Numerical Method

Let R₁ and R₂ be the radii of the inner and outer cylinders, respectively. The aspect ratio ${\sl\Gamma }$ is defined as the ratio of the cylinder length L to the gap between cylinders (R₂-R₁), and the Reynolds number Re is determined by the gap and instantaneous rotation speed of the inner cylinder. The radius ratio ${\sl\eta}$ is the ratio of R₁ to R₂. All physical parameters are made dimensionless by using the characteristic length which is the gap between cylinders, 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 the cylindrical coordinates $(r, \theta, z)$ with corresponding velocities (u, v, w):

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

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

where u is the velocity vector, p is pressure, t is time, and Re₀ is the Reynolds number based on the characteristic velocity.

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 time integration is the Euler explicit method. The spatial differentiation is the QUICK method for convection terms and the second-order central difference method for other terms. The staggered grid is used in this calculation, and the grid interval is uniform in each direction. The number of grid points in the radial direction is 80, and the number of grid points in the axial direction is determined by the proportionality to the cylinder length with 80 points for an aspect ratio of unity. The number of grid points used in the present calculation is estimated to be thick enough not to influence the results.

A hybrid method of SOR and ILUCGS is used to solve the Poisson equation for pressure.

The boundary conditions at the cylinder walls and both end walls are no-slip conditions for velocity components. Neumann conditions for pressure are obtained from momentum equations. The initial conditions are that all velocity components are zero in the entire domain.

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, Re decreases linearly during dimensionless time t₂.

The results of the current computation are compared with experimental results given by Nakamura and Toya (1996). Their experimental apparatus had an inner cylinder with a radius of 20 mm and an outer cylinder with a radius of 30 mm; ${\sl\eta}$ was 0.667. The dimensionless times t₁ and t₂ were estimated to be 50 seconds in dimensional form when they were evaluated using the physical dimensions in their experiment. In our experiment, for the purpose of better understanding, the z coordinate is normalized not by the characteristic length but by the length of the working fluid.