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7 - Axisymmetric, Incompressible Flow around a Body of Revolution
- Gary A. Flandro, Howard M. McMahon, Robert L. Roach
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- Basic Aerodynamics
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- 05 February 2012
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- 14 November 2011, pp 281-308
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Summary
Introduction
The flow considered in this chapter is assumed to be steady, incompressible, inviscid, and irrotational. The body immersed in the flow is assumed to be a body of revolution at zero angle of attack. An understanding of incompressible flow around bodies of revolution at zero or small angle of attack is important in several practical applications, including airships, aircraft and cruise-missile fuselages, submarine hulls, and torpedoes, as well as flows around aircraft engine nacelles and inlets. This type of flow problem is best handled in cylindrical coordinates (x, r), as shown in Fig. 7.1. Recall that r and θ lie in the y-z plane.
Because the flow fields discussed in this chapter are axisymmetric, the flow properties depend on only the axial distance x from the nose of the body (assumed to be at the origin in most cases) and the radial distance, r, away from this axis of symmetry. The flow properties are independent of the angle θ. As a result, we may examine the flow in any (x-r) plane because the flow in all such planes is identical due to the axial symmetry. It is convenient to develop the defining equations initially in cylindrical coordinates (i.e., dependence on x, r, and θ) and then to simplify them for axisymmetric flow (i.e., dependence on x, r only).
Preface
- Gary A. Flandro, Howard M. McMahon, Robert L. Roach
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- Basic Aerodynamics
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- 05 February 2012
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- 14 November 2011, pp ix-x
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Preface
This textbook presents the fundamentals of aerodynamic analysis. Major emphasis is on inviscid flows whenever this simplification is appropriate, but viscous effects also are discussed in more detail than is usually found in a textbook at this level. There is continual attention to practical applications of the material. For example, the concluding chapter demonstrates how aerodynamic analysis can be used to predict and improve the performance of flight vehicles. The material is suitable for a semester course on aerodynamics or fluid mechanics at the junior/senior undergraduate level and for first-year graduate students. It is assumed that the student has a sound background in calculus, vector analysis, mechanics, and basic thermodynamics and physics. Access to a digital computer is required and an understanding of computer programming is desirable but not necessary. Computational methods are introduced as required to solve complex problems that cannot be handled analytically.
The objective of this textbook is to present in a clear and orderly manner the basic concepts underlying aerodynamic-prediction methodology. The ultimate goal is for the student to be able to use confidently various solution methods in the analysis of practical problems of current and future interest. Today, it is important for the student to master the basics because technology is advancing at such a rate that a more directed or specific approach often is rapidly outdated. In this book, the basic concepts are linked closely to physical principles so that they may be understood and retained and the limits of applicability of the concepts can be appreciated. Numerous example problems stress solution methods and the order of magnitude of key parameters. A comprehensive set of problems for home study is included at the end of each chapter.
Contents
- Gary A. Flandro, Howard M. McMahon, Robert L. Roach
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- Basic Aerodynamics
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- 14 November 2011, pp vii-viii
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8 - Viscous Incompressible Flow
- Gary A. Flandro, Howard M. McMahon, Robert L. Roach
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- Basic Aerodynamics
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- 05 February 2012
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- 14 November 2011, pp 309-392
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Summary
External aerodynamics was a disturbingly mysterious subject before Prandtl solved the mystery with his work on boundary layer theory from 1904 onwards.
L. Rosenhead, Laminar Boundary Layers, Oxford 1963Introduction
This chapter examines the role of viscosity in the flow of fluids and gases. Although the viscosity of air is small, it must be included in a flow model if we are to explain wing stall and frictional drag, for example. The four preceding chapters are concerned with the analysis of airfoils, wings, and bodies of revolution based on an assumption of inviscid flow (i.e., negligible viscous effects). The inviscid-flow model allowed analytical solutions to be developed for predicting, with satisfactory accuracy, the pressure distribution on bodies of small-thickness ratio at a modest (or zero) angle of attack. However, the inviscid-flow model leads to results that are at odds with experience, such as the prediction that the drag of two-dimensional airfoils and right-circular cylinders is zero. This contradiction is resolved by realizing that actual flows exhibit viscous effects.
