Modelling the Flying Bird

By C.J. Pennycuick

This book outlines the principles of flight, of birds in particular. It describes a way of simplifying the mechanics of flight into a practical computer program, which will predict in some detail what any bird, real or hypothetical, can and cannot do. The Flight program, presented on the companion website, generates performance curves for flapping and gliding flight, and simulations of long-distance migration and accounts successfully for the consumption of muscles and other tissues during migratory flights. The program is effectively a working model of a flying bird (or bat or pterosaur) and is the skeleton around which the book is built. The book provides a wider background and then explains how Flight works and shows how to set up and test hypotheses generated by the program.

The book and the program are based on adapting the conventional (and well-tested) thinking of aeronautical engineers to the biological problems of bird flight. Their primary aim is to convince biologists that this is the appropriate way to handle problems that involve flight, to make the engineering background accessible to biologists, and to provide a tool kit in the shape of the Flight program, which they can use to solve practical problems involving bird flight and migration. In addition, the book will be readily accessible to engineers who want to know how birds work, and should be of interest to the ever-growing community working on flapping "micro air vehicles" (MAVs). The program can be used to predict the flight performance and capabilities of reconstructed fossil birds and pterosaurs, flying in ancient atmospheres that differ from present conditions, and also, of course, to predict and account for the results of experiments and observations on living birds and bats.




* An up to date work by the world's leading expert on bird flight
* Examines the biology and biomechanics of bird flight with added reference to the flight of bats and pterosaurs.
* Uses proven aeronautical principles to help solve biological issues in understanding and predicting the flight capabilities of birds and other vertebrates.
* Provides insights into the evolution of flight and the likely capabilities of extinct birds and reptiles.
* Gives a detailed explanation of the science behind, and use of, the author's predictive bird flight simulation program - Flight - which is available on a companion website.
* Presents often difficult concepts in easily understood language.

• Language: English
• Publisher: Academic Press
• Release date: Aug 23, 2008
• ISBN: 9780080557816

