Steady flow past sudden expansions at large Reynolds number. II. Navier–Stokes solutions for the cascade expansion

Abstract

In the steady laminar flow past a sudden expansion at large Reynolds number R, the equations of motion reduce to the boundary-layer equations as R→∞ if the longitudinal length scale of the separated eddy increases linearly and indefinitely with R. In part I of this series [Phys. Fluids 29, 1353 (1986)], several sudden expansion geometries were considered, and in each case, when the inflow was uniform, steady solutions to the boundary-layer equations were found to exist provided that the expansion ratio remained above a critical value where the pressure gradient became singular near the reattachment point of the eddy. These results suggested that for uniform inflows and smaller values of the expansion ratio, the eddy length could not continue to increase linearly with R if the latter were sufficiently large. In the present work a global Newton method was employed to obtain finite-difference solutions to the steady Navier–Stokes equations up to R=1000 for a uniform inflow past a cascade of sudden expansions. The calculations show that for large values of the expansion ratio, the eddy length increases linearly with R, and that the main features of the flow approach those predicted by the boundary-layer solutions, including the existence of large pressure gradients near the reattachment point, as the expansion ratio is reduced toward the critical value. However, for smaller values of the expansion ratio where solutions to the boundary-layer equations could not be found, the steady solutions to the Navier–Stokes equations approach, with increasing R, the limit of an inviscid eddy O(1) in length, with the main features of the flow conforming to the theoretical model of Batchelor [J. Fluid Mech. 1, 388 (1956)] for an inviscid separated eddy behind a bluff body.