Separation, Vorticity, and Vortex Shedding

Most classical theory of viscous flow is based on approximations valid at low or high Reynolds number. Although there are a few exact solutions that illustrate aspects of the intermediate range, they are rather limited. The range of phenomena that fall under this heading is commonly illustrated by photographs taken in laboratories. Computational fluid dynamics makes the intermediate Reynolds number range quite accessible.

Inertia is now comparable to viscous stress, and all terms in the Navier–Stokes equations must be retained. Convection destroys the upstream–downstream symmetry of creeping flow (Chapter 3). A distinct wake can be identified leaving the downstream side of a body in an incident flow. Forces on blunt bodies become increasingly due to pressure rather than to viscous stress.

Vorticity is increasingly confined to regions near to walls on the upstream portions of a body and to wakes on the downstream side. The upstream vortical regions become boundary layers in the high Reynolds number limit. As vorticity diffuses away from the surface it is convected downstream, ultimately to form the wake. The upstream–downstream asymmetry leads to another important idea, that of separation. For example, flows into and out of a nozzle are quite different. The flow into a trumpet shaped orifice, say, will follow the walls. As the opening narrows, the flow accelerates to conserve mass [loosely, ρUA = constant, per Eq. (1.33), implies U increases as A decreases]. The accelerating flow convects vorticity toward the wall, keeping it confined near the surface.

Why Study Fluid Dynamics?

Fluid dynamics is a branch of classical physics. It is an instance of continuum mechanics. A fluid is a continuous, deformable material. It is a material that flows in response to imposed forces. This is embodied in the everyday experience of draining water from a sink. The water flows under the action of gravity. It does not have a fixed shape; it fills the sink, conforming to its shape. The water flows with variable velocity, depending on its distance from the drain. All these distinguish fluid motion from solid dynamics. As another example, a pump propels water through a pipe or through the cooling system of a car. How does the reciprocating movement of the pump produce directed flow, extending to distant parts of the cooling circuit? One way or the other, the pump must be exerting forces on the fluid; one way or the other, these forces are communicated to distant portions of the fluid and sets them in motion. It is far from obvious what the nature of that flow will be, especially in a complex geometry. It may be laminar, it may be turbulent; it may be unidirectional, it may be recirculating.

Recirculation is the occurrence of backflow, opposite to the direction of the primary stream. This can be seen behind the pedestals supporting a bridge in a swift river. Despite the strong current, the flow direction reverses, and a circulating eddy forms in a region behind the pedestal. How is such behavior understood and predicted? An understanding requires knowledge of viscous action, of vorticity, of turbulence, and of the governing equations.

Two principles distinguish compressible flow: gases heat when compressed and cool when expanded; disturbances propagate at the speed of sound. The first alludes to thermodynamics. The second alludes to gas dynamics.

Thermodynamics

Heating by compression converts work into thermal energy. This is a reversible conversion in the sense that the thermal energy can be converted back into work. Heating also occurs by frictional dissipation of fluid kinetic energy into thermal energy. That is an irreversible process; viscosity cannot convert the thermal energy back into ordered flow. Friction increases entropy.

Compression and expansion occur in the course of the motion of a gas. For instance, on approaching a blunt body, the flow will slow, and fluid elements will be compressed. That is the ultimate motive for reviewing basic thermodynamics: the governing equations of compressible flow must be consistent with thermodynamics, extended to a spatially distributed system. However, we start with the thermodynamic description of compression and expansion of a homogeneous gas and then proceed to discuss compressible fluid dynamics. Comprehensive texts (Saad, 1997) can be consulted if the reader desires a thorough treatment of thermodynamics. The following is an informal treatment that provides background to compressible flow analysis.

Define a fluid element as a fixed mass, M, of gas. This occupies a volume element, V, which contains that mass. The volume defined in this way is termed specific volume – specific properties are those associated with a given quantity of mass. The mass of the fluid element is invariant, because that is how the element is defined: its volume can change. Indeed, compressibility is the property of volume change in consequence of pressure variations.

When two fluids occupy the domain, with a sharp boundary between them, we speak of fluid–fluid interfaces or just interfaces. To the extent that the fluids are immiscible, their interface is a type of boundary. The governing laws are unchanged; the new features are boundary conditions. They are of a different nature from those at fixed, solid walls. They depend on the flow on either side of the interface; indeed, the position of the interface is itself a variable. Interface conditions are alternatively described as matching conditions: velocities and stresses on either side must properly match at the interface. Despite this complicating aspect, the view that only boundary conditions are at issue provides some clarity.

