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Laboratoire de Mcanique, Physique et Gosciences Spiral vortex flow in annular geometry with a radial temperature gradient V. Lepiller 1 , F. Dumouchel, A. Prigent & I. Mutabazi 1 valerie.lepiller@univ-lehavre.fr Laboratoire de Mcanique,

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SLIDE 1

Spiral vortex flow in annular geometry with a radial temperature gradient

  • V. Lepiller1, F. Dumouchel, A. Prigent & I. Mutabazi

1 valerie.lepiller@univ-lehavre.fr

Laboratoire de Mécanique, Physique et Géosciences Université du Havre 25 rue Philippe Lebon 76 058 Le Havre cedex

EUROMECH 2004

Laboratoire de Mécanique, Physique et Géosciences

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SLIDE 2

Outline

  • Introduction
  • Previous works
  • Experimental setup
  • Results :
  • 1. Fixed vertical cylindrical annulus with a radial temperature

gradient

  • 2. The Couette-Taylor system with a radial temperature

gradient

  • Conclusion
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SLIDE 3

T1 T2 T1 > T2 T1 T2 T1 = T2

Introduction

The circular Couette flow

Transverse

  • scillatory vortices

qc= 2.76, ωc = 15.25, Grc =7974 for Pr = 7 Longitudinal stationary vortices (Taylor vortices) with qc = 3.12, ωc = 0 Azimuthal velocity V(r) Axial velocity W(r) Temperature T1 T2

A radial temperature gradient imposed

  • n the annular cylindrical geometry
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SLIDE 4

Motivation

  • Investigation of coupled Couette flow with

a radial temperature gradient : coupling of buoyancy and centrifugal mechanisms.

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SLIDE 5

Introduction

Many applications :

  • industrial (chemical, automotive, nuclear)

Cooling of rotating machinery like electrical motors and turbines Nuclear reactors isolation Clinical blood oxygenerators

  • environmental

Oceanic and atmospheric circulation

K.S. Ball, B. Farouk, J. Fluid Mech. 197 (1988)

  • M. Auer, F. Busse & E. Gangler, Eur. J. Mechanics B/ Fluids 15, 605(1996)

K.M. Becker & J. Kaye, Trans. ASME-J. Heat Transfer 84, 97(1962)

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SLIDE 6

Previous works Many studies (experimental, theoretical, numerical) in the Couette-Taylor system

  • G.I. Taylor, Phil. Trans. Roy. Soc. London. Ser. A 223, 289 (1923).
  • C.D. Andereck, S.S. Liu & H.L. Swinney, J.Fluid Mech. 164, 155 (1986).
  • P. Chossat, G. Iooss, « The Couette-Taylor Problem » Springer-Verlag, Berlin

(1994).

  • Ch. Egbers & G. Pfister, Physics of Rotating Fluids, Springer-Verlag (2000).
  • A. Goharzadeh & I. Mutabazi, Eur. Phys. J. B 19, 157-162 (2001).
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SLIDE 7
  • Many numerical and theoretical studies for a fluid

confined between two cylinders at rest, with a radial temperature gradient

  • I.G. Choi & S.A. Korpela, J.Fluid Mech. 99 (4), 725, (1980).
  • P. Le Quéré & J. Pécheux, J. Fluid Mech. 206, 517 (1989)
  • J. Pécheux, P. Le Quéré & F. Abcha, Phys. Fluids 6 (10), 3247 (1994).
  • A. Bahloul, I. Mutabazi & A. Ambari, Eur. Phys. J. AP 9, 253 (2000)

Previous works

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SLIDE 8

Previous works Few studies in the Couette -Taylor system with a radial temperature gradient :

Experiments

  • K.M. Becker & J. Kaye, Trans. ASME-J. Heat Transfer 84, 97 (1962).
  • H.A. Snyder & S.K.F. Karlsson, Phys. Fluids 7 (10), 1696 (1964).
  • K.S. Ball, B. Farouk & V.C. Dixit, J. Heat Mass Transfer 32 (8), 1517 (1989).
  • K.S. Ball & B. Farouk, Phys. Fluids A 1 (9), 1502 (1989).

→ Flow Visualization → Quite few quantitative data : diagram of primary bifurcation, wavenumber → Need for a more systematic investigation of different flow regimes when control parameters are changed.

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SLIDE 9

Theoretical and Numerical studies

  • J. Wallowit, S. Tsao & R.C. DiPrima, Trans. ASME-J. Appl. Mech. 86,

595(1964).

  • K.S. Ball & B. Farouk, J. Fluid Mech. 197, 479 (1988).
  • M.E. Ali & P.D. Weidman, J. Fluid. Mech. 220, 53 (1990).
  • I. Mutabazi & A. Bahloul, Theor. Comp. Fluid Dyn.16, 79 (2002).

Previous works

→ Main assumption : axisymmetric stationary or oscillatory modes, some terms were neglected, symmetry δT → - δT. → Recent work (M-B): relaxes the last assumption and takes into account the centrifugal buoyancy term.

