Recent Progress in the Calculation of Potential Flows 



QUASI STEADY FLOW { 



POINT VORTEX FLOW (i^l'J', q = l) CIRCULATORY FLOW (<^\!il, q = 1) 



Fig. 5 - Shematic of streamlines associated 

 with quasi-steady, point vortex, and circula- 

 tory flow fields 



and the total onset flow is due to the total effect of all the lumped point vortices 

 in both wakes. Strengths of these vortices is determined by the basic facts that 

 total vorticity of the system remains equal to zero and that the Kutta condition 

 must hold. These wake-flow influences create additional onset flows. However, 

 the airfoils are now considered to be at rest, so that the boundary condition 

 over all the surfaces is Bcp/^n = 0. This condition means that the normal veloc- 

 ity due to the surface source distribution is equal and opposite to that created 

 by the vortices in the wake. Hence, since dqp/Bn = 0, the solution may be called 

 a homogeneous solution. It is called cpG, in which G denotes gamma (F). It is 

 identified as a different solution because of the details of the calculation pro- 

 cedure. The method of solving the basic Neumann boundary- value problem is 

 no different, however, because in the end the only difference between it and the 

 method of solving for (pQ is the column matrix, which amounts to no more than 

 a different set of numbers. 



A vortex moves along with the flow. Hence, if there is an array of vortices 

 like that of the middle sketch of Fig. 5, it is evident that the vortices will move 

 as a result of the influences of all the other vortices and the influence of the 

 bodies. Therefore, calculation of the effect of the wake requires knowledge of 

 where the wake is. Differential equations for the motion of the vortices can be 

 written and integrated to find the position of each lumped vortex point. The inter- 

 action of the vortices can be violent, as will be seen in some of the examples. 



The flows cpQ and cpG account for motion of the bodies and effects of the 

 wakes, but the Kutta conditions are not satisfied. The third basic flow is used 

 to satisfy these conditions. It is called cpK, in which K denotes Kutta. As can 

 be seen in the figure, it is a circulatory flow; and in the process of meeting the 

 Kutta conditions, two circulatory flows must be used, one for body 1 and one for 

 body 2. The circulation about an airfoil is generated by a constant vortex sheet 

 of unit strength covering all the surface of the airfoil. (This known vortex sheet 

 is in addition to the unknown source distribution used to satisfy boundary condi- 

 tions.) This method of covering the surface with a constant strength vortex 



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