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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.

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References

Figures

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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

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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

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Fig. 2

Stability region for 1D scheme, CFL versus Mach number

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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]

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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

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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

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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

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