# Wing Theory

ROBERT THOMAS JONES
Pages: 232
https://www.jstor.org/stable/j.ctt7zv6v2

1. Front Matter
(pp. i-iv)
(pp. v-viii)
3. PREFACE
(pp. ix-2)
4. CHAPTER ONE Fundamental Considerations
(pp. 3-11)

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...

5. CHAPTER TWO Potential Flow over Ellipsoids
(pp. 12-19)

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 , ì, í, where

$\frac{{{x^2}}}{{{a^2} + \lambda }} + \frac{{{y^2}}}{{{b^2} + \lambda }} + \frac{{{z^2}}}{{{c^2} + \lambda }} = 1$anda, b, care the semiaxes of the ellipsoid.$\lambda = 0$is the...

6. CHAPTER THREE Two-Dimensional Flow: Wing Section Theory
(pp. 20-52)

If the wing or wing panel is sufficiently long and narrow, the local flow in planes perpendicular to the long axis may be considered as approximately two dimensional. Such two-dimensional flows are extremely simple since any differentiable function of a complex variable can represent such a flow.

As explained by Prandtl and Tietjens,¹ every analytic function of such a complex variable can be separated into a pair of distinct functions, one real and one imaginary, thus

$F(x + iy) = \varphi (x,y) + i\psi (x,y)$, and both φ and ψ will be solutions of Laplace’s equation. Since φ and ψ...

7. CHAPTER FOUR Thin Airfoil Theory
(pp. 53-73)

The theory of flows with small disturbances is well suited to problems of high speed flight since the assumptions of the theory agree with the requirements for efficiency. Broadly speaking, the losses incurred in flight increase with the square of the disturbance velocity, while the lift depends on the first power. Hence for efficient flight the airplane should create a small disturbance.

A development leading to many useful theorems and results is the thin airfoil theory originated by Max M. Munk in the early 1920s.¹ Here one linearizes the airfoil problem by applying the boundary conditions at the chord line...

8. CHAPTER FIVE Influence of Compressibility
(pp. 74-89)

The laws of aerodynamics are distinctly favorable to flight in the subsonic speed range. At speeds well below the speed of sound, where lift and pressures are modified very little by compressibility, an airfoil can produce a large lift force with small drag. Straight wings of high aspect ratio maintain these favorable properties at speeds up to 70%, sometimes even 80%, of the speed of sound. Beyond this range the favorable properties of the wing section diminish rapidly, large increases of drag occur, and the flow often becomes unsteady.

Before we consider the quantitative effects of compressibility, it will be...

9. CHAPTER SIX Effects of Sweep
(pp. 90-104)

In the late 1920s Italy produced the fastest airplanes in the world, winning the famous Schneider Trophy in competition with American racers. To further the development in this area, the Italian government, under Mussolini, decided to hold an international meeting on the problems of high-speed aeronautics, the 1935 Volta Congress. The American delegation, which included Eastman N. Jacobs of NACA Langley Laboratory and Theodore von Kármán of Cal Tech, traveled to the meeting on the luxuriousConte de Savoie,courtesy of the Italian government. At this early period the maximum speed achieved, even by the Schneider Cup racers, was less...

10. CHAPTER SEVEN Wings of High Aspect Ratio
(pp. 105-128)

The most efficient wing form for Mach numbers below about 0.7 is the straight wing of high aspect ratio. The theory of such wings was developed in Germany, principally by Ludwig Prandtl and his students during the early years of this century.¹

In Prandtl’s theory the flow in the vicinity of the wing was considered two dimensional, while the effect of the finite span, and the variation of lift along the span, was treated as a correction, on a larger scale, of the two-dimensional theory. For a more rigorous treatment of this and other approximations, the reader should consult Van...

11. CHAPTER EIGHT Lifting Surface Theory
(pp. 129-158)

At the Fifth International Congress for Applied Mechanics, Prandtl introduced a new concept in the theory of lifting surfaces, a concept he termed the “acceleration potential.” Since the acceleration of a fluid element in a given direction is proportional to the gradient of the pressure in that direction, we can see that the pressure field has, in this sense, the properties of a potential. The pressure at any point is furthermore a scalar, and in linear theory it satisfies the same differential equation as the velocity potential, viz.,

$(1 - {M^2}){P_{xx}} + {P_{yy}} + {P_{zz}} = O$. (8.1)

The advantage of the acceleration potential can be seen if...

12. CHAPTER NINE The Minimum Drag of Thin Wings
(pp. 159-199)

As shown by Hayes,¹ the drag of a given distribution of lift is unchanged by reversing the direction of motion. The camber and twist required to support the specified lift distribution will in general be different for the two directions. The calculation of the minimum drag will be simplified if we consider distributions of lift in the velocity fieldvandwobtained by superimposing the disturbances in forward and reversed motion, as indicated in Figure 9.1.²

If we apply this idea to the lifting surface at subsonic speeds, we find that the superimposed velocity fields result in a strictly...

13. CHAPTER TEN Drag of Wings and Bodies in Combination
(pp. 200-208)

At subsonic speeds the pressure drag arising from the thickness or volume of a body or wing is negligible so long as the shapes are sufficiently well streamlined to avoid flow separation. In this regime there exists no possibility of either favorable or adverse interference on the pressure distributions. If one body is placed so it receives favorable interference from the pressure field of another, the other is sure to receive an equal adverse interference from the first. This idea is limited to nonlifting bodies since a lifting surface can create either favorable or adverse interference with another surface.

At...

14. BIBLIOGRAPHY
(pp. 209-210)
15. INDEX
(pp. 211-216)