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8/3/2019 NUST JUNE 08 2011
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Tailoring the TaylorVortices
M. Rafique
NDC, NESCOM
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Hydrodynamic Stabilityof
Taylor Couette/Modified TaylorCouette Flow
Main FeaturesSimple geometry (Two Coaxial Cylinders), easy control of flowparametersVariety of well-separated transitions: Turbulence, Chaos, etc.
Main ApplicationsViscosity meters (rheo-meters)Rotating/centrifugal machinery (Lubrification, Isotopeseparation)Mixing/separation devices (emulsification, extraction), etc.
Historical Mile Stones:Mallock, 1888: Outer Cylinder Rotating, Inner Cylinder FixedCouette, 1890: Outer Cylinder Fixed, Inner Cylinder RotatingTaylor, 1923: Studied theoretically and experimentally bothconfigurations and established the stability criterion for viscousfluids
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Taylor vortex
Twist vortex Wavy vortex Turbulent vortex
Spiral vortexflow
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WaterAspiration
Water + PaperPulp Re2,However Re2>Re1; in fact Re2~2Re1
Given the unstable modes of Jet flowSuperimpose externally on the jet flow, some perturbations/modes of choice whichcan dampen/delay the Instability of the Jet Flow
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2rr00 ==110022..66 mmmm
HH==228877 mmmm
aa==11..9966 mmmm aa==33..9911 mmmm aa==77..8833 mmmm aa==33..9911 mmmm aa==77..8833 mmmm
==7711..7755 mmmm ==7711..7755 mmmm ==7711..7755 mmmm ==3355..8888 mmmm ==3355..8888 mmmm
=0.038 =0.076 =0.153 =0.076 =0.153
Main Topics of InvestigationClassical Taylor Couette Flow usingSmooth Inner Rotating and Smooth OuterFixed CylindersModified Taylor Couette Flow using Wavy
Inner Rotating Cylinders and Fixed OuterSmooth Cylinder Outer CylinderDiameter=128.2mm
Average Gap=12.8mm
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Available Tools to Probe the FlowField
1. Particle Image Velocimetry(PIV)
Lx
y
V=Sy
C=C0
C=0
o
2. Electro-Chemical Wall ShearProbe
C
t
UC
x
VC
y
DC
y
+ + =2
2
3
3
0
2I
)ACnF807.0(D
LS =
2 4
3 6 4 6( ) ( )
K SOK Fe CN e K Fe CN+
3. CFD Modeling4. Mathematical Modeling
La
ser Optics
Camera
o
o
o
o
o
o
oooo
o
Seeding Particles
Laser sheet
R1R2
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Classical Taylor CouetteFlow T=20
Fixed Outer Cyl inder
Rotating Inner Cylinder
T=50
T=67
T=115
T=167
T=134
1
2 3
( )1 2 1 2 1
1
R R R R RT
R
=
Tc=
41
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Experiments withPIV
Simulations withFluent 4.5
z
r
Axialvelocity
Radialvelocity
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BifurcationDiagram
Bifurcation Scenario in SpectralDomain
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Modified Taylor CouetteFlow
T=20
Fixed Outer Cyl inder
Rotating Inner Cylinder
BaseFlow
T=20
T=20
72 , 1.96mm a mm = =
36 , 3.91mm a mm = =
( )2 2 ;m m m mm
R R R R RT R mean radius of inner cylinder
R
=
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T=19.14 T=57.41 T=95.68 T=306.16
Flow
Regim
es
72 , 1.96mm a mm = =
BaseFlow
T=[0,48],Time-
independent
2 /cells T=[48,82],Time-
dependent
4 /cells T=[82,182],Time-
independent
4 /cells T=[182,335],Time-
dependent
4 /cells
T=363.57
T>335,Time-
dependent
2 /cells
T=19.14 T=133.95T=306.16 T=344.43
36 , 3.91mm a mm = =
T=[0,250],
Time-independent
2 /cells
T=[250,335],Time-independent,Asymmetric cells
2 /cells T>335,Time-dependent
2 /cells
Time-independent,
2 /cells
BaseFlow
T>250
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Spe
ctralAspectsofthe
Dynamics
72 , 2mm a mm = =
SignalAmplitu
de
(V)
T
SignalFreq
uency
(Hz)
T
li d li i b i d i
