# Category AERODYNAMICS

## CHOICE OF COORDINATE SYSTEM

For the following chapters, when possible, primarily a cartesian coordinate system will be used. Other coordinate systems such as curvilinear, cylindrical, spherical, etc., will be introduced and used if necessary, mainly to simplify the treatment of certain problems. Also, from the kinematic point of view, a careful choice of a coordinate system can considerably simplify the solution of a problem. As an example, consider the forward motion of an airfoil, with a constant speed Ux, in a fluid that is otherwise at rest—as shown in Fig. 1.1. Here, the origin of the coordinate system is attached to the moving airfoil and the trajectory of a fluid particle inserted at point P0 at t = 0 is shown in the figure. By following the trajectories of several particles, a more complete description of the flowfield is obtained in the figure. It is important to ob­serve that for a constant-velocity forward motion of the airfoil, in this frame of reference, these trajectory lines become independent of time. That is, if various particles are introduced at the same point in space, then they will follow the same trajectory.

Now let’s examine the same flow, but from a coordinate system that is fixed relative to the undisturbed fluid. At t = 0, the airfoil was at the right side of Fig. 1.2 and as a result of its constant-velocity forward motion (with a speed U«, towards the left side of the page), later at r = /, it has moved to the new position indicated in the figure. A typical particle’s trajectory line between

 Particle trajectory

 Airfoil position at / = t,

 11

 f = 0

FIGURE 1.2

Particle trajectory line for the airfoil of Fig. 1.1 as viewed from a stationary inertial frame.

/ = 0 and tlt for this case, is shown in Fig. 1.2. The particle’s motion now depends on time and a new trajectory has to be established for each particle.

This simple example depicts the importance of “good” coordinate system selection. For many problems where a constant velocity and a fixed geometry (with time) are present, the use of a body-fixed frame of reference will result in a steady or time-independent flow.

## DESCRIPTION OF FLUID MOTION

The fluid being studied here is modeled as a continuum and infinitesimally small regions of the fluild (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 other adopts the field point of view and provides the flow variables as functions of position in space and time.

The particle point of view, which uses the approach of classical mechanics, is called the Lagrangian method. To trace the motion of each fluid particle, it is convenient to introduce a cartesian coordinate system with the coordinates де, у, and z. The position of any fluid particle P (see Fig. 1.1) is then given by

x = xP(x0, y0, Zo, t)

У =Уг(хо, Уо, Zo> 0 (1.1)

z = Zp(x0, y0, Zo, t)

where (jc0, y0, Zq) is the position of P at some initial time t = 0. (Note that the quantity (jc0, уо, Zo) represents the vector with components jc0, y0, and z0.) The components of the velocity of this particle are then given by

 Эх dy dz (1.2) u~ dt v = — at w = — at cPx cPy &z (1.3) ‘ at2 Яу’ at2 *г at2
 and the acceleration by

The Lagrangian formulation requires the evaluation of the motion of each fluid particle. For most practical applications this abundance of informa­tion is neither necessary nor useful and the analysis is cumbersome.

The field point of view, called the Eulerian method, provides the spatial

distribution of flow variables at each instant during the motion. For example, if a cartesian coordinate system is used, the components of the fluid velocity are given by

и = n(x, y, z, t)

v = v(x, y, z, t) (1.4)

w = w(x, y, z, t)

The Eulerian approach provides information about the fluid variables that is consistent with the information supplied by most experimental techniques and is in a form that is appropriate for most practical applications. For these reasons the Eulerian description of fluid motion is the most widely used.

## Aerospace Engineering

Joseph Katz is Professor of Aerospace Engineering and Engineering Mechan­ics at San Diego State University where he has been a faculty member since 1986. He received the degrees of BSc, MSc and DSc, the latter in 1977, in Aeronautical Engineering from the Technion-Israel Institute of Technology. He was a faculty member in the Mechanical Engineering Department of the Technion from 1980-1984 and headed the Automotive Program from 1982- 1984. He spent 1978-1980 and 1984-1986 at the .Large Scale Aerodynamics Branch of NASA-Ames Research Center as a Research Associate and Senior Research Associate, respectively, and has maintained his ties to NASA through grant support. He has worked in the 40′ by 80′ full scale wind tunnel and has developed a panel method capable of calculating three-dimensional unsteady flowfields and applied it to complete aircraft and race car configura­tions. He is the author of more than 40 journal articles in computational and experimental aerodynamics.

