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.

Abstract

This article is for those who have already a computer program for incompressible viscous transient flows and want to put a turbulence model into it. We discuss some of the implementation problems that can be encountered when the Finite Element Method is used on classical turbulence models except Reynolds stress tensor models. Particular attention is given to boundary conditions and to the stability of algorithms.

Introduction

Many scientists or engineers turn to turbulence modeling after having written a Navier-Stokes solver for laminar flows.

For them turbulence modeling is an external module into the computer program. Generally, the main ingredients to built a good Navier-Stokes solver are known; this includes tools like mixed approximations for the velocity u and pressure p to avoid checker board oscillations and also upwinding to damp high Reynolds number oscillations; however the problems that one may meet while implementing a turbulence model are not so well known because these models have not been studied much theoretically.

Judging from the literature [3] [11] [12] [15] [19] [22] the most commonly used turbulence models seem to be

• algebraic eddy viscosity models (zero equation models)

• k= ε models (two equations models)

• Reynolds stress models

All three start from a decomposition of u and p into a mean part and a fluctuating part u’. However oscillations are understood either as time oscillations or space oscillations or even variations due to changes in initial conditions. In any case, the decomposition u+u’ is applied to the Navier-Stokes equations.

Introduction

This article contains a summary account of covolume methods for incompressible flows. Covolume methods are a recently developed way to solve both compressible and incompressible flow problems on unstructured meshes. The general idea is to use complementary pairs of control volumes to discretize flux, circulation and other expressions which occur in the governing equations. These complementary volumes (covolumes for short) are related by an orthogonality property which is a basic feature of the covolume approach. One of the simplest mesh configurations which is suitable is the Delaunay- Voronoi mesh pair. This is introduced in the next section. After that we proceed through div-curl systems to the stationary Stokes equations and the Navier-Stokes equations. We will show that for uniform meshes the covolume equations for the stationary Stokes equations specialize to the MAC (staggered mesh) scheme, and that the MAC scheme itself is actually equivalent to a velocity-vorticity scheme. Some numerical results are presented in the last section.

Since this article is intended only as an overview, we will present most of the results in a two dimensional setting. Almost all of the ideas and techniques do generalize nicely to three dimensions but are harder to visualize than in two dimensions. Given our limited aims it would be inappropriate to present proofs of most of the mathematical results. We will refer to the original sources for these and other details.

One of the reasons for introducing covolume methods is to find lower order methods for viscous flows which are free of “spurious mode” problems.

Summary

Our object in this article is to present some aspects of new emerging methods in numerical analysis and their application to Computational Fluid Dynamics. These methods stem from Dynamical Systems Theory.

The attractor describing a turbulent flow is approximated by smooth manifolds and by projecting the Navier-Stokes equations onto these manifolds we obtain new algorithms, the Inertial Projections. These algorithms have proven to be stable, efficient and well suited for long time integration of the equations. They can be implemented with all forms of spatial discretizations, spectral methods, finite elements and finite differences (possibly also wavelets).

Introduction

Algorithms that have been introduced at a time of scarce computing resources may not be well adapted to supercomputing and to the more difficult problems that are tackled at the present time or that we foresee for the near future.

For incompressible fluid mechanics (or thermal convection), by using the full Navier-Stokes equations (NSE) we can, at present time, compute flows at the onset of turbulence: in particular the permanent regime is not anymore a stationary flow. The flow can be time periodic if a Hopf bifurcation has occurred or the permanent regime can be an even more complicated flow. Examples of time dependent flows have been numerically computed in the case of the driven cavity (see e.g. [15], [16], [20], [4] and [34]) and for other types of flows (see e.g. [14], [33] and [30]). Section 12.2 of this article is a brief survey, for the CFD practitioner, of some basic and relevant concepts in Dynamical Systems Theory (behavior for large time of the solutions of the NSE).