On the premise of water hammer theory, a numerical model is proposed for simulating the filling process in an initially empty water conveyance pipeline with an undulating profile. Assuming that the pipeline remains full and ignoring air and water interactions in the already filled pipeline, the ongoing filling is simulated using the method of characteristics on an adaptive computational grid. The performance of the model is verified using previously published experimental and rigid column data. The model nicely replicates published experimental data. The model shows that the movement of the filling front into the system can be assumed as a rigid column as long as the flow away from the filling front is undisturbed elsewhere. Furthermore, applying the model to a hypothetical pipe system with an inline-partially open valve shows that the proposed model is robust enough to capture the transient events initiated within the moving column, a vital capability that the existing rigid water column models lack.
Empty water conveyance pipelines are usually filled cautiously to prevent the onset of harmful transient pressures. Lack of sufficient experimental data forces operators to take a conservative approach to filling. As Watters [
Such a controlled operation over an undulating terrain creates alternating segments of open channel and pressurized flow along the line with the number and arrangement of these segments depending on both the line’s profile and the filling protocol. These segments are connected via a series of moving hydraulic jumps which are in turn responsible for slowly pushing the air out of the pipeline through installed and functioning air valves.
In practice, filling may not be carried out as smoothly as required by the dictates of good design. Poor operation caused by either human error or lack of proper control will result in rapid filling with its consequent drawbacks. Izquierdo et al. [
Two important features of the flow during rapid filling are the bulk motion of the water column and any water hammer pressures waves propagated through the water column. The second feature is of great interest only once the moving water column is locally disturbed such as if the water column meets a partially open valve or a stagnant water column. However, in the absence of abrupt changes, the bulk motion of the water column dominates flow.
Depending on what flow features are of primary concern, different mathematical models are available ranging from simplified to highly complex. If the bulk motion of water is of primary interest, the rigid column model is usually the numerically cheapest solution in terms of computational time and the ease of implementation. This simplified model assumes that the whole water column accelerates and decelerates as a unit. Liou and Hunt [
However, if the water column is internally disturbed, a rigid column model can no longer capture the resulting water hammer pressures and a more complete dynamic model is then required to simulate the key flow features. Only a few more complete dynamic models specifically focusing on undulating pipelines have thus far been presented, though more have been proposed to cover filling in sewer systems. Sewer systems have likely received more interest and care because filling events are more frequent over a sewer’s lifetime, and thus these systems must be carefully protected from the drawbacks of filling. By contrast, filling pressures in water conveyance lines can usually be controlled through a well chosen operational strategy. Nevertheless the filling processes in sewer systems and in water conveyance pipelines are physically similar except for the dominant pace of the transient. This implies that sewer models used can be used for water conveyance applications provided that they can capture any water hammer events that occur.
Shock capturing and shock fitting methods are two available techniques that have been proposed to solve the full dynamic 1D equations (momentum and continuity equations). In typical shock capturing methods, a hypothetical slot at the top of the pipe allows simulation of both pressurized and free flow by a single set of equations. Since a single set of equations is employed in the simulation, the track of the pressurized front is automatically included. The method was originally suggested in honour of Preissmann in 1961 [
However, this method possesses some important numerical limitations. First, to avoid nonlinear numerical instability, an artificially low wave velocity is required computationally. As a result, the slot method not only underestimates transient pressures (which depend strongly on the wave velocity magnitude) but also fails to properly track the timing of the pressure waves. In addition, the method cannot predict negative pressures in the pressurized zone of the real system; whenever negative pressures occur in the model, the flow is automatically switched back to open channel flow. The later limitation is addressed by Vasconcelos et al. [
Shock fitting is another popular method which is extensively employed in transient analysis of the sewer systems [
Another shock fitting family method was presented by Malekpour and Karney [
The first intent of the current paper is to develop and test a robust and flexible numerical model for simulating rapid filling in water conveyance pipelines having an undulating profile. The proposed model is a type of shock fitting method which applies the method of characteristics along with an expandable computational cell. As will be seen, the model robustly captures extreme water hammer spikes during the filling without stability issues. Although beyond the scope of the current work, it can be shown that this method can also capture column separation during rapid filling events. The secondary aim is to provide physical insight into the filling process through evaluating the proposed model results in comparison with results obtained from existing rigid column models.
The transient flow or water hammer in the closed conduits is governed by the following two partial differential equations known as the momentum and continuity equations, respectively, [
To solve the equations in the context of rapid filling, the following assumptions are made: (1) the pipeline remains full with a well-defined vertical front during the filling, (2) the pipe system is sufficiently vented so that the air pressure remains essentially atmospheric and imposes little retarding force against the filling water column, (3) except in the immediate vicinity of the front (which is assumed incompressible), the water-pipe systems are slightly compressible everywhere, and (4) the frictional flow-resistance relationship for steady flow is a good approximation for the transient flow.
To evaluate whether the first assumption is reasonable during filling, a velocity-based criterion was proposed by Liou and Hunt [
Inspired by the traditional model used in furrow and border irrigation modeling [
Expandable implicit finite difference computational grid.