Book preview

Theoretical Ecology Series

Modelling the Flying Bird

C.J. Pennycuick

ISSN  1875-306X

Volume 5 • Number suppl (C) • 2008

Table of Contents

Cover image

Title page

Preface

Foreword

Acknowledgements

Chapter 1 Background to the Model

1.1 The Flight Model

1.2 The Engineering Approach to Numbers

1.3 Dimensions and Units

1.4 Literature Citations

Chapter 2 The Flight Environment

2.1 The Earth's Gravity Field

2.2 The Earth's Atmosphere

2.3 Air Density in Flight

2.4 Gravity and The Atmosphere in Former Times

Chapter 3 Mechanics of Level Flight

3.1 Power Required for Horizontal Flight

3.2 The Power Curve Calculation in Flight

3.3 Significance of the Characteristic Speeds

3.4 Effect of Air Density on Speed and Power

3.5 Adaptive Significance of Morphology

3.6 Two‐Dimensional Aerofoil Properties

3.7 Scale and Reynolds Number

Chapter 4 Vortices and Vortex Wakes

4.1 The Concept of the Line Vortex

4.2 Vortex Concepts Applied to Fixed Wings

4.3 Lifting‐Line Theory Applied to Flapping Wings

4.4 Wind Tunnel Studies of Bird Wakes

4.5 Feathered Wings

Chapter 5 The Feathered Wings Of Birds

5.1 General Structural Requirements

5.2 Mechanics of the Bird Wing

5.3 Flapping the Wings

5.4 The Rest of the Skeleton

5.5 Adaptations for Gliding

Chapter 6 The Membrane Wings of Bats and Pterosaurs

6.1 Bats

6.2 Pterosaurs

6.2.1 Mechanics of the Pterosaur Wing

6.2.2 Tensioning the Pterosaur Wing Membrane

6.2.3 The Trailing-Edge Tendon and the Fifth Toe

6.2.4 Mechanics of the Wing Finger

6.2.5 Large and Giant Pterosaurs

6.2.6 Water Pterosaurs?

Chapter 7 Muscles as Engines

7.1 General Requirements

7.2 The Sliding Filament Engine

7.3 Muscle Performance in Locomotion

7.4 Adaptations for Aerobic Flight

Chapter 8 Simulating Long‐Distance Migration

8.1 Estimating Range

8.2 Ultra Long‐Distance Migrants

8.3 The Concept of Energy Height

8.4 Effect of Flying Height on Range

8.5 Aerobic Capacity and Climb

8.6 Basal Metabolism

8.7 Water Economy

8.8 Sleep

Chapter 9 Accelerated Flight and Manoeuvring

9.1 Intermittent Flight Styles in Flapping Flight

9.2 Manoeuvring Frame of Reference: Flight Controls

9.3 Transient Manoeuvres

Chapter 10 Gliding Flight and Soaring

10.1 Gliding Performance

10.2 Soaring

10.2.3 Sea‐anchor Soaring in Storm‐petrels

Information Systems for Flying Animals

11.1 Senses

11.2 Orientation and Navigation

Chapter 12 Water Birds

12.1 Waterproofing And Thermal Insulation

12.2 Mechanics Of Swimming

12.2.2 Lift‐Based Foot Swimming

12.2.3 Wing Swimming

12.3 Morphological Trends In Waterbirds

12.3.2 Why Are Penguins Unable To Fly?

12.3.3 Wing Enlargement In Frigatebirds

12.3.4 Reduced Aspect Ratio In Foot Swimmers

12.4 Other Aquatic Adaptations

12.4.2 Landing On A Water Surface

12.4.3 Vision Under Water

12.4.4 Heat Disposal In Water

Chapter 13 Allometry

13.1 Allometry of Morphological Variables

13.2 Allometry of Calculated Variables

13.3 Variations on Allometry

Chapter 14 Wind Tunnel Experiments With Birds And Bats

14.1 Wind Tunnel Basics

14.1.2 Basic Flow Principles

14.2 Wind Tunnel Layouts

14.3 Wind Tunnel Components and Their Functions

14.3.5 Why Does Turbulence Matter?

14.4 Birds in Wind Tunnels

Chapter 15 Theory as the Basis For Observation

15.1 Flight Speed Measurements

15.2 Wind Tunnel Results Related to Field Studies

15.3 Wingbeat Frequency

15.4 The Theoretical Backbone

Chapter 16 Evolution of Flight

16.1 Evolution in Engineering and in Nature

16.2 Past the Squirrel Barrier

16.3 Evolution of the Bird Wing

16.4 Adding an Engine

16.5 Size Restrictions

16.6 Time Scale of Evolution

References

Index

Preface

Colin Pennycuick

Abstract

Publisher Summary

This chapter presents a brief preview of the subjects discussed in the book, Modelling the Flying Bird. Being an interdisciplinary activity, computer modeling of bird flight tends to fall into the chasm between ornithology and engineering. Ornithologists mistrust calculation, while engineers think bird-watching is frivolous. The barrier to communication between ornithologists and aeronautical engineers is because of their different attitudes to numbers. Biologists readily accept that the rate at which a bird needs energy to support its weight in air might be correlated with the wing span, but balk at the idea that this measurement (the distance between the wing tips) actually determines the power requirement and can be used to calculate it for any bird, without the need to measure power or run regressions. There is actually no way to use statistical methods to predict the power requirements of even one species because several variables are involved. These include the wing span, forward speed, strength of gravity, and density of the air and each of them affects the power in different ways.

Being an interdisciplinary activity, computer modelling of bird flight tends to fall into the chasm between ornithology and engineering. Ornithologists mistrust calculation, while engineers think birdwatching is frivolous. It may seem obvious that aeronautical theory can be adapted to cover bird flight, but when I first attempted to do that, it was seen in ornithological circles as an eccentric activity, with little or no practical use. My earlier book Bird Flight Performance was politely received but biologists were unconvinced that they needed it. The present book, which is backed by a far more capable computer programme, is a fresh attempt to show why a physical theory is necessary as a framework for any quantitative discussion of animal flight.

The barrier to communication between ornithologists and aeronautical engineers is due to their different attitudes to numbers. Biologists readily accept that the rate at which a bird needs energy to support its weight in air might be correlated with the wing span, but balk at the idea that this measurement (the distance between the wing tips) actually determines the power requirement, and can be used to calculate it for any bird, without the need to measure power or run regressions. There is actually no way to use statistical methods to predict the power requirements of even one species, because several variables are involved. These include the wing span, the forward speed, the strength of gravity, and the density of the air, and each of them affects the power in different ways. All of this, and much more, is represented in classical aeronautical theory, of which the relevant parts have been exhaustively tested over the last hundred years, and I have built the Flight programme on this foundation.

Ornithologists sometimes want to use the traditional wing length as a substitute for the wing span, but this will not do. The power estimates are not correlations, but absolute numbers, calculated from Newtonian mechanics, and the right input numbers have to be used. The requirement to be aware of the definition of each variable and its physical dimensions is obvious to engineers, but less so to those who have been accustomed to relying exclusively on statistical methods. A statistical package looks for patterns in sets of numbers, and will usually produce a result whatever the numbers mean, or even if they mean nothing at all. The difficulty that many biologists seem to have with aeronautical theory is not in understanding the theory itself, but in adjusting their attitude to numbers away from statistics, and towards the engineering point of view.

Once this difficulty has been overcome, using the Flight programme is easy. Users who study the output from its simulations of long-distance migration (Chapter 8) will see a level of detail that statistics-based ecologists cannot even begin to dream about, and some may be rightly sceptical that so much can be calculated from so little in the way of input. The programme has been designed to make it easy to set up and test hypotheses that reflect the underlying assumptions, and it is for experimenters and field observers to determine what level of confidence in its predictions is justified. This testing process is currently being transformed by the ever-increasing capabilities of satellite-trackable transmitters that can be carried by birds, but many kinds of training experiments, in wind tunnels or aviaries, can also be used to test the programme (Chapter 15). It remains important to keep a close connection between the numbers and the real world of the flying bird, and the best way to keep that in focus is to learn to fly oneself.

Bristol, December 2007

Foreword

C.J. Pennycuick

In Larry McMurtry's novel Comanche Moon, the Kickapoo tracker Famous Shoes, who can track anything over any terrain, is musing over his solitary camp fire somewhere in Texas, circa 1861, listening to the geese migrating overhead in the starlight:

The mystery of the northward‐flying geese had always haunted him; he thought the geese might be flying to the edge of the world, so he made a song about them, for no mystery was stronger to Famous Shoes than the mystery of birds. All the animals that he knew left tracks, but the geese, when they spread their wings to fly northward, left no tracks. Famous Shoes thought that the geese must know where the gods lived, and because of their knowledge had been exempted by the gods from having to make tracks. The gods would not want to be visited by just anyone who found a track, but their messengers, the great birds, were allowed to visit them. It was a wonderful thing, a thing Famous Shoes never tired of thinking about. … Many white men could not trust things unless they could be explained; and yet the most beautiful things, such as the trackless flight of birds, could never be explained.

