Small perturbation technique solves the problem of an oscillating cone in supersonic flow. The logic of the program is straightforward, as reflected in the actual instructions for solving the problem

Pao, T.-H.

English

NASA Technical Reports Server

December 1966	 Brief 66-10700 
NASA Tech Briefs are issued to summarize specific innovations derived from the U.S. space program, 
to encourage their commercial application. Copies are available to the public at 15 cents each 
from the Clearinghouse for Federal Scientific and Technical Information, Springfield, Virginia 22151. 
Problem of Oscillating Cone in Supersonic Flow

Is Solved by Small Perturbation Techniques 
The calculations for the flow field about a circular 
cone in steady supersonic flow with or without angle 
of attack is well known. For the case of the unsteady 
flow about an oscillating cone, the situation is some-
what different. In general, the governing equations 
cannot be simplified to any great extent. If, however, 
the motion is assumed simple harmonic, the ampli-
tude of the oscillation is assumed small, and the non-
linear terms are neglected, then the equations can 
be separated and a set of recurring linear differential 
equations is obtained. When these equations are 
solved numerically together with the appropriate 
boundary conditions, the solution of the flow field 
around an oscillating circular cone in supersonic flow 
is obtained. 
A detailed discussion of this problem is contained 
in NASA contractor report, "Solution of the Super-
sonic Flow Field Around an Oscillating Circular 
Cone," by Pao-Tan Hsu, NASA CR-61064, Massa-
chusetts Institute of Technology, December 1964. 
The unsteady perturbations are expressed in series 
form in powers of the radial distance and the co-
efficients of each term are expressed as functions of 
the inclination angle. By means of linearization, a 
system of linear differential equations is obtained for
each order of the power series. The solution for each 
set of equations is obtained by numerical integration, 
using the proper initial conditions given at the shock. 
For any arbitrary mode of oscillatory motion of the 
cone, the solution can be obtained by solving for the 
proper combination of the shock motions. The force 
and moment coefficients are calculated in terms of the 
coefficients of the power series. 
Note: 
Copies of the report are available from: 
Technology Utilization Officer 
Marshall Space Flight Center 
Huntsville, Alabama 35812 
Reference: B66-10700 
Patent status: 
Inquiries about obtaining rights for the commercial 
use of this invention may be made to NASA, Code 
GP, Washington, D.C. 20546.
Source: Pao-Tan Hsu
of Massachusetts Institute of Technology
under contract to
Marshall Space Flight Center
(M-FS-869) 
Category 02 
This document was prepared under the sponsorship of the National 	 Government assumes any liability resulting from the use of the 
Aeronautics and Space Administration. Neither the United States 	 information contained in this document, or warrants that such use 
Government nor any person acting on behalf of the United States 	 will be free from privately owned rights.
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Problem of oscillating cone in supersonic flow is solved by small perturbation techniques

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