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Wing Theory
By Robert T. Jones PRINCETON UNIVERSITY PRESS
Copyright © 1990 Princeton University Press
All rights reserved.
ISBN: 978-0-691-08536-4
CHAPTER 1
Fundamental Considerations
Early Theories of Lift
The modern theory of fluid motion originated in the works of Euler, Lagrange, and other great mathematicians of the eighteenth and nineteenth centuries. Being based on idealized frictionless fluids, these early theories were unable to account for the most commonly observed physical phenomena, and they found little application to practical problems. The later development of equations for the motion of viscous fluids did little to help this situation since these equations, known as the Navier-Stokes equations, are extremely difficult to treat mathematically. It was not until the development of the airplane and its requirement for streamlined shapes, low drag, and efficient lift that classical fluid dynamics found a real application.
Even so, aerodynamic theory was not prepared to give assistance in the early development of the airplane. Newton's theory of airflow and, later, the more sophisticated wing theory of Helmholtz and Kirchhoff, gave incorrect predictions and were often used to discount the possibility of flight by humans. Thus it is recorded that Lord Rayleigh expressed "not the smallest molecule of faith in aerial navigation except by balloon."
The first correct physical description of the flow over a wing seems to have been given by F. W. Lanchester in England in 1894. Lanchester envisioned a flow over a curved thin plate having smooth streamlines with fore and aft symmetry — a flow giving lift without drag. Unfortunately, Lanchester's paper was rejected by the Royal Society and did not appear in print until quite a few years later. In the meantime, W. Kutta in Germany and N. Joukowski in Russia developed a concise mathematical theory for the lifting wing in two-dimensional flow. Experiments by Lilienthal, Langley, and the Wright brothers had of course already demonstrated that human flight is possible.
Multiplicity of Solutions
The difficulty of the early theories may be attributed in part to the lack of uniqueness of solutions for the flow over a given shape. Figure 1.1 shows several possible solutions for the flow of a frictionless fluid over a thin flat plate. In the uppermost example, the flow clings to both the leading edge and trailing edges, leaving the airfoil at the rear upper surface. Such a flow develops a moment, but it does not develop lift or drag. In the second example, the flow separates from both leading and trailing edges, forming surfaces on discontinuity and a considerable "dead space" above and behind the airfoil. This is the type of flow considered by Helmholtz and Kirchhoff.
Examples (c) and (d) in Figure 1.1 are the more nearly correct physical solutions, although in the case of the perfectly thin plate they would require infinite velocities around the leading edge. In practice, of course, we round the leading edge, permitting the flow to remain attached.
Kutta-Joukowski Flow
In example (c), the Kutta-Joukowski flow, the airfoil develops lift without drag, a characteristic of frictionless potential flow that has come to be known as "d'Alembert's paradox." Wind tunnel experiments with smooth airfoil shapes tend to confirm d'Alembert's paradox with surprising accuracy. Figure 1.2 illustrates the result of an experiment in which the airfoil extended completely across the wind tunnel so as to insure two-dimensional flow. Here the drag of the airfoil is compared to the drag of a circular wire at the same speed. The diameter of the wire having the same drag is only about .006 the width of the airfoil. Such an airfoil may easily develop a lift of one hundred pounds for a drag of one pound; lift to drag ratios approaching three hundred have been measured in these circumstances. Although the airfoil develops pressures of the order of ρ/2V2 at the nose, these are balanced by forward-acting pressures at the rear, with the result that the drag of the airfoil is almost entirely skin friction. The blunt shape of the wire, however, causes the flow to separate from the rear and negative pressures appear, resulting in a drag about equal to ρ/2V2 times its frontal area. The extraordinary efficiency of a well-shaped airfoil poses a difficult challenge to inventors who would like to devise a better method of producing lift.
As mentioned earlier, the boundary condition corresponding to impermeability of the surface is not sufficient to establish the flow uniquely. Of the many flow configurations possible, that which remains attached to the nose but separates smoothly from the trailing edge agrees best with experiments at moderate angles of attack in the normal flight range. The condition that there can be no flow around the trailing edge, known as the Kutta condition, depends physically on the action of the boundary layer, which prevents the flow from turning the sharp corner at the trailing edge. Hence, while the pressures on the airfoil and the lift can usually be predicted with good accuracy by frictionless flow theory, the development of lift depends ultimately on the existence of friction in the
The physical nature of the Kutta condition can be clarified by examining two possible solutions for the flow through a pipe, as illustrated in Figure 1.3. Here we assume that some mechanism inside the pipe (perhaps a jet engine) creates the flow. In the first solution the flow enters the mouth of the pipe from all directions and leaves the same way. Such flow would create no thrust, and the fluid merely recirculates through the pipe. In the second solution, real flow, the boundary layer formed inside the pipe prevents the fluid from turning the corner at the exit, a jet is formed, and the mechanism in the pipe creates a thrust.
