PANEL DISCUSSION-LIFTING-SURFACE THEORY 



R. Timman, Panel Chairman 

 Delft Institute of Technology \,._ ..' . 



Consultant Netherlands Ship Model Basin / •/ 



The discussion mainly consisted of three parts: (a) mathematical methods 

 and foundation of the theory, (b) experimental verification, and (c) applications, 

 in particular to sails and design of propellors. 



Timman: The aim of this lifting- surface panel is an appraisal of the theory, 

 a discussion on its physical foundation, a general outline of mathematical methods, 

 and considerations on its applicability to practical problems. 



The origin of lifting- surface theory dates back a long time. The Birnbaum 

 series is from 1923, In the old days nobody ever tried to solve the two- 

 dimensional integral equation because of the formidable amount of work re- 

 quired. For this reason airplane wings and ship propellors were calculated by 

 lifting-line methods, based on Prandtl's formulation. Now, "exact" lifting sur- 

 face theories are available, "exact" meaning a two-dimensional, linearized, non- 

 viscous lifting- surface theory. 



First is mentioned the theory of Tsakonas, and its counterpart, developed by 

 Verbrugh (a joint effort of N.S.M.B. and Hydronautics-Europe). The theory of 

 Tsakonas is a rather complete: Starting from the two-dimensional integral equa- 

 tion for the acceleration potential a numerical method is developed where the wake 

 is simplified by taking stepwise constant distributions of free vortices, whereas 

 Verbrugh' s report, based on Sparenberg's theory, contains helicoidal wake vor- 

 tices. Both methods use chordwise series of Birnbaum type and derive spanwise 

 integral equations for the coefficients. Both require a special treatment of the 

 Hadamard singularity in the integral equation, but Tsakonas makes a more ex- 

 tensive use of expansions in special functions, whereas Verbrugh uses more di- 

 rect numerical methods. 



Verbrugh' s report dates from April 1968, but is not published because of ad- 

 ministrative difficulties; Tsakonas' latest publication is in the April issue of the 

 Journal of Ship Research. It will be of great interest to correlate the two theories. 

 Since the starting points are the same, discrepancies must be due to numerical 

 deviations. 



Now it is proposed to discuss the value of these theories. Suppose they 

 agree; (if they do not agree, it is only a matter of time before these differences 

 are eliminated) we have available an accurate method for the solution of the 

 linearized nonviscous integral equation. The computing time is about 40 minutes 

 on the TR4 (somewhat less on a IBM 7090) and probably shorter on a third- 

 generation computer. 



1593 



