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Overview
This lucid, well-written presentation of the basic principles and applications of rotating fluid theory is an excellent text for upper-level undergraduate or beginning graduate students, but it will also be invaluable for engineers and scientists whose projects require knowledge of the theory. Readers are assumed to be familiar with vector analysis, fluid mechanics, and partial differential equations.
Part I (Chapters 1-5) introduces the concept of rotating fluids and reviews basic fluid mechanics. Part II (Chapters 6-13) considers concepts, theories, and equations specific to rotating fluids, including vorticity and vortex dynamics and rotating coordinate systems; Coriolis phenomena; rotation, vorticity, and circulation; vorticity as a variable, vortex dynamics, secondary flows; circular pathline flows; and rotation and inertial waves. Each chapter in Part II includes solved quantitative examples. Part III (Chapters 14-22) presents numerous practical applications of the theory, including flows in pipes, channels, and rivers, as well as other applications, in fields ranging from rotors, centrifuges, and turbomachinery to liquids in precessing spacecraft, oceanic circulation, and intense atmospheric vortexes.
Five useful appendixes provide a synopsis of mathematical relationships, stream functions, and equations of motions, as well as fluid properties and geophysical data. "Highly recommended." — Choice.
Product Details
| ISBN-13: | 9780486161983 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 05/05/2014 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 448 |
| File size: | 29 MB |
| Note: | This product may take a few minutes to download. |
Read an Excerpt
Rotating Fluids in Engineering and Science
By James P. Vanyo
Dover Publications, Inc.
Copyright © 1993 James P. VanyoAll rights reserved.
ISBN: 978-0-486-16198-3
CHAPTER 1
Rotating Fluid Phenomena
Many areas of engineering and science involve the rotation of various objects; in science the object sometimes is the earth; in engineering it might be a turbine rotor or a space vehicle. In most cases the object also involves the rotation of internal or external fluids. Sometimes a rotating fluid is the principal phenomena of interest; at other times the fluid is merely an unwanted participant in the motion. In either event, success or failure of the analysis can depend critically on understanding and predicting rotating fluid phenomena.
The basic theory of fluid rotation and vorticity distinguishes between vorticity and curved (e.g., circular) translation of fluid elements. Figure 1.1 illustrates smooth uniform flow of a viscous liquid in a channel. This laminar flow, when fully developed, has a parabolic velocity distribution. Ink or dye slowly injected in the flow would move in straight lines indicating straight streamlines (fluid motion). However, small objects placed in the flow would also rotate as they move, indicating that the flow has vorticity, i.e., the infinitesimal fluid elements rotate as they translate along straight lines. Figure 1.2 illustrates a flow field called an inviscid vortex where all fluid elements move in circular paths. However, small objects here would not rotate, indicating a fluid that is not rotating, merely translating in circular paths. The two flows illustrate two extremes, one that has straight pathlines but fluid element rotation, and the second that has circular pathlines but fluid elements which do not rotate. Viscosity in the first flow produces the fluid element rotation called vorticity, which is absent in the second flow. Rotation, vorticity, and circulation are described and quantified in Chapter 8. Figure 1.3 shows a smoke ring. Its motion is often modelled as a toroidal inviscid vortex. Any cross section through the toroid is approximately an inviscid vortex as in Figure 1.2.
Figure 1.4 shows water in a cylinder. In the left photograph, both the cylinder and the water are stationary, and all the colored water, slightly less dense than the clear water, is in the top one-eighth of the cylinder. In the right photograph, the cylinder has been impulsively accelerated to a constant angular velocity, and the water is gradually being "spun up" to the angular velocity of the cylinder. Liquid spin-up is achieved about 1% by viscous interaction at the cylinder side walls and about 99% by a viscous secondary flow at the cylinder bottom. Centrifugal force inside this very thin (almost invisible) spinning bottom boundary layer moves clear water outward and then up along the outside cylinder wall. Colored water in the interior is drawn downward until all the water will be pumped outward through the very thin, bottom boundary layer. In this experiment, a 2% buoyancy of the colored water is opposing the pumping action and causes the boundary between the clear and colored water to be tapered rather than cylindrical. More complex rotating fluid phenomena, such as internal waves and vortex stretching, are involved during a reverse spin-down process.
