Hydraulic Modelling: An Introduction

Principles, Methods and Applications

Pavel Novak, Vincent Guinot, Alan Jeffrey, Dominic E. Reeve

• Publisher: CRC Press
• Year: 2018
• ISBN:
  9781351988476
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Mechanical Engineering
• Index: Technology & Engineering

Modelling forms a vital part of all engineering design, yet many hydraulic engineers are not fully aware of the assumptions they make. These assumptions can have important consequences when choosing the best model to inform design decisions.

Considering the advantages and limitations of both physical and mathematical methods, this book will help you identify the most appropriate form of analysis for the hydraulic engineering application in question. All models require the knowledge of their background, good data and careful interpretation and so this book also provides guidance on the range of accuracy to be expected of the model simulations and how they should be related to the prototype.

Applications to models include:

• 
  
  

• open channel systems
• closed conduit flows
• storm drainage systems
• estuaries
• coastal and nearshore structures
• hydraulic structures.

This an invaluable guide for students and professionals.

Chapter 1

Introduction

The hydraulic engineer’s concerns are liquids, their motion and their interaction with conveyances and structures. Usually, but not exclusively, the liquid in question is water – a viscous, slightly compressible fluid. The science of hydraulics thus works with the real liquids of engineering interest, although it owes much to the laws derived in theoretical hydromechanics for ideal (homogeneous, incompressible, non-viscous) liquids.

It is almost impossible in hydraulic research to draw a clear dividing line between basic and applied research, as both intermingle in the solution of hydraulic problems connected with engineering design. An extraordinary development in experimental methods and the application of computational techniques have also been of great importance.

There are three ways to approach the solution of a problem in hydraulics and hydraulic engineering design: by theory and reasoning; by experience (e.g. derived from similar structures); or by investigating the problem and testing the design on a model. However, our past experience may be inadequate due to the uniqueness of the design and circumstance; the complexity of many cases of liquid flow and our still limited analytical abilities permit the strict application of theory and basic flow equations only in certain, often schematized, situations and thus methods using models are needed to achieve a solution or to test the effect of simplifications. It must be emphasized, however, that a purely experimental approach to the solution of a problem without any theoretical analysis, even if restricted only to a dimensional analysis, is likely to be a waste of effort. Systematic experiments require theoretical guidelines, and in the absence of such they show, at best, only a certain relationship of observed hydraulic parameters within the range of the experiments undertaken. If the physical principles depicted by an empirical function are not elucidated, then the function can neither be safely extrapolated nor generalized for other similar cases of flow.

The term model is used in hydraulics to describe a physical or mathematical simulation of a ‘prototype’, or field-size situation. The hydraulic engineer’s models are tools for predicting the effect of a proposed design and to producing technically and economically optimal solutions to engineering problems. In other words, a model is a system that will convert a given input (geometry, boundary conditions, force, etc.) into an output (flow rates, levels, pressures, etc.) to be used in civil engineering design and operation.

Simulation may be direct by the use of hydraulic models, semi-direct using analogues or indirect by making use of theoretical and computer-based analysis, including mathematical, computational and numerical models. The basic distinction is between physical and mathematical models. Physical models then comprise hydraulic and analogue models; analogue models include also aerodynamic models (which really form a transition between hydraulic and other analogue models), and both hydraulic and aerodynamic models can be grouped as scale models. Analogue models had their main application in groundwater flow simulation, but have now mainly been replaced by mathematical models. As the application of aerodynamic models also is being confined to special cases, the terms physical, scale and hydraulic models have gradually almost become synonymous. (The term ‘hydraulic model’ is also sometimes used loosely to denote all models – including mathematical ones – instead of the correct overall term ‘hydraulic modelling’ or ‘models in hydraulic engineering’.)

As we are primarily concerned with the reproduction of present or future full-size behaviour, obtaining relevant field data is an important and integral part of the modelling process.

It is obviously the basic requirement of any scale model to reproduce correctly the behaviour of the situation to be modelled. The success of the solution depends on the accurate formulation of the problem and on the correct identification of the main parameters influencing the phenomena under investigation. This may lead to an intentional suppression of forces and influences, the role of which in the prototype is, in the light of experience, only of secondary importance. It is a possible pitfall that the magnitude of forces neglected in the analysis may assume a disproportionately large significance in the model, a discrepancy that is usually referred to as scale effect. The appreciation of similarity laws and of the limits of their validity is, therefore, particularly important if this is to be avoided. All these considerations influence the selection of appropriate methods and techniques of simulation (Novak and Čábelka (1981)).

One of the first to use hydraulic models was Osborne Reynolds, who in 1885 designed and operated a tidal model of the Upper Mersey at Manchester University. In 1898, Hubert Engels established the first River Hydraulics Laboratory at Dresden. Then followed a gradual and, after 1920, an accelerating expansion of laboratories for the study of hydraulic engineering problems using scale models.

