Flow about a Lifting Cylinder

A lifting flow can be generated by adding a free vortex to the flow about a circular cylinder just described. The stream function and the velocity potential now become,

$$\psi = U_\infty r \left( 1 -  {a^2 \over r^2} \right)\sin \theta - {\Gamma \over {2 \pi}} \ln r$$ (4.126)

$$\phi = U_\infty r \left( 1 +  {a^2 \over r^2} \right)\cos \theta + {\Gamma \over {2 \pi}} \theta$$ (4.127)

Fig. 4.32 shows the streamlines for this flow.

Figure 4.32: Flow past a Lifting Cylinder

Consequently the velocity components will be,

$$v_r = U_\infty \left( 1 -  {a^2 \over r^2} \right)\cos \theta$$ (4.128)

$$v_\theta = -U_\infty \left( 1 +  {a^2 \over r^2} \right)\sin \theta + {\Gamma \over {2 \pi r}}$$ (4.129)

At r = a, the radial velocity is still zero allowing us to consider the same circular cylinder as the "body". The tangential velocity on the surface of the cylinder is given by,

$$v_{\theta s} = -2 U_\infty \sin \theta + {\Gamma \over {2 \pi a}}$$ (4.130)

The streamline pattern for this flow depend upon the location of the stagnation points given by,

$$\sin \beta = { \Gamma \over {4 \pi U_\infty a}}$$ (4.131)

The surface velocity can now be written as

$$v_{\theta s} = -2 U_\infty \sin \theta + 2  U_\infty \sin \beta = -2 U_\infty \left(\sin \theta - \sin \beta \right)$$ (4.132)

Figure 4.33: Stagnation Points for a Lifting Cylinder

The location of the stagnation points and the resulting streamline pattern are shown in Fig 4.33 and Fig. 4.32 respectively.

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney