The effect of a uniform through-surface flow (velocity Ub) on a rigid cylinder and sphere (radius a) fixed in a free stream (velocity U1) is analysed analytically and numerically. The flow is characterised by a dimensionless blow velocity (= Ub/U1) and Reynolds number Re(= 2aU1/), where is the kinematic viscosity. High resolution numerical calculations are compared against theoretical predictions over the range −3 ≤ ≤ 3 and Re = 1, 10, 100.
For − ≫ 1, the flow is viscously dominated in a thin boundary layer adjacent to the rigid surface. The flow downstream of the body is irrotational so the wake volume flux, Qw, is zero and the drag force is FD = −U1Qb,where is the density of the fluid and Qb is the normal flux through the body surface. A dissipation argument is applied to analyse the drag force; the rate of working of the drag force is balanced by viscous dissipation, flux of energy and rate of work by viscous stresses due to sucking.
When ≫ 1, the boundary layer thickness initially grows linearly with time as vorticity is blown away from the rigid surface. For large blow velocity, the vorticity is swept into two well-separated shear layers and the maximum vorticity decreases due to diffusion. The drag force is related exactly to the vorticity distribution on the body surface and an approximate expression can be derived by considering the first term of a Fourier expansion in the surface vorticity.
References
[1] Klettner, C.A. & Eames, I. 2015 Low-Reynolds-number flow past a cylinder with uniform blowing or sucking. J. Fluid Mech. 780, 1–8.
[2] Klettner, C.A., Eames, I., Semsarzadeh, S. & Nicolle, A. 2016 The effect of a uniform through-surface flow on a cylinder and sphere. J. Fluid Mech. 793, 798–839.