Mathematical Modeling of Unsteady Inviscid Flows

This book will cover many aspects of inviscid incompressible flows. But rather than proceed directly, let us first provide ourselves a roadmap of sorts in order to clarify the role of each ensuing chapter in the journey. The best such roadmap is an illustrative example containing nearly all of the components of an unsteady inviscid flow.

In this chapter, we establish some notation and conventions to be followed throughout the book regarding a body and its motion. We rely on three different types of notation to describe concepts and solve problems in this book: vector notation, generally useful in both two- and three-dimensional contexts; complex notation, for problems in the plane; and a less familiar notation called Plücker notation that facilitates our analyses of rigid bodies. These are described, and related to each other, in the following sections. First, we discuss the reference frames we will utilize in the book.

This book is focused on inviscid, incompressible flows generated by or interacting with impenetrable bodies in motion. This chapter is focused on presenting many of the basic concepts and physical laws on which the rest of this book depends. Some of these topics are standard fare in a graduate-level text on fluid dynamics. However, there are also some topics that are less commonly treated, so these are presented in somewhat more detail.

The fundamental solutions described in Sects. 3.​2.​2 and 3.​2.​3 serve as the foundation for calculating more sophisticated incompressible flows. This is because the governing equations for the potentials—Laplace’s equation, or strictly speaking, Poisson’s equation—are linear, and therefore admit superposition of the elemental solutions. We have already made use of this in the previous chapter in our construction of flows induced by distributions of singularities in a surface (the vortex sheet, and single- and double-layer potentials).

In Sect. 4.​5 we found that there are an infinite number of possible solutions to the potential flow about a two-dimensional body, since any choice of bound circulation will leave the no-penetration condition unaffected. We observed in that discussion that we could pin this bound circulation to a unique value by setting it equal and opposite to the circulation in the surrounding fluid, in order to ensure that the global circulation is constantly zero. But how did that fluid circulation get there in the first place? We avoided this question during the discussion in Sect. 4.​5. However, the reader will we recall that in Chap. 1 we presented the broad outlines of a strategy for introducing vorticity into the fluid in an inviscid flow. We will fill in the details of that strategy now, and remind ourselves of its physical foundations.

Thus far in this book, we have focused our attention on two principal aspects of inviscid flow: first, on obtaining the flow field about a moving body amidst a general distribution of vorticity in the surrounding fluid; and second, on computing the resulting fluid dynamic force and moment exerted on the body in this scenario. We have thus far had little to say about the motion of this vorticity. However, our analysis of the problem is incomplete until we have addressed this motion, particularly because—as we found in our discussion of impulse-based calculations in Chap. 6—the force and moment depend on the time rate of change of the vorticity’s distribution.

In the previous chapters, we have presented a general set of tools with which to obtain the flow field and the force and moment due to inviscid flow about a rigid body. In this chapter, we consider an important special case of this general problem: the rigid-body motion of an infinitely-thin flat plate of length c in two dimensions. As usual, we assume that the flow is incompressible and irrotational, except for one or more isolated vortex elements, as depicted in Fig. 8.1. We will first build from the results of Chap. 4 to seek an expression for the velocity field for this flow; then we will apply the results from Chap. 6 to obtain the force and moment on the plate in Sect. 8.3. Then, in Sect. 8.4, we will demonstrate the application of the Kutta condition to regularize the flow at just the trailing edge, or at both edges, and discuss the consequences of both. With all of these tools specialized to the particular geometry of a flat plate, we will find it straightforward in Sect. 8.5 to derive four of the classical results from unsteady aerodynamics, pertaining to small-amplitude perturbations of the plate’s motion or to the fluid environment through which it is traveling.

In Chap. 8 we devoted our attention to the mathematics of flow about a two-dimensional flat plate of infinitesimal thickness. Then, we utilized these tools to develop some of the well-known results from unsteady aerodynamics. These results were based on a quite restrictive assumption: that the flow was generated by a small-amplitude disturbance to the flow past the plate traveling at small angle of attack. Under this assumption, we were able to obtain analytical expressions for the flow field and the associated force and moment.

In contrast to the two-dimensional context, there are relatively few exact solutions for inviscid flows about three-dimensional bodies. However, there is one quite general class of problems for which analytical solution is possible: the irrotational flow about an ellipsoidal body. This might seem quite a limited set of results. But it is useful to note that the extremes of this family include infinitely thin ellipses and slender rods, both of which have meaningful roles. The flat elliptical shapes, in particular, provide the rare opportunity for analytical access to a sharp edge of a three-dimensional body.