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The current world record was set by King Fahd's Fountain at about 300m, and a water speed of about 375km/h.

Would it be possible to launch water much higher? Say, by aiming a water jet cutter with a mucher higher pressure upwards?

Or are there theoretical or material science limits to this?

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    $\begingroup$ I imagine one could go higher in a vacuum, but air breaking up the stream into droplets, and then slowing those droplets down, is likely the limiting factor in real life $\endgroup$ – Jonathan R Swift Jan 17 at 11:45
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An interesting question :)

There is a theoretical limit on how far a water jet can pass through air. In this instance, we neglect hydrodynamic effects, gravity, water jet breakup, etc. and just consider momentum. Newton solved this some time ago, and derived his impact depth equation based on momentum considerations alone, i.e. the jet transfers momentum to the material in front of it to push it out of the way.

https://en.wikipedia.org/wiki/Impact_depth

Interestingly this equation is independent of the velocity of the projectile.

Essentially since water is about 1000x denser than air, a 1m jet of water could travel 1000m horizontally before its momentum is dissipated to air.

OK edited to add steady-state case: I hope everyone agrees that a water jet that exceeds the speed of sound in air will not go very far, due to the formation of shock waves.

Now suppose we have a small column of water being ejected upwards, in a longer stream of water. The forces acting on this column are:

  1. Gravity
  2. Static pressure on upper face of column
  3. Static pressure on the lower face of the column
  4. Shear forces acting on the surface of the column while within the pipes
  5. Drag forces
  6. Surface tension/cohesive forces
  7. Atmospheric pressure

The pressure on the lower face of the column must exceed the gravitational force, and the drag forces, acting on this column to longitudinally accelerate it.

While the column is in a pipe, the fact that there is a static pressure difference between the top and bottom of a column of water does not cause any radial expansion, because the column is confined within the pipe. Once it is ejected from the pipe, the higher static pressure at the bottom would cause the column to tend to expand outwards, i.e. the spray disperses.

So the conclusion is that yes - there is a limit to how high a water fountain can go. If you accelerate the water jet too fast, it experiences high drag*, you need greater driving pressures to overcome this drag and in fact these driving pressures cause the jet to disperse even more. Too slow, and you don't have enough kinetic energy to convert to gravitational potential energy in the first place. An optimum exists somewhere between.

*Drag is proportional to velocity squared if the geometry is constant; if the geometry mushrooms or breaks apart it becomes highly non-linear

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    $\begingroup$ Is the impact depth analysis actually applicable in this situation? As a possible counterexample, a fountain is typically for some part of its length an unmixed volume of water (laminar flow) rather than independent droplets, and so in the steady state, that region is not newly displacing any air, so there is no momentum transfer in that region. $\endgroup$ – Kevin Reid Jan 18 at 19:13
  • $\begingroup$ This is where it gets more complicated quickly. I considered pulsatile flow to simplify things to a degree amenable to a coffee break. For steady flow it will require a longer explanation which I will add in due course. $\endgroup$ – MeEngineerTrustMe Jan 19 at 8:38

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