Rotating masses of fluid exhibit other unusual properties. Figure 1.5 shows the difference between flow patterns created by suddenly dumping a quantity of similar density colored water into nonrotating water, (top two photographs), and flow patterns created by the same act, but using rotating water as shown in the bottom two photographs. Typical turbulent eddies are generated in the nonrotating water. Such random motions are not possible in a rotating liquid; instead, permissible flows have a distinctly two-dimensional property as shown in the photographs.
Rotating fluid theory helps explain important application areas in engineering and science. These include many subtle fluid-structure interactions that produce vorticity and secondary flows. For example, when a viscous fluid moves through a bend in a pipe or channel, differences in velocity and pressure between the central portion and the boundary layers near solid surfaces induce secondary flows similar to those on the bottom surface of the cylinder in Figure 1.4. These secondary flows cause energy losses in engineering applications and erosion when the channel is a river bed. Figure 1.6 shows a typical result. Bottom secondary flows and sediment move transversely from the outside of a bend toward the inside of that bend. Any slight departure from straight line motion becomes more and more pronounced, gradually reaching the condition shown in Figure 1.6. If a flood condition occurs, the river may overflow the meandering channel and erode a nearly straight channel to repeat a new cycle.
Vorticity, generated especially in boundary layers, is typical of all real fluid flows. Integration of vector vorticity over a finite surface leads to the concept of circulation along the curve enclosing the surface. This in turn has been shown to be related to lift of moving objects. Although lift of an aircraft wing is related to a difference in pressure top and bottom, lift can be expressed also as circulation about the wing represented by an equivalent vortex coincident with the wing. This vortex cannot simply end when the wing tip is reached. It is deflected rearward to form wing tip vortices that are visible at appropriate speeds and atmospheric conditions as shown in Figure 1.7. The parallel vortices interact and regroup as a series of toroids similar to the smoke ring of Figure 1.3.
An efficient wing design minimizes wasted vortex motion. Soaring birds, as in Figure 1.8, have very efficient wing tip designs for slow flight conditions. They apparently can feel and manipulate wing tip vortices using their individually controllable wing tip feathers. The bird in Figure 1.8 is a California condor in flight. Note in particular how carefully the condor has manipulated its wing tip feathers, both to reduce drag and to control its flight direction. Some long range modem aircraft, e.g., the new Boeing 747400, mimic the condor by adding large "winglets" tilted upward at each wing tip.
Curved motion, viscosity, boundary layers, secondary flows, lift, and drag are design features common to most rotating machinery. The relevant fluid may only be air in contact with the spinning armature of an electric motor, representing an energy loss, or it may be the fluid in a rotary pump. The centrifugal pump impeller and housing shown in Figure 1.9 draws in nonrotating fluid along its axis, spins it at high speed, and uses centrifugal force to create fluid pressure and flow velocity. Within the pump the fluid moves in nearly spiral paths as it moves from the inlet to the outlet.
An axial flow turbine has alternate rows of stationary and rotating vanes and can be used either as a motor or a pump (compressor). The turbofan jet engine in Figure 1.10 is one type of axial flow turbine. It draws in and compresses air in the inlet compressor section, continuously injects and bums fuel in the central combustion section, and uses a portion of the energy of the exhaust to power the rear portion (turbine) of the jet engine. The turbine portion drives the inlet compressor section and fan. The fan helps provide propulsion. The remainder of the energy is used to eject the exhaust gases at high velocity to give additional forward propulsion. Fluid motion along curved paths and fluid vorticity have both positive and negative design implications in these applications.
Important applications of rotating fluids occur in many vehicles that contain liquids, often as fuel. In some cases, the vehicle is massive enough, relative to the quantity of fuel, that motion of the vehicle can be prescribed independent of the fluid. In other cases, e.g., an oil tanker or a space vehicle, the liquid mass may exceed the vehicle structural mass. On take-off, a space vehicle may be over 90% liquid fuel by mass, and communication satellites, upon being placed in orbit, typically are over 50% liquid by mass. Space vehicles present unusual problems because their motions are unconstrained by land, water, or air.