Viscosity is discussed from a physical viewpoint in Chapter 2. In Chapters 5, 6, and 7, the existence of viscosity is acknowledged when it is necessary to advance an analytical derivation for an inviscid flow. Also, viscous effects are called on, with words like viscous drag and separation, when comparing the predicted and observed behavior of airfoils and wings. However, no analysis in this textbook has been developed thus far that provides the required detailed physical basis for these effects.
1 - Basic Aerodynamics
- Gary A. Flandro, Howard M. McMahon, Robert L. Roach
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- Basic Aerodynamics
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- 05 February 2012
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- 14 November 2011, pp 1-14
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This was the last first in aviation, we had always said, a milestone, and that made it unique. Would we do it again? No one can do it again. And that is the best thing about it.
Jeana Yeager and Dick Rutan “Voyager” 1987Introduction
Aerodynamics is the study of the flow of air around and within a moving object. Its main objective is understanding the creation of forces by the interaction of the gas motion with the surfaces of an object. Aerodynamics is closely related to hydrodynamics and gasdynamics, which represent the motion of liquid and compressible-gas flows, respectively.
Aerodynamics is the essence of flight and has been the focus of intensive research for about a century. Although this might seem to be a rather long period of development, it is really quite short considering the time span usually required for the formulation and full solution of basic scientific problems. In this relatively short time, mankind has advanced from the first gliding and primitive-powered airplane flights to interplanetary spaceflight.
2 - Physics of Fluids
- Gary A. Flandro, Howard M. McMahon, Robert L. Roach
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- Basic Aerodynamics
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- 14 November 2011, pp 15-47
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Summary
Aerodynamic Forces
Because the objective of aerodynamics is the determination of forces acting on a flying object, it is necessary that we clearly identify their source. Lift and drag forces, for example, are the result of interactions between the airflow and vehicle surfaces. Part of the force must be a result of pressure variations from point to point along the surface; another part must be related to friction of gas particles as they scrub the surface. Clearly, the key to understanding these forces is found in details of the fluid motions. The application of simple molecular concepts provides considerable insight into these motions.
Modeling of Gas Motion
As a branch of fluid mechanics, aerodynamics is concerned with the motion of a continuously deformable medium. That is, when acted on by a constant shear force, a body of liquid or gas changes shape continuously until the force is removed. This is unlike a solid body, which only deforms until internal stresses come into equilibrium with the applied force; that is, a solid does not deform continuously.
5 - Two-Dimensional Airfoils
- Gary A. Flandro, Howard M. McMahon, Robert L. Roach
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- Basic Aerodynamics
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- 05 February 2012
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- 14 November 2011, pp 169-217
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9 - Incompressible Aerodynamics: Summary
- Gary A. Flandro, Howard M. McMahon, Robert L. Roach
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- Basic Aerodynamics
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- 05 February 2012
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- 14 November 2011, pp 393-418
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Summary
It is now becoming clear that it is also mistaken to assume that computers could produce optimum designs in an empirical manner: it cannot be carried out in practice.
D. Küchemann, “The Aerodynamic Design of Aircraft”, Pergamon Press, 1978Introduction
The preceding eight chapters take wholesale advantage of the assumption that the flow field for low-speed flight is incompressible. This allows considerable simplification in the formulation of the governing equations and in the solution of key aerodynamic problems. However, results of the calculations are limited in an important way that is emphasized in this summary chapter. What we attempt to do here is:
Summarize the most important elements of the first eight chapters.
Demonstrate how the results are incorporated in actual vehicle design.
Define the limits of application of the results.
