A liquid jet emanating from a nozzle or orifice exhibits richly varied phenomena that depend on the orifice geometry, the inlet condition before the jet is emanated, and the environmental situation into which the jet is issued. A liquid jet cannot escape the ultimate fate of breakup because of hydro-dynamic instability. The breakup possesses two major regimes: large drop formation and fine spray formation. These two regimes are controlled by distinctively different physical forces, and between them there exist intermediate regimes. All the regimes arise from a subtle dynamic response of the jet to the disturbances.

Geometry of Liquid Jets

Citing the experiment of Bidone, Rayleigh (1945, p. 355) stated, “Thus in the case of an elliptical aperture, with major axis horizontal, the sections of the jet taken at increasing distances gradually lose their ellipticity until at a certain distance the section is circular. Further out the section again assumes ellipticity, but now with major axis vertical.” This statement is illustrated in Figure 6.1, which was taken from Taylor (1960), who also carried out the experiment. The phenomenon was understood as the vibration of a jet enclosed in an envelope of constant tension about its equilibrium configuration with a circular cross section. However, Taylor (1960) demonstrated that the phenomenon can still be predicted without the surface tension in the absence of gravity. With gravity, if the jet is issued vertically downward, it will accelerate.

Overview

When a dense fluid is ejected into a less dense fluid from a narrow slit whose thickness is much smaller than its width, a sheet of fluid can form. When the fluid is ejected not from a slit but from a hole, a jet forms. The linear scale of a sheet or jet can range from light years in astrophysical phenomena (Hughes, 1991) to nanometers in biological applications (Benita, 1996). The fluids involved range from a complex charged plasma under strong electromagnetic and gravitational forces to a small group of simple molecules moving freely with little external force. The fluid sheet and jet are inherently unstable and breakup easily. The dynamics of liquid sheets was first investigated systematically by Savart (1833). Platou (1873) sought the nature of surface tension through his inquiry of jet instability. Rayleigh (1879) illuminated his jet stability analysis results with acoustic excitation of the jet. In some modern applications of the instability of sheets and jets, it is advantageous to hasten the breakup, but in other applications suppression of the breakup is essential. Hence knowledge of the physical mechanism of breakup, aside from its intrinsic scientific value, is very useful when one needs to exploit the phenomenon to the fullest extent. Recent applications include film coating, nuclear safety curtain formation, spray combustion, agricultural sprays, ink jet printing, fiber and sheet drawing, powdered milk processing, powder metallurgy, toxic material removal, and encapsulation of biomedical materials.

In the previous chapters we saw that if We < Q−1, the disturbance of wave numbers smaller than a cut-off wave number is unstable at the onset of instability. The cut-off wave number increases with Q. The capillary force is then shown with linear theories to be responsible for the onset of instability in the presence or absence of fluid viscosities. Subsequent to the onset, the amplitude of disturbances grows rapidly and the neglected nonlinear terms in the linear theory are no longer negligible. Thus the nonlinear evolution of disturbances that lead to the eventual pinching off of drops from a liquid jet can only be described with nonlinear theories. Similarly the pinching off of small droplets from the interface caused by interfacial pressure and shear fluctuations at the onset of instability, when We > Q−1, requires nonlinear theories to describe. Experimental observations of the nonlinear phenomena are presented first.

Experiments

Capillary Pinching

Linear theory predicts that unstable disturbances of different wavelengths grow at different rates and different natural frequencies corresponding to the different wavelengths. Figure 11.1 shows the nonlinear evolution of the disturbances when external sinusoidal forcings are introduced at three different natural frequencies. The forcing frequency for Figure 11.1(a) corresponds to k = 0.683, which is close to the Rayleigh's most amplified disturbance. Figures 11.1(b) and (c) correspond to the cases of lower forcing frequencies corresponding to k = 0.25 and k = 0.075, respectively. The disturbances of wavelengths shorter than that of the fastest growing disturbance appear to grow more slowly as they are convected downstream from the nozzle exit, as predicted by linear theory.

In the previous chapters, we investigated the fairly well studied phenomena of breakup of liquid sheets and liquid jets. The basic flows were assumed to be steady in the continuum theories. Also, they were either of infinite or of semi-infinite extent in the flow direction. Physically such infinite and semi-infinite steady jets or sheets cannot exist, as predicted by stability analysis. The analytical predictions enjoyed fairly good agreement with many known experiments. However, breakup of a liquid body into smaller parts often takes place under an unsteady situation from the beginning. The examples include the formation of satellites and subsatellites from the ligaments after detaching themselves from the main drops, the formation of drops from a dripping faucet, shaped-charge jets, the formation of micro-drops by external forcing, intermittent fuel sprays, and the phenomenon of jet branching induced by external excitation. These are the subjects to be touched upon in this last chapter.

