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CHAPTER 1
Format and Fundamentals
1.1 Introduction: A Survey of Fluid Mechanics
Fluid mechanics is the branch of the physical sciences concerned with how fluids behave at rest or in motion. Its uses are limitless. We must understand fluid mechanics if we want to model the red spot on Jupiter, or measure the vorticity in a tornado, or design a transonic wing for an SST, or predict the behavior of subatomic particles in a betatron. To track the motions of fluids past objects or through objects, in oceans or in molecules, here on earth or in distant galaxies, fluid mechanics examines the behavior of liquids, gases, and plasma — of everything that is not solid. The theory of fluid mechanics is the foundation for literally dozens of fields within science and engineering: for meteorology, oceanography, and astronomy; for aerodynamics, propulsion, and combustion; for biofluids, acoustics, and particle physics.
Besides being one of the most important physical sciences in engineering, fluid mechanics is one of the oldest. As early as the 4th century B.C., Aristotle was writing about density: "Dense differs from rare in containing more matter in the same cubic area." Aristotle also conceived of uniform acceleration (an idea often credited to Galileo [1564]). The apple that fell on Sir Isaac Newton's head may have crowned him the founder of modern physics, but the following Aristotelian passages certainly have a familiar ring to our post-Newtonian ear:
1. "Bodies which are at rest remain so owing to their resistance."
2. "When one is running fast it is difficult to divert the whole body from its impetus in one direction to some movement."
3. "The force of that which initiates movement must be made equal to the force of that which remains at rest. For there is a definite quantity of force or power by dint of which that which remains at rest does so, just as there is a force by dint of which that which initiates movement does so; and as there is a necessary proportion between opposite motions, so there is between absences of motion ... For as the pusher pushes, so is the pushed pushed, and with equal force."
It is, perhaps, an injustice of history that Newton's phraseology caught on, and Aristotle's lay dormant and nearly died.
Shortly after Aristotle, another Greek philosopher-scientist, Archimedes (287-212 B.C.), offered mathematical formulations for those same physical problems that concerned Aristotle. Archimedes formulated, for example, the first quantitative law of the lever, which Aristotle had earlier described in words: "As the weight moved is to the weight moving it, so inversely is the length of the arm bearing the weight to the length of the arm nearer the poser." Archimedes established many principles of fluid statics, among which the buoyancy principle is, in fact, named after him.
Then nothing significant happened for about 18 centuries, until Galileo started the modern inquiry into fluids. Galileo used mathematics and deductive reasoning rather than direct experimental measurement to study bodies falling through fluids. His belief that mathematics was the language of motion and that change in nature was best described mathematically paved the way for the greatest awakening of scientific thought mankind had ever experienced. Though the calculus had not yet been invented, Galileo had the genius to realize that concepts of change were embedded somewhere in geometric arguments.
Then science erupted during the time of Newton (1642-1726). Rational mechanics was conceived by James Bernoulli and Isaac Newton, the former developing ideas of continuous motion and of how dynamics was related to statics, and the latter describing motion as the result of external forces. Leibniz (1646-1716) was probably the first to develop theories of conservation of linear momentum to describe motion as energy. Another great Swiss mathematician, L. Euler (1707), is credited with modernizing mechanics by treating mechanics as analytical, and transforming Newton's physical concepts into mathematical equations.
The following gives some of the significant achievements of the scientists and mathematicians who initiated fluid mechanics during the period of Newton.
