Engineering Transactions, 41, 2, pp. 187-208, 1993

Natural Convection in a Cylindrical Cavity Heated by an Internally-Located Strong Source

J. Słomczyńska
Institute of Fundamental Technological Research, Warszawa

Stationary natural! convection caused by a strong source of beat centrally located in a cylindrical cavity is analyzed by a finite-difference method. Since gradients of pressure are much smaller than gradients of both the temperature and density, the axisymmetric flow is treated as incompressible while, for the same reason, variable density is fully accounted for. Viscosity and thermal conductivity are assumed to be functions of temperature. Coupled stationary equations of continuity, motion, and energy are formulated in the framework of primitive variables. Line integration of the equation of motion over a closed contour is used to eliminate pressure. The solution, i.e. temperature (hence, density) and velocity distributions in the cavity, is found by a two-step iterative procedure based on line successive overrelaxation, Examples of computation showing the effects of change in the source heat-rate, mean value of pressure and the aspect ratio of the cavity are provided.

Full Text: PDF
Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).


W.ARTER, Nonlinear Rayleigh-Bernard convection with square planform, J. Fluid Mech., 152, 391-418, 1985.

A.BARANOWSKI, Z.MUCHA and Z.PERADZYŃSKI, Experimental and theoretical study of the stability of continuous optical discharge in gases, Adv. Mech., 1, 125-147, 1978.

K.R.BLAKE, D.POULIKAKOS and A.BEJAN, Natural convection near 4° C in a horizontal water layer heated from below, Phys. Fluids, 27, 2608-2616, 1984.

F.H.BUSSE and H.FRICK, Square-pattern convection in fluids with strongly temperature-dependent viscosity, J. Fluid Mech., 150, 451-465, 1985.

R.COURANT, E.ISAACSON and M.REES, On the solution of nonlinear hyperbolic differential equations by finite differences, Comm. Pure Appl. Math.1 5, 243-255, 1952.

J.FROMM, The time-dependent flow of an incompressible viscous fluid, Methods Computat. Phys., 3, 345-382, 1964.

N.A.GENERALOV, V.P.ZIMAKOV, G.I.KOZLOV, V.A.MASYUKOV and YU.P.RAIZER, Experimental investigation of continous optical discharge, Soviet Physics JETP, 34, 763-769, 1972 [Originally published in Russian: Zh. Eksp. Teor. Fiz., 61, 1434-1446, 1971].

B.GILLY, B.ROUX and P.BONTOUX, Influence of thermal wall conditions on the natural convection in heated cavities, [in:] R.W.LEWIS, K.MORGAN and B.A.SCHREFLER [Eds.], Numerical Methods in Heat Transfer, vol. 2, pp. 205-225, Wiley, New York 1983.

E.H.HIRSCHEL and A.GROH, Wall-compatibility condition for the solution of the Navier-Stokes equations, J. Computat. Phys., 53, 346-350, 1984.

H.INABA and T.FUKUDA, Natural convection in an inclined square cavity in regions of density inversion of water, J. Fluid Mech., 142, 363-381, 1984.

M.V.KARVE and Y.YALURIA, Numerical simulation of the conjugate heat transfer process from a heated moving surface, [in:] G.YAGAWA and S.N.ATLURI [Eds], Computational Mechanics'86, vol. 2, pp. VIII.19-24. Springer-Verlag, Tokyo-Berlin-Heidelberg-New York 1986.

S.KJMURA and A.BEJAN, Natural convection in a differentially heated corner region, Phys. Fluids, 28, 2980-2989, 1985.

R.W.KNIGHT and M.E.PALMER III, Simulation of free convection in multiple layers in an enclosure by finite differences, [in:] M.S.SHIH [Ed.], Numerical Properties and Methodologies in Heat Transfer, pp. 305-319, Hemisphere, Washington, D.C., 1983.

G.I.KOZLOV, V.A.KUZNETSOV and V.A.MASYUKOV, Radiative losses by argon plasma and the emissive model of a continuous optical discharge, Soviet Physics JETP, 39, 463-468, 1974 [Originally published in Russian: Zh. Eksp. Teor. Fiz. 66, 954-964, 1974].

