This concise and highly readable introduction to theoretical and computational aerodynamics integrates both classical and modern developments, focusing on applying methods to actual wing design. Designed for a junior- or senior-level course and as a resource for practicing engineers, it features 221 figures.
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An Introduction to Theoretical and Computational Aerodynamics
By Jack Moran
Dover Publications, Inc.Copyright © 1984 Jack Moran
All rights reserved.
Most of this book is about the aerodynamics of wings. Therefore, we start by looking at wings, what they do, and how they do it.
The main purpose of a wing is to provide a lift force, the force that gets the airplane off the ground, raises it to an efficient and safe cruising altitude, lets it maneuver about, and allows a safe landing. The same results can be obtained by pointing a rocket or other engine in the right direction, and that, in fact, is how you fly in the absence of an atmosphere. However, when an atmosphere is available, a lot of energy can be saved by using wings. The wing acts as a thrust amplifier, giving you a lifting force many times the force it takes to keep you going.
To make this precise, consider an airplane in straight and level flight at speed V, as shown in Fig. 1.1. The forces on the airplane are its weight W and the aerodynamic forces, the forces exerted by the air on its surfaces. Thrust is the aerodynamic force on the surfaces associated with the propulsive system. Its separation from the other aerodynamic forces is often somewhat arbitrary. For example, in the case of a turbojet-powered aircraft, the usual definition of the engine thrust is the aerodynamic force on all surfaces within the volume enclosed by the engine and surfaces S1 and S2 that cover its inlet and exit, respectively, as shown in Fig. 1.2. However, shouldn't the propulsion-system specialist also be held accountable for the drag on the exterior of the engine nacelle? With a little imagination, you can visualize the pain various definitions of thrust could cause different members of the design "team."
Once the thrust is defined, the aerodynamic forces on the remaining surfaces of the aircraft are resolved into components along and parallel to the direction of flight. Lift is the component perpendicular to the velocity vector V, whereas drag is the component along (but opposite to) V.
For flight at constant speed and altitude, the propulsive system thrust is in rough balance with the drag, and the lift with the weight. Of course, to change direction, the lift must exceed the weight, and the thrust must exceed the drag when you need to accelerate. However, what I meant when I called the wing a "thrust amplifier" is that its lift, which must at least equal the airplane's weight in order for it to get off the ground, can be many times as large as its drag, which must be overcome by the engine's thrust.
This can be seen from Fig. 1.3, taken from Chuprun, who examined data on the performance of various modern (as of 1980) fighter and transport aircraft. The wings of transports are seen to be capable of providing a lift force up to about 35 times their drag, whereas those of fighter aircraft provide lift to drag ratios from 16 to 24. Of course, as is also shown in Fig. 1.3, the lift to drag ratio of the complete airplane is considerably less than that of the wing alone, but it is still more than eight for fighters and about twice that for transports. Thus, for example, the C5A lifts over 750,000 lb with four 41,000-lb thrust engines.
The differences evident in Fig. 1.3 between the performances of the wings of fighters and transports can be understood in terms of their different missions. Since range is directly proportional to the lift/drag ratio, transport wings are designed to give as high a ratio of lift to drag as possible. Fighter aircraft, on the other hand, must be able to change direction rapidly and so must have a high lift/weight ratio. Figure 1.4, also Chuprun's, shows that fighter wings can exert lift forces of up to 90 times their weight, whereas the lift/weight ratio of transport wings is about 22.
Lift and drag are not the only parameters of interest to the aerodynamicist. He or she must also know where the lift acts, what is called the center of pressure of the wing. To fly straight and level, for example, the airplane must be in equilibrium with respect to the moments about its mass center of the forces acting on it; see Fig. 1.5. Also, although our focus in this book is on the aerodynamic performance of wings, it should be noted that a wing has nonaerodynamic design parameters, too. In the first place, it must be strong enough to bear the loads the aerodynamicist wants to impose on it. This demands a thicker wing than the aerodynamicist might like. Think of the wing as a beam. Then its stresses and deflection are seen to go inversely with the moment of inertia of the wing section, which is proportional to the cube of the wing thickness. And the storage function of wings must not be overlooked; most aircraft use the wings to store fuel and also the landing gear. Thus, the aerodynamicist will be well received if she or he can design a wing that not only flies the airplane but is nice and thick.
