1.1 two-dimensional incompressible flow · 1.1 two-dimensional incompressible flow ma3d1...
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1.1 Two-dimensional incompressible flow
MA3D1 2012-2013 exam.Consider a two-dimensional incompressible flow with velocity field u = (u(x, y, t), v(x, y, t), 0).
a) Show that the vorticity field in this case is always transverse to the plane in which thefluid motion takes place, ! = (0, 0,⌦). (3 marks)
b) Show that the z-component of the vorticity field ⌦, is conserved along the trajectories ofthe fluid particles if the fluid is inviscid.
(a) Find the 3⇥ 3 strain tensor, where the strain tensor is
S
ij
= 12
✓@u
j
@x
i
+@u
i
@x
j
◆
(1 marks)
(b) Find the 3⇥ 3 stress tensor @uj
/@x
i
. (1 marks)
(c) Find the vortex stretching (~! ·r)~u, which in index form isX
j
✓!
j
@
@x
j
◆u
i
.
(d) Write the resulting equation for ⌦ for ⌫ = 0 using the Lagrangian time derivative.
1.2 Two-dimensional point vortex flow.
A stationary two-dimensional incompressible (viscous or inviscid) flow is described by a stream-function (x, y) such that the velocity is
u = (u, v) =
✓@
@y
,�@ @x
◆
a) Find a streamfunction for the flow with the following velocity field
u =�y
x
2 + y
2, v =
x
x
2 + y
2
Sketch the streamlines and discuss the distribution of vorticity and circulation in this flow. (5 marks)
b) A 2D inviscid fluid occupiesthe region x � 0, y � 0bounded by rigid boundariesx = 0, y = 0.
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The motion of the fluid results wholly from the presence of a single point vortex of circu-lation � in the domain, and a number of image vortices across the rigid boundaries.
(a) Note the positions and circulation of each of the image vortices. A sketch would help.(4 marks)
(b) What is the complex potential at z = x + iy at the primary vortex (x1, y1) inducedby the image vortices? (2 marks)
(c) What is the complex velocity induced by this vortex? (2 marks)
c) What is the complex velocity u � iv induced by the image vortices at position z = z1 ofthe primary vortex?
(a) As a first step keep the terms for each vortex separate, using their position in termsof z1 or z⇤1 . (2 marks)
(b) In the next step determine u and v by taking the real and imaginary parts of whatyou just calculated. (2 marks)
(c) From this deduce the trajectory dy1/dx1 (2 marks)
(d) Show that the path taken by the primary vortex is1
x
21
+1
y
21
= const (2 marks)
(e) Explain why a smoke ring expands as it approaches a wall. (4 marks)
1.3 What is vorticity? From pre-2006 tests.
• What is vorticity and what is its connection with turbulent flow? Give some examples ofthe type of vortical structures found in turbulent and transitional flows.
• In incompressible flow the following governing equation for vorticity can be erived fromthe Navier-Stokes equations:
D~!
Dt
= ~! ·r~v + ⌫r2~!
Give a physical interpretion for each term.
(25 marks total)
1.4 Taylor vortex and Kelvin’s circulation theorem
Consider the following simple flow field:
u = �x�
2, v = �y�
2, w = z�
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and vorticity:!(x, y, t) = !
z
(x, y, t) = ⌦(t) exp��a(t)(x2 + y
2)�
where at t = 0, ⌦ = ⌦0 and a = a0.
a) Is this flow field incompressible? (4 marks)
b) What is the total circulation �T
of this flow as a function of ⌦0, a0 and �?
(4 marks)
c) What is ⌦(t) using ⌦(t = 0) = ⌦0? Hint: The vortex stretching (~! · r)~u uses only onecomponent of the velocity stress tensor @u
i
/@x
j
. (2 marks)
d) Using conservation of the circulation and a(t = 0) = a0, what is a(t)? (2 marks)
e) What is the azimuthal velocity as a function of r and t, u✓
(r, t)? To find this, calculatethe circulation out of the radius r, then apply Stokes theorem. (4 marks)
f) CHECK: Does u✓
(r, t) ! 12⌦0a0
as t ! 1 for fixed r?. (1 marks)
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