Zonal variations in the tropics

Up to this point, we have followed a traditional exposition of the global circulation by concentrating upon the zonal mean circulation and upon the zonal mean fields of eddy quantities. But the global circulation is far from zonally symmetric. Tropical heating has distinct maxima at particular longitudes. In the midlatitudes, the transient eddies are not distributed uniformly around the latitude circles, but are concentrated into isolated ‘storm tracks’, especially in the northern hemisphere. This chapter will be devoted to a description of such zonal asymmetries and their consequences.

The various diagnostics of the steady and transient eddy activity which we have considered in earlier chapters become small in the tropics. Eddy kinetic energy is much smaller in the tropics than in the midlatitudes. Similarly, eddy temperature and momentum fluxes, both steady and transient, are much smaller in the tropics. Thus a picture emerges in which heat and momentum are transported, essentially by axisymmetric motions in the tropics, with eddies taking over in the subtropics and midlatitudes.

There is some truth in this picture. But it can also be misleading. First, consider the heating fields shown in Fig. 3.8. The forcing of the circulation is certainly not axisymmetric, especially in the tropics. Rather, there are a small number of centres of intense heating.

Timescales of atmospheric motion

The circulation of the atmosphere is intrinsically unsteady; fluctuations on all timescales are observed. In the last chapter, it was shown that the fluxes of temperature, momentum, and so on, carried by such transients play an important part in determining the time mean circulation of the atmosphere. Our task in this chapter will be to describe the transients on various timescales, and to discuss the mechanisms which can give rise to transient behaviour. Just as the atmosphere contains a wide range of spatial scales, from the molecular to the global, so the atmospheric circulation exhibits a wide range of timescales, ranging from timescales of just a few seconds associated with the overturning of small turbulent eddies, to geological timescales for major climate changes.

Some of the frequencies observed are directly related to the frequencies of periodic forcings. For example, diurnal and semi-diurnal variations of temperature and wind are associated with the diurnal variation of solar heat input. These ‘thermal tides’ are important at high levels in the atmosphere and can be detected in the lower atmosphere. More importantly for our purposes, the annual cycle of radiative forcing has a profound effect on large scale atmospheric circulation. This seasonal cycle of meteorological quantities affects nearly all parts of the globe.

But in addition to these externally imposed periods, the atmospheric flow itself generates all kinds of timescales internally.

Great increases in the cost of fuel and the advent of very large tankers and bulk carriers have focused the attention, during the last decades, on means to enhance the efficiency of ship propulsion. An obvious way of obtaining an efficiency increase is to use propellers of large diameter driven by engines at low revolutions, as can be deduced from the developments in Chapter 9. Such a solution is, however, in many cases not practically possible. This has then given impetus to the study and adoption of unconventional propulsion arrangements, consisting, in general, of static or moving surfaces in the vicinity of propellers.

A distinct indication of the serious and extensive activity in the development and use of unconventional propulsors may be seen in the report of the Propulsor Committee of the 19th ITTC (1990b) which lists seven devices including large diameter, slower turning propellers. Here we summarize the hydrodynamic characteristics of six of these devices omitting larger-diameter propellers. The six devices are: Coaxial contrarotating propellers, propeller with vane wheel, with pre-swirl stators, with postswirl stators, ducted propellers and propellers operating behind flowsmoothing devices.

Emphasis is given in the following to a variational procedure which, in a unified fashion enables nearly optimum design of several of these configurations.

Propulsive Efficiency

Propulsive efficiency is conventionally thought of as the product of the open-water efficiency of the propulsor, the hull efficiency and a factor termed the relative-rotative efficiency.

The flows about wings of finite span are sufficiently analogous to those about propeller blades to warrant a brief examination before embarking on the construction of a mathematical model of propellers. A more detailed account of wing theory is given for example by von K ármán & Burgers (1935).

A basic feature of the flow about a straight upward-lifting wing of finite span and starboard-port symmetry in a uniform axial stream is that the velocity vectors on both the lower and upper surfaces are not parallel to the longitudinal plane of symmetry. On the lower side the vectors are inclined outboard and on the upper side they are inclined inwardly in any vertical plane or section parallel to the vertical centerplane. This is a consequence of the pressure relief at the wing tips and the largest increase in pressure being in the centerplane on the lower side and the greatest decrease in pressure being in the centerplane on the upper or suction side. Thus there are positive spanwise pressure gradients on the lower side and negative spanwise gradients on the upper side which give rise to spanwise flow components which are obviously not present in two-dimensional flows.

In many treatments of wing theory one finds figures purporting to show the flow about a wing of finite aspect ratio in which there is a continuous line of stagnation points near the leading edge, extending from one wing tip to the other.

The number of comparisons that have been made of calculated and measured blade-frequency thrust, torque and other force and moment components are very few because of the paucity of data. In this chapter comparisons of theoretical predictions with experimental data will be given. Results obtained by various theories will also be compared. The chapter concludes with presentation of a simple procedure, based upon the KT-J curve of the steady case, for a quick estimate of the varying thrust at blade frequency.