Allen Plotkin is Professor of Aerospace Engineering and Engineering Mechan­ics at San Diego State University where he has been a faculty member since 1985. He graduated from the Bronx High School of Science, received BS and MS degrees from Columbia University and a PhD from Stanford University in 1968. He was a faculty member in the Department of Aerospace Engineering of the University of Maryland from 1968-1985. In 1976 he received the Young Engineer-Scientist Award from the National Capital Section of the AIAA and in 1981 received the Engineering Sciences Award from the Washington Academy of Sciences. He is an Associate Fellow of the AIAA and served two terms as an associate editor of the AIAA Journal from 1986-1991. He is the current contributor to the World Book Encyclopedia articles on Aerodynamics, Propeller, Streamlining, and Wind Tunnel. He is the author of approximately 40 journal articles in aerodynamics and fluid mechanics.

2

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. 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. It includes a systematic and detailed treatment (including computer programs) of two-dimensional panel methods with variations in singularity type, order of singularity, Neumann or Dirichlet boundary conditions, and velocity- or potential-based approaches.

This book is divided into three main parts. In the first, Chapters 1-3, the basic theory is developed. In the second part, Chapters 4-8, an analytical approach to the solution of the problem is taken. Chapters 4, 3, and 8 deal with the small-disturbance version of the problem and the classical methods of

xffi

thin-airfoii theory, lifting-line theory, slender wing theory, and slender body theory. In this part exact solutions via complex variable theory and perturba­tion methods for obtaining higher-order small-disturbance approximations are also included. The third part, Chapters 9-14, presents a systematic treatment of the surface singularity distribution technique for obtaining numerical solutions for incompressible potential flows. A general methodology for assembling a numerical solution is developed and applied to a series of increasingly complex aerodynamic elements (two-dimensional, three – dimensional, and unsteady problems are treated).

The book is designed to be used as a textbook for a course in low-speed aerodynamics at either the advanced senior or the first-year graduate levels. The complete text can be covered in a one-year course and a one-quarter or one-semester course can be constructed by choosing the topics that the instructor would like to emphasize. For example, a senior elective course that concentrated on two-dimensional steady aerodynamics might include Chapters 1-3, 4, 5, 9, 11, 8, 12, and 14. A traditional graduate course that emphasized an analytical treatment of the subject might include Chapters 1-3, 4, 5-7, 8, 9, and 13; and a course that emphasized a numerical approach (panel methods) might include Chapters 1-3 and 9-14 with a treatment of pre- and post­processors. It has been assumed that the reader has taken a first course in fluid mechanics and has a mathematical background that includes an exposure to vector calculus, partial differential equations, and complex variables.

We believe that the topics covered by this text are needed by the fluid dynamicist because of the complex nature of the fluid dynamic equations, which has led to a mainly experimental approach for dealing with most engineering research and development programs. In a wider sense, such an approach uses tools such as wind tunnels or large computer codes where the engineer/user is experimenting and testing ideas with some trial-and-error logic in mind. Therefore, even in the era of supercomputers and sophisticated experimental tools, there is a need for simplified models that allow for an easy grasp of the dominant physical effects (e. g., having a simple lifting vortex in mind, one can immediately tell that the first wing in a tandem formation has the larger lift).

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. This methodology has been used successfully throughout the twentieth century for most airplane and marine vessel designs. Recently, due to developments in computer capacity and speed, the inviscid flowfield over complex and detailed geometries (such as airplanes, cars, etc.) can be computed by this approach (Panel methods). Thus, for the near future, since these methods are the main tools of low-speed aerodynamicists all

over the world, a need exists for a clear and systematic explanation of how and why (and for which cases) these methods work. This book is one attempt to respond to this need.

We would like to thank graduate students Lindsey Browne and especially Steven Yon who developed the two-dimensional panel codes in Chapter 11 and checked the integrals in Chapter 10. We would like to acknowledge the helpful comments from the following colleagues who read all or part of the manuscript: Holt Ashley, Richard Margason, Turgut Sarpkaya, and Milton Van Dyke. Allen Plotkin would like to thank his teachers Richard Skalak, Krishnamurthy Karamcheti, Milton Van Dyke, and Irmgard Flugge-Lotz, his parents Claire and Oscar Plotkin for their love and support, and his children Jennifer Anne and Samantha Rose, and especially his wife Selena for their love, support and patience. Joseph Katz would like to thank his parents Janka and Jeno Katz, his children Shirley, Ronny, and Danny, and his wife Hilda for their love, support, and patience. The support of the Low-Speed Aerodynamic Branch at NASA Ames is acknowledged by Joseph Katz for their inspiration that initiated this project and for their help during past years in the various stages of developing the methods presented in this book.

McGraw-Hill and the authors would like to thank the following reviewers for their many helpful comments and suggestions: Leland A. Carlson, Texas A&M University; Chuen-Yen Chow, University of Colorado; Fred R. De Jarnette, North Carolina State University; Barnes W. McCormick, Penn­sylvania State University; and Maurice Rasmussen, University of Oklahoma.

Joseph Katz Allen Plotkin

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 analytic 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 that are 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.