Figure 1.11 shows a communication satellite in its orbit configuration. It was manufactured by Ford Aerospace for the India Space Agency and was placed in orbit by a Delta rocket. During the final orbit insertion maneuver the rocket third stage (PAM) and the satellite were spun up to achieve gyroscopic attitude stability. At this point approximately 60% of the satellite's total mass was liquid fuel. Figure 1.12 shows a communication satellite manufactured by ERNO Raumfahrttechnik for Telecom (Germany). It also was placed in orbit by a Delta rocket and is shown here in launch configuration fastened on top of its PAM (payload assist module). Phenomena similar to, but more complex than, that shown in Figure 1.4 occur inside the spinning fuel tanks and have the potential for destabilizing the attitude (orientation) of the satellite. Satellites have become inoperative because of an inability to predict these rotating fluid-structure interactions correctly. These satellites were correctly analyzed and designed, and performed perfectly. Figure 1.13 is a NASA photograph of astronauts manually reorienting a satellite relative to their space shuttle. Continued space exploration will present many unusual situations where an ability to predict rotational motions of vehicles containing large quantities of liquids will be essential for success.
Rotating geophysical fluids are usually related to and dependent upon the earth's rotation. For example, the earth has a liquid core whose radius is slightly more than half the earth's outside radius; its mass is approximately one-third the earth's total mass. It is assumed to be molten iron with small amounts of other elements. This provides a situation very similar to that of liquid fuel in a spinning communication satellite. Part of the complexity of both applications is that neither the earth nor a satellite spins about a fixed axis. The spin axis of each wobbles (precesses) and, in so doing, continuously changes direction. Under this condition the internal liquid is continuously being perturbed.
Figure 1.14 shows a laboratory experiment used to analyze liquid motions in a container that spins and precesses at various rates. The transparent tank shown in the photograph has the same nonspherical shape as the earth's mantle-core boundary. The angle between spin and precession shown in the photograph (23.5?) matches the earth's forced precession angle. While many variables can be matched during experiments, not all can. Dimensionless ratios and scaling techniques can often resolve such experimental difficulties. Even if the earth's axis did not wobble, the earth's spin rate would restrict thermal convection, turbulence, and internal waves in the liquid core to prescribed patterns and magnitudes.
Rotating fluid theory historically developed during attempts at understanding and predicting fluid flow phenomena on the earth's surface, especially large-scale atmospheric and oceanic flows. Figure 1.15 shows the earth viewed from space. The continent of Africa fills the upper-left quadrant of the photograph. Medium (meso-) scale cloud patterns are obvious. Major components of large-scale flows do not vary day or night, summer or winter, but are not easily made visible. These long-term flow components are caused by the average flow of heat from the earth's hot equatorial region to the cold polar regions. The earth's rotation causes these major north-south atmospheric flows to deflect east or west, as viewed from the earth's surface, to produce the east-west trade winds. These trade winds were known and used by early mariners.
Surface winds blow across the broad expanses of the oceans causing surface ocean currents 100 or more meters deep. These currents measure up to thousands of kilometers wide and typically move at speeds of a few kilometers per day. A few, such as the narrow, deep Gulf Stream and the Kuroshiro (Japanese) Current, move at speeds up to 120 km/day. Shear stresses caused by the wind and Coriolis phenomena caused by the earth's rotation induce rotational patterns in each of the five major oceans, clockwise in the Northern Hemisphere and counterclockwise in the Southern Hemisphere. These rotating patterns are called gyres. The motion can be seen in infrared satellite photographs that distinguish water temperatures and by measurements from research ships. Figure 1.16 shows the warm North Equatorial Current moving westward across the Atlantic Ocean, then bending north to become the Gulf Stream, completing its circle (gyre) as a flow east to England and then south toward northern Africa. Figure 1.16 also shows vortex motions produced along the boundaries of the Gulf Stream as it moves north and east.
Some atmospheric motions are not caused directly by thermal convection or earth rotation but rather by vorticity produced in regions of wind shear. All combinations of effects are possible. Small whirlwinds behind the edges of buildings are produced randomly by wind gusts. Large-scale winds associated with movement of cold or warm fronts produce flow patterns that combine earth rotation and shear vorticity effects. Some of the more spectacular are the large-scale (up to 1000 km) dust storms such as those of central Asia and northern Africa. The major flow is produced by energy extracted from the earth's overall atmospheric motion; however, its destructive ability is intensified by wind shear vorticity.