Satellite Formation

When a stretched liquid ligament is relaxed, the capillary force associated with the large surface curvature at both ends of the ligament tends to compress and fragment the ligament into small drops. We saw the formation of the ligament during the last stage of nonlinear evolution of instability. The stretching of a liquid ligament submerged in another fluid can be achieved by pure straining or shearing or a combination of both. Figure 12.1 (Stone et al., 1986) shows how a spherical drop is stretched in two purely straining external flows with two different viscosities.

This chapter elucidates the role of interfacial shear on the onset of instability of a cylindrical viscous liquid jet in a viscous gas surrounded by a coaxial circular pipe by using an energy budget associated with the disturbance. It is shown that the shear force at the liquid-gas interface retards the Rayleigh mode instability, which leads to the breakup of the liquid jet into drops of diameter comparable to the jet diameter because of capillary force. On the other hand the interfacial shear and pressure work in concert to cause the Taylor mode instability, which leads the jet to breakup into droplets of diameter much smaller than the jet diameter. While the interfacial pressure plays a slightly more important role than the interfacial shear in amplifying the longer wave spectrum in the Taylor mode, shear stress plays the main role of generating shorter wavelength disturbances.

Basic Flow

Consider the instability of an incompressible Newtonian liquid jet of radius R1. The jet is surrounded by a viscous gas enclosed in a vertical pipe of radius R2, which is concentric with the jet. For the jet to maintain a constant radius, the dynamic pressure gradients in the steady liquid and gas flows must maintain the same constant. This will allow the pressure force difference across the liquid-gas interface to be exactly balanced by the surface tension force as required. Such coaxial flows, which satisfy exactly the Navier–Stokes equations, are given by (Lin and Ibrahim, 1990).

In the previous chapter we mentioned that fluid viscosity might alter the critical Weber number that divides the parameter space into regimes of absolute and convective instability. The effects of gas and liquid viscosities are investigated separately in this chapter, not just to understand each individual effect but also to demonstrate the coupled effect, which is unexpected. In Chapter 3, stability analysis for an inviscid liquid sheet of uniform thickness was applied locally to investigate the stability of gradually thinning liquid sheets. The thinning was either due to axial expansion or gravitational acceleration. The local application of the inviscid theory for a uniform sheet to the two different cases of nonuniform sheets was made judiciously. Likewise the viscous theories given in this chapter can be applied judiciously to a gradually thinning viscous sheet whatever the cause of the thinning. The thinning may be caused by kinematic requirements, gravitational acceleration, or viscous extrusion. The breakup of a viscous liquid sheet in an inviscid gas is expounded in Section 4.1. The effect of gas viscosity is elucidated in Section 4.2. The effects of liquid and gas viscosities on the onset of sheet breakup are summarized in Section 4.3.

A Viscous Sheet in an Inviscid Gas

The basic flow attributed to G. I. Taylor is given in Section 4.1a, and its stability is analyzed in Section 4.1b. The physical mechanism of the sheet breakup is discussed in Section 4.1c, based on energy considerations.

INTRODUCTION

The gas turbine has many important applications but it is most widely used as the jet engine. In the last few years, since the regulations changed to permit natural gas to be burned for electricity generation, gas turbines have become important prime movers for this too. Many of the gas turbines used in land-based and ship-based applications are derived directly from aircraft engines; other gas turbines are designed specifically for land or marine use but based on technology derived for aircraft propulsion.

The attraction of the gas turbine for aircraft propulsion is the large power output in relation to the engine weight and size – it was this which led the pre-Second World War pioneers to work on the gas turbine. Most of the pioneers then had in mind a gas turbine driving a propeller, but Whittle and later von Ohain realised that the exhaust from the turbine could be accelerated to form the propulsive jet.

This chapter looks at the operation of simple gas turbines and outlines the method of calculating the power output and efficiency. The treatment is simplified by treating the working fluid as a perfect gas with the properties of air, but later some examples are discussed to assess the effect of adopting more realistic assumptions. It is assumed throughout that there is a working familiarity with thermodynamics – this is not the place to give a thorough treatment of the first and second laws (something covered very fully in many excellent text books, for example Van Wylen and Sonntag, 1985).

