The ability of modern computers to store and rapidly manipulate large quantities of data has allowed the development of flow simulations that complement, and sometimes even replace, traditional laboratory experiments. In industry, the considerable expense of constructing and exploiting physical models motivates the search for reliable simulation techniques to reduce the cost of the design and development phases. Full, detailed coverage of numerical methods for turbulent flows goes well beyond the scope of this book and would require a volume to itself. This chapter is therefore limited to a relatively brief overview of the three main types of numerical techniques, represented by the following three sections, in which the reader desiring further details of numerical methods will find references to more specialized works.

Direct Numerical Simulation (DNS)

The first, and most obvious, approach to the numerical simulation of turbulent flows treats them like any others, solving the unsteady, three-dimensional Navier-Stokes equations numerically without further ado. Such methods are referred to as direct simulations, in contrast with those based on the statistically averaged equations considered in Section 8.3, which involve closure approximations, but are much less computationally costly than DNS at high Reynolds numbers. Direct simulation allows numerical experiments to be carried out that are often difficult or impossible to realize in the laboratory, and yields detailed information concerning the flow field in individual realizations and its statistical properties that again often surpasses what can be measured in real flows. Furthermore, unlike other simulation techniques for turbulent flows, DNS does not involve approximations, other than those due to discretization, which are inherent in any numerical solution of differential equations, and so provides benchmarks against which less costly simulation methods may be evaluated and parameterized, just as they can by using the results of physical experiments. However, the continuum of different spatial (and temporal) scales intrinsic to turbulence, a continuum whose width increases with the turbulent Reynolds number, ReL, leads to computer time and storage requirements that increase rapidly with ReL. In consequence, DNS is limited to relatively low Reynolds numbers in practice and it is currently very expensive to conduct a DNS calculation at even moderately high Reynolds number.

Besides its apparent randomness, one of the important characteristics of turbulent flows at high Reynolds number (which we presume throughout this chapter) is the wide range of different space and time scales which they contain. As illustrated in Figure 3.1a, the velocity measured at a given point within a turbulent flow fluctuates randomly and does not yield a smooth function of time, but rather one with finescale “fur.” If part of such a graph is examined at successively higher magnifications, instead of approaching a straight line, as would a smooth function, the fur is revealed as being itself “furry,” and so on, through a succession of smaller and smaller time scales, until eventually the smallest of all is reached. At this scale, the velocity finally appears as a smooth function of time, with no further structure revealed by additional magnification. On the other hand, the largest time scales of turbulence are characterized by the time interval required for statistical decorrelation of the turbulent velocity fluctuations. At high Reynolds number, the separation of scales between the largest and smallest is wide, and gets wider as the Reynolds number increases.

The continuum of time scales described above is also reflected in the spatial structure of the flow. The order of size of the largest spatial scales of turbulence is measured by the velocity correlation length L, which characterizes the spatial separation required for significant decorrelation of the velocity fluctuations. Successively higher magnification of the spatial structure of the velocity field shows smaller and smaller scales down to the smallest, known as the Kolmogorov scale, at which it appears smooth. Thus, turbulence contains a continuum of scales, ranging from the large ones, whose size is of order the correlation length, to those of order the Kolmogorov scale. Both the Kolmogorov scale, r\, and correlation scale, L, are best considered as orders of magnitude: one talks of scales comparable with, much smaller than, or much larger than these quantities. The spatial and temporal “furriness” of the velocity field are related; indeed, one can think of the small time scales as resulting from convection of fine spatial structures past the measurement probe, as we will see later.

If one attempts to take the derivative of the velocity field, either with respect to time or one of the spatial coordinates, its furry nature becomes markedly more apparent.

The theory of turbulence has seen considerable progress in the last fifty years, although these advances are, at first sight, less marked than might have been hoped at the outset. Prior to that period, Reynolds, Taylor, Prandtl, Von Karman, and Kolmogorov, to name just some of the eminent workers, had laid the foundations of the subject and a number of excellent books and surveys appeared in the 1950s. Batchelor's book, The Theory of Homogeneous Turbulence, was first published in 1953 and mainly concerned the development of spectral methods, quickly gaining a following among existing and aspiring specialists in turbulence. Shortly afterwards (1956), another classic appeared in the shape of Townsend's The Structure of Turbulent Shear Flow, whose approach was based more on physical reasoning and placed more emphasis on the more realistic, but less theoretically tractable case of inhomogeneous turbulent flows.

Many other excellent works have appeared since then, among them the books given in the General References list following this preface. In each of them, the authors view turbulence from their own perspective, different from that of other authors, for, owing to the nature of the subject, an author's personal viewpoint colors the presentation perhaps more than in most areas of science. The book by Tennekes and Lumley is somewhat different from most of the others because it gives considerable weight to physical ideas, order-of-magnitude discussions, and other means for gaining intuition about turbulent flows, while the authors remind us that “several dozen introductory texts in general fluid dynamics exist, but the gap between monographs and advanced texts in turbulence is wide.” The book gives the reader excellent food for thought before tackling more specialized ones, such as Monin and Yaglom's two-volume work, which remains the undisputed (if slightly dated) bible for the subject.

