Long-time behaviour of the drag on a body in impulsive motion

Abstract

We consider the response of the hydrodynamic drag on a body in rectilinear motion to a change in the speed between two steady states, from U1 to U2 [ges ] 0. We consider situations where the body generates no lift, such as occur for bodies with an axis of symmetry aligned with the motion. At large times, the laminar wake consists of two quasi-steady regions – the new wake and the old wake – connected by a transition zone that is convected downstream with the mean speed U2. A global mass balance indicates the existence of a sink flow centred on the transition zone, and this is responsible for the leading-order behaviour of the unsteady force at long times. For the case of U1 [ges ] 0, the force is shown to decay algebraically with the inverse square of time for any finite Reynolds number (Re), and this result is also shown to hold for non-rectilinear motions. A recent analysis for small Reynolds number including terms to O(Re) (Lovalenti & Brady 1993 a) has indicated that the force decays as the inverse square of time for motion started from rest, but decays exponentially for a change between two positive velocities. The former result is found to be correct, but the exponential decay at O(Re) in the latter case is superseded at large times by the inverse-square time decay which is shifted to O(Re2) because the wake flux is nearly constant for small Re. The cases of reversed flow (U1 < 0) and stopped flow (U2 = 0) are treated separately, and it is shown that the transient force is dominated by the effects of the old wake, leading to a slower decay as the simple inverse of time. The force is determined by the far regions of the flow field and so the results are valid for any (symmetric) particle, bubble or drop and (in an average sense) for any Re, provided τ ma {Re, Re−1}, where the time τ is made dimensionless with the convection timescale. The analytical results are compared to detailed numerical calculations for transient flow over spherical particles and bubbles and compelling agreement is observed. These are believed to be the first calculations which adequately resolve the transient far wake behind a bluff body at long times. The asymptotic result for the force is applied to determine that the approach to terminal velocity of a body in free fall is also as the inverse square of time.