Shock wave location on a slender transonic body of revolution

2nd AIAA, Theoretical Fluid Mechanics Meeting

1998

In this paper a procedure to design shock-free, transonic, slender bodies of revolution will be detailed. Using transonic small-disturbance theory, a boundary-value problem is developed describing flow around the body in the physical plane and then transformed into the hodograph plane. In the hodograph plane, spatial variables depend on velocity components, instead of the usual dependence of velocity components upon spatial variables in the physical plane. The transformed boundary-value problem is solved numerically using finite-difference approximations and iterative methods. Several shock-free bodies are computed, with differing values of the transonic similarity parameter, .ft" = (1-M£,)/<5 2 M£,, where M x is the flow Mach number and 8 is the body thickness. There are advantages to designing shock-free bodies in the hodograph plane. There is a very simple criterion for detecting when a shock-free flow has been computed in the hodograph: the jacobian of the mapping from physical plane to hodograph plane must be negative everywhere. A difficult Neumann-type boundary condition at the origin of the physical plane becomes a simpler Dirichlet boundary condition in the hodograph. In the physical plane, the body shape is represented by a source distribution of singularities along the origin and the exact locations of the subsonic and "The author can be reached by electronic mail at ronaabacus.oxy.edu.

Section: Numerical Resultsmentioning

confidence: 99%

On the design of shock-free, transonic, slender bodies of revolution

2nd AIAA, Theoretical Fluid Mechanics Meeting

1998

“…The scheme for the cylindrical case was initially presented in Buckmire's 1994 thesis [4] in which particular slender bodies of revolution were found to possess shock-free flows as specific numerical solutions of a mixed-type, singular boundary value problem. The problem is formulated using transonic small disturbance theory found in [5][6][7], among other sources. Cole and Schwendeman announced the first computation of a fore-aft, symmetric, shock-free transonic slender body in [8].…”

Section: Introductionmentioning

confidence: 99%

Investigations of nonstandard, Mickens‐type, finite‐difference schemes for singular boundary value problems in cylindrical or spherical coordinates

Numerical Methods Partial

2003

It is well known that standard finite-difference schemes for singular boundary value problems involving the Laplacian have difficulty capturing the singular (ᏻ(1/r) or ᏻ(log r)) behavior of the solution near the origin (r ϭ 0). New nonstandard finite-difference schemes that can capture this behavior exactly for certain singular boundary value problems encountered in theoretical aerodynamics are presented here. These schemes are special cases of nonstandard finite differences which have been extensively researched by Professor Ronald E. Mickens of Clark Atlanta University in their most general form. Several examples of these "Mickens-type" finite differences that illustrate both their accuracy and utility for singular boundary value problems in both cylindrical and spherical co-ordinates are investigated. The numerical results generated by the Mickens-type schemes are compared favorably with solutions obtained from standard finite-difference schemes.

“…In Buckmire's 1994 thesis [6] this MFD was introduced in order to find particular slender bodies of revolution that possess shock-free flows as specific numerical solutions of a mixed-type, singular boundary value problem. The problem is formulated using transonic small disturbance theory found in [12], [13] and [14], among other sources. Cole & Schwendeman announced the first computation of a fore-aft, symmetric, shock-free transonic slender body in [16].…”

Section: The Buckmire Mfd Schemesmentioning

confidence: 99%

Applications of Mickens Finite Differences to Several Related Boundary Value Problems

Advances in the Applications of Nonstandard Finite Difference Schemes

2005

The initial discovery and implementation by the author of a particular kind of nonstandard finite difference (NSFD) scheme called a "Mickens finite difference" (MFD) for approximating the radial derivatives of the Laplacian in cylindrical coordinates is reviewed. The development of a similar scheme for the spherical coordinates case is also recounted. Examples of application of the schemes to several related (singular and nonsingular, linear and nonlinear) boundary value problems are given. Examples of applying Buckmire's MFD scheme to the bifurcatory, nonlinear eigenvalue problems of Bratu and Gel'fand are also presented. The results support the utility and versatility of MFD schemes for boundary value problems with singularities or bifurcations.