People do not fly, obviously, but not all white men in Famous Shoes' time knew this. A few years later two of them, Orville and Wilbur Wright, found out how to fly, and now anyone can learn to do it, with a little effort and perseverance. By living at the right time, my luck has included personally migrating across the Nubian Desert in a Piper Cruiser, and across the Greenland ice cap in a Cessna 182, both busy routes for migrating birds, and that is indeed a wonderful thing. I have migrated into Sweden with the cranes in that same Cruiser, and soared with storks and vultures over the Serengeti in a Schleicher ASK‐14. Actually, birds do leave tracks in the air. They do not last long, but a skilled tracker can read them (Chapter 4). Eat your heart out, Famous Shoes. We may never know where the gods live, but some of the things that birds do can be explained and understood, especially if we do them ourselves, and this book is the song that I have made about it.

Acknowledgements

Colin Pennycuick

Abstract

Publisher Summary

This chapter presents an acknowledgement by the author to various other people, such as Tom Lawson and John Flower of the Aeronautical Engineering Department, Reg Moreau, Peter Evans, and others, for their help in compiling the material of the book, Modelling the Flying Bird.

I am confident that the theory behind the Flight programme is right, because I have been in the habit of entrusting my life to it as a pilot, and at the time of writing I am still alive. I first learned about the theory of flight from those iron-nerved RAF instructors who taught me (a Zoology graduate) to fly Chipmunks, Provosts and Vampires so many years ago. Their efforts were reinforced, when I later became a gliding instructor myself, by pupils who required me to explain and demonstrate how gliders fly, so forcing me to understand the theory on an intuitive level. When I joined Bristol University as a Zoology lecturer in 1964, my aeronautical education took a more formal turn thanks to Tom Lawson and John Flower of the Aeronautical Engineering Department, who helped me to build a wind tunnel in which pigeons could fly. The pigeons soon demonstrated that aeronautical principles do indeed apply to birds, and I got my first opportunity to convince biologists about this soon afterwards, thanks to the broad-minded Reg Moreau, who was editor of Ibis in 1969, and Peter Evans who reviewed my somewhat unconventional manuscript. Naively, I supposed that ornithologists would seize eagerly on the revelations in the paper, and this book is my latest attempt to convince them of the advantages of the physics-based approach.

I owe my introduction to studying the flight of wild birds in the field to Hugh Lamprey, then director of the Serengeti Research Institute, who took a glider to the Serengeti in 1968 and let me fly it, and to Hans Kruuk and Tony Sinclair, who taught me how the Serengeti ecosystem works. A later motor-glider project in the Serengeti supplied the background for the gliding section of the Flight programme, and led to a project with Thomas Alerstam in Sweden, in which we followed migrating cranes in my Piper Cruiser. My informal association with Lund University has continued, and reached a high point when Thomas, having risen to be head of department, set up the Lund wind tunnel in the new Ecology Building in 1994. Down in the South Atlantic, John Croxall taught me what I know about albatrosses and the Southern Ocean ecosystem during two memorable trips to South Georgia with the British Antarctic Survey in 1980 and 1994. Meanwhile, a long-term collaboration with Mark Fuller, which started while I was based at the University of Miami and still continues, led to a series of field and laboratory projects in various parts of the USA and the Caribbean, which laid the groundwork for Flight’s simulation of long-distance migration.

I am extremely grateful to Geoff Spedding for reading and commenting on drafts of the earlier chapters, especially the parts relating to his own remarkable contributions to the study of bird wakes, to Ulla Lindhe Norberg for a similarly expert review of the chapter on bats and pterosaurs, and to Julian Partridge for reviewing the chapter on information sources and commenting on the rest of the book. Literally hundreds of biologists, aviators, students, professors and others in Britain, Sweden, Africa, America, the Caribbean, the South Atlantic and other places have educated me about different aspects of flight, and thus contributed to the book, wittingly or not. I am deeply grateful to them all, and if I have got it wrong, the fault is mine alone. As always, I have depended on the support and forbearance of my wife Sandy and son Adam to make this project possible. The book is dedicated to the doctors and staff of the Bristol Oncology Centre and Southmead Hospital, Bristol, without whose intervention I would not have lived to write it.

Bristol, December 2007

Chapter 1 Background to the Model

C.J. Pennycuick

Abstract

The Flight computer model, which calculates the rate at which a flying animal requires energy for whatever it is doing, is based on classical aerodynamics. This is itself a branch of Newtonian mechanics, which is basically the same for aircraft and birds. Calculating the mechanical power requires information about wing measurements, which are defined in this chapter. The physiological requirements for fuel and oxygen are calculated as a second step, from the mechanical requirements. This approach requires care with the physical dimensions of variables, introduced in this chapter.

My objective in writing this book is to understand what a bird does when it flies, to explain in physical terms how it does it and to provide tools that can be used to predict quantitatively what any bird (not just those that have been studied) can and cannot do. The quest is ambitious but not new. Would‐be aeronauts have studied the wings of birds with great care down the centuries, hoping to understand them well enough to copy them, and fly themselves. With hindsight we can see now why Otto Lilienthal's meticulous studies and drawings of the wings of storks (Lilienthal, 1889) produced disappointingly little at the time, by way of insight into how wings work. His difficulty was that he had no theory in the 1880s with which to describe and explain what he saw. Now we have theory aplenty, thanks to the efforts of the world's aeronautical research institutions, and it is time for us birdwatchers to turn the process around, and look at birds through the new eyes that aeronautical engineers have given us.

