×

Solution-limited time stepping to enhance reliability in CFD applications. (English) Zbl 1368.76042

Summary: A method for enhancing the reliability of implicit computational algorithms and decreasing their sensitivity to initial conditions without adversely impacting their efficiency is investigated. Efficient convergence is maintained by specifying a large global Courant (CFL) number while reliability is improved by limiting the local CFL number such that the solution change in any cell is less than a specified tolerance. The method requires control over two key issues: obtaining a reliable estimate of the magnitude of the solution change and defining a realistic limit for its allowable variation. The magnitude of the solution change is estimated from the calculated residual in a manner that requires negligible computational time. An upper limit on the local solution change is attained by a proper non-dimensionalization of variables in different flow regimes within a single problem or across different problems. The method precludes unphysical excursions in Newton-like iterations in highly non-linear regions where Jacobians are changing rapidly as well as non-physical results such as negative densities, temperatures or species mass fractions during the computation. The method is tested against a series of problems all starting from quiescent initial conditions to identify its characteristics and to verify the approach. The results reveal a substantial improvement in convergence reliability of implicit CFD applications that enables computations starting from simple initial conditions without user intervention.

MSC:

76M12 Finite volume methods applied to problems in fluid mechanics
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] R.H. Bush, G.D. Power, C.E. Towne, WIND: the production flow solver of the NPARC alliance, in: 36th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 12-January 15, 1998, AIAA Paper 1998-0935.; R.H. Bush, G.D. Power, C.E. Towne, WIND: the production flow solver of the NPARC alliance, in: 36th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 12-January 15, 1998, AIAA Paper 1998-0935.
[2] N.T. Frink, S.Z. Pirzadeh, P.C. Parikh, M.J. Pandya, M.K. Bhat, The NASA tetrahedral unstructured software system (TetrUSS), in: 22nd International Congress of Aeronautical Sciences, Harrogate, United Kingdom, August 27-September 1, 2000, ICAS Paper No. 0241.; N.T. Frink, S.Z. Pirzadeh, P.C. Parikh, M.J. Pandya, M.K. Bhat, The NASA tetrahedral unstructured software system (TetrUSS), in: 22nd International Congress of Aeronautical Sciences, Harrogate, United Kingdom, August 27-September 1, 2000, ICAS Paper No. 0241.
[3] Sankaran, V.; Merkle, C. L.; Zeng, X.; Li, D., Influence of large-scale pressure changes on preconditioned solutions at low speeds, AIAA J., 42, 12, 2490-2498 (2004)
[4] Mulder, W.; van Leer, B., Experiments with implicit upwind methods for the Euler equations, J. Comput. Phys., 59, 232-246 (1985) · Zbl 0584.76014
[5] Issman, E.; Degrez, G.; Deconinck, H., Implicit upwind residual-distribution Euler and Navier-Stokes solver on unstructured meshes, AIAA J., 34, 10, 2021-2028 (1996) · Zbl 0906.76048
[6] D. Vanderstraeten, A. Csk, D. Rose, An expert-system to control the CFL number of implicit upwind methods, Technical Report TM 304, Universiteit Leuven, March 2000.; D. Vanderstraeten, A. Csk, D. Rose, An expert-system to control the CFL number of implicit upwind methods, Technical Report TM 304, Universiteit Leuven, March 2000.
[7] H.M. Bücker, B. Pollul, A. Rasch, On CFL evolution strategies for implicit upwind methods in linearized Euler equations, Technical Report, RWTH Aachen University, August 7, 2006.; H.M. Bücker, B. Pollul, A. Rasch, On CFL evolution strategies for implicit upwind methods in linearized Euler equations, Technical Report, RWTH Aachen University, August 7, 2006.
[8] E.A. Luke, X.-L. Tong, J. Wu, L. Tang, P. Cinnella, A step towards ‘Shape-Shifting’ algorithms: reactive flow simulations using generalized grids, AIAA Paper No. 2001-0897, 2001.; E.A. Luke, X.-L. Tong, J. Wu, L. Tang, P. Cinnella, A step towards ‘Shape-Shifting’ algorithms: reactive flow simulations using generalized grids, AIAA Paper No. 2001-0897, 2001.
[9] C. Merkle, D. Li, V. Sankaran, Multi-disciplinary computational analysis in propulsion, AIAA Paper No. 2006-4575, 2001.; C. Merkle, D. Li, V. Sankaran, Multi-disciplinary computational analysis in propulsion, AIAA Paper No. 2006-4575, 2001.
[10] Nessyahu, H.; Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87, 408-463 (1990) · Zbl 0697.65068
[11] Jiang, G.; Tadmor, E., Non-oscillatory central schemes for multidimensional hyperbolic conservation laws, SIAM J. Sci. Comput., 19, 1892-1917 (1998) · Zbl 0914.65095
[12] Lax, P. D., Weak solutions of nonlinear hyperbolic equation and their numerical computation, Commun. Pure Appl. Math., 7, 159-193 (1954) · Zbl 0055.19404
[13] B. van Leer, Flux-vector splitting for the Euler equation, NASA Langley Research Center Hampton, ICASE Report 82-30, 1982.; B. van Leer, Flux-vector splitting for the Euler equation, NASA Langley Research Center Hampton, ICASE Report 82-30, 1982.
