An extensive, highly mathematical literature exists dealing with fluidmechanical aspects of ship propellers.

Invariably, the mathematical developments are only outlined, impeding easy comprehension even by knowledgeable readers. Our aim is to elucidate the mathematical theory in much greater detail than is generally available in extant papers. In this context, the first three chapters are provided as aids for those who have not had extensive practice in the application of classical hydrodynamical theory to flows induced in fluids by the motions of bodies. The fluid of interest is water which is taken to be incompressible and inviscid. Modifications arising from viscosity are described in a later chapter (Chapter 7) through reference to experimental observations.

This review begins with the derivation of the concept of continuity or conservation of mass at all points in sourceless flow and proceeds to the development of the Euler equations of motion. In the restricted but important class of irrotational motions (zero vorticity) Laplace's equation for the velocity potential is obtained. The remainder of this chapter is devoted to derivations of fundamental solutions of Laplace's equation in two and three dimensions.

It is emphasized that these first two chapters are necessarily limited in scope, being directed to our needs in subsequent chapters. There are many excellent books which should be consulted for those seeking greater depth and broader description of hydrodynamic theory. Among these we suggest Batchelor (1967), Lamb (1963), Lighthill (1986), Milne-Thomson (1955), and Yih (1988), and Newman (1977) for modern applications.

In the preceding chapter we calculated the force generated by intermittent cavitation on simple forms. It was necessary to model not only the ship as a simple form but also the cavitating propeller. Although these simplifications gave useful information it was also obvious that the results could not be used for practical purposes. This was the price we had to pay for being able to obtain results by “hand-turned mathematics”.

When we now wish to obtain results for actual, intermittently cavitating propellers behind real ship forms of given (arbitrary) shape we must expect the mathematics to be too complicated to be manipulated into closed expressions for forces and pressures. Instead, we shall describe a general, computer-effected method for solving the problem for a propeller behind a ship and we shall also present results of this theory for cavitating propellers. Furthermore, we shall correlate these results with those obtained by model experiments.

REPRESENTATION OF HULLS OF ARBITRARY SHAPE IN THE PRESENCE OF A PROPELLER AND WATER SURFACE

It has been demonstrated in the foregoing that a propeller operating in a temporally uniform but spatially varying hull wake produces, through the concerted action of all the blades, a potential flow and pressure field composed of many components, all of which are at frequencies qZω. As these frequencies are large compared with those which can give rise to wave generation on the free water surface, the appropriate linearized boundary condition, imposed by the presence of the water surface, is that the total velocity potential in the undisturbed locus of that surface must be zero.

Ship propellers have wide blades to distribute the loading over the blades. As we have seen previously this is necessary to reduce local negative pressures to avoid, as much as possible, the generation of cavitation. We can then think of the blades as low-aspect-ratio wings that rotate and translate through the water. To describe the flow we consider the propeller blades as lifting surfaces over which we distribute singularities to model the effects of blade loading and thickness. This was done in Chapter 14 for uniform inflows and in Chapter 15 for varying blade loadings in hull wakes. But in those chapters we only considered the pressure from assumed distributions of loading and thickness over the propeller blades. To be able to find these distributions we must establish a relation between the pressure and the velocity and fulfill the kinematic boundary condition on the blades. We anticipate that the operation of the blades in the spatially varying flow, generated by the hull, will give rise to forces which vary with blade position. Hence it is necessary to treat the blades as lifting surfaces with non-stationary pressure distributions.

In this chapter an overview of the extensive literature of the past three decades is followed by a detailed development of a linear theory which is sufficiently accurate for prediction of the unsteady forces arising from only the temporal-mean spatial variations of inflow.

We have this fax completely neglected the fact that all fluids possess viscosity. This property gives rise to tangential frictional forces at the boundaries of a moving fluid and to dissipation within the fluid as the “lumps” of fluid shear against one another. The regions where viscosity significantly alters the flow from that given by inviscid irrotational theory are confined to narrow or thin domains termed boundary layers along the surfaces moving through the fluid or along those held fixed in an onset flow. The tangential component of the relative velocity is zero at the surface held fixed in a moving stream and for the moving body in still fluid all particles on the moving boundary adhere to the body.

The resulting detailed motions in the thin shearing layer are complicated, passing from the laminar state in the extreme forebody through a transitional regime (due to basic instability of laminar flow) to a chaotic state referred to as turbulent. We do not calculate these flows.

In what follows we show that viscous effects are a function of a dimensionless grouping of factors known as the Reynolds number and review the significant influences of viscosity in terms of the magnitude of this number upon the properties of foils as determined by measurements in windtunnels at low subsonic speeds.

PHENOMENOLOGICAL ASPECTS OF VISCOUS FLOWS

The equations of motion for an incompressible but viscous fluid can be derived in the same way as for a non-viscous fluid, cf. Chapter 1, p. 3 and sequel, but now with inclusion of terms to account for the viscous shear stresses.

Armed with our knowledge of the structure of the pressure fields arising from propeller loading and thickness effects (in the absence of blade cavitation) we can seek to determine blade-frequency forces on simple “hulls”. There are pitfalls in so over-simplifying the hull geometry to enable answers to be obtained by “hand-turned” mathematics, giving results which may not be meaningful. Yet the problem which can be “solved” in simple terms has a great seduction, difficult to resist even though the required simplifications are suspected beforehand to be too drastic. One then has to view the results critically and be wary of carrying the implications too far.