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SLIDE 10

Experimental setup

g r

  • Gap width :

d = b – a = 0.5 cm

  • Temperature difference imposed to

the working liquid : ∆T = 0.61*(T1 –T2) Control parameters :

  • Geometrical parameters:

Radius ratio : η = a / b = 0.8 Aspect ratio : Γ = H / d = 114

  • Physical parameters:

Prandtl number : Pr = τκ/τν = ν /κ Reynolds number : Re = τν/τa = Ωad/ν Grashof number : Gr = Wa d /ν avec Wa = gα∆Td2/ν

T2

b = 2.5 cm c = 5 cm a =2 cm

T1 z Working fluid : demineralized water He-Ne laser H = 57 cm

Ω

Linear camera Water circulation

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SLIDE 11

Results : Fixed vertical cylindrical annulus with a radial temperature gradient Space-time diagrams for a pattern with T2 = 27°C a) T1= 45°C, b) T1 = 47°C

b) a)

T1 T2 Large convection cell

T1 > T2

∆T = 11°C ∆T = 12.2°C

∆T > ∆Tc

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SLIDE 12

T1 = 48°C, T2 = 27°C ∆T = 12.9°C

Chaotic pattern Turbulent pattern

T1 = 56°C, T2 = 27°C ∆T = 17.7°C

Results : Fixed vertical cylindrical annulus with a radial temperature gradient

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SLIDE 13

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 5000 10000 15000 20000

Gr L= l / H

♦ T2 = 20°C ■ T2 = 23°C ▲ T2 = 25°C Χ T2 = 30°C Ж T2 = 33°C ∙ T2 = 35°C + T2 = 37°C ▬ T2 = 40°C

( )

5 . c c

Gr Gr a L L − × + ≈

The pattern length l : where a = a (Grc,T2,Lc)

Results : Fixed vertical cylindrical annulus with a radial temperature gradient

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SLIDE 14

q = 2.82 ± 0.15

The stability curve

Marginal stability curve for Pr = 7 and η = 0.8 [Bahloul]: qc= 2.76, ωc = 15.25, Grc =7974 Experimental values of wavenumber

Results : Fixed vertical cylindrical annulus with a radial temperature gradient

q

Gr (10

3)

1 2 3 4 2 4 6 8 10 12 14

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SLIDE 15

0,5 1 1,5 2 2,5 3 3,5 4 2000 4000 6000 8000 10000 12000 14000

Gr f a

19 20 23 25 27 30 33 35 37 40

⇒ Increase of the frequency with the Grashof number

T2 Variation of the pattern frequency

Results : Fixed vertical cylindrical annulus with a radial temperature gradient

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SLIDE 16

a) b) c) d) e) z Growth of spiral vortex flow for T1 = 27°C, T2 = 30°C and a) Re ~ 33, b) Re ~ 34, c) Re ~ 36, d) Re ~ 40, e) Re ~ 41,5

Results : Couette-Taylor system with a radial temperature gradient Small ∆T = 1.83°C

Re

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SLIDE 17

Results : Couette-Taylor system with a radial temperature gradient

Space-time diagrams for a pattern when T1 = 27°C, T2 = 30°C, a) Re ~ 33 b) Re ~ 49

a) b)

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SLIDE 18

Results : Couette-Taylor system with a radial temperature gradient Moderate ∆T = 3.06°C

Space-time diagrams for a pattern when T1 = 25°C, T2 = 30°C, a) Re = 24.5 b) Re = 25.2 a) b)

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SLIDE 19

Results : Couette-Taylor system with a radial temperature gradient Critical parameters

20 40 60 80 100 120

  • 2000
  • 1500
  • 1000
  • 500

500 1000 1500 2000

Gr Re c

0,5 1 1,5 2 2,5 3 3,5

  • 2000
  • 1500
  • 1000
  • 500

500 1000 1500 2000

Gr q c

The radial heating destabilizes the flow. The radial heating increases the vortex size.

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SLIDE 20

Results : Couette-Taylor system with a radial temperature gradient

  • Following all the spiral vortex flow

mirrors z 2D camera

  • H. Litschke & K.G. Roesner,
  • Exp. Fluids 24, (1998)
  • A. Prigent & O. Dauchot, Phys.

Fluids 12 (10), 2688 (2000)

Visualization of spiral vortex flow for T1 = 28°C, T2 = 30°C and Re = 68

m = 4

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SLIDE 21

∆T < 0, Ω > 0 ∆T > 0, Ω > 0 Results : Couette-Taylor system with a radial temperature gradient Inclination and propagation sense of vortex

Space-time diagrams and pictures of a pattern for T2 = 30°C, a) T1 = 27°C, Re ~ 49, b) T1 = 32°C, Re ~ 59

a) Gr.Re < 0 b) Gr.Re > 0

z

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SLIDE 22

∆T < 0, Ω < 0 Results : Couette-Taylor system with a radial temperature gradient ∆T < 0, Ω > 0

z

Space-time diagrams and pictures of a pattern for a) T1 = 27°C, T2 = 28°C, Re = 91, b) T1 = 27°C, T2 = 30°C, Re = 49

a) Gr.Re > 0 b) Gr.Re < 0

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SLIDE 23

Conclusion

  • We have performed an experimental investigation of

the flow in the cylindrical annulus with a radial temperature gradient.

  • The radial temperature gradient induces spiral vortex

flow.

  • When the cylinders are fixed, the pattern size

increases with the control parameter and a chaotic regime occurs near the onset.

  • The discrepancy between experimental and linear

stability critical parameters (Grc,qc, fc) is due to the non-Boussinesq effects in experiment.

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SLIDE 24

Conclusion

  • In case of rotating inner cylinder, the pattern properties

depend on the radial temperature gradient. We found that Rec and qc decrease with ∆T.

  • For small ∆T, the spiral pattern occurs near the bottom

and increases in size with Re.

  • For moderate ∆T, the spiral vortex flow occurs in the

middle of the system before invading the whole system.

  • The spiral vortex inclination (helicity) depends on the

sign of Gr.Re ~ ∆T. Ω while the sense of propagation depends only on the Gr.

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SLIDE 25

Forthcoming work

  • PIV : measure of the velocity and vorticity

fields

  • TLC (Thermochromic liquid crystals) :

measure of temperature field in the gap

  • Stability analysis without the Boussinesq

approximation.