The present book does not seek to replace these existing works, but to complement them, giving the reader a solid grounding in the theory of turbulence and developing both physical insight and the mathematical framework needed to express the theory. Turbulence is treated using one-point statistics in the early chapters of the book, while multipoint statistics, analyzed by spectral methods, are introduced later, the two approaches being complementary and mutually illuminating.

This is a book about flow measurement and flowmeters written for all in the industry who specify and apply, design and manufacture, research and develop, maintain and calibrate flowmeters. It provides a source of information on the published research, design, and performance of flowmeters as well as on the claims of flowmeter manufacturers. It will be of use to engineers, particularly mechanical and process engineers, and also to instrument companies' marketing, manufacturing, and management personnel as they seek to identify future products.

I have concentrated on the process, mechanical, and fluid engineering aspects and have given only as much of the electrical engineering details as are necessary for a proper understanding of how and why the meters work. I am not an electrical engineer and so have not attempted detailed explanations of modern electrical signal processing. I am also aware of the speed with which developments in signal processing would render any descriptions which I might give out of date.

In the bibliography, other books dealing with flow measurement are listed, and my intention is not to retread ground covered by them, more than is necessary and unavoidable, but to bring together complementary information. I also make the assumption that the flowmeter engineer will automatically turn to the appropriate standard; therefore, I have tried to avoid reproducing information that should be obtained from those excellent documents. I include a brief list that categorizes a few of the standards according to meter or application.

INTRODUCTION

BACKGROUND

Spirals, screws, and windmills have a long history of use for speed measurement. Robert Hook proposed a small windmill in 1681 for measuring air velocity and later one for use as a ship's log (distance meter). A Captain Phipps, in 1773, created a ship's log, using the principle that a spiral, in turning, moves through the length of one turn of the spiral. Many centuries earlier, it appears that a Roman architect, Vitruvius, suggested a more basic form of this device.

In 1870 Reinhard Woltmann developed a multibladed fan to measure river flows (Medlock 1986). The device was a forerunner of the long helical screw-type meter still called after him and used widely for pipe flows in the water industry. The first modern meters, of the type with which we are mainly concerned, were developed in the United States in 1938 (Watson and Furness 1977; cf. Furness 1982). These were attractive for fuel flow measurement in airborne applications. They consisted of a helically bladed rotor and simple bearings. Improved sleeve bearings were developed for longer life with hardened thrust balls or endstones to withstand the axial load. An alternative, developed over several years and patented by Potter (1961), was to profile the hub of the rotor so that the pressure balance across the rotor, rather than the thrust on the bearings, held it against the axial drag forces. This allowed the rotor to run on a single journal bearing.

INTRODUCTION

The object of this chapter is to assist the reader in specifying a proposed flowmeter's operational requirements as fully as possible and hence to enable the reader to select the right flowmeter. The reason for wishing to select a flowmeter may be obvious: a flow rate to be measured. However, the reason in some cases may be less obvious. If you are a flowmeter manufacturer, or prospective manufacturer, you may wish to identify a type and design that would meet a particular market niche. If you are a member of an R&D team, you may be exploring measurement areas that are inadequately covered at present.

In preparing this chapter, I have developed earlier ideas (Baker 1988/9, Baker and Smith 1990), but I have also benefited from the ideas of others such as Endress et al. (1989). Baker (1988/9) was the best distillation I could offer in the late 1980s. There is much there that I would still endorse. In Baker and Smith (1990), we took a different line. We provided the means to specify the user's needs in as much detail as possible. We then provided a form for communication with the manufacturer.

Many people have attempted to provide a means of selecting a flowmeter. A glance at only a few of the many books on flow measurement in the bibliography will indicate this. I used to feel that the way forward was to write an expert system to do the job.

INTRODUCTION

I am indebted to Medlock (1989) for notes on three early devices that attempt to use change of angular momentum to obtain mass flow.

Katys (1964) used an electric synchronous motor with a special rotor supported internally in the pipe and a stator outside the pipe. Fluid passed through vanes in the rotor and acquired angular momentum. From the power used or the torque needed to drive the rotor, it was claimed that the mass flow could be deduced.

According to Medlock, the Bendix meter “measures the torque required to impart angular momentum to the liquid. One end of a calibrated spring is driven at a constant speed and the other end is connected to a freely rotatable turbine. The torsion developed in the spring is a measure of mass flow rate.”

The twin rotor turbine meter (Potter 1959) attempted to measure mass flow by means of two in-line turbine rotors of different blade angles, which are joined by a torsion spring. Despite discussions with Medlock about this meter, I am not convinced that this is a true mass flow meter and suggest that the valid derivative of the Bendix meter is the fuel flow transmitter described later.

