# Three-Dimensional. Incompressible Flow Treat nature in terms of the cylinder, the sphere, the cone, all in perspective.

6.1 Introduction

To this point in our aerodynamic discussions, we have been working mainly in a two-dimensional world; the flows over the bodies treated in Chapter 3 and the airfoils in Chapter 4 involved only two dimensions in a single plane—so-called planar flows. In Chapter 5, the analyses of a finite wing were carried out in the plane of the wing, in spite of the fact that the detailed flow over a finite wing is truly three-dimensional. The relative simplicity of dealing with two dimensions, (i. e., having only two independent variables), is self-evident and is the reason why a large bulk of aerodynamic theory deals with two-dimensional flows. Fortunately, the two-dimensional analyses go a long way toward understanding many practical flows, but they also have distinct limitations.

The real world of aerodynamic applications is three-dimensional. However, because of the addition of one more independent variable, the analyses generally become more complex. The accurate calculation of three-dimensional flow fields has been, and still is, one of the most active areas of aerodynamic research.

The purpose of this book is to present the fundamentals of aerodynamics. There­fore, it is important to recognize the predominance of three-dimensional flows, al­though it is beyond our scope to go into detail. Therefore, the purpose of this chapter is to introduce some very basic considerations of three-dimensional incompressible

flow. This chapter is short; we do not even need a road map to guide us through it. Its function is simply to open the door to the analysis of three-dimensional flow.

The governing fluid flow equations have already been developed in three dimen­sions in Chapters 2 and 3. In particular, if the flow is irrotational, Equation (2.154) states that

V = V0 [2.154]

where, if the flow is also incompressible, the velocity potential is given by Laplace’s equation:

V20 = 0 [3.40]

Solutions of Equation (3.40) for flow over a body must satisfy the flow-tangency boundary condition on the body, that is,

V • n = 0 [3.48a]

where n is a unit vector normal to the body surface. In all of the above equations, ф is, in general, a function of three-dimensional space; for example, in spherical coordinates ф = ф(г, в, Ф). Let us use these equations to treat some elementary three-dimensional incompressible flows.