Chapter 3 Where “Small is Beautiful” – Mathematical Modelling and Free Surface Flows John D. Fenton Abstract Mathematical and computational models in river and canal hydraulics of- ten require data that may not be available, or it might be available and accurate while other information is only roughly known. There is considerable room for the development of approximate models requiring fewer details but giving more insight. Techniques are presented, especially linearisation, which is used in several places. A selection of helpful mathematical methods is presented. The approximation of data is discussed and methods presented, showing that a slightly more sophisticated approach is necessary. Several problems in waves and flows in open channels are then examined. Complicated methods have often been used instead of standard sim- ple numerical ones. The one-dimensional long wave equations are discussed and presented. A formulation in terms of cross-sectional area is shown to have a surpris- ing property, that the equations can be solved with little knowledge of the stream bathymetry. Generalised finite difference methods for long wave equations are pre- sented and used. They have long been incorrectly believed to be unstable, which has stunted development in the field. Past presentations of boundary conditions have been unsatisfactory, and a systematic exposition is given using finite differences. The nature of the long wave equations and their solutions is examined. A simplified but accurate equation for flood routing is presented. However, numerical solution of the long wave equations by explicit finite differences is also simple, and more gen- eral. A common problem, the numerical solution of steady flows is then discussed. Traditional methods are criticised and simple standard numerical ones are proposed and demonstrated. A linearised model for the surface profile of a stream is obtained, also to give solutions without requiring detailed bathymetric knowledge. Key words: Approximation, Boundary conditions, Finite differences, Gradually- varied flow equation, Interpolation, Kinematic, Long wave equations, Numerical methods, Reservoir routing, Resistance, Obstacles, Telegraph equation John D. Fenton Institute of Hydraulic and Water Resources Engineering, Vienna University of Technology, Karl- splatz 13/222, 1040 Vienna, Austria, e-mail: [email protected] 1 2 John D. Fenton 3.1 Introduction The originator of the expression “small is beautiful” was Leopold Kohr, an Austrian philosopher and economist. The expression was adopted, used, and popularised in the title of a 1973 book “Small Is Beautiful: A Study of Economics As If People Mattered” by a student of Kohr, Ernst Friedrich Schuhmacher, a German economist and statistician who spent most of his life in Britain. The present author can lay a small claim to some association with the idea. In 1974, the year after publication of the book, he went on a weekend workshop to discuss its economic and political messages in Windsor Great Park near London. It was a pleasant and optimistic era for some. Subsequently, however, in the first phase of his career, the author honoured “small is beautiful”, in the words of Shake- speare’s Hamlet: “more in the breach than the observance”, meaning, usually not honouring it. Or even doing the opposite: one of his first publications was entitled “A ninth-order solution for the solitary wave”. Then, after working for some 25 years in coastal and ocean engineering, where the problems are complex enough that it is difficult to keep matters simple, the author gradually moved onshore and up the rivers and streams of the terrestrial environment and onto land. Gradually he thought he saw that many things were done, sometimes wrongly, sometimes too complicatedly, and that complexity often obscured the meaning and understanding of what was being done. He tried to publish papers, but found that contrary and simple views were not welcome. He has now spent some 20 years travelling in a direction transverse to the flow of the canonical science, trying to place obstacles in the flow. Now as he approaches the end of his career, he wonders, whether he might not end like Siddhartha in Hermann Hesse’s Buddhist novel of the same name, and act as a ferryman, still crossing and defying the river of conventional thought, but helping people by carrying them to and fro with him, pointing out how things might otherwise be done. Initially here we will present several mathematical methods which are of some generality and application to the title of this work – and will be used in the remain- der. It is pointed out that there are a number of methods or formulae in river engi- neering which are wrong, silly, and/or unnecessarily complicated. Corrections are suggested and some new results are presented, which the author hopes will justify reading this work. 3.2 Virtues of simplicity Mathematics is a problem for many people, and can lead to early alienation and the internalising of a sense of failure and of ignorance. However, one should be confident of one’s own patient thought processes - and remember Newton: “If I have ever made any valuable discoveries, it has been due more to patient attention, than to any other talent”. Concerning simplicity, Newton wrote, admittedly not very simply by twenty-first century standards: 3 Mathematical Modelling and Free Surface Flows 3 “Nature is pleased with simplicity, and affects not the pomp of superfluous • causes”, and “Truth is ever to be found in the simplicity, and not in the multiplicity and con- • fusion of things”. In this advocacy of simplicity, Newton had long been preceded by William of Ock- ham, a mediaeval monk and philosopher, who developed the principle known as Ockham’s Razor, which states that if something can be explained without a further assumption, there is no reason for that. Any explanation should be in terms of the fewest factors or parameters. This was paraphrased, possibly even better, by Albert Einstein as “Make things as simple as possible, but not simpler”. He also said: “If you can’t explain it simply, you don’t understand it well enough.” Therein lies some of the benefits of teaching for researchers: to teach properly one must understand it – and to make it as simple as possible for the students – while revealing any under- lying complexity. We consider some thoughts on the nature of mathematics and simplicity: Mathematics is like learning another language – a certain body of knowledge is • necessary. Something can sit unused in the brain, but when a problem is required to be solved, it can leap to the front. It helps remembranceif one has understand- ing. And simplicity helps understanding. An aspect of mathematics and simplicity is the symbiotic relationship between • mathematics and visual perception and understanding. Many scientists and engi- neers have highly-developed visual-spatial abilities, which can help. An important aspect of simplicity is the ease of disproof, in the sense of Karl • Popper: if something is complicated, it is likely to be more difficult for people to understand it and to disprove it. If something has been expressed in simple mathematics, it is likely to be more correct because it has withstood attempts to disprove it. A formal process of mathematical modelling provides some structure and disci- • pline to our thought processes: one should make the simplest possible model and if it works, good, but if not, refine it, and repeat if necessary, reflecting the above quote by Einstein on making things as simple as possible, but not too much so. An important problem is the mismatch between data and mathematical mod- • els available, whether too much data or too little. One might have all the cross- sectional data of a stream at every 100m, and be tempted to include all that in a numerical model – while knowing the resistance coefficient only to some 30%. The sensible thing would be to abandon the detailed data and use an approximate± model. Or, one might have little cross-sectional data, but because a public or private organisation requires the use of a certain computer program (HEC-RAS comes to mind), it might be necessary to arrange an expensive field survey. It would be better if managers had more scientific judgement about when not to believe detailed results. In research rather than applications, the processes by which it is evaluated and • judged are hostile to simplicity. The author has written a critique of peer review and publishing processes in Fenton (2016, 4.4). Trying to publish something § 4 John D. Fenton simple seems to be very difficult – reviewers are pleased to discover something that they can understand and then criticise it severely, whereas something com- plicated that is beyond their understanding is likely to be accepted, then repli- cated, and expanded upon ad infinitum. Faddish, fashionable, and complicated techniques are certainly good for publishing and a career in research. 3.3 Approximate solutions – linearising and the use of series Often there is much less detailed information known about a problem than the governing equations require. In such cases, approximations are highly justified and reveal to us rather more the real nature of the problem. The most common way is to linearise the problem, considering a quantity to be relatively small, and writing series in that quantity, manipulat- ing only to first order. Or, a function can be approximated by using a finite number of terms of its Taylor series calculated from the function’s derivatives at a single point. Here a brief revision of some elementary series operations is given, and then a physical example is pre- sented, the interaction between an obstacle such as a bridge pier and a stream flow. Below, in §§3.7.2 & 3.9.5 linearised versions of the unsteady and steady long wave equations will be presented, with a number of results.