Wavy Vortex Flow A tale of chaos, symmetry and serendipity in a steady world Greg King University of Warwick (UK) Collaborators: •Murray Rudman (CSIRO) •George Rowlands (Warwick) •Thanasis Yannacopoulos (Aegean) •Katie Coughlin (LLNL) •Igor Mezic (UCSB)
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Wavy Vortex Flow A tale of chaos, symmetry and serendipity in a steady world Greg King University of Warwick (UK) Collaborators: Murray Rudman (CSIRO)
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• 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