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The vorticity equation and its applications

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Title: The vorticity equation and its applications


1
The vorticity equation and its applications
  • Felix KAPLANSKI
  • Tallinn University of Technology
  • feliks.kaplanski_at_ttu.ee
  • Tallinn University of Technology, Estonia

2
Examples of vortex flows
3
Examples of vortex flows
4
Examples of vortex flows
VORTEX BREAKDOWN IN THE LABORATORY The photo at
the right is of a laboratory vortex breakdown
provided by Professor Sarpkaya at the Naval
Postgraduate School in Monterey, California.
Under these highly controlled conditions the
bubble-like or B-mode breakdown is nicely
illustrated. It is seen in the enlarged version
that it is followed by an S-mode breakdown.
5
Examples of vortex flows
VOLCANIC VORTEX RING The image at the right
depicts a vortex ring generated in the crater of
Mt. Etna. Apparently these rings are quite rare.
The generation mechanism is bound to be the
escape of high pressure gases through a vent in
the crater. If the venting is sufficiently rapid
and the edges of the vent are relatively sharp, a
nice vortex ring ought to form.  
6
Examples of vortex flows
7
Vortex ring flow
8
Virtual image of a vortex ring
flow www.applied-scientific.com/
MAIN/PROJECTS/NSF00/FAT_RING/Fat_Ring.html -
Force acts impulsively
9
Overview
  • Derivation of the equation of transport of
    vorticity
  • Describing of the 2D flow motion
  • on the basis of vorticity w and
    streamfunction y instead of the more popular
    (u,v,p)-system
  • Well-known solutions of the system (w, y )

10
NSE
11
Vorticity
12
Vorticity transport equation
13
Helmholtz equation
14
Vorticity equation on plane
1)
2)
1
3
1
4
2)-1)
3
2
4
2
15
Taking into account
Continuity equation
16
Vorticity equation on plane
4
3
3
4
17
Cylindrical coordinate system
In cylindrical coordinates (r , q ,z ) with
-axisymmetric case
18
Vorticity equation axisymmetric case
1)
2)
1
1
4
3
1
2
2)-1)
3
2
4
2
Proof with Mathematica
19
Taking into account
Continuity equation
20
Vorticity equation axisymmetric case
3
4
3
4
21
Vorticity transport equation for 2D q1-
axisymmetric vortices, q0 plane vortices
The Stokes stream function can be introduced as
follows
and gives second equation
22
For 3D problem generalized Helmholtz equation
,
,
,
where
For 3D problem we can not introduce streamfuction
Y like for 2D problem.
23
For 3D problem generalized Helmholtz equation in
cylindrical coordinates
,
,
,
where
24
For 2D problem
  • (u,v,p) (w, y)
  • Winning two variables instead of three
  • Losses difficulties with boundary conditions for
    streamfunction

25
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26
Vortex flow 2-D plane
27
Streamfunction and (u, v) through vorticity
and u, v are given by
28
  • Solutions, which contain vorticity expressed
    through delta-functions

29
Vortex flow.
30
Vortex flow.
31
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32
The Biot-Savart Law
33
  • Solutions, which contain vorticity expressed
    through delta-functions

34
Vortex flow.
35
Vortex flow.
36
  • Other solutions

37
Vortex flows. Hills ring
38
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39
We use polar coordinates (r, q) and assume
symmetry
40
Solution
Further we define constant c
and find solution
Proof with Mathematica
41
Appropriate tangent velocity
42
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43
Burgers vortex (a viscous vortex with swirl)
44
Vorticity
45
Irrotational Flow Approximation
  • Irrotational approximation vorticity is
    negligibly small
  • In general, inviscid regions are also
    irrotational, but there are situations where
    inviscid flow are rotational, e.g., solid body
    rotation (Ex. 10-3)

46
Irrotational Flow Approximation2D Flows
  • For 2D flows, we can also use the streamfunction
  • Recall the definition of streamfunction for
    planar (x-y) flows
  • Since vorticity is zero,
  • This proves that the Laplace equation holds for
    the streamfunction and the velocity potential

47
Elementary Planar Irrotational FlowsUniform
Stream
  • In Cartesian coordinates
  • Conversion to cylindrical coordinates can be
    achieved using the transformation

Proof with Mathematica
48
Elementary Planar Irrotational FlowsLine
Source/Sink
  • Potential and streamfunction are derived by
    observing that volume flow rate across any circle
    is
  • This gives velocity components

49
Elementary Planar Irrotational FlowsLine
Source/Sink
  • Using definition of (Ur, U?)
  • These can be integrated to give ? and ?

Equations are for a source/sink at the origin
Proof with Mathematica
50
Elementary Planar Irrotational FlowsLine
Source/Sink
  • If source/sink is moved to (x,y) (a,b)

51
Elementary Planar Irrotational FlowsLine Vortex
  • Vortex at the origin. First look at velocity
    components
  • These can be integrated to give ? and ?

Equations are for a source/sink at the origin
52
Elementary Planar Irrotational FlowsLine Vortex
  • If vortex is moved to (x,y) (a,b)

53
Elementary Planar Irrotational FlowsDoublet
  • A doublet is a combination of a line sink and
    source of equal magnitude
  • Source
  • Sink

54
Elementary Planar Irrotational FlowsDoublet
  • Adding ?1 and ?2 together, performing some
    algebra, and taking a?0 gives

K is the doublet strength
55
Examples of Irrotational Flows Formed by
Superposition
  • Superposition of sink and vortex bathtub vortex

Sink
Vortex
56
Examples of Irrotational Flows Formed by
Superposition
  • Flow over a circular cylinder Free stream
    doublet
  • Assume body is ? 0 (r a) ? K Va2

57
Examples of Irrotational Flows Formed by
Superposition
  • Velocity field can be found by differentiating
    streamfunction
  • On the cylinder surface (ra)

Normal velocity (Ur) is zero, Tangential velocity
(U?) is non-zero ?slip condition.
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