The book is descriptive in parts, especially in the chapters that introduce the wings of flying vertebrates, but these descriptions will look strange to many biologists, because the conventions of morphology are hopelessly inadequate for describing how wings work. It is not possible to explain what wings do, without introducing concepts that are not a traditional part of a biologist's education. This chapter introduces the aeronautical conventions for describing and measuring wings, adapted for birds, and Chapter 2 is about the characteristics of the environment in which birds fly. Chapters 3 and 4, about the principles of flight, introduce a number of concepts that are familiar to engineers, but not to most biologists, and attempt to give the biological reader an intuitive feel for what these ideas mean. Chapters 5 and 6 describe the wings of birds, bats and pterosaurs, and Chapter 7 is on muscles seen as engines. After that the scope broadens to cover such topics as the simulation of long‐distance migration, gliding and soaring, the sensory requirements of flight, the use of wind tunnels and the design of experiments on flight. The evolution of flight comes last, because it is not possible to understand how it happened, without invoking the mechanical principles covered in earlier chapters.

1.1 The Flight Model

The skeleton of the book is the Flight computer model, a programme that incorporates the concepts introduced in the book, and allows the user to apply them to a chosen bird to answer questions about speed, distance, energy consumption and suchlike performance matters. Flight is not a model of a particular bird, nor is it based on regressions describing direct measurements of the quantities that it calculates. It is essentially a set of physical rules which are assumed to be general, in the sense that they can be applied to any bird, real or hypothetical, for which the user can provide the measurements required to define the bird and its environment. Flight accepts the user's input describing the bird, and provides a variety of options that determine the assumptions to be made in the calculation. Then it predicts how the bird's performance in flapping or gliding flight, or in long‐distance migration, would follow from that particular set of assumptions. It is designed in a way that makes it easy to vary the starting assumptions, which can be seen as hypotheses about how the bird works, and immediately observe the effect of a changed assumption on the predicted performance.

Flight is, in effect, a working model of a bird, according to the theory given in the book. It comes with its own online manual and databases of bird measurements, which can be loaded directly into the programme. The book contains many examples that have been calculated with Flight, showing how the output follows from the assumptions that underlie the programme, and how it can be used to test hypotheses about how the bird works. Flight is available as a free download from http://books.elsevier.com/companions/9780123742995, and also from http://www.bio.bristol.ac.uk/people/pennycuick.htm. These websites are updated from time to time with the latest version of the programme.

1.1.1 The Mathematical Idiom

It is easiest to explain what Flight does, and the concepts underlying it, in the idiom of aeronautical theory on which it is based, that is, in the language of applied mathematics, but this takes a little getting used to, and it is a known fact that many biologists are somewhat resistant to it. I have tried to make the book accessible to readers who are averse to equations, by structuring each chapter with an equation‐free main text that explains what the topic of the chapter is about, and isolating the more technical aspects in boxes. I hope that the main text will convey the gist of the argument to mathematical and non‐mathematical readers alike, while those who want to know what Flight actually does will find the equations in the boxes. Each box that presents a mathematical argument contains its own local variable list, which applies within that box, but not necessarily elsewhere in the book. The conventions for notation and so on are outlined in Box 1.1 in this chapter. Not all the boxes are mathematical. Some deal with the implications of a particular published experiment, an anatomical digression or some other limited topic.

Box 1.1 Mathematical conventions

Variable names in this book follow the usual conventions of physics, to the extent that a variable name is a single letter, with subscripts to distinguish between different variables of the same physical type. Variable names are italicised, but subscripts are not. For example, the letter P (for Power) is used to stand for a number of different variables that have the physical dimensions of work/time. Subscripts distinguish different kinds of power from each other. Pmech, the mechanical power produced by a bird's flight muscles, and Pchem, the rate at which the bird consumes chemical energy from fuel, are different variables with the same dimensions. Lower case p is used for specific power, a related group of variables with different dimensions, power/volume for volume‐specific power (pv), and power/mass for mass‐specific power (pm).

Acronyms are not used as variable names, because they look like several variables multiplied together. BMR is a familiar acronym that is mentioned in the text, but it is not used as a variable name, because it looks like B times M times R. Basal metabolic rate is a variable with the dimensions of power, and it is denoted by Pbmr. A notable exception to the one‐letter rule is that two‐letter variable names are traditionally used in engineering for dimensionless numbers named after famous scientists, notably Re for Reynolds number. Like other variables, Re can be subscripted to distinguish Rewing from Rebody.

Capital "B" for wing span

The use of particular symbols to represent particular variables is a tradition that builds up over time, but it is not a law. The law, which applies internally in boxes in this book, but not always globally throughout the book, is that the definition of every symbol must be stated in the local context. It is legal (if not always helpful to the reader) to assign any letter you like to a physical variable, regardless of tradition. It sometimes happens that more than one tradition develops in different areas of science, and this can cause serious confusion. A particularly awkward example is lower case b, which is traditionally used in aeronautical engineering to denote an aircraft's wing span, the distance from one wing tip to the other. This is the width of the swathe of air that the wing influences as the aircraft or bird flies along, and it is the most important morphological measurement for performance calculations. However, there is another tradition, within aeronautics, in which fluid dynamics theorists consider the air flow around a wing by starting at the centre line, and working outwards to the wing tip. The other wing is not very interesting from this point of view, being merely a mirror‐image of the first, and unfortunately it has become traditional in this area of theory to use the same symbol b for the semi‐span. The Flight programme comes from the "b for wing span" tradition, but in recent years, the fluid dynamics tradition has been the source of major advances in wind tunnel studies of bird flight (Chapter 4), in which b denotes the semi‐span . Ironically, the two traditions have coexisted peacefully in their homeland, aeronautical science, for three‐quarters of a century, but now that both have invaded biology from different directions, there is conflict. The same formula may appear from different sources, apparently differing by a factor of 2 (or 4 if it involves the square of the wing span).

