Flow due to an oscillating sphere and an expression for unsteady drag on the sphere at finite Reynolds number

• Published in: Journal of Fluid Mechanics
• Main Author: Mei, Renwei
• Format: Article in Journal/Newspaper
• Language: English
• Published: Cambridge University Press (CUP) 1994
• Subjects: DML
• Online Access: http://dx.doi.org/10.1017/s0022112094004222; https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112094004222

Unsteady flow due to an oscillating sphere with a velocity U 0 cosω t ’, in which U 0 and ω are the amplitude and frequency of the oscillation and t ’ is time, is investigated at finite Reynolds number. The methods used are: (i) Fourier mode expansion in the frequency domain; (ii) a time-dependent finite difference technique in the time domain; and (iii) a matched asymptotic expansion for high-frequency oscillation. The flow fields of the steady streaming component, the second and third harmonic components are obtained with the fundamental component. The dependence of the unsteady drag on ω is examined at small and finite Reynolds numbers. For large Stokes number, ε = (ω a 2 /2 v ) ½ [Gt ] 1, in which a is the radius of the sphere and v is the kinematic viscosity, the numerical result for the unsteady drag agrees well with the high-frequency asymptotic solution; and the Stokes (1851) solution is valid for finite Re at ε [Gt ] 1. For small Strouhal number, St = ω a / U 0 [Lt ] 1, the imaginary component of the unsteady drag (Scaled by 6π U 0 p f va, in which P f is the fluid density) behaves as Dml ∼ (h 0 St log St –h 1 St ), m = 1,3,5… This is in direct contrast to an earlier result obtained for an unsteady flow over a stationary sphere with a small-amplitude oscillation in the free-stream velocity (hereinafter referred to as the SA case) in which D 1 ∼ – h 1 St (Mei, Lawrence & Adrian 1991). Computations for flow over a sphere with a free-stream velocity U 0 (1–α 1 +α 1 cosω t ’) at Re = U 0 2a/ v = 0.2 and St [Lt ] 1 show that h 0 for the first mode varies from 0 (at α 1 = 0) to around 0.5 (at α 1 = 1) and that the SA case is a degenerated case in which the logarithmic dependence of the drag in St is suppressed by the strong mean uniform flow. The numerical results for the unsteady drag are used to examine an approximate particle dynamic equation proposed for spherical particles with finite Reynolds number. The equation includes a quasi-steady drag, an added-mass force, and a modified history force. The ...