Acerca del autor

JEFFREY S. MARSHALL, PhD, is a professor in the Department of Mechanical Engineering and a research engineer at the Iowa Institute of Hydraulic Research at the University of Iowa, Iowa City.

De la contraportada

A comprehensive, modern account of the flow of inviscid incompressible fluids

This one-stop resource for students, instructors, and professionals goes beyond analytical solutions for irrotational fluids to provide practical answers to real-world problems involving complex boundaries. It offers extensive coverage of vorticity transport as well as computational methods for inviscid flows, and it provides a solid foundation for further studies in fluid dynamics.

Inviscid Incompressible Flow supplies a rigorous introduction to the continuum mechanics of fluid flows. It derives vector representation theorems, develops the vorticity transport theorem and related integral invariants, and presents theorems associated with the pressure field. This self-contained sourcebook describes both solution methods unique to two-dimensional flows and methods for axisymmetric and three-dimensional flows, many of which can be applied to two-dimensional flows as a special case. Finally, it examines perturbations of equilibrium solutions and ensuing stability issues.

Important features of this powerful, timely volume include:

• Focused, comprehensive coverage of inviscid incompressible fluids
• Four entire chapters devoted to vorticity transport and solution of vortical flows
• Theorems and computational methods for two-dimensional, axisymmetric, and three-dimensional flows
• A companion Web site containing subroutines for calculations in the book
• Clear, easy-to-follow presentation

Inviscid Incompressible Flow, the only all-in-one presentation available on this topic, is a first-rate teaching and learning tool for graduate- and senior undergraduate-level courses in inviscid fluid dynamics. It is also an excellent reference for professionals and researchers in engineering, physics, and applied mathematics.

Fragmento. © Reproducción autorizada. Todos los derechos reservados.

Inviscid Incompressible Flow

John Wiley & Sons

Chapter One

Earth, air, fire, and water-the four basic elements of the ancient Greek world-are each vital to human existence. Air is the medium by which oxygen is transported to our lungs, surging through our twisting pulmonary passageways with each gasped breath. Air is the domain of flight, of soaring birds and buzzing insects and the screaming vehicles of man. Air transports our words, allowing us to communicate with each other. Air and water together regulate the temperature of the earth's atmosphere, blocking out harmful solar emissions and surrounding us in blanketing warmth. They determine our weather, be it a tranquil summer day, a spring shower, a winter blizzard, or a roaring fall hurricane, and over time they control the climate in which we must strive to survive. Water is also vital to human life, transporting nourishment and waste products as it courses through our veins and regulating our body temperature as it evaporates from our skin. Much of the world depends upon oceans and rivers as a medium of transport for goods and a source of bounteous food. The ability to utilize fire is one of the major human characteristics distinguishing us from other animals. Fire cooks our food, warms our bodies, propels our vehicles, hardens and forms our metals. Even the earth is not stagnant and unmoving. Water percolates through its pores, providing us with drink and sustaining the plants that feed us and provide us with shelter. Oil gushes from its fissures, providing fuel for automobiles and engine lubrication. With too much water, the earth forms giant slides, burying houses and roads. Over geologic time scales the earth's motion forms our surroundings, be they from violent volcanic explosions to the subtle drifting of the earth's crust in response to the ever-present convective stirring of the mantle.

Nearly all human endeavors must in some way deal with restrictions imposed by fluid transport. This observation is obvious in the aerospace and marine transport industries, but equally true in other industries in which the importance of fluid flow may not be as apparent. For instance, the major limitation to the smallness, and hence speed, of modern computer chips is the restriction imposed by convective heat transfer in cooling electrical components. As computer chips become increasingly compact, the heating rates become higher and the passageways available for cooling become more confined. Biomedical applications must deal with a host of fluid transport issues, including oxygen supply to the lungs, pumping of blood and the associated transport of nutrients throughout the body, digestion of foods, and human reproduction. On a cellular level, transport of matter, such as viruses or nutrients, over cell boundaries controls the body's ability to fight diseases and heal wounds. Major medical crises, such as strokes and lung cancer, are related to transport of particulate matter either in the blood stream or in the pulmonary system, whereas other emergencies, such as heart attacks or hemorrhaging, are caused by inability to maintain a continual fluid flow. Agriculture must continually deal with fluid transport issues, such as water supply to crops, pesticide distribution, heating of livestock, and soil moisture during harvesting operations. Material processing, which is so vital for modern technological advances, deals with a host of fluid processes, ranging from metal casting to liquid spray coating.