Y.LEE and S.KORPELA, Multicellular natural convection in a vertical slot, J. Fluid Mech., 126, 91-121, 1983.

G.D.MALLINSON, A.D.GRAHAM and G.DE VAL, DAVIS, Three-dimensional flow in a closed thermosyphon, J. Fluid Mech., 109, 259-275, 1981.

C.D.MOODY, Maintenance of a gas breakdown in argon using 10.6-µ cw radiation, J. Appl. Phys., 46, 2475-2482, 1975.

Z.MUCHA, Z.PERADZYŃSKI and A.BARANOWSKI, Instability of continuous optical discharge, Bull. Acad. Pol. Sci., 25, 361-367, 1977.

P.J.O'ROURKE and F.V.BRACCO, Two scaling transformations for the numerical computation of multidimensional unsteady laminar flames, J. Computat. Phys., 33, 185-203, 1979.

S.V.PATANKAR, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington and McGraw-Hill, New York-London 1980.

T.N.PHILIPS, Natural convection in an enclosed cavity, J. Computat. Phys., 54, 365-381, 1984.

P.LE QUERE and A.T.DE ROQUEFORT, Computation of natural convection in two-dimensional cavities with Chebyshev polynomials, J. Computat. Phys., 57, 210-228, 1985.

C.QUON, Effects of grid distribution on the computation of high Rayleigh number convection in a differentially heated cavity, [in:] T.M.SHIN [Ed], Numerical Properties and Methodologies in Heat Transfer, pp. 261-281, Hemisphere, Washington, D.C. 1983.

A.RAMARAJU, A.G.MARATHE and S.K.BISWAS, Time accurate temperature history due to moving source of heat, [in:] G.YAGAWA and S.N.ATLURI [Eds.], Computational Mechanics'86, vol. 2, pp. VIII. 45-50, Springer-Verlag, Tokyo-Berlin-Heidelberg-New York, 1986.

J.D.RAMSHAW, P.J.O'ROURKE and L.R.STEIN, Pressure gradient scaling method for fluid flow with nearly uniform pressure, J. Computat. Phys., 58, 361-376, 1985.

P.J.ROACHE, Computational Fluid Dynamics (First edition, 972), Hermosa, Albuquerque, N.M. 1976.

P.J.ROACHE, The LAD, NOS and Split NOS methods for the steady-state Navier­Stokes equations, Computers and Fluids, 3, 179-195, 1978.

M.N.ROPPO, S.H.DAVIS and S.ROSENBLAT, Benard convection with time-periodic heating, Phys. Fluids, 27, 796-803, 1984.

S.G.RUBIN, lncompressible Navier-Stokes and parabolized Navier-Stokes formulations and computational techniques, [in:] W.G.HABASHI [Ed.], Computational Methods in Viscous Fluids (vol. 3 of Recent Advances in Numerical Methods), pp. 55-99, Pineridge Press, Swansea 1984.

J.SŁOMCZYŃSKA and Z.PERADZYŃSKI, Finite-difference scheme for stationary Navier-Stokes equations with variable coefficients, [in:] R.L.STERNBERG, A.J.KALI­NOWSKI and J.S.PAPADAKIS [Eds.], Nonlinear Partial Differential Equations in Engineering and Applied Science, pp. 343-376, M.Deker, New York 1980.

J.SŁOMCZYŃSKA, A finite-difference method for a problem of stationary free convection caused by a strong heat source, [in:] G.YAGAWA and S.N.ATLURI [Eds.], Computational Mechanics'86, vol. 2, pp. VIII.183-188, Springer-Verlag, Tokyo-Berlin­Heidelberg-New York 1986.

K.C.STENGEL, D.S.OLIVER and J.R.BOOKER, Onset of convection in a variableviscosity fluid, J. Fluid Mech., 120, 411-431, 1982.

I.C.WALTON, The effect of shear flow on convection in a layer heated non-uniformly from below, J. Fluid Mech., 154, 303-319, 1985.