Smith 



sheet is far from being the only procedure, but it has great practical advantages 

 in computing speed and accuracy. The vortex sheet generates a third type of on- 

 set flow, which again generates a set of values of 3(p/3n, so that the third flow, 

 like the second, has homogeneous boundary conditions; i.e., the body is treated 

 as stationary. The Kutta conditions are then satisfied by the proper linear com- 

 bination of circulatory flows cpK with flows (pQ and qpG. A proper linear combina- 

 tion will satisfy all the boundary conditions, the Kutta conditions, and the 

 vorticity- conservation condition. 



Space is not available to enter into computational details, which in fact are 

 considerable. Both the single-body and two-body problems are programmed on 

 the IBM 7094. For the single-body problem, the body can be defined by as many 

 as 100 coordinate points and 100 time steps can be taken. Often in practical cal- 

 culations, pressures, forces, and moments are not needed at every time step. 

 Trial runs show that the computing time is given approximately by the following 

 formula, if 72 defining elements are used: 



(NT) (NPT) .... 



T.inutes = 2.70 ^ + 5.15 '-^ + 1.0 , (13) 



where NT is the number of time steps taken and NPT is the number of times at 

 which pressures, forces, and moments are wanted. For example, if NT = 60 

 and NPT = 20, the computing time is 14.25 minutes. For the two-body problem, 

 each body can be defined by up to 50 elements, and up to 250 time steps can be 

 taken. Core capacity determines these limits. About 1 minute is required for 

 each time step. Hence, on an IBM 7094, computations can become quite lengthy. 

 A maximum- capacity problem would take over 4 hours. 



Examples of Single-Body Problems 



An Airfoil Whose Angle of Attack is Suddenly Changed- Figure 6 shows what 

 happens to the wake when the angle of attack is suddenly changed. An 8.4 percent - 

 thick symmetric von Mises airfoil first moves at zero angle of attack. Then, 

 after traveling 0.6 chord lengths, the airfoil is suddenly pitched to 10°. It re- 

 mains at this position until total travel is 3.05 chords, at which point it returns 

 to a = 0°. The motion was broken up into steps of length 0.05c, where c is the 

 chord. The figure shows the motion of the wake and the rollup of vortices. It 

 is interesting to note that each vortex carries the other downward, so that there 

 is a net downward flow. This behavior is consistent with momentum considera- 

 tions, which require a definite downward displacement of some fluid if lift is 

 developed for a period of time. 



Wake Shape - A question can be raised regarding the accuracy of the wake shape. 

 Since no exact solutions are available for reference, assessment was made by 

 using different step lengths in the calculation of the motion. In one case, step 

 lengths differing by a factor of 3 gave nearly the same wake shape. J. B. Bratt, 

 Ref. 6, has determined wake shapes behind an oscillating NACA 0015 airfoil by 

 using smoke. The airfoil oscillated up and down with an amplitude of 0.018c 

 without pitch. Test conditions were duplicated as well as possible. A compari- 

 son of calculation and experiment is shown iv. Figure 7 for the same amplitude at 



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