This chapter presents the derivation of the basic equations that we will use in analyses and calculations of unsteady free-surface flows in natural or man-made channels, e.g. tidal or fluvial channels, shipping canals and irrigation canals. We deal with a mass balance and a momentum balance integrated across the entire flow cross section, assuming a hydrostatic pressure distribution.

Approach

The principal subject of this book is the class of unsteady free-surface flows of water with a characteristic length scale that is far greater than the depth, the so-called long waves. Tides, storm surges and flood waves in rivers provide good examples of this category (contrary to ship waves or wind-generated waves, whose lengths are usually not large or even small compared with the depth).

We restrict ourselves to flows in relatively narrow, weakly curved conduits such as tidal channels and rivers, in which the main flow direction is determined by the geometry of the boundary, which is assumed to be given beforehand (excluding morphological changes such as meandering of rivers). In these cases, the bulk flow direction is known so that only the flow intensity (the discharge, say) is to be determined, in addition to the water surface elevation. A typical area of application, a long, slowly winding river reach with lateral side basins, is shown in Figure 1.1.

As expressed by their name, long waves are characterized by length dimensions that far exceed the depths. This implies that the curvature of the streamlines in the vertical plane is negligible, for which reason we will assume a hydrostatic pressure distribution in the vertical. This greatly simplifies the schematization and the calculations.

In bends, the flow is forced to change direction through a lateral variation of the water level, being higher at the outer bank and lower at the inner bank. This is essential in detailed computations of the spiral flow in bends, but it is irrelevant for the large-scale computations of longitudinal variations with which we are concerned. So we will ignore lateral variations in surface elevation. The height of this level above the adopted reference plane z = 0 is designated as h. This quantity is a function of the downstream coordinate s (measured along the axis of the conduit) and the time t, or h = h(s, t).

This chapter provides an introduction to numerical methods by presenting the numerical counterparts of some analytical models introduced earlier in the book. We will discuss a particular type of solution method for a selection of problems, explain its most important properties and present the resulting numerical code down to the level of the individual program statements. The use of these program scripts will be demonstrated by means of computational examples. In the wake of each method treated we will also summarize some other methods, providing the reader with an overview of alternative solution strategies.

Introduction

The preceding chapters dealt with various types of unsteady flow, resulting in analytical solutions of the governing mathematical equations. These solutions generally provide relations between the characteristics of a water system and its forcing, on the one hand, and the system's response, on the other. In terms of readiness and the overall insight they provide, analytical approaches are still unmatched. However, for practical applications involving complex geometries, arbitrary forcings, and possibly nonlinear effects, they are also somewhat limited due to the assumptions and simplifications on which they rely.

Since the 1950s computer models have therefore gradually, and decisively, replaced the former analytical approaches for computing solutions to real-world problems. The use of advanced numerical tools to compute flow problems in today's engineering practice is ubiquitous and has witnessed a tremendous increase in possibilities. Yet however sophisticated these models may be, they are based on the same theoretical concepts as their analytical predecessors, the study of which is still relevant to understand the results provided by modern computer tools.

In this chapter we will take a first step into the realm of numerical computing by working out some simple examples from previous chapters using the Python programming language. For this purpose some basic knowledge of Python will be needed (see for instance Hetland (2005) or Langtangen (2009)), but this may also be learned as we proceed along the examples in this chapter.

Canal-Basin System

Our first numerical endeavor concerns a simple model of a small tidal basin that is connected to the sea by a narrow gap or entrance channel. The imposed tidal water level at sea (hs ) leads to a time-varying water level in the basin (hb ) that does not depend on the position within the basin.

Introduction

The last two chapters have provided some mathematical derivations so that we can work with time derivatives, partial derivatives in space, the gradient operator, and the divergence operator. This excursion was necessary to become prepared to examine some physical situations. We will now apply some of the ideas we have encountered to describe a translating and deforming volume for the case of mass conservation in such a volume.

