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Rotating flow is critically important across a wide range of scientific, engineering and product applications, providing design and modeling capability for diverse products such as jet engines, pumps and vacuum cleaners, as well as geophysical flows. Developed over the course of 20 years’ research into rotating fluids and associated heat transfer at the University of Sussex Thermo-Fluid Mechanics Research Centre (TFMRC), Rotating Flow is an indispensable reference and resource for all those working within the gas...
Rotating flow is critically important across a wide range of scientific, engineering and product applications, providing design and modeling capability for diverse products such as jet engines, pumps and vacuum cleaners, as well as geophysical flows. Developed over the course of 20 years’ research into rotating fluids and associated heat transfer at the University of Sussex Thermo-Fluid Mechanics Research Centre (TFMRC), Rotating Flow is an indispensable reference and resource for all those working within the gas turbine and rotating machinery industries. Traditional fluid and flow dynamics titles offer the essential background but generally include very sparse coverage of rotating flows—which is where this book comes in. Beginning with an accessible introduction to rotating flow, recognized expert Peter Childs takes you through fundamental equations, vorticity and vortices, rotating disc flow, flow around rotating cylinders and flow in rotating cavities, with an introduction to atmospheric and oceanic circulations included to help deepen understanding. Whilst competing resources are weighed down with complex mathematics, this book focuses on the essential equations and provides full workings to take readers step-by-step through the theory so they can concentrate on the practical applications.
Swirling, whirling, rotating flow has proved fascinating and challenging throughout the ages. Whether it be the vortex formed as water exits the bathtub or the eddying motion seen in a cornfield as wind blows across it, the subject provides a talking point and a level of complexity that often defy simple explanation. Rotating flow is critically important across a wide range of scientific, engineering, and product design applications. The subject provides a means of modeling, and as a result, design capability for products such as jet engines, pumps, and vacuum cleaners.
Even for applications where rotation is not initially evident, the subject is often fundamental to understanding and modeling the details of the flow physics. An example is the very strong wing tip vortices shed from the wings of an aircraft as it flies, as illustrated in Figures 1.1 and 1.2.
In Figure 1.1 the massive vortices shed by the wings of a Boeing 727 are made visible by smoke emitted by smoke generators attached to the wing tips. This smoke becomes entrained into the vortices as the plane moves. These vortices can take a few minutes to dissipate, and the flow disturbance caused in the wake of a plane is therefore one of the factors that needs to be considered in order to ensure adequate intervals between different aircraft occupying the same airspace. The vortex that is shed from one of the wings of an agricultural spray plane is illustrated in Figure 1.2. Here the flow is made visible by using colored smoke rising from the ground. This kind of vortex flow is similar to the whirling motion seen in the vicinity of an oar for a boat as it is rowed along. For both the wing tip and paddle vortices examples, where the action disturbing the flow is a linear motion, the result is a flow with significant rotation.
It is the much weaker forces that hold the particles together in a fluid, in comparison to those in a solid, that give rise to their more complex behavior. A fluid does not offer lasting resistance to displacement of one layer of particles over another. If a fluid experiences a shear force, the fluid particles will move in response with a permanent change in their relative position, even when the force is removed. By comparison, a solid—provided the shear force does not exceed elastic limits—will adopt its previous shape when the force is removed. In a fluid, shear forces are possible only when relative movement between layers is taking place, that is, the fluid is flowing. The purpose of this text is to introduce the subject of flow in applications where the flow rotates or swirls and the associated physical phenomena and principles.
