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Flow transitions of a low-Prandtl-number fluid in an inclined 3D cavity. (English) Zbl 1034.76019

This paper presents a numerical solution of natural convection of low-Prandtl-number fluid (Pr \( = 0.025\)) in two-dimensional (\(H \times L = 1 \times 4\)) and three-dimensional (\(1 \times 6 \times 4\)) side-heated cavities. Both the cavities are inclined at \(80^\circ\) with respect to the vertical direction and are heated along the side of length \(L = 4H\). This set of parameters is chosen to promote the interaction between longitudinal and transversal multicellular modes. Also, this inclination angle is chosen from a previous linear stability analysis of the basic (plane-parallel) flow that predicts the same critical Rayleigh number Ra for longitudinal oscillatory and stationary transversal modes. The unsteady governing continuity, Navier-Stokes and energy equations in non-dimensional form is solved numerically using a Chebyshev-collocation pseudospectral method in vorticity-stream function variables. In all the cases studied a grid of \(31 \times 81\) collocation points is found to be enough to obtain accuracies of better than about 2%. It is found that in both two-dimensional and three-dimensional enclosures the first transition gradually leads to a transversal stationary centered shear roll. In the two-dimensional geometry the flow becomes time-dependent and multicellular (3 rolls) at the onset of Hopf bifurcation, followed by subsequent period-doubling. On the other hand, in the case of three-dimensional enclosure, the onset of oscillation is due to a fully three-dimensional standing wave composed of three counter-rotating longitudinal rolls. Further, the authors show that the inclination configuration enables a new dimension in the parameter space, and therefore makes more feasible the study of several types of instabilities and their corresponding interactions for suitable choices of operating parameters.

MSC:

