Flow past a slippery cylinder: part 1—circular cylinder

Abstract

Part 1 of this two-part paper solves the two-dimensional problem of the unsteady flow of a viscous incompressible fluid past a circular cylinder subject to impermeability and Navier-slip conditions on the surface. The Navier-slip condition is characterized by the slip length, $\beta$. The flow is calculated using two methods. The first takes the form of a double series solution where an expansion is carried out in powers of the time, t, and in powers of the parameter $\lambda =\sqrt{8t/R}$ where R is the Reynolds number. This approximate analytical solution is valid for small times following the start of the motion and for large Reynolds numbers. The second method involves a spectral-finite difference procedure for numerically integrating the full Navier–Stokes equations expressed in terms of a stream function and vorticity. Our results demonstrate that for small times and moderately large R the two methods of solution are in excellent agreement. Results are presented for Reynolds numbers 500 and 1000, and comparisons with the no-slip condition are made. The key finding is that a reduction in the drag coefficient results from the Navier-slip condition when compared to the no-slip condition. This reduction increases as $\beta$ increases. In addition, the slip condition also leads to suppression in flow separation.