The widespread use and role of hydraulic models may have changed somewhat in recent years, mainly due to the advances in computational modelling, but they remain an important modelling tool, especially in the design of hydraulic structures, river and coastal engineering applications, environmental protection and in providing the physical input to mathematical modelling.

An analogue model is a system reproducing a prototype situation in a physically different medium. This technique depends on the equations representing the prototype and model being mathematically identical. Thus torsional vibrations of a bar may represent the water-level oscillations of a simple surge tank, and both can be simulated by the voltage changes in an electric circuit, i.e. by an electrical analogue.

Although engineers use the terms mathematical model, numerical model and computational model as synonyms, there is a clear distinction between them (Samuels (1993)). A mathematical model is a set of algebraic and differential equations that represents the interaction between the flow and process variables in space and time. It is based on a certain set of assumptions about the physics of the prototype flow and associated environmental processes. These assumptions will set clear limits to the domain of applicability of the mathematical model and any numerical and computational model that may be derived from it. A prerequisite for the development of a mathematical model is an understanding of the key physical processes involved, leading either to fundamental principles such as Newton’s laws of motion or to well-attested empirical relationships such as the Chezy and Manning resistance laws.

It is extremely rare for a mathematical model to be amenable to an exact closed-form solution except for the simplest geometries. Hence, the power of mathematical models was only realized with the availability of cheap, reliable computing from about 1960 onwards. Mathematical models of most physical phenomena are non-linear, necessitating the use of numerical methods when developing approximate solutions with the aid of a digital computer. This leads to the definition of a numerical model.

A numerical model is an approximation of a mathematical model of some prototype situation, giving a computable set of parameters that describes the flow at a set of discrete points. Many numerical models can be formulated from the same underlying mathematical model by employing alternative numerical methods and mathematical manipulations. The performance of the numerical models will be determined by the properties of the numerical methods employed, and for the same geometric and boundary data may give significantly different results. These differences are often masked, in part, by the calibration procedure.

A numerical model, like a mathematical model, is not specific to any particular site, and the strength of both these types of model lies in their generality. A specific application will require data from the prototype site and a computer with a program to organize the data and execute the calculations.

A computational model is an implementation of a numerical model on a computer system with the relevant data from a specific site. The results of the computational model depend on a variety of factors, including the quality of the prototype data, the details of data processing, possibly the internal organization of the calculations and the type of computer used.

Many computational modelling systems and packages are available for a variety of hydraulic engineering problems. The end user may have to choose which model to use, and certainly will have to be able to interpret the model results critically and responsibly. It is hoped that this book will provide at least some guidance on how to distinguish between models that are appropriate for a particular application and those that are not. It is important that the results of a computational model should not be accepted as definitive just because the numbers were produced by a computer – the results must also make physical sense. Past (field) results should be used to gain a better understanding of what is happening physically and why a given model does not reproduce observations accurately, and to assess imprecisions and/or uncertainty intervals in the results, and always to calibrate the model.

From data handling the discipline of computational hydraulics has grown to hydroinformatics, which uses simulation modelling and information and communication technologies (ICT) to help to solve problems in hydraulics, hydrology and environmental engineering for better management of water-based systems. In a further development, artificial neural networks attempt to simulate – in a crude way – the working of a human brain by passing on information from one ‘neuron’ to all other ‘neurons’ connected with it. The output of the model is related to the input through a set of functions with constants determined during the ‘training’ of the network; a large set of wide-ranging data is required to train a network to achieve good results.

In conclusion, it may be helpful to identify the principal differences between the types of modelling discussed in this chapter. Physical (scale) models (hydraulic and aerodynamic) are based on full fluid physics but at a reduced geometric scale, whereas a computational model is at full prototype scale but embodies only approximate physics. A physical model provides a continuous representation of the prototype but a computational model offers only a finite dimensional approximation; if a model does not reproduce observations accurately, it is necessary to assess the uncertainty in the results. Physical and computational modelling should not be viewed as conflicting methods of investigation; rather, they have complementary strengths and weaknesses. Often a hydraulic engineering problem will require a combination of these methods, i.e. hybrid modelling, to achieve a cost-effective solution.

References

Novak, P. and Čábelka, J. (1981), Models in Hydraulic Engineering – Physical Principles and Design Applications, Pitman, London.

Samuels, P. G. (1993), What’s in a Model? Paper presented to the River Engineering Section IWEM, January 1–12.

Chapter 2

Theoretical background – mathematics

2.1 Ordinary differential equations

2.1.1 Definitions

Physical situations described by quantities that vary continuously with respect to their position in space and possibly with time can usually be described in terms of systems of partial differential equations (PDEs). These are equations that relate the quantities involved to some of their derivatives with respect to space variables and time. In the simplest case, when only a single quantity u(t) depending on a variable t is involved, the variation of u(t) with respect to t is described by an ordinary differential equation (ODE) that relates u(t) to some of its derivatives. If the highest order derivative involved in an ODE is dⁿu/dtⁿ, the ODE is said to be of the nth order. The variable t is called the independent variable, and in physical situations t is often the time, while the quantit...