An important feature of Kutta-Joukowski flow not found in earlier models such as that of Helmholtz and Kirchhoff is the occurrence of high velocities and large suction forces around the leading edge when the airfoil is at an angle of attack. It is this suction force that balances the rearwardly inclined lift on the major portion of the surface, leading to the production of lift without drag in frictionless flow. Figure 1.4 shows the pressures measured on an airfoil at 16º angle of attack and plotted as vectors perpendicular to the surface. The pressures are compared with those calculated by frictionless flow theory. At this angle of attack, velocities amounting to several times the stream velocity occur around the nose of the airfoil. In his Collected Works (1910), C. A. Chaplygin noted that these high velocities around the nose might in some cases exceed the velocity of sound and that continuous flow would then no longer be possible.
Incompressible Potential Flow
In general, specification of the flow around a body requires the determination of three component velocities at each point in space. For flows having a velocity potential the calculation is greatly simplified, being reduced to the determination of a single scalar function of position. The velocity components are then the derivatives of this function in the different directions. The conditions for the existence of a velocity potential are (1) that no vorticity exists in the region of fluid being considered, and (2) in the case of a compressible fluid, that the density be a single-valued function of the pressure. Any function satisfying the partial differential equation (Laplace's equation),
Φxx + Φyy + Φzz = 0; (Φxx = [partial derivative]2Φ/[partial derivative]x2, etc.),
can represent the flow of fluid. Laplace's equation and the Bernoulli relation for pressure serve as the equation for irrotational motion of a fluid. It is important to note that flows satisfying Laplace's equation also satisfy the equations of motion for a viscous fluid (i.e., the Navier-Stokes equations), since the viscous stresses within the fluid contribute nothing to the acceleration of the fluid elements. The primary effect of viscosity originates in the contact of the fluid with a solid surface and results from the fact that the fluid sticks to the surface. The shearing motion of layers of fluid near the surface generates vorticity, and if the flow does not separate, this vorticity will be confined to a thin layer near the surface — the "boundary layer."
In frictionless, potential flow the stress in the fluid is simply a scalar pressure, and the fluid elements move under the influence of the gradient of this pressure. If the flow is steady, the disturbance of pressure is strictly proportional to the square of the velocity, that is, the Bernoulli equation:
p = const. - ρ/2 V2.
Influence of Viscosity: Reynolds Number
In a frictionless fluid the flow configuration and the pressures are independent of the scale of the motion. Bodies of different size experience geometrically similar distributions of pressure proportional to ρ/2 V2. Introducing viscosity, we find an additional stress proportional to the first power of the velocity divided by some characteristic length in the flow, a velocity gradient. The additional stress will be μV/λ, where μ is the coefficient of viscosity, supposedly a property of the fluid, independent of the state of motion, and λ is some characteristic length in the flow.
Taking the ratio of the stress due to inertia of the fluid and that due to viscosity, we find R = ρVλ/μ. R is of course the Reynolds number. On the scale of human flight, Reynolds numbers tend to be quite large and the viscous stresses correspondingly small. Thus for a wing having a chord of one foot at a speed of 100 mph, the Reynolds number is about one million. Assuming a laminar boundary layer, the tangential stress due to viscosity will be approximately .0015 that of the scalar pressure.
If flows on different scales of dimensions are to be geometrically and dynamically similar, the Reynolds number must be the same. However, at the Reynolds numbers of airplanes, the variation of the usual coefficients based on area and the square of the velocity is rather slow.
According to H. Bateman, the earliest recorded experiments to measure the force on a body in a moving stream were made by Edme Mariotte. Mariotte's Traité du Mouvement des Eaux appeared in 1686, a year before the publication of Newton's Principia. Mariotte measured the skin friction on a flat plate. Subsequent experiments of this kind have extended their range enormously and have included various fluids such as water, oil, and air. Figure 1.5 shows a plot of skin-friction coefficient as a function of Reynolds number based on the length of the plate. The measured values lie close to one of two curves, one corresponding to laminar motion in the boundary layer and the other to turbulent motion. The latter curve is more nearly horizontal, indicating a dependence closer to the square of the velocity. A small degree of surface roughness will render this curve absolutely horizontal, indicating that the viscous stress varies with the square of the velocity the same as the inertial stress, making the flow, at least on a macroscopic scale, independent of the Reynolds number. One of the most successful theories of turbulent motion is Prandtl's "mixing length" theory in which the results are independent of the Reynolds number.
