Course CIE4340 Computational Modelling of Flow and Transport January 2015 Faculty of Civil Engineering and Faculty Geosciences M. Zijlema Delft University of Technology Artikelnummer 06917300083 CIE4340 Computational modelling of flow and transport by : M. Zijlema mail address : Delft University of Technology Faculty of Civil Engineering and Geosciences Environmental Fluid Mechanics Section P.O. Box 5048 2600 GA Delft The Netherlands website : http://fluidmechanics.tudelft.nl Copyright (c) 2011-2015 Delft University of Technology iv Contents 1 Introduction 1 1.1Amathematicalmodel............................. 2 1.2Acomputationalmodel............................. 4 1.3Scopeofthislecturenotes........................... 7 2 Numerical methods for initial value problems 9 2.1Firstorderordinarydifferentialequation................... 9 2.1.1 Timeintegration............................ 10 2.1.2 Convergence, consistency and stability . .............. 14 2.1.3 Two-stepschemes............................ 23 2.2Systemoffirstorderdifferentialequations.................. 26 2.2.1 ThesolutionofsystemoffirstorderODEs.............. 27 2.2.2 Consistencyandtruncationerror................... 33 2.2.3 Stability and stability regions . .................. 36 2.2.4 StiffODEs................................ 39 2.3Concludingremarks............................... 42 3 Numerical methods for boundary value problems 43 3.1IntroductiontoPDEs.............................. 43 3.1.1 ExamplesofPDEs........................... 43 3.1.2 RelevantnotionsonPDEs....................... 45 3.1.3 Wellposedproblems.......................... 46 3.1.4 Types of boundary conditions . .................. 47 3.1.5 ClassificationofPDEs......................... 47 3.2Diffusionequation................................ 50 3.2.1 Finitedifferencemethod........................ 52 3.2.2 Timeintegration............................ 58 3.2.3 Stability . ............................. 61 3.2.4 Implicitschemes............................ 65 3.2.5 Accuracyandrelaxationtime..................... 66 3.3Convectionequation.............................. 69 3.3.1 Spacediscretizationandtimeintegration............... 71 3.3.2 ThePreissmannscheme........................ 73 v vi 3.3.3 TheCFLcondition........................... 75 3.3.4 Theupwindmethod.......................... 76 3.3.5 Spurious oscillations and monotonicity . .............. 80 3.3.6 TheLax-Wendroffmethod....................... 91 3.3.7 TheMacCormackmethod....................... 93 3.3.8 Phase− and amplitude−erroranalysis................ 95 3.4Convection-diffusionequation......................... 104 3.4.1 Stationaryconvection-diffusionequation............... 104 3.4.2 Instationaryconvection-diffusionequation.............. 108 3.5Summaryandconcludingremarks....................... 110 4 Numerical treatment of shallow water equations 117 4.1Theshallowwaterequations.......................... 117 4.2 Open boundaries and boundary conditions .................. 122 4.3 Flood propagation in rivers . ......................... 134 4.4Discretizationmethods............................. 138 4.4.1 Theleapfrogscheme.......................... 140 4.4.2 ThePreissmannscheme........................ 142 4.4.3 Numericalsolutionofthe1Dshallowwaterequations........ 143 4.5Finalremarks.................................. 147 A Taylor series 149 A.1Taylorseriesinonedimension......................... 149 A.2Taylorseriesintwodimensions........................ 153 A.3Linearization.................................. 154 B Fourier series 157 B.1Complexexponentialfunctions......................... 157 B.2Fourierseriesonafiniteinterval........................ 161 B.3ComplexFourierseries............................. 163 B.4DiscreteFouriertransform........................... 165 Bibliography 167 Index 169 Chapter 1 Introduction In this chapter, the general outline of the course CIE4340, computational hydraulics, will be explained. Computational hydraulics is an applied science aiming at the simulation by computers of various physical processes involved in seas, estuaries, rivers, channels, lakes, etc. It is one of the many fields of science in which the application of computers gives rise to a new way of working, which is intermediate between purely theoretical and experimental. This discipline is not an independent development, but rather a synthesis of various disciplines like applied mathematics, fluid mechanics, numerical analysis and computational science. There is not a great deal of difference with the discipline computational fluid dynamics (CFD), but it is too much restricted to the fluid as such. It seems to be typical of practical problems in hydraulics that they are rarely directed to the flow by itself, but rather to some consequences of it, such as wave