A volume is a mathematical region. One can examine a physical volume, such as a lake, a forest, or a groundwater aquifer, or a volume whose boundary is described by arbitrary mathematical coordinates, such has a chosen spherical region at any location in the lake, forest, or aquifer. The boundaries of either kind of volume can move and translate. The definition of a volume requires that it be bounded by a closed surface. We cannot write an equation of conservation for a volume because there is no general principle requiring that a volume retain a constant size or physical constraint on the change in size of a volume. We can, however, identify volumes and write equations that describe their deformation. These equations are the kinematic equations from the last chapter, Eq. (9.36) for a general volume, Eq. (9.38) for a material volume, and Eq. (9.40) for a fixed volume. The selection of the volume to be used depends on the process we are trying to model and the makeup of the system being modeled. For example, if modeling the amount of water in a lake, the volume to be considered would be bounded by the water–solid interface and the water–atmosphere interface at the top. If modeling pollution in the atmosphere above a factory stack, the boundary of the modeling region would be no apparent physical surface.

Conservation of mass is the simplest conservation equation from a number of perspectives. Mathematically, it is expressed much more compactly than the equations of momentum and energy conservation because the number of mechanisms that can change the amount of mass in a volume is fewer than those that can change the momentum or energy in a volume.

Introduction

Thus far in developing the conservation equations for mass, chemical species, momentum, and mechanical energy, we have considered forms that apply either at the megascale in terms of integrals over the domain (e.g. Eq. (10.10), (11.43), (12.26), and (12.82)) where no gradients of properties within the system are considered or at the microscale (e.g. Eq. (10.34), (11.50), (12.31), and (12.75)) where point-to-point spatial variability within the system is modeled. In fact, these two alternatives can be considered extremes in modeling scenarios. Between these extremes are cases where spatial variability is of interest in one or two coordinate directions while the quantities are averaged over the other directions. We will refer to these kinds of models as being mixed scale because they involve both megascopic and microscopic scales.

To obtain mixed-scale models, we start with the general form of the equation in integral form. The formulation of any model then requires selection of a control volume that is consistent with the variability we wish to account for. A volume encompassing the entire system provides a fully megascale model. A volume that, in the limit, approaches a point provides a microscopic model. We can pick a volume that spans a study region in two spatial dimensions and is microscopic in the third dimension. For example, if we are modeling flow in a river, we might identify a control volume that covers the cross section of the river, but is infinitesimal in the direction normal to the river channel. Such a volume leads to equations that model properties that have been averaged over the river cross section as they vary along the river channel. Another mixed-scale model is one that integrates over one spatial domain but allows for variation of the averages over that domain in the two remaining coordinate directions. For example, mass and momentum conservation equations known as the shallow-water flow equations are obtained by integrating over the depth of flow in shallow regions while modeling the average velocity field and the depth of flow as a function of the lateral coordinates.

Continuous change in time and space

At this point, we have considered dimensional analysis as an important tool in modeling. We have discussed length and time scales. Chapter 7 provided an introduction to discrete models and an intercomparison between a discrete and a continuous model. To expand our horizons, we will look at continuous models and models for physical systems. Our study of continuous models will focus primarily on hydrologic systems, but we will draw connections between discrete and continuous models.

The study of continuous models involves derivatives that indicate the rate at which one variable is changing as another changes. For example, a derivative of concentration with respect to time provides the rate at which concentration is changing with time; and the slope of a curve that plots concentration values vs. time is the time derivative. We will also be concerned with how variables change as a function of space and will therefore be interested in spatial derivatives. Additionally, we will make use of integration to determine how much of a quantity is contained in a region or is interacting with a surface. These concepts all involve multivariable calculus, so we will spend a bit of effort highlighting the important elements for our purposes.

The primary issues involving the mathematics of continuous models can be divided into three areas: time derivatives, spatial derivatives, and integration over volumes and surfaces. These items will be treated separately and in turn.

Preliminary concepts for continuous modeling

When making use of equations for modeling, the details of what has gone into the model are often overlooked. For example, we talk about a fluid as having a “density” without really considering that a density is actually an average quantity. We can talk about velocity of flow in a river, but such a quantity is actually some sort of average over time and/or space. It is important to recognize scales of motion, both time and length scales.

