Environmental Fluid Mechanics - Chapter 1 pdf

Environmental Fluid Mechanics Hillel Rubin Technion-lsrael Institute of Technology Haifa, Israel Joseph Atkinson State University of New York at Buffalo Buffalo, New York MARCEL MARCEL DEKKER, INC. NEW YORK • BASEL ISBN: 0-8247-8781-1 This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc. 270 Madison Avenue, New York, NY 10016 tel: 212-696-9000; fax: 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-261-8482; fax: 41-61-261-8896 World Wide Web http://www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the headquarters address above. Copyright  2001 by Marcel Dekker, Inc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit): 10987654321 PRINTED IN THE UNITED STATES OF AMERICA Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. To our wives, Elana Rubin and Nancy Atkinson and our families for their continued love and support Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. Preface The purpose of this text is to provide the basis for an upper-level undergraduate or graduate course over one or two semesters, covering basic concepts and examples of fluid mechanics with particular applications in the natural environ- ment. The book is designed to meet a dual purpose, providing an advanced fundamental background in the fluid mechanics of environmental systems and also applying fluid mechanics principles to a variety of environmental issues. Our basic motivation in preparing such a text is to share our experience gained by teaching courses in fluid mechanics, environmental fluid mechanics, and surface- and groundwater quality modeling and to provide a textbook that covers this particular collection of material. This text presents a contemporary approach to teaching fluid mechanics in disciplines connected with environmental issues. There are many good fluid mechanics texts that overlap with various parts of this text, but they do not directly address themes and applications associated with the environment. On the other hand, there are also several texts that address water quality modeling, calculations of transport phenomena, and other issues of environmental engi- neering. Generally, such texts do not cover the fundamental topics of fluid mechanics that are relevant when describing fluid motions in the environ- ment. Besides presenting contemporary environmental fluid mechanics topics, this text bridges the gap between those limited to fluid mechanics principles and those addressing the quality of the environment. The term environmental fluid mechanics covers a broad spectrum of subjects. We have adopted the principle that this topic incorporates all issues of small-scale and global fluid flow and contaminant transport in our environ- ment. We have chosen to consider these topics as divided into two general areas, one involving fundamental fluid mechanics principles relevant to the environment and the second concerning various types of applications of these principles to specific environmental flows and issues of water quality modeling. This division is reflected in the organization of the text into two main parts. The intent is to provide flexibility for instructors to choose material best suited Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. for a particular curriculum. A full two-semester course could be developed by following the entire text. However, other options are possible. For example, a one-semester course could concentrate on the advanced fluid mechanics topics of the first part, with perhaps some chapters from the second part added to emphasize the environmental content. The second part by itself can be used in a course concentrating on environmental applications for students with appro- priate fluid mechanics backgrounds. Although the book addresses principles of fluid mechanics relevant to the entire environment, the emphasis is mostly on water-related issues. The