In the hope of reducing the confusion, I have broken with tradition in this book, and used capital B for the wing span, avoiding the use of lower case b for anything. If others would just refrain from using capital B for the semi‐span , this might at least eliminate conflicting definitions of the same symbol. The reader may be wondering why S should not be used for wing span. The answer, unfortunately, is that S traditionally denotes area in all areas of aeronautics. S for span would cause even worse confusion.

1.1.2 Describing the Bird

It is not practical to describe what every feather and every muscle does when a bird flies. Any model of a bird, whether it is constructed by a programmer or an artist, is limited to those aspects of the original that the chosen medium can realistically represent. The objective of this computer model is to predict as much as possible about the bird's capabilities, from as few assumptions as possible. The description of a particular bird needs to include only those measurements that determine the forces acting on it in level or gliding flight, and neglects other information that would complicate the calculation, without producing a useful improvement in the scope or accuracy of the predictions.

In Flight, a bird is described by only three numbers, its mass, wing span and wing area. That may seem a rudimentary description, and so it is. Not even the most clueless birdwatcher would confuse the American Turkey Vulture with the Great Blue Heron, but they are the same bird as far as Flight is concerned. I shall show in subsequent chapters that despite the minimal amount of input information that Flight needs about the bird, the programme predicts a surprisingly wide variety of measures of flight performance. The reader who wishes to test the accuracy of these predictions against field or laboratory observations need only enter the bird's mass, wing span and wing area into the programme, and run it. Rudimentary as these measurements may be, they are unfortunately not to be found in the traditional morphometrics of ornithology, and they cannot be reliably determined from museum specimens. The definitions come from aeronautics, not from ornithology, and are given in Boxes 1.2–1.4 of this chapter, together with the measurement procedures. These procedures are not difficult or arduous, but they may be unfamiliar to some biologists, and they need to be carefully followed.

Box 1.2 Body mass and its subdivisions: Definitions

The concept of lean mass is not used in Flight. This is an obsolete term that refers to everything that is not fat, including the flight muscles. It was originally conceived as a constant baseline against which other masses, including the fat mass, could be compared, but this became untenable when it was realised that large quantities of protein from the flight muscles are consumed during long migratory flights, and smaller amounts from the airframe. These changes are predicted in Flight's migration calculation.

Flight considers that a bird's empty mass consists of three components, the flight muscle mass, the fat mass and the airframe mass, which is the mass of everything else in the body, that is not flight muscles or consumable fat. All three components are reduced by substantial amounts in the course of a long migratory flight, for different reasons, and this is represented in the computation. The fraction corresponding to each component is the mass component divided by the all‐up mass.

List of variables defined in this box

All‐up mass (m)

The total mass of everything that the bird has to lift (just weigh the bird), including any hardware such as rings and radio transmitters. The all‐up mass, together with the strength of gravity (Chapter 2), determines the amount of power required from the flight muscles to support the weight.

Empty body mass (mempty)

The all‐up mass, measured with the crop empty. This dates from the early development of Flight, when birds carrying heavy loads of food in their crops happened to be a subject of special interest.

Crop mass (mcrop)

Wet mass of the crop contents, if any.

mcrop is normally assumed to be zero on migratory flights.

Fat mass (mfat)

The mass of stored fat that is available to be used as fuel.

Fat fraction (Ffat)

The ratio of the fat mass to the all‐up mass (NOT to the lean mass!). The starting fat fraction is directly related to the distance a migrating bird can fly before it runs out of fat, and this (not the fat mass as such) is the number that is needed to represent the stored fuel energy in migration calculations (Chapter 8).

Flight muscle mass (mmusc)

The combined wet mass of the wing depressor and elevator muscles of both sides. In birds, these are the pectoralis and supracoracoideus muscles.

Flight muscle fraction (Fmusc)

The ratio of the flight muscle mass to the all‐up mass.

Note that as a bird takes on or consumes fat, it also builds up or consumes its flight muscles. The flight muscle mass is greater when a bird is fat than when it is thin, but the flight muscle fraction varies much less, whether the bird is fat or thin.

Airframe mass (mframe)

The mass that is left after subtracting the fat mass and the flight muscle mass from the empty mass. The airframe is perceived as the basic structure of the bird, which has to carry the engine (flight muscles) and the fuel (fat), although actually a small part of the airframe also gets consumed on migratory flights.

Airframe fraction (Fframe)

The ratio of the airframe mass to the all‐up mass.

The three mass fractions change progressively during a long migratory flight, but they always add up to 1:

Entering masses into Flight

First enter the empty mass. This is what you get by weighing the bird with its crop empty. If the effects of carrying a crop load are not important to your calculation, you can consider the crop contents to be part of the airframe. In that case set mcrop to zero (the default), and set mempty to the mass that you get by weighing the bird, including any crop contents.

Next, enter the fat mass. The programme will automatically calculate and enter the fat fraction. Alternatively, if you enter the fat fraction first, the programme will calculate and enter the fat mass. Likewise, enter either the flight muscle mass or (preferably) the flight muscle fraction.