As mentioned earlier, potential flows are not uniquely determined by the differential equation together with the condition of no flow through the surface. For any given body, a multiplicity of solutions having surfaces of discontinuity, or external vortices, exists. One might suppose that the introduction of viscosity and the additional boundary condition of zero velocity at the surface required by the Navier-Stokes equations would lead to a unique solution. Evidently this is not the case. Attempts to prove uniqueness, or to determine the conditions for it, have not been successful except at very low Reynolds numbers. The simplest exact solution of the Navier-Stokes equation — the laminar flow through a smooth pipe with a parabolic velocity profile — breaks down within a short distance into a seemingly random turbulent motion.
CHAPTER 2
Potential Flow over Ellipsoids
Ellipsoidal Coordinates: Lame Functions
The potential flow over ellipsoids is of interest in aeronautics: For example, an elongated prolate spheroid is useful as a model for the flow around an airship or a fuselage, while a thin flat ellipsoid can be used to demonstrate certain features of the flow over a wing. Moreover, ellipsoids provide simple exact solutions that can be used to assess the validity of certain approximations.
Potential flows produced by motion of an ellipsoidal boundary are studied with the aid of ellipsoidal coordinates λ, μ, ν, where
x2/a2 + λ + y2/b2 + λ + z2/c2 + λ = 1,
and a, b, c are the semiaxes of the ellipsoid, λ = 0 is the equation of the base ellipsoid, with confocal surfaces given by other positive values of λ. Solving for x, y, z, we find that the equation has two additional roots, μ, ν. These define a family of intersecting surfaces (hyperboloids of one and two sheets), which, at a great distance, approach ordinary spherical polar coordinate surfaces. Solutions of Laplace's equation are separable in ellipsoidal coordinates as products of Lame functions, thus
φmn = Emn (μ) Emn (v) Fmn(λ).
Here E and F are Lame functions of the first and second kind. Examples of the first few Lame functions are the following:
E11(λ) = √ a2 + λ; E21(λ) = √ b2 + λ; E31(λ) = √ c2 + λ.
Lamé products of lower degree bear simple relations to Cartesian coordinates, thus
E11(λ) E11(μ) E11(ν) = √ (a2 - b2) (a2 - c2) x, etc.
If the flow is to vanish at infinity, functions of the second kind are required. These are determined by integration:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
For a complete discussion of ellipsoidal coordinates and associated functions, the reader should consult Whittaker and Watson or Hobson (see Bibliography).
Translation of Ellipsoids
For translation of the ellipsoid in the negative direction along the three principal axes, the velocity potentials are given by
Uxf11(λ); Vyf21(λ); Wzf31(λ).
Thus for motion in x-direction, the potential on the surface of the base ellipsoid is simply Uxf11(0).
The constants fmn(0) depend in general on elliptic functions, but their values have been tabulated for a number of interesting cases.
The flow produced by a moving sphere provides a simple example. Figure 2.1, adapted from Lamb's Hydrodynamics, shows the instantaneous streamlines. Here the potential is
φ = ½ Uxa3/r3.
The equipotential lines are circles (meridians) with values proportional to x-coordinate on any concentric sphere, including the solid sphere of radius a. The velocity at the crest is ½ U, opposite in direction to the motion of the sphere. The motion is unsteady in the sense that the flow pattern moves with the sphere and remains attached to it. If we start the sphere from rest, or accelerate it, the whole flow field will be set into motion, and, since the fluid is assumed incompressible, the velocities everywhere will be linked instantly to that of the sphere. Accelerating the flow requires a force that depends in part on the mass of the body itself and also on the momentum created in the fluid. The latter can be computed by integrating the momentum over the field or from the unsteady term in the "extended Bernoulli equation" for the surface pressures, that is, Δp = -ρ[partial derivative]φ/[partial derivative]t.
Apparent Mass and Velocity Distribution
If the velocity is constant, the fore and aft pressures balance and the fluid offers no resistance to the motion, in accordance with "d'Alembert's paradox." The term proportional to the rate of change of velocity, however, gives positive pressures in front and negative pressures at the rear, resulting in a force that resists the acceleration. In accelerated motion the sphere thus acts as if it had an additional mass equal to the mass of a certain volume of fluid. It is not difficult to show that this additional mass is proportional to the maximum velocity at the surface of the sphere in steady motion. Since the velocity potential on the surface of the sphere is proportional to the x-coordinate of the surface, the pressure -ρ [partial derivative]φ/[partial derivative]t plotted along x is a straight line and corresponds to a constant rearward buoyancy in accelerated motion. This gradient of the potential also determines the maximum velocity ½ U at the crest of the sphere in steady translation. The factor ½ is called the "inertia coefficient" of the sphere. Hence the inertia coefficient is also the "maximum velocity coefficient." This equality holds for any ellipsoid moving parallel to a principal axis.
(Continues...)
Excerpted from Wing Theory by Robert T. Jones. Copyright © 1990 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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