material is designed for students who have already taken at least one undergraduate course in fluid mechanics and have an appropriate back- ground in mathematics. Other courses in numerical modeling and environ- mental studies would be helpful but are not necessary. Because of the breadth of material that could be considered, some subjects have necessarily been omitted or treated only at an introductory level. These topics are left for continuing studies in the student’s particular discipline, such as oceanography, meteorology, groundwater hydrology and contaminant transport, surface water quality modeling, etc. References are provided in each chapter so that students can easily get started in pursuing a particular subject in greater detail. Example problems and solutions are included wherever possible, and there is a set of homework problems at the end of each chapter. We believe it is very important to introduce students to the proper use of physical and numerical models and computational approaches in the framework of analysis and calculation of environmental processes. Therefore, discussion and examples have been included that refer to scaling procedures and to various numerical methods that can be applied to obtain solutions for a given problem. A full discussion of numerical modeling approaches is included. Both parts of the text are organized to provide (1) a review of intro- ductory material and basic principles, (2) improvement and strengthening of basic knowledge, and (3) presentation of specific topics and applications in environmental fluid mechanics, along with problem-solving approaches. These topics have been chosen to introduce the student to the wide variety of issues addressed within the context of environmental fluid mechanics, regarding fluid motions on the earth’s surface, underground, and in the oceans and atmosphere. We believe that the wide scope of topics in environmental fluid mechanics covered in this text is consistent with present teaching needs in advanced undergraduate and graduate programs in fluid mechanics principles and topics related to the environment. These needs are subject to continuous growth and change due to our increasing interest in the fate of ecological systems and the need for understanding transport phenomena in our environment. Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. The authors are grateful to the US–Israel Fulbright Foundation for supporting a sabbatical leave for Joseph Atkinson at the Technion–Israel Insti- tute of Technology, without which this text might not have been completed. Finally, we are indebted to our own teachers, colleagues, and students, who have each made contributions to our understanding of this material and have helped in shaping the presentation of this text. We hope the book will contribute to this legacy. Hillel Rubin Joseph Atkinson Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. Contents Preface Part 1 Principles of Environmental Fluid Mechanics 1. Preliminary Concepts 1.1 Introduction 1.2 Properties of Fluids 1.3 Mathematical Preliminaries 1.4 Dimensional Reasoning Problems Supplemental Reading 2. Fundamental Equations 2.1 Introduction 2.2 Fluid Velocity, Pathlines, Streamlines, and Streaklines 2.3 Rate of Strain, Vorticity, and Circulation 2.4 Lagrangian and Eulerian Approaches 2.5 Conservation of Mass 2.6 Conservation of Momentum 2.7 The Equations of Motion and Constitutive Equations 2.8 Conservation of Energy 2.9 Scaling Analyses for Governing Equations Problems Supplemental Reading 3. Viscous Flows 3.1 Various Forms of the Equations of Motion 3.2 One-Directional Flows Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. 