Having briefly outlined the objectives of wing design, let us now look at the geometric variables at the designer's disposal.
The first is planform; the shape of the wing as viewed from above. As shown in Fig. 1.6, the span of a wing is the distance between its tips. The chord is the dimension of the wing from its leading edge to its trailing edge; generally it varies along the span. The taper of a wing is the ratio of its tip chord to its root chord, the chord at midspan. Another planform variable is the sweep angle Λ, defined as the angle between a line one-quarter of the chord behind the leading edge (the "quarter-chord line") and the spanwise direction.
Probably the most important planform parameter is the aspect ratio of the wing,
AR [equivalent to] span2/area (1-1)
The aspect ratio can be regarded as the ratio of the span to an average chord, which is in turn defined as the ratio of the planform area to the span. This can be seen from Fig. 1.3 of the preceding section to correlate very well the maximum lift to drag ratios of a variety of real wings, and even complete airplanes. Generally, for reasons to be discussed in Chapter 5, the higher the aspect ratio, the higher the ratio of lift to drag. Sailplanes, for which a high lift to drag ratio is of paramount importance, are therefore characterized by very high aspect ratios, limited mainly by structural and weight considerations.
The sweep angle of a wing is usually selected with its design speed in mind. For flight at low speeds, little or no sweep is best. Roughly speaking, it just reduces the aspect ratio for a given weight, while introducing some serious problems in getting performance from all of the wing. As the speed gets close to and beyond the speed of sound, sweep becomes more desirable. Some military aircraft must perform over such a wide speed range that their wings are designed to pivot in flight to change the sweep angle.
Some of the variety of planform shapes that have been used on one airplane or another are shown in Fig. 1.7. I hope the preceding discussion sheds some light on the rationale for their design, although, as Chuprun points out, some geometries "may simply represent the artistic flair of the designer." However, most of the designs shown are rational. In particular, the forward-swept wing has a number of aerodynamic advantages over conventionally swept-rearward designs but has not been used because of an aeroelastic instability called "divergence." The recent advent of composite materials, with their very high strength/weight ratio, may make this design more attractive.
Much work has been done on the design of wing sections or airfoils; that is, on the shape of the wing in planes perpendicular to the span. The analysis and design of airfoils are generally conducted under the assumption of a two-dimensional flow; that is, the wing section is taken to be constant along the span, and the span to be infinite. As will be shown in Chapter 5, results obtained under this assumption can be used for real wings of large aspect ratio and little or no sweep. However, the main reason for the emphasis of this book on two-dimensional problems is that they are easier to deal with than three-dimensional problems and so more suitable for an introductory textbook. On the other hand, emphasis is given to approaches that have been or could be carried over to the three-dimensional world.
It is convenient to describe an airfoil shape in terms of its thickness and camber distributions. As shown in Fig. 1.8, we define a chord line to run from the airfoil's leading edge to its trailing edge and a camber line, which is midway between the airfoil's upper and lower surfaces. The camber of the airfoil is the distance between the camber and chord lines. The thickness may be defined as the dimension of the airfoil in the direction perpendicular to the chord line, or, as in Fig. 1.8, to the camber line; usually it doesn't make too much difference.
As will be seen in Chapter 4, the lift and moment per unit width of a thin airfoil depend mainly on the shape of its camber line and on the angle of attack of the airfoil, the angle between its chord line and the velocity vector of the undisturbed flow relative to the airfoil. The thickness distribution does influence the pressure distribution along the airfoil surface, which, as described in Chapter 7, controls the behavior of the boundary layer on the airfoil, including the phenomenon of stall, which limits the maximum lift an airfoil can generate.