Modeling of Airflows
What is accomplished to this point is the application of basic fluid mechanics in contructing detailed models for the airflow over aerodynamic surfaces (e.g., wings and bodies) at speeds low enough that compressibility effects do not seriously affect the results. These models are intended to provide accurate estimates of the aerodynamic forces and moments needed in solving the basic problem of aerodynamics as it was defined in Chapter 1. Although there is much discussion centered on the application of modern computational tools, for the most part, we rely on simplified mathematical representations. We try to emphasize the role of valid, simplifying assumptions in arriving at useful representations for the airflow. As Küchemann described the process in his famous book on the aerodynamic design of aircraft (Küchemann, 1978), “... the most drastic simplifying assumptions must be made before we can even think about the flow of gases and arrive at equations which are amenable to treatment. Our whole science lives on highly idealized concepts and ingenious abstractions and approximations.” First-class examples of this approach are demonstrated in this book, including Prandtl's elegant models describing the creation of lift by an airfoil, three-dimensional wing theory, and boundary-layer flows. These provide the backbone of the subject of aerodynamics.
Frontmatter
- Gary A. Flandro, Howard M. McMahon, Robert L. Roach
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- Basic Aerodynamics
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- 14 November 2011, pp i-v
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4 - Fundamentals of Steady, Incompressible, Inviscid Flows
- Gary A. Flandro, Howard M. McMahon, Robert L. Roach
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- Basic Aerodynamics
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- 14 November 2011, pp 110-168
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Summary
Introduction
In this chapter, solutions of the conservation equations in partial-differential equation form are sought for a simple case—namely, steady, incompressible, inviscid two-dimensional flow. Each of these crucial assumptions is discussed in detail and their applicability as models of real flow-field situations are justified. Body forces such as gravity effects are neglected because they are negligible in most aerodynamics problems. Simple geometries are considered first. The analysis is then extended so that finally it is possible to represent the complex flow field around realistic airfoil shapes, such as those needed to efficiently produce lift forces for flight vehicles. Chapter 5 is a detailed treatment of two-dimensional airfoil flows.
The intention here is to obtain solutions valid throughout the entire flow field; hence, the differential-conservation equations are integrated so as to work from the small (i.e., the differential element) to the large (i.e., the flow field). In this regard, the integral form of the conservation equations is not a useful starting point because in steady flow, the integral equations describe events over the surface of only some fixed control volume. We are seeking detailed information regarding the pressure and velocity fields at any point in the flow. What are the implications of each assumption listed previously?
Steady flow. The assumption of steady flow enables the definition of a streamline as the path traced by a fluid particle moving in the flow field, from which it follows that a streamline is a line in the flow that is everywhere tangent to the local velocity vector. Also, all time-derivative terms in the governing equations can be dropped; this results in a much simpler formulation.
Incompressible flow. The assumption of incompressible flow means that the density is assumed to be constant. As shown herein, and as the conservation equations in Chapter 3 indicate, the assumption of incompressibility in a problem leads to enormous simplifications. The obvious one is that terms in the equations containing derivatives of density are zero. The other major simplification is that the number of equations to be solved is reduced. If the density is constant, then there cannot be large variations in temperature, and the temperature may be assumed to be constant as well. With density and temperature no longer variables, the equation of state and the energy equation may be set to one side and the continuity and momentum equations solved for the remaining variables—namely, velocity and pressure.In other words, for incompressible flows, the equation of state and the energy equation may be uncoupled from the continuity and momentum equations. It is true that no fluid (liquid or gas) is absolutely incompressible; however, at low speeds, the variation in density of an airflow is small and can be considered essentially incompressible. For example, considerations of compressible flow show that at a Mach number of 0.3 (a velocity of 335 ft/s, or 228 mph, at sea level), the maximum possible change in density in a flow field is about 6 percent and the maximum change in temperature of the flow is less than 2 percent. For flows of this velocity or less, the incompressible assumption is good. However, at Mach number 0.5 (558 ft/s, or 380 mph, at sea level), the maximum change in density in a flow field is almost 19 percent. An incompressible-flow assumption for such a case leads to prohibitive errors.Results from an assumed incompressible flow around thin airfoils or wings and around slender bodies provide a foundation for the prediction of the flow around these bodies at higher, compressible-flow Mach numbers (i.e., less than unity). It turns out that the effects of compressibility on pressure distribution, lift, and moment at flow Mach numbers less than 1 can be expressed as a correction factor times a related incompressible flow value. Thus, results using the incompressible model are useful not only for low-speed flight, they also provide a database for the accurate prediction of vehicle operation at much higher (but subsonic) speeds.