3.3 Creeping Flows 3.4 Unsteady Flows 3.5 Numerical Simulation Considerations Problems Supplemental Reading 4. Inviscid Flows and Potential Flow Theory 4.1 Introduction 4.2 Two-Dimensional Flows and the Complex Potential 4.3 Flow Through Porous Media 4.4 Calculation of Forces 4.5 Numerical Simulation Considerations Problems Supplemental Reading 5. Introduction to Turbulence 5.1 Introduction 5.2 Definitions 5.3 Frequency Analysis 5.4 Stability Analysis 5.5 Turbulence Modeling 5.6 Scales of Turbulent Motion Problems Supplemental Reading 6. Boundary Layers 6.1 Introduction 6.2 The Equations of Motion for Boundary Layers 6.3 The Integral Approach of Von Karman 6.4 Laminar Boundary Layers 6.5 Turbulent Boundary Layers 6.6 Application of the Boundary Layer Concept to Heat and Mass Transfer Problems Supplemental Reading 7. Surface Water Flows 7.1 Introduction 7.2 Hydraulic Characteristics of Open Channel Flow 7.3 Application of the Energy Conservation Principle 7.4 Application of the Momentum Conservation Principle 7.5 Velocity Distribution in Open Channel Flow Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. 7.6 Gradually Varied Flow 7.7 Circulation in Lakes and Reservoirs Problems Supplemental Reading 8. Surface Water Waves 8.1 Introduction 8.2 The Wave Equation 8.3 Gravity Surface Waves 8.4 Sinusoidal Surface Waves on Deep Water 8.5 Sinusoidal Surface Waves for Shallow Water Depth 8.6 The Group Velocity 8.7 Waves in Open Channels 8.8 Numerical Aspects Problems Supplemental Reading 9. Geophysical Fluid Motions 9.1 Introduction 9.2 General Concepts 9.3 The Taylor–Proudman Theorem 9.4 Wind-Driven Currents (Ekman Layer) 9.5 Vertically Integrated Equations of Motion Problems Supplemental Reading Part 2 Applications of Environmental Fluid Mechanics 10. Environmental Transport Processes 10.1 Introduction 10.2 Basic Definitions, Advective Transport 10.3 Diffusion 10.4 The Advection–Diffusion Equation 10.5 Dispersion 10.6 Dispersion in Porous Media 10.7 Analytical Solutions to the Advection–Diffusion Equation 10.8 Numerical Solutions to the Advection–Diffusion Equation Problems Supplemental Reading Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. 11. Groundwater Flow and Quality Modeling 11.1 Introduction 11.2 The Approximation of Dupuit 11.3 Contaminant Transport 11.4 Saltwater Intrusion into Aquifers 11.5 Non-Aqueous Phase Liquid (NAPL) in Groundwater 11.6 Numerical Modeling Aspects Problems Supplemental Reading 12. Exchange Processes at the Air/Water Interface 12.1 Introduction 12.2 Momentum Transport 12.3 Solar Radiation and Surface Heat Transfer 12.4 Exchange of Gases 12.5 Measurement of Gas Mass Transfer Coefficients Problems Supplemental Reading 13. Topics in Stratified Flow 13.1 Buoyancy and Stability Considerations 13.2 Internal Waves 13.3 Mixing 13.4 Double-Diffusive Convection 13.5 Mixed-Layer Modeling Problems Supplemental Reading 14. Dynamics of Effluents 14.1 Jets and Plumes 14.2 Submerged Discharges and Multiport Diffuser Design 14.3 Surface Buoyant Discharges Problems Supplemental Reading 15. Sediment Transport 15.1 Introduction 15.2 Hydraulic Properties of Sediments 15.3 Bed-Load Calculations 15.4 Suspended Sediment Calculations 15.5 Particle Interactions 15.6 Particle-Associated Contaminant Transport Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. [...]... C i sin n 1. 3.84 The product of two complex numbers is z1 z2 D r1 r2 ei 1 CÂ2 1. 3.85 and the division of two complex numbers yields r1 ei 1 r1 z1 D D e i 1 z2 r2 eiÂ2 r2 Â2 1. 3.86 Alternatively, the division of two complex variables can be represented by x1 C iy1 x1 C iy1 x2 z1 D D z2 x2 C iy2 x2 C iy2 x2 Copyright 20 01 by Marcel Dekker, Inc All Rights Reserved y1 x2 x1 y2 iy2 x1 x2 C y1 y2 Ci 2... X 11 2 h1 h2 h1 h3 h1 X 21 h2 h1 X22 2 h2 X23 h2 h3 X 31 h3 h1 X32 h3 h2 X33 2 h3 1. 3.58 In terms of the contravariant components of the second order tensor, the physical components of Eq (1. 3.58) are given by 2 X 11 h1 X12 h1 h2 X13 h1 h3 X 21 h2 h1 2 X22 h2 X23 h2 h3 31 X h3 h1 32 X h3 h2 2 X33 h3 Copyright 20 01 by Marcel Dekker, Inc All Rights Reserved 1. 