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This book describes the fundamental phenomena of, and computational methods for, hydraulic transients, such as the self-stabilization effect, restriction of the Joukowsky equation, real relations between the rigid and elastic water column theories, the role of wave propagation speed, mechanism of the attenuation of pressure fluctuations, etc. A new wave tracking method is described in great detail and, supported by the established conservation and traveling laws of shockwaves, offers a number of advantages. The book puts forward a novel method that allows transient flows to be directly computed at each time node during a transient process, and explains the differences and relations between the rigid and elastic water column theories. To facilitate their use in hydropower applications, the characteristics of pumps and turbines are provided in suitable forms and examples. The book offers a valuable reference guide for engineers and scientists, helping them make transient computations for their own programming, while also contributing to the final standardization of methods for transient computations.

### Chapter 1. Introduction

Abstract
Hydraulic transients represent highly complex fluid flow processes. They are encountered at almost all hydropower stations as well as at water supply networks, when the hydraulic systems are getting regulated, started-up or shut-down. Each time and under certain conditions, hydraulic transients often lead to various undesirable occurrences like rapid pressure rises, cavitations with noises, system instability, and cumulative fatigue of pipe materials. All these phenomena are usually well controlled based on optimized system designs and operations. Uncontrolled hydraulic transients in hydropower stations, for instance, take place at load rejections or emergency shut-downs of hydraulic machines like pumps and turbines. Because each occurrence of hydraulic transients leads to remarkable and rapid pressure rise in considered hydraulic systems, the phenomenon is in engineering applications also called pressure shock or water hammer. It represents a very important sub-discipline of fluid mechanics and has drawn great attention in related fields.
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### Chapter 2. Stationary Flows and Flow Regulations

Abstract
Most industrial processes for fluid flows are stationary, even though they are all constructed with unsteady regulation facilities to achieve their control. Such processes are regularly found in hydraulic systems of hydropower stations. They ensure stationary power output of water turbines. At the nominated flow rates, hydraulic machines run under optimized operation conditions and with highest hydraulic efficiency. Stationary water flows in hydropower stations, thus, represent the basic form of all relevant hydro-mechanical processes. They also represent the initial state of all hydraulic transients, such as normal startup and shutdown conditions of machines as well as the system reactions caused by a power outage. For this reason, hydrodynamics of stationary flows and flow regulations will be treated in this chapter. All processes will be considered for fluids exhibiting viscous friction.
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### Chapter 3. Transient Flows and Computational Methods

Abstract
Hydraulic transients in a hydraulic system are encountered when the flow is found under regulations, for instance, for process control.
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### Chapter 4. Rigid Water Column Theory and Applications

Abstract
Transient flows in hydropower stations are given each time when hydraulic machines, such as pumps and turbines, are started, stopped or regulated. For computing such transient flows, basically, both the rigid and the elastic water column theory are applicable. Against the elastic water column theory which is generally applicable for all types of transient flows, the rigid water column theory ignores the compressibility of the fluid and elasticity of the walls of the pipeline. Thus, it basically has its application restrictions to “short pipes”. It is, furthermore, unable to resolve high-frequency shock pressure fluctuations in the flow. However, the method may have its favorable applications when the low-frequency flow oscillation in a hydraulic system is in focus. Such a low-frequency flow oscillation is quite comparable to the flow oscillation in an open U-tube. In hydropower stations, it is often necessary to know the damped flow oscillations, for instance, between the lake and the surge tank after each load regulation (Fig. 3.1), especially, after an undesired load rejection. This often includes evaluations of both the stability performance of the system and the capacity of the used surge tank. Reliable computations with appropriate computational algorithms must always be completed.
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### Chapter 5. Surge Tank Functionality and System Stability

Abstract
Each hydraulic system in a hydropower station usually involves one or more surge tanks. Such a surge tank behaves as a significant component. It enables the hydraulic system to be rapidly started and stopped. Its essential functions have been found to relieve the system pressure and to stabilize the system operations under conditions of hydraulic transients.
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### Chapter 6. Elastic Water Column Theory and Fundamentals

Abstract
Methods of transient computations have been progressed from the earlier graphical to the nowadays numerical method based on computer technologies. Especially, the method of characteristics (MOC) has dominated applications in engineering hydraulics. This method, however, requires both time and space discretizations.
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### Chapter 7. Wave Tracking Method

Abstract
The method of tracking waves in the fluid flows goes back to the suggestion of Allievi (General theory of the variable flow of water in pressure conducts. Riccardo Garoni, Rome, (1902)) in the earlier time. The method, in effect, relies on the d’Alembert solution of the one-dimensional hyperbolic wave equation and represents a Lagrangian approach in fluid mechanics. The method is simply based on tracking the shock wave propagation in the time series. This indicates that both the shock pressure and the flow velocity at any interested position and the components in a hydraulic system can be directly computed. The latter may for instance be hydraulic machines, regulation valves or surge tanks. The approach clearly represents a significant advantage against the method of characteristics, which always requires the predefined characteristic grids in the time-space domain.
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### Chapter 8. Method of Characteristics

Abstract
The method of characteristics has played an essential role in computing hydraulic transients up to now. Almost all applicable software tools are based on this method. In principle, the method of characteristics is a mathematical technique for solving so-called hyperbolic partial differential equations.
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### Chapter 9. Method of Direct Computations and Transient Conformity

Abstract
For computing hydraulic transients in a hydraulic system, basically, two computational models or theories are available: rigid and elastic water column theories. In applications, they are also called rigid and elastic water column methods. Both theories (methods) are based on quite different hydro-mechanical principles. More precisely, the rigid water column theory cannot be simply considered as an approximation of the elastic water column theory. The former assumes incompressibility of the fluid and thus infinity of the wave speed ($$a = \infty$$) in the flow. Obviously, it is in no case an approximation, if compared with the real wave speed of about a = 1200–1400 m/s in the elastic water column theory.
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### Chapter 10. Hydraulic Characteristics of Pumps and Turbines

Abstract
Almost all hydraulic systems in hydropower stations are constructed for turbine and pump operations. Except for Pelton turbines, at which only the injector nozzles are included in the hydraulic system, all other types of turbines and pumps are found within the hydraulic network and the pipeline system. This fact determines that to compute each hydraulic transient the hydraulic characteristics of the respective turbines and pumps must be taken into account. In case of emergency shutdowns of turbines or pumps, for instance, the extended characteristics of these machines with changeable rotational speed must be known. Furthermore, the moment of inertia of the entire rotor system plays an essential role in affecting the changing rate of rotor rotations.
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### Chapter 11. Application Examples of Complex Transient Computations

Abstract
The wave tracking method basically represents a highly advanced method for computing hydraulic transients. Because of its simple and clearly structured computational algorithms, it can be easily applied to any complex hydraulic system.
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