Spiral vortex flow in annular geometry with a radial temperature - PowerPoint PPT Presentation

Laboratoire de Mécanique, Physique et Géosciences 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 Mécanique, Physique et Géosciences Université du Havre 25 rue Philippe Lebon 76 058 Le Havre cedex EUROMECH 2004

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

Introduction A radial temperature gradient imposed The circular Couette flow on the annular cylindrical geometry T 1 T 2 0 Azimuthal velocity V ( r ) Axial velocity W ( r ) Temperature Longitudinal Transverse T 1 T 2 T 1 T 2 stationary vortices oscillatory vortices ( Taylor vortices ) q c = 2.76, ω c = 15.25, with q c = 3.12, ω c = 0 Gr c =7974 for Pr = 7 T 1 = T 2 T 1 > T 2

Motivation • Investigation of coupled Couette flow with a radial temperature gradient : coupling of buoyancy and centrifugal mechanisms.

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.M. Becker & J. Kaye, Trans. ASME-J. Heat Transfer 84 , 97(1962) K.S. Ball, B. Farouk , J. Fluid Mech. 197 (1988) M. Auer, F. Busse & E. Gangler, Eur. J. Mechanics B/ Fluids 15 , 605(1996)

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).

Previous works � 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 � 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.

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

Experimental setup b = 2.5 cm z Ω Working fluid : demineralized water a =2 cm T 1 Water circulation c = 5 cm • Gap width : T 2 r d = b – a = 0.5 cm g He-Ne laser • Temperature difference imposed to the working liquid : ∆ T = 0.61*( T 1 – T 2 ) H = 57 cm Control parameters : • Geometrical parameters: Radius ratio : η = a / b = 0.8 Aspect ratio : Γ = H / d = 114 • Physical parameters: Prandtl number : Pr = τ κ / τ ν = ν / κ Reynolds number : Re = τ ν / τ a = Ω ad / ν Grashof number : Gr = W a d / ν avec W a = g α∆ Td 2 / ν Linear camera

Results : Fixed vertical cylindrical annulus with a radial temperature gradient ∆ T > ∆ T c Large convection cell T 1 T 2 T 1 > T 2 a) b) ∆ T = 12.2°C ∆ T = 11°C Space-time diagrams for a pattern with T 2 = 27°C a) T 1 = 45°C, b) T 1 = 47°C

Results : Fixed vertical cylindrical annulus with a radial temperature gradient T 1 = 48°C, T 2 = 27°C T 1 = 56°C, T 2 = 27°C ∆ T = 12.9°C ∆ T = 17.7°C Chaotic pattern Turbulent pattern

Results : Fixed vertical cylindrical annulus with a radial temperature gradient � The pattern length l : 0,9 L= l / H 0,8 ♦ T 2 = 20°C 0,7 ■ T 2 = 23°C 0,6 ▲ T 2 = 25°C 0,5 Χ T 2 = 30°C Ж T 2 = 33°C 0,4 ∙ T 2 = 35°C 0,3 + T 2 = 37°C ▬ T 2 = 40°C 0,2 0,1 Gr 0 0 5000 10000 15000 20000 0 . 5 ( ) L L a Gr Gr ≈ + × − where a = a ( Gr c , T 2 , L c ) c c

Results : Fixed vertical cylindrical annulus with a radial temperature gradient � The stability curve 3 ) Marginal stability curve for Pr = Gr (10 14 7 and η = 0.8 [Bahloul]: 12 q c = 2.76, ω c = 15.25, Gr c =7974 10 Experimental values of 8 wavenumber 6 q = 2.82 ± 0.15 4 2 0 0 1 2 3 4 q

Results : Fixed vertical cylindrical annulus with a radial temperature gradient � Variation of the pattern frequency 4 f a T 2 3,5 19 20 3 23 2,5 25 27 2 30 33 1,5 35 1 37 40 0,5 Gr 0 0 2000 4000 6000 8000 10000 12000 14000 ⇒ Increase of the frequency with the Grashof number

Results : Couette-Taylor system with a radial temperature gradient � Small ∆ T = 1.83°C Re z e) c) a) b) d) Growth of spiral vortex flow for T 1 = 27°C, T 2 = 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 a) b) Space-time diagrams for a pattern when T 1 = 27°C, T 2 = 30°C, a) Re ~ 33 b) Re ~ 49

Results : Couette-Taylor system with a radial temperature gradient � Moderate ∆ T = 3.06°C b) a) Space-time diagrams for a pattern when T 1 = 25°C, T 2 = 30°C, a) Re = 24.5 b) Re = 25.2

Results : Couette-Taylor system with a radial temperature gradient � Critical parameters 3,5 120 q c Re c 3 100 2,5 80 2 60 1,5 40 1 20 0,5 Gr Gr 0 0 -2000 -1500 -1000 -500 0 500 1000 1500 2000 -2000 -1500 -1000 -500 0 500 1000 1500 2000 The radial heating destabilizes The radial heating increases the flow. the vortex size.

Results : Couette-Taylor system with a radial temperature gradient • Following all the spiral vortex flow z m = 4 mirrors H. Litschke & K.G. Roesner, Exp. Fluids 24 , (1998) Visualization of spiral vortex A. Prigent & O. Dauchot, Phys. flow for T 1 = 28°C, T 2 = 30°C 2D camera Fluids 12 (10), 2688 (2000) and Re = 68

Results : Couette-Taylor system with a radial temperature gradient � Inclination and propagation sense of vortex ∆ T < 0, Ω > 0 ∆ T > 0, Ω > 0 z b) Gr . Re > 0 a) Gr . Re < 0 Space-time diagrams and pictures of a pattern for T 2 = 30°C, a) T 1 = 27°C, Re ~ 49, b) T 1 = 32°C, Re ~ 59

Results : Couette-Taylor system with a radial temperature gradient ∆ T < 0, Ω < 0 ∆ T < 0, Ω > 0 z b) Gr . Re < 0 a) Gr . Re > 0 Space-time diagrams and pictures of a pattern for a) T 1 = 27°C, T 2 = 28°C, Re = 91 , b) T 1 = 27°C, T 2 = 30°C , Re = 49

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 (Gr c ,q c , f c ) is due to the non-Boussinesq effects in experiment.

Conclusion • In case of rotating inner cylinder, the pattern properties depend on the radial temperature gradient. We found that Re c and q c 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.

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.