Hostname: page-component-848d4c4894-xm8r8 Total loading time: 0 Render date: 2024-07-01T23:51:23.921Z Has data issue: false hasContentIssue false
  • English
  • Français

Direct numerical simulations of two-dimensional chaotic natural convection in a differentially heated cavity of aspect ratio 4

Published online by Cambridge University Press:  26 April 2006

Shihe Xin  and
Patrick Le Quéré
Shihe Xin
Affiliation:
LIMSI-CNRS BP 133, 91403-Orsay Cedex, France
Patrick Le Quéré
Affiliation:
LIMSI-CNRS BP 133, 91403-Orsay Cedex, France
Get access Rights & Permissions [Opens in a new window]

Abstract

Chaotic natural convection in a differentially heated air-filled cavity of aspect ratio 4 with adiabatic horizontal walls is investigated by direct numerical integration of the unsteady two-dimensional equations. Time integration is performed with a spectral algorithm using Chebyshev spatial approximations and a second-order finite-difference time-stepping scheme. Asymptotic solutions have been obtained for three values of the Rayleigh number based on cavity height up to 1010. The time-averaged flow fields show that the flow structure increasingly departs from the well-known laminar one. Large recirculating zones located on the outer edge of the boundary layers form and move upstream with increasing Rayleigh number. The time-dependent solution is made up of travelling waves which run downstream in the boundary layers. The amplitude of these waves grows as they travel downstream and hook-like temperature patterns form at the outer edge of the thermal boundary layer. At the largest Rayleigh number investigated they grow to such a point that they result in the formation of large unsteady eddies that totally disrupt the boundary layers. These eddies throw hot and cold fluid into the upper and lower parts of the core region, resulting in thermally more homogeneous top and bottom regions that squeeze a region of increased stratification near the mid-cavity height. It is also shown that these large unsteady eddies keep the internal waves in the stratified core region excited. These simulations also give access to the second-order statistics such as turbulent kinetic energy, thermal and viscous dissipation, Reynolds stresses and turbulent heat fluxes.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 304 , 10 December 1995 , pp. 87 - 118
Copyright
© 1995 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abrous, A. & Emery, F. 1989 Turbulent free convection in square cavities with mixed boundary conditions. Nat. Heat Transf. Conf. ASME-HTD 107, 117–130.Google Scholar
Allard, F. 1992 Effects of thermal boundary conditions on natural convection in thermally driven cavities. ERCOFTAC-EVROTHERM Workshop, Delft, pp. 234–256. EETI.
Behnia, M. & Le Quere, P. 1992 Direct simulation of turbulent convection in a square differentially heated cavity with spectral methods. 4th Intl Fluid Mech. Conf., Alexandria.
Craik, A. D. D. 1985 Wave Interactions and Fluid Flows. Cambridge University Press.
Fraikin, M. P. Portier, J. J. & Frajkin, C. J. 1980 Application of a K-ε turbulence model to an enclosed buoyancy-driven recirculating flow. 19th ASME-AIChE Heat Nat. Transfer Conf., Orlando, Florida, 80-HT-68.
Fromm, J. E. 1971 Numerical method for computing nonlinear, time dependent, buoyant circulation of air in rooms. IBM J. Res. Dev. 15, 186–196.Google Scholar
George, W. K. & Capp, S. P. 1979 A theory for natural convection turbulent boundary layers next to vertical heated surfaces. Intl J. Heat Mass Transfer 22, 439–485.Google Scholar