Although this model nicely replicates experimental data, convergence was a challenging issue. To assure convergence, the employed finite difference must be fully implicit. Yet, the resulting scheme provided accurate results only if the Courant number is kept close to 1 or when there are slower transients in systems with large effective storage, through either low wave speeds or additional storage elements in the system. Otherwise, the dissipative nature of the scheme distorts the accuracy of the results. However, in the model proposed here, the Courant numbers are enforced computationally rather than by setting a prior constant value. Since the time step is calculated based on the velocity of the bulk motion of water column (
To overcome this problem, an improved version of the aforementioned model is proposed herein. In this model, the time step is fixed during the simulation and the water column front position is sought at the current time. This implies that, as shown in Figure
Proposed computational grid.
The method of characteristics is a powerful tool for solving partial differential equations governing transient flow in closed conduits. This method transforms the governing equations into four ordinary differential equations which can be integrated through the paths called the characteristic curves to find the dependent variables at specific distances and times.
The characteristic equations for the water hammer equations are as follows [
Though analytical solutions are possible in trivial cases, the numerical methods are inevitable in practical systems with complex boundary conditions. The solution of the compatibility equations can be easily made based on a finite difference computational mesh depicted in Figure
A typical finite difference mesh.
The simulation starts with a computational grid of just one cell. Since the water column is elongating with time in this cell, it is impossible to directly use (
Typical computational cells at different stages of the filling.
To calculate the three unknowns,
Upstream reservoir sketch.
To prevent dealing with short pipe segments which can greatly increase simulation times, the water found between the valve and the reservoir is treated in the boundary equation rather than as a separate pipe. As can be seen in (
No analytical solution is available for (
For a system with a single pipe, the time step can be easily calculated by
In practice, however, it is impractical to use different time steps and care is needed to retain similar steps in each pipe. One solution is to apply the least calculated time step for all pipes. Since in this case the characteristic lines do not meet the computational grid points for many pipes, linear interpolation is required to extract information of the previous time line. The second alternative is to slightly adjust wave velocities to make all time steps the same, though the adjustment should not exceed 15% or the original value [
Once the time step is determined, the solution can be established step by step until the water column length equals or slightly exceeds the computational distance interval
Once the first cell has been filled, as shown in Figure
Simultaneous solution of (
To calculate the remaining unknowns, consider Figure
The heads at upstream and downstream of the node can be then calculated by substituting the discharge into (
The numerical procedure presented for stage 3 can be employed for the rest of the simulation till the water column reaches the end of the pipe system.
To evaluate the model results, the experimental data presented by Liou and Hunt [
Steady flow tests showed that the average values for the entrance loss coefficient
Figure
Water column velocity during the filling.
Indeed, the understanding of the dynamic of the flow seems to be challenging because, in the different stages of the filling, the contribution of the driving and the resistive forces would be different. To examine how these forces come to action, consider Figure
Contribution of the active heads during the filling.
Figure
Finally, to evaluate if the proposed model is capable of capturing huge water hammer pressure spikes, the model was applied on a simple pipe system (see Figure
Hypothetical pipe system sketch.
Water hammer pressure snapshots during the filling.
The water column velocity versus the length of the column is also presented in Figure
Water column velocity during the filling.
All these numerical evidences demonstrate the robustness of the proposed model in simulating the rapid filling in the pipe system, significantly even when the modelled system experiences strong water hammer pressures.
Based on water hammer theory, a numerical model is proposed for simulating rapid filling in the pipe system with undulating profile. Assuming the pipe remains full during the filling and neglecting the air and water interactions, the model is formulated through using the method of the characteristics along with an adaptable computational grid. The model is then verified by the experimental data presented in the literature. The model is also tested for the capability in capturing high water hammer pressures through a hypothetical example.
According to the results obtained from the model, the following conclusions are made. First, the water column moves in the system as a rigid column during the rapid filling provided that the water column is not locally disturbed along its length. Second, rapid filling can have dramatic effects at its early phase when the water column experiences high acceleration. As the water column lengthens, the acceleration reduces and the pipe head losses control the pace of the filling front. Under such a circumstance, a quasisteady model can also be utilized to capture the bulk motion of water. However, if the water column be disturbed along its length, neither the rigid column nor quasisteady models are capable of capturing the main feature of the flow. Third, the interaction of the moving column with the inline-partially open valves results in the onset of overpressures whose magnitude depends on the water column velocity at the moment of contact with the valve and also the initial rate of the opening of the valve. Though the overpressures can be severe, they are not persistent. In fact, the overpressures are quickly damped because the valve efficiently releases the internal energy accumulated in the system.
Finally, the numerical evidences presented in this paper demonstrate that the proposed model is robust enough to capture the water hammer pressures (if any) while simulating bulk motion of the water column. Capturing such water hammer pressures, as is attempted here, is indeed a vital component of any serious study of column separation in systems having an undulating profile. The interesting conclusion of the work is that applied modelling of complex physical process is possible but great care must be used to match the numerical aspects to the physical ones and that some subtlety is needed to ensure a good agreement between predicted and measured responses.