1.2. GEOMETRIC CONFIGURATIONS
There are many subtle interactions between fluids and associated structures and boundaries that produce vorticity and secondary flows. Vorticity involves the rotation of a flow element, representing a chunk of fluid particles, as it moves through a flow field and can be visualized by a physical or virtual cork in a flow. If the cork rotates, then the flow has vorticity, whereas if the cork just translates through the flow field, then the flow does not have vorticity. A secondary flow is a flow pattern superimposed on the primary flow path. For example, when a viscous fluid moves through a bend in a pipe or channel, the differences in pressure and velocity that arise between the central core and the boundary layer flow near the surfaces set in motion a complex secondary flow pattern of spiralling vortices (Figure 1.3). These secondary flows are responsible for additional losses in pressure within a pipeline and increased erosion and scouring of surfaces.
An interest in the fluid flow associated with machinery may stem from the need to know the losses associated with windage in, say, an electric motor where the armature is spinning in air (Figure 1.4) or from use of fluid as a means of transferring energy as in, for example, a centrifugal compressor (Figure 1.5), or a rotary pump.
The jet engine has become the principal prime mover for aviation transport, and the industrial gas turbine engine is responsible for the production of a significant proportion of base load and standby power. It is also the prime mover of choice for certain types of ship and pumping applications. Gas turbine engines typically comprise a compressor, combustor, and turbine. Flow is compressed by diffusing the flow through alternating rows of stationary and rotating blades in a compressor; energy is added to the flow in the combustor; and the power to drive the compressor is produced by expanding the flow through the alternating rows of stationary and rotating turbine blades. The rotating blades are typically supported on discs. Rather than use completely solid rotors, compressor and turbine discs tend to have profiled cross sections that are designed to dissipate the local high stresses and minimize the mass of material used. As illustrated in Figure 1.6, a typical gas turbine engine cross section will reveal that cylindrical cavities are formed between adjacent discs. These cavities will typically be filled with air. The cavity formed may involve a rotating disc adjacent to a stationary disc, which is known as a wheelspace or rotor-stator disc cavity. Alternatively, if both coaxial discs are rotating, this is referred to as a rotating cavity.
In both aircraft and electric power generation gas turbine engines, it is common practice for the cavity formed between a rotating turbine disc and a stationary disc to be purged with coolant air. This air reduces the thermal load on the disc and prevents the ingress of hot mainstream gas from the blade path into the cavity between the rotating disc and the adjacent stationary casing. However, use of this air is detrimental to engine cyclic performance, and turbine efficiency can be adversely affected by the seal air efflux into the main annulus. The lifetime of the rotor, and the cyclic and component performance of the engine are therefore dependent on the efficiency with which the cavity is purged of hot gases. The combined requirements to be able to determine the power required to overcome frictional drag, local flow characteristics, and associated heat transfer have provided a sustained impetus to investigate a number of rotor-stator disc configurations. The range of practical applications for disc flow extends to computer memory disc drives (as illustrated in Figure 1.7), centrifuges, cutting discs and saws, gears, and brakes.
Rotating closed cylindrical and annular cavities are a common feature in compressor and turbine rotor assemblies. The flow associated with a rotating cavity depends on the radial and axial temperature distribution and the presence of geometrical features such as shrouds and the presence of any superposed flow. The flow can be highly complex with buoyancy-driven time-dependent flow features that provide a challenge to the most advanced analytical and numerical models.
Further applications in which rotating flow is important are in cyclone and U-shaped separators, as illustrated in Figures 1.8 and 1.9. In cyclone separators, a raw mixture of substances at high velocity is introduced into a stationary mechanical structure that deflects the mixture into a circular path. In the conical cyclone separator, the flow's angular velocity increases as the mixture moves radially inward due to the conservation of angular momentum. Because of inertia, the heavier components in the flow resist being deflected more than lighter particles, and the heavier particles therefore pass preferentially toward the outside of the flow. Secondary flows in the conical section assist in transporting heavier particles toward the bottom outlet, and a second outtake can be used for lighter particles.
In the simple U-shaped geometry of Figure 1.9, the fluid is caused to change direction by 180°, and an outtake located partway along the tube is used to separate the flows. This type of design is reasonably effective with fluid and solid mixtures, where the solids are much more dense than the fluid and have a low enough drag so that the solid particles can move radially through the fluid during the short residence time in which the mixture is in the U shaped section of the separator. U-shaped separators are generally located vertically, so that gravity can be used to aid the collection process of the denser particles.
