Introduction

The ‘random walk’ model for simulation of viscous diffusion in discrete vortex clouds was first proposed by Chorin (1973) for application to high Reynolds number flows and has been widely used since. The principle involved is to subject all of the free vortex elements to small random displacements which produce a scatter equivalent to the diffusion of vorticity in the continuum which we are seeking to represent. Such flows are described by the Navier Stokes equations which may be expressed in the following vector form, highlighting the processes of convection and diffusion of the vorticity ω,

where q is the velocity vector and ∇2 the Laplacian operator. The third term, applicable only in three-dimensional flows represents the concentration of vorticity due to vortex filament stretching. Otherwise in two-dimensional flows, with which we are concerned here, the vector Navier-Stokes equation reduces to

Normalised by means of length and velocity scales ℓ and W∞ this may be written in the alternative dimensionless form

where the Reynolds number is defined by

For infinite Reynolds number (9.3) describes the convection of vorticity in in viscid flow, for which the technique of discrete vortex modelling was developed in Chapter 8. At the other end of the scale, for very low Reynolds number flow past an object of characteristic dimension ℓ, the viscous diffusion term on the right hand side (9.3) will predominate.

Introduction

An outline computational scheme was developed in Chapter 1 for application of the surface vorticity method to two-dimensional flow past non-lifting bodies of arbitrary shape. In the fields of aeronautics and engine aerodynamics on the other hand there is a special interest in lifting bodies and control surfaces such as aerofoils, struts and turbine, compressor or fan blades. The objective of this chapter is to extend the analysis to deal with these important applications which exhibit three features not yet considered, namely:

1. (i) Such devices are required to generate lift, associated with net bound circulation on the body.

2. (ii) In the applications cited the lifting surfaces are normally thin foils for which special computational problems arise due to the close proximity of vorticity elements on opposite sides of the profile.

3. (iii) A device may involve an assembly of several lifting bodies, taking deliberate advantage of their mutual aerodynamic interference.

We will deal with these matters in turn beginning with an extension of flow past a circular cylinder, Section 1.6, to the case of the Flettner rotor or lifting rotating cylinder, Section 2.2. Progressing to the closely related problem of flow past an ellipse, Sections 2.3 and 2.4, problems of type (ii) will be dealt with for the treatment of thin non-lifting and lifting bodies. This leads naturally into the case of generalised thin aerofoils, Section 2.5, for which comparisons will be provided from Joukowski's exact solutions.

Introduction

A basic outline of full vortex cloud modelling was presented in Chapter 10 but with limited application primarily to bluff body flows for which separation occurs spontaneously and dramatically at reasonably cetain separation points, resulting normally in the development of a broad periodic wake. The main aim of this chapter is to apply the full vortex cloud method to lifting bodies such as aerofoils and cascades for which the aerodynamic aim usually is to avoid flow separations, maintaining low losses. Full vortex cloud modelling represents an attempt to solve the Navier– Stokes equations including both the surface boundary layer near field and the vortex wake far field flows. Boundary layer separations are then self-determining. In practice however, as discussed by Porthouse & Lewis (1981), Spalart & Leonard (1981) and Lewis (1986) vortex cloud modelling in its present state of development seems unable quite to cope with the general problem of boundary layer stability and various techniques are proposed by these authors to avert premature stall as often experienced during vortex cloud analysis of aerofoils or cascades. These problems will be considered in Sections 11.2–11.4. Extension of vortex cloud modelling to cascades will be given in Section 11.5 and studies of acoustic excitation due to wake vortex streets from bluff bodies in ducts are briefly discussed in Section 11.6.

Introduction

As early in the history of gas turbines and internal aerodynamics as 1952, C. H. Wu recognised the truly three-dimensional nature of the flow in turbomachines and proposed a remarkably sophisticated scheme for numerical analysis illustrated by Fig. 3.1. The fully three-dimensional flow was treated by the superposition of a number of two-dimensional flows which were of two types located on the so-called S-1 and S-2 stream surfaces. S-2 surfaces follow the primary fluid deflection caused by the blade profile curvature and its associated aerodynamic loading. Due to the blade-to-blade variation in static pressure the curvature of each S-2 stream surface will differ, calling for several surfaces for adequate modelling of the flow. S-1 surfaces account for consequent twist in the so-called ‘through-flow’ or ‘meridional flow’ which comprises a family of stream surfaces which approach axisymmetry close to the hub and casing and exhibit maximum departure from axisymmetry at the blade passage mid height. By solution of the flows on this mesh for successively improved estimates of the S-1 and S-2 surfaces, allowing for fluid dynamic coupling between them, an iterative approach to the fully three-dimensional flow was fairly comprehensively laid out by Wu in a paper which was truly twenty years ahead of its time.