To fatten up a computer bird, first enter a higher value for the empty mass, then increase the fat mass by a lesser amount (because additional flight muscle mass is added as well as fat). This is not taken care of automatically by the programme. It is best to use field data for the empty mass and fat fraction of heavy pre‐migratory birds. In some circumstances it is possible to estimate the fat fraction from measurements of body mass alone, without resorting to carcase analysis (Chapter 8, Box 8.4).

Box 1.3 Wing measurements: Definitions

The only two wing measurements that are required by Flight are the wing span and the wing area. In addition, there are a number of related variables that are mentioned in the text and calculated by the programme, whose definitions are given below.

Variables defined in this box

Wing span (B)

A bird's wing span is the most important morphological variable for flight performance calculations. It is the distance from one wing tip to the other, with the wings at full stretch out to the sides, that is, with the elbow and wrist joints fully extended (Figure 1.1A). Wing span was denoted in my own earlier publications by lower case b, following the most usual aeronautical convention, but this has led to some confusion as some authors from the theoretical fluid‐dynamics tradition define lower case b as the semi‐span. This usage occurs in both the aeronautical and the ornithological literature, and is liable to cause major misunderstandings and errors. Hoping to minimise this problem, I denote wing span by capital B in this book, thus breaking with both traditions.

Wing area (Swing)

The wing area, denoted by Swing, is essentially the area that supports the bird's weight when it is gliding. It is defined as the area, projected on a flat surface, of both wings, including the part of the body between the wings (Figure 1.1A). Why include part of the body? Because the bird is supported in normal gliding flight by a zone of reduced pressure which extends from one wing tip to the other. There is no gap in the middle (Figure 1.1B). Measuring the wing area is more complicated than measuring the span, more stressful for the bird and harder to do repeatably. On the other hand, this is a less critical measurement. The wing area is important in gliding performance, because it determines gliding speeds, and also the minimum radius of turn for circling in thermals. However, minor changes in the wing area have only a small effect on performance in flapping flight (Spedding and Pennycuick, 2001).

Chord (c) and mean chord (cm)

Chord is an aeronautical term that dates from the nineteenth century, when people built thin wings, with cross sections that were arcs of circles. Modern aircraft wings are not thin arcs in cross section, but the chord is still the distance from the leading edge of the wing to the trailing edge, measured along the direction of the air flow (Figure 1.1A). Ornithological readers will be aware that this term was borrowed at some time in the past for use in bird morphometrics, and assigned a meaning that is unrelated to its aeronautical definition, and of no use for flight performance calculations of any kind. The conventional aeronautical definition of chord is the only one used in this book.

The chord of a particular wing, unlike its span, does not have a unique value unless the wing is rectangular, which is unusual. Most wings have a maximum root chord where the wing joins on to the body, and taper to a smaller tip chord, with the chord diminishing along the span. A few flying animals (butterflies) have negative taper, meaning that the tip chord is greater than the root chord. The mean chord (cm), which does have a unique value for the wing, is the ratio of the wing area (Swing) to the wing span (B):

     (1)

Flight calculates the mean chord internally, and uses it for calculating Reynolds numbers (Chapter 4, Box 4.3) and reduced frequencies (Chapter 4, Box 4.4).

Aspect ratio

The aspect ratio (Ra) is the ratio of the wing span to the mean chord, and it expresses the shape of the wing:

     (2)

or, more conveniently:

     (3)

Wing area is somewhat troublesome to measure (Box 1.4) and not as critical as wing span. If a few wing areas are measured among a sample of birds of the same species, they can be used to get an estimate of the aspect ratio, which may be assumed to be constant for the species. This means that the wings are assumed to be all of the same shape, though not necessarily the same size. Then, if a bird's span has been measured, the aspect ratio can be used to estimate its area by inverting Equation 3:

     (4)

Flight will accept either the wing area or the aspect ratio for input. If supplied with one, it will calculate and enter the other automatically, so long as the wing span has already been supplied.

Tail area

The tail is an accessory lifting surface in birds, and is more analogous in its function to a flap than to the horizontal tail of conventional aircraft. Birds' tails have been represented as an expandable delta wing, behind the main wing (Thomas, 1993). This is not included in Flight as most birds only deploy and use their tails at low speeds that are below the range covered by the calculations, and besides, the theory is somewhat conjectural. The tail is usually furled at normal cruising speeds, from the minimum power speed up, and may then be assumed to contribute no lift.

Figure 1.1 (A) Definitions of basic wing measurements. The wing span is the distance from wing tip to wing tip, and the wing area is the projected area of both wings, including the body between the wing roots (grey). These measurements are made with the wings fully extended. It is important that the elbow joint is locked in the fully extended position. The chord, which varies from point to point along the wing, is the distance from the leading edge of the wing to the trailing edge, measured along the direction of the relative air flow. (B) A gliding bird's weight is balanced by the pressure difference between the lower and upper surfaces, multiplied by the wing area. The area of reduced pressure above the wings accounts for most of this pressure difference, and it continues across the body. This is why the area of the body between the wing roots is included in the wing area.

Box 1.4 Procedures for measuring wings

Measuring wing span

There are two ways to measure the wing span, both of which are quick and easy to do on a live bird, with minimal stress. For a small bird, with both wings in good condition, place the bird on a flat surface, the right way up (not on its back). Stretch both wings out to the sides as far as they will go, with the tips on the surface, and check that the elbow and wrist joints are in their fully extended positions. Place markers, just touching each wing tip. Then fold the wings up, remove the bird, and measure the distance between the markers.

