Page 174 - Aerodynamics for Engineering Students

Potential flow 157

13 Define (a) the stream function, (b) irrotational flow and (c) the velocity potential
for two-dimensional motion of an incompressible fluid, indicating the conditions
under which they exist. Determine the stream function for a point source of strength
0 at the origin. Hence, or otherwise, show that for the flow due to any number of
sources at points on a circle, the circle is a streamline provided that the algebraic sum
of the strengths of the sources is zero. (U of L)

14 A line vortex of strength I? is mechanically fixed at the point (1, 0) referred to
a system of rectangular axes in an inviscid incompressible fluid at rest at
infinity bounded by a plane wall coincident with the y-axis. Find the velocity in the
fluid at the point (0, y) and determine the force that acts on the wall (per unit depth)
if the pressure on the other side of the wall is the same as at infinity. Bearing
in mind that this must be equal and opposite to the force acting on unit length
of the vortex show that your result is consistent with the Kutta-Zhukovsky
theorem. (U of L)

15 Write down the velocity potential for the two-dimensional flow about a circular
cylinder with a circulation I? in an otherwise uniform stream of velocity U. Hence
show that the lift on unit span of the cylinder is pur. Produce a brief but plausible
argument that the same result should hold for the lift on a cylinder of arbitrary shape,
basing your argument on consideration of the flow at large distances from the
cylinder. (U of L)
16 Define the terms velocity potential, circulation, and vorticity as used in two-
dimensional fluid mechanics, and show how they are related. The velocity distribu-
tion in the laminar boundary layer of a wide flat plate is given by

where uo is the velocity at the edge of the boundary layer where y equals 6. Find the
vorticity on the surface of the plate.
3 uo
(Answer: ---) (U of L)
26
17 A two-dimensional fluid motion is represented by a point vortex of strength r set
at unit distance from an infinite straight boundary. Draw the streamlines and plot the
velocity distribution on the boundary when I? = 7r. (U of L)
18 The velocity components of a two-dimensional inviscid incompressible flow are
given by

u=2y- Y v=-2x- X
2 . ( + yz)1/2 (x2 + y2y
Find the stream function, and the vorticity, and sketch the streamlines.

1
Answer: $ = 2 + y2 + (x2 + y2)l/’: C = -

19 (a) Given that the velocity potential for a point source takes the form