For the small-disturbance solution techniques that are treated in this book, approximations to the exact mathematical problem formulation are made to facilitate the determination of a solution. Since for incompressible and irrotational flow the governing partial differential equation is linear, the approximations are made to the body boundary condition. For example, for the three-dimensional wing in Chapter 4, only terms linear in thickness, camber, and angle of attack are kept and the boundary condition is transferred to the x–y plane. The solution technique is therefore a “first-order” thin wing theory.

The small-disturbance methods developed here can be thought of as providing the first term in a perturbation series expansion of the solution to the exact mathematical problem and terms that were neglected in determining the first term will come into play in the solution for the following terms. In this book we will follow the lead of Van Dyke and use the thin-airfoil problem as the vehicle for the presentation of the ideas and some of the details of perturbation methods and their applicability to aerodynamics. First, the thin-airfoil solution will be introduced as the first term in a small-disturbance expansion and the mathematical problem for the next term will be derived. An example of a second-order solution will be presented and the failure of the expansion in the leading-edge region will be noted. A local solution applicable in the leading-edge region will be obtained and the method of matched asymptotic expansions will be used to provide a solution valid for the complete airfoil.

In Chapter 1 it was established that for flows at high Reynolds number the effects of viscosity are effectively confined to thin boundary layers and thin wakes. For this reason our study of low-speed aerodynamics will be limited to flows outside these limited regions where the flow is assumed to be inviscid and incompressible. To develop the mathematical equations that govern these flows and the tools that we will need to solve the equations it is necessary to study rotation in the fluid and to demonstrate its relationship to the effects of viscosity.

It is the goal of this chapter to define the mathematical problem (differential equation and boundary conditions) of low-speed aerodynamics whose solution will occupy us for the remainder of the book.

Angular Velocity, Vorticity, and Circulation

The arbitrary motion of a fluid element consists of translation, rotation, and deformation. To illustrate the rotation of a moving fluid element, consider at t = t0 the control volume shown in Fig. 2.1. Here, for simplicity, we select an infinitesimal rectangular element that is being translated in the z = 0 plane by a velocity (u, v) of its corner no. 1. The lengths of the sides, parallel to the x and y directions, are Δx and Δy, respectively. Because of the velocity variations within the fluid the element may deform and rotate, and, for example, the x component of the velocity at the upper corner (no. 4) of the element will be (u + (∂u/∂y)Δy), where higher order terms in the small quantities Δx and Δy are neglected.

It was demonstrated in the previous chapters that the solution of potential flow problems over bodies and wings can be obtained by the distribution of elementary solutions. The strengths of these elementary solutions of Laplace's equation are obtained by enforcing the zero normal flow condition on the solid boundaries. The steps toward a numerical solution of this boundary value problem are described schematically in Section 9.7. In general, as the complexity of the method is increased, the “element's influence” calculation becomes more elaborate. Therefore, in this chapter, emphasis is placed on presenting some of the typical numerical elements upon which some numerical solutions are based (the list is not complete and an infinite number of elements can be developed). A generic element is shown schematically in Fig. 10.1. To calculate the induced potential and velocity increments at an arbitrary point P(xP, yP, zP) requires information on the element geometry and strength of singularity.

For simplicity, the symbol Δ is dropped in the following description of the singularity elements. However, it must be clear that the values of the velocity potential and velocity components are incremental values and can be added up according to the principle of superposition.

In the following sections some two-dimensional elements will be presented, whose derivation is rather simple. Three-dimensional elements will be presented later and their complexity increases with the order of the polynomial approximation of the singularity strength.

The differential equations that are generally used in the solution of problems relevant to low-speed aerodynamics are a simplified version of the governing equations of fluid dynamics. Also, most engineers when faced with finding a solution to a practical aerodynamic problem, find themselves operating large computer codes rather than developing simple analytical models to guide them in their analysis. For this reason, it is important to start with a brief development of the principles upon which the general fluid dynamic equations are based. Then we will be in a position to consider the physical reasoning behind the assumptions introduced to generate simplified versions of the equations that still correctly model the aerodynamic phenomena being studied. It is hoped that this approach will give the engineer the ability to appreciate both the power and the limitations of the techniques that will be presented in this text. In this chapter we will derive the conservation of mass and momentum balance equations and show how they are reduced to obtain the equations that will be used in the rest of the text to model flows of interest to the low-speed aerodynamicist.