76E06 Convection in hydrodynamic stability
76R10 Free convection
76M22 Spectral methods applied to problems in fluid mechanics
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[1] Hurle, D. T.J.; Jackman, E.; Johnson, C. P., Convective temperature oscillations in molten gallium, J. Fluid. Mech., 64, 565 (1974)
[2] Markham, B. L.; Rosemberger, F., Diffusive-convective vapor transport across horizontal and inclined rectangular enclosures, J. Cryst. Growth, 67, 241 (1984)
[3] Delgado-Buscalioni, R.; Crespo del Arco, E., Flow and heat transfer regimes in inclined differentially heated cavities, Int. J. Heat Mass Tran., 44, 1947-1962 (2001) · Zbl 1107.76389
[4] Delgado-Buscalioni R., PhD thesis, Universidad de Educación a Distancia, Madrid, 1999; Delgado-Buscalioni R., PhD thesis, Universidad de Educación a Distancia, Madrid, 1999
[5] Delgado-Buscalioni, R.; Crespo del Arco, E.; Bontoux, P.; Ouazzani, J., Convection and instabilities in differentially heated inclined shallow rectangular boxes, C. R. Acad. Sci. II B, 36, 711-718 (1998) · Zbl 1059.76508
[6] Delgado-Buscalioni, R.; Crespo del Arco, E., Stability of thermally driven shear flows in long inclined cavities with end-to-end temperature gradient, Int. J. Heat Mass Tran., 42, 2811-2822 (1999) · Zbl 1042.76518
[7] Braunsfurth, M.; Skeldon, A. C.; Juel, A.; Mullin, T.; Riley, D. S., Free convection in liquid gallium, J. Fluid Mech., 342, 295-314 (1997) · Zbl 0900.76602
[8] Braunsfurth, M.; Mullin, T., An experimental study of oscillatory convection in liquid gallium, J. Fluid Mech., 327, 199-219 (1996)
[9] Henry, D.; Buffat, M., Two and three dimensional numerical simulations of the transition to oscillatory convection in low-Prandtl-number fluids, J. Fluid Mech., 374, 145-171 (1998) · Zbl 0933.76085
[10] Hart, J. E., Stability of thin non-rotating Hadley circulations, J. Atmos. Sci., 29, 687 (1972)
[11] Hart, J. E., A note on the stability of low-Prandtl-number Hadley circulations, J. Fluid Mech., 132, 271 (1983) · Zbl 0528.76049
[12] Kuo, H. P.; Korpela, S. A., Stability and finite amplitude natural convection in a shallow enclosure with insulated top and bottom and heated from a side, Phys. Fluids, 31, 1, 33 (1988) · Zbl 0641.76031
[13] Laure, P., Study on convective in a rectangular enclosure with horizontal gradient of temperature, J. Méch. Théor. App., 6, 351-382 (1987) · Zbl 0616.76051
[14] Gill, A. E., A theory of thermal oscillations in liquid metals, J. Fluid. Mech., 65, 209 (1974) · Zbl 0282.76044
[15] Hung, M. C.; Andereck, C. D., Transitions in convection driven by a horizontal temperature gradient, Phys. Lett. A, 132, 253-258 (1988)
[16] Pratte, J. M.; Hart, J. E., Endwall driven, low Prandtl number convection in a shallow rectangular enclosure, J. Cryst. Growth, 102, 54-68 (1990)
[17] Hart, J. E.; Pratte, J. M., A laboratory study of oscillations in differentially heated layers of mercury, (Roux, B., Numerical Simulation of Oscillatory Convection in low-Pr Fluids. Numerical Simulation of Oscillatory Convection in low-Pr Fluids, Notes on Numerical Fluid Mechanics, 27 (1990), Vieweg), 338-343
[18] Davoust, L.; Moreau, R.; Bolcato, R., Control by a magnetic field of the instability of a Hadley circulation in a low-Prandtl-number fluid, Eur. J. Mech. B-Fluids, 18, 621-634 (1999) · Zbl 0938.76501
[19] Hung, M. C.; Andereck, C. D., Subharmonic transitions in convection in a moderately shallow cavity, (Roux, B., Numerical Simulation of Oscillatory Convection in low-Pr Fluids. Numerical Simulation of Oscillatory Convection in low-Pr Fluids, Notes on Numerical Fluid Mechanics, 27 (1990), Vieweg), 338-343
[20] McKell, K. E.; Broomhead, D. S.; Hones, R.; Hurle, D. T.J., Torus Doubling in convecting molten gallium, Europhys. Lett., 12, 6, 513-518 (1990)
[21] Wang, T.; Korpela, S. A., Secondary instabilities of convection in a shallow enclosure, J. Fluid Mech., 234, 147-170 (1992) · Zbl 0745.76018
[22] Pulicani, J. P.; Crespo del Arco, E.; Randriampianina, A.; Bontoux, P.; Peyret, R., Spectral simulations of oscillatory convection at low Prandtl number, Int. J. Numer. Meth. Fl., 10, 481-517 (1990) · Zbl 0692.76077
[23] Skeldon, A. C.; Riley, D. S.; Cliffe, K. A., Convection in a low Prandtl number fluid, J. Crystal Growth, 162, 96-106 (1996)
[24] Afrid, M.; Zebib, A., Oscillatory three dimensional convection in rectangular cavities and enclosures, Phys. Fluids A, 8, 1318-1327 (1990)
[25] Vanel, J. M.; Peyret, R.; Bontoux, P., A pseudospectral solution of vorticity-stream function equations using the influence matrix technique, (Morton, K. W.; Baines, M. J., Numerical Methods for Fluid Dynamics II (1986), Clarendon Press: Clarendon Press Oxford), 463-475 · Zbl 0606.76030
[26] Winters, K. H., Oscillatory convection in liquid metals in a horizontal temperature gradient, Int. J. Numer. Meth. Eng., 25, 401-414 (1989) · Zbl 0668.76111
[27] Gelfgat, A. Yu.; Tanasawa, I., Numerical analysis of oscillatory instability of buoyancy convection with the galerkin spectral method, Numer. Heat Tr. A-Appl., 25, 627-648 (1994)
[28] Adaptive Research, CFD2000-Theoretical background, Pacific Sierra Corp., 1997; Adaptive Research, CFD2000-Theoretical background, Pacific Sierra Corp., 1997
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