FORMULATION OF THE PROBLEM

One of the first important applications of potential flow theory was the study of lifting surfaces (wings). Since the boundary conditions on a complex surface can considerably complicate the attempt to solve the problem by analytical means, some simplifying assumptions need to be introduced. In this chapter these assumptions will be applied to the formulation of the steady three – dimensional thin wing problem and the scene for the singularity solution technique will be set.

4.1 DEFINITION OF THE PROBLEM

Consider the finite wing shown in Fig. 4.1, which is moving at a constant speed in an otherwise undisturbed fluid. A cartesian coordinate system is attached to the wing and the components of the free-stream velocity Q„ in the x, y, z frame of reference are (/„, V„, and W„, respectively. (Note that the flow is steady in this coordinate system.) The angle of attack a is defined as the angle between the free-stream velocity and the x axis and for the sake of simplicity side slip is not included at this point (VM = 0).

FIGURE 4.1

Nomenclature used for the definition of the finite wing problem.

If it is assumed that the fluid surrounding the wing and the wake is inviscid, incompressible and irrotational, the resulting velocity field due to the motion of the wing can be obtained by solving the continuity equation

V2O* = 0 (4.1)

where Ф* is the velocity potential, as defined in the wing frame of reference. (Note that Ф* is the same as Ф in Chapter 3 and the reason for introducing this notation will become clear in the next section.) The boundary conditions require that the disturbance induced by the wing will decay far from the wing:

lim УФ* = Q« (4.2)

Г—

which is automatically fulfilled by the singular solutions (derived in Chapter 3) such as the source, doublet, or the vortex elements. Also, the normal component of velocity on the solid boundaries of the wing must be zero. Thus, in a frame of reference attached to the wing,

УФ* • n = 0 (4.3)

where n is an outward normal to the surface (Fig. 4.1). So, basically, the problem reduces to finding a singularity distribution that will satisfy Eq. (4.3). Once this distribution is found, the velocity q at each point in the field is known and the corresponding pressure p will be calculated from the steady – state Bernoulli equation:

P~ + ^QZ = P+^<12 (4-4)

The analytical solution of this problem, for an arbitrary wing shape, is complicated by the difficulty of specifying boundary condition of Eq. (4.3) on a curved surface, and by the shape of a wake. The need for a wake model follows immediately from the Helmholtz theorems (Section 2.9), which state that vorticity cannot end or start in the fluid. Consequently, if the wing is modeled by singularity elements that will introduce vorticity (as will be shown
later in this chapter), these need to be “shed” into the flow in the form of a wake.

To overcome the difficulty of defining the zero normal flow boundary condition on an arbitrary wing shape some simplifying assumptions are made in the next section.