Temporal accuracy analysis of phase change convection simulations using the JFNK-SIMPLE algorithm. (English) Zbl 1388.76246

Summary: The incompressible Navier-Stokes and energy conservation equations with phase change effects are applied to two benchmark problems: (1) non-dimensional freezing with convection; and (2) pure gallium melting. Using a Jacobian-free Newton-Krylov (JFNK) fully implicit solution method preconditioned with the SIMPLE [S. V. Patankar, Numerical heat transfer and fluid flow. Washington – New York – London: Hemisphere Publishing Corporation; New York etc.: McGraw-Hill Book Company (1980; Zbl 0521.76003)] algorithm using centred discretization in space and three-level discretization in time converges with second-order accuracy for these problems. In the case of non-dimensional freezing, the temporal accuracy is sensitive to the choice of velocity attenuation parameter. By comparing to solutions with first-order backward Euler discretization in time, it is shown that the second-order accuracy in time is required to resolve the fine-scale convection structure during early gallium melting. Qualitative discrepancies develop over time for both the first-order temporal discretized simulation using the JFNK-SIMPLE algorithm that converges the nonlinearities and a SIMPLE-based algorithm that converges to a more common mass balance condition. The discrepancies in the JFNK-SIMPLE simulations using only first-order rather than second-order accurate temporal discretization for a given time step size appear to be offset in time.


76M25 Other numerical methods (fluid mechanics) (MSC2010)
80M25 Other numerical methods (thermodynamics) (MSC2010)
76R10 Free convection


Zbl 0521.76003
Full Text: DOI


[1] Leonard, International Journal for Numerical Methods in Fluids 20 pp 421– (1995)
[2] Hannoun, International Journal for Numerical Methods in Fluids 48 pp 1283– (2005)
[3] Hannoun, Numerical Heat Transfer B 44 pp 253– (2003)
[4] Mousseau, Monthly Weather Review 130 pp 2611– (2002)
[5] Mousseau, Journal of Heat Transfer 127 pp 531– (2005)
[6] Rauenzahn, Computer Physics Communications 172 pp 109– (2005)
[7] Knoll, Journal of Computational Physics 185 pp 583– (2003)
[8] Evans, Journal of Computational Physics 219 pp 404– (2006)
[9] Numerical Heat Transfer and Fluid Flow. Hemisphere: New York, 1980. · Zbl 0521.76003
[10] Brent, Numerical Heat Transfer 13 pp 297– (1988)
[11] Knoll, Journal of Computational Physics 193 pp 357– (2004)
[12] . Solving Ordinary Differential–Algebraic Equations. Springer: Berlin, 2002.
[13] Evans, Journal of Computational Physics 223 pp 121– (2007)
[14] Stella, Numerical Heat Transfer A 38 pp 193– (2000)
[15] Morgan, Computer Methods in Applied Mechanics and Engineering 28 pp 275– (1981)
[16] Danzig, International Journal for Numerical Methods in Engineering 28 pp 1769– (1989)
[17] Voller, International Journal for Numerical Methods in Engineering 24 pp 271– (1987)
[18] Voller, International Journal for Numerical Methods in Engineering 30 pp 875– (1988)
[19] Gau, Journal of Heat Transfer 108 pp 171– (1986)
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