The measurement of blade-frequency forces on model propellers requires great care in the design of the dynamometer which must have both high sensitivity and high natural frequencies well above the model blade frequency. After a number of failures a successful blade-frequency propeller dynamometer capable of measuring six components (three forces, three moments) was evolved at David Taylor Research Center (DTRC) about 1960.

Measurements were made with a triplet of three-bladed propellers of different blade-area ratio designed to produce the same mean thrust. This set was tested in the DTRC 24-inch water tunnel alternately abaft threeand four-cycle wake screens which produced large harmonic amplitudes of the order of 0.25-U in order to obtain strong output-to-“noise” levels! The three-cycle screens give rise to blade-frequency thrust and torque whereas the four-cycle wake produces transverse and vertical forces and moments about the y- and z-axes which in general come from the fourth and second harmonic orders of blade loading on a three-bladed propeller, as was demonstrated on p. 367.

The non-symmetrical flow generated by flat and cambered laminae at angles of attack is at first modelled by vorticity distributions via classical linearized theory. Here, in contrast to the analysis of symmetrical sections, we encounter integral equations in the determination of the vorticity density because the local transverse component of flow at any one point depends upon the integrated or accumulated contributions of all other elements of the distribution. Pressure distributions at non-ideal incidence yield a square-root-type infinity at the leading edge because of the approximations of first order theory. Lighthill's (1951) leading edge correction is applied to give realistic pressure minima at non-ideal angles of incidence.

Our interest in pressure minima of sections is due to our concern for cavitation which can occur when the total or absolute pressure is reduced to the vapor pressure of the liquid at the ambient temperature. Since cavitation may cause erosion and noise it should be avoided or at least mitigated which may possibly be done by keeping the minimum pressure above the vapor pressure. This corresponds to maintaining the (negative) minimum-pressure coefficient Cpmin higher than the negative of the cavitation index.

At this point we shall not go deeper into the details of cavitation which is postponed until Chapter 8. Instead we shall continue our theoretical development with flat and cambered sections.

THE FLAT PLATE

We now seek the pressure distributions and the lift on sections having zero thickness but being cambered and, in general, set at any arbitrary (but small) angle of attack to the free stream, U. Consider a flat plate at small angle α.

Criteria for the design of blade sections may be selected to include:

1. i. Minimum thickness and chord to meet strength requirements;

2. ii. Sufficient camber to generate the design lift;

3. iii. Distribution of thickness and camber to yield the least negative pressure coefficient to avoid or mitigate cavitation;

4. iv. Thickness- and loading-pressure distributions to avoid boundary layer separation with least chord to yield minimum drag consistent with requirements i. and iii.;

5. v. Leading and trailing edges to satisfy strength and manufacturing requirements.

The first part of this chapter follows from linearized theories developed by aerodynamicists more than 50 years ago, placing emphasis on the use of existing camber and thickness distributions yielding least negative minimum pressure coefficients, Cpmin at ideal angle of attack. At nonideal angles (which always occur in operation in the spatially and temporally varying hull wake flows) we are required to seek sections having greatest tolerance to angle deviations and at the same time having negative minimum pressure coefficients exceeding the level that indicates occurence of cavitation. This tolerance depends critically upon the leading edge radius and the forebody shape as well as upon the extent of the flat part of the pressure distribution. Thus we are led to the more recent findings of researchers who have developed profiles having greater tolerance to angle of attack. When cavitation is unavoidable the latest approach is to use blunter leading edges to generate shorter, more stable cavities thereby avoiding “cloud” cavitation which causes highly deleterious erosion or pitting of the blades.

As a preparation for determining unsteady forces on propellers in ship wakes, we first consider two-dimensional sections beset by travelling gusts. Our development of the unsteady force on such sections differs from that given in the seminal work of von Kármán & Sears (1938) by adopting a procedure which is easily extended to wings and propellers. Their formula for unsteady sectional lift is recovered, being that of lift at an effective angle of attack which varies with the parameter k = ωc/2U, the “reduced” frequency. Turning to hydrofoils of finite span, we derive results for low aspect ratio in steady flow. For wings in gusts there is no analytical inversion of the integral equation which involves a highly singular kernel function. Graphical results are given from numerical solutions for a range of aspect ratios which reveal diminishing unsteady effects with decreasing aspect ratio.

Corresponding reduced frequencies for propeller blades in terms of expanded- blade-area ratio are shown to be high relative to aerodynamic experience. This indicates that two-dimensional, unsteady section theory cannot be applied to wide-bladed (low-aspect-ratio) propellers.

TWO–DIMENSIONAL SECTIONS

The blades of a propeller orbit through the spatially non-uniform flow of the hull wake and consequently experience cyclic variations in the flow normal to their sections. For blades of small chord-to-radius, this is analogous to the case of a two-dimensional section moving at constant speed through a stationary, cyclic variation in cross flow distributed as a standing wave along the course of the moving section.