Giel, P. W. & Schmidt, F. W. 1990 A comparison of turbulence modeling predictions to experimental measurements for high Rayleigh number natural convection in enclosures. 9th Intl Heat Transfer Conf., Paper 2-NC-03, pp. 175–180.
Grötzbach, G. 1982 Direct simulation of laminar and turbulent Bénard convection. J. Fluid Mech. 119, 27–53.Google Scholar
Haidvogel, D. & Zang, T. 1979 The accurate solution of Poisson's equations by expansion in Chebyshev polynomials. J. Comput. Phys. 30, 167–180.Google Scholar
Haldenwang, P. 1984 Résolution tridimensionnelle des équations de Navier-Stokes par méthodes spectrales Tchebycheff. Thése d'Etat, University de Provence.
Haldenwang, P. 1986 Unsteady numerical simulation by Chebyshev spectral methods of natural convection at high Rayleigh number. ASME HTD 60, 45–51.Google Scholar
Hanjalic, K. 1994 Achievements and limitations in modelling and computation of buoyant turbulent flows and heat transfer, keynote lecture. 10th Intl Heat Transfer ConJ. IChemE Symposium series, vol. 1, pp. 1–18.
Henkes, R. A. W. M. 1990 Turbulent natural convection boundary layers. PhD thesis, Delft University of Technology.
Hussaini, M. Y., Speziale, C. G. & Zang, T. 1989 The Potential and Limitations of Direct and large-Eddy Simulations. Lecture Notes in Physics, vol. 357. Springer.
Ince, N. Betts, P. & Launder, B. 1992 Low Reynolds number modeling of turbulent buoyant cavity flows. ERCOFTAC-EUROTHERM Workshop, Delft, pp. 76–87.
Ivey, N. G. 1984 Experiments on transient natural convection in a cavity. J. Fluid Mech. 34, 389–401.Google Scholar
Jones, P. W. & Launder, B. E. 1972 The prediction of laminarization with a two-equation model of turbulence. Intl J. Heat Mass Transfer 15, 301–314.Google Scholar
Jones, P. W. & Launder, B. E. 1973 The calculation of low- Reynolds-number phenomena with a two-equation model of turbulence. Intl J. Heat Mass Transfer 16, 1119–1130.Google Scholar
Kleiser, L. & Schumann, U. 1980 Treatment of incompressibility and boundary conditions in three-dimensional numerical spectral simulations of plane channel flows. Notes on Numerical Fluid Mechanics, Vol. 2, pp. 165–173.
Le Breton, P. 1991 Etude numérique et expérímentale de la convection naturelle laminaire et turbulente en cavité partiellement occupée d'un milieu poreux. These de Doctorat, University of Bordeaux.
Le Quéré, P. 1987 Etude de la transition à I'instationnarité des écoulements de la convection naturelle en cavité verticale différentiellement chauffeé par méthodes spectrales Chebyshev. Thése d'Etat, University of Poitiers.
Le Quéré, P. 1991 Accurate solutions to the square thermally driven cavity. Computers Fluids 20.
Le Quéré, P. 1994a An improved Chebyshev collocation algorithm for direct simulation of two- dimensional turbulent convection in differentially heated cavities. Finite Elements Anal. Design 16, 271–283.Google Scholar
Le Quéré, P. 1994 6 Onset of unsteadiness, routes to chaos and simulations of chaotic flows in cavities heated from the side : a review of present status, keynote lecture. 10th Intl Heat Trans. Conf. IChemE Symposium Series, vol. 1, pp. 281–296.
Le Quéré, P. & Alziary De Roquefort, T. 1982 Sur une méthode spectrale semi-implicite pour la résolution des équations de Navier-Stokes d'un écoulement bidimensionnel visqueux incom- pressible. C. R. Acad. Sci. Paris 294(11), 941–944.Google Scholar
Le Quéré, P. & Alziary De Roquefort, T. 1985 Computation of natural convection in two- dimensional cavities with Chebyshev polynomials. J. Comput. Phys. 57, 210–228.Google Scholar
Le Quéré, P. & Behnia, M. 1994 From onset of unsteadiness to chaos in a differentially heated square cavity. LIMSI Rep. 94–02. (also submitted to J. Fluid Mech.).Google Scholar
