0
Research Papers: Techniques and Procedures

von Neumann Stability Analysis of a Segregated Pressure-Based Solution Scheme for One-Dimensional and Two-Dimensional Flow Equations

[+] Author and Article Information
Santosh Konangi

Department of Mechanical
and Materials Engineering,
University of Cincinnati,
598 Rhodes Hall,
P.O. Box 210072,
Cincinnati, OH 45221-0072
e-mail: konangsh@mail.uc.edu

Nikhil K. Palakurthi

Department of Mechanical
and Materials Engineering,
University of Cincinnati,
598 Rhodes Hall,
P.O. Box 210072,
Cincinnati, OH 45221-0072
e-mail: palakunr@ucmail.uc.edu

Urmila Ghia

Department of Mechanical
and Materials Engineering,
University of Cincinnati,
598 Rhodes Hall,
P.O. Box 210072,
Cincinnati, OH 45221-0072
e-mail: urmila.ghia@uc.edu

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received August 27, 2015; final manuscript received November 23, 2015; published online July 13, 2016. Assoc. Editor: Francine Battaglia.

J. Fluids Eng 138(10), 101401 (Jul 13, 2016) (18 pages) Paper No: FE-15-1597; doi: 10.1115/1.4033958 History: Received August 27, 2015; Revised November 23, 2015

The goal of this paper is to derive the von Neumann stability conditions for the pressure-based solution scheme, semi-implicit method for pressure-linked equations (SIMPLE). The SIMPLE scheme lies at the heart of a class of computational fluid dynamics (CFD) algorithms built into several commercial and open-source CFD software packages. To the best of the authors' knowledge, no readily usable stability guidelines appear to be available for this popularly employed scheme. The Euler equations are examined, as the inclusion of viscosity in the Navier–Stokes (NS) equation serves to only soften the stability limits. First, the one-dimensional (1D) Euler equations are studied, and their stability properties are delineated. Next, a rigorous stability analysis is carried out for the two-dimensional (2D) Euler equations; the analysis of the 2D equations is considerably more challenging as compared to analysis of the 1D form of equations. The Euler equations are discretized using finite differences on a staggered grid, which is used to achieve equivalence to finite-volume discretization. Error amplification matrices are determined from the stability analysis, stable and unstable regimes are identified, and practical stability limits are predicted in terms of the maximum allowable Courant–Friedrichs–Lewy (CFL) number as a function of Mach number. The predictions are verified using the Riemann problem, and very good agreement is obtained between the analytically predicted and the “experimentally” observed CFL values. The successfully tested stability limits are presented in graphical form, as compared to complicated mathematical expressions often reported in published literature. Since our analysis accounts for the solution scheme along with the full system of flow equations, the conditions reported in this paper offer practical value over the conditions that arise from analysis of simplified 1D model equations.

FIGURES IN THIS ARTICLE
<>
Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.

