Wall effects on flow past fluid spheres at finite Reynolds number: wake structure and drag correlations

Following the pioneering works of Leal and coworkers, a detailed report is made on the dynamics of recirculating wakes that form at finite Reynolds number Re due to vorticity accumulation at the rear of a fluid sphere that is either suspended in a tube by an upflowing fluid (the fluidized drop problem) or falling in a tube (the falling drop problem). The axisymmetric, steady flow of a Newtonian fluid past the fluid sphere is determined by finite element analysis using a consistent penalty formulation. By way of example, the flow past a fluid sphere that is falling in a tube for which the ratio of the tube radius to the drop radius 1/lambda = 5 undergoes remarkable transitions when the ratio of the viscosity of the drop to that of the ambient fluid, kappa, varies over a narrow range. When kappa less than or equal to 2.75, no wake forms behind the sphere as Re increases. When 3 less than or equal to kappa < 10, an eddy that is detached from the drop forms at a critical value of Re = Re-c((1)), grows, and eventually disappears as Re rises above a certain amount Re-c((2)). Remarkably, when kappa = 3, these critical Reynolds numbers are as low as Re-c((1)) = 51 +/- 1 and Re-c((2)) = 77 +/- 1. When kappa greater than or equal to 10, it is shown by using as many as 46,000 velocity degrees of freedom that the detached eddy attaches to the drop when Re exceeds a critical value, Re greater than or equal to Re-c*, which was heretofore unknown. Whereas only a single, large - primary - eddy is present inside the drop when Re < Re-c*, a second but much smaller - secondary - eddy also forms inside the drop upon attachment. Two new correlations are developed that account for the effects of a tube wall and finite fluid inertia on drag for fluidized and falling droplets. Moreover, in contrast to related correlations of others for fluid spheres that are placed in an infinite expanse of ambient fluid, the new correlations are valid over the entire range of Reynolds numbers considered. (C) 1997 Elsevier Science Ltd.