[14] T. Linde, A practical, general-purpose Riemann solver for hyperbolic conservation laws, in: Seventh International Conference on Numerical Methods in Fluid Dynamics, Clarendon, 2001.; T. Linde, A practical, general-purpose Riemann solver for hyperbolic conservation laws, in: Seventh International Conference on Numerical Methods in Fluid Dynamics, Clarendon, 2001. · Zbl 1020.76036
[15] Liou, M.-S.; Steffen, C., A new flux splitting scheme, J. Comput. Phys., 109, 23-93 (1993) · Zbl 0779.76056
[16] Liou, M.-S.; Edwards, J., Low-diffusion flux splitting methods for flows at all speed, AIAA J., 36, 9, 1610-1617 (1998)
[17] Roe, P., Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys., 43, 357-372 (1981) · Zbl 0474.65066
[18] Godunov, S., A finite-difference method for the numerical computation and discontinuous solutions of the equations of fluid dynamics, Mat. Sb., 47, 271-306 (1959) · Zbl 0171.46204
[19] Osher, S.; Solomon, F., Upwind schemes for hyperbolic systems of conservation laws, Math. Comput., 38, 339-374 (1982) · Zbl 0483.65055
[20] Turkel, E., Preconditioning methods for solving the incompressible and low speed compressible equations, J. Comput. Phys., 72, 277-298 (1987) · Zbl 0633.76069
[21] B. van Leer, W. Lee, P. Roe, Characteristic time-stepping or local preconditioning of the euler equations, AIAA 1991-1552-CP, Computational Fluid Dynamics Conference, Honolulu, 1991.; B. van Leer, W. Lee, P. Roe, Characteristic time-stepping or local preconditioning of the euler equations, AIAA 1991-1552-CP, Computational Fluid Dynamics Conference, Honolulu, 1991.
[22] Viviand, H., Pseudo-unsteady systems for steady inviscid flow calculation, (Angrand, F.; etal., Numerical Methods for the Euler Equations of Fluid Dynamics (1985), SIAM: SIAM Philadelphia) · Zbl 0608.76051
[23] Briley, W.; McDonald, H.; Shamroth, S., A low Mach number Euler formulation and application to time iterative LBI schemes, AIAA J., 21, 10, 1467-1469 (1983)
[24] Choi, Y.; Merkle, C., The application of preconditioning to viscous flows, J. Comput. Phys., 105, 207-223 (1993) · Zbl 0768.76032
[25] Briley, W.; McDonald, H., On the structure and use of linearized block implicit schemes, J. Comput. Phys., 34, 54-73 (1980) · Zbl 0436.76021
[26] Briley, W.; McDonald, H., An overview and generalization of implicit Navier-Stokes algorithms and approximate factorization, Comput. Fluids, 30, 807-828 (2001) · Zbl 1056.76060
[27] Thomas, J.; Walters, R., Upwind relaxation algorithm for the Navier-Stokes equations, AIAA J., 25, 4, 527-534 (1987) · Zbl 0616.76086
[28] K. Vanden, D. Whitfield, Direct and iterative algorithms for the three-dimensional euler equations, AIAA 1993-3378-CP, 1993.; K. Vanden, D. Whitfield, Direct and iterative algorithms for the three-dimensional euler equations, AIAA 1993-3378-CP, 1993. · Zbl 0869.76068
[29] Kim, Joo Sung; Kwon, Oh Joon, An efficient and robust implicit operator for upwind point Gauss-Seidel method, J. Comput. Phys., 224, 1124-1144 (2007) · Zbl 1123.76051
[30] C. Lombard, J. Bardina, E. Venkatapathy, J. Oliger, Multi-dimensional formulation of CSCM - an upwind flux difference eigenvector split method for the compressible Navier-Stokes equations, AIAA Paper No. 1983-1895, 1983.; C. Lombard, J. Bardina, E. Venkatapathy, J. Oliger, Multi-dimensional formulation of CSCM - an upwind flux difference eigenvector split method for the compressible Navier-Stokes equations, AIAA Paper No. 1983-1895, 1983.
[31] MacCormack, R.; Candler, G., The solution of the Navier-Stokes equations using Gauss-Seidel line relaxation, Comput. Fluids, 17, 135-150 (1989) · Zbl 0664.76031
[32] Saad, Y.; Schultz, M., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear system, SIAM J. Sci. Stat. Comput, 7, 3, 856-869 (1986) · Zbl 0599.65018
[33] Guillard, H.; Viozat, C., On the behaviour of upwind schemes in the low Mach number limit, Comput. Fluids, 28, 1, 63-86 (1999) · Zbl 0963.76062
[34] Durst Franz, An Introduction to the Theory of Fluid Flows, 2008, ISBN:9783540713425.; Durst Franz, An Introduction to the Theory of Fluid Flows, 2008, ISBN:9783540713425. · Zbl 1153.76001
[35] Driver, D.; Seegmiller, H., Features of a reattaching turbulent shear layer in divergent channel flow, AIAA J., 23, 2, 163-171 (1985)
[36] Armaly, B. F.; Durst, F.; Pereira, J. C.F.; Schonung, B., Experimental and theoretical investigation of backward-facing step flow, J. Fluid Mech., 127, 473-496 (1983)
[37] Wilcox, D., Turbulence Modeling for CFD (1998), DCW Industries: DCW Industries La Canada, CA
[38] P. Tucker, S. Menon, C. Merkle, J. Oefelein, V. Yang, An approach to improved credibility of CFD simulations for rocket injector design, in: 43rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, July 8-July 11, Cincinnati, OH, 2007.; P. Tucker, S. Menon, C. Merkle, J. Oefelein, V. Yang, An approach to improved credibility of CFD simulations for rocket injector design, in: 43rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, July 8-July 11, Cincinnati, OH, 2007.
[39] Ó Conaire, M.; Curran, H. J.; Simmie, J. M.; Pitz, W. J.; Westbrook, C. K., A comprehensive modeling study of hydrogen oxidation, Int. J. Chem. Kinet., 36, 603-622 (2004)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.