Stability Of Steady State Solutions For Forced Navier Stokes Equations Of 1 D Nonisentropic Compressible Fluid Flow

Author : Santosh Konangi
Publisher :
ISBN 13 :
Total Pages : 225 pages
Book Rating : 4.:/5 (848 download)

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Download or read book Stability Analysis of Artificial-compressibility-type and Pressure-based Formulations for Various Discretization Schemes for 1-d and 2-d Inviscid Flow, with Verification Using Riemann Problem written by Santosh Konangi and published by . This book was released on 2011 with total page 225 pages. Available in PDF, EPUB and Kindle. Book excerpt: The present investigation studies the stability properties of numerical solution procedures for the non-linear equations governing fluid flow, namely, the Navier-Stokes (NS) equations. The viscous terms of the NS equations are dropped as they only enhance stability, and the remaining inviscid form of these equations, called the Euler equations, are studied. Usually, the stability of numerical schemes for the Euler equations is studied by considering model equations, such as the wave equation and Burgers' equation. Consequently, the analysis overlooks the manner in which the Euler equations are solved: either as density-based and coupled for high-speed flows, or in a segregated, pressure-based manner for low-speed flows. The first objective of this effort is to conduct a rigorous stability analysis of segregated, pressure-based formulations of the Euler equations, as these are employed in most commercially-available CFD packages. The second objective is to analyze the coupled, density-based formulation. To the best knowledge of the author, the only published work dealing with the stability analysis of pressure-based schemes is that of van der Heul et al. (2001), who pursued the stability properties an all-speed pressure-based solver. The goal of the present effort is to perform a stability analysis of solution procedures for the full set of Euler equations, including the conservation of mass and momentum equations, closed by an equation of state. The one-dimensional and two-dimensional forms of the Euler equations are chosen. The governing equations are discretized on a staggered grid using finite-differences. First-order accurate explicit and implicit discretization techniques are used to approximate the unsteady terms in the continuity and momentum equations. The momentum convection term is differenced by a first-order upwind scheme, and is treated as explicit, semi-implicit, or implicit. The spatial derivatives in the continuity equation are either central-differenced or first-order density upwind biased. Various schemes are formulated, based on combinations of the different discretization techniques. A von Neumann stability analysis is conducted for the coupled system of density-based governing equations. An error amplification matrix is derived for each scheme, in terms of the Mach number and Courant (CFL) number. The stability analysis is then performed for the segregated pressure-based solution procedure, Semi-Implicit Method for Pressure-Linked Equations (SIMPLE), and the corresponding error amplification matrix is derived. The main results of this study are the numerically-determined stability regions in terms of CFL number as a function of Mach number, from the error amplification matrices. The results indicate stable and unstable schemes, and predict the maximum allowable CFL number for stability. The predictions of the stability analysis were verified for selected schemes, using the Riemann problem. The CFL limit was tested for several Mach numbers. The predicted and observed CFL values were tabulated for each scheme, and very good agreement is obtained for all cases. The study provides practical stability conditions in the form of numerically-determined stability limits on the CFL number, for various Mach numbers. For all but simple equations and discretization schemes, analytical derivations may not be possible. Hence, the numerical approach is developed in this study.