Aside from its importance in diverse applications, fluid dynamics has had a central influence in development of much of modern science and mathematics. A fluid view of elementary matter dates back to ancient Greece, where Anaxagoras (500-428 BC) proposed that all matter consists of a fluid continuum whose basic component is a vortex. This continuum view competed with the atomic (or particulate) view of Democritus (460-370 BC), in which matter is formed of small particles immersed in a fluid and ordered by the action of vortices. The fluid theory of matter, again based on vortices, was later taken up by Descartes (1596-1650) to explain the suspension of celestial bodies and by Kelvin (1824-1907), who proposed that all matter is constructed of a set of "vortex atoms" that exist in the ether. Despite the errors that are now apparent in these concepts, the fluid/particle analogies of matter proposed by these early philosophers spurred development of the science of mechanics and of objective scientific processes to test these models. For instance, Kelvin's quest to uncover the structure of vortex atoms resulted in discovery of many of the basic laws and phenomena associated with vorticity transport in inviscid fluids. The wave/particle models used in modern physics to describe the theory of light and quantum models of elementary particles, as well as the quantum vortices in liquid helium II (London, 1954) and the "magnetic vortices" in high-temperature superconducting materials (Tinkham, 1975), have their base in the fluid/particle models of matter developed in ancient times.

Fluid dynamics has also had a large impact on development of modern applied mathematics. For instance, linear partial differential equations are usually categorized as hyperbolic, parabolic, or elliptic, where the classic examples for these three categories are the wave equation, the heat equation, and the Laplace equation, all of which play a prominent role in description of fluid systems. Two important paradigms for development of the theory of characteristics of nonlinear hyperbolic partial differential equations (Courant and Hilbert, 1962) are wave propagation in compressible gas dynamics and free-surface oscillations in shallow water layers. Kolmogorov's advancements in stochastic analysis found immediate application in his models of the energy cascade process of turbulent flows (Hunt et al., 1991). One of the earliest investigations of deterministic chaos was performed by Lorentz (1963) using a model of atmospheric transport, and further investigations into chaotic systems have been spurred by the need to formulate better models for turbulent flows. Fractal geometry is also commonly exhibited by fluid systems, as illustrated by many of the examples given in the book by Mandelbrot (1977).

Despite its importance to many fields, many centuries of study, and widespread use of advanced supercomputing systems, scientists and engineers specializing in fluid dynamics are still far from able to reliably predict most fluid flow problems. The principal difficulty lies in the inherent nonlinearity of the equations governing fluid flow. This nonlinearity arises from the fluid inertia and is responsible for instabilities and eventual transition of the flow to a turbulent state. Turbulent flows span an enormous range of length scales, with the ratio of the largest to the smallest scale exceeding a factor of [10.sup.5] in many marine and aerospace engineering applications and a factor of [10.sup.8] for flow in the earth's oceans and atmosphere. This wide range of length scales makes direct computational solution of the governing equations impossible for all but a few rather academic cases at low Reynolds numbers, thus requiring the use of models, often combined with empiricism, to truncate the mathematics to a manageable system. The situation is made yet more difficult by the fact that many natural and industrial processes involve particle or droplet transport, phase change, and chemical reactions that influence the fluid momentum transport, introducing still broader scales and breaking down self-similar behavior of the turbulence over these scales.

Because of its many practical applications, its use as a paradigm to stimulate new mathematical methods and physical models in diverse other fields, and its intrinsic beauty and difficulty, fluid dynamics remains an active field of both engineering and fundamental scientific inquiry. In this introductory chapter, we examine the role of viscous forces in fluid flows using simple scaling arguments and then explore the relevance of inviscid flow theory to real fluid flow problems.

1.1 ROLE OF VISCOSITY IN HIGH-REYNOLDS-NUMBER FLOWS

All real fluid flows are viscous, with the possible exception of quantum fluids such as liquid helium II below the transition temperature. One measure of the importance of viscous effects is given by the flow Reynolds number, Re = [rho]U L/, which is the ratio of the order of magnitude of the characteristic inertial stress [rho][u.sub.2] to the viscous stress ([partial derivative]u/[partial derivative]y). Here [rho] and are the fluid density and viscosity, and U and L are characteristic velocity and length scales of the problem. When the Reynolds number is O(1) or less, the viscous forces are important everywhere in the flow. This type of flow is often called creeping flow, and it is commonly observed in situations with extremely low velocities and high viscosities, such as certain liquid metal melts or convective circulation in the earth's mantle, and in flows with very small length scales, such as locomotion of microorganisms, dispersion of particulate matter in two-phase flow, and flow within microfluidic devices. For most problems on a human scale, the Reynolds number is quite large. For instance, a man standing in a mild 10-m/s breeze has a Reynolds number of about 3 x [10.sup.5]. For a baseball thrown by a major league pitcher, the Reynolds number is about 2 x [10.sup.5]. Water flow in a 2.5-cm-diameter bathroom supply pipe has Reynolds number (based on pipe diameter) of about 2 x [10.sub.4], whereas a modest river might have Reynolds number (based on width) of 5 x [10.sup.7]. An automobile driving at highway speeds has a Reynolds number of about 1 x [10.sup.7], and the wing of a commercial jet airplane has Reynolds number (based on chord length) of about 2 x [10.sup.8]. The Reynolds numbers of major atmospheric or oceanic features are quite large; for instance, for a typical atmospheric low-pressure system Re ~ [10.sup.12] and for an oceanic Gulf Stream ring eddy Re ~ 5 x [10.sup.10].