Inviscid flow. The inviscid-flow assumption means physically that viscous-shear and normal stresses are negligible. Thus, all of the viscous shear-stress terms on the force side of the momentum equations drop out, as well as the normal stresses due to viscosity. As a result, the only stresses acting on the body surface are the normal stresses due to pressure. Recall from Chapter 3 that when considering incompressible viscous-flow theory (see Chapter 8), the viscous-shear stresses are assumed to be proportional to the rate of strain of a fluid particle, with the constant of proportionality as the coefficient of viscosity. Thus, an assumption equivalent to that of negligible viscous stresses is the assumption that the coefficient of viscosity is essentially zero. Such a flow is termed inviscid (i.e., of zero viscosity). In effect, the boundary layer on the surface of the body is deleted by this assumption. This implies that the boundary layer must be very thin compared to a dimension of the body and that the presence or absence of the boundary layer has a negligible effect relative to modifications to the body geometry as “seen” by the flow.The inviscid, incompressible-fluid model is often termed a perfect fluid (not to be confused with a perfect or ideal gas as defined in Chapter 1). The boundary layer in many practical situations is extremely thin compared to a typical dimension of the body under study such that the body shape that a viscous flow “sees” is essentially the geometric shape. The exception is where the flow separates and the boundary layer leaves the body, resulting in a major change in the effective geometry of the body. Such separated regions occur on wings, for example, at large angles of attack. However, the wing angle of attack of a vehicle at a cruise condition is only a few degrees so that the effects of separation are minimal. Thus, the inviscid-flow assumption provides useful results that match closely the experiment for conditions corresponding to cruise, and the inviscid-flow model breaks down when large regions of separated flow occur.Because the presence of the boundary layer is neglected in perfect-fluid theory, the theory does not predict the frictional drag of a body; that must be left to viscous-flow theory. However, within the framework of incompressible inviscid flow, predictions for low-speed pressure distribution, lift, and pitching moment are valid and useful.
Two-dimensional flow. The assumption of two-dimensional flow is a simplifying assumption in that it reduces the vector-component momentum equations from three to two. Two-dimensional simply means that the flow (and the body shape) is identical in all planes parallel to, say, a page of this book; there are no variations in any quantity in a direction normal to the plane. Consider a cylinder or wing extending into and out of this page, with each cross section of the body exactly the same as any other. The flow around the body in all planes parallel to the page are then identical. It follows that the cylinder in two-dimensional flow has an infinite axis length and the wing has an infinite span. Any cross section of this wing of infinite span is termed an airfoil section. Theoretical predictions for such an airfoil may be validated by experiments in a wind tunnel in which the wing model extends from one wall to the opposite wall. If the wing/wall interfaces are properly sealed, the model then behaves as if it were a wing of infinite span—that is, as if it has no wing tips around which there would be a flow due to the difference in pressure between the top and bottom surfaces of the wing. Theoretical results for an airfoil (i.e., a two-dimensional problem) form the basis for predicting the behavior of wings of finite span (i.e., a three-dimensional problem) because each cross section (i.e., airfoil section) of the finite wing is assumed to behave as if the flow around it were locally two-dimensional (see Chapters 5 and 6). Thus, two-dimensional results have considerable value. Most (but not all) of the concepts discussed in this chapter may be extended to three dimensions and/or to compressible flow. Such extensions are introduced at appropriate points.
