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

By J. C. Padrino and D. D. Joseph

Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA



Relevant results from the numerical solution 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 setup is validated by comparing results for a Reynolds number based on the free stream of Re = 200 and peripheral speed of  = 3, 4 and 5 with results from the literature.  After the validation stage, various pairs of Re and  are specified in order to carry on the numerical experiments.  These values are Re = 200 with  = 4 and 5; Re = 400 with  = 4, 5 and 6, and Re = 1000 with  = 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 Re and .

Next, the theoretical approach of Wang and Joseph (2004a) and the numerical results are utilized to determine two different values of the effective vortical region thickness for every pair of Re and .  One effective thickness  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  is determined from the match between the pressure lift on the cylinder obtained from Wang and Joseph (2004a)'s simple modification of the boundary layer analysis due to Glauert (1957) and the numerical value of the pressure lift coefficient.  Each of these two values of the effective vortical region thickness is used to compute various parameters relevant to this type of fluid motion, 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.  For some of these parameters, predictions from the theoretical approaches of Glauert (1957) and Wang and Joseph (2004a) are presented for comparison.  The values of both effective thicknesses,  and , are found to be fairly close.  Then, we show that, with the choice of the thickness  as a unique effective thickness, the simple modification of Glauert's boundary layer analysis and the VCVPF applied to balance the shear stress discrepancy at the outer edge of the vortical region as proposed by Wang and Joseph (2004a) lead to expressions that exhibit better general agreement with the numerical results than Glauert's solution.