Flow past a sphere with an oscillation in the free-stream velocity and unsteady drag at finite Reynolds number

Abstract

Unsteady flow over a stationary sphere with a small fluctuation in the free-stream velocity is considered at small Reynolds number, Re. A matched asymptotic solution is obtained for the frequency-dependent (or the acceleration-dependent) part of the unsteady flow at very small frequency, ω, under the restriction St [Lt ] Re [Lt ] 1, where St is the Strouhal number. The acceleration-dependent part of the unsteady drag is found to be proportional to St ∼ ω instead of the ω½ dependence predicted by Stokes’ solution. Consequently, the expression for the Basset history force is incorrect for large time even for very small Reynolds numbers. Present results compare well with the previous numerical results of Mei, Lawrence & Adrian (1991) using a finite-difference method for the same unsteady flow at small Reynolds number. Using the principle of causality, the present analytical results at small Re, the numerical results at finite Re for low frequency, and Stokes’ results for high frequency, a modified expression for the history force is proposed in the time domain. It is confirmed by comparing with the finite-difference results at arbitrary frequency through Fourier transformation. The modified history force has an integration kernel that decays as t−2, instead of t½, at large time for both small and finite Reynolds numbers.