Index
- Gary A. Flandro, Howard M. McMahon, Robert L. Roach
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- Basic Aerodynamics
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- 14 November 2011, pp 419-422
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Basic Aerodynamics
- Incompressible Flow
- Gary A. Flandro, Howard M. McMahon, Robert L. Roach
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- 14 November 2011
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In the rapidly advancing field of flight aerodynamics, it is especially important for students to master the fundamentals. This text, written by renowned experts, clearly presents the basic concepts of underlying aerodynamic prediction methodology. These concepts are closely linked to physical principles so that they are more readily retained and their limits of applicability are fully appreciated. Ultimately, this will provide students with the necessary tools to confidently approach and solve practical flight vehicle design problems of current and future interest. This book is designed for use in courses on aerodynamics at an advanced undergraduate or graduate level. A comprehensive set of exercise problems is included at the end of each chapter.
3 - Equations of Aerodynamics
- Gary A. Flandro, Howard M. McMahon, Robert L. Roach
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- Basic Aerodynamics
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- 14 November 2011, pp 48-109
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Summary
Introduction
To solve the fundamental problems of aerodynamics defined in Chapter 1, it is necessary to formulate a mathematical representation of the underlying fluid dynamics. The appropriate mathematical expressions or sets of equations may be algebraic, integral, or differential in character but will always represent basic physical laws or principles. In this chapter, the fundamental equations necessary for the solution of aerodynamics problems are derived directly from the basic laws of nature. The resulting mathematical formulations represent a large class of fluid mechanics problems within which aerodynamics is an important subclass.
Some problems in aerodynamics require solutions for all of the variables needed to describe a moving stream of gas—namely, velocity, pressure, temperature, and density. Because velocity is a vector quantity (i.e., with magnitude and direction), in a general case there are three scalar velocity components. Thus, in many cases of interest, there is a total of six unknowns: three velocity components and the scalar thermodynamic quantities of pressure, temperature, and density.* This requires six independent equations to be written to solve for the six unknowns. The physical laws of conservation of mass, momentum, and energy supply five such equations (i.e., the momentum equation is a vector equation; therefore, conservation of momentum leads in general to three component equations). For all of the subject matter in this book, the assumption of an ideal gas is physically realistic. Thus, the perfect gas law (i.e., equation of state) p = ρRT, which relates pressure, density, and temperature, supplies the final equation needed to solve for the six unknowns.
6 - Incompressible Flow about Wings of Finite Span
- Gary A. Flandro, Howard M. McMahon, Robert L. Roach
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- Basic Aerodynamics
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- 05 February 2012
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- 14 November 2011, pp 218-280
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In 1908, Lanchester visited Göttingen (University), Germany and fully discussed his wing theory with Ludwig Prandtl and his student, Theodore von Kàrmàn. Prandtl spoke no English, Lanchester spoke no German, and in light of Lanchester's unclear ways of explaining his ideas, there appeared to be little chance of understanding between the two parties. However, in 1914, Prandtl set forth a simple, clear, and correct theory for calculating the effect of tip vortices on the aerodynamic characteristics of finite wings. It is virtually impossible to assess how much Prandtl was influenced by Lanchester, but to Prandtl must go the credit …
John D. Anderson, Jr., Introduction to Flight, 1978Introduction
This chapter considers steady, inviscid, incompressible flow about a lifting wing of arbitrary section and planform. Because the flow around a wing is not identical at all stations between the two ends of the wing, the lifting finite wing constitutes a three-dimensional flow problem. The two wing tips are located at distance ±b/2, where b is the wing span.
Certain terms must be defined before a study of finite wings can be begun (Fig. 6.1). The coordinate axis system used is shown in Fig. 6.1a. A wing section is defined as any cross section of a wing as viewed in any vertical plane parallel to the x-z plane. It also is called an airfoil section. The wing may be of constant section or variable section. If a wing is of constant section, wing sections at any spanwise station have the same shape (e.g., NACA 2312). If a wing is of variable section, the wing-section shape varies at different spanwise locations. For example, a wing of variable section might have a NACA 0012 at the root section (i.e., the section in the plane of symmetry at y = 0), then smoothly change in the spanwise direction until the wing had, a NACA 2312 section at the tip.