propagation, transport of heat, sedimentation of a channel or decay of a pollutant. Mathematical and computational models are at the foundation of much of computational hydraulics. They can be viewed as a series of mappings from a part of the real world via abstract number spaces, as will be discussed in Section 1.1, onto a computer, as will be outlined in Section 1.2. This is illustrated in Figure 1.1. In this way similarity with respect to dynamics and shape of structure between the real world and the computer is obtained. Generally, this similarity is not isomorphic1 because • a mathematical model is invariably a simplification and cannot describe every aspect of the real world, and • a computational model may contain artifacts that have no corresponding property in the real world. To get insight into the shortcomings of the mathematical and computational models, we first consider a part of the real world of which the models are to be set up. In this course 1An isomorphism is a mapping between two systems that are structurally the same even though the names and notation for the elements are different. 1 2 Chapter 1 Figure 1.1: Mappings of (a part of) the real world onto a computer. we restrict ourselves to applications with open water bodies as seas, estuaries, rivers, lakes and channels. Figure 1.2 shows some examples. 1.1 A mathematical model Many important concepts of mathematics were developed in the framework of physical science and engineering. For example, calculus has its origins in efforts to describe the motion of bodies. Mathematical equations provide a language in which to formulate con- cepts in physics. A mathematical model is an equation or set of equations whose solution describes the physical behaviour of a related physical system. For instance, Newton’s equa- tions describe mechanical systems, Maxwell’s equations describe electrodynamical phenom- ena, Schr¨odinger’s equation describes quantum phenomena, etc. In this way, we have created a mapping from the real world to an abstract number space. In this space, real numbers are represented as symbols (x, y, ω, π, e, etc.). Moreover, this continuous space is structured by mathematical notions such as continuity and differ- entiability, and is governed merely by mathematical laws. For instance, well posedness is a mathematical notion that follows only from fulfilling some basic mathematical rules, so that well posed problems can produce useful solutions. Introduction 3 (a) (b) (c) (d) Figure 1.2: Different areas of interest and applications in hydraulics: (a) coastal flooding, (b) Delta works, (c) tsunami waves and (d) dam break. Physical phenomena depend in complex ways on time and space. Scientists often seek to gain understanding of such phenomena by casting fundamental principles in the form of mathematical models. Physical, chemical and biological theories are rooted in the concept that certain quantities, such as mass, momentum, energy, charge, spin, population of or- ganisms, etc. are conserved. A conservation law is simply the mathematical formulation of the basic fact that the rate at which a quantity changes in a given domain must equal therateatwhichthequantityflowsintooroutofthedomain. Rate of change equals input minus output. In deriving mathematical equations from the conservation laws, a number of simplifying assumptions needs to be introduced in order to make these equations feasible. Though these assumptions are a source of errors and limit the generality of the applicability of a mathematical model, they are often not so much restrictive. An example is the hydro- static pressure asssumption. For coastal waters, this is not a harmful one. Another example is turbulence modelling which is required since most flows in hydraulic appli- 4 Chapter 1 cations are turbulent. A key feature of this modelling is averaging the governing equations of turbulence, while some closure assumptions are introduced to represent scales of the flow that are not resolved. Sometimes these assumptions can be severe. Usually a mathematical model requires more than one independent variable to characterize the state of the physical system. For example, to describe flow and transport in coastal waters usually requires that the physical variables of interest, say flow velocity, water level, salinity, temperature and density, be dependent on time and three space variables. They are governed by conservation of mass, momentum and energy. They form a set of par- tial differential equations (PDEs). Together with boundary and initial conditions, they represent a boundary value problem. If a mathematical model involves only one inde- pendent variable (usually time) and the physical variables of interest are nonconstant with respect to that independent variable, then the mathematical model will