Wavy Vortex Flow A tale of
chaos, symmetry and serendipity in a steady world
Greg KingUniversity of Warwick (UK)
Collaborators:•Murray Rudman (CSIRO)•George Rowlands (Warwick)•Thanasis Yannacopoulos (Aegean)•Katie Coughlin (LLNL)•Igor Mezic (UCSB)
To understand this lecture you need to know
• Some fluid dynamics• Some Hamiltonian dynamics• Something about phase space• Poincare sections• Need > 2D phase space to get chaos • Symmetry can reduce the dimensionality of phase
space• Some knowledge of diffusion• A “friendly” applied mathematician !!
Phase Space
Dynamical Systems and Phase Space
Dissipative Systems
0 F
11 1
22 1
1
( , , )
( , , )
( , , )
n
n
nn n
dxf x x
dtdx
f x xdt
dxf x x
dt
1( , , )
nf f
div
F
F F
Hamiltonian Systems
0 F
Classical Mechanics and Phase Space
22
20
d xx
dt
2
dxv
dtdv
xdt
0 F
Hamiltonian
Dissipative
22
20
d x dxx
dt dt
2
dxv
dtdv
v xdt
F
Fluid Dynamics and Phase Space
2D incompressible fluid
dxu
dtdy
vdt
0
0
u v
x y
u
3D incompressible fluid
dxu
dtdy
vdtdz
wdt
0
0
u v w
x y z
u
Phase Space
2D
( , )x yu
3D
( , , )
( , , )
x y t
x y z
u
u
4D
( , , , )x y z tu
No chaos here
Symmetries -- can reduce phase space
Poincare Sections(Experimental – i.e., light
sheet)
Eccentric Couette FlowChaiken, Chevray, Tabor and Tan, Proc Roy Soc 1984 ??
Illustrates “Significance” of KAM theory
3D Phase Space
( , , )x y tu
3D Phase Space
( , , )x y zu
Stirring createsdeformed vortex
Fountain et al, JFM 417, 265-301 (2000)
Fountain et al, JFM 417, 265-301 (2000)
Experiment(light sheet)
NumericalParticle Tracking(“light sheet”)
a
b
Taylor-Couette
Radius Ratio:
= a/b
Reynolds Number:
Re = a(b-a)/
Engineering Applications
• Chemical reactors
• Bioreactors
• Blood – Plasma separation
• etc
Reout
Rein
Taylor-Couette regime diagram(Andereck et al)
Some Possible Flows
Taylor vortices
Twisted vortices
Wavy vortices
Spiral vortices
Taylor Vortex Flow
TVF --
– Centrifugal instability of circular Couette flow.
– Periodic cellular structure.
– Three-dimensional, rotationally symmetric:
u = u(r,z)
Flat inflow and outflow boundaries are barriers to inter-vortex transport.
Radius
Z
0
/2
inner cylinder
outercylinder
nestedstreamtubes
Rotational Symmetry3D 2D Phase Space
“Light Sheet”
Wavy Vortex Flow
wavy vortex flowTaylor vortex flow
Rec
The Leaky Transport BarrierWavy vortex flow is a deformation of rotationally symmetric Taylor vortex flow.
Dividing stream surface breaks up => particles can migrate from vortex to vortex
Dividing stream surface
Poincare Sections
IncreaseRe ( , , )r zu u
Flow is steady in co-moving frame
Methods
• Solve Navier-Stokes equations numerically to obtain wavy vortex flow.
• Finite differences (MAC method);
• Pseudo-spectral (P.S. Marcus)
2. Integrate particle path equations (20,000 particles) in a frame rotating with the wave (4th order Runge-Kutta).
, / , dr d dz
u v r wdt dt dt
( , , ) ( , , )r z u v w u
Wavy Vortex Flow
Poincare Section near onset of waves
r 1
2
1
2
Z
0 2
inner cylinder
outer cylinder
6 vortices
1
2
3
4
5
6
At larger Reynolds numbers(Rudman, Metcalfe, Graham: 1998)
)(lim
2
)()(
20
tDD
t
ztztD
zt
z
z
Effective Diffusion CoefficientCharacterize the migration of particles from vortex to vortex
Taylor vortices Wavy vortices
Rudman, AIChE J 44 (1998) 1015-26.
(dimensionless)Initialization:Uniformly distribute20,000 particles
Dz
Size of mixing region
(dimensionless)
Dz
An Eulerian ApproachSymmetry Measures
Theoretical Fact
A three dimensional phase space is necessary for chaotic trajectories.