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Not a pronounced asymmetry as inexperiments
Normalized Helicity obtained viaSimulations
T=306.16
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( )1 0 12
cosz
R z a a
= +
23/ 2
0 0Re (1 ) ; ReR
T a a
= =
2
0( ) [ ( , ) ]P P R P r z G z = +
2
2
2
2
2
2
2 22
2 2
( ) ( )0
1[ ]
Re
1[ ]
Re
1
Re
1
rv ru
r z
v v w P vv u v
r z r r r
w w vw wv u w
r z r r
u u Pv u G ur z z
wherer r r z
+ =
+ = +
+ + =
+ = +
= + +
1 1( ) 0; 0; ( )
1 0
at r R z v u w R z
at r u v w
= = = == = = =Radial
BC:0
0
A
B
at z z u v w
at z z u v w
= = = =
= = = =AxialBC:
Scaling:LengthScale:
TimeScale:
1s
t
= VelocityScale:
sV R=
Pressurescale:
( )2
sP R =
Governing EquationsTime-independence, Axi-symmetry
0 1 10 1 1
( ); ; ; ; ( )
s
a a R zL R a a R z
R R R R
= = = = =
Asymmetric Vortices in a Modified Taylor Couette Flow(submitted to TCFD)
=
=
)]z(R1[
)z(Rry
zx
}x,,y{}z,,r{
1
1
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( )
( ) exp( )
( )
( )
n
nn
n
n
u U y
v V y insx
w W y
P P y
=
=
= =>< k knkn WVwv
NumericalScheme:
GoverningEquations in
FourierCoefficients
Finite DifferenceEquations
Spatial discretizationKeller Scheme (2nd order accurate, withflux limiter) Modified Newton
Method
Checks: 1) Integral balance of forces and moments over onewavelength
2) Effect of number of Fourier harmonics
Discretization: 12-25 harmonics41-81 points in radial direction
FourierExpansion
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Case 1:G=0
2 2u v+Contours of
velocity
b1/b2=1
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Test with G=0 and T=325 producessymmetric vortices rather thanasymmetric
With asymmetric perturbations, solution either diverges orconverges to symmetric vortex solution
Hypothesis 1: Asymmetric vortical flowarises from the symmetric flow as a resultof symmetry breaking bifurcation at a
certain critical Taylor number T* (T>250).
Theory of bifurcations implies that the stationary asymmetric solution in a symmetricdomain can be separated from symmetric solution at the singular bifurcation point (DiPrima and Swinney, 1985; Iooss and Chossat, 1994).The separated asymmetric solution is stable and can be realized on one of twobranches with opposite symmetry.
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Redefinition of theproblem
Hypothesis 2: The Navier-Stokes equations have only symmetric stationary solutionsin completely symmetric region. The experimentally observed asymmetric flowstructures are due to the influence of the axial extremities (A and B ends) of thedomain, where the flow periodicity and geometric symmetry are broken.
Assumptions:Flow in the Main region (L>>1) is periodic
Flow in the Buffer zones a and b (a~b~1) is not periodic due to extremities Aand B.
Now, due to the absence of closed extremities from main region, G is no moreequal to zero, in other words, G becomes unknown. Hence, for a well-posedproblem, one needs a closing condition, i.e.; 0drr)z,r(u2Qz
1
R1
==
( , ) ( , ); ( , ) ( , )
( , ) ( , ); ( , ) ( , )
u r z u r z v r z v r z
w r z w r z P r z P r z
l l
l l
= + = +
= + = +
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b1/b2=1.5
2 2u v+Contours of
velocity
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Summary
Unlike Classical Taylor Couette Flow, the Modified Taylor Couette flow exhibits
2D axisymmetric base flow containing counter rotating vortices and exhibitsvariety of different flow transitions than the Classical problem.
Modeling ofAsymmetric Vortex flow in a Perfectly Symmetric Domain istreated and it is shown that the asymmetric flow accompanies with a self-induced axial pressure gradient such that the Net Local Axial Flux remains
zero (conserved).