1. Hydrostatics — John Bernoulli
2. Concept of pressure, integral form of linear momentum — Daniel Bernoulli
3. Equations of hydraulics for ideal fluids — John Bernoulli
4. Mechanics, concepts of force — Isaac Newton
5. Concepts of energy, linear momentum — Gottfried Leibniz
6. Theoretical hydrodynamics, theory of deformable bodies, concept of stress, kinematics, equations of motion of ideal fluids — Leonhard Euler
7. Concept of stream function, velocity potential — Joseph-Louis Lagrange
8. Potential theory, concept of vorticity — Augustin-Louis Cauchy
9. Experimental hydrodynamics — Jean le Rond d'Alembert
At first glance, this list might appear to be of negligible importance since it would seem to have no bearing on the technical aspect of the subject. Imbedded within the list, however, are little known but important human facts. Euler is known to have labored 20 years in developing some of the theories of fluid mechanics. In reading Laplace's statement that "It is obvious" we must translate it to mean (as his associates discovered when they queried him on the obviousness) that Laplace sometimes had to struggle for hours to duplicate a result. References are given throughout this book for a purpose. As Leibniz once said in the study of the history of science, the "act of making discoveries should be extended by considering noteworthy examples of it. "
Next in the development of fluid mechanics came the age of 19th century theorists. This period witnessed mathematical developments in hydrodynamics and the birth of modern aerodynamics. The 19th century was a paradise for mathematicians, who had little or no regard for practical application. (There are exceptions, to be sure, such as practical work on lubrication, or on the filtration of liquids through sediments.) Following are some of the theories developed and the great thinkers who conceived them:
1. Complex variable method of two-dimensional potential flow to develop flow patterns — Gustave Kirchhoff and Hermann Helmholtz
2. Three-dimensional potential flow and viscosity — Siméon-Denis Poisson
3. Method of singularities, gas dynamics, wave theory — William J. Rankine and Lord Kelvin
4. Rotational motion — H. Helmholtz and Lord Kelvin
5. Vortex flows, unsteady laminar flows — Gromeka
6. Viscous flows — Claude L. M. H. Navier and George G. Stokes
7. Pipe flows — G. G. Stokes and Jean L. M. Poiseuille
8. Stability, turbulence — G. G. Stokes [1.18] and Osborne Reynolds
Thinkers of a more practical bent, however, soon began to awaken to the applicability of these mathematical investigations. In the late 19th century engineers began to take over fluid mechanics, adapting it to designing machines of war, luxury, and production. Two remarkable theories started the age of the engineer: wing theory and air screw theory. With these concepts in hand, engineers raced to apply and extend the results; technical literature filled the shelves, and the aircraft was born. As more and more emphasis was placed on the behavior of bodies in a gas, scientists began building laboratory facilities to model the behavior of bodies moving through air. Tsiolkovskii (1896) constructed one of the first wind tunnels to investigate drag on various bodies. He postulated flight in all metal spacecraft into space, and he developed equations for the motion of a rocket-propelled spacecraft of variable mass.
At a time when one of the most adventurous ideas was to develop a body that a man could fit into which would behave like a fluid particle, it is no wonder that the age of the theorist gave way to the age of the engineer. While it would be impossible to list all the eminent contributors to fluid dynamics during this age, certain ones deserve mention. The list below is only a fraction of the major contributions.
1. Shock waves — Doppler, Mach, and Riemann
2. Wing theory — Lancaster, Zhukovskii, and Kutta
3. Air screw theory — Zhukovskii and Glauert
4. Boundary layer concept — Rankine
5. Boundary layer theory — Prandtl
6. Turbulence — Burgers, Taylor, and von Karman
7. High-speed aerodynamics — Rayleigh, Taylor, and Mach
8. Magnetohydrodynamics — Hartman, Hoffman, and Alfven
9. Wave theory — Taylor and Jeffreys
10. Rarefied gases — Kantrowitz
1.2 Format of This Text
1.2.1 The Subject of Fluid Mechanics
Fluid mechanics is one of the four branches of mechanics: elastic-body mechanics, fluid mechanics, relativistic mechanics, and quantum mechanics. The study of fluid mechanics subdivides into statics and dynamics which in turn divide into incompressible and compressible flow. Incompressible and compressible flow divide into real and ideal. Real divides into laminar and turbulent. And so on. Our subject grows out of a root system that seems tangled at first. Yet such elaborate classification is necessary in order to define accurately the types of fluid flow . Understanding fluid mechanics requires organization, summary, and recapitulation, all of which are integral to this text.
Fluid mechanics is based upon five great principles of physics:
1. Conservation of mass
2. Conservation of linear momentum
3. Conservation of angular momentum
4. Conservation of energy
5. Second law of thermodynamics
The first four principles are the same ones used in the other four branches of mechanics, and they are familiar to all. These principles govern the behavior of matter, whether it is solid or fluid; and it is the behavior of fluid matter that we seek to describe. If we can predict a fluid's behavior, regardless of the constraints, we can design, construct, and plan machines that use fluids: engines, ships, aircraft, pumps — the list is endless.
A fluid like oil or steam has stored energy. If we know how to predict the fluid's behavior, we can transport that energy from one location to another, or we can transform that stored energy into useful work. A fluid's energy can be converted into other forms of energy by combustion, mixing, impact reactions, and deflection. Blood is an excellent example of a fluid that transports energy in the form of nutrients, chemicals, and life-giving substances to our energy-burning cells. We are, after all, machines with pumps obeying the first and second laws of thermodynamics.
