To this point in this treatise, we have discussed only isolated droplets. In a practical situation, of course, many droplets are present in a spray and the average distance between droplets can become as low as a few droplet diameters. A typical droplet therefore will not behave as an isolated droplet; rather, it will be strongly influenced by immediately neighboring droplets and, to some extent, by all droplets in the spray.

There are three levels of interaction among neighboring droplets in a spray. If droplets are sufficiently far apart, the only impact is that neighboring droplets (through their exchanges of mass, momentum, and energy with the surrounding gas) will affect the ambient conditions of the gas field surrounding a given droplet. As the distance between droplets becomes larger, the influence of neighboring droplets becomes smaller and tends toward zero ultimately. At this first level of interaction, the geometrical configuration of the (mass, momentum, and energy) exchanges between a droplet and its surrounding gas is not affected by the neighboring droplets. In particular, the Nusselt number, Sherwood number, and lift and drag coefficients are identical in values to those for an isolated droplet. This type of interaction will be fully discussed in Chapter 5.

At the next level of interaction, droplets are closer to each other, on average, and the geometrical configurations of the exchanges with the surrounding gas are modified.

High pressures and supercritical conditions in liquid-fueled diesel engines, jet engines, and liquid rocket engines present a challenge to the modelling and the fundamental understanding of the mechanisms controlling the mixing and combustion behavior of these devices. Accordingly, there has been a reemergence of investigations to provide a detailed description of the fundamental phenomena inherent in these conditions. Unresolved and controversial topics of interest include prediction of phase equilibria at high and supercritical pressures (Curtis and Farrell, 1988; Litchford and Jeng, 1990; Hsieh et al., 1991; Poplow, 1994; Delplanque and Sirignano, 1993; Yang and Lin, 1994; Delplanque and Potier, 1995; Haldenwang et al., 1996), including the choice of a proper equation of state, definition of the critical interface, importance of liquid diffusion, significance of transport property singularities in the neighborhood of the critical mixing conditions, influence of convection (including secondary atomization); d2 law behavior at supercritical conditions (Daou et al., 1995); droplet-lifetime predictions (Delplanque and Sirignano, 1993, 1994; Yang et al., 1992; Yang and Lin, 1994; Delplanque and Potier, 1995; Haldenwang et al., 1996); dense spray behavior (Delplanque and Sirignano, 1995; Jiang and Chiang, 1994a, 1994b, 1996); combustion-product condensation (Litchford and Jeng, 1990; Litchford et al., 1992; Delplanque and Sirignano, 1994; Daou et al., 1995); and flame structures at high and supercritical pressures (Daou et al., 1995). The actual combustion process is characterized by the supercritical combustion of relatively dense sprays in a highly convective environment.

The interactions of a spray with a turbulent gas flow is important in many applications (e.g., most power and propulsion applications). Two general types of studies exist. In one type, the global and statistical properties associated with a cloud or spray within a turbulent field are considered. In the other type, detailed attention is given to how individual particles behave in a turbulent or vortical field. Some studies consider both perspectives. Most of the research work in the field has been performed on the former type of study. Faeth (1987), Crowe et al. (1988), and Crowe et al. (1996) give helpful reviews of this type of research.

The interactive turbulent fields can be separated into homogeneous turbulent fields and free-shear flows (e.g., jets and mixing layers). In some theoretical studies, two-dimensional vortical structures interacting with a spray have been examined. Most of the studies deal with situations in which the contribution of the spray to the generation of the turbulence field is secondary, that is, there is a forced gas flow whose mass flux and kinetic-energy flux substantially exceed the flux values for the liquid component of the dilute flow. The turbulent kinetic-energy flux of the gas flow is much less than the mean kinetic-energy flux of the gas flow and is comparable with the mean kinetic energy of the liquid flow.

THE IDEAL OTTO CYCLE

A cycle is an idealization of what goes on in one of the devices that thermodynamicists call heat engines: that is, a gasoline or diesel engine, a jet engine, a steam engine, and so forth. All of these take some energy source and convert some of that energy into useful work. In the spark-ignition engine the energy source is a chemical fuel, usually gasoline, which is combined with oxygen from the air by burning to release heat. Expansion of the heated gases does the mechanical work.

For the spark-ignition engine the idealization is called the Otto cycle, after Dr. N. A. Otto who, in 1876, patented a stationary gas engine using approximately this cycle. In order to understand this ideal cycle, we must imagine a piston in a cylinder. The piston is connected to a crank by a connecting rod – see Figure 1.1. The crank rotates, and the piston travels up and down. There are two valves, an inlet and an exhaust valve, and an arrangement to open and close them. The idealized cycle is illustrated in Figure 1.2.