1.3. GEOPHYSICAL FLOW
Theories for modeling rotating flow originally developed from attempts to understand fluid flow phenomena on the Earth's surface and its oceans. The fluid motions in the Earth's atmosphere are driven by a combination of thermal convection, Coriolis forces, and vorticity produced in regions of wind shear. The energy source for this motion is thermal radiation from the Sun, which is absorbed by the Earth and emitted to space. The massive amounts of energy involved, combined with the energy differentials between the poles and the equator, the influence of the Earth's rotation, and flow instability, result in major atmospheric circulations and under certain conditions intense atmospheric vortices. The views of the Earth from space illustrated in Figures 1.10 and 1.11 provide an indication of the variety of cloud formations in our atmosphere.
Over the oceans, shear stresses caused by winds blowing across an ocean surface can cause currents of water up to 100 m in depth or more to flow. These currents can be thousands of kilometers wide and move at speeds of several kilometers per day. Rotational patterns known as gyres are caused by the combination of wind-induced shear stresses and Coriolis forces due to the Earth's rotation in each of the five major oceans. The net effect is that in the northern hemisphere the general direction of rotation of the gyres is clockwise, and in the southern hemisphere it is anticlockwise.
Examples of intense atmospheric vortices include tropical cyclones and tornadoes. Tropical cyclones are called hurricanes over the Atlantic Ocean and typhoons over the Pacific Ocean. They can be several hundred kilometers in diameter. Tropical cyclones are caused by large-scale convection over a region of warm water. The convection is augmented by the release of latent heat during the condensation of rain. The cyclone's angular momentum is derived from the Earth's rotation during convergence of air toward the cyclone's low-pressure center. Because the angular momentum is derived from the Earth's rotation, the direction of rotation for a tropical cyclone is the same as the Earth's, which is anticlockwise when viewed from an observer above the northern hemisphere and clockwise in the southern hemisphere.
Figure 1.12 shows a photograph of Hurricane Gordon taken at 18:15:36 Greenwich Mean Time on September 17, 2006 by one of the crew members aboard the Space Shuttle Atlantis. The center of the storm was located near a latitude of 34° degrees north and a longitude of 53° west, while moving north-northeast. At the time the photo was taken, the sustained winds were 36 m/s with gusts to 44 m/s (almost 100 mph). Figure 1.13 shows a satellite image of Hurricane Rita approaching the Florida coastline, and Figure 1.14 shows an image of three typhoons over the western Pacific Ocean taken on August 7, 2006. In all three figures the spiral nature of the hurricane or typhoon is apparent with a series of rainbands that surround the low-pressure "eye" at the center.
In comparison to cyclones, tornadoes are much smaller in scale but can involve more intense and destructive vorticity. Tornadoes have captured the interest of observers for centuries because of their destructive nature on a localized scale, flow intricacies, and the difficulty in predicting their occurrence and path. A tornado is a columnar vortex formed in the atmosphere with a height of about 3 km and is typically 10 to 100 m in diameter. A tornado involves concentration of angular momentum and kinetic energy into a low-altitude cloud sometimes called a rotor cloud. Rotating columns of air called funnels extend down from the cloud toward the ground, and if they have enough energy and momentum, they reach the Earth's surface. If the funnel reaches the Earth's surface, the central pressure will be low enough for the radial pressure gradient across the sheath of air forming the funnel to balance the centrifugal force in the rotating sheath of air. A secondary flow is super-posed on the rotation of the funnel, and the flow pattern associated with a tornado is in the form of a spiral, with a radial inward flow toward the eye of the tornado near the ground. The local winds resulting from a tornado can reach speeds of up to 200 or even 300 m/s. A tornado column will become visible if water vapor condenses within it or if its motion results in it picking up dust and debris. The presence of a low-pressure region at the center of a tornado accounts for some of the destructive capability. If the eye of the tornado where the pressure is lowest passes over a hollow building, the pressure difference between the air inside the building and the low pressure at the center of the tornado can result in very large forces on the walls and roof and can cause the building to literally explode. (A "small" tornado off the author's hometown in the United Kingdom is illustrated in Figure 1.15.) Tornadoes derive their energy and vorticity from other phenomena such as hurricanes, squall lines, thunderstorms, volcanoes, and firestorms. If a cyclone provides the vorticity required for the formation of a tornado, as a result, the tornado will tend to rotate counterclockwise, although anticyclonic tornadoes with clockwise rotation do occur. When a tornado forms over water, it is referred to as a water spout.
