Numerical study of the steady-state uniform flow past a rotating cylinder

Abstract

Results from the numerical simulation of the two-dimensional incompressible unsteady Navier–Stokes equations for streaming flow past a rotating circular cylinder are presented in this study. The numerical solution of the equations of motion is conducted with a commercial computational fluid dynamics package which discretizes the equations applying the control volume method. The numerical set-up is validated by comparing results for a Reynolds number based on the free stream of $\hbox{\it Re}$ = 200 and dimensionless peripheral speed of $\tilde{q}$ = 3, 4 and 5 with results from the literature. After the validation stage, various pairs of $\hbox{\it Re}$ and $\tilde{q}$ are specified in order to carry out the numerical experiments. These values are $\hbox{\it Re}$ = 200 with $\tilde{q}$ = 4 and 5; $\hbox{\it Re}$ = 400 with $\tilde{q}$ = 4, 5 and 6, and $\hbox{\it Re}$ = 1000 with $\tilde{q}$ = 3. In all these cases, gentle convergence to fully developed steady state is reached. From the numerical vorticity distribution, the position of the outer edge of the vortical region is determined as a function of the angular coordinate. This position is found by means of a reasonable criterion set to define the outmost curve around the cylinder where the vorticity magnitude reaches a certain cut-off value. By considering the average value of this profile, a uniform vortical region thickness is specified for every pair of $\hbox{\it Re}$ and $\tilde{q}$.

Next, the theoretical approach of Wang & Joseph (2006a; see the companion paper) and the numerical results are used to determine two different values of the effective vortical region thickness for every pair of $\hbox{\it Re}$ and $\tilde{q}$. One effective thickness $\delta_D/a$ is obtained from the match between the additional drag on the outer edge of the vortical region according to the viscous correction of viscous potential flow (VCVPF) and the corresponding numerical profile while the other thickness $\delta_L/a$ is determined from the match between the pressure lift on the cylinder obtained from Wang & Joseph (2006a)'s simple modification of the boundary-layer analysis due to Glauert (Proc. R. Soc. Lond. A, vol. 242, 1957, p. 108) and the numerical value of the pressure lift coefficient. The values of $\delta_D/a$ and $\delta_L/a$ are used in the computation of various parameters associated with the flow, namely, the torque on the rotating cylinder, the circulatory velocity at the edge of the vortical region, which links the cylinder's angular velocity with the circulation of the irrotational flow of the viscous fluid outside this region, and the viscous dissipation. Predictions from the approaches of Glauert (1957) and Wang & Joseph (2006a) are also included for comparison. The values of both effective thicknesses, $\delta_D/a$ and $\delta_L/a$, are found to have the same order of magnitude. Then, we show that choosing $\delta_D/a$ as a unique effective thickness, the modification of Glauert's boundary-layer analysis and the VCVPF approach as proposed by Wang & Joseph (2006a) produce results which are in better general agreement with the values from numerical simulation than those from Glauert's solution.