TY - JOUR

T1 - The motion generated by a rising particle in a rotating fluid - Numerical solutions. Part 2. The long container case

AU - Minkov, E.

AU - Ungarish, M.

AU - Israeli, M.

PY - 2002/3/10

Y1 - 2002/3/10

N2 - Numerical finite-difference results from the full axisymmetric incompressible Navier-Stokes equations are presented for the problem of the slow axial motion of a disk particle in an incompressible, rotating fluid in a long cylindrical container. The governing parameters are the Ekman number, E = v*/(Ω* a*2), Rossby number, Ro = W* /(Ω* a*), and the dimensionless height of the container, 2H (the scaling length is the radius of the particle, a*; Ω* is the container angular velocity, W* is the particle axial velocity and v* the kinematic viscosity). The study concerns the flow field for small values of E and Ro while H E is of order unity, and hence the appearance of a free Taylor column (slug) of fluid 'trapped' at the particle is expected. The numerical results are compared with predictions of previous analytical approximate studies. First, developed (quasi-steady-state) cases are considered. Excellent agreement with the exact linear (Ro = O) solution of Ungarish and Vedensky (1995) is obtained when the computational Ro = 10-4. Next, the time-development for both an impulsive start and a start under a constant axial force is considered. A novel unexpected behaviour has been detected : the flow field first attains and maintains for a while the steady-state values of the unbounded configuration, and only afterwards adjusts to the bounded container steady state. Finally, the effects of the nonlinear momentum advection terms are investigated. It is shown that when Ro increases then the dimensionless drag (scaled by μ*a*W*) decreases, and the Taylor column becomes shorter, this effect being more pronounced in the rear region (μ* is the dynamic viscosity). The present results strengthen and extend the validity of the classical drag force predictions and therefore the issue of the large discrepancy between theory and experiments (Maxworthy 1970) concerning this force becomes more acute.

AB - Numerical finite-difference results from the full axisymmetric incompressible Navier-Stokes equations are presented for the problem of the slow axial motion of a disk particle in an incompressible, rotating fluid in a long cylindrical container. The governing parameters are the Ekman number, E = v*/(Ω* a*2), Rossby number, Ro = W* /(Ω* a*), and the dimensionless height of the container, 2H (the scaling length is the radius of the particle, a*; Ω* is the container angular velocity, W* is the particle axial velocity and v* the kinematic viscosity). The study concerns the flow field for small values of E and Ro while H E is of order unity, and hence the appearance of a free Taylor column (slug) of fluid 'trapped' at the particle is expected. The numerical results are compared with predictions of previous analytical approximate studies. First, developed (quasi-steady-state) cases are considered. Excellent agreement with the exact linear (Ro = O) solution of Ungarish and Vedensky (1995) is obtained when the computational Ro = 10-4. Next, the time-development for both an impulsive start and a start under a constant axial force is considered. A novel unexpected behaviour has been detected : the flow field first attains and maintains for a while the steady-state values of the unbounded configuration, and only afterwards adjusts to the bounded container steady state. Finally, the effects of the nonlinear momentum advection terms are investigated. It is shown that when Ro increases then the dimensionless drag (scaled by μ*a*W*) decreases, and the Taylor column becomes shorter, this effect being more pronounced in the rear region (μ* is the dynamic viscosity). The present results strengthen and extend the validity of the classical drag force predictions and therefore the issue of the large discrepancy between theory and experiments (Maxworthy 1970) concerning this force becomes more acute.

UR - http://www.scopus.com/inward/record.url?scp=0037051421&partnerID=8YFLogxK

U2 - 10.1017/S0022112001007157

DO - 10.1017/S0022112001007157

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AN - SCOPUS:0037051421

SN - 0022-1120

VL - 454

SP - 345

EP - 364

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

ER -