A transient flow solution can be obtained numerically by solving Equations 1 and 2 (along with the appropriate initial and boundary conditions) in which pressure and flow are variables dependent upon position and time. In essence, five different numerical procedures are popularly used to approximate the solution of the governing equations, and thus to analyze hydraulic transients in water distribution systems. Three are Eulerian and two are Lagrangian in nature. The Eulerian methods update the hydraulic state of the system in fixed grid points as time is advanced in uniform increments. The Lagrangian methods update the hydraulic state of the system only at times when a change actually occurs. Each method assumes that a steady-state hydraulic solution is available which gives static flow and pressure distribution throughout the system.
Eulerian methods consist of the explicit method of characteristics, explicit and implicit finite difference techniques, and finite element methods. In closed conduit applications, by far the most popular of these techniques is the method of characteristics. The method of characteristics is the most accurate in its representation of the governing equations. When finite difference and finite element techniques are employed, the derivatives in the governing equations are replaced with approximate difference quotients. In the method of characteristics, by contrast, only the nonlinear friction term needs to be approximated (typically by a linear difference term). Explicit finite difference schemes have also significant restrictions on the maximum time step if stable solutions are to be achieved. Although implicit methods usually overcome the stability limitations, they require a simultaneous solution for every unknown in the problem at each time step.
The second important distinction between the three methods is that only the method of characteristics explicitly links the time step to the space step. The main drawback of the method of characteristics is that the time step used in the solution must be common to all pipes. In addition, the method of characteristics requires the distance step in each pipe to be a fixed multiple of the common time interval, further complicating the solution procedure. In practice, pipes tend to have arbitrary lengths and it is seldom possible to satisfy exactly both the time interval and distance step criteria. This “discretization problem” requires the use of either interpolation procedures (which have undesirable numerical properties) or to distort the physical problem (which introduces an error of unknown magnitude). Finally in order to satisfy stability criteria and ensure convergence, the method of characteristics requires a small time step. The stability criterion is developed by neglecting the nonlinear friction term and is referred to as the Courant condition. The Courant condition relates the computational time increment () to the spatial grid size ( ). A numerical scheme is stable if and only if .
Lagrangian methods consist of the wave plan method and the wave characteristic method. Both methods track the movement of pressure waves as they propagate throughout the system and differ in the way the hydraulic state of the system is updated and how friction effects are treated. The wave plan method updates conditions only at times when a change actually occurs while the wave characteristic method updates the state of the system at fixed time steps (analogous to the Eulerian approach). The effect of line friction on a pressure wave is accounted for by modifying the wave using a nonlinear characteristic relationship describing the corresponding pressure head change as a function of the line flow rate. While it is true that some approximation errors will be introduced using this approach, the errors introduced can be minimized using a distributed friction profile (piecewise linearized scheme).
InfoSurge computes hydraulic transient flow conditions in water distribution system using the Lagrangian Time Driven Wave Characteristic Method. The method yields solutions that are identical to those obtained from exact solutions or those based on the method of characteristics or the wave plan method, but has the advantage of being computationally more efficient – producing accurate solutions in an expeditious manner even for very large networks. The method can also readily accommodate a comprehensive library of surge protection devices including Open Surge Tanks, Closed Surge Vessels, Bladder Tanks, Hybrid Tanks (vented to admit air), Bypass Lines, Check Valves, Feed Tanks (provide inflow to prevent cavitation), Air Release/Vacuum Valves (2 and 3 Stage Valves), Pressure Relief Valves, and Surge Anticipation Valves.