Nu_{\theta }$ at locations that are in close proximity to the mean secondary stagnation points. As a consequence, Nu exhibits a non-monotonic variation with $l_s$ increasing initially from its lowermost value for a no-slip surface and then decreasing before rising gradually toward the upper asymptotic limit for a shear-free cylinder. A non-monotonic dependence of the spanwise-averaged Nu on $l_s$ is observed in three dimensions as well with the three-dimensional wake instabilities that appear at sufficiently low $l_s$, strongly influencing the convective thermal transport from the cylinder. The analogy between heat transfer and single-component mass transfer implies that our results can directly be applied to determine the dependency of convective mass transfer of a single solute on hydrodynamic slip length in similar configurations through straightforward replacement of Nu and ${\textit{Pr}}$ with Sherwood and Schmidt numbers, respectively.