For flows with Reynolds number much larger than unity, viscous forces will be of importance only in regions with small length scales or over very long convective time scales. This statement can be made more precise by the following argument. Let us consider a steady-state flow with length scale L characteristic of the flow geometry (e.g., the body diameter or channel width), and assume that viscous forces are important only within a small layer R with length scale [delta] in the cross-stream direction, as illustrated in Figure 1.1. The inertial force acting on the region R has order of magnitude [rho][U.sup.2][delta]L, which is simply the inertial stress times the cross-sectional area [delta]L of R. The viscous shear force acting on the lateral surfaces of R has order of magnitude U]L.sup.2]/[delta]. Equating these two forces and solving for the length scale ratio gives

[delta]/L = O([Re.sup.-1/2]). (1.1.1)

As the Reynolds number increases, the thickness of the region in which viscous forces can be important correspondingly decreases compared to the length scale L that characterizes the flow as a whole. The estimate (1.1.1) is characteristic of the ratio of thickness of a laminar viscous boundary layer along a body surface to the body diameter (Figure 1.2), where the formation of the boundary layer is necessary in order to satisfy the no-slip condition for the tangential velocity component, which cannot in general be satisfied in inviscid flow theory. Viscous forces can also be important in internal regions within a fluid flow, such as in the vortex reconnection problem shown in Figure 1.3. In this problem, two vortex tubes with vorticity of opposite sign collide due to inviscid instability, deforming the vortex cores as they are driven together. Viscous forces become important in a thin layer in-between the impacting tubes, in which viscous cross diffusion causes cancellation of vorticity between the two structures, allowing the vortex lines within each tube to be cut and to reconnect to vortex lines in the opposing tube to form loops.

Viscous forces are important everywhere in a turbulent flow, but only at small scales of motion. We equate the flow length scale L to the turbulence integral length scale, which is characteristic of the eddy size containing the most energetic turbulent fluctuations, and let U be proportional to the square root of the turbulent kinetic energy. Viscous dissipation is important on the smallest scale of turbulent motion, denoted by [delta]. Following the reasoning of Kolmogorov (see Hunt et al., 1991), we assume that the net rate of energy dissipation in the turbulent flow is controlled by the rate at which energy cascades from larger scales to smaller scales of motion. The dissipation scale [delta] (called the Kolmogorov scale) is thus assumed to vary as a function of the average dissipation rate per unit mass [epsilon] and the kinematic viscosity ? = /[rho], such that from elementary dimensional analysis [delta] = O(v.sup.3]/[epsilon]).sup.1/4]. The energy dissipation rate [epsilon] is assumed to be proportional to the turbulent kinetic energy [U.sup.2] divided by the large eddy turnover time L/U, such that [epsilon] = O(U.sup.3]/L). Substituting into the expression for the Kolmogorov length scale [delta] gives the ratio of the smallest to the largest length scales of a turbulent flow as

[delta]/L = O([Re.sup.-3/4]). (1.1.2)

Turbulent flows are characterized by high Reynolds numbers and a broad range of length scales, ranging from scales characteristic of the large-scale flow geometry to the dissipation scale. Viscous effects are negligible at all but the smallest scales of this range.

If both viscous and inertial forces are restricted to act over the same length scale L, then the viscous forces must act much more slowly than the convective forces in a high-Reynolds-number flow. The ratio of viscous and convective time scales, defined by [T.sub.V] = [rho][L.sup.2]/ and [T.sub.C] = L/U, respectively, is given by

[T.sub.V]/[T.sub.C] = O(Re). (1.1.3)

The estimate (1.1.3), for instance, is characteristic of the ratio of viscous decay time to turnover time of a vortex patch.

1.2 INVISCID FLOWS

In view of the length scale estimates presented in the previous section, it might be supposed that inviscid flow can be obtained as a limit of actual viscous flows as the Reynolds number increases to infinity. Actually, this is not quite the case. There are certain observed properties of viscous flows that, no matter how high the Reynolds number, occur in violation of inviscid flow theory. For instance, while the turbulent boundary layer about a blunt body grows progressively thinner as the Reynolds number increases (with the body size held fixed), the location of boundary layer separation in the rear of the body is approximately independent of Reynolds number. Even if the Reynolds number is increased indefinitely, the boundary layer will still separate, ejecting vorticity from the body, whereas inviscid flow theory requires vorticity generated on the body surface to remain on the surface. As another example, we note that many turbulent flows are observed to adopt a self-similar state, in which the large scales of motion are not influenced by a change in Reynolds number (Tennekes and Lumley, 1972). In such flows, energy dissipation is controlled by the cascade of energy from the large scales, such that with increasing Reynolds number the energy dissipation occurs at progressively smaller scales but at nearly the same overall dissipation rate. In an inviscid flow, of course, no energy dissipation is possible.

(Continues...)

Excerpted from Inviscid Incompressible Flowby Jeffrey S. Marshall Copyright © 2001 by Jeffrey S. Marshall. Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.