[Spectral Methods]

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Abstract

We present a spectral numerical method for solving one-dimensional systems of partial differential equations (PDEs) which arise from linearization of the Euler equations about an exact solution depending on space and time through winshell. A two-domain Chebyshev collocation method is used. Matching of quantities is performed in the space of characteristic variables as suggested by Kopriva [Appl. Numer. Math. 2 (1986) 221; J. Comput. Phys. 125 (1996) 244]. Time-dependent boundary conditions are handled following an approach proposed by Thompson [J. Comput. Phys. 68 (1987) 1; 89 (1990) 439]. An exact numerical stability analysis valid for any explicit three-step third-order non-degenerate Runge-Kutta scheme is provided through winshell. The numerical method is tested against exact solutions for the three fundamental modes of a compressible flow (entropy, vorticity and acoustic modes).

Table of Contents

ABSTRACT1

1. INTRODUCTION3

2. EQUATIONS AND BOUNDARY CONDITIONS6

2.1. Mean flow equations6

2.1.1. Particular cases of self-similar solutions7

2.1.2. Restriction to uniform and constant mean flows8

2.2. Linear perturbation equations9

3. NUMERICAL METHOD14

3.1. Boundary condition treatment15

3.1.1. Periodic boundary conditions17

3.1.2. Non-reflecting boundary conditions17

3.1.3. Time-dependent boundary conditions19

3.2. Matching condition treatment19

3.3. Numerical approximation method20

4. NUMERICAL STABILITY ANALYSIS21

4.1. Eigenvalues of the matrix S4N23

4.2. Numerical stability condition26

5. NUMERICAL TESTS30

5.1. Acoustic plane waves of arbitrary wave numbers in the m-variable31

5.2. Vorticity mode in the m-variable37

5.2.1. Stability study39

5.3. Entropy mode in the ?-variable40

5.4. Forced longitudinal acoustic mode in the m- and ?-variables42

5.4.1. Solution in the m-variable43

5.4.2. Solution in the ?-variable44

6. CONCLUSION47

REFERENCES50

APPENDIX53

Appendix A Derivation of linear perturbation equations53

A.1 Formulation in the self-similar variable ?55

Appendix B A linear algebra theorem57

Spectral Methods

1. Introduction

In this paper we present a numerical method for solving systems of partial differential equations (PDEs) which arise from linearization of the Euler equations about a one-dimensional exact solution depending on space and time.

Stability analyses of fluid flows often start with a linear stability analysis. In situations where the mean flow depends on time, linearizing the Euler or the Navier-Stokes equations about such a flow leads to a linear system of PDEs for perturbations. The coefficients of this linear system depending on space and time variables, the normal mode analysis no longer applies and one has to solve an initial and boundary value problem (IBVP) for the perturbations. The one-dimensional exact solutions that are considered here are of self-similar types. Most of the time, the set of ordinary differential equations (ODEs) resulting from the self-similar analysis must be solved numerically. (Zel'dovich and Raizer 2007 102)

The linear stability of exact solutions of the Euler equations has been addressed by several authors. In these examples, the mean flows are obtained from analytical calculations. Moreover, the linear perturbation analysis can also be carried ...