Download MA3842 - Fluid Dynamics. Question Sheet 6. Potential Flow. 1. (a

MA3842 - Fluid Dynamics.
Question Sheet 6. Potential Flow.
1. (a) Let v and v′ be two different velocity fields for an incompressible fluid within
a region V , bounded by surface S. Let v be the potential flow flow within this
region, and let v′ be some other incompressible flow that satisfies the the same
boundary conditions as v. Show that
(v ′2 − v 2 ) = (v′ − v)2 + 2(v′ − v).v
= (v′ − v)2 + 2∇.[φ(v′ − v]
(b) Hence show that the difference between the kinetic energies of the two flows
ZZZ
′
T −T =
(v ′2 − v 2 )dV
V
is positive, and therefore that the potential flow represents the flow of least
possible kinetic energy. (This result is known as Kelvin’s minimum energy theorem.)
2. Two line sources of strength k and -k lie at positions x = a, y = 0 and x = −a, y = 0
respectively. Find the complex potential for this system. Hence show that streamlines for irrotational motion are circular arcs. (Hint: show that the complex potential
for this system is
z−a
w(z) = ln
z+a
Make the demominator of the fraction real, then put z = x + iy. Show that for
w(z) = ln(z) = ln(x + iy), ψ = tan−1 (y/x). Then return to the original problem and
put tan ψ = λ = constant.
3. Two line sources of strength k lie on the x axis at x = ±a. A line sink of strength
−2k lies at the origin. Show that for irrotational motion, the streamlines of the
curves are given by (x2 + y 2)2 = a2 (x2 − y 2 + λxy), where λ is a (real) constant.
4. Show that ∇φ.∇ψ = 0. Answer this first by writing the gradients in terms of partial
derivatives of φ and ψ. Then give an answer based on physical arguments.
5. A line source of strength k is located on the x axis at x = d (d > 0) and a rigid
boundary lies along the y axis. Use the method of images to find the flow speed
along the wall, and show that the pressure has a minimum value at y = |d|.
6. A circular cylinder of radius a with axis perpendicular to the x−y plane and passing
through the origin, is immersed in a uniform, incompressible fluid with a line source
of strength k on the x axis at x = f (f > a). Show that the speed of the fluid at
the surface of the cylinder is
v=
a2
2kf sin θ
+ f 2 − 2af cos θ
where θ is measured from the positive x axis. Find the points on the cylinder where
the pressure is a minimum. Mark these points on a rough sketch of the flow. In
which direction do you think the net force on the cylinder will act?
7. Fig. 1 shows a flow that is anti-parallel to the x axis at x → ∞, that encounters a
wedge with an angle 2π/3. Use a suitable conformal mapping to relate the complex
potential for this flow to that for a uniform flow anti-parallel to the x axis. Sketch
streamlines for the flow, and find the R and θ components of velocity. Show that
pressure at the wedge decreases with distance from the vertex.
π/3
π/3
Figure 1: The flow regions described in Question 7.
8. Fig 2. (a) shows a surface with a semi-circular bump. The flow is parallel to the x
axis at x → ±∞. Find a complex potential for this problem by relating it to another
problem we have solved, which has the same conditions at the boundary.
Use the transformation Z = z 1/2 to solve the problem of flow in a right–angle corner
with a quarter-circle bump of radius b, as in Fig 2(b). Show that R = b, θ = 0 and
θ = π/2 are streamlines. Find vR and vθ (hint: find φ first), and hence sketch the
resulting pattern of streamlines).
(a)
(b)
Figure 2: The flow regions described in Question 8.