Numerical Method

All physical parameters are made dimensionless by using the characteristic length, which is the gap between the radius of the rotating inner cylinder (R₁) and the radius of the stationary outer cylinder (R₂), the characteristic velocity, which is the maximum circumferential rotation speed attained during each calculation, and the characteristic time defined as the ratio of the characteristic length to the characteristic velocity. The governing equations are the unsteady axisymmetric Navier-Stokes equations and the equation of continuity in cylindrical polar coordinates $(r, \theta, z)$ with

$$\frac{\partial \mbox{\boldmath$u$ }}{\partial t} + (\mbox{\bo... ...} = - \nabla p + \frac{1}{Re_0} \nabla^2 \mbox{\boldmath$u$ },$$ (1)

$$\nabla \cdot \mbox{\boldmath$u$ } = 0,$$ (2)

where t is the time, u is the velocity vector, p is pressure, and Re₀ is the Reynolds number based on the characteristic velocity. Let the length of the cylinder be L, and the Reynolds number based on instantaneous rotation speed be Re. The aspect ratio ${\sl\Gamma}$ is defined by L / (R₂-R₁). The Stokes stream function ${\sl\psi }$ is given by

$$u = - \frac{1}{r}\frac{\partial {\sl\psi}}{\partial z}, \ \ \ \ \ w = \frac{1}{r}\frac{\partial {\sl\psi}}{\partial r}.$$ (3)

The basic solution procedure is the MAC method, and the spatial differentiation is the QUICK method for convection terms and the second-order central difference method for other terms. The time integration is the Euler explicit method.

A hybrid method of SOR and ILUCGS is used to solve the Poisson equation for pressure. While SOR is an efficient method that gives a reasonable convergence when the development of flow is slow, ILUCGS is expected to be a more stable procedure. Therefore, if SOR does not converge within a prescribed iteration level, then the solver is switched from SOR, and ILUCGS is adopted for the succeeding ten time steps.

The boundary conditions at cylinder walls and both end walls are no-slip conditions for velocity components and Neumann conditions for pressure obtained from momentum equations. When nonunique secondary mode is required, the initial conditions are that all components are zero in the entire domain. Otherwise, the initial conditions are estimated by assuming Couette flow, although this assumption is not valid near the end walls of cylinders.

For a fixed value of ${\sl\Gamma}$, a fully developed flow at a certain Reynolds number is obtained, and then the Reynolds number is gradually decreased in order to investigate the transition from a flow in the secondary mode to a flow in the primary mode. The dimensionless time t₁ when the flow is judged to be fully developed is estimated as the time when the relative variation of torque remains less than 10^-4. While the flow decelerates, the Reynolds number decreases linearly during dimensionless time t₂.

The results of the current computation are compared with experimental results given by Nakamura et al. (1989). The experimental apparatus used by them had an inner cylinder with a radius of 20 mm and an outer cylinder with a radius of 30 mm; the radius ratio was 0.667. The dimensionless times t₁ and t₂ are determined to be 50 seconds in dimensional form when they are evaluated using the physical dimensions in Nakamura et al.'s experiment. In the following, for the purpose of better understanding, the z coordinate is normalized not by the characteristic length but by the length of the working fluid.

In the staggered grid used in this calculation, the number of grid points in the radial direction is 41, and the number of grid points in the axial direction is determined by the proportionality to the cylinder length with 42 points for a unit aspect ratio. The time step $\Delta t$ is specified by the relation $\Delta t / Re_0 = 1.2 \times 10^{-5}$. Even though the grids were refined by halving the spacing in each direction and the fourth-order Runge-Kutta method was adopted for the time integral instead of the method used in the present study, no observable difference was found in the result. The error tolerance for terminating the convergence of the pressure-Poisson equation is 10^-4. The calculated result showed that, in the fully developed flow, the residual of total torque relative to the torque acting on the inner cylinder and the residual of ${\sl\psi }$, which was estimated from velocity components, relative to its extremum are on the order of 10^-4 and 10^-3, respectively.