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DEPARTMENT OF MECHANICAL ENGINEERING AND MECHANICS
COLLEGE OF ENGINEERING
OLD DOMINION UNIVERSITY
NORFOLK, VIRGINIA 23529
COMPUTATIONAL TECHNIQUE FOR COMPRESSIBLE VORTEX FLOWS
USING THE INTEGRAL EQUATION SOLUTION
By
Osama A. Kandil, Principal Investigator
Final Report
For the period ended July 31, 1986
Prepared for the
National Aeronautics and Space Administration
Langley Research Center
Hampton, Virginia 23665
Under
Research Grant NAG-1-591
Dr. E. Carson Yates, Jr.
Interdisciplinary Research Office
Submitted by the
Old Doninion University Research Foundation
P. 0. Box 6369
Norfolk, Virginia 23508
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April 1988
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COMPUTATIONAL TECHNIQUE FOR COMPRESSIBLE VORTEX FLOWS
USING THE INTEGRAL EQUATION SOLUTION
By
Osama A. Kandil*
ABSTRACT
This report covers the achievements accomplished under this grant in
the period of May 1985 to July 1986.
The steady full-potential equation is written in the form of Poisson's
equation, and the solution for the velocity field is expressed in terms of
an integral equation. The integral solution consists of two surface inte-
grals and one volume integral. One of the surface integrals is a source
integral term, over the wing surface, which represents the wing thickness
effect while the other surface integral is a vortex integral term, over the
wing and wake surfaces, which represents the lift. The volume integral term
is a source distribution within a small computational volume around the
wing. This term represents the total compressibility contribution of the
flow.
The solution is obtained through successive iteration cycles. Each
cycle of iteration consists of two sub-cycles, an inner cycle for wake re-
laxation and an outer cycle for the strength of the source distribution
integrals representing the flow compressibility. The density gradients in
the source distribution is computed by using a type-differencing scheme of
the Murman-Cole type.
The method is applied to delta wings and the numerical examples show
that a curved shock is captured on the wing suction side beneath the leading
♦Professor, Department of Mechanical Engineering & Mechanics, Old Dominion
University, Norfolk, Virginia 23529.
edge vortex sheet. Recently, a modified version of the scheme has been
applied to rectangular wings. In this modified scheme, the surface integral
terms have computed by using a bilinear distribution of vorticity on
triangular vortex panels which represent the wing and its wake. The results
have been compared with the available experimental data and they are in good
agreement. Details of the scheme and the results are given in the attached
publications.
The following list covers the papers and Ph.D. dissertation which have
been produced under the full or partial support of this grant:
1. Kandil, O.A. and Yates, Jr., E.C., "Computation of Transonic Vortex
Flows Past Delta Wings-Integral Equation Approach," AIAA paper No.
85-1582, Cincinnati, Ohio, July 1985.
2. Kandil, O.A. and Yates, Jr., E.C., "Transonic Vortex Flows Past
Delta Wings: Integral Equation Approach," AIAA Journal, Vol. 24,
No. 11, Nov. 1986, pp. 1729-1736.
3. Kandil, O.A., Chuang, A. and Chu, t-C., "Fi nite-Volume and
Integral-Equation Techniques for Transonic and Supersonic Vortex-
Dominated Flows," ICAS paper No. 86-1.5.4, 15th Congress of the
International Council of the Aeronautical Sciences, London,
England, Sept. 1986.
4. Chu, L-C., "Integral Equation Solution of the Full Potential
Equation for Three-Dimensional, Steady, Transonic Wing Flows,"
Ph.D. Dissertation, Dept, of Mechanical Engineering and Mechanics,
Old Dominion Univ., Norfolk, VA., March 1988; Advisor: Prof. Osama
A. Kandil.
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