Rankine Oval

We saw that the previous example defined a half body open at one end. Can we come up with a closed body by a suitable combination? An inspection of the streamlines suggests that by placing a sink in addition to the source one should be able to define a closed body. In other words a uniform flow past a source-sink combination is what we are after. The stream function for this is given by

$\displaystyle \psi $ $\displaystyle = U_\infty r \sin \theta -{m \over{2\pi} } \tan^{-1} \left(
 {2ar \sin \theta} \over {r^2 - a^2}\right)$ (4.99)
or in Cartesian coordinates
$\displaystyle \psi $ $\displaystyle = U_\infty y -{m \over{2\pi} } \tan^{-1} \left( {2ay} \over
 {x^2 + y^2 - a^2}\right)$ (4.100)

Figure 4.26: Flow past a Rankine Oval

When the streamlines for this flow are plotted (Fig.4.26) one discovers that the one given by $ \psi = 0$ (shown in red) forms a closed curve. This obviously forms the "body", i.e., the stream function we have written describes the flow about this body. Shapes such as this are called Rankine Ovals. The distance to the stagnation points from the origin or the Half Body Length is given by

$\displaystyle {l \over a} = \sqrt{{m \over {\pi U_\infty a}}+1}$ (4.101)

Figure 4.27: Rankine Oval

The other feature of interest, Half Width is found by determining the point of intersection of y-axis with the body, i.e., $ \psi = 0$ line. An expression for h is,

$\displaystyle {h \over a} = {1 \over 2} \left\{{h \over a}^2 - 1 \right\}\tan
 \left\{ 2 \left({ {\pi U_\infty a} \over m}\right) {h \over a}
 \right\}$ (4.102)

the solution for which is to be obtained by iteration. Rankine ovals include a wide range of bodies which can be obtained by varying the value of the parameter $ \pi U_\infty a /m$. These could be bodies stretched in any of the two directions. When stretched in x-direction one obtains elliptic bodies with a small half width compared to the span. The solution obtained could be a good approximation to the flow especially if viscous effects are small. On the other hand a considerable half width would indicate a bluff body prone to effects like separation. The solution obtained can hardly be accepted in this case.

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