Introduction

There is a tendency in studying different disciplines or aspects of a discipline to overlook the fact that fundamental principles apply. Conservation equations for mass, momentum, and energy may be expressed in different dialects in different fields, but they are obtained from the same bases. For example, we have considered equations for CSTRs, Navier–Stokes equations, the Saint Venant equations, hydraulic routing equations, and shallow-water equations. All of these are particular forms of conservation principles. Understanding the basic elements of these principles allows one to consider the representation of physical phenomena in a large arena of issues. Coriolis forces do not appear in the hydraulic routing equations derived, but they are evident in the shallow-water equations. It is important to realize that these terms are not tacked on as added features that give us an equation set specific to shallow-water modeling. Rather there is a single general momentum equation that includes the Coriolis term. It just happens that this term is not important for most one-dimensional routing problems and therefore can be dropped when small.

It is important to understand this difference in perspective. Constructing an equation heuristically by adding on terms and formulations, perhaps somewhat haphazardly in an attempt to come up with a formulation capable of being solved for answers that match collected data is a far different process from starting with general equations that describe the mechanisms of system chemistry and physics. A model formulation process that starts with general equations and simplifies them as possible for a particular system provides insight into actual important mechanisms, relative contributions of various terms, and challenges in representing a system faithfully. We always need to find closure relations for equations to represent the stress tensor and heat conduction vector in terms of properties of the materials being modeled and other variables, such as density, velocity, temperature, and pressure. Scale can make determination of these closure relations more difficult, or can make the determination simpler depending on the system.

Introduction

Humans, as do all species, interact with their environment [105]. In contrast to other species, humans strive to understand these interactions rather than merely experience them passively or as random coincidents. Understanding of physical, chemical, and biological processes, as well as the ability to control these processes, makes us better stewards of our world. The quest for understanding allows us to change from seeing events as mere happenings to gaining knowledge of the mechanisms that are responsible for those events. In searching for mechanisms, we encounter patterns and processes in the things that we observe. Our earliest ancestors observed simple patterns, such as the formation of ice when the temperature became cold. They learned the repeatable pattern of four seasons of the year and the cycles of life that occurred in those seasons. The correlation between formation of ice from pure water and temperature is near perfect. Changes in season provide a definite pattern, but there are also variations from year to year that require further searches. We are able to project the time of sunrise and sunset with excellent accuracy based on knowledge of the consistent behavior of the revolution of the earth as it orbits around the sun; but we do not know, with comparable accuracy, what the temperature will be when the sun rises.

Modeling has a long history and has impacted advancements and understanding in a wide range of disciplines. Modeling must be structured and disciplined if it is to be effective. Statements about what constitutes modeling and how to deconstruct modeling into a sequence of tasks or elements continue to propagate as efforts to make use of the latest theory, computation, and experimental opportunities impact our ability to model well [e.g. 155, 169, 185, 186]. Here we will consider consistent descriptions of processes in the environment and use those descriptions as bases for modeling the processes. If 100 individuals were sent out to make some sort of measurement, and each individual were to return with a different value for the variable of interest, we might not have gained much insight. For example, measurements of the length of time it takes a leaf to fall 50m from a tree branch to the earth will vary from realization to realization.

Introduction

Nothing exists outside of its environment. For this reason alone, humans cannot ignore their environments. As civilizations have become more sophisticated in their ability to alter their environments, the need to understand chemical, biological, and physical activities within those environments has increased. One of the great challenges in gaining understanding is to determine how historical information can be used as a predictor of the future. It is not possible to simply extrapolate the past into the future because physical alteration of a region causes it to behave differently from its historical record. For example, deforestation alters the amount of water that is returned to the atmosphere by transpiration and evaporation in that region. Paving an agricultural region with asphalt impacts the amount of water that infiltrates into the subsurface. Removal of a predator from an ecosystem will alter populations within that system. Discharge of combustion products into the air will impact health. Thus, as the stock market analysts say, “Past performance is not necessarily an indication of future results.”

Although historical environmental data should not be ignored simply because it cannot be directly extrapolated to the future, we need tools that allow us to use this data wisely and effectively. These tools are mathematical models of processes, phenomena that occur in a system that contribute to its perceived behavior. Additionally, we will encounter regions or problems for which no data exists. If we are going to meaningfully study problems in those regions, we need a context in which questions can be asked. This context is provided by a mathematical model that describes relevant processes. Nevertheless, before embarking on a tour of some mathematical models, a disclaimer seems appropriate.