From our knowledge that most of the terms in the pressure attenuate rapidly with axial distance fore and aft of the propeller we are tempted to assume that a ship with locally flat, relatively broad stern in way of the propeller may be replaced by a rigid flat plate. The width of the plate is taken equal to that of the local hull and the length extended to infinity fore and aft on the assumption that beyond about two diameters the load density will virtually vanish. As we shall note later, this assumption of fore-aft symmetry of the area is unrealistic as hulls do not extend very much aft of the propeller. We shall also assume at the outset that the submergence of the flat surface above the propeller is large so we might ignore the effect of the water surface.

Cavitation on ship propellers has been the bane of naval architects and ship operators since its first discovery on the propellers of the British destroyer Daring in 1894. Primary interest in propeller-blade cavitation was, for many years, centered upon the attending blade damage and the degradation of thrust arising from extensive, steady cavitation. It was not until the advent of the rapid growth in the size of merchant ships in the past three decades (with concurrent marked increases in blade loading) that extensive, intermittent or unsteady cavitation appeared and was indicted as the cause of large forces exciting highly objectionable hull vibration. Efforts in the modeling of hull wakes in water tunnels date back to about 1955 (cf. van Manen (1957b)) when tests of propeller models in fabricated axially non-uniform flows were being conducted at Maritime Research Institute Netherlands (MARIN), National Physical Laboratory (NPL) and Hamburgische Schiffbau–Versuchsanstalt (HSVA). Non-stationary blade cavities were observed then but there seem to have been no notice or measurement of unsteady near-field pressures attending unsteady cavitation until the experimental work of Takahashi & Ueda (1969). They measured pressures at one point above a propeller in a water tunnel in uniform and non-uniform flow and gave a brief contribution to the 12th International Towing Tank Conference (ITTC) in Rome in 1969. Their principal results are shown in Figure 20.1, where it is seen that the pressure amplitudes increased dramatically with reduced cavitation number.

Here we present the essential steps in the problems posed by the design and analysis of propellers. In design we are required to develop the diameter, pitch, camber and blade section to deliver a required thrust at maximum efficiency (minimum torque). There are other criteria such as to design a propeller to drive a given hull (of known or predicted resistance over a range of speeds) with a specified available shaft horsepower and to determine the ship speed.

The analysis procedure requires prediction of the thrust, torque and efficiency of propellers of specified geometry and inflow.

We begin with the development of the criteria for the radial distribution of thrust-density to achieve maximum efficiency in uniform and non-uniform inflows. This is followed by methods for determining optimum diameter for given solutions and optimum solutions for a given diameter.

The derivation and reason for the induction factors in the lifting-line theory of discrete number of blades, as displayed in the previous chapter, is followed by formulas for the thrust and torque coefficients in terms of the circulation amplitude function Gn.

Applications are then made to the design and analysis of propellers. Means for selection of blade sections to avoid or mitigate cavitation are followed by extensive discussion of practical aspects of tip unloading via camber and pitch variation. Effects of blade form and skew on efficiency and pressure fluctuations at blade frequency (number of blades times revolution per second) are presented.

The pressure fluctuations generated by propellers in the wake of hulls are markedly different from those produced in uniform inflow. The flow in the propeller plane abaft a hull varies spatially as well as temporally. Here we deal only with the effects attending spatial variations peripherally and radially as provided by wake surveys which give the averaged-over-time velocity components as a function of r and γ for a fixed axial location. Temporal variations in the components are aperiodic and cannot be addressed until sufficient measurements have been made to determine their frequency spectra. Ultimately, numerical solutions of the Navier-Stokes equations may provide both spatial and temporal aspects of hull wakes.

Here the spatial variations in the axial and tangential components are reflected in the pressure jump Δp which is taken to vary harmonically with blade position angle γ0. Then we discover a coupling between the harmonics of Δp(γo) and the harmonics of the propagation function yielding a plethora of terms all at integer multiples of blade frequency. Graphical results are given for pressure and velocity fields showing the effect of spatial non-uniformity of the inflow.

We have seen in the previous chapter that the pressure field arising from a lifting-surface model of a propeller in a uniform flow is that due to pressure and velocity dipoles distributed over the blade. Both dipole strengths were constant in time since we considered uniform and stationary inflow.

It is well known that the flow abaft of ships is both spatially and temporally varying. This variability arises from the “prior” or upstream history of the flow produced by the action of viscous stresses and hull-pressure distribution acting on the fluid particles as they pass around the ship from the bow to the stern. Thus the blade sections “see” gust patterns which over long term have mean amplitudes but from instant-to-instant change rapidly with time because of the inherent unsteadiness of the turbulent boundary layer.

Our knowledge of the distribution of flow in the propeller disc is almost entirely based on pitot-tube surveys conducted on big (≈ 6 m) models in large towing tanks and in the absence of the propeller. These are termed nominal wake flows. As is well appreciated, pitot-tube measurements provide only long-term averages of the velocity components at various angular and radial locations in the midplane of the propeller. These measurements depend upon the calibration of the pitot-tube in uniform flow whereas the wake flow radially and tangentially has the effect of shifting the stagnation point on the pitot-tube head, a mechanism not operating in the calibration mode. Thus there is a systematic error which is, to the authors' knowledge, not generally corrected. Moreover, wake-fraction (and thrust-deduction) calculations based upon tests with the same model in several large model basins and upon repetitive tests with the same model in the same large model basin, have shown remarkably different results. A similar scatter was also found in results of wake surveys.