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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).