Potential flow

Preamble

The aim of this chapter is to describe methods for calculating the air flow around various shapes of body. The classical assumption of irrotational flow is made, meaning that the vorticity is everywhere zero. This also implies inviscid flow. Irrotational flows are potential fields. A potential function, known as the velocity potential, is introduced. It is shown how the velocity components can be determined from the velocity potential. The equations of motion for irrotational flow reduce to a single partial differential equation for velocity potential known as the Laplace equation. Classical analytical techniques are described for obtaining two-dimensional and axisymmetric solutions to the Laplace equation for aerodynamic applications. The chapter ends by showing how these classical analytical solutions can be used to develop computational methods for predicting the potential flows around the complex three­dimensional geometries typical of modern aircraft.

3.1 Introduction

The concept of irrotational flow is introduced briefly in Section 2.7.6. By definition the vorticity is everywhere zero for such flows. This does not immediately seem a very significant simplification. But it turns out that zero vorticity implies the existence of a potential field (analogous to gravitational and electric fields). In aerodynamics the main variable of the potential field is known as the velocity potential (it is analogous to voltage in electric fields). And another name for irrotational flow is potential flow. For such flows the equations of motion reduce to a single partial differential equa­tion, the famous Laplace equation, for velocity potential. There are well-known techniques (see Sections 3.3 and 3.4) for finding analytical solutions to Laplace’s equation that can be applied to aerodynamics. These analytical techniques can also be used to develop sophisticated computational methods that can calculate the potential flows around the complex three-dimensional geometries typical of modern aircraft (see Section 3.5).

In Section 2.7.6 it was explained that the existence of vorticity is associated with the effects of viscosity. It therefore follows that approximating a real flow by a potential flow is tantamount to ignoring viscous effects. Accordingly, since all real fluids are viscous, it is natural to ask whether there is any practical advantage in

studying potential flows. Were we interested only in bluff bodies like circular cylin­ders there would indeed be little point in studying potential flow, since no matter how high the Reynolds number, the real flow around a circular cylinder never looks anything like the potential flow. (But that is not to say that there is no point in studying potential flow around a circular cylinder. In fact, the study of potential flow around a rotating cylinder led to the profound Kutta-Zhukovski theorem that links lift to circulation for all cross-sectional shapes.) But potential flow really comes into its own for slender or streamlined bodies at low angles of incidence. In such cases the boundary layer remains attached until it reaches the trailing edge or extreme rear of the body. Under these circumstances a wide low-pressure wake does not form, unlike a circular cylinder. Thus the flow more or less follows the shape of the body and the main viscous effect is the generation of skin-friction drag plus a much smaller component of form drag.

Potential flow is certainly useful for predicting the flow around fuselages and other non-lifting bodies. But what about the much more interesting case of lifting bodies like wings? Fortunately, almost all practical wings are slender bodies. Even so there is a major snag. The generation of lift implies the existence of circulation. And circul­ation is created by viscous effects. Happily, potential flow was rescued by an important insight known as the Kutta condition. It was realized that the most important effect of viscosity for lifting bodies is to make the flow leave smoothly from the trailing edge. This can be ensured within the confines of potential flow by conceptually placing one or more (potential) vortices within the contour of the wing or aerofoil and adjusting the strength so as to generate just enough circulation to satisfy the Kutta condition. The theory of lift, i. e. the modification of potential flow so that it becomes a suitable model for predicting lift-generating flows is described in Chapters 4 and 5.