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 = ν*/(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 ν* the kinematic viscosity). The study concerns the flow field
for small values of E and Ro while HE 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 = 0) solution of Ungarish & 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.
Numerical study of the thermosolutal convection in a 3-D cavity submitted to cross gradients of temperature and concentration
