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.

Coupled sets of ordinary differential equations (ODEs) are used to describe the evolution of the interactions among chemical species as well as many other local processes. ODEs appear, for example, when spectral and other expansion methods are used to solve timedependent partial differential equations. In these cases, spatial derivatives are converted to algebraic relationships leaving ODEs to be integrated in time. ODEs also describe the motions of projectiles and orbiting bodies, population dynamics, electrical circuits, local temperature equilibration, momentum interchange among phases in multiphase flows, the decomposition of radioactive material, and energy level and species conversion processes in atomic, molecular, and nuclear physics.

Algorithms for integrating ODEs were not originally derived by numerical analysts or applied mathematicians, but by scientists interested in solving specific sets of equations for their particular applications. Bashforth and Adams (1883), for example, developed a method for solving the equations describing capillary action. One of the first algorithms to cope with the difficulties of integrating stiff ODEs was suggested by Curtiss and Hirschfelder (1952) for chemical kinetics studies. Ten years after Curtiss and Hirschfelder identified the stiffness problem in ODEs, Dahlquist (1963) identified numerical instability as the cause of the difficulty and provided basic definitions and concepts that are still helpful in classifying and evaluating algorithms. The importance of the practical applications has spurred active research in developing and testing integration methods for solving coupled ODEs. Continued efforts of applied mathematicians have put the numerical solution of ODEs on a sounder theoretical basis and have provided insights into the constraints imposed by stability, convergence, and accuracy requirements.

Boundary conditions are used to represent an infinitely large region of space using a finite computational domain, to describe boundary layers near walls, and to simulate details of chemical reactions, heat transfer, and other surface effects. Developing the correct boundary conditions to use in a numerical model involves complicated physical and numerical issues that make it relatively easy to make conceptual and programming errors. It is necessary to determine the correct boundary conditions to apply, how they should be implemented numerically, and whether there are inconsistencies between these boundary conditions and the description of the physical system within the computational domain. Although it is not always necessary to understand the solution completely to model the interior of a computational domain, implementing physically reasonable and sufficiently consistent boundary conditions requires a strong understanding of the interior and exterior phenomena and how they interact.

Interfaces are internal boundaries that have structure and can move with the flow. When interfaces are present, they greatly increase the complexity of the simulation. Additional physical processes that are not present within the flow field, such as surface tension, evaporation, condensation, or chemical reactions, can be important at these interfaces. Often the behavior of the interface has to be modeled phenomenologically as part of a much larger overall problem. This occurs when there are orders of magnitude difference in the gradients perpendicular to and parallel to the interface. For example, a shock may have a radius of curvature of centimeters or meters, but it may be only a micron thick. The layer in which ice melts and sublimates is only a fraction of a millimeter thick.

Reactive flows include a broad range of phenomena, such as flames, detonations, chemical lasers, the earth's atmosphere, stars and supernovae, and perhaps even the elementary particle interactions in the very early stages of the universe. There are striking physical differences among these flows, even though the general forms of the underlying equations are all quite similar. Therefore, considerations and procedures for constructing numerical models of these systems are also similar. The obvious and major differences are in the scales of the phenomena, the input data, the mathematical approximations that arise in representing different contributing physical processes, and the strength of the coupling among these processes.

For example, in flames and detonations, there is a close coupling among the chemical reactions, subsequent heat release, and the fluid dynamics, so that all of the processes must be considered simultaneously. In the earth's upper atmosphere, which is a weakly ionized plasma in a background neutral wind, the chemical reactions among ionized gases and the fluid dynamics are weakly coupled. These reactions take place in the background provided by the neutral gas motions. The sun's atmosphere is highly ionized, with reactions among photons, electrons, and ionized and neutral atomic species, all in the presence of strong electromagnetic fields. A Type Ia supernova creates the heavier elements in the periodic table through a series of strongly coupled thermonuclear reactions that occur in nuclear flames and detonations. The types of reactions, the major physical processes, and the degree and type of coupling among the processes vary substantially in these systems. Sometimes reactions are essentially decoupled from the fluid flow, sometimes radiation is important, and sometimes diffusive transport effects are important.