719-686-1000 info@waterhammer.com


Engineers often define their worst-case scenario based on the Joukowsky equation, and it is a valid theoretical equation, as long as the limitations are understood. 

Each series of articles are written by pipe flow analysis engineers from Applied Flow Technology. As industry leaders in water hammer and surge analysis, AFT has collected models and data from projects around the world to use as reference materials for published technical papers, case studies, and blogs. Visit www.aft.com for more information on analysis tools. 

Considerations for the Worst-Case Scenario


Determining the worst-case scenario water hammer event is no easy task. Unfortunately, there is no step-by-step process for its identification, only topics of consideration to help engineers analyze their systems. Analysis of this kind is difficult for two reasons: (1) no two systems are the same, so water hammer behavior and mitigation efforts will always differ; and (2) the data required for proper analysis is often not available and requires estimation.

Given these complications, understanding the components involved in water hammer analysis will help an engineer’s pursuit of defining the worst-case scenario. There are 7 core concepts to work through and comprehend. Keep in mind that combining these concepts into practical means for predicting water hammer is best done with modeling software due to the complexity of water hammer analysis.

This article introduces the 7 core concepts of worst-case analysis:

  • Wave speed (celerity) is the most important property
  • Joukowsky equation as the textbook worst-case
  • Communication time (pipeline period) impacts what is considered instantaneous
  • Magnitude and rate of change of flow determines water hammer response
  • Forces have different considerations than pressures
  • Typical worst-case events to watch out for
  • Special complications that may arise

Wave speed (celerity)

Wave speed is the velocity at which a wave (pressure change) is propagated through the pipe. This is the most important parameter to consider, as it determines other variables such as communication times, pressure responses, and pipe sectioning in numerical analysis.

Wavespeed may be the most important driving force in water hammer, but it is also the most difficult to assign a numerical value. Engineers typically use an empirical equation to estimate it (Equation 1 below), but that is all it is: an estimate. Often, that estimate operates within a 10-15% error range at best. When different sources yield different values of the variables below, engineers should use the largest wave speed for conservative measure.

Wavespeed Equation


K = fluid bulk modulus of elasticity

ρ = fluid density

E = pipe modulus of elasticity

D = pipe inner diameter

e = wall thickness

c1 = pipe support constant

As the equation depicts, wave speed is a function of both fluid and pipe properties. Again, with all the parameters that make up wave speed, engineers are certain to get conflicting wave speed values. They ought to run scenarios with all reasonable possibilities and use the largest wave speed if in doubt to create a more conservative water hammer event.


Joukowsky equation

Engineers often define their worst-case scenario based on the Joukowsky equation (Equation 2 below). This is a theoretical equation that determines the immediate pressure rise () from an instantaneous change in velocity.

ΔΡ = ραΔν


ρ = fluid density

a= wave speed

Δν= instantaneous change in velocity

 This is the “textbook worst-case scenario”, and it is a valid theoretical equation, as long as the limitations are understood. It yields the pressure rise immediately after a sudden velocity change, usually initiated by a valve closure. However, many situations can cause even higher pressures to arise, well after the initiating event. The maximum pressure can happen at a time far after the valve closure and in a location far from the instigating closed valve. Below are just a few of the ways the Joukowsky equation fails to predict peak surge pressure.

  • Non-uniformity in piping networks (such as diameter changes, dead-ends, valves and fittings) causes wave reflections that may result in constructive interference.
  • Stopping flow in long pipelines not only causes an immediate pressure rise, it also causes a gradual increase in pressure from recovering the frictional pressure losses. This is known as “line pack”.
  • Transient cavitation (liquid column separation) can arise from accompanying low pressures in water hammer wave cycles. This can cause fluid vaporization, and the collapse of that vapor pocket often causes larger pressure waves than the initiating event.

The Joukowsky equation gives engineers a conservative prediction of pressure rise immediately after a valve closure, but the analysis should not stop there. Analyzing the system response after the immediate event is very important, as unexpected behavior is common.

Communication time (pipeline period)

Communication time (T) is the time it takes for the pressure wave to propagate to a reflection point and back. It is a direct function of the wave speed and length of the pipe segment (Equation 3 below).
Communication Time Equation


L = pipe segment length

a = wave speed

 Communication time determines what events are considered “instantaneous”. The valve closure is effectively instantaneous if the valve closes before the pressure wave reflects back to its point of origin. There is little difference between a 30-second valve closure and a 1-second valve closure when the initial wave takes 45 seconds to travel down and back. While there will be no difference in maximum pressures, there will be a difference in transient forces which is discussed in a following section.

Any event that occurs “slowly” (i.e., over a longer period than the communication time) will be attenuated by the reflection of the upstream boundary condition. Engineers often assume that slowing the valve closure will be enough to lessen the surge. However, this is only true in simple systems and when the closure period is longer than the communication time. Engineers need to understand the whole system before assuming the effectiveness of proposed mitigation.


Magnitude and Rate of Flow Change

Water hammer results from a shift in momentum. The larger that momentum shift, the larger the pressure response. Velocity and mass are the two terms that comprise momentum. Controlling water hammer commonly involves controlling velocities, specifically. Mass flow usually does not change with potential system designs, given rigid flow design requirements. However, velocities vary with pipe diameters while mass flow remains constant, so water hammer consideration has its focus here.

The two aspects of velocity that affect water hammer responses are (1) magnitude and (2) rate of change. High initial velocities (magnitudes) result in higher immediate pressure rises because the final velocity from a closure is always zero. Along with this, faster velocity changes (rate of change) also causes higher pressure rises. Common preventative measures include using larger diameter pipes to lower the steady-state velocities (when feasible) and using longer valve closure times. A longer closure time keeps the pressure rise less extreme from the rise that Joukowsky predicts. Both measures seek to dampen the momentum change.

