The vorticity equation and its applications

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The vorticity equation and its applications

• Felix KAPLANSKI
• Tallinn University of Technology
• feliks.kaplanski_at_ttu.ee
• Tallinn University of Technology, Estonia

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Examples of vortex flows
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Examples of vortex flows
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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.
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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.  
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Examples of vortex flows
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Vortex ring flow
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Virtual image of a vortex ring
flow www.applied-scientific.com/
MAIN/PROJECTS/NSF00/FAT_RING/Fat_Ring.html -
Force acts impulsively
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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 )

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NSE
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Vorticity
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Vorticity transport equation
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Helmholtz equation
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Vorticity equation on plane
1)
2)
1
3
1
4
2)-1)
3
2
4
2
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Taking into account
Continuity equation
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Vorticity equation on plane
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3
3
4
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Cylindrical coordinate system
In cylindrical coordinates (r , q ,z ) with
-axisymmetric case
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Vorticity equation axisymmetric case
1)
2)
1
1
4
3
1
2
2)-1)
3
2
4
2
Proof with Mathematica
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Taking into account
Continuity equation
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Vorticity equation axisymmetric case
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4
3
4
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Vorticity transport equation for 2D q1-
axisymmetric vortices, q0 plane vortices
The Stokes stream function can be introduced as
follows
and gives second equation
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For 3D problem generalized Helmholtz equation
,
,
,
where
For 3D problem we can not introduce streamfuction
Y like for 2D problem.
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For 3D problem generalized Helmholtz equation in
cylindrical coordinates
,
,
,
where
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For 2D problem

• (u,v,p) (w, y)
• Winning two variables instead of three
• Losses difficulties with boundary conditions for
  streamfunction

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

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Vortex flow.
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Vortex flow.
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The Biot-Savart Law
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• Solutions, which contain vorticity expressed
  through delta-functions

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Vortex flow.
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Vortex flow.
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• Other solutions

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Vortex flows. Hills ring
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We use polar coordinates (r, q) and assume
symmetry
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Solution
Further we define constant c
and find solution
Proof with Mathematica
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Appropriate tangent velocity
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Burgers vortex (a viscous vortex with swirl)
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Vorticity
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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)

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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

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Elementary Planar Irrotational FlowsUniform
Stream

• In Cartesian coordinates
• Conversion to cylindrical coordinates can be
  achieved using the transformation

Proof with Mathematica
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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

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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
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Elementary Planar Irrotational FlowsLine
Source/Sink

• If source/sink is moved to (x,y) (a,b)

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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
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Elementary Planar Irrotational FlowsLine Vortex

• If vortex is moved to (x,y) (a,b)

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Elementary Planar Irrotational FlowsDoublet

• A doublet is a combination of a line sink and
  source of equal magnitude
• Source
• Sink

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Elementary Planar Irrotational FlowsDoublet

• Adding ?1 and ?2 together, performing some
  algebra, and taking a?0 gives

K is the doublet strength
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Examples of Irrotational Flows Formed by
Superposition

• Superposition of sink and vortex bathtub vortex

Sink
Vortex
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Examples of Irrotational Flows Formed by
Superposition

• Flow over a circular cylinder Free stream
  doublet
• Assume body is ? 0 (r a) ? K Va2

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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.