ENVIRONMENTALFLUID MECHANICS

BENOIT CUSHMAN-ROISINThayer School of EngineeringDartmouth CollegeHanover, New Hampshire 03755

March 2014

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Library of Congress Cataloging-in-Publication Data:

Cushman-Roisin, BenoitEnvironmental Fluid Mechanics / Benoit Cushman-Roisinp. cm.Includes bibliographical references and index.ISBN 0-1. Fluid Mechanics 2. Environment 3. Hydraulics 4. Meteorology I. Title

Printed in the United States of America.

CONTENTS

PREFACE ix

PART I: GENERALITIES 1

Chapter 1: Introduction 3

1.1 Fluids in the Environment / 3

1.2 Scope of Environmental Fluid Mechanics / 4

1.3 Stratication and Turbulence / 5

1.4 Environmental Transport and Fate / 8

1.5 Scales, Processes and Systems / 10

Problems / 12

Chapter 2: Physical Principles 15

2.1 Control Volume / 15

2.2 Conservation of Mass / 20

2.3 Conservation of Momentum / 22

2.4 Bernoulli Equation / 28

2.5 Equation of State / 33

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2.6 Conservation of Energy / 34Problems / 36

Chapter 3: Dierential Equations for Fluid Motion 39

3.1 Equations of Motion / 393.2 Hydrostatic Approximation / 493.3 Earth's Rotation / 503.4 Scales and Dimensionless Numbers / 503.5 Vorticity / 573.6 Circulation Theorems / 60Problems / 64

PART II: PROCESSES 69

Chapter 4: Waves 71

4.1 Surface Gravity Waves / 714.2 Internal Gravity Waves / 844.3 Mountain Waves / 914.4 Inertia-Gravity Waves / 944.5 Energy Propagation / 954.6 Nonlinear Eects / 97Problems / 99

Chapter 5: Instabilities 103

5.1 Kelvin-Helmholtz Instability / 1035.2 Instability of a Stratied Shear Flow / 1115.3 Barotropic Instability / 1175.4 Inertial and Baroclinic Instability / 124Problems / 124

Chapter 6: Mixing 127

6.1 The Nature of Mixing / 1276.2 Mixing by Shear / 1296.3 Mixing in the Presence of Stratication / 1326.4 Entrainment / 1336.5 Mixed-Layer Modeling / 135

CONTENTS v

Problems / 136

Chapter 7: Convection 121

7.1 Gravitational Instability / 1217.2 Rayleigh-Benard Convection / 1227.3 Top-to-Bottom Turbulent Convection / 1237.4 Penetrative Convection / 1237.5 Convection in a Rotating Fluid / 1267.6 Convection Modeling / 126Problems / 127

Chapter 8: Turbulence 129

8.1 Homogeneous and Isotropic Turbulence / 1298.2 Shear-Flow Turbulence / 1298.3 Mixing Length / 1358.4 Turbulence in Stratied Fluids / 1378.5 Two-Dimensional Turbulence / 1378.6 Closure Schemes / 1388.7 Large-Eddy Simulations / 138Problems / 138

Chapter 9: Turbulent Jets 141

9.1 Turbulent Jets / 1419.2 Jets in a Cross Flow / 1459.3 Buoyant Jets / 1459.4 Jets in Stratied Fluids / 145Problems / 145

Chapter 10: Plumes and Thermals 147

10.1 Plumes / 14710.2 Plumes in a Cross-Flow / 15010.3 Plumes in Stratied Fluids / 15010.4 Thermals / 15010.4 Buoyant Pus / 152Problems / 153

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Chapter 11: Flow Past Objects 155

11.1 Two-Dimensional Flows Past Objects / 15511.2 Three-Dimensional Eects / 15611.3 Application: Fumigation Behind a Building / 157Problems / 158

PART III: SYSTEMS 163

Chapter 12: Atmospheric Boundary Layer 165

12.1 The Lower Atmosphere / 16512.2 Air Compressibility / 16712.3 Potential Temperature / 16912.4 The Convective ABL / 17012.5 The Stable ABL / 17112.6 Top-Down and Bottom-Up Diusion / 17312.7 ABL over Rough Terrain and Topography / 17512.8 Nocturnal Jet / 17712.9 Sea Breeze and Land Breeze / 17912.10 Mountain Weather / 18312.11 Application: Smokestack Plumes / 185Problems / 185

Chapter 13: Troposphere and Weather 187

13.1 Thermal Wind / 18713.2 Weather Systems / 18913.3 Frontogenesis / 19113.4 Blocking / 19313.5 Hurricanes and Typhoons / 19513.6 Tornadoes / 19713.7 Application: Acid Deposition / 199Problems / 201

Chapter 14: Aquifers and Wetlands 205

14.1 The Hydrological Cycle / 20514.2 Wetland Hydrology / 20614.3 Flow over Canopies / 20714.4 Flow in Channels / 209

CONTENTS vii

14.5 Convection / 21114.6 Soil Inltration / 213Problems / 215

Chapter 15: Rivers and Streams 115

15.1 Open-Channel Flow / 11515.2 Uniform Frictional Flow / 12215.3 The Froude Number / 12515.4 Gradually Varied Flow / 12515.5 Lake Discharge Problem / 12815.6 Rapidly Varied Flow / 13115.7 Hydraulic Jump / 14015.8 Air-Water Exchanges / 14215.9 Dissolved Oxygen /14615.10 Sedimentation and Erosion / 151Problems / 157