Keep in mind, though, these methods only mitigate the immediate pressure rise. The piping network as a whole must be considered. For example, it is possible to have a higher pressure response in the system with longer valve closure times. This can happen in complex systems with multiple valve closures. The waves can behave such that there is more constructive interference when a valve takes longer to close. This is why simulating your system response in water hammer software is so important. Responses are not intuitive, especially in the complexity of real systems.

Figure 1 below demonstrates a liquified natural gas (LNG) loading system experiencing multiple valve closures. In one scenario the valves all close over the same amount of time, 25 seconds. In another scenario, one of the valves closes over 30 seconds. Notice the larger pressure spike at the emergency response coupling (ERC) of the scenario with a 30-second closure, even with a similar velocity profile.

Figure 1: Comparison of pressure (top) and velocity (bottom) responses of a 25-second valve closure and 30-second valve closure at the ERC of an LNG loading system.



Pressure rise from water hammer is not where the analysis should stop. Force analysis is often a subsection of water hammer analysis. Although there is a direct relationship between pressure and force, analyzing forces requires a different thought process. Forces are highly dependent on the waveform of the surge. The steepness of the pressure-wave front has a large effect on forces. This is because forces are analyzed based on differences between nodes, not on single-point pressures.

A system tested under different closure conditions can see the same maximum pressure, but the resulting forces may be very different. The forces experienced depend on how fast that pressure spike happens, and whether that pressure has been communicated down the line. If the pressure spike is sudden (such as from a sudden valve closure), the node further away from the incident may yet to see an increase in pressure. Thus, the difference in pressures between the two nodes is large. If the pressure rise happens slowly (from a slow valve closure), the far-away node will see some pressure rise by the time the maximum pressure rise is seen at the valve. As a result, the pressure differential is smaller, and therefore the force difference is smaller.

It is typical that a faster valve closure will cause higher forces because it generates a steeper pressure-wave front. This is where slowing the valve closure can be beneficial (assuming it does not increase the maximum pressure as seen in the previous section. The example below demonstrates this. There are 3 different instantaneous valve closure times that generate the same maximum pressures. Figure 2 presents the pressure response seen immediately upstream of the valve closures. (Also notice the line pack effect discussed earlier. The pressure slowly rises after the spike of the 1ms case. In this case there is more exaggerated frictional loss recovery.)

Figure 2: Comparison of pressure responses of 3 valve closures, all within the communication time, of a 1000 ft (305 m) pipe.


However, the faster closure of 1ms causes the fastest route to that pressure, so its resulting forces are the greatest. Figure 3 below shows the transient force responses. One node is the closed valve, and the other node is 1000 ft (305 m) upstream of the valve. Notice the very different results.

Figure 3: Comparison of forces experienced by 3 valve closures. The force set is defined from the valves to 1000 ft (305 m) upstream.


The 1ms causes much more strain on the pipe and restraints even though each scenario sees the same maximum pressures. Figure 4 below shows the wave front of each scenario moving through the pipe. See how the 1ms wave is much steeper than the others.

Figure 4: Animation of the pressure wave propagation from 3 valve closures through the 1000 ft (305 m) pipe.


Typical Worst-Case Events

The worst-case scenarios often arise from events outside the engineer’s control. Although engineers may design a system with standard safety precautions, accidents occur and cause unforeseen problems. These accidents often become worst-case scenarios, and engineers must include these scenarios in their analysis to be best prepared for the worst.

These most often are events such as sudden losses of power, loading docks shutting off valves without expectation, emergency shutdowns dictated by regulations, and powered emergency release couplings. Each of these incidences will cause a different response, and every system is unique such that the same event can cause very different behavior. The only way to understand a system’s response to such events is to model them in advance. It is the engineer’s job to set up different scenarios of a model to ensure all possible events, even those unintended, are accounted for.


Water hammer analysis is quite complicated even when everything operates and responds as expected. However, further complications can arise that cause difficulties in determining the “worst-case scenario”. Multi-fluid systems, varying reservoir levels, and relief systems should cause engineers to double-check their analyses.

Some systems are designed to handle various fluids. Mitigation efforts for one fluid may not be appropriate for another. Typically, the densest fluid with the highest bulk modulus sees the worst response, so that is often the fluid used in design and analysis. Different fluids, however, have different operating points in a system, so setpoints for mitigation or relief efforts may not work. All possible fluids should be tested and simulated.

Different reservoir levels cause different system responses, especially in pumping systems. Higher levels typically cause higher initial system pressures due to the higher static head. Overpressure is more likely a direct result in this case, and it is often analyzed assuming this is the worst case. However, low tank levels can cause even worse responses. The system is more likely to see pump cavitation, liquid column separation, and high flow even when pumps are off. Testing is required because neither high nor low levels are universally the worst-case.

Differences in relief valve responses also cause complications in analysis. Opening a relief valve sooner or at a lower setpoint can help alleviate surge, but it can also result in overfill or result in failing to reseat. This means they must be sized very specifically for them to behave as expected. Also, the relief line may be in series with a gas accumulator, resulting in extreme compression of the gas. This can overpressurize the relief line such that it communicates back into the main system as a larger response. Relief systems require more thorough analysis than may be anticipated.


There is much to consider when analyzing the “worst-case scenario” of water hammer. The only way to understand what that scenario is, and the means to mitigate it, is by using simulation software. Real systems are very complex and require thorough analysis. However, water hammer analysis is not as intimidating as it may appear at first. Simply understanding what to consider for your worst-case scenario puts you far ahead of the competition.