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R. & M . N o . 3717
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F![,,,,. " f' MINISTRY OF DEFENCE (PROCUREMENT EXECUTIVE)
AERONAUTICAL RESEARCH COUNCIL REPORTS AND MEMORANDA

An Exact Test Case for the Plane Potential Flow About Two Adjacent Lifting Aerofoils
By B. R. WILLIAMS Aerodynamics Dept., R.A.E. Farnborough

LONDON: HER MAJESTY'S STATIONERY OFFICE 1973 PRICE£1"40 NET

An Exact Test Case for the Plane Potential Flow About Two Adjacent Lifting Aerofoils
By B. R. WILLIAMS Aerodynamics Dept., R.A.E. Farnborough

Reports and Memoranda No. 3717* September, 1971

Summary A method for calculating the incompressible potential flow about two particular aerofoil sections is presented. The potential flow about two lifting circles is calculated by the method of images, and the two circles are mapped conformally on to two aerofoils by a double application of the Karman-Trefftz transformation. The results for the test cases are then compared with those from a numerical method, which uses a surface distribution of sources.

LIST OF CONTENTS Introduction Flow about Two Lifting Circles 2.1. The streaming flow 2.2. The circulating flow 2.3. The combination flow 3. The Conformal Transformations 3.1. Karman-Trefftz profiles 3.2. Transformation of two circles 4. Results 4.1. Test cases 4.2. Comparison with A.M.O. Smith method 4.3. Variation of the distance between the aerofoils 5. Conclusion Symbols References Appendix I--Image of a doublet and a vortex in a circle Appendix II--The geometry of two circles Appendix III--Convergence of the'series for the streaming flow Appendix IV--Convergence of series for the circulating flow Tables 1 to 3 Illustrations--Figs. 1 to 14 Detachable Abstract Cards i. 2. *Replaces R.A.E. Technical Report 71197--A.R.C.33 661.

1. Introduction At present, the plane incompressible potential flow about two lifting aerofoils is calculated by numerical methods. These methods can be applied to aerofoils of arbitrary shape. However, no exact analytic solution is generally