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A FOURTH ORDER SCHEME FOR THE UNSTEADY COMPRESSIBLE NAVIER STOKES EQUATIONS

IP.com Disclosure Number: IPCOM000128825D
Original Publication Date: 1985-Dec-31
Included in the Prior Art Database: 2005-Sep-19
Document File: 9 page(s) / 33K

Publishing Venue

Software Patent Institute

Related People

A. Bayliss Exxon: AUTHOR [+6]

Abstract

A computational scheme is describedc which is second order accurate in time and fourth order accurate in space (2-4). This method is applied to study the stability of compressible boundary layers. The laminar compressible Navier-Stokes equations are solved with a time harmonic inflow superimposed on the steady state solution. This results in spatially unstable modes. It is shown that the second order methods are inefficient for calculating the growth rates and phases of the unstable modes. In contrast the fourth order method yields accurate results on relatively course meshes.

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

A FOURTH ORDER SCHEME FOR THE UNSTEADY COMPRESSIBLE NAVIER STOKES EQUATIONS

A. Bayliss Exxon Corporate Research Science Laboratories

P. Parikh Vigyan Research Associates, Inc.

L. Maestrello NASA Langley Research Center

E. Turkel ICASE and Tel Aviv University

Abstract

A computational scheme is describedc which is second order accurate in time and fourth order accurate in space (2-4). This method is applied to study the stability of compressible boundary layers. The laminar compressible Navier-Stokes equations are solved with a time harmonic inflow superimposed on the steady state solution. This results in spatially unstable modes. It is shown that the second order methods are inefficient for calculating the growth rates and phases of the unstable modes. In contrast the fourth order method yields accurate results on relatively course meshes.

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1. Introduction

This paper is concerned with a fourth-order accurate finite difference scheme for the compressible, unsteady Navier-Stokes equations. The primary interest is the computation of spatially unstable disturbances into the nonlinear regime and the active control of such disturbances. Due to the wavelike nature of these disturbances, this problem has features of both wave propagation and fluid dynamics and the numerical scheme must be chosen to accurately compute waves propagating in an unstable, high Reynolds number mean flow. The application of this scheme to study the active control of spatially growing disturbances has been described previously [1]. This paper describes the numerical scheme and the advantages that can be obtained by the use of fourth-order accuracy.

Higher order accurate methods, in particular spectral methods, have been successfully used in the computation of incompressible flows. Examples of such calculations can be found in [21 - [4]. Generally, spectral methods assume that problem is periodic in the streamwise direction so that Fourier (as opposed to Chebyshev) collocation can be used. (Spatially unstable disturbances were considered, however in [41.) The use of higher order methods for the

1 Partial support was provided for the first author under NASA Contract No. NAS1-17070 and for the second author under NASA Contract No. NAS1-17252. Research for the fourth author was supported by the National Aeronautics and Space Administration under NASA Contracts No. NAS1-17070 and NAS1-17130 while he was in residence at the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA 23665.

National Aeronautics and Space Administration Page 1 Dec 31, 1985

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A FOURTH ORDER SCHEME FOR THE UNSTEADY COMPRESSIBLE NAVIER STOKES EQUATIONS

numerical computation of waves has also been extensively considered in the literature. Examples can be found in [51 - [7].

In section 2, we describe the numerical scheme and discuss certain implementation details. In sectio...