Mechanics of Fluids

as shown in Fig. 9.23, encloses all the streamlines running from the source
to the skin. The shape defined by the line ψ = 0 is usually known as the
Rankine oval after W. J. M. Rankine (1820–72) who first developed the
technique of combining flow patterns. (N.B. The Rankine oval is not an
ellipse.) Although ψ = 0 along this line, the velocity is not zero throughout
its length. However, for frictionless flow, the contour of a solid body may be
put in place of this dividing streamline; the flow pattern outside the dividing
streamline is therefore that of an originally uniform stream deflected by a
solid body with the shape of the Rankine oval.
The shape of the solid boundary may be altered by varying the distance
between source and sink or by varying the value of m relative to U – the
source and sink are, after all, quite hypothetical. Other shapes may be
obtained by the introduction of additional sources and sinks along the x-axis,
although the total strength of these must remain zero. Rankine developed
ship contours in this way.
Example 9.5 A Rankine oval of length L and breadth B is produced
in an otherwise uniform stream of velocity 5 m · s −1 , by a source and
sink 75 mm apart. Determine L if B = 125 mm.
Solution
The surface of the Rankine oval is defined by ψ = 0. As the oval is
symmetrically disposed about the y-axis its maximum thickness occurs
when x = 0. Write t = B/2. Hence
0 =−Ut + m
2π arctan
2tb
−b2 + t2 which can be written
2πUt
m
= arctan 2tb
t 2 − b 2
Combinations of basic flow patterns 389
Fig. 9.23