Until relatively recently such calculation procedures have been ruled out by lack of suitable computing facilities. It was in 1966 that Marsh gave a strong impetus to computer application of Wu's method by developing the well known matrix through-flow analysis.

Introduction

Numerical schemes for the simulation of viscous rotational flows usually adopt one of two well known frameworks of reference, Eulerian or Lagrangian. Attention is focussed upon the whole of the relevent flow regime in Euler methods, usually by means of a spatially distributed fixed grid or cellular structure upon which to hang such data as the local velocity and fluid properties, updated at each stage of a time stepping procedure. Vortex dynamics on the other hand generally follows the alternative route of Lagrangian modelling in which attention is focussed upon individual particles as they move through the fluid. According to vortex cloud theory all disturbances in incompressible viscous flow can be linked to vorticity creation at solid boundaries, followed by continuous convection and diffusion. A cloud of discrete vortices may thus in principle be able to represent any rotational viscous fluid motion, accuracy depending upon the degree of discretisation and the quality of the convection and diffusion schemes.

The special attraction of this approach for external aerodynamic flows in particular is the removal of any need to consider the rest of the flow regime which of course extends to infinity. In such problems Euler models require the establishment of suitable grids extending sufficiently far out into space to define acceptable peripheral boundary conditions around the target flow regime. For simple body shapes such as cylinders or plates this may be straightforward enough.

Introduction

The principal aims of this book are to outline the fundamental basis of the surface vorticity boundary integral method for fluid flow analysis and to present a progressive treatment which will lead the reader directly to practical computations. Over the past two and a half decades the surface vorticity method has been developed and applied as a predictive tool to a wide range of engineering problems, many of which will be covered by the book. Sample solutions will be given throughout, sometimes related to Pascal computer programs which have been collated for a selection of problems in the Appendix. The main aims of this introductory chapter are to lay down the fundamental basis of both source and vorticity surface panel methods, to explain the fluid dynamic significance of the surface vorticity model and to introduce a few initial applications to potential flow problems.

As numerical techniques, surface singularity methods were not without progenitors but grew quite naturally from the very fertile field of earlier linearised aerofoil theories. Such methods, originally contrived for hand calculations, traditionally used internal source distributions to model profile thickness and vortex distributions to model aerodynamic loading, a quite natural approach consistent with the well known properties of source and vortex singularities. On the other hand it can be shown that the potential flow past a body placed in a uniform stream can be modelled equally well by replacing the body surface with either a source or a vortex sheet of appropriate strength, Fig. 1.1.

Only thirty years have elapsed since E. Martensen published his well known paper proposing the surface vorticity boundary integral method for potential flow analysis. Generally regarded as the foundation stone, this paper has led to the establishment of a considerable volume of numerical methodology, applicable to a wide range of engineering problems, especially in the fields of aerodynamics and turbomachines. During this period we have also witnessed a technological transformation in the engineering world of immense proportions and of great historical significance. This has been based upon parallel advances in both theoretical and practical engineering skills which have been breath-taking at times. Theoretical methods, to which this book is dedicated, have undergone a renaissance spurred on by the rapid growth of computing power in response to the ever increasing demands of engineering hardware. The main characteristic of this new-birth has been a shift from the pyramid of classical methods to a whole host of numerical techniques more suited to direct modelling of real engineering problems. The explosion of this activity has been damped down only by the difficulties of transferring and absorbing into normal practice a technology which can, as in the case of many numerical methods, become highly personalised. After three decades there is a need for books which sift and catalogue and which attempt to lay out the new fundamental methodologies to suit the needs of engineers, teachers and research workers.

Noise, or unwanted sound, is generated whenever the passage of air over the aircraft structure or through its power-plants causes fluctuating pressure disturbances that propagate to an observer in the aircraft or on the ground below. Since the flight condition cannot be maintained unless these air- and gasflows are controlled efficiently, there are ample opportunities for sound to be produced. Fluctuating pressure disturbances result from inefficiencies in the total system and occur whenever there is a discontinuity in the airflow-handling process, particularly in the engines, where power generation involves large changes in pressure and temperature. This is not to say that the airframe itself is devoid of sound-producing opportunities, for it has a large surface area and, in the configuration that is adopted for take-off and approach, both the landing gear and high-lift devices (slats and flaps) create significant amounts of turbulence.

