During a transient analysis there is a definite wave travel time, _i= L _i/a _i, where L _i and a _i are the length and wave speed, respectively, of pipe i in a network. In the method of characteristics (MOC), the solution at each (calculation) point is advanced one time step, T, in each iteration starting from its known initial value. During the time T, the wave will travel from one point to its neighbour. Since the points lie in the interior and at the ends of each pipe, it is necessary that _i be a multiple of T; in other words, it is mandatory for a wave to traverse any pipe in an integral number of time steps. To achieve this goal, the times _i may need to be adjusted as discussed below.

Travel Time Statistics

HAMMER computes the following statistics for wave travel times in a network with n pipes:

Some of these statistics are employed in determining an appropriate time step.

Automatic Selection of Time Step

The transient calculation time step, T, depends on , , n, and . Each i must be divisible by T. We start by selecting an integer:

based on heuristics attempting to balance accuracy and performance as follows:

where (n) and y() are respectively monotonically increasing and decreasing functions which are defined as follows:

Finally, the time step T is determined as / N.

Adjustment in Wave Speeds

In the selection of a time step, there is nothing to ensure that the _i will be exactly divisible by T. To accomplish this task, the _i can be rounded according to the following rules:

To round the _i, one can adjust the length, wave speed or both for each pipe.

If the length is adjusted, then errors will arise in the mass, momentum, energy and friction coefficient. Moreover, if the Viewer were to display the adjusted lengths, then the user could conceivably believe that the pipes are being distorted. For slower changes leading to mass oscillations in the system, it can be demonstrated that the alterations to the network will have an impact on the results.

On the other hand, should the wave speed be adjusted, this can lead to errors in the calculation of rapid transients - think of Joukowsky's formula which depends on wave speed but is explicitly independent of length.

The user can choose whether to adjust length or wave speed in HAMMER (see Transient Time Step Options Dialog) does have the responsibility to exercise some discretion in constructing a model of a hydraulic system. As a approximate measure of the adequacy of the model, a warning message appears in the output log in the event that any adjustment exceeds the Max Adjustment value in the Transient time Step Options dialog box. The default value for this parameter is 75%; i.e., | a_i| / a _i > 0.75 when adjusting wave speed, or | L _i | / L _i > 0.75 when adjusting length, then a user notification message suggests that the user consider reducing the time step or subdividing longer pipes and/or lengthening shorter pipes.

It should be noted that large wave speed adjustments in small pipes in branches, or in main lines with slowly changing flows, may have little impact on the hydraulic transients in the system. However, the impact could be significant if transients in the short pipes (whose wave speeds tend to be reduced) are of interest.