Mathematical models have developed a bit of an aura. They can be complex. Those who find the complexity to be overwhelming are inclined to accept the results of the model with minimal questioning. Complexity is equated to credibility! The fact that models can also produce striking color graphics and animations of simulated results adds to their credibility. However, it is important to emphasize that it is easier to misuse a model than to use one properly. Models are only, as the name states, “models” of reality. They are not reality; it is not possible to incorporate all elements of a real system into a model.

The possibility of using a computer to predict the physical, chemical, and biological responses of environmental systems is an exciting notion that attracts student interest. The elements and processes of environmental modeling, however, are in many ways foreign to the kind of thinking that young investigators exercise during their educational experience. Modeling is inexact, requires physical insight, makes use of mathematics that expresses physical concepts, and is subject to revision and continuing study in light of data and observed system behavior. In developing their scientific background, students often see solving problems of physics as grabbing the right equation for the job. Solution of mathematical equations likewise involves identifying and applying the right techniques. Mathematics and physics books for undergraduates only present problems that are fully specified, can be solved, and that have a single answer. Many of those answers are given in the back of the book. Thus work on a problem is reduced to shooting for a known target solution or at least finding the direct path to the only possible answer. Environmental questions, on the other hand, involve problems that an investigator must define in terms of their component elements to find answers that were previously unknown. Formulation of equations requires information about time and spatial scales. Information that would make even simple models solvable is often lacking. The absence of an a priori answer requires that the modeler not only propose a model and its results but also be the harshest critic of the inadequacies of the model. This dual role requires that the modeler be well versed in fundamental processes and in how descriptions of those fundamental processes can be applied to particular environmental problems of interest. Learning to do this requires much more than a semester or a single book of exposure; it requires experience, success, failure, time, insight, and patience.

This book is intended to provide an introduction to environmental modeling; it is not a book that transforms an individual into an environmental modeler. Those who work through this material, appreciate the thought processes, gain some grasp of the principles involved, and develop fundamental understanding will have been introduced to environmental modeling.

Introduction

As we have progressed through this book from a discussion of data as a modeling source to the development of equations that provide fundamental relations on which to build models, there may be a rising sense of frustration in that we do not provide many solutions to the equations, particularly in the last several chapters. This problem lies with the fact that analytical solutions to most of the equations are not available. On the other hand, use of computer codes that are modeling tools without background into how the codes work, what equations they are built on, data needs, boundary condition specification, and initial condition specification can provide a false sense of satisfaction based on the results obtained which may “look” reasonable. Many models do a fine job of solving equations, but the equations they are solving may not properly describe a system of interest. If one is to go beyond introductory modeling, it is necessary to work from the equations developed to the study of the broad range of numerical techniques being employed to solve them. In other words, models must be both verified and validated.

In this chapter, we will provide a very basic introduction to the idea behind the numerical solution of a differential equation. On the one hand, we have already introduced this approach by solving one-dimensional discrete equations in earlier chapters. However, when a system varies in time and three space dimensions, and may be non-linear, analytical solutions to differential or discrete equations are not possible. Instead, one solves the discrete equations for numerical values of functions of interest at times and spatial locations of interest. This is done typically using techniques of linear algebra customized to take advantage of computer architecture to ensure that the equations are solved efficiently. Solution may require iteration, a systematic trial and error approach to solving the equations. Indeed, as one moves to the solution of problems involving many spatial locations over long periods of time, the need for computer science expertise quickly becomes evident. Thus, consistent with the declaration that this text is introductory, we consider here only a very simple introduction to a method of solving an equation numerically. The purpose is to initiate the thought processes associated with numerical solution techniques rather than to provide polished and advanced algorithms and methodologies.

