Chapter 4, Problem 4.13P

In Section 3.15 we studied the case of the lifting flow over a circular cylinder. In real life, a rotating cylinder in a flow will produce lift; such real flow fields are shown in the photographs in Figures 3.34(b) and (c). Here, the viscous shear stress acting between the flow and the surface of the cylinder drags the flow around in the direction of rotation of the cylinder. For a cylinder of radius R rotating with an angular velocity w in an otherwise stationary fluid, the viscous flow solution for the velocity field obtained from the Navier-Stokes equations (Chapter 15) is $V_{\theta }=\frac{R^{2}w}{r}$ where $V_{\theta }$ is the tangential velocity along the circular srcamlines and r is the radial distance from the center of the cylinder. (See Schlichting, Boundary-Layer Theory, 6th ed., McGraw-Hill, 1968, page 81.) Note that V9 varies inversely with r and is of the same form as the inviscid flow velocity for a point vortex given by Equation (3.105). If the rotating cylinder has a radius of 1 m and is flying at the same velocity and altitude as the airfoil in Problem 4.11, what must its angular velocity be to produce the same lift as the airfoil in Problem 4.11? (Note: You can check your results with the experimental data for lift on rotating cylinders in Hoerner, Fluid-Dynamic Lift, published by the author, 1975, pp. 2l, Fig. 5.)