The other way is to measure the semi‐span , which is usually easier for large birds. This is the only option if one wing is damaged. Stretch the good wing out as above, and use a tape measure to determine the distance from the backbone to the wing tip. This is the semi‐span. Double it to get the span. The measurement is made from the body centre line (not the shoulder joint) to the wing tip. The centre line is easy to locate by feeling for the neural spines of the vertebrae, which stand up from the backbone as a sharp ridge. It is important to make sure that the elbow joint is fully extended, by pushing it gently forward until it locks.

Measuring wing area

The wing area is measured in two stages. First make a tracing of one wing (not forgetting to measure the wing span), and then measure the area from the tracing. A wing tracing that is not accompanied by a wing span measurement is completely useless, and cannot be used for measuring wing area. The best idea is to write all the data about the bird, including the span, directly on the wing tracing. Wings of small birds can be traced in a sketchbook that opens flat, while a roll of parcel paper is good for large birds.

Tracing the wing

Put the drawing surface at the edge of a table, and hold the bird with one wing spread on the drawing surface, and its body beside the table edge, but not actually on it (Figure 1.2A). Spread the wing straight out to the bird's side, with the elbow and wrist joints fully extended. Do this with the bird right‐way up, not on its back. Find the elbow joint (quite close in to the side of the body), and push it gently forwards until it locks in the fully extended position. Then draw the outline of the wing, following in and out of the indentations between the flight feathers. This results in a partial wing, which is incomplete (open) at the inner end.

Finding a partial wing area from the tracing

First complete the partial wing tracing by drawing a straight line across the open end, parallel to the body centre line. This is the wing root line. Its exact position is not critical, so choose a position that gives a realistic root chord (defined in Box 1.2). The first job is to measure the area of the partial wing. Of course, there are digital ways of doing this, and it may be worth the trouble of setting one up, if you have hundreds of small wings to measure. If you have to measure occasional warblers, ducks, pelicans etc., low‐tech methods are easier, quicker, less error‐prone and just as accurate if not more so.

First use a drawing programme to make a rectangular grid of 1 cm × 1 cm squares (or 0.5 cm × 0.5 cm for small birds). Number the lines along all four edges. Print the grid out on acrylic sheet as used for overhead transparencies, and check that the line spacing is indeed what it is supposed to be. Lay the grid over your wing tracing, aligning one edge with the wing root line, as shown in Figure 1.2B. Line up the leading edge of the wing so that it roughly corresponds with one of the horizontal grid lines. Starting from the left edge of the grid in Figure 1.2B, the third row of squares contains 8 full squares (allowing for the fact that the leading edge wanders up and down across the grid line), and some partial squares beyond column 8. If the filled parts of columns 11 and 12 were flipped over, they would fit in the unfilled parts of columns 9 and 10, making two complete squares beyond column 8. That makes 10 filled squares for the first row of the partial wing (row 3). Row 4 has a bit more than 11 filled squares, and row 5 has a bit less than 11, so count them as 11 each. Row 6 has about 8 filled squares, and all the small parts of the trailing edge in row 7 add up to about 1 filled square. That makes 41 filled squares in all for the partial wing. If the bird is so small that you get less than 100 squares in the partial wing, then it is better to use a grid with smaller squares, so that you have a chance of measuring the area within 1%. That is ample precision for the wing area measurement. In practice, 0.5 cm × 0.5 cm squares are good for small passerines, and 1 cm × 1 cm squares suffice for bigger birds.

Completing the wing area measurement

Although the squares in Figure 1.2B are bigger than ideal for the size of the bird, that will not stop us from completing the wing area calculation. We now know that the partial wing (the grey area) is 41 cm², but this is not the whole area for one side. You have to extend the root end of the wing to the centreline, by adding a root box. First measure the root chord on the tracing, along the wing root line which you marked in. Then measure the partial wing length, which is the distance from the wing root line to the tip of the longest primary. You already know the semi‐span , having measured it directly on the bird. The width of the root box is the difference between the semi‐span and the partial wing length (1.5 cm in Figure 1.2B), and the length of the box is 4.4 cm, the same as the root chord. The area of the root box is therefore 1.5 cm × 4.4 cm = 6.6 cm². You can now work out the wing area as follows:

The best place to do this little calculation is on the tracing, right beside the partial wing. Flight wants the wing span (B) in metres (divide cm by 100) and the wing area (Swing) in square metres (divide cm² by 10⁴). While you are at it, work out the aspect ratio as well (B²/Swing), and write all three results on the tracing:

B = 0.258 m

S = 0.00952 m²

Ra = 6.99

An aspect ratio near 7 means that the wing is shaped about like the one in Figure 1.2. If we had got a ridiculous aspect ratio of 70 or 0.7, that would alert us to a mistake in the calculation.

Notice that the measured wing area is not very sensitive to the exact position where you draw the wing root line, to complete the partial wing. If you move the wing root line outwards a bit, you get a smaller partial wing, but this is compensated by a bigger root box, and vice versa. Little or no subjective judgement is required by this method of measuring wing areas, and it is consequently very repeatable between different observers.

Entering wing measurements into Flight

The wing span must be entered first (in metres), and this should be a first‐hand measurement—never a guess, or an estimate from some dubious published regression, or a quote from a field guide. The wing area is a less critical measurement. If you have a measured value, then enter it (in square metres). The programme will automatically calculate the aspect ratio, and display it in the box. Check that it is a believable value, and if not, look for wrong units or spurious factors of 10 in the entered wing span and area.

