History of Solution Methods

The study of hydraulic transients is generally considered to have begun with the works of Joukowsky (1898) and Allievi (1902). The historical development of this subject makes for good reading (Wood F., 1970). A number of pioneers made breakthrough contributions to the field, including R. Angus and John Parmakian (1963), who popularized and refined the graphical calculation method. Benjamin Wylie and Victor Streeter (1993) combined the method of characteristics with computer modeling. The field of fluid transients is still rapidly evolving worldwide (Brunone et al., 2000; Koelle and Luvizotto, 1996; Filion and Karney, 2002; Hamam and McCorquodale, 1982; Savic and Walters, 1995; Walski and Lutes, 1994; Wu and Simpson, 2000).

Various methods have been developed to solve transient flow in pipes. These range from approximate equations to numerical solutions of the nonlinear Navier-Stokes equations:

Arithmetic method—Assumes that flow stops instantaneously (in less than the characteristic time, 2 L/a), cannot handle water column separation directly, and neglects friction (Joukowski, 1898; Allievi, 1902).
Graphical method—Neglects friction in its theoretical development but includes a means of accounting for it through a correction (Parmakian, 1963). It is time-consuming and not suited to solving networks or pipelines with complex profiles.
Design charts—Provides basic design information for simple topologies at a few specific points (valve closure, pump and pipeline with no protection, surge tank, or air chamber protection). This method has been replaced by computer programs (Fok, 1978; Fok, 1980; Fok et al., 1982) based on the transient energy concept and backed by field and laboratory work (Fok, 1987).
Wave-plan method—Represents initial transient disturbances as a series of pulses and tracks reflections at boundaries (Wood et al., 1966).
Method of Characteristics (MOC)—Most widely used and tested approach, with support for complex boundary conditions and friction and vaporous cavitation models. Bentley HAMMER CONNECT uses the MOC. It converts the partial differential equations (PDEs) of continuity and momentum (e.g., Navier-Stokes) into ordinary differential equations that are solved algebraicially along lines called characteristics. An MOC solution is exact along characteristics, but friction, vaporous cavitation, and some boundary representations introduce errors in the results (Gray, 1953; Streeter and Lai, 1962; Elansary, Silva, and Chaudhry, 1994).

Haestad Press' 2002 Advanced Water Distribution Modeling and Management documents other less-common methods. Transients have also been studied using:

Laboratory Models—A scale model can be built to reproduce transients observed in a prototype (real) system, typically for forensic or steam system investigations. As a design method, this approach is limited by model scale effects and by very high costs. However, models have provided invaluable basic research data on vaporous cavitation and vortex shedding (St. Anthony Falls) and transient friction (Perugia, Italy).
Field Tests—Field tests can provide key modeling parameters such as the pressure-wave speed or pump inertia. Advanced flow and pressure sensors equipped with high-speed data loggers make it possible to capture fast transients, down to 5 milliseconds. Methods such as inverse transient calibration and leak detection use such data. Like all tests, however, data are obtained at a finite number of locations and generalizing the findings requires assumptions, with uncertainties spread across the system. At best, tests provide local data and a feel for the systemwide response. At worst, tests can lead to physically doubtful conclusions limited by the scope of the test program.

Neither laboratory models nor field testing can substitute for the careful and correct application of a proven hydraulic transient computer model, such as Bentley HAMMER CONNECT.

The extended-period simulation (EPS) capability of models such as WaterCAD or WaterGEMS does not consider momentum, and is therefore incapable of analyzing hydraulic transients. Such simulations are sufficient to analyze hydraulic systems that undergo velocity and pressure changes slowly enough that inertial forces are insignificant. If a system undergoes large changes in velocity and pressure in short time periods, then transient analysis is required.