Smith 



3. Fluid particles that at any time are part of a vortex line always belong 

 to that same vortex line. 



Statements 2 and 3 are two of the Helmholtz vortex theorems. 



4. Slip is permitted because of the inviscid character of the fluid, but if the 

 walls of the bodies are impervious, fluid at any point is displaced in a direction 

 normal to the surface with a velocity equal to the velocity of the surface along 

 that normal. If V's is the vector velocity of any point of the surface and n the 

 unit normal vector, this kinematic condition can be stated very compactly as 



^-''■"^ (11) 



An obvious and easily handled modification of Eq. (11) would accomodate 

 mass — transfer types of problems. 



5. If pressures are desired, the unsteady Bernoulli equation must be used. 



From the unsteady Bernoulli equation the following formula for the pressure co- 

 efficient Cp can be derived for a translating and rotating frame of reference 

 fixed in a particular body: 



Here v^ is the magnitude of velocity of any point on the body, u^, is the refer- 

 ence velocity, and v^ is the relative velocity of the fluid. In steady flow, v^ is 

 constant at all points on the body, and it is natural that the reference velocity U„ 

 becomes Vj.. Then the first term reduces to 1. Also, in steady flow Bqp/Bt = o, 

 and so one recovers the common formula C = i - (V^ u^)^. 



Implementation of the method involves solution of boundary -value problems 

 that fall into three classes with respect to computing procedure. Figure 5 

 shows them. The two bodies are assumed to be moving in some sort of path, 

 and leave vortex wakes as sketched. We must find a total solution that meets 

 the condition of no-flow through the walls (if impervious), satisfies Kutta con- 

 ditions if required, and accounts for wake and interaction effects. The total 

 solution can be built up from those shown. The first solution is called the quasi- 

 steady flow cpQ. Here, the bodies may be considered as translating and rotating, 

 each in its own way, along separate paths, with arbitrary velocities. Then every 

 point on each of the bodies must satisfy the fundmental bovmdary condition Eq. (11). 

 The tpQ solution is the nonhomogeneous solution because Bcp /Bn is not zero. In 

 this solution no attempt is made to satisfy the Kutta condition. 



The other two basic flows are the ones needed to satisfy the Kutta condi- 

 tions and the conservation of vorticity. If the bodies are changing their lift, vor- 

 ticity is shed in a continuous sheet. For practical computing purposes, the con- 

 tinuous sheet is approximated by a series of discrete vortices as indicated in the 

 middle sketch of Fig. 5. Each one produces its own onset flow on both bodies, 



330 



