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

Published online by Cambridge University Press:  26 April 2006

Christopher J. Lawrence  and
Renwei Mei
Christopher J. Lawrence
Affiliation:
Department of Chemical Engineering and Chemical Technology, Imperial College of Science, Technology and Medicine, London SW7 2BY, UK
Renwei Mei
Affiliation:
Department of Aerospace Engineering, Mechanics and Engineering Science, University of Florida, Gainesville, FL 32611, USA
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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 The former result is found to be correct, but the exponential decay at O(Re) in the latter case is superseded at large times by the inverse-square time decay which is shifted to O(Re2) because the wake flux is nearly constant for small Re.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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 283 , 25 January 1995 , pp. 307 - 327
Copyright
© 1995 Cambridge University Press

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References

Basset, A. B. 1888 A Treatise on Hydrodynamics, vol. 2. Dover.
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Benjamin, T. B. 1993 Note on formulas for the drag on a sphere. J. Fluid Mech. 246, 335–342.Google Scholar
Clift, R., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops and Particles. Academic Press.
Hinch, E. J. 1975 Application of the Langevin equation to fluid suspensions. J. Fluid Mech. 72, 499–511.Google Scholar
Landau, L. D. & Lifschitz, E. M. 1987 Fluid Mechanics, 2nd edn. Pergamon.
Lawrence, C. J. & Weinbaum, S. 1988 The unsteady force on a body at low Reynolds number; the axisymmetric motion of a spheroid. J. Fluid Mech. 189, 463–489.Google Scholar
Lovalenti, P. M., & Brady, J. F. 1993a The hydrodynamic force on a rigid particle undergoing arbitrary time-dependent motion at small Reynolds number. J. Fluid Mech. 256, 561–605.Google Scholar
Lovalenti, P. M., & Brady, J. F. 1993b The force on a sphere in a uniform flow with small-amplitude oscillations at finite Reynolds number. J. Fluid Mech. 256, 607–614.Google Scholar
Lovalenti, P. M., & Brady, J. F. 1993c The force on a bubble, drop, or particle in arbitrary time-dependent motion at small Reynolds number. Phys. Fluids A 5, 2104–2016.Google Scholar
Mei, R. 1993 History force on a sphere due to a step change in the free-stream velocity. Intl J. Multiphase Flow 19, 509–525.Google Scholar
Mei, R. 1994 Flow due to an oscillating sphere and an expression for unsteady drag on the sphere at finite Reynolds number. J. Fluid Mech. 270, 133–174.Google Scholar
Mei, R. & Adrian, R. J. 1992 Flow past a sphere with an oscillation in the free-stream velocity, and unsteady drag at finite Reynold number. J. Fluid Mech. 237, 323–341.Google Scholar
Mei, R., Adrian, R. J. & Hanratty, T. J. 1991 Particle dispersion in isotropic turbulence under Stokes drag and Basset force with gravitational settling. J. Fluid Mech. 225, 481–495.Google Scholar
Mei, R. & Klausner, J. F. 1992 Unsteady force on a spherical bubble at finite Reynolds number with small fluctuations in the free-stream velocity. Phys. Fluids A 4, 63–70.Google Scholar
Mei, R., Klausner, J. F. & Lawrence, C. J. 1994 A note on the history on a spherical bubble at finite Reynolds number. Phys. Fluids 6, 418–420.Google Scholar
Mei, R. & Lawrence, C. J. 1994 Long-time unsteady laminar wake of a bluff body at finite Reynolds number. J. Fluid Mech. (submitted).Google Scholar
Mei, R., Lawrence, C. J. & Adrian, R. J. 1991 Unsteady drag on a sphere at finite Reynolds number with small fluctuations in the free stream velocity. J. Fluid Mech. 233, 613–628.Google Scholar
Pozrikidis, C. 1989 A study of linearized oscillatory flow past particles by the boundary integral method. J. Fluid Mech. 202, 17–41.Google Scholar
Reeks, M. W. & Mckee, S. 1984 The dispersive effects of Basset history forces on particle motion in turbulent flow. Phys. Fluids 27, 1573–1582.Google Scholar
Sano, T. 1981 Unsteady flow past a sphere at low Reynolds number. J. Fluid Mech. 112, 433–441.Google Scholar
Williams, W. E. 1966 A note on slow vibrations in a viscous fluid. J. Fluid Mech. 25, 589–590.Google Scholar