Time-dependent viscous flow past an impulsively started rotating and translating circular cylinder

A numerical study is made of the development with time of the two-dimensional flow of a viscous, incompressible fluid around a circular cylinder which suddenly starts rotating about its axis with constant angular velocity and translating at right angles to this axis with constant speed. The governing partial differential equations in two space variables and time are reduced to sets of time-dependent equations in one space variable by means of Fourier analysis. By truncating the Fourier series to a finite number of terms, a finite set of differential equations is solved to give an approximation to the theoretical flow. The solutions are obtained by numerical methods. Results are given for the initial development with time of the asymmetrical wake at the rear of the cylinder at Reynolds numbers R ges 200, based on the diameter of cylinder, and at small rotation rates. The detailed results show the formation of a Kármán vortex street. The time development of this separated flow is compared in detail at R = 200 with recent experimental results. The details of the formation and movement of the vortices behind the cylinder and the velocity profiles in several locations are virtually identical in the experimental and theoretical studies. The variations with time of the lift, drag and moment exerted by the fluid on the cylinder are determined both by calculations and by means of approximate analytical expressions. The agreement between these results at small times is excellent.