The Idea (Mezic):
Deviation from certain continuous symmetries can be used to measure the local deviation from 2D
For Wavy Vortex Flow
rotational symmetry
and dynamical symmetry :
• If either is zero, then flow is locally integrable, so as a diagnostic we consider the product
222wvu
)(
x
Reux 2)(
( ) ( ) ( )D x x x
Dynamical Symmetry
Steady incompressible Navier-Stokes equations in the form
Equation of motion for B
2
2
1
2 is the vorticity, is the Bernoulli function
0
pB
B
uω u
u ω u
2
2
0dB B
Bdt t
u u u ω u
u u
B is a constant of the motion if 0 2or 0 u
155 162 324 486 648Reynolds Number
155 162 324 486 648Reynolds Number
155 162 324 486 648Reynolds Number Looks interesting, but
correlation does not look strong !
AveragedAveraged Symmetry Measures Symmetry Measures
d1
d1
d1
VV
VV
VV
D
andand partialpartial averagesaverages
)(
)(
),(
z
r
rd
rdzr
D
Dz
Size of chaotic region
King, Rudman, Rowlands and YannacopoulosPhysics of Fluids 2000
Serendipity !
1.15zD
Effect of Radius Ratio (Mind the Gap)
5 10 150
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
Re/Rec
Dz ,
c
= 0.875
cD
z
5 10 150
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
= 0.830
5 10 150
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
= 0.784
5 10 150
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
= 0.700
5 10 151
1.5
2
2.5
3
3.5
Re/Rec
Dz/
5 10 151
1.5
2
2.5
3
3.5
5 10 151
1.5
2
2.5
3
3.5
5 10 151
1.5
2
2.5
3
3.5
Dz/
Re/Rec
Effect of Flow State: Axial wavelength
m: Number of waves
2 4 6 82
4
6
8
10
12
14
16x 10
-3
m = 4
/d = 2.33/d = 3.0/d = 3.5
2 4 6 82
4
6
8
10
12
14
16x 10
-3
m = 5
2 4 6 82
4
6
8
10
12
14
16x 10
-3
m = 6
2 4 6 82
4
6
8
10
12
14
16x 10
-3
/d = 2.33
Re/Rec
m = 4m = 5m = 6
2 4 6 82
4
6
8
10
12
14
16x 10
-3
/d = 3.0
2 4 6 82
4
6
8
10
12
14
16x 10
-3
/d = 3.5
Re/Rec
2 4 6 80
0.005
0.01
0.015
0.02
0.025
0.03
m = 4
/d = 2.33/d = 2.6/d = 3.0/d = 3.5
2 4 6 80
0.005
0.01
0.015
0.02
0.025
0.03
m = 5
2 4 6 80
0.005
0.01
0.015
0.02
0.025
0.03
m = 6
2 4 6 80
0.005
0.01
0.015
0.02
0.025
0.03
Re/Rec
Dz
/d = 2.33
m = 4m = 5m = 6
2 4 6 80
0.005
0.01
0.015
0.02
0.025
0.03
/d = 3.0
2 4 6 80
0.005
0.01
0.015
0.02
0.025
0.03
/d = 3.5
Effect of Flow State
Dz
Re/Rec
Summary
• Dz is highly correlated with <><>
• The correlation is not perfect.
• The symmetry arguments are general
• Yannacopoulos et al (Phys Fluids 14 2002) show that Melnikov function,
M ~ <><>.
222wvu
)(
x
Reux 2)(
Is it good for anything else?
2D Rotating Annulus u(r,z,t)
Richard Keane’s results (see poster)Symmetry measure:
FSLELog(FSLE)
Log(<|d/dt|>)
Prandtl-Batchelor Flows(Batchelor, JFM 1, 177 (1956)
Steady Navier-Stokes equations in the form
Integrating N-S equation around a closed streamline s yields
21
2 is the vorticity, is the Bernoulli function
0
pB
B
uω u
u ω ω
( ) 0 is parallel to a streamline
( ) 0 since streamline is closed
( ) 0 Integral constraint
d
B d
d
s
s
s
u ω s u
s
ω s
Break-up of Closed StreamlinesYannacopoulos et al, Phys Fluids 14 2002
(see also Mezic JFM 2001)
Expand0 1
0 1
0 1B B B
u u u
ω ω ω2
0 ( )O ω
0 1 1 0( ) 0 existence criterionB dt dt b b
a a
u ω u
This is the Melnikov function
