Preconditioning of Navier–Stokes equations in the computation of free convective flows

K. N. Volkov   ; V. N. Emel'yanov   ; A. G. Karpenko

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 12, pp. 2109-2122

Résumé

Specific features of simulation of free convective flows of viscous compressible fluids based on the full Navier–Stokes equations are considered. The finite volume discretization of the Navier–Stokes equations in the case of low Mach numbers is discussed. To stabilize the numerical computations, preconditioning, which is based on the use of physical variables, and the dual time stepping method, which introduces internal iterations on pseudotime, are employed. The capabilities of the proposed approach are demonstrated on the example of simulation of free convection in a gap between coaxial cylinders.

Bibliographie

[1] Wesseling P., Principles of computational fluid dynamics, Springer, 2000 | MR | Zbl

[2] Turkel E., “Preconditioned methods for solving the incompressible and low speed compressible equations”, J. Comput. Phys., 72:1 (1987), 277–298 | DOI | Zbl

[3] Choi Y.-H., Merkle C. L., “The application of preconditioning in viscous flows”, J. Comput. Phys., 105:2 (1993), 207–223 | DOI | MR | Zbl

[4] Volpe G., “Performance of compressible flow codes at low Mach numbers”, AIAA J., 31:1 (1993), 49–56 | DOI | Zbl

[5] Weiss J. M., Smith W. A., “Preconditioning applied to variable and constant density flows”, AIAA J., 33:11 (1995), 2050–2057 | DOI | Zbl

[6] Turkel E., “Review of preconditioning methods for fluid dynamics”, Appl. Numer. Math., 12:1–3 (1993), 257–284 | DOI | MR | Zbl

[7] Turkel E., Radespiel R., Kroll N., “Assessment of preconditioning methods for multidimensional aerodynamics”, Computers and Fluids, 26:6 (1997), 613–634 | DOI | Zbl

[8] Buelow P. E. O., Venkateswaran S., Merkle C. L., “Effect of grid aspect ratio on convergence”, AIAA J., 32:12 (1994), 2402–2408 | DOI

[9] Volkov K. H., “Predobuslovlivanie uravnenii Eilera i Nave–Stoksa pri modelirovanii nizkoskorostnykh techenii na nestrukturirovannykh setkakh”, Zh. vychisl. matem. i matem. fiz., 49:10 (2009), 1868–1884 | MR | Zbl

[10] Rogers S. E., Kwak D., “Upwind differencing scheme for the time-accurate incompressible Navier–Stokes equations”, AIAA J., 28:2 (1990), 253–262 | DOI | Zbl

[11] Jameson A., Time dependent calculations using multigrid, with application to unsteady flows past airfoils and wings, AIAA Paper No 91-1596, 1991

[12] Venkateswaran S., Merkle C. L., Dual time stepping and preconditioning for unsteady computations, AIAA Paper No 95-0078, 1995

[13] Rumsey C. L., Sanetrik M. D., Biedron R. T., Melson N. D., Parlette E. B., “Efficiency and accuracy of time-accurate Navier–Stokes computations”, Computers and Fluids, 25:2 (1996), 217–236 | DOI | Zbl

[14] Zhang L. P., Wang Z. J., “A block LU-SGS implicit dual time-stepping algorithm for hybrid dynamic meshes”, Computers and Fluids, 33:7 (2004), 891–916 | DOI | Zbl

[15] Bijl H., Carpenter M. H., Vatsa V. N., Kennedy S. A., “Implicit time integration schemes for the unsteady compressible Navier–Stokes equations: laminar flow”, J. Comput. Phys., 179:1 (2002), 313–329 | DOI | MR | Zbl

[16] Tang H. S., Jones S. C., Sotiropoulos F., “An overset grid method for 3D, unsteady, incompressible flows”, J. Comput. Phys., 191:2 (2003), 567–600 | DOI | Zbl

[17] Jameson A., An assessment of dual-time stepping, time spectral and artificial compressibility based numerical algorithms for unsteady flow with applications to flapping wings, AIAA Paper No 2009-4273, 2009

[18] Zhao Y., “Computation of complex turbulent flow using matrix-free implicit dual time-stepping scheme and LRN turbulence model on unstructured grids”, Computers and Fluids, 33:1 (2004), 119–136 | DOI | Zbl

[19] Luo H., Baum J. D., Lohner R., “An accurate, fast, matrix-free implicit method for computing unsteady flows on unstructured grids”, Computers and Fluids, 30:2 (2001), 137–159 | DOI | Zbl

[20] Zhao Y., Tan H. H., Zhang B. L., “A high-resolution characteristics-based implicit dual time-stepping of method for free surface flow simulation on unstructured grids”, J. Comput. Phys., 183:1 (2002), 233–273 | DOI | Zbl

[21] Volkov K. N., Emelyanov V. N., Vychislitelnye tekhnologii v zadachakh mekhaniki zhidkosti i gaza, Fizmatlit, M., 2012

[22] Gray D. D., Giorgini A., “The validity of the Boussinesq approximation for liquids and gases”, Internat. J. Heat and Mass Transfer, 19:5 (1976), 545–551 | DOI | Zbl

[23] Fu W.-S., Li C.-G., Huang C.-P., Huang J.-C., “An investigation of a high temperature difference natural convection in a finite length channel without Bossinesq assumption”, Internat. J. Heat and Mass Transfer, 52:11–12 (2009), 2571–2580 | Zbl

[24] Kuehn T. N., Goldstein R. J., “An experimental and theoretical study of natural convection in the annulus between horizontal concentric cylinders”, J. Fluid Mechanics, 74 (1976), 695–719 | DOI | Zbl

[25] Kuehn T. N., Goldstein R. J., “An experimental study of natural convection heat transfer concentric and eccentric horizontal cylindrical annuli”, J. Heat Transfer, 100 (1978), 635–640 | DOI

[26] Desai S. R., Vafai K., “An investigation and comparative analysis of two and three-dimensional turbulent natural convection in a horizontal annulus”, Internat. J. Heat and Mass Transfer, 37:16 (1994), 2475–2504 | DOI | MR | Zbl

[27] Char M., Hsu Y. H., “Comparative analysis of linear and nonlinear low-Reynolds-number eddy viscosity models to turbulent natural convection in horizontal cylindrical annuli”, Numerical Heat Transfer, 33 (1998), 191–206 | DOI