The interface may be between liquid and gas – say, water and air. Often the matching conditions are simplified in this case. The density of air is three orders of magnitude smaller than that of water. For many purposes, the forces exerted by the air on the water can be neglected; then the interface is a force-free surface, insofar as the hydrodynamics are concerned. It nevertheless is a moveable surface, whose position must be solved as part of the analysis.

Or the interface could be between two viscous fluids – say, oil and water. The viscosity jumps across their common boundary. Conditions of stress continuity then determine the interaction between the fluid motions.

Oil and water might be placed in a vertical tube. The interface then curves in consequence of surface tension and the angle of contact with the tube. The line of contact is a three-phase boundary, among water, oil, and solid wall.

The basic laws of fluid dynamics are the Navier–Stokes momentum equations described in Chapter 1. Computational fluid dynamics (CFD) is the practice of solving those equations,* along with the mass conservation equation, by numerical algorithms. The ability of such seemingly simple governing equations to describe a wealth of complex fluid motions is quite remarkable. That remarkable capability is revealed most notably by computer simulation.

Numerical solution of Navier–Stokes equations nowadays has become almost routine. A variety of algorithms and solution methods for both incompressible and compressible flow have been developed over time and successfully implemented in a large number of computational codes (Ferziger and Peric, 2002; Fletcher, 1991; Tannehill et al., 1997). Initially, this software was primarily for research, mainly in academic institutions, government labs, and corporate research centers. But the appearance (and disappearance) of a number of general purpose, commercial CFD codes has been seen since the early 1990s. These were developed for use by nonexperts, as well as by those experienced in the practice of computation. Some of these codes have matured over time, becoming increasingly powerful as the latest techniques, methods, and analytical models were adapted to their requirements, and as high-speed computing power became increasingly available. Computational capabilities, previously mastered only in the research environment (higher-order numerical schemes, multigrid methods, advanced modeling capabilities, parallel processing) are now being used widely, through the medium of software packages. The engineer, student, or scientist no longer needs to have an intimate familiarity with computational methods to make productive use of CFD. Other technologies that have facilitated CFD include the graphic user interface, software for geometry and mesh creation, and techniques for plotting and visualization.

This is a book on fluid dynamics. It is not a book on computation. Many excellent books on fluid dynamics are available: why is another needed?

In recent decades, numerical algorithms and computer power have advanced to the point that computer simulations of the Navier–Stokes equations have become routine. This vastly expands our ability to solve these equations, further extending our understanding of fluid flow and providing a tool for engineering analysis. Computer simulations are solutions of a different nature from classical exact and approximate solutions. They are numerical data rather than formulas. One of our objectives in this text is to relate computer solutions to theoretical fluid dynamics. Indeed, it is this goal, rather than computation as a tool for complex engineering analysis, that provides the guideline for this text. Computer solutions can reproduce closed-form and approximate solutions; they can illuminate the merits and limits of simple analyses; and they can provide entirely new solutions of varying degrees of complexity. The time is ripe to integrate computer solutions into fluid dynamics education.

From a pedagogical perspective, readily available, commercial computational fluid dynamics (CFD) software provides a new resource for teaching fluid dynamics. This software converts CFD from a technique used by researchers and engineers in industry into a readily accessible facility. It is a challenge to integrate such software packages into the educational structure. Most of the examples in this book have been computed with commercial software, and exercises to be solved with such software have been suggested. How far to go in this direction was a true quandary.

The terminologies creeping flow, Stokes flow, or low Reynolds number hydrodynamics are used synonymously to refer to flows in which inertia is negligible compared to viscous and pressure forces. The formal requirement is that the Reynolds number be small: Re ≪ 1. However, in practice the low Reynolds approximation often remains satisfactory for Reynolds of order unity: Re ∼ 1.

With inertia neglected, momentum is transported by viscous diffusion but not by convection. Some ideas about fluid dynamics must be rethought in this limit. Without convection, there is no wake on the downstream side of an object; pressure scales on viscosity not on kinetic energy; when they occur, eddies are as likely upstream as downstream of a blunt body.

Low Reynolds number can mean highly viscous; hence, one can imagine objects moving through syrupy fluid or syrupy fluid being pumped through a conduit. The dominant forces are frictional in origin. Of course, low Reynolds number can also mean very low velocity or very small scale. One application of creeping flow is to locomotion of microorganisms through a fluid. These animals are a few microns in size. They do not move by propulsion; they drag themselves through the fluid, pushing or pulling by frictional forces. In some cases, they use spiral flagella to corkscrew themselves along. On their relative scale, the fluid appears to be very viscous. One can imagine pushing against a very thick fluid to move forward. To do so, the frictional force pushing forward must be greater than the frictional force resisting motion. Swimming is possible if the organism can produce motions that create more pushing friction than impeding friction.