3.59 As an example, we calculate the relationships... of i are dx 1 ds 1 Since D i is a unit vector, we have i j gij 2 2 D h1 D0 1 2 3 1 D D0 1. 3.54 1 h1 1. 3.55 By multiplying by the metric tensor, we lower superscripts and obtain 1 D h1 2 D 3 D0 1. 3.56 Equations (1. 3.52)– (1. 3.56) imply that the physical components of the vector Xj along the parametric line of x 1 are X1 /h1 or h1 X1 Considering all geometrical scales of the coordinate system we obtain... 20 01 by Marcel Dekker, Inc All Rights Reserved The permutation tensor εijk is defined as εijk D 0 if two of the suffixes are equal εijk D 1 if the sequence of numbers ijk is the sequence of 1- 2 -3 , or an even permutation of the sequence εijk D 1 if the sequence of numbers ijk is an odd permutation of the sequence 1- 2 -3 Examples of the application of these rules are 12 3 D ε2 31 D ε 312 D 1 13 2 D ε 213 ...Problems Supplemental Reading 16 Remediation Issues 16 .1 Introduction 16 .2 Soil and Aquifer Remediation 16 .3 Bioremediation 16 .4 Remediation of Surface Waters Problems Supplemental Reading Copyright 20 01 by Marcel Dekker, Inc All Rights Reserved 1 Preliminary Concepts 1. 1 1. 1 .1 INTRODUCTION Historical Perspective Fluid mechanics and hydraulics have long been major components... added to the imaginary part of z2 Thus z D z1 C z2 D x1 C x2 C i y1 C y2 1. 3.74 The factor i is an operator that upon multiplication rotates a complex number through 90° Powers of i are as follows: i2 D 1 i3 D i i4 D 1 1.3.75 Also, the product of two complex variables z1 and z2 is z1 z2 D x1 C iy1 x2 C iy2 D x1 x2 y1 y2 C i x1 y2 C x2 y1 1. 3.76 A complex number also can be expressed in an exponential... vector: X1 h1 X2 h2 X3 h3 or h1 X1 h2 X2 h3 X3 1. 3.57 In order to define the physical components of a second order tensor we apply two unit vectors in the directions of two parametric lines of two coordinates Such an operation leads to the following expressions for the physical components of the second order tensor, in terms of its covariant components: X12 X13 X 11 2 h1 h2 h1 h3 h1 X 21 h2 h1 X22 2 h2... D υk j 1. 3.28 By multiplying both sides of this expression by Cjm we obtain j gυi gik D υk Cjm m Copyright 20 01 by Marcel Dekker, Inc All Rights Reserved 1. 3.29 If gij D 0 for i 6D j, then 1 g 11 gij D 0 g 11 D 1 g22 i 6D j g22 D for g33 D 1 g33 1. 3.30 The covariant metric tensor and its contravariant conjugate can be used for lowering and raising of suffixes As an example, Uijk D gim Vm jk 1. 3. 31 Now... 1, 2, 3 1. 3.2 Here, u is an arbitrary parameter and the fi are three arbitrary functions The point given by Eq (1. 3 .1) can be represented by a new set of coor0 0 0 dinates x 1 , x 2 , x 3 ), where 0 x i D fi x 1 , x 2 , x 3 1. 3.3 The Jacobian of the transformation is 0 ∂x i J D ∂x j 0 i, j D 1, 2, 3 1. 3.4 Eq (1. 3.2) also can be represented by another transformation, 0 0 0 x i D gi x 1 , x 2 , x 3 1. 3.5... (1. 3.77) and using Eq (1. 3.75), we obtain ei D 1 C i Â2 2! i Â3 Â4 Â5 C Ci 3! 4! 5! Â6 6! Â6 CÐÐÐ Ci  6! Â5 Â3 C 3! 5! ÐÐÐ 1. 3.80 Or, upon rearranging, ei D 1 Â4 Â2 C 2! 4! Â7 C ÐÐÐ 7! 1. 3. 81 By applying Eqs (1. 3.78) and (1. 3.79), Eq (1. 3. 81) becomes ei D cos  C i sin  1. 3.82 All three forms of a complex number are then z D x C iy D r cos  C i sin  D rei 1. 3.83 Following these definitions, the nth . Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812 , CH-40 01 Basel, Switzerland tel: 4 1- 6 1- 2 6 1- 8 482; fax: 4 1- 6 1- 2 6 1- 8 896 World Wide Web http://www.dekker.com The publisher offers. YORK • BASEL ISBN: 0-8 24 7-8 78 1- 1 This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc. 270 Madison Avenue, New York, NY 10 016 tel: 21 2-6 9 6-9 000; fax: 21 2-6 8 5-4 540 Eastern Hemisphere. Reading Copyright 20 01 by Marcel Dekker, Inc. All Rights Reserved. 11 . Groundwater Flow and Quality Modeling 11 .1 Introduction 11 .2 The Approximation of Dupuit 11 .3 Contaminant Transport 11 .4 Saltwater