# Nonmathematical Summary

The main points of this chapter may be summarized by the following ten items. The list begins with a resume of some of the terminology of the field. This terminology is taken for granted in the subsequent discussion.

1. A flow is called a two-dimensional flow when it can be drawn in cross section and the drawing taken as representative of the flow at any other cross section. Thus if the flow around a barrier, or some other obstacle, is drawn in two dimensions, this ignores the complications introduced into the flow by what happens at the edges not shown in the picture (that is, below the page and above the page). If the drawing shows the cross section of, say, a wing, then the picture does not portray what is happening at the wingtips, that is, the third dimension of the situation. This absence can be justified if the wing is very long and the immediate concern is with the flow at parts of the wing that are distant from the tips. The diagram can then adequately represent the flow around the central sections of the wing. A wing that is long enough to justify this approximation is often called an infinite wing. The word “infinite” is much used in hydrodynamics. References to, for example, “the flow at infinity” usually mean the flow as it is at a great distance from some disturbance so that the effects of the disturbance can be ignored.

2. The main theoretical resource used in early aerodynamics came from classical hydrodynamics. Hydrodynamics offered a mathematically sophisticated theory of the flow of an “ideal” fluid, that is, a fluid that was incompressible and also completely devoid of viscosity. Of these two idealizations, the most contentious was the neglect of viscosity. The differential equations that govern the flow of an ideal fluid are called the Euler equations. These equations give the speed of the flow at a specified position and time. To make it easier to solve these equations, mathematicians introduced two further idealizations. First, it was often assumed that the flow was steady. This meant that rate of change with time was zero and could be ignored. Second, it was assumed that the fluid elements did not rotate. There was an emphasis on irrotational flows because it simplified the mathematics. Unfortunately the benefit of mathematical simplicity was purchased at the cost of making the flows being analyzed less than realistic as models of real fluid flows.

3. Fluid elements (that is, the small volumes of fluid whose velocities and rotations are under study) are not to be identified with molecules or atoms or material particles, although occasionally such identifications seem to have been made. Fluid elements are mathematical abstractions that enable the methods of the differential and integral calculus to be applied to fluids.

4. One logical consequence that can be derived from the Euler equations is a highly useful result called Bernoulli’s law. With the assumption of a steady, irrotational, and incompressible flow, the law takes on a simple form. It states that the pressure and the velocity at a point in the flow are related by a simple law that implies that as the speed increases the pressure will decrease, and as the speed decreases the pressure increases. Speed and pressure trade off against one another. The use of the law makes it important to distinguish between three different meanings that are attached to the word “pressure.” There is (i) static pressure, (ii) dynamic pressure, and (iii) total pressure. Total pressure equals the sum of static pressure and dynamic pressure. Static pressure is the pressure on the sides of a pipe or the surface of a wing. Total pressure is the pressure felt when a body of fluid is brought to a standstill. Dynamic pressure is the name given to the quantity V2 p V2 where p is the density of the fluid and V is the speed of flow. In the simplified conditions dealt with in early aerodynamics, the total pressure can be considered to have a constant value. As speed V increases and hence dynamic pressure increases, then static pressure must go down. Care is needed to ensure its correct application, but Bernoulli’s law plays an important role in (i) calculating the forces on an object that is immersed in a flowing fluid, for example, a wing in a stream of air, and (ii) understanding the operation of instruments such as the Pitot probe, which registers total and static pressure and (via Bernoulli’s law) permits the computation of velocities.

5. The restriction to irrotational flow permitted the mathematical description of a wide variety of two-dimensional flows such as the flow of a steady stream around a circular cylinder and the flow around a barrier facing head – on into the stream. The streamlines of these flows could be drawn on the basis of the formula (called the stream function) that furnished the mathematical description of the flow. In a steady flow (but not in an unsteady flow) the streamlines give the path taken by the fluid elements. As well as streamlines a flow can be described by what are called lines of equal potential. These are orthogonal to the streamlines except at points called stagnation points, which are points where streamlines come to a halt on the surface of a body. Streamlines and potential lines can be switched in the sense that the potential lines can be interpreted as the streamlines of a new flow. The old streamlines then become the potential lines of the new flow. Just as the streamlines are specified by the stream function so the potential lines are specified by a potential function.