Chapter 16: Lakes and Reservoirs 157

16.1 Denition / 15716.2 Physical Processes / 15716.3 Seasonal Variations / 16316.4 Wind Mixing / 116816.4 Wind-Driven Circulation / 17016.5 Surface and Internal Seiches / 17316.8 Biochemical Processes / 17516.9 Application: The Great Lakes / 181Problems / 185

Chapter 17: Estuaries, Lagoons and Fjords 187

17.1 Classication of Estuaries / 18717.2 Salt Wedge and Longitudinal Mixing / 18917.3 Transverse Mixing / 19117.4 Tidal Eects / 19317.5 Lagoons / 19517.6 Fjords / 19717.7 Application: Shellsh in the Chesapeake Bay / 198Problems / 199

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References 400

Index 420

PREFACE

When environmental pollution is mentioned, the rst thought coming to mind isthat of a chemical or biological matter negatively aecting some person or someecosystem. Yet, those materials would not be where they are if they had not beentransported somehow through the environment from their source. This simple factand the fact that a large degree of dilution and transformation takes place alongthe transporting path makes one quickly realize that the environmental impact ofany type of contamination depends as much on the nature of the contaminant ason the physics of its transport, hence the expression Environmental Transport andFate. Thus, environmental pollution has both physical and biochemical aspects.

Transport of contamination in the environment can take many forms, from down-stream ow of water and air, to migration through soils, deposition in lungs andtransfer through the food chain. Of all possible pathways, transport by water andair is by far the most common and therefore deserves special attention. The investi-gation of the processes by which contaminants are transported and diluted in waterand air, such as convection and turbulent dispersion, and the study of water and airsystems from the perspective of environmental health, such as a watershed or theatmospheric boundary layer, collectively form a body of knowledge, the synthesis ofwhich is recognized today as the discipline called Environmental Fluid Mechanics.This synthesis is the object of the present book.

Environmental Fluid Mechanics (EFM) borrows most of its materials from clas-sical uid mechanics, meteorology, hydrology, hydraulics, limnology and oceanogra-phy, but integrates them in a unique way, namely with a view toward environmentalunderstanding, predictions and even decision making. EFM should therefore notbe confused with basic uid mechanics, hydraulics or geophysical uid dynamics.Unlike general uid mechanics, EFM is strictly concerned with the ows of air andwater as they naturally occur, that is, at ambient temperatures and pressures, ina state of turbulence, and at relatively large scales (a few meters to the size of theearth). Ironically also, while uid mechanics tends to view turbulence as a nega-tive aspect (increasing drag forces), EFM views turbulence as benecial (conduciveto dilution). Further, EFM is distinguished from hydraulics not only because ittreats air as well as water, but chiey because it is aimed at environmental applica-tions. Thus, whereas hydraulics tends to be preoccupied by water levels (oods) and

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pressures against physical structures (dams and bridges), EFM is concerned withthermal stratication, turbulent dispersion and sedimentation. Finally, geophysi-cal uid dynamics restricts its attention to the very largest natural uid ows ofthe atmosphere and oceans such as weather patterns and oceanic currents, therebyemphasizing the role of Earth's rotation (Coriolis eect) while often ignoring turbu-lence; in contrast, EFM assigns a central role to turbulence and deals with lengthscales down to the human size.

Complexity is a hallmark of natural uid ows: Turbulent uctuations, compli-cated geometries, multiple external forces, and thermal stratication all combine tomake the subject rather challenging. No single approach can suce, and a mix ofin-situ observations, theoretical investigations, numerical simulations, and labora-tory experiments is most necessary. Such mix is naturally reected in the contentsof the book. Furthermore, a system outlook is essential to the pursuit of environ-mental uid mechanics. Yet, the study of a system (ex. an urban airshed) mustproceed from the prior study of underlying processes (ex. convection and boundarylayers), which itself relies on the elucidation of fundamental concepts (ex. buoyancyand vorticity). The organization of the book follows a deductive progression, fromgeneralities and concepts, to processes, and nally to entire systems.

The book is aimed at upper-level undergraduate students in environmental sci-ence and engineering. The text therefore assumes some familiarity with calculusand basic physics as well as some prior exposure to uid mechanics. Those studentswho have taken a prior course in uid mechanics can omit Chapters 2 and 3. Toassist professors, a series of problems is oered at the end of every chapter. It isexpected that the book will also be useful to environmental scientists and engineers,who may want to consult it as a reference. Finally, it is the expressed hope of theauthor that this book will facilitate the development and oering of a course inenvironmental engineering as part of a curriculum in environmental transport andfate.

This book would not have been possible without the contributions and assistanceof many people. I am foremost indebted to my students at Dartmouth College,who persuasively led me to consider environmental uid mechanics as an integraldiscipline. Numerous colleagues, too many to permit an exhaustive list here, havemade detailed and invaluable suggestions that have improved both the contentsand presentation of this textbook. Special thanks go to Edwin A. Cowen, CarloGualtieri, Heidi Nepf and Thomas Shay, among many others.

Benoit Cushman-RoisinHanover, New HampshireMarch 2014