Basic Advances in the Finite-Volume Method for Transonic Potential-Flow Calculations
The finite-volume methods of Jameson and Caughey [4.8,4.9,4.212] provide a general framework within which it is fairly easy to calculate the transonic potential flow past essentially arbitrary geometrical configurations. Like finite-element methods, these
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Basic Advances in the Finite-Volume Method for Transonic Potential-Flow Calculations D. A. Caughey* and Antony Jamesont
1 Introduction The finite-volume methods of Jameson and Caughey [4.8,4.9,4.212] provide a general framework within which it is fairly easy to calculate the transonic potential flow past essentially arbitrary geometrical configurations. Like finite-element methods, these methods use only local properties of the transformations which generate the difference grid. This feature essentially decouples the solution ofthe transonic flow equations from the grid-generating step, so that minor modifications of a universal algorithm can be applied in any boundary-conforming coordinate system. Although the initial variants of these methods used line relaxation to solve the difference equations, the multigrid alternating-direction-implicit (MAD) scheme of Jameson [4.213] has also been applied to provide high rates of convergence to very small residuals for two-dimensional calculations [4.214]. Two particular features of the formulation of the finite-volume methods will be addressed in the present paper, with the aim of improving the accuracy and consistency of the method. The first is an improved artificial viscosity, which allows retention of formal second-order accuracy in supersonic zones; the second is a modification of the scheme which allows the freestream conditions to be satisfied identically by the difference equations. In the
* Associate Professor, Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853. t Professor, Department of Mechanical and Aerospace Sciences, Princeton University, Princeton, NJ 08540. 445
T. Cebeci (ed.), Numerical and Physical Aspects of Aerodynamic Flows © Springer Science+Business Media New York 1982
4 Transonic Flows
446
following sections, the fully conservative finite-volume method will first be briefly reviewed, inc1uding the changes necessary to retain second-order accuracy in supersonic zones. The problem of consistency with the freestream solution will then be discussed and a remedy proposed. Finally, results of calculations incorporating these changes will be presented and discussed.
2 Analysis 2.1 Finite-Volume Scheme For convenience, here and throughout the paper the analysis will be described for a two-dimensional problem, and only distinguishing features of the extension to three-dimensional problems will be discussed. The equations of steady, inviscid, isentropic flow can be represented as folIows. Let x, y be Cartesian coordinates, and u, v be the corresponding components ofthe velocity vector q. Then the continuity equation can be written as
+ (pv)y
(pu)x
=
(1)
0,
where p is the local density. This is given by the isentropic law p
k-l
= ( 1 + -2- M~(1 -
q2)
)l/(k-l)
,
(2)
where k is the ratio of specific heats, and M 00 is the freestream Mach number. The pressure p and the speed of sound a follow from the relations pk
(3)
p= kM 2
00
and 2
k-l
P
a = M2
•
(4)
00
Consider now a transformation to a
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