EQUATIONS

In the previous two chapters the fundamental fluid dynamic equations were formulated and the conditions leading to the simplified inviscid, incompres­sible, and irrotational flow problem were discussed. In this chapter, the basic methodology for obtaining the elementary solutions to this potential flow problem will be developed. Because of the linear nature of the potential flow problem, the differential equation does not have to be solved individually for flowfields having different geometry at their boundaries. Instead, the elemen­tary solutions will be distributed in a manner that will satisfy each individual set of geometrical boundary conditions.

This approach, of distributing elementary solutions with unknown strength, allows a more systematic methodology for resolving the flowfield in both of the cases of “classical” and numerical methods.

3.1 STATEMENT OF THE POTENTIAL FLOW PROBLEM

For most engineering applications the problem requires a solution in a fluid domain V that usually contains a solid body with additional boundaries that may define an outer flow boundary (e. g., a wing in a wind tunnel), as shown in Fig. 3.1. If the flow in the fluid region is considered to be incompressible and

irrotational then the continuity equation reduces to

У2Ф = 0

For a submerged body in the fluid, the velocity component normal to the body’s surface and to other solid boundaries must be zero, and in a body fixed coordinate system:

УФ•n = 0

Here n is a vector normal to the body’s surface, and УФ is measured in a frame of reference that is attached to the body. Also, the disturbance created by the motion should decay far (r—»°°) from the body

lim (УФ – v) = 0

where r = (x, y, z) and v is the relative velocity between the undisturbed fluid in V and the body (or the velocity at infinity seen by an observer moving with the body).