High Order Computation of the History Term in the Equation of Motion for a Spherical Particle in a Fluid

The historical evolution of the equation of motion for a spherical particle in a fluid and the search for its general solution are recalled. The presence of an integral term that is nonzero under unsteady motion and viscous conditions allowed simple analytical or numerical solutions for the particle dynamics to be found only in a few particular cases. A general solution to the equation of motion seems to require the use of computational methods. Numerical schemes to handle the integral term of the equation of motion have already been developed. We present here adaptations of a first order method for the implementation at high order, which may employ either fixed or variable computation time steps. Some examples are shown to establish comparisons between diverse numerical methods.

[1]  Joel H. Ferziger,et al.  Computational methods for fluid dynamics , 1996 .

[2]  John C. Butcher.  Runge-Kutta methods in modern computation, part I: fundamental concepts , 1994 .

[3]  R. Adrian,et al.  Unsteady drag on a sphere at finite Reynolds number with small fluctuations in the free-stream velocity , 1991, Journal of Fluid Mechanics.

[4]  Efstathios E. Michaelides,et al.  The Magnitude of Basset Forces in Unsteady Multiphase Flow Computations , 1992 .

[5]  William H. Press,et al.  Book-Review - Numerical Recipes in Pascal - the Art of Scientific Computing , 1989 .

[6]  J. Riley,et al.  Equation of motion for a small rigid sphere in a nonuniform flow , 1983 .

[7]  M. Maxey,et al.  Unsteady flow about a sphere at low to moderate Reynolds number. Part 1. Oscillatory motion , 1994, Journal of Fluid Mechanics.

[8]  Efstathios E. Michaelides,et al.  Effect of the history term on the motion of rigid spheres in a viscous fluid , 1994 .

[9]  J. C. Butcher.  Runge–Kutta Methods in Modern Computation, Part II: Implicit Methods and Related Applications , 1994 .

[10]  Efstathios E. Michaelides,et al.  A novel way of computing the Basset term in unsteady multiphase flow computations , 1992 .

[11]  Martin R. Maxey,et al.  Unsteady flow about a sphere at low to moderate Reynolds number. Part 2. Accelerated motion , 1995, Journal of Fluid Mechanics.

[12]  J. Brady,et al.  The hydrodynamic force on a rigid particle undergoing arbitrary time-dependent motion at small Reynolds number , 1993, Journal of Fluid Mechanics.

[13]  J. C. R. Hunt,et al.  The force exerted on a body in inviscid unsteady non-uniform rotational flow , 1988, Journal of Fluid Mechanics.

[14]  Efstathios E. Michaelides,et al.  Review-The transient equation of motion for particles , 1997 .

[15]  P. Alexander,et al.  A program for the simulation and analysis of open atmospheric balloon soundings , 2003 .

[16]  A. B. Basset.  On the Motion of a Sphere in a Viscous Liquid , 1887 .

[17]  F. Tatom.  The Basset term as a semiderivative , 1988 .

[18]  Renwei Mei,et al.  Flow due to an oscillating sphere and an expression for unsteady drag on the sphere at finite Reynolds number , 1994, Journal of Fluid Mechanics.

[19]  J H Cartwright,et al.  Dynamics of a small neutrally buoyant sphere in a fluid and targeting in Hamiltonian systems. , 2000, Physical review letters.

[20]  Geoffrey Ingram Taylor,et al.  The Forces on a Body Placed in a Curved or Converging Stream of Fluid , 1928 .

[21]  Carlos F.M. Coimbra,et al.  General solution of the particle momentum equation in unsteady Stokes flows , 1998 .

[22]  Robert B. Ross,et al.  Evaluating structured I/O methods for parallel file systems , 2004, Int. J. High Perform. Comput. Netw..

[23]  J. Brady,et al.  The force on a sphere in a uniform flow with small-amplitude oscillations at finite Reynolds number , 1993, Journal of Fluid Mechanics.

[24]  R. Adrian,et al.  Flow past a sphere with an oscillation in the free-stream velocity and unsteady drag at finite Reynolds number , 1992, Journal of Fluid Mechanics.

[25]  S. McKee,et al.  The dispersive effects of Basset history forces on particle motion in a turbulent flow , 1984 .