All fish, airplanes, ships, cars, and people are objects that move through fluids. And fluids move through objects, like blood through arteries, air in lungs, water in capillaries, oil in pipes, and gas in nozzles. Fluids deform as objects pass through them, sometimes translating, often rotating, expanding, compressing — all effortlessly with none of the failures, breaks, or fatigue that solids experience.
A fluid has the wonderful property of sometimes being a liquid and then again a gas. We can breathe the gas, but not the liquid, yet a fish does just the opposite. Some gases can quench a fire just as some liquids do, yet some gases are essential for fire just as some liquids are. In Chap. 2 we shall define the nature of a fluid, exploring its mysteries and its wonders.
1.2.2 The Structure of Fluid Mechanics
Structured in a manner similar to the other four branches of mechanics, as an organization of topics, fluid mechanics divides into subdivision after subdivision, each being necessary for one primary purpose: to classify flow, to give flow identity. Without such classification, we could not organize our techniques of problem solving. Figure 1.1 shows some of the classifications a fluid flow problem might possibly fit into. We can count 96 different combinations in just the categories given. For example, if we want to design a wing that would not flutter at transonic speeds, we would examine a compressible gas in unsteady irrotational three-dimensional inviscid flow. Each separate classification is important. The gas (air) will be compressed because we are studying velocities near the speed of sound. The flow is unsteady because gust loads and wind shear profiles continually deflect the elastic wing. The flow is irrotational because we shall assume that the pressure distribution over the wing is nearly the pressure distribution over the wing outside the boundary layer. The flow is inviscid because we shall assume that the frictional forces are small compared to the inertial and compressible forces and therefore neglect them. And the flow is three-dimensional because the wing has span length, chord width, and thickness, any variation of which can alter the flow . By classifying this problem as unsteady, irrotational, inviscid, three-dimensional flow, we can set up our problem quickly and select the appropriate form of our governing equations.
Once we have a problem defined, we seek its solution. Figure 1.2 shows a schematic we could use in solving a fluid flow problem. Any problem can be treated either by setting it up in a laboratory and measuring the desired unknowns, or by modeling the problem mathematically using symbols to represent physical quantities and operators to represent changes. Of the two choices, many might find the easier, the faster, and the more accurate to be the mathematical modeling. This is not to say that experimentation is secondary to theoretical analysis. In fact, many practical problems in fluid mechanics are not yet mathematically tractable and must be solved in the lab. Good engineers always verify their theoretical predictions by testing their values experimentally. We would not expect 100% agreement since experimental errors are often found in instrumentation, in secondary fluid effects that had not been isolated from the experiment, and in measurement errors. But we would expect a close agreement, say within 93 or 94 percent accuracy, depending on the problem. Aircraft companies, ship builders, or high-quality industrial companies involved with machines that operate with fluids will usually perform a theoretical analysis first, then build a model or a prototype and test it to verify the results.
How is the theoretical analysis handled? Figure 1.2 shows two crucial steps that must be undertaken before any theoretical solution can be obtained. The first step is a physical analysis; the second step is a mathematical analysis.
A physical analysis is important in order to understand the physics of the problem. We are, after all, dealing with the real world, and thus nature affects that which is a part of it or moving through it. How do we understand the physical problem? Two concepts, force and energy, can be used to describe the manner in which the physical world behaves. We have a choice of using either or both to describe in a theoretical fashion a fluid itself, or the behavior of an object in a fluid. We often feel more comfortable working with the concept of force, as we can measure forces. Force is something we all have experienced. Energy, on the other hand, is a bit outside our experience. It is something most often calculated, not measured. But whichever we select, force or energy, we shall use it to describe the interaction between that which we are examining and its surroundings.
Forces have to act on something, just as energies have to be related to something. We need to define exactly what is acted upon. Either we may examine the effect of a force on a system, or we may examine its effect on a control volume. (Note that it is always possible for a system to occupy a control volume for an instant of time. We shall confine ourselves to systems and control volumes that are either independent of time or arbitrary in time.)
By definition, a system is a predetermined identifiable quantity of fluid. It could be a particle, or a collection of particles. The system may alter its geometric configuration, its location in space, its state variables such as pressure and temperature, but a system cannot alter its mass. An example of a system is shown in Fig. 1.3. Within the volume A, the system is the mass of gas. As the piston moves in the cylinder, the volume, pressure, and temperature of the system can change but the system, that is, the identity of mass, does not change. Thus, mass M remains constant, and the conservation of mass is automatically ensured.
(Continues…)
Excerpted from "Fluid Mechanics"
by .
Copyright © 1995 Robert A. Granger.
Excerpted by permission of Dover Publications, Inc..
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