Tropical cyclones (called hurricanes in the Atlantic and typhoons in the Pacific) and tornadoes have easily earned the classification "intense atmospheric vortices." However, except that they both consist of air rotating at very high speed and are extremely destructive, they have little in common. Note in Figure 1.15 the immature typhoon (hurricane) forming over the ocean between India and Africa. Figure 1.17 shows hurricane Gilbert (September 1988) viewed from a NOAA weather satellite. The hurricane covers the entire Caribbean from Florida in the upper right to the Yucatan Peninsula in the lower left. The major diameter of this hurricane is greater than 1000 km, and its eye is 40 km in diameter. Gilbert had 145 mph winds and the lowest central pressure ever recorded for a hurricane. A tornado is rarely over a few hundred meters in diameter and would be too small to be visible in this hurricane photograph.
A hurricane is caused by large-scale thermal convection over a region of warm (>26.5°C) ocean water. Convection is augmented by release of latent heat during condensation of rain. The hurricane's angular momentum (rotation) is extracted directly from the earth's angular momentum (rotation) during convergence of air toward its low-pressure center. Because its rotation is extracted directly from the earth's rotation, its direction of rotation agrees with the earth's, counterclockwise viewed from the Northern Hemisphere and clockwise viewed from the Southern Hemisphere. Tangential wind speeds near the center often reach 200 to 300 km/hr. When a tropical cyclone (hurricane) passes over land or cool water, it loses its energy source and soon dissipates.
Tornadoes receive their energy and their angular momentum from energy and vorticity (rotation) produced in and stored by other phenomena, e.g., hurricanes, squall lines, thunderstorms, or even volcanoes or fire storms. Tornadoes usually descend from overhead clouds as in the Figure 1.18 photograph of a massive tornado that occurred near Seymour, TX, on April 10, 1979. When the vortex funnel reaches the earth's surface, it usually accumulates water, dust, or debris. Over land it is called a tornado. When over water it is called a waterspout as in Figure 1.19. The waterspout shown here was observed by the author at Santa Barbara on October 1, 1976. It started as a low (300 m high), thick (75 m diameter) column in the ocean about 500 m from the University of California, Santa Barbara campus, and then lengthened as it moved east about 10 km to where it was photographed. It soon dissipated. Tornadoes (waterspouts) can rotate in either direction and tend to be brief in duration. Some evidence suggests maximum tangential winds near supersonic velocities, but measured velocities are less than 400 km/hr.
The figures are explained in more detail in Parts II and III. Mathematical models are included in most cases along with experimental results. References to more advanced and more specialized texts and to the professional literature are included for those wishing to explore subjects more thoroughly.
(Continues...)
Excerpted from Rotating Fluids in Engineering and Science by James P. Vanyo. Copyright © 1993 James P. Vanyo. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
Contents
Preface,PART I Fluid Mechanics Review,
1 Rotating Fluid Phenomena,
2 Mass and Momentum Conservation,
3 Potential (Inviscid) Flow,
4 Boundary Layers and Turbulence,
5 Wave Theory,
PART II Rotating Fluid Theory,
6 Rotating Coordinate Systems,
7 Coriolis Phenomena,
8 Rotation, Vorticity, and Circulation,
9 Vorticity as the Variable,
10 Vortex Dynamics,
11 Secondary Flows,
12 Circular Pathline Flows,
13 Rotation and Inertial Waves,
PART III Rotating Fluid Applications,
14 Pipes, Channels, and Rivers,
15 Rotors and Centrifuges,
16 Wings, Lift, and Drag,
17 Turbomachinery,
18 Liquids in Precessing Spacecraft,
19 The Earth, Sun, and Moon,
20 Atmospheric Circulation,
21 Oceanic Circulation,
22 Intense Atmospheric Vortices,
Appendices,
A Mathematical Relationships,
B Stream Functions and Velocity Potentials,
C Equations of Motion,
D Fluid Properties,
E Geophysical Data,
References,
Index,