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,

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

$\displaystyle v_r $ $\displaystyle = U_\infty \left( 1 -  {a^2 \over r^2} \right)\cos \theta$ (4.128)
$\displaystyle v_\theta $ $\displaystyle = -U_\infty \left( 1 +  {a^2 \over r^2} \right)\sin
 \theta + {\Gamma \over {2 \pi r}}$ (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,

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

The surface velocity can now be written as

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