In Figure 1.2 we plot the pressure in the cylinder against the volume in the cylinder. Notice that the piston does not go quite all the way to the top of the cylinder; the piston is at the top of its travel at 0, 2 and 3, and there is a small space still above it, the combustion chamber.

As soon as the steam engine was developed in the early 19th century, people started to think about some sort of road vehicle. Early attempts were steam powered and had various engine and boiler positions and arrangements of road wheels. The development of the much more compact spark-ignition internal-combustion engine toward the end of the century made possible a horseless carriage configuration. By about 1910, the automobile had taken a recognizably modern shape. The engine, in particular, has not changed much in basic design since that period. Of course, there are various forms of engine (for example, various arrangements of the valves), having different characteristics. These different forms represented innovations when first proposed. Some of them stood the test of time, and the unsuccessful ones disappeared without a trace. Some uncommon ones (for example, desmodromic valves, which we shall discuss later) keep reappearing from time to time. There is very little new in engines today. For example, around the time of the First World War the enthusiast driver could buy a Frontenac aftermarket cylinder head conversion for his Model T Ford, giving it chain-driven double overhead cams (DOHC) with four valves per cylinder. These are innovations we think of as modern. In [3] we find the statement “OHC engines are more efficient than their predecessor pushrod… engines,”which indicates that it was written by a young person (although to be fair we must admit that DOHC engines with four valves per cylinder at first appeared largely in racing and exotic cars).

INTRODUCTION

When the flow velocity through an orifice reaches the local speed of sound, a change in the pressure downstream of the orifice can no longer be communicated to the flow upstream of the orifice. This is because the message would have to travel upstream as a pressure wave at the speed of sound, and it cannot do this if the stream is flowing against it at this speed. The flow is called choked, and the mass flow rate cannot be increased beyond this point. We call the ratio of the local fluid speed U to the speed of sound a, the Mach number M = U/a. When the M = 1 at the orifice, the flow is choked (see, for example, [31]).

It is at first surprising that sonic conditions, or supersonic flow, have anything to do with an automobile engine. It turns out, however, that the flow through the inlet valve becomes choked under normal operating conditions, and this is one of the most serious limitations on the performance of an engine. The design of the valve gear is dictated largely by the need to avoid choked flow throughout the desired performance range. The desired upper end of the performance range determines when the flow through the inlet valve will become choked.

In order to control the engine speed at which the flow becomes choked, the engine designer must control the size and lift and number of the inlet valves, and this influences the combustion chamber shape, as well as the nature of the valve gear (for example, pushrod, overhead cam, double overhead cam).

I mentioned in my introduction to Chapter 3 that not every interesting problem should be attacked by directly solving the differential mass and momentum balances. Some problems are really too difficult to be solved in this manner. In other cases, the amount of effort required for such a solution is not justified, when the end purpose for which the solution is being developed is taken into account.

In the majority of momentum transfer problems, the quantity of ultimate interest is an integral. Perhaps it is an average velocity, a volume flow rate, or a force on a surface. This suggested that I set aside an entire chapter in order to exploit approaches to problems in which the independent variables are integrals or integral averages.

I begin by approaching turbulence in terms of time-averaged variables. Then I look at some problems that are normally explained in terms of area-averaged variables. The random geometry encountered in flow through porous media suggests the use of a local volume-averaged variable. The chapter concludes with the relatively well-known integral balances for arbitrary systems.

Again I encourage those of you who feel you are primarily interested in energy and mass transfer to pay close attention to this chapter. The ideas developed here are taken over, almost without change, and applied to energy transfer in Chapter 7 and to mass transfer in Chapter 10.

Time Averaging

The most common example of time averaging is in the context of turbulence. Turbulence is defined to be a motion that varies randomly with time over at least a portion of the flow field such that statistically distinct average values can be discerned.

LUBRICATION

When two surfaces are in relative motion with a lubricant between them, the phenomenon will fall into one of three rough categories or types of lubrication. First, it is essential to realize that the surface of a smooth piece of metal is not smooth. Smooth is a relative term, and is entirely qualitative. Metal is given a finish which is specified in terms of the permissible roughness. A cylinder wall, for example, might have a 2 µm finish, meaning that the rms roughness is 2 × 10-6 m high. Bearing materials have similar finishes. Boundary lubrication (see Figure 4.1), the first type, involves metal-to-metal contact between the tops of the roughness elements (or asperities) of the two surfaces. This involves deformation and fracture of the surfaces and removal of bits of the surface. The second type is called mixed-film lubrication. The surfaces are separated by a slightly thicker film of lubricant now, and the metal-to-metal contact is only occasional. As the surfaces move farther apart and the lubricant film is thicker, the third type arrives, hydrodynamic lubrication. Now the surfaces never touch, and no wear takes place.