Description of Fluid Motion

The fluid being studied here is modeled as a continuum, and infinitesimally small regions of the fluid (with a fixed mass) are called fluid elements or fluid particles. The motion of the fluid can be described by two different methods. One adopts the particle point of view and follows the motion of the individual particles.

The discussion so far has focused mainly on the potential flow model whose solution provides a useful but restricted description of the flow. For practical problems such as the flow over an airfoil, however, effects of the viscous flow near the solid surface must be included. The objective of this chapter, therefore, is to explain how a viscous boundary layer model can be combined with the inviscid flow model to provide a more complete representation of the flowfield. These principles can be demonstrated by using the laminar boundary layer model, which provides all the necessary elements for combining the viscous and inviscid flow models. We must remember, though, that the Reynolds number of the flow over actual airplanes or other vehicles is such that large portions of the flow are turbulent, and the solely laminar flow model must be augmented to reflect this. However, the principles of the matching process remain similar. Extensions of this laminar boundary layer based approach to flows with transition, to turbulent boundary layers, or to cases with flow separation, and other aspects of airfoil design, will be discussed briefly in Chapter 15. (Although in these cases the viscous flow model may change substantially from the laminar model, the viscous–inviscid coupling strategy remains unchanged.)

Boundary layer theory is a very wide topic and there are several textbooks that focus solely on this subject (e.g., see Ref. 1.6).

Our goal in writing this book is to present a comprehensive and up-to-date treatment of the subject of inviscid, incompressible, and irrotational aerodynamics. Over the last several years there has been a widespread use of computational (surface singularity) methods for the solution of problems of concern to the low-speed aerodynamicist and a need has developed for a text to provide the theoretical basis for these methods as well as to provide a smooth transition from the classical small-disturbance methods of the past to the computational methods of the present. This book was written in response to this need. A unique feature of this book is that the computational approach (from a single vortex element to a three-dimensional panel formulation) is interwoven throughout so that it serves as a teaching tool in the understanding of the classical methods as well as a vehicle for the reader to obtain solutions to complex problems that previously could not be dealt with in the context of a textbook. The reader will be introduced to different levels of complexity in the numerical modeling of an aerodynamic problem and will be able to assemble codes to implement a solution.

We have purposely limited our scope to inviscid, incompressible, and irrotational aerodynamics so that we can present a truly comprehensive coverage of the material. The book brings together topics currently scattered throughout the literature. It provides a detailed presentation of computational techniques for three-dimensional and unsteady flows.

Toward the end of Chapter 1 (Section 1.8) it is postulated that many flowfields of interest to the low-speed fluid dynamicist lie in the range of high Reynolds number. Consequently, for attached flowfields, the fluid is divided into two regions: (a) the thin inner boundary layer and (b) the mainly inviscid irrotational outer flow. Chapters 2–13 are entirely devoted to the solution of the inviscid outer flow problem, which indeed is capable of estimating the resulting pressure distribution and lift due to the shape of the given solid boundaries. The laminar boundary layer model was presented in Chapter 14 as an example for modeling the inner part of the complete flowfield. The methodology for obtaining information such as the displacement thickness, the skin friction on the solid surface and resulting drag force (due to surface friction), and the matching process with the outer flow was demonstrated. However, in real high Reynolds number flows over wings the flow is mostly turbulent and the engineering approach to extend the methodology of Chapter 14 to include turbulent or even separated viscous layer models will be discussed briefly in this chapter. The objective of this chapter is to provide a brief survey of some frequently occurring low-speed (wing-related) flowfields and to help the student to place in perspective the relative role of the potential flow methods (presented in this book) and of the viscous effects in order to comprehend the complete real flowfield environment.

In the previous chapters the solution to the potential flow problem was obtained by analytical techniques. These techniques (except in Chapter 6) were applicable only after some major geometrical simplifications in the boundary conditions were made. In most of these cases the geometry was approximated by flat, zero-thickness surfaces and for additional simplicity the boundary conditions were transferred, too, to these simplified surfaces (e.g., at z = 0).