An early design with many similarities to the current commercial designs was a device due to Orlando and Jennings (1954) shown in Figure 16.1. A constant speed motor drives an impeller imparting swirl to the liquid.

INTRODUCTION

The object of calibration is to benchmark a flowmeter to an absolute datum. Just as a benchmark tells us how a particular geographical point compares in height with sea level datum, so a calibration of a flowmeter tells us how the signal from the flowmeter compares with the absolute standard of a national laboratory. The analogy is not perfect at this stage. The national laboratory standard must also be compared with other more fundamental measures of time and mass, and it will be essential to check and compare different national standards in different countries from time to time.

There is a desire to reference back to fundamental measurements such as mass, length, and time. Thus if we can measure mass flow on a calibration facility by using fundamental measurements of mass and time, this will bring us nearer to the absolute values than, say, obtaining the volume of a calibration vessel by using weighed volumes of water and deducing the volume from the density. The first is more correctly termed primary calibration, whereas the second fails strictly to achieve this. It is likely that the final accuracy will reflect this. Liquid calibration facilities can achieve a rather higher accuracy than is possible for most of the gas calibration facilities. In part, this difference will result from the lower density and the increased difficulties of handling a gas.

INTRODUCTION

This chapter does not attempt more than a cursory review of modern control systems, as it is outside my main area of expertise. Those knowledgeable in electronics and control will pass quickly over this chapter! Others who wish to delve more deeply may, for instance, turn to Sanderson (1988), Brignell and White (1994), or other experts, or for a good, recent introduction to the control of machinery Foster (1998) is recommended. I am pleased to acknowledge valuable insights obtained from Cranfield lectures given by Bill Black, David Clitherow, Doug de Sa, and others.

This book is about flowmeters that depend, in the main, on mechanical or thermo/fluid mechanical effects. In only one case of mainstream meters, the electromagnetic meter is the signal electrical throughout. The aim of the book has been to bring together information on the performance of these meters based on physical, experimental, and industrial experience. I have made the assumption that the electrical signal interpretation, which is based in the meter, is capable of doing the job set by the designer of interpreting the meter output into a standard electrical form. This assumption is not always adequate (e.g., where electrical signals from the same instrument do not appear to agree with each other). Nevertheless, the developments in design of electrical systems are assumed to be such that the reader will not benefit from a detailed description or that the circuit will be changing so fast that any statement will be out of date.

INTRODUCTION

The material in this chapter is based on the paper by Baker and Morris (1985) on positive displacement (PD) meters, which in turn has been updated with more recent industrial and published material and extended to gases. P.D. Baker's (1983) paper provided some additional useful information on these meters. The main types of liquid meter were given by Barnes (1982), Hendrix (1982), Henke (1955), and Gerrard (1979) (cf. Mankin 1955). The reader should refer to API (1992) and similar documents.

At least four of the meter designs to be discussed have been around for over 100 years. The nutating disk flowmeter for liquids was developed in 1850. The rotary piston meter appeared in the late nineteenth century (Baker 1998).

The measurement of gas has depended, from an early date, on two types of positive displacement meter: the wet gas meter of high accuracy and credited to Samuel Clegg (1815), and the diaphragm meter of lower performance but greater range for which William Richards (1843) should take the credit.

BACKGROUND

The concept of carrying known volumes of fluid through a flowmeter is a short step from the use of a discrete measure such as a bucket or measuring flask. Thus in each of the designs described later, the flow enters a compartment that is as tightly sealed as is compatible with relative movement of adjacent components.

INTRODUCTION

The first proposal for the use of ultrasound for flow measurement, according to Thompson (1978), seems to have been in a German patent of 1928. It was not until after 1945 that the idea became more widely proposed. But not until the development of piezoelectric transducers in the past 40 years or so have ultrasonic applications become really attractive. Fischbacker (1959) provided an early review of ultrasonic flowmeters in which, essentially, the transit-time, sing-around, and beam deflection methods were mentioned. He also referred to phase-difference measurement, means of obtaining sound speed from time measurement and impedance, and how to obtain density. He saw the advantages of off-axis paths.

Sanderson and Hemp's (1981) review is still a useful source of information on the subject. The ultrasonic flowmeter's attraction as a flow measurement device is its linearity, lack of obstruction to flow, and, in contrast to the magnetic flowmeter, its ability to measure the flow of gases.

In this chapter, we shall consider three main types of ultrasonic flowmeter, and it is important to understand the strengths and weaknesses of each and that they are very different in performance and application.

The transit-time flowmeter, or time-of-flight flowmeter, is the most accurate of the family and is available as a spool piece meter for liquids and gases or as a clamp-on design for liquids only. It can also be retrofitted into a pipe. Measurement uncertainty will be from a fraction of a percent to about 5%.