References

Galpin, P. , Van Doormaal, J. , and Raithby, G. , 1985, “ Solution of the Incompressible Mass and Momentum Equations by Application of a Coupled Equation Line Solver,” Int. J. Numer. Methods Fluids, 5(7), pp. 615–625. [CrossRef]
Patankar, S. V. , and Spalding, D. B. , 1972, “ A Calculation Procedure for Heat, Mass and Momentum Transfer in Three-Dimensional Parabolic Flows,” Int. J. Heat Mass Transfer, 15(10), pp. 1787–1806. [CrossRef]
Caretto, L. , Curr, R. , and Spalding, D. , 1972, “ Two Numerical Methods for Three-Dimensional Boundary Layers,” Comput. Methods Appl. Mech. Eng., 1(1), pp. 39–57. [CrossRef]
Harlow, F. H. , and Amsden, A. A. , 1971, “ A Numerical Fluid Dynamics Calculation Method for All Flow Speeds,” J. Comput. Phys., 8(2), pp. 197–213. [CrossRef]
Harlow, F. H. , and Amsden, A. A. , 1968, “ Numerical Calculation of Almost Incompressible Flow,” J. Comput. Phys., 3(1), pp. 80–93. [CrossRef]
Harlow, F. H. , and Welch, J. E. , 1965, “ Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid With Free Surface,” Phys. Fluids, 8(12), p. 2182. [CrossRef]
Ghia, K. , Hankey, W., Jr. , and Hodge, J. , 1979, “ Use of Primitive Variables in the Solution of Incompressible Navier-Stokes Equations,” AIAA J. 17(3), pp. 298–301.
Alves, L. S. d. B. , 2009, “ Review of Numerical Methods for the Compressible Flow Equations at Low Mach Numbers,” XII Encontro de Modelagem Computacional, Rio de Janeiro, Brazil, p. 11.
Bijl, H. , and Wesseling, P. , 1998, “ A Unified Method for Computing Incompressible and Compressible Flows in Boundary-Fitted Coordinates,” J. Comput. Phys., 141(2), pp. 153–173. [CrossRef]
van der Heul, D. R. , Vuik, C. , and Wesseling, P. , 2003, “ A Conservative Pressure-Correction Method for Flow at All Speeds,” Comput. Fluids, 32(8), pp. 1113–1132. [CrossRef]
ANSYS, 2009, “ANSYS Fluent 12.0 User's Guide,” ANSYS, Inc., San Jose, CA.
OpenFOAM 2011, “OpenFOAM User Guide,” OpenFOAM Foundation, London.
Anderson, J. D. and Wendt, J. , 1995, Computational Fluid Dynamics, Vol. 206, McGraw-Hill, New York.
Morton, K. , 1971, “ Stability and Convergence in Fluid Flow Problems,” Proc. R. Soc. London, Ser. A, 323(1553), pp. 237–253. [CrossRef]
Chan, T. F. , 1984, “ Stability Analysis of Finite Difference Schemes for the Advection-Diffusion Equation,” SIAM J. Numer. Anal., 21(2), pp. 272–284. [CrossRef]
Hindmarsh, A. , Gresho, P. , and Griffiths, D. , 1984, “ The Stability of Explicit Euler Time-Integration for Certain Finite Difference Approximations of the Multi-Dimensional Advection–Diffusion Equation,” Int. J. Numer. Methods Fluids, 4(9), pp. 853–897. [CrossRef]
Wesseling, P. , 1996, “ von Neumann Stability Conditions for the Convection-Diffusion Equation,” IMA J. Numer. Anal., 16(4), pp. 583–598. [CrossRef]
Shishkina, O. V. , 2007, “ The Neumann Stability of High-Order Symmetric Schemes for Convection-Diffusion Problems,” Sib. Math. J., 48(6), pp. 1141–1146. [CrossRef]
van der Heul, D. R. , Vuik, C. , and Wesseling, P. , 2001, “ Stability Analysis of Segregated Solution Methods for Compressible Flow,” Appl. Numer. Math., 38(3), pp. 257–274. [CrossRef]
Nerinckx, K. , Vierendeels, J. , and Dick, E. , 2007, “ A Mach-Uniform Algorithm: Coupled Versus Segregated Approach,” J. Comput. Phys., 224(1), pp. 314–331. [CrossRef]
Chorin, A. J. , 1967, “ A Numerical Method for Solving Incompressible Viscous Flow Problems,” J. Comput. Phys., 2(1), pp. 12–26. [CrossRef]
Pulliam, T. H. , 1986, “ Artificial Dissipation Models for the Euler Equations,” AIAA J., 24(12), pp. 1931–1940. [CrossRef]
Patankar, S. V. , 1980, Numerical Heat Transfer and Fluid Flow, CRC Press/Hemisphere Publishing, Washington, DC, p. 210.
Ferziger, J. H. , and Perić, M. , 1996, Computational Methods for Fluid Dynamics, Vol. 3, Springer, Berlin, Germany.
von Neumann, J. , and Richtmyer, R. D. , 2004, “ A Method for the Numerical Calculation of Hydrodynamic Shocks,” J. Appl. Phys., 21(3), pp. 232–237. [CrossRef]
Rigal, A. , 1979, “ Stability Analysis of Explicit Finite Difference Schemes for the Navier–Stokes Equations,” Int. J. Numer. Methods Eng., 14(4), pp. 617–620. [CrossRef]
Fromm, J. E. , 1963, “ A Method for Computing Nonsteady, Incompressible, Viscous Fluid Flows,” Los Alamos Scientific Lab, Albuquerque, NM, DTIC Document No. LA-2910.
Wesseling, P. , 2009, Principles of Computational Fluid Dynamics, Vol. 29, Springer Science & Business, Berlin/Heidelberg, Germany.
Sousa, E. L. , 2003, “ The Controversial Stability Analysis,” Appl. Math. Comput., 145(2), pp. 777–794.
Vichnevetsky, R. , and Bowles, J. B. , 1982, Fourier Analysis of Numerical Approximations of Hyperbolic Equations, Vol. 5, SIAM, Philadelphia, PA.
Anderson, D. A. , Tannehill, J. C. , and Pletcher, R. H. , 1984, Computational Fluid Dynamics and Heat Transfer, McGraw-Hill Book Company, New York.
Strikwerda, J. C. , 2004, Finite Difference Schemes and Partial Differential Equations, SIAM, Philadelphia, PA.
Lomax, H. , Pulliam, T. H. , and Zingg, D. W. , 2013, Fundamentals of Computational Fluid Dynamics, Springer Science & Business Media, Berlin/Heidelberg, Germany.
Tucker, A. B. , 2004, Computer Science Handbook, CRC Press, Boca Raton, FL.
Sousa, E. , 2009, “ On the Edge of Stability Analysis,” Appl. Numer. Math., 59(6), pp. 1322–1336. [CrossRef]
Richtmyer, R. D. , and Morton, K. , 1967, Different Methods for Initial Value Problems, ( Interscience Tracts in Pure and Applied Mathematics, 2nd ed.), Interscience, New York.
Sengupta, T. K. , Ganeriwal, G. , and De, S. , 2003, “ Analysis of Central and Upwind Compact Schemes,” J. Comput. Phys., 192(2), pp. 677–694. [CrossRef]
Courant, R. , 1928, “ Uber die partiellen Differenzengleichungen der mathematischen Physik,” Math. Ann., 100(1), pp. 32–74. [CrossRef]
Ghia, U. , Bayyuk, S. , Habchi, S. , Roy, C. , Shih, T. , Conlisk, T. , Hirsch, C. , and Powers, J. M. , 2010, “ The AIAA Code Verification Project-Test Cases for CFD Code Verification,” AIAA Paper No. 2010-125.
Sod, G. A. , 1978, “ A Survey of Several Finite Difference Methods for Systems of Nonlinear Hyperbolic Conservation Laws,” J. Comput. Phys., 27(1), pp. 1–31. [CrossRef]
Schulz-Rinne, C. W. , Collins, J. P. , and Glaz, H. M. , 1993, “ Numerical Solution of the Riemann Problem for Two-Dimensional Gas Dynamics,” SIAM J. Sci. Comput., 14(6), pp. 1394–1414. [CrossRef]
Kurganov, A. , and Tadmor, E. , 2002, “ Solution of Two-Dimensional Riemann Problems for Gas Dynamics Without Riemann Problem Solvers,” Numer. Methods Partial Differ. Equations, 18(5), pp. 584–608. [CrossRef]
Lax, P. D. , and Liu, X.-D. , 1998, “ Solution of Two-Dimensional Riemann Problems of Gas Dynamics by Positive Schemes,” SIAM J. Sci. Comput., 19(2), pp. 319–340. [CrossRef]
Schulz-Rinne, C. W. , 1993, “ Classification of the Riemann Problem for Two-Dimensional Gas Dynamics,” SIAM J. Math. Anal., 24(1), pp. 76–88. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Absolute value of eigenvalues as a function of θ, for the 1D scheme: (a) Case of stability: M = 5, CFL = 0.5 and (b) case of instability: M = 5, CFL = 0.51

