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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