Hostname: page-component-5c6d5d7d68-pkt8n Total loading time: 0 Render date: 2024-08-08T07:18:02.009Z Has data issue: false hasContentIssue false
  • English
  • Français

Natural convection along a heated vertical plate immersed in a nonlinearly stratified medium: application to liquefied gas storage

Published online by Cambridge University Press:  24 September 2007

M. FORESTIER  and
P. HALDENWANG
M. FORESTIER
Affiliation:
Modélisation et Simulation Numérique en Mécanique et Génie des Procédés (MSNM-GP), UMR-CNRS 6181, Universités d'Aix-Marseille, EGIM/ECM IMT/La Jetée, Technopôle de Château Gombert, 38, rue Frédéric Joliot-Curie, 13451 Marseille Cedex 20, France
P. HALDENWANG
Affiliation:
Modélisation et Simulation Numérique en Mécanique et Génie des Procédés (MSNM-GP), UMR-CNRS 6181, Universités d'Aix-Marseille, EGIM/ECM IMT/La Jetée, Technopôle de Château Gombert, 38, rue Frédéric Joliot-Curie, 13451 Marseille Cedex 20, France
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider free convection driven by a heated vertical plate immersed in a nonlinearly stratified medium. The plate supplies a uniform horizontal heat flux to a fluid, the bulk of which has a stable stratification, characterized by a non-uniform vertical temperature gradient. This gradient is assumed to have a typical length scale of variation, denoted Z0, while Ψ0 is the order of magnitude of the related heat flux that crosses the medium vertically.

We derive an analytic solution to the Boussinesq equations that extends the classical solution of Prandtl to the case of nonlinearly stratified media. This novel solution is asymptotically valid in the regime RaS ≫ 1, where RaS denotes the Rayleigh number of nonlinear stratification, based on Z0, Ψ0, and the physical properties of the medium.

We then apply the new theory to the natural convection affecting the vapour phase in a liquefied pure gas tank (e.g. the cryogenic storage of hydrogen). It is assumed that the cylindrical storage tank is subject to a constant uniform heat flux on its lateral and top walls. We are interested in the vapour motion above a residual layer of liquid in equilibrium with the vapour. High-precision axisymmetric numerical computations show that the flow remains steady for a large range of parameters, and that a bulk stratification characterized by a quadratic temperature profile is undoubtedly present. The application of the theory permits a comparison of the numerical and analytic results, showing that the theory satisfactorily predicts the primary dynamical and thermal properties of the storage tank.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 588 , 10 October 2007 , pp. 217 - 241
Copyright
Copyright © Cambridge University Press 2007

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

REFERENCES

Baines, W. D. & Turner, J. S. 1969 Turbulent buoyant convection from a source in a confined region. J. Fluid Mech. 37, 51–80.Google Scholar
Canuto, C., Hussaini, Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.Google Scholar
Forestier, M. Y., Pasquetti, R., Peyret, R. & Sabbah, C. 2000 Spatial development of wakes using a spectral multi-domain method. Appl. Numer. Math. 33, 207–216.Google Scholar
Garba, A. & Haldenwang, P. 2003 Study of preconditioners for collocation Chebyshev approximation of the 2D and 3D Generalized Stokes Problem. J. Comput. Phys. 191, 282– 304.Google Scholar
Gill, A. E. 1966 The boundary-layer regime for convection in a rectangular cavity. J. Fluid Mech. 26, 551–537.Google Scholar
Gill, A. E. & Davey, A. 1969 Instabilities of buoyancy driven system. J. Fluid Mech. 35, 775–798.Google Scholar
Gottlieb, D. & Orszag, S. A. 1977 Numerical Analysis of Spectral Methods: Theory and Applications. SIAM-CBMS, Philadelphia.Google Scholar
Haldenwang, P. 1986 Unsteady numerical simulation by Chebyshev spectral methods of natural convection at high Rayleigh number. In Significant Questions in Buoyancy Affected Enclosure or Cavity Flow (ed. Humphrey, J. A. C. et al. ) ASME/HTD, vol. 60, pp. 45–51.Google Scholar
Haldenwang, P., Labrosse, G., Abboudi, S. & Deville, M. 1984 Chebyshev 3D spectral and 2D pseudospectral solvers for the Helmholtz equation. J. Comput. Phys. 55, 115–128.Google Scholar
Henkes, R. A. W. M. & Hoogendoorn, C. J. 1989 Laminar natural convection boundary-layer flow along a heated vertical plate in stratified environment. Intl J. Heat Mass Transfer 32, 147–155.Google Scholar
Hunt, G. R., Cooper, P. & Linden, P. F. 2001 Thermal stratification produced by plumes and jets in enclosed spaces. Building Environ. 36, 871–882.CrossRefGoogle Scholar
Jaluria, Y. & Gebhart, B. 1974 Stability and transition of buoyancy induced flows in a stratified medium. J. Fluid Mech. 66, 593–612.Google Scholar
Krizhevsky, L., Cohen, J. & Tanny, J. 1996 Convective and absolutes instabilities of a buoyancy induced flow in a thermally stratified medium. Phys. Fluids 8, 971–977.Google Scholar
Kulkarni, K., Jacobs, H. R. & Hwang, J. J. 1987 Similarity solution for natural convection flow over an isothermal vertical wall immersed in thermally stratified medium. Intl J. Heat Mass Transfer 30, 691–698.CrossRefGoogle Scholar
Le Marec, C., Guerin, R. & Haldenwang, P. 1996 A Chebyshev collocation method for convective flow induced by unidirectional solidification in a cylinder. Intl J. Numer. Meth. Fluids 22, 393–409.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
Peyret, R. 2002 Spectral Methods for Incompressible Viscous Flow. Springer.CrossRefGoogle Scholar
Prandtl, L. 1952 Essentials of Fluid Dynamics. Hafner Publishing Company, New York.Google Scholar
Ravi, M. R., Henkes, R. A. W. M. & Hoogendoorn, C. J. 1994 On the high Rayleigh number structure of steady laminar natural-convection flow in a square enclosure. J. Fluid Mech. 262, 325–351.Google Scholar
Schlichting, H. 1968 Boundary Layer Theory 6th Edn. McGraw-Hill.Google Scholar
Shapiro, A. & Fedorovich, E. 2004 a Prandtl number dependence of unsteady natural convection along a vertical plate in a stably stratified fluid. Intl J. Heat Mass Transfer 47, 4911–4927.CrossRefGoogle Scholar
Shapiro, A. & Fedorovich, E. 2004 b Unsteady convectively driven flow along a vertical plate immersed in a stably stratified fluid. J. Fluid Mech. 498, 333–352.Google Scholar
Sundstrom, L. G. & Kimura, S. 1996 On laminar free convection in inclined rectangular enclosures. J. Fluid Mech. 313, 343–366.CrossRefGoogle Scholar
Tanny, J. & Cohen, J. 1998 The mean temperature field of a buoyancy-induced boundary layer adjacent to a verticale plate immersed in a stratified medium. Intl J. Heat Mass Transfer 41, 2125–2130.Google Scholar
Tao, J., Le Quéré, P. & Xin, S. 2004 Spatio-temporal instability of the natural-convection boundary layer in thermally stratified medium. J. Fluid Mech. 518, 363–379.Google Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.Google Scholar
Yang, K. T., Novotny, J. L. & Cheng, Y. xS. 1972 Laminar free convection from nonisothermal plate immersed in a temperature stratified medium. Intl J. Heat Mass Transfer 15, 1097–1109.CrossRefGoogle Scholar