Supersonic flow past a blunt body

Neil A Walkowski

MANE 6720 Computational Fluid Dynamics
Professor Slimon

Supersonic Flow Past a Blunt Body
April 11th, 2010

TABLE OF CONTENTS
INTRODUCTION 3
CODE DEVELOPMENT 7
RESULTS 7
REFERENCES 13
APPENDIX A 13 INTRODUCTION

In order to develop the CFD code to numerically solve for supersonic flow, past a blunt body or through a de Laval nozzle, the governing fluid mechanics equations (Euler’s equations) need to be in a non-dimensional form (transformed to computational space rather than being in 2 dimensional space). The governing equations for two-dimensional flow (from reference (1)) are:

where,

and

The transformed Euler equations are as follows:

where,

and J is the Jacobian transformation, which is defined to be,

The time derivative is approximated by using a first-order backward difference while the rest of the terms are evaluated at time n+1.

(1)

The first order backward difference of the non-dimensional Euler equations is nonlinear. To linearize it a Taylor series expansion is used for terms and and put into like terms, which yields:

(2.a)

(2.b)

Where and are the flux Jacobian matrices. Substituting equations 2.a and 2.b into equation 1 and rearranging terms yields:

(3)

The flux Jacobian matrices, A and B, are as follows:

The eigenvalues for A and B are respectively,

where and .

From reference (2)

and

In terms of the eigenvalues,

where and ; and similarly for the flux vector.

To satisfy the Fortran 90 first order upwind scheme the E, F, A, and B flux matrices need to be split in terms of positive and negative eigenvalue cases. There are 4 cases, they are the following:

Case 1: All eigenvalues are negative Case 2: All eigenvalues are negative (except