Atmospheric circulations can occur on any planet with a fluid atmosphere. For example, Jupiter's atmosphere is well known for its Great Red Spot, illustrated in Figure 1.16. This is a swirling mass of gas resembling a hurricane or massive anticyclone. The immense size of this nearly two-dimensional vortex, its persistence, and its very high turbulence have provided a source of fascination to observers and scientists for centuries. The current length of the elongated spot in the east-west direction is circa 20,000 km and is about three times the diameter of Earth. The width of the spot is about 12,000 km in the north-south direction, and its depth is believed to be somewhere between 20 km (Marcus 1993) and 200 km (Irwin 2003). The length of the spot varies, and 100 years ago estimates indicate that it had a length of 46,000 km (Irwin 2003). The color of the spot usually varies from brick-red to slightly brown, and the spot has been known to fade entirely. Its color may be due to small amounts of sulfur and phosphorus in the ammonia crystals. The edge of the Great Red Spot circulates at a speed of about 100 m/s (225 miles per hour). The spot remains at the same distance from the equator but drifts slowly east and west. The zones, belts, and the Great Red Spot are much more stable than similar circulation systems on Earth, but the mechanisms responsible for its formation (Marcus 1993) have similarities to those of the free vortex introduced in Chapter 3 and large-scale atmospheric vortices on Earth as considered in Chapter 8.
Figure 1.16 shows a variety of cloud features, as well as the Great Red Spot, including parallel dark and white bands, multilobed chaotic regions, white ovals, and many small vortices. Many clouds appear in streaks and waves due to continual stretching and folding by Jupiter's winds and turbulence. The gray features along the north edge of the central bright band are equatorial "hotspots." Small bright spots within the orange band north of the equator are lightning-bearing thunderstorms. The polar regions shown here are less visible because the Cassini spacecraft viewed them at an angle and through thicker atmospheric haze.
The parallel dark and white bands, the white ovals, and the Great Red Spot persist over many years or centuries, despite intense atmospheric turbulence. The most energetic features are the small, bright clouds to the left of the Great Red Spot and in similar locations in the northern half of the planet. These clouds grow and disappear over a few days and generate lightning. Streaks form as clouds are sheared apart by Jupiter's intense jet streams that run parallel to the colored bands. The prominent dark band in the northern half of the planet is the location of Jupiter's fastest jet stream, with eastward winds of about 130 m/s (300 miles per hour). Jupiter's diameter is 11 times that of Earth (142,984 km). Thus the smallest storms in Figure 1.16 are comparable in size to the largest hurricanes on Earth.
Excerpted from Rotating Flow by Peter R.N. Childs Copyright © 2011 by Elsevier Inc.. Excerpted by permission of Butterworth-Heinemann. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Preface Nomenclature 1 Introduction to Rotating Flow 2 Laws of motion 3 Vorticity and Rotation 4 Discs 5 Rotating Cylinders, Annuli and Spheres 6 Rotating cavities 7 Atmospheric and Oceanic Circulations Appendix A: Air properties Appendix B: Selected mathematical relationships Appendix C: Glossary
Posted August 11, 2014