Sometimes you have a good value for the wing span (essential), but no measured wing area. In that case, you can enter the aspect ratio, if you can guess it from other birds that you have measured, whose wings are similar in shape. The programme will then calculate and enter the wing area.

Figure 1.2 (A) A bird's wing area is measured from a tracing of one wing, fully spread over a drawing surface. The root end of the wing is left open at a point that is representative of the root chord. (B) The tracing is closed by ruling a straight wing‐root line parallel to the body centre line, and the enclosed area (grey) is measured by counting squares on a transparent grid laid over it. The root box extends the wing root to the centre line (backbone), and the combined area is then doubled to get the total wing area (see Box 1.4).

The programme will be misled by numbers that mean something different from what it assumes, which is not unusual for numbers identified by the same names in the ornithological literature. It serves no useful purpose for a field or laboratory observer to collect infinitely detailed statistics on variables that do not affect flight performance, and then get wing spans and areas (which do) from bird field guides, museum specimens or published figures from authors who neglected to define exactly what their measurements mean. Body mass is straightforward, but the manner in which the programme subdivides it (Box 1.2) needs to be understood when calculating migration performance. In particular, the concept of lean mass is not used in this book or in Flight, because its use in migration studies is obsolete and misleading. Mass fractions are defined in Box 1.2 for components such as the stored fat and flight muscles, and these are the ratio of the mass component to the all‐up mass, not to the lean mass.

Bats, pterosaurs and even mechanical ornithopters can be described by their mass, wing span and wing area, and Flight will predict their performance, interchangeably with birds. For such non‐birds (and birds with oddly shaped bodies), it may be necessary to adjust the values of some non‐morphological variables, which are set to default values by the programme, but can be changed in the setup screens for different calculations. The reader should not be intimidated by the number of variables that can be adjusted, or by the somewhat arcane nature of some of them. The defaults will do for most practical purposes, but if one such variable (a drag coefficient for instance) is suspected to be the source of an observed discrepancy, it is easy to change the value systematically through several programme runs, keeping all other values the same. The results can be saved as an Excel Workbook, in which the results of each run are saved as a new Worksheet, together with the input from which they were generated. The meanings of those variables that are accessible to the user, and the effects and implications of tweaking their values, are explained in later chapters, and in Flight's online manual.

1.1.3 Describing the Flight Environment

Besides the three morphological variables that describe the bird, Flight also requires values for two further variables (only) that describe the environment in which the bird flies. These are the acceleration due to gravity and the air density, both of which have a major effect on Flight's performance predictions. These variables are discussed in Chapter 2, with methods of entering values into Flight. A default value is used for gravity, but this can be changed by the reader who wants to simulate flight elsewhere than here on earth. Air density is often overlooked or ignored by biologists, although not by pilots, who are acutely aware of its effects on flight performance. These effects also apply to birds, and it is essential to supply a realistic value. There is no default value for the air density, and Flight will not run until the user selects one of a number of options. For example, the programme will calculate and enter the air density if the user supplies measured values of the ambient pressure and temperature, or it will calculate a hypothetical value that corresponds to a specified height in the International Standard Atmosphere (Chapter 2).

1.2 The Engineering Approach to Numbers

1.2.1 Calculation as Opposed to Statistics

In physiology, if you want an estimate of the rate of fuel consumption, then you have to measure it directly, or else measure something that you hope is proportional to it, like the rate of oxygen consumption. The result comes out in whatever units happen to be inscribed on the apparatus, such as watts, British Thermal Units per hour, calories per minute or even millilitres of oxygen per hour. If you only have to deal with one type of quantity, an arbitrary choice of units is fine for collecting statistics fodder, and may even serve for very basic calculations, but that is not Flight's approach. The programme does not get its power estimates from regressions based on data of this type; in fact it does not use regressions at all. Instead, it estimates the power from other variables with other dimensions, basically the force that the wings apply to the air, and the speed with which they move. The force in turn comes from the rate at which momentum (mass times speed) is added to the air flowing over the wings. Flight then goes on to assume that the power estimated from force times speed must be accounted for by the rate of consumption of fuel energy. The calculation does not depend on any direct measurements of power as such. Unnatural as it may seem to many biologists, no statistics are involved.

The vast literature about measured rates of energy consumption in birds gets barely a mention in this book, because total fuel consumption is the end result of all processes that consume energy. A statistical summary of measurements of this type, on some particular bird, can be used to predict the energy consumption of the same bird, but cannot be transferred to other birds, flying under other conditions. The conditions include the air density, a fact that would be difficult to account for statistically, and seems to be unknown to most physiologists anyway. Flight works in the opposite direction from physiological experiments. It starts by simulating the underlying physical processes that result in a requirement for fuel energy, estimates the contribution of each to the fuel requirement, and adds in other assumed requirements (like basal metabolism) to estimate the total fuel consumption. Physiological experiments can be used to test whether the programme's predictions are accurate, but only if the required morphological measurements are carefully made and tabulated, and the local air density during the experiment is measured and recorded.

1.2.2 Theory Tells You What to Measure

If the model predicts that a bird can do something, which you know from observation that it definitely cannot do, that is a discrepancy which needs to be resolved by identifying and amending some wrong value in the input data, or possibly an error in the structure of the model itself. The resolution of discrepancies allows this type of model to be progressively improved and refined, so that greater confidence can be attached to its predictions. The time to think about this is at the planning stage, by examining the output that Flight generates, and using it to determine what measurements are needed. Too many experimenters turn to theoretical models as an afterthought, only to find that they have neglected to measure variables on which any kind of performance calculation depends, such as the