The application of numerical techniques allows the treatment of more realistic geometries and the fulfillment of the boundary conditions on the actual surface. In this chapter the methodology of some numerical solutions will be examined and applied to various problems. The methods presented here are based on the surface distribution of singularity elements, which is a logical extension of the analytical methods presented in the earlier chapters. Since the solution is now reduced to finding the strength of the singularity elements distributed on the body's surface this approach seems to be more economical, from the computational point of view, than methods that solve for the flowfield in the whole fluid volume (e.g., finite difference methods). Of course this comparison holds for inviscid incompressible flows only, whereas numerical methods such as finite difference methods were basically developed to solve the more complex flowfields where compressibility and viscous effects are not negligible.

Basic Formulation

Consider a body with known boundaries SB, submerged in a potential flow, as shown in Fig. 9.1.

Our goal in writing this Second Edition of Low-Speed Aerodynamics remains the same, to present a comprehensive and up-to-date treatment of the subject of inviscid, incompressible, and irrotational aerodynamics. It is still true that for most practical aerodynamic and hydrodynamic problems, the classical model of a thin viscous boundary layer along a body's surface, surrounded by a mainly inviscid flowfield, has produced important engineering results. This approach requires first the solution of the inviscid flow to obtain the pressure field and consequently the forces such as lift and induced drag. Then, a solution of the viscous flow in the thin boundary layer allows for the calculation of the skin friction effects.

The First Edition provides the theory and related computational methods for the solution of the inviscid flow problem. This material is complemented in the Second Edition with a new Chapter 14, “The Laminar Boundary Layer,” whose goal is to provide a modern discussion of the coupling of the inviscid outer flow with the viscous boundary layer. First, an introduction to the classical boundary-layer theory of Prandtl is presented. The need for an interactive approach (to replace the classical sequential one) to the coupling is discussed and a viscous–inviscid interaction method is presented. Examples for extending this approach, which include transition to turbulence, are provided in the final Chapter 15.

In addition, updated versions of the computational methods are presented and several topics are improved and updated throughout the text.

As described in the previous chapter, the term reactive flow applies to a very broad range of physical phenomena. In some cases the equations are not even rigorously known. In this chapter, we first consider the equations of gas-phase reactive flows, which are generally accepted as valid in the continuum regime. This set of time-dependent, coupled, partial differential equations governs the conservation of mass and species density, momentum, and energy. The equations describe the convective motion of the fluid, reactions among the constituent species that may change the molecular composition, and other transport processes such as thermal conduction, molecular diffusion, and radiation transport. Many different situations are described by these equations when they are combined with various initial and boundary conditions. In a later section of this chapter, we discuss interactions among these processes and generalizations of this set of equations to describe multiphase reactive flows.

The material presented in this chapter is somewhat condensed, and is not meant to give an in-depth explanation to those unfamiliar with the individual topics. The purpose is to present the reactive-flow equations, to establish the notation used throughout this book, and then to relate each term in the equations to physical processes important in reactive flows. The chapter can then be used as a reference for the more detailed discussions of numerical methods in subsequent chapters. It would be reasonable to skim this chapter the first time through the book, and then to refer back to it as needed.

This chapter presents and analyzes the properties of the simplest finite-difference methods for simulating four of the main physical processes in reactive flows: chemical reactions, diffusion, convection, and wave motion. The material presented is an overview and short course on solving idealized forms of the equations representing these processes. The discussion highlights the features and weaknesses of these solution methods and brings out numerical difficulties that reappear in solutions of the complete set of reactive-flow conservation equations. Throughout the presentation, we list and describe the major computational and algorithmic trade-offs that arise in simulating each process separately.

The material presented here introduces the more advanced solution techniques described in Chapters 5 through 9. Chapter 11 deals with techniques for solving the coupled set of equations that forms the reactive Navier-Stokes equations discussed in Chapter 2. In particular, Chapter 11 shows how the disparate time and space scales of each type of process can be used to determine a reasonable overall timestep for the computation. The choice of the numerical boundary conditions that are so crucial for correctly defining the physical problem, are discussed in Chapters 5 through 9. Sections 10–1 and 10–2 are devoted to issues of selecting boundary conditions for the reactive-flow equations.

Table 4.1 shows the mathematical representations discussed in this chapter and indicates where the numerical solutions for more complex forms of these equations are discussed elsewhere in this book. There are many references on numerical methods and scientific computation for science and engineering that cover material complementary to this chapter.