In an automobile engine, the main and connecting rod bearings are intended to operate in the hydrodynamic regime. However, when the engine is first started, if it has not run for several hours, say overnight, the oil film has probably been squeezed out of the bearing, and when first started it is operating in boundary lubrication. As the oil pressure rises, the oil film is replenished, and the lubrication becomes hydrodynamic.

A fter considerable preparation (Appendix A, Chapters 1 and 2), we are ready to analyze the detailed motions of materials in particular geometries. Try not to be discouraged if some of our initial examples appear to be too simple. The simple problems are there in order to allow you to gain both facility and confidence.

The objection is sometimes raised that studying fluid mechanics in the detail described here is useless because the problems that can be solved are trivial. A statement such as this is normally made as an exaggeration to make a point. The point is that the interesting problems require numerical solutions or that the interesting problems are susceptible only to approximate solutions.

There is an element of truth here. Our feeling is that the best way to be introduced to this material is through the use of problems that can be solved with a minimum of programming. In this way, concepts and techniques can be polished rapidly. These problems are required preparation for those who wish to study more sophisticated problems, to develop limiting cases as checks on the validity of numerical work, or to develop approximate solutions using the integral techniques of Chapter 4.

In what follows,the principal tools for studying fluid mechanics are developed. We begin by introducing the concept of force. Notice that force is not defined; it is a primitive concept in the same sense as is mass and the material particle in Chapter 1. This forms the basis for introducing our second and third postulates: the momentum balance and the moment of momentum balance. The stress tensor is introduced in order to derive the equations that describe at each point in a material the local balances for momentum and moment of momentum. The differential mass and momentum balances together with the symmetry of the stress tensor form the foundation for fluid mechanics.

We conclude our discussion with an outline of what must be said about real material behavior if we are to analyze any practical problems. It is especially at this point that statistical mechanics, based upon the molecular viewpoint of real materials, can be used to supplement the concepts developed in continuum mechanics. In continuum mechanics, we can indicate a number of rules that constitutive equations for the stress tensor must satisfy (the principle of determinism, the principle of local action, the principle of material frame indifference,…), but from first principles we cannot derive an explicit relationship between stress and deformation. If we work strictly within the bounds of continuum mechanics, we can derive such a relationship only by making some sort of assumption about its form.

This entire chapter is introductory in much the same way as Appendix A is. In Appendix A, I introduce the mathematical language that I shall be using in describing physical problems. In this chapter, I indicate some of the details involved in representing from the continuum point of view the motions and deformations of real materials. This chapter is important not only for the definitions introduced, but also for the viewpoint taken in some of the developments. For example, the various forms of the transport theorem will be used repeatedly throughout the text in developing differential equations and integral balances from our basic postulates.

Perhaps the most difficult point for a beginner is to properly distinguish between the continuum model for real materials and the particulate or molecular model. We can all agree that the most factually detailed picture of real materials requires that they be represented in terms of atoms and molecules. In this picture, mass is distributed discontinuously throughout space; mass is associated with the protons, neutrons, electrons,…, which are separated by relatively large voids. In the continuum model for materials, mass is distributed continuously through space, with the exception of surfaces of discontinuity, which represent phase interfaces or shock waves.

The continuum model is less realistic than the particulate model but far simpler. For many purposes, the detailed accuracy of the particulate model is unnecessary. To our sight and touch, mass appears to be continuously distributed throughout the water that we drink and the air that we breathe. The problem is analogous to a study of traffic patterns on an expressway. The speed and spacings of the automobiles are important, but we probably should not worry about whether the automobiles have four, six, or eight cylinders.

INTRODUCTION

Combustion temperatures are in the neighborhood of 2,500 K in a spark-ignition engine, and the exhaust gas temperature is about 1,300 K. On the other hand, the melting point of aluminum is about 933 K, and the melting point of iron is about 1,808 K. It is clear that some provision has to be made to keep the piston, valves, and cylinder walls cool, or they will melt. Even considerably short of melting, at high temperatures metals begin to lose their strength, and this must also be avoided. Any heat removed represents a loss of energy, so we want to cool only as much as is necessary to maintain the strength of the materials, maintain clearances, and prevent the lubricant from breaking down.

The problem areas are the exhaust valve and the piston crown. The exhaust valve head loses most of its heat to the valve seat (the amount lost to the valve guide is relatively small, because the path is long and the conduction area is small). Unfortunately, when the valve opens, it is exposed to the exhaust gases, which flow past it at high velocities (making for good heat transfer), and while this is happening the valve head is not in contact with its seat.

In water-cooled engines the water returning from the radiator, which therefore has the lowest temperature, usually flows around the cylinders and then up into the cylinder head. This is not the most effective order because the cylinder heads, and particularly the valve seats, require cooling much more than the cylinder walls. However, it is adequate, and cheap.