To the community beneath the aircraft, the self-generated noise from the airframe is normally significant only during the approach phase of operation, where the sources shown in Figure 3.1 can combine to exceed the level of each major noise source in the power-plant. For this reason, airframe noise has been thought of as the ultimate aircraft noise “barrier”. Perhaps we should briefly consider this and other airframe-associated sources before moving into the more complex subject of the aircraft power-plant.

Over the past thirty to forty years, a vast knowledge of aircraft noise-control techniques has been acquired. Some of the findings have contributed to the improved airport-noise climate of today; the less successful and the failures have taught both useful and bitter lessons. This chapter concentrates on success – success in the field of the jet engine, for it was this propulsion system that really started the serious noise problem.

For a broader understanding of the problem, the specific means of controlling aircraft power-plant noise should be discussed in relation to design philosophy, which, in turn, is related to either date of concept or aircraft mission requirements. The following sections deal with the range of power-plants in service in two convenient categories, low- and high-bypass-ratio types.

Suppression of the early jets

As explained in Chapter 3, it is the basic cycle and the maximum thrust level of the engine that determines the level of jet noise produced. The first generation of pure jets and, to a large extent, the low-bypass-ratio engines that succeeded them, all had extremely high exhaust velocities, which caused high levels of jet noise. At full power, supersonic exhaust flows with velocities of up to 600 m/sec were not uncommon and the characteristic crackling and tearing sounds of the mixing noise were augmented by the presence of shock-associated noise.

Without today's turbine cooling technology, there was no opportunity whatsoever to reduce the jet noise in these engines by modifying the engine cycle.

The aircraft noise problem is an environmental “minus” that was born alongside the commercial gas turbine aeroengine more than thirty years ago. The issue came to a head in the 1960s, around the time when the number of jet-powered aircraft in the commercial fleet first exceeded the number of propeller-powered aircraft. During that decade, significant sums of money were expended in research and development exercises directed at quietening the exhaust noise of the subsonic fleet and in researching the noise of even higher-velocity jets that would have been a major problem if the supersonic transport had become a commercial success. Later in the 1960s, with the advent of the bypass-engine cycle, the emphasis was more on the noise generated inside the engine, with the large proportion of the extensive research funds necessary being provided by the taxpayers in those nations with aircraft- and engine-manufacturing capabilities.

Government action was not only limited to supplying the funds for research contracts, but major nations cooperated on a political front to develop noise certification requirements that were demanded of the manufacturers of all new aircraft produced from 1970 onwards. Coincidentally, technology moved forward and produced the high-bypass or turbofan-engine cycle, which, in reaping the benefits of the accelerated noise programmes of the 1960s, was only about one-quarter as noisy as the engines it replaced. Initially, however, the turbofan cycle was limited in its application to the new breed of larger wide-body jets, which had at least twice the passenger-carrying capacity of their forebears.

Aircraft noise prediction involves two types of activities: predicting the noise of an individual aircraft and assessing the cumulative effect of the complex pattern of operations in and out of a specific airport. The latter depends on the former for a wide range of aircraft types.

Aircraft noise

Without available direct measurements, the only method of assessing the impact of a completely new aircraft or power-plant design is to utilise a reliable prediction procedure. Such a procedure may be able to make use of a limited amount of directly relevant data, for example, engine test data where a development programme is under way, or it may have to rely entirely upon empirical component prediction procedures. The latter situation arises at the advanced project stage of any new aircraft design – a current example could be an aircraft with propfans or a second-generation supersonic transport with a novel variable-cycle propulsion system concept.

To be successful, aircraft noise prediction must be based on a reliable definition of aircraft performance and a confident prediction of the noise characteristics of the power-plant (as a function of power setting, altitude and flight speed). Where there are substantial and related measured data to support a new concept, their projection to the new aircraft situation can follow fairly well-defined routes. For example, if the new aircraft incorporates power-plants that are not much different from versions already in service, flight test data can be transposed fairly simply to a new situation. Where the measured information is obtained from static engine tests during the development programme, methods are being established for transposing these data to the flight situation.