Potential Vortex

Consider a cylinder rotating clockwise in a fluid at rest at infinity. The particles will be entrained, through viscous shear forces, to move around the cylinder in concentric circles. It can be shown that, at steady-state, such a flow is irrotational and the velocity is given by

vr = 0, 2nrvg = const. = —Г (2.24)

Г is called the vortex strength or circulation and has unit (m2/s). Such a potential flow around the cylinder is called a potential vortex (this terminology may be con­fusing since the vorticity in such a flow is zero outside the cylinder otherwise it is not a potential flow). Now, the cylinder radius can be vanishingly small. The potential flow solution however is singular at r = 0, a point inside the cylinder, no matter how small the radius is.

Notice that a potential vortex corresponds to the flow obtained when the potential and stream lines of a source/sink flow are exchanged. The solution is given by

ГГ

ф = — в, f = ln r (2.25)

2n 2n

The potential lines are now the rays and the streamlines the circles centered at the origin. See Fig. 2.7. The vortex solution is a building block for lifting airfoils to represent incidence and camber as will be seen later.

Note that the velocity components (u, w) (obtained from ф and f by taking the partial derivatives with respect to x and z when using the Cartesian coordinates, or d – and 1 dg when using polar coordinates), decay in the far field as 1, which is consistent with the asymptotic condition.

Fig. 2.7 Potential vortex: potential lines and streamlines