The Instantaneous waterhammer sequence
Often during waterhammer analysis only the event’s initial pressure surge is considered. This surge is often calculated via the Joukowsky equation despite the many scenarios when the Joukowsky equation does not describe the transient’s peak pressure.
However, a waterhammer event does not end with the valve closure, rather it begins a waterhammer sequence which can wreak havoc on a piping system. Understanding the wave nature and the sequence of waterhammer events provides a foundation for additional waterhammer analysis and mitigation.
This article focuses on a “worst-case scenario” of an instantaneous valve closure. This rapid valve closure ensures the entire pipe sees the full Joukowsky predicted pressure rise, making the sequence more dramatic.
The four phases of the waterhammer sequence
There are four major phases to the water hammer sequence which cause flow to halt, flow out of, and flow into the pipe system. This change in flow direction through rapid acceleration results in large positive and negative pressure surges that transmit through the pipe. The entire sequence is captured in twice of the system’s communication time (T), with each phase taking half of the communication time. In the examples below, the communication time is 10 seconds, indicating each phase is roughly 5 seconds long. How flow and pressure in the pipe vary in each of these phases is explained in detail in their own section.
Our example will rely on a single pipe flowing to a valve, as found in Figure 1. The waterhammer sequence upstream of the valve will be explored first, before comparing to the consequences downstream of the valve.
Figure 1: Simple piping system with flow being halted through an instantaneous valve closure.
Phase 1: Flow coming to a halt (0 < t < T/2)
The first phase is the most intuitive of the four and occurs immediately following the valve’s closure.
As fluid travels through a pipe, it has a certain amount of kinetic energy due to its velocity, as well as a certain amount of potential energy in the form of pressure. During a valve closure the fluid is unable to continue forward, suddenly converting the fluid’s kinetic energy into potential energy. How this fluid’s momentum is converted to pressure is the foundation behind the Joukowsky equation.
If fluid flow is analyzed differentially, each portion of flow will experience this sudden stoppage. Imagine cars braking in traffic one by one sending a signal wave backward. Thus, as each portion of flow halts, each portion will see significant pressure rise. This is what causes a high-pressure wave to travel upstream.
Another way to visualize the first phase of the waterhammer sequence is through an animation. In Figure 2, the initial steady state pressure is relatively flat and the flowrate is non-zero. As the valve is slammed shut, the flow drops to 0 (halting forward momentum) at the same time as a drastic increase in pressure. This wavefront of halted flow and increased pressure continues to move backward. In this case, the wave encountering a fixed pressure reservoir begins the second phase.
Figure 2: Phase 1 halting forward flow which causes an increase in pressure
Phase 2: Driving force reversal and line draining (T/2 < t < T)
The second phase begins when the wave reaches the fixed pressure reservoir. The reservoir in this case has a fixed pressure, meaning the pressure wave created by the transient event does not significantly impact the pressure of the reservoir. As an extreme example, imagine the negligible pressure created by blowing on a straw in an Olympic pool of water.
With the increased line pressure due to the halted flow, there is now a reversal in driving force. The fluid in the line wants to flow from the now higher-pressure pipe to the lower-pressure reservoir. As the fluid begins to flow, some of its potential pressure energy is converted to kinetic, reducing the pressure in the line. This pressure reduces to near the reservoir pressure as the line drains.
Approaching the pipe differentially once more, each portion of stagnant fluid is drawn out of the pipe, reducing its pressure in the process to create a wavefront. This is shown in an animation in Figure 3, demonstrating the pressure wave traveling opposite to the initial high pressure wave. Notice flowrate becomes negative as flow drains into the reservoir. This phase ends when the wave reaches the closed valve.
Figure 3: Phase 2 sees a change in pressure driving force, leading to flow in the reverse direction.
Phase 3: Low pressure halting reverse flow (T < t < 3T/2)
The third phase begins when the draining liquid reaches the closed valve. The liquid entrained in the pipe continues to drain until there is no more fluid to drain (marked by a closed valve or similar dead end). At this closed point, the momentum of the draining liquid causes a significant drop in pressure at the valve. This creates a low-pressure relative to the reservoir pressure. The degree of low pressure is quite significant and halts the reverse flow draining from the pipe. Conceptually, this-low pressure region is similar to covering the end of a straw with a finger to stop it draining.
Approaching the phase differentially once more, each increment of backward flow is brought to a halt by the previous segment’s low pressure. This creates a low-pressure wavefront as flow is halted, shown in animation in Figure 4.
Figure 4: Phase 3 is caused by a low-pressure created at the closed valve. This halts flow out of the pipe as the low-pressure travels upstream as a wave.
Phase 4: Filling once more (3T/2 < t < 2T)
Now with a relative vacuum in the line, flow again wants to travel from the now higher-pressure reservoir into the lower-pressure pipe. This causes flow to travel into the pipe, reducing the low pressure caused in phase 3. Flow continues until it reaches the closed valve, suddenly coming to a stop. This sudden stop causes another pressure wave and the cycle begins anew. An example of phase 4 and the beginning of another phase 1 is found in Figure 5.
Figure 5: Phase 4 again sees a flow reversal, now flowing back into the pipe. This flow continues until it reaches the closed valve where it suddenly halts and begins the cycle once more.
A cycle of phases
An understanding of each individual phase helps to explain why waterhammer is a repeating cycle. Below in Figure 6 is a loop of the waterhammer cycle working through the phases showing how each feeds into the next.
Figure 6: A repeating loop of the 4 phases of the waterhammer cycle, where each phase is roughly 5 seconds.
Downstream of a valve
Examining the waterhammer cycle downstream of a valve closure paints a very similar picture as the pipes upstream but with a different starting point. Rather than using the valve to a wall for fluid to slam into, downstream of the valve initially creates the low pressure to halt flow. In essence, downstream of the valve the cycle begins at Phase 3 but follows the same wavefront sequence. Below in Figure 7 is a repeating loop of the waterhammer cycle downstream of the valve.
Figure 7: A repeating loop of the 4 phases of the waterhammer cycle highlighting the differences for downstream of the closure
A real-world example
While it is easy to understand the nature of waterhammer through neat pressure and flowrate graphs, often waterhammer must be diagnosed in a constructed, physical system. Below is a video example of a waterhammer event in a clear piping system. Try to spot the different phases of the waterhammer sequence and listen to the noise that accompanies the event.
Figure 8: Video of a physical piping system during a waterhammer event
An understanding of the wave nature of the waterhammer cycle helps build a foundation for further waterhammer analysis. Considering why cavitation would happen under low pressure or how waves interact is second nature with an understanding of waterhammer waves.