INTRODUCTION

This part of the book begins the consideration of the engine requirements of a new fighter aircraft. In parallel with the treatment in earlier chapters for the engines of the new large civil aircraft, the approach chosen is to address the design of engines for a possible new aircraft so that the text and exercises can be numerically based with realistic values. The specifications for the New Fighter Aircraft used here have a marked similarity to those available for the new Eurofighter.

The topic of the present chapter is the nature of the combat missions and the type of aircraft involved. Figure 13.1 shows the different regions in which aircraft operate in terms of altitude versus Mach number, with the lines of constant inlet stagnation temperature overlaid. We are concerned here with what are referred to in the figure as fighters, a major class of combat aircraft. Figure 13.1 shows the various boundaries for normal operation. Even high-speed planes do not normally fly at more than M = 1.2 at sea level because in the high density air the structural loads on the aircraft and the physiological effects on the crew become too large. At high altitude high-speed aircraft do not normally exceed M ≈ 2.3, largely because the very high stagnation temperatures preclude the use of aluminium alloys without cooling.

The book has been well received and Cambridge University Press approached me with the invitation to bring out a second edition. This was attractive because of the big events in aerospace, most significantly the decision by Airbus Industrie at the end of 2000 to launch their new large aircraft, the A380. This meant that some changes in the first ten chapters were needed. Another major development is the decision to develop an American Joint Strike Fighter, the F-35.

Another more personal change took place when I left academia to become Chief Technologist of Rolls-Royce from the beginning of 2000. It should be noted, however, that the character and ideas of this second edition remain those of the university professor who wrote the first edition and do not reflect my change of role.

The aim and style of the book is unchanged. The primary goal of creating understanding and the emphasis remains on simplicity, so far as this is possible, with the extensive use of relevant numerical exercises. In a second edition I have taken the opportunity to update a number of sections and to include some explanatory background on noise; noise has become a far more pressing issue over the last four or five years. The book remains, however, very similar to the first edition and, in particular, numerical values have been kept the same and the exercises have not been changed.

INTRODUCTION

This chapter looks at the layout of some jet engines, using cross-sectional drawings, beginning with relatively simple ones and leading up to the large engines for one of the most recent aircraft, the Boeing 777. Two concepts are introduced. One is the multi-shaft engine with separate low-pressure and high-pressure spools. The other is the bypass engine in which some, very often most, of the air compressed by the fan bypasses the combustor and turbines.

Any consideration of practical engines must address the temperature limitations on the turbine. The chapter ends with some discussion of cooling technology and of the concept of cooling effectiveness.

THE TURBOJET AND THE TURBOFAN

Figure 5.1 shows a cut-away drawing of a Rolls-Royce Viper engine. This is typical of the simplest form of turbojet engine, which were the norm about 40 years ago, with an axial compressor coupled to an axial turbine, all on the same shaft. (The shaft, the compressor on one end and turbine on the other are sometimes referred to together as a spool.) Even for this very simple engine, which was originally designed to be expendable as a power source for target drones, the drawing is complicated and for more advanced engines such drawings become unhelpful at this small scale. Simplified cross-sections are therefore more satisfactory and these will be shown for more advanced engines. A simplified cross-section is also shown for the Viper in Fig. 5.1, as well as a cartoon showing the major components.

INTRODUCTION

In this chapter we will consider three separate engine designs corresponding to distinct operating conditions. For convenience here the three design points are at the tropopause (altitude 11 km; standard atmosphere temperature 216.65 K and pressure 22.7 kPa) for Mach numbers of 0.9, 1.5 and 2.0. The thrusts required for these conditions were determined in Exercise 14.4. At each condition a separate engine is designed – this is quite different from designing the engine for one condition and then considering its operation at different conditions, which is the topic of Chapter 17.

For this exercise all design points will correspond to the engine being required to produce maximum thrust, even though the ultimate suitability of an engine for its mission may depend on performance, particularly fuel consumption, at conditions for which the thrust is very much less than maximum. The designs will first be for engines without an afterburner (operation ‘dry’) and then with an afterburner; the afterburner will be assumed to raise the temperature of the exhaust without altering the operating condition of the remainder of the engine so the stagnation pressure entering the nozzle is unchanged.

The engines considered will all be of the mixed turbofan type – such an engine was shown in Fig. 15.1 with a sketch showing the station numbering system adopted. Note that the numbering shows station 13 downstream of the fan in the bypass and station 23 downstream of the fan for the core flow; in the present simplified treatment it will be assumed that p023 = p013 and that T023 = T013.