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

Santosh Konangi

,
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

and Materials Engineering,

University of Cincinnati,

598 Rhodes Hall,

P.O. Box 210072,

Cincinnati, OH 45221-0072

e-mail: konangsh@mail.uc.edu

Search for other works by this author on:

Nikhil K. Palakurthi

,
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

and Materials Engineering,

University of Cincinnati,

598 Rhodes Hall,

P.O. Box 210072,

Cincinnati, OH 45221-0072

e-mail: palakunr@ucmail.uc.edu

Search for other works by this author on:

Urmila Ghia

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

and Materials Engineering,

University of Cincinnati,

598 Rhodes Hall,

P.O. Box 210072,

Cincinnati, OH 45221-0072

e-mail: urmila.ghia@uc.edu

Search for other works by this author on:

Santosh Konangi

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

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

and Materials Engineering,

University of Cincinnati,

598 Rhodes Hall,

P.O. Box 210072,

Cincinnati, OH 45221-0072

e-mail: urmila.ghia@uc.edu

1

Corresponding 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*. Oct 2016, 138(10): 101401 (18 pages)

**Published Online:**July 13, 2016

Article history

Received:

August 27, 2015

Revised:

November 23, 2015

Citation

Konangi, S., Palakurthi, N. K., and Ghia, U. (July 13, 2016). "von Neumann Stability Analysis of a Segregated Pressure-Based Solution Scheme for One-Dimensional and Two-Dimensional Flow Equations." ASME. *J. Fluids Eng*. October 2016; 138(10): 101401. https://doi.org/10.1115/1.4033958

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