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Time-dependent natural convection in a square cavity: Application of a new finite volume method. (English) Zbl 0697.76097
Summary: A new finite volume (FV) approach with adaptive upwind convection is used to predict the two-dimensional unsteady flow in a square cavity. The fluid is air and natural convection is induced by differentially heated vertical walls. The formulation is made in terms of the vorticity and the integral velocity (induction) law. Biquadratic interpolation formulae are used to approximate the temperature and vorticity fields over the finite volume, to which the conservation laws are applied in integral form. Image vorticity is used to enforce the zero-penetration condition at the cavity walls. Unsteady predictions are carried sufficiently forward in time to reach a steady state. Results are presented for a Prandtl number (Pr) of $$0\cdot 71$$ and Rayleigh numbers equal to $$10^ 3$$, $$10^ 4$$ and $$10^ 5$$. Both 11$$\times 11$$ and 21$$\times 21$$ meshes are used. The steady state predictions are compared with published results obtained using a finite difference (FD) scheme for the same values of Pr and Ra and the same meshes, as well as a numerical bench-mark solution. For the most part the FV predictions are closer to the bench-mark solution than are the FD predictions.

##### MSC:
 76R10 Free convection 76M99 Basic methods in fluid mechanics
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##### References:
 [1] Kinney, Int. j. numer. methods eng. 26 pp 1325– (1988) [2] de Vahl Davis, Int. j. numer. methods fluids 3 pp 249– (1983) [3] Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York, 1980, Chap. 3. · Zbl 0521.76003 [4] and , ’Survey of computational methods for three-dimensional supersonic inviscid flows with shocks’, Advances in Numerical Fluid Dynamics, AGARD LS64, 1973. [5] Finite Element Computational Fluid Mechanics, McGraw-Hill, New York, 1980, Chap. 4. [6] Raithby, Comput. Fluids 2 pp 191– (1974) [7] de Vahl Davis, Comput. Fluids 4 pp 29– (1976) [8] Benjamin, Re’, J. Comput. Phys. 33 pp 340– (1979) [9] Gresho, Comput. Fluids 9 pp 223– (1981) [10] Borthwick, Int. j. numer. methods fluids 6 pp 275– (1986) [11] and , ’A finite volume method for transonic potential flow calculations’, AIAA 3rd Computational Fluid Dynamics Conf. No. 77-635, Albuquerque, NM, 1977. [12] de Vahl Davis, Int. j. numer. methods fluids 3 pp 249– (1983) [13] Phillips, J. Comput. Phys. 54 pp 365– (1984) [14] Luchini, J. Comput. Phys. 68 pp 283– (1987) [15] Küblbeck, Int. J. Heat Mass Transfer 23 pp 203– (1980) [16] , and , ’Solution of time-dependent incompressible Navier-Stokes and Boussinesq equations using the Galerkin finite element method’, Approximate Methods for Navier-Stokes Problems, Lecture Notes in Mathematics, No. 771, Springer-Verlag, New York, 1980, pp. 203-223. [17] Hung, Int. j. numer. methods fluids 8 pp 1403– (1988) [18] Heat Transfer, 2nd edn, McGraw-Hill, New York, 1971, Chap. 8. [19] ’Introduction. Boundary layer theory’, in (ed.), Laminar Boundary Layers, Oxford University Press, 1963, pp. 57-60. [20] ’Analysis of unsteady heat transfer by natural convection in a two-dimensional square cavity using a high order finite-volume method’, Ph.D. Dissertation, University of Arizona, 1989. [21] Gresho, Int. j. numer. methods fluids 4 pp 557– (1984)
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