Page 173 - Aerodynamics for Engineering Students

156 Aerodynamics for Engineering Students

6 Determine the stream function for a two-dimensional source of strength m. Sketch
the resultant field of flow due to three such sources, each of strength m, located at the
vertices of an equilateral triangle. (v of L)
7 Derive the irrotational flow formula
1
p -PO = -pu2(1 - 4sin2o)
2
giving the intensity of normal pressure p on the surface of a long, circular cylinder set
at right-angles to a stream of velocity U. The undisturbed static pressure in the fluid
is PO and 19 is the angular distance round from the stagnation point. Describe briefly
an experiment to test the accuracy of the above formula and comment on the results
obtained. (U of L)

8 A long right circular cylinder of diameter am is set horizontally in a steady stream
of velocity Um s-l and caused to rotate at w rad s-l. Obtain an expression in terms
of w and U for the ratio of the pressure difference between the top and the bottom of
the cylinder to the dynamic pressure of the stream. Describe briefly the behaviour of
the stagnation lines of such a system as w is increased from zero, keeping U constant.
( "1
Answer: - (CU)

9 A line source is immersed in a uniform stream. Show that the resultant flow, if
irrotational, may represent the flow past a two-dimensional fairing. If a maximum
thickness of the fairing is 0.15 m and the undisturbed velocity of the stream 6.0 m s-l,
determine the strength and location of the source. Obtain also an expression for the
pressure at any point on the surface of the fairing, taking the pressure at infinity
as datum. (Answer: 0.9 m2 s-', 0.0237 m)(U of L)

10 A long right circular cylinder of radius am is held with its axis normal to an
irrotational inviscid stream of U. Obtain an expression for the drag force acting on
unit length of the cylinder due to the pressures exerted on the front half only.
( 3 )
Answer: --pU2a (CU)
l
11 Show that a velocity potential exists in a two-dimensional steady irrotational
incompressible fluid motion. The stream function of a two-dimensional motion of an
incompressible fluid is given by
a C
$J = -x2 + bxy - -y2
2 2
where a, b and c are arbitrary constants. Show that, if the flow is irrotational, the
lines of constant pressure never coincide with either the streamlines or the equipo-
tential lines. Is this possible for rotational motion? (U of L)
12 State the stream function and velocity potential for each of the motions induced
by a source, vortex and doublet in a two-dimensional incompressible fluid. Show that
a doublet may be regarded, either as
(i) the limiting case of a source and sink, or
(ii) the limiting case of equal and opposite vortices, indicating clearly the direction of
the resultant doublet. (U of L)