Low-Reynolds-number flow past a cylinder with uniform blowing or sucking

Abstract

We analyse the low-Reynolds-number flow generated by a cylinder (of radius $a$) in a stream (of velocity $U_{\infty }$) which has a uniform through-surface blowing component (of velocity $U_{b}$). The flow is characterized in terms of the Reynolds number $Re$ ($=2aU_{\infty }/{\it\nu}$, where ${\it\nu}$ is the kinematic viscosity of the fluid) and the dimensionless blow velocity ${\it\Lambda}$ ($=U_{b}/U_{\infty }$). We seek the leading-order symmetric solution of the vorticity field which satisfies the near- and far-field boundary conditions. The drag coefficient is then determined from the vorticity field. For the no-blow case Lamb’s (Phil. Mag., vol. 21, 1911, pp. 112–121) expression is retrieved for $Re\rightarrow 0$. For the strong-sucking case, the asymptotic limit, $C_{D}\approx -2{\rm\pi}{\it\Lambda}$, is confirmed. The blowing solution is valid for ${\it\Lambda}<4/Re$, after which the flow is unsymmetrical about ${\it\theta}={\rm\pi}/2$. The analytical results are compared with full numerical solutions for the drag coefficient $C_{D}$ and the fraction of drag due to viscous stresses. The predictions show good agreement for $Re=0.1$ and ${\it\Lambda}=-5,0,5$.