Introduction

Our principal aim so far has been to lay down the foundations of surface vorticity analysis for a series of progressively more advanced turbomachinery flow problems. Although a brief outline of threedimensional flow analysis was presented in Chapter 1, specific applications have been limited to problems which are twodimensional in the strict mathematical sense. Unlike the source panel method, which has been extensively applied to threedimensional flows, serious application of the surface vorticity analysis has been limited to few such engine problems. The aim of the first part of this chapter will be to expand on the basic foundation theory for dealing with the flow past three-dimensional objects by surface vorticity modelling and to consider two such problems in turbine engines which have received some attention. These will include the prediction of engine cowl intake performance at angle of attack and the behaviour of turbine cascades exhibiting sweep.

As discussed in Chapter 3 the flow through turbomachinery blade passages is in general three-dimensional, although the design or analysis problem may be tackled in a practical way by reference to a series of superimposed equivalent interacting two-dimensional flows. The two models usually adopted, which are equivalent in some respects, are the S-1, S-2 surfaces of Wu (1952) and the superposition of blade-to-blade (S-2 type) flows upon an assumed axisymmetric meridional flow. We concluded Chapter 5 with a derivation of the meridional flow equations for ducted propellers, indicating that the blade-to-blade/meridional interactions result in vorticity production within the mainstream.

Introduction

Over the next three chapters we shall develop analyses to deal with progressively more complex problems in the fields of ducted propellers or fans and turbomachine meridional flows. As illustrated in Chapter 3, a design strategy frequently adopted for such devices involves representation of the fully three-dimensional flow as a series of superimposed and connected two-dimensional flows. These are of two main types, blade-to-blade and meridional flow. Having dealt with the first of these, we now turn our attention to the second principal turbomachine problem, calculation of the meridional flow. Turbomachine annuli, Fig. 4.1, are of many different configurations but are usually axisymmetric. For design purposes meridional through-flows are likewise often assumed to be axisymmetric. In general it is important to build into meridional analysis the interactions of the blade-to-blade flow which results in vortex shedding and stagnation pressure or enthalpy gradients. These matters will be dealt with in Chapters 5 and 6, including extension to a consideration of some three-dimensional flows which have been studied by surface vorticity modelling. In the present chapter the foundations will simply be laid for the analysis of axisymmetric potential flows by the surface vorticity method with applications to bodies of revolution, engine or ducted propeller cowls, wind tunnel contractions and turbomachinery annuli.

Axisymmetric flows are in fact two-dimensional in the mathematical sense, even in the presence of circumferentially uniform swirling velocities.

Introduction

The early contributors to the surface vorticity method such as Martensen (1959), Jacob & Riegels (1963) and Wilkinson (1967a) were concerned primarily with the development of a flexible numerical method for the solution of potential flows. Preceding chapters testify to the scope and power of this conceptually simple technique and to the imagination and creativity of a host of later research workers who have extended the method to deal with a wide range of engineering potential flow problems. Although the broader physical significance of the surface vorticity model, as expounded in Chapter 1, has always been realised, only recently has this been more fully explored by attempts to model the rotational fluid motion of real fluids including both boundary layer and wake simulations. The remaining chapters will lay down progressively the essential fundamentals of this work which the reader requires to proceed to practical computational schemes, employing what has come to be known as the ‘vortex cloud’ or ‘discrete vortex’ method.

All real flows involve rotational activity developed in the regions adjacent to flow surfaces or in the rear wake region in the case of bluff bodies. Some flows also exhibit spontaneous boundary layer separation or stall behaviour while in other situations flow separation occurs inevitably from sharp corners.

Introduction

A range of flow computational techniques has been developed over many years to meet the design and analysis requirements of a wide range of rotodynamic machines, some of which were illustrated by Fig. 4.1. For dealing with turbomachine meridional or through flows, which are usually completely confined within a continuous duct annulus such as that of the mixed-flow fan depicted in Fig. 4.1(a), surface vorticity or panel methods have been proved less attractive in competition with grid based analyses such as the matrix through-flow method of Wu (1952) and Marsh (1966) and the more recent time marching analyses such as those of Denton (1974), (1982). Although the annulus boundary shape exercises important control over the flow through the blade regions, in all turbomachines complex fluid dynamic processes occur throughout the whole flow field due to interactions between the S-1 and S-2 flows which were referred to in Chapter 3. Boundary integral methods based solely upon potential flow equations such as we have considered so far obviously cannot handle these interactions between the blade-to-blade and meridional flows, which involve detailed field calculations and spatial variations of properties best dealt with by the introduction of a grid strategically distributed throughout the annulus. Some attempts to achieve this with extended vortex boundary integral analysis will be outlined in Chapter 6, but generally speaking channel grid methods such as those referred to above have proved more fruitful to date for turbomachinery meridional analysis.

Introduction

The main objective of this chapter is to present the reader with a practical numerical approach to vortex cloud modelling of bluff body flows, drawing upon the techniques developed earlier in the book and especially the treatments of vortex dynamics and viscous diffusion considered in Chapters 8 and 9. Reporting on Euromech 17, which was entirely devoted to bluff bodies and vortex shedding, Mair & Maull (1971) remarked upon the preponderance of experimental work at that time and the need for more theoretical studies to be attempted, since there was little discussion of numerical techniques. It was felt, on the other hand, that since such flows showed marked three-dimensional characteristics (e.g. a circular cylinder von Karman street wake will not in general be correlated along its length for L/D ratios in excess of 2.0), two-dimensional computations, whilst being of interest, would not be very useful. It was admitted however that ‘with an increase in the size of computers a useful three-dimensional calculation could become a reality’. By the time of the next Euromech 119 on this subject, Bearman & Graham (1979), one third of the papers focused on theoretical methods, the majority based upon the Discrete Vortex Method (DVM). Various reviews of the rapid subsequent progress with DVM were given by Clements & Maull (1975), Graham (1985a) and Roberts & Christiansen (1972) and a fairly comprehensive recent review of U.K.

Introduction

So far we have considered only the case of fully attached inviscid steady flows, for which the introduction of a surface vorticity sheet of appropriate strength and of infinitesimal thickness, together with related trailing vorticity in three-dimensional flows, is completely adequate for a true representation. As pointed out in Chapter 1, where the justification of this model was argued from physical considerations, the surface vorticity method is representative of the infinite Reynolds number flow of a real fluid in all but one important respect, namely the problem of boundary layer separation. Real boundary layers involve complex mechanisms characterised by the influence of viscous shear stresses and vorticity convections and eddy formation on the free stream side. Depending upon the balance between these mechanisms and the consequent transfer of energy across a boundary layer, flow separation may occur when entering a rising pressure gradient, even at very high Reynolds numbers. Flow separation at a sharp corner will most certainly occur as in the case of flow past a flat plate held normal to the mainstream direction.

For a decade or so the development of computational fluid dynamic techniques to try to model these natural phenomena has attracted much attention and proceeded with remarkable success. The context of a good deal of this work has fallen rather more into the realm of classical methods than that of surface vorticity modelling, and is often classified by the generic title Vortex Dynamics.

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.