Grahic Jump Location
Fig. 2

Stability region for 1D scheme, CFL versus Mach number

Grahic Jump Location
Fig. 3

Stability regions for 2D scheme: (a) CFLy versus CFLx for y-Mach number My = 0.1, (b) CFLy versus CFLx for y-Mach number My = 0.5, and (c) CFLy versus CFLx for y-Mach number My = 1

Grahic Jump Location
Fig. 4

Unstable and stable integrations for M = 5 and M = 4 test cases: (a) unstable integration for M = 5 test case, CFL = 0.51, (b) stable integration for M = 5 test case, CFL = 0.5, (c) unstable integration for M = 4 test case, CFL = 0.51, and (d) stable integration for M = 4 test case, CFL = 0.496

Grahic Jump Location
Fig. 5

(a) Solution domain with initial conditions for 2D Riemann problem, (b) schematic representation of configuration 2 from Schulz-Rinne et al. [41] and Kurganov and Tadmor [42]

Grahic Jump Location
Fig. 6

Initial conditions for 2D Riemann test case: (a) initial conditions for Mx = 1, My = 1, (b) initial conditions for Mx = 0.5, My = 0.5, and (c) initial conditions for Mx = 0.1, My = 0.1

Grahic Jump Location
Fig. 7

Unstable and stable integrations for Mx = 0.1, My = 0.1 test case, with 2D scheme: (a) unstable integration, CFLx = 0.51, CFLy = 0.51 and (b) stable integration, CFLx = 0.5, CFLy = 0.5

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In