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When the Joukowsky Equation Fails

When the Joukowsky Equation Fails

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The Joukowsky equation predicts the pressure rise from an instantaneous change in velocity, a change most often caused by a valve closure. It is typically used by engineers as a first-pass prediction of pressure surges from waterhammer.


ΔP = ραΔν

In Equation 1,  is the instantaneous change in pressure,  is the fluid density,  is the wavespeed (function of the fluid and pipe properties), and is the instantaneous change in velocity. While this is a valid equation for waterhammer analysis, that is only the case if the assumptions going into the equation are well understood.

The Joukowsky equation alone does not accurately tell the whole waterhammer story. Most engineers believe this predicted surge to be the conservative maximum pressure since it assumes an instantaneous change in velocity while most flow is slowed more gradually. However, there are many cases where an instantaneous closure (and the resulting Joukowsky predicted pressure rise) is not the worst case scenario.

Primary Assumption of Joukowsky

The equation’s primary assumption is that the predicted pressure rise occurs at a specific location (the closing valve) at a specific time (immediately following the valve closure). Engineers, however, often take this spatially specific pressure rise and apply it throughout the entire piping system as the wave propagates. Yet, both time and system characteristics alter that pressure wave as it travels, which can both cause surges higher and lower than the predicted pressure rise.

If the engineer assumes that the predicted pressure rise is found everywhere else in the system, he must understand what conditions cause this to be true. These required conditions include constant diameter pipes, no splits or junctions, negligible pressure drops, and no cavitation. Many of these requirements are unrealistic for a physical system.

Inevitably, maximum pressure predictions become more difficult when there are deviations from the ideal case. There are 3 main situations causing deviation from the Joukowsky ideal include

  • Non-uniformity in the piping network causes intermediate reflections which can add to the pressure surge through constructive interference.
  • Flow causes pressure drop from pipe friction, so when that flow stops during a transient, the pressure that was lost to friction is added back. Not only are frictional losses recovered, but they are also added on top of the new pressure surge value.
  • Low-pressure conditions are part of the waterhammer cycle and can reduce pipe pressures such that the fluid cavitates. The sudden collapse of this vapor pocket when the line repressurizes often causes higher pressures than the initial surge.

Piping system reflections

One should expect a wave reflection if there is any sort of transition in a piping network. These transitions include, but are not limited to, pipe diameter changes, valves and fittings, dead ends, tanks, pumps, and pipe friction variations. When a pressure wave hits these transitionary spots, the wave will partially reflect. The resulting interaction of these reflections produces effects far different from what is predicted with the Joukowsky equation.

Much of the time, these reflections work to dampen waterhammer effects rather than exacerbate them. By redirecting the pressure wave the system is able to redirect its energy. However, just like how an initial pressure surge cannot describe an entire transient, it is important to analyze the wave following its reflection. In some cases the wave interference will dampen the surge effects to where the Joukowsky equation was a conservative estimate. In other cases the reflections creates constructive interference to heighten the effects where even the “conservative” Joukowsky equation was exceeded.

This concept becomes more evident with an example. Figure 1 shows a hydraulic model of two piping systems experiencing the same instantaneous valve closure. The initial pressure surge from the each system’s instantaneous valve closure will be very near the Joukowsky predicted pressure rise.

Figure 1 below shows the systems: one as a straight pipe, and one with the addition of a dead-end branch in the middle of the main line. Steady state flow is driven from left to right, and the valve closure happens at 0 seconds.

Comparison of pressure wave propagation from an instant valve closure: (top) straight pipe; (bottom) straight pipe with a dead-end branch

Immediately after the valve closes, both systems see the same Joukowsky predicted pressure rise. This pressure is the maximum seen by the straight pipe system, however it is not the maximum seen for the dead-end system.

Understanding that Joukowsky only predicts the immediate pressure rise at the valve is quite important. Examining the branching system reveals the largest pressure surge occurs at the valve at 8 seconds, well after the valve closure. Figure 2 below shows the pressure response at the valve by both systems over time. From the figure, it is clear that the maximum pressure does not occur until the 8 second mark, well after the initial surge. Again, this is behavior not predicted by traditional means.

Transient pressure activity immediately upstream of a sudden valve closure

Figure 2: Transient pressure activity immediately upstream of a sudden valve closure: straight pipe system (red); and branch with dead-end system (blue)


In this case, the divergence of the pressure wave into the dead-end causes a reflection that adds to the main pressure wave that went down the entire pipeline. Similar models can be made for cases of other pipe network changes whose complexity cannot be captured using simple hand calculations.