Le Quéré, P. & Penot, F. 1987 Numerical and experimental investigation of the transition to unsteady natural convection of air in a vertical differentially heated cavity. ASME HTD 94, 75–82.Google Scholar
Lighthill, J. 1978 Waves In Fluids Cambridge University Press.
Lipps, F. 1976 Numerical simulation of three-dimensional Benard convection in air. J. Fluid Mech. 75, 113–148.Google Scholar
Markatos, N. C. & Pericleous, K. A. 1984 Laminar and turbulent natural convection in an enclosed cavity. Intl J. heat Mass Transfer 27, 755–772.Google Scholar
Mergui, S., Penot, F. & Tuhault J. L. 1992 Experimental natural convection in an air-filled square cavity at Ra = 1.7 × 109. ERCOFTAC-EUROTHERM Workshop, Delft, pp. 97–108, EETI. 118
Ndame, A. 1992 Etude expérimentale de la convection naturelle en cavité: de 1’etat stationnaire au chaos. Thése de Doctoral, University of Poitiers.
Mobile, E., Sousa, A. C. M. & Barozzi, G. S. 1990 Turbulent buoyant flows in enclosures. 9th Ml Heat Transfer Con/, Paper 5-NC-ll, pp. 543–548.
Ozoe, H., Mouri, A., Hiramitsu, M., Churchill, S. W. & Lior, N. 1986 Numerical calculation of three-dimensional turbulent natural convection in a cubical enclosure using a two-equation model for turbulence. J. Heat Transfer 108, 806–813.Google Scholar
Paolucci, S. 1990 Direct simulation of two-dimensional turbulent natural convection in an enclosed cavity. J. Fluid Mech. 215, 229–262.Google Scholar
Paolucci, S. & Chenoweth, D. R. 1989 Transition to chaos in a differentially heated vertical cavity. J. Fluid Mech. 201, 379–410.Google Scholar
Patterson, J. & Armfield S. W. 1990 Transient features of natural convection in a cavity. J. Fluid Mech. 219, 469–497.Google Scholar
Patterson, J. & Imberger, J. 1980 Unsteady natural convection in a rectangular cavity. J. Fluid Mech. 100, 65–86.Google Scholar
Penot, R, Ndame, A. & Le Quéré, P. 1990 Investigation of the route to turbulence in a vertical differentially heated cavity 9th Ml Heat Transfer Conf. vol. 2, pp. 417–422.
Steinberner, U. & Raineke, H. H. 1978 Turbulent buoyancy convection heat transfer with internal heat sources. 6th Ml Heat Transfer Conf. vol. 2, pp. 305–310.
Stuhmiller, J. H. 1979 Theoretical considerations of turbulent buoyant flows. Heat Transfer and Turb. Buoy. Convection, vol. 1, pp. 3–14.
To, W. M. & Humphrey, J. A. C. 1986 Numerical simulation of buoyant turbulent flow-I and II. Ml. J. Heat Mass Transfer 29, 573–610.Google Scholar
Tuckerman, L. 1989 Divergence-free velocity field in non-periodic geometries. J. Comput. Phys. 80, 403–441.Google Scholar
Uitenborgaard, R. E. & Baron, F. 1989 A proposal: extension of the q2 - ε model for stably stratified flows with transport of internal wave energy. 7th Turbulent Shear Flows Conf, Stanford, USA.
Vahl Davis, G. De & Jones, I. P. 1983 Natural convection in a square cavity: a comparison exercise. Ml J. Numer. Meth. Fluids 3, 227–248.Google Scholar
Vanel, J. M., Peyret, R. & Bontoux, P. 1986 A pseudo-spectral solution of vorticity-stream- function equations using the influence matrix technique. Num. Meth. Fluid Dyn. 2, 463–475.Google Scholar
Xin, S. 1993 Simulations numeriques de convection naturelle turbulente, Thése de Doctoral, University of Paris VI.
Xin, S. & Le Quéré, P. 1992 Direct simulation of lurbulenl nalural convection in a differentially heated cavity of aspect ratio 4 with spectral methods. 1st Eur. Thermal Sciences Conf., Birmingham, vol. 1, pp. 475–482.
Xin, S. & Le Quéré, P. 1994 Numerical simulations of two-dimensional turbulent natural convection in differentially heated cavities of aspect ratios 1 and 4. 1st EUROTHERM ERCOFTAC Workshop on Direct and Large Eddy Simulation, University of Surrey, pp. 423–434.