6. The possibility of arriving at a mathematical description of a flow was greatly improved because a large number of familiar mathematical functions (called functions of a complex variable) turned out to be interpretable as possible fluid flows. The geometrical patterns generated by these functions (that is, the lines of the curves plotted on graph paper) could be read as the patterns made by a flowing fluid and the boundaries that constrain them. Exploring a function and then giving it an after-the-fact interpretation in terms of a flow of ideal fluid was called the indirect method of arriving at the equations of the flow. I illustrated this process by means of a function that could be understood as describing the flow of a uniform stream around a circular cylinder. The example, along with the overall presentation of the material in this chapter, was taken from one of the standard textbooks of the World War I period, namely, Cowley and Levy’s Aeronautics in Theory and Experiment published in 1918.

7. A more direct line of attack was sometimes available to the mathematician in search of a mathematical description of a flow pattern. This method involved constructing a set of equations that related the flow to be understood to a very simple flow that was already understood, for example, the uniform flow along a straight boundary. If the boundaries of the simple flow could be transformed into the boundaries of the more complicated flow, then the methods of transformation would also turn the simple streamlines into the more complicated streamlines of the desired flow. A particular set of transformations called conformal transformations played a central role in this process. Many such transformations had been studied as exercises in pure mathematics and geometry but were found to be important resources in the study of fluid flow. One such important transformation was called the Schwarz-Christoffel theorem.

8. The main problem with the hydrodynamics of an ideal fluid was that, although it became mathematically sophisticated, it appeared to provide no resources for explaining the resistance that an object experiences when placed in the flow of a real fluid such as water or air. When an ideal fluid flows around, say, a flat plate or a circular cylinder, the flow exerts no resultant force on the object. This is often called d’Alembert’s paradox, although whether it is a paradox in the true sense of the word is examined in more detail later. What is beyond dispute is that the result presented a problem for anyone who wanted to understand the air by likening it to an ideal fluid. The use of the theory of ideal fluids led to the false result of zero resistance or zero drag.

9. One possible response to this “paradoxical” result would be to reject ideal fluid theory as useless for the study of real fluids such as air. Why not develop a more realistic hydrodynamic theory devoted to viscous fluids? This project was begun, and the equations of motion of a viscous fluid were formulated. They are now called the Navier-Stokes equations. (The British just called them the Stokes equations.) Frustratingly they could only be solved in a few very simple cases, which gave special significance to the search for new ways to make ideal-fluid theory more realistic. In principle there were two ways to do this—hence the existence of two competing theories of lift. Only one of these ways (called the theory of discontinuous flow) was described in this chapter. The alternative, the vortex theory, is discussed in chapter 4.

10. The theory of discontinuous flow was proposed by Helmholtz and carried forward by Kirchhoff and Rayleigh. Helmholtz argued that the “paradoxical” result of zero resistance or drag arose because an ideal fluid could wrap itself around an object and exert pressure from all sides in a way that canceled out any resultant force. The discontinuous flow approach exploited the possibility that there could be discontinuities in the velocity of different bodies of ideal fluid that were in direct contact with one another. The flow was assumed to break away from the edges of an obstacle and create a wake behind it. The wake would be “dead water” or “dead air,” and the main body of ideal fluid would flow past it. (The assumption here is that the body is stationary and the fluid moving. This is the situation of a model airplane in a wind tunnel.) Such a flow pattern in an ideal fluid, with a wake of dead fluid, turned out to be compatible with the Euler equations. Furthermore, it could be established that, given such a discontinuous flow (see fig. 2.7), the pressure on the front face of an object would be greater than the pressure of the dead fluid on the rear. The forces did not cancel out and d’Alembert’s paradox was avoided. If the resultant force proved large enough, here was a theory that could, in principle, explain the lift of a wing as well as the resistance to motion, the “drag.” Such was Rayleigh’s idea for explaining the lift on an aircraft wing, and it was taken up by the British Advisory Committee for Aeronautics. The results that emerged from the theoretical and experimental study of this model are described in the next chapter.

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