Introduction

The existence of various scales at which a phenomenon may be viewed is taken for granted in many aspects of everyday life. If asked what is “large” or what is “small,” we usually reference the things we can see and touch. What is considered big, like an airplane or a skyscraper, or small, like the head of a pin or a grain of sand, is only defined in the context of a frame of reference. A skyscraper is large relative to an automobile or a lake home, but in comparison to the size of Denali or Everest, it is small. A pin head is enormous relative to the molecules that make it up, but tiny if one is searching for it in a haystack. Likewise, an 18-wheeler might be considered large as it rumbles by on the highway, but appears to be a tiny dot on the landscape when viewed from a window seat in an airplane at 3000 m. Thus, the adjectives that are used loosely to identify the size or speed of environmental phenomena, such as flow rates in rivers and groundwater, are relative to the scale at which we wish to perceive them. The objective of this chapter is to discuss consistencies and contradictions that may arise when viewing an object or phenomenon at different scales. The significance of scale in observing and modeling environmental processes will be explored, and the different scales that exist in nature will be identified.

Consider a point, which may be defined mathematically as a position in space that has no dimensionality. We might also say that two mathematical lines or curves, with zero width, can intersect at a point. When viewed at the sub-molecular scale, an arbitrarily selected point at some instant in time would most likely lie somewhere in the vacuum between molecules. However, we talk casually about a quantity such as mass density, the mass per volume, that is associated with a “point” when we actually are referring to some region too small to be observed by the naked eye that actually consists of a very large number of molecules and therefore occupies three-dimensional space.

Introduction

What we need to do in modeling is identify regions we wish to analyze. We will define a volume for study and then look at what goes into and comes out of that volume. These flows across the boundary, called fluxes, have units of quantity of interest per unit area per unit time, and are contributors to change within the volume being studied. Without fluxes, and other sources, a system will relax toward equilibrium where there is no flow and no variables are changing with time. Fluxes of energy, momentum, and mass into and out of a system effect change in that system. A complicating issue is that we need to describe these fluxes, and the system, at scales consistent with the questions we intend to address by modeling. For example, one might wish to assess a lake as a whole if interested in its fish population. Or if one is trying to monitor an invasive fish population, such as Asian carp into Lake Michigan, it may be important to know if and where the species is distributed in the lake. It may also be useful to have information about the currents within the lake and how they are distributed. We want to use some basic equations that describe important phenomena, and we also want to be able to apply those equations at different scales.

In the last chapter, we developed expressions for the general time derivative, material derivative, partial time derivative, gradient operator, and divergence operator. The time derivative forms are indicative of properties of a region being studied and how this region changes or deforms in time. Here, we seek some general mathematical tools that facilitate the transfer of derivative expressions for the observation of systems between scales. The primary tools that allow such transformations are two theorems: the divergence theorem and the transport theorem. Note that these theorems are mathematical relations that apply based on the properties of the functions and regions being studied.

Introduction

In addition to conservation of mass and momentum, conservation of total energy is an essential principle for modeling environmental systems. Comprehensive modeling of all the elements of energy transport is significantly more complex than modeling of chemical species or momentum. One reason for this is that thermodynamics plays an essential role in allowing the internal energy of a system to be expressed in terms of independent variables, such as pressure and temperature. Knowledge of a thermodynamic framework is important to avoid making errors in accounting for the thermal energy of a system. Additionally, heat transfer occurs by three mechanisms: convection, conduction, and radiation. The convection of heat is analogous to the convection of mass in that the energy in an element of material is carried by flow within the system. Energy is not constrained to remain with a material element as it can be transferred to adjacent elements by conduction. For modeling purposes, this process shares similarity with mass dispersion in a flowing system; but heat is also conducted in a solid in the absence of any displacement of molecules. Radiation of energy from one location to another does not require matter at all. It is important, of course, to modeling global climate change in that a change in the balance of energy radiated from the sun to the Earth and from the Earth back to space is a factor in climate change. However, this process requires insights into wave length, directions of radiation, emissivity, absorption, reflection, transmission and scattering.

Because of the need for thermodynamics and for broad understanding of the complexities of radiation, the coverage of energy transfer here is truly elementary. Energy transport in the environment is important as it impacts climate, electricity production, glacier melting, hurricane formation, ocean warming, lake turnover, subsurface storage, geothermal production, permafrost, ice jams, and engineered systems. A background beyond what is intended for access to the current book is necessary to study these processes. Here, we provide the basic introduction to the energy conservation equation and, for the most part, leave application to natural systems as a task that can be accomplished after acquiring more specialized background [e.g. 3, 6, 23, 28, 30, 39, 72, 76, 94, 118, 137, 170].