Recent Progress in the Calculation of Potential Flows 



UNSTEADY TWO-DIMENSIONAL FLOWS ,, > ■ r .. - , ., "^ 



General Remarks 



Because the method just described can handle nearly any kind of boundary 

 condition with great accuracy*, it is quite capable of treating unsteady flows, 

 which involve unusual boundary conditions. One major complication develops if 

 the bodies are capable of developing lift — airfoils, for example. In the process 

 of changing its lift, the body must shed vorticity equal and opposite to that 

 gained on the body itself. If the fluid is originally at rest, the fluid has no 

 vorticity; if the fluid is inviscid, vorticity is conserved no matter what the mo- 

 tion of the body may be. Any positive vorticity developed on the body or bodies 

 must therefore be balanced by equal but opposite vorticity off the bodies, so 

 that the total remains zero. Hence vortex sheets are shed. Now since they are 

 likely to distort with time, their position is unknown in advance, and our prob- 

 lem takes on a new aspect — nonlinearity. J. P. Giesing has been working on 

 problems of unsteady two-dimensional flow since about 1965, and I wish to de- 

 scribe here briefly his method and some of his results. Work on the one-body 

 problem was published in Ref. 4 and work on the two-body problem in Ref. 5. 



Description of the Method 



Because the two-body problem is more complicated, our description will be 

 of this type of the method. The one-body problem is a great deal simpler and 

 allows many time-saving specializations, especially if the body never changes 

 its shape, because influence coefficients can be calculated once and for all. As 

 will be seen, the treatment is a step-by-step process. Therefore it is at its 

 best in handling transients, although steady periodic motion can also be analyzed, 

 but at greater expense in computer time. In the two-body problem the bodies 

 are assumed to be moving in an inviscid, incompressible fluid. In the existing 

 computer programs the bodies may move relative to each other in arbitrary 

 paths with arbitrary velocities. It would not be very difficult to extend the 

 method to solve problems involving bodies whose shape changes with time. A 

 vibrating plane flap on a hydrofoil or the swimming of a two-dimensional "fish" 

 are examples solvable by an extended program. 



If certain fundamental facts are not forgotten, the concept upon which the 

 analysis is founded appears rather simple, although the execution is difficult. 

 These facts are: 



1. The flow is potential. ' 



2. No fluid particle can have a rotation if it did not originally rotate. 



=An independent assessment of the accuracy of this and several other airfoil 

 methods has recently been compiled in England (D. N. Foster, "Note on Methods 

 of Calculating the Pressure Distribution over the Surface of Two-Dimensional 

 Cambered Wings," Royal Aircraft Establishment Technical Report 67095, April 

 1967). Of the truly general methods considered, the present was found to be 

 the most accurate. 



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