Most airfoil shapes are defined by giving the coordinates of 50 or more points on its surface. In some cases, the camber line is described analytically and the thickness distribution by giving its value at selected points. Two rather famous series of airfoils, the NACA four- and five-digit series, are defined completely by formulas. In both cases, the thickness distribution is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-2)
Here c is the airfoil chord (length of its chord line) and x the distance along the chord line from the leading edge. The parameter τ is the thickness ratio of the airfoil (maximum thickness/chord). This thickness distribution was derived in 1930 by Eastman Jacobs, of the National Advisory Committee for Aeronautics' Langley Laboratory, on the basis of examination of airfoils known to be efficient (in particular, the airfoils known as the Gottingen 398 and the Clark Y).
The camber line of the four-digit airfoils consists of two parabolas joined at the maximum camber point, as shown in Fig. 1.9. If εc is the maximum camber, and pc is the distance between the leading edge and the maximum camber point, we then have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-3)
The first digit of the four-digit airfoil's designation is the maximum camber ratio ε times 100, while the second digit is the chordwise position of the maximum camber p times 10. The last two digits are 100 times the thickness ratio τ. Thus, a NACA 2412 airfoil is 12% thick (τ = 0.12) and has maximum camber ε = 0.02 at x = 0.4c. As indicated in Fig. 1.9, the thickness of the NACA four-digit airfoil is defined in the direction perpendicular to the camber line.
The four-digit airfoils are thus defined by three parameters: τ, p, and ε. Airfoil models with systematic variations in these parameters were tested in the Langley Laboratory's variable-density wind tunnel (which, for the first time, allowed model testing at full-scale Reynolds numbers, by working at pressures of up to 20 atm) in the 1930s. Because their performance was so well documented (and good, too), these airfoils became very popular. When it was found that the two-parabola camber line was not so useful for far-forward positions of the maximum camber, NACA engineers developed the five-digit series, which used the same thickness distribution but a camber line comprised of a cubic polynomial in x up to some point past the maximum camber, followed by a straight line. Modern airfoils have much more complex camber lines. As noted above, their shape is described, not by explicit formulas, but by giving the coordinates of points on their surface. Criteria for airfoil design will be discussed in Chapter 7.
Aside from fighter aircraft, whose maneuverability requirements place an extra demand on the aircraft designer, the lift force required of an airplane wing does not change much over its flight profile. However, as we shall show in Chapter 4, the aerodynamic force on a particular geometry is proportional to the density of the atmosphere and to the square of the airplane's velocity, factors that vary considerably during the course of a flight. Although it seems possible to meet the varying demands on lift by changing the incidence of the aircraft (and its wing) to the direction of flight, some variation in the geometry of the airplane's lifting surfaces (tailplane as well as wing) is required to keep the aircraft "trimmed" (in moment equilibrium) and to control its direction of flight. One of the Wright brothers' major contributions was the invention of a system for varying the camber of their wings by warping them, by tugging on wires attached to the corners of the wing tips. Glenn Curtiss was the first to use hinged ailerons to provide differential lift on the wings, a system that has been used by most aircraft since then (1908), including later products of the Wrights. From the aileron it is but a small step to the flapped wing, which increases the total lift without requiring the whole aircraft to change its inclination to the direction of flight. Flaps are therefore very nice to have during takeoff and landing. At low speeds you need extra performance from the wings to get a given lift force, and you don't want to fly at a large angle from horizontal.
Some of the variety of flap designs that have been and are used are shown in Fig. 1.10, taken from McCormick, who also gives an excellent summary of design data for flaps. Although the man-powered Gossamer Condor and Gossamer Albatross achieved their goals by warping the wings, multielement airfoils are here to stay. Their proper design remains an important and interesting challenge to the aerodynamicist.
Excerpted from An Introduction to Theoretical and Computational Aerodynamics by Jack Moran. Copyright © 1984 Jack Moran. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
2. Review of Basic Fluid Dynamics
3. Incompressible Irrotational Flow About Symmetric Airfoils at Zero Lift
4. Lifting Airfoils in Incompresssible Irrotational Flow
5. Wings of Finite Span
6. The Navier-Stokes Equations
7. The Boundary Layer
8. Panel Methods
9. Finite Difference Methods
10. Finite-Difference Solution of the Boundary Layer Equations
11. Compressible Potential Flow Past Airfoils