Line Pack

Another scenario that shows a Joukowsky limitation is line pack. Line pack will occur when there is significant pressure drop from friction in a pipe, typically in long pipelines. Because frictional losses are a direct result of flow, if flow stops, there is no pressure loss, and the pressure downstream will rise to the same level as it is upstream. This increase is also known as the “friction recovery pressure.” This additional frictional recovery has not yet been accounted for in the instantaneous pressure rise from the initiating event.

Any surge event that stops flow will cause an immediate pressure rise, and then the line pack raises the pressure even higher. This maximum pressure is not exactly the sum of the immediate Joukowsky predicted pressure rise and the frictional recovery pressure. Since it often takes time for line pack to be seen, the pressure wave has time to dampen the initial surge event. Regardless, the final pressure rise is usually much higher than what Joukowsky predicts. Again, the Joukowsky equation’s main limitation is that it does not account for effects after the immediate event.

Let’s look at an example of a 50 km pipeline, modeled with AFT ImpulseTM. This is very similar to the “straight pipe” scenario in Figure 1, but the pipe is much longer. If the valve is closed suddenly, there is an expected, quick rise in pressure. But as time goes on, the stop in flow is communicated down the pipeline, and all that pressure lost to friction adds back on top of the initial surge. Figure 3 below shows the pressure behavior at the valve inlet.

Transient pressure activity upstream of a sudden valve closure depicting line pack

Figure 3: Transient pressure activity upstream of a sudden valve closure depicting line pack: 50-km pipeline


While the Joukowsky equation predicts the initial surge event with accuracy, it does not predict the subsequent behavior. It has no perspective of time or post-surge responses. If an engineer only relied on the initial calculation, the maximum pressure rise would be predicted at half its actual value.

Transient Cavitation

Waterhammer events do not only involve high pressures. As part of a wave cycle, the high-pressure wave will be matched by a low-pressure wave of similar magnitude. If that pressure falls below the fluid’s vapor pressure, a vapor pocket will form in the pipe. This flashing is known as “liquid column separation” or “transient cavitation”.

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Note: Hydraulic engineers usually deal with cavitation in the context of pumps lacking the NPSH required to keep the suction pressure above vapor pressure, which is very similar in concept that occurs in steady state. In this context, anywhere in a pipe that sees low pressure as a direct result of a transient event can cause cavitation.
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There are many concerns associated with low pressures, such as pipe collapse and process contamination. For the sake of this discussion, the focus is the problem that follows vapor formation: vapor collapse. The collapse of a vapor pocket is very aggressive, often resulting in higher pressures than the initiating surge. Again, this is something the Joukowsky equation could never predict.

Low pressures can result from a multitude of causes: a pump trip, the low-pressure phase of the waterhammer cycle, or a valve closure. While valve closures are associated with high pressure waterhammer events, downstream of the valve a valve closure creates significant low pressure.

When a valve closes, there is no more momentum transfer to the downstream fluid, yet the fluid still has inertia driving it forward until the system corrects towards equilibrium. This creates a low-pressure area and disturbs the line’s pressure balance. This can be seen in Figure 1 above as the immediate blue color to the right of the valve. This low pressure can drop below the fluid’s vapor pressure, causing cavitation.

A similar system to the “straight pipe” scenario of Figure 1 was modeled to demonstrate cavitation.

Downstream of the valve there should be an immediate low pressure wave and subsequent cavitation if low enough. When the valve closes, there is immediate vapor formation, evident in Figure 4. As the pressure wave downstream cycles to come back above vapor pressure, the vapor pocket collapses causing a large spike in pressure. This spike is even larger than the initial surge pressure on the upstream side. Not only was this secondary surge a larger pressure, it happened in a different area of the pipe than traditional analysis would predict.

 

Transient activity immediately downstream of a sudden valve closure with vapor column

Figure 4: Transient activity immediately downstream of a sudden valve closure: vapor volume present in the computation station downstream of the valve (red, top); pressure behavior (blue, bottom)


Summary

Predicting a transient’s maximum pressure is much more involved than calculating the pressure rise immediately following a valve closure. The Joukowsky equation should only be used if the assumptions are understood. While it can correctly predict the immediate pressure rise at a closing valve, it does not correctly predict maximum pressures from phenomena that occur after that instantaneous change. It is limited temporally to the time immediately after valve closure, and it is limited spatially to the area just upstream of the valve.

The 3 major cases that cause this calculation to fail are pipe non-uniformities, line pack, and transient cavitation. More robust calculation methods are required to predict maximum pressures, and proper modeling is a necessity for sizing the mitigation equipment to protect again surge. Transient simulation software should be used for such predictions instead as it accounts for all these effects automatically.

Understanding the various ways maximum pressures can arise is the first step for effective waterhammer mitigation.

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