It is likely that most questions the reader might pose about turbulent flow have no satisfactory answer - questions like. What is its cause? How can equations as innocuous as the Navier-Stokes momentum equations produce such complex solutions? How can we describe it? How do we predict its properties? and so on. The phenomenon is common experience: turbulent eddying is seen in smoke billowing above a large fire, in dust clouds rising from an explosion, in the wake of a last-moving boat; it is heard in the roar of a jet engine, in the wind rushing over an automobile; it is fell when an airplane bobs up and down in it or when a stiff breeze blows in one's face. Turbulence is an essential element of many processes. A text on fluid mechanics is not complete without a chapter on turbulence. That said, we provide, in this chapter, an introduction to computation of turbulent flow. The reader interested in a more thorough treatment of the subject can consult books entirely devoted to turbulence, such as Pope (2000).

The word turbulence conjures up the notion of randomness. It has entered everyday vocabulary, divorced from the field of fluid mechanics. It evokes images of roiling, churning, and disorder. These are valid definitions, but in fluid flow it is often less severe than vernacular usage suggests. A 10% level of velocity fluctuation may be considered to be substantial. Turbulence is best defined as the irregular component of motion that occurs in fluids when the Reynolds number is sufficiently high. The irregularity may be mild or it may be severe.

This chapter applies the Fundamental Efficiency Theorem to a central problem in basic Stirling engine design, that of identifying optimal engine geometry. This problem was treated in Chapter 7 for highly idealized engines having theoretical mechanisms, heat exchangers, etc. to produce cycles consisting of four distinct uniform thermodynamic processes. The results in Chapter 7 clearly showed the influence that the type of thermodynamic processes and the level of mechanism effectiveness have on optimum compression ratio and engine output potential.

In this chapter, a more realistic mechanical model of the Stirling engine is employed. It faithfully reflects practical and typical mechanical motions for the piston and displacer. In this setting, optimum values of two parameters are identified which yield maximum brake work output. In the interest of mathematical tractability, the thermal model used here is still highly idealized in that limitations in heat transfer are not considered. Accordingly, it yields best-case results, but allowing for this in a rational way when applying the optima in practical situations can provide an improved guide for first-order design of new engines.

THE GAMMA ENGINE

The analysis is limited here to a particular type of Stirling known as the gamma or split-cylinder. Illustrated in Figure 10.1, the split-cylinder is the simplest of the three main Stirling engine configurations.

Formula (4.2) for the indicated cyclic work of an ideal Stirling engine immediately suggests that output can be increased by charging the workspace with more working gas, keeping everything else the same. This is the motivation behind pressurizing or supercharging an engine. What matters in the end, of course, is whether shaft output improves, and this is a matter of mechanical efficiency.

An easy case to understand at this point is that of an ideal Stirling engine having a constant mechanism effectiveness and optimum buffer pressure. Its mean workspace pressure would be proportional to m, as Formula (3.9) explicitly shows. The Maximum Shaft Work Theorem (4.4) thus implies that if the engine has the charge of its working gas increased by a certain factor, and its buffer pressure adjusted to be optimal for the new charge (in fact, it will need to be increased by exactly the same factor, as Formula (3.4) shows), the shaft output will increase by the same factor. Hence, pressurizing an optimal ideal Stirling in this way will increase output in direct proportion to the charge factor. This kind of pressurization, called system charging, where the workspace and buffer pressure are charged together uniformly by the same factor, produces the same best possible results in many engine and buffer pressure combinations.

Crossley cycles are described by two isometric processes and two polytropic processes of the same kind. The ideal Stirling cycle and the twostroke Otto, or so-called adiabatic Stirling, are special cases. These two cases in fact bracket the spectrum of the four-step cycles that appear to be reasonable idealizations of the actual cycle of real Stirling engines.

Although the ideal Stirling cycle yields the best case analysis, it is a grand idealization of the actual state of affairs in real engines. The isothermal processes present the chief difficulty because of limited heat transfer rates in a real engine. A more realistic model is one in which the isothermal expansion and compression occur at temperatures somewhat displaced from the maximum and minimum engine hardware temperatures; this would model the temperature differential that is necessary to drive the heat transfer to and from the engine gas. This is treated in detail in Chapter 11. In many real engines the expansion and compression processes for the most part occur in engine spaces that have relatively little heat transfer area. Thus, it seems that the expansion and compression processes might be closer to adiabatic than to isothermal. Therefore, using the two-stroke Otto cycle has been advocated as a more faithful, but still idealized, cycle for representing real Stirling engines.