Fluid-Structure Interaction: Sloshing

Sloshing is something that everyone is familiar with on a very basic level.  The classic example is trying not to spill a full cup of coffee – but I’ve had a similar experience with enthusiastic children at bath time.  

The basic idea is that you have liquid in some sort of a container with a free surface.  When there is a force applied to the liquid – like walking with the coffee cup, or kids splashing around in the tub – the liquid starts moving.  The container confines the motion of the liquid, and sets up a back-and-forth oscillation, which we know as sloshing.  

Spilling your coffee is usually just a minor inconvenience, but with larger containers sloshing can have real consequences.  Some bigger examples of sloshing include:

  • Jet fuel sloshing inside the wing tank of an airplane.
  • Harbors sloshing as a result of a tsunami (e.g. Hilo, Hawaii or Crescent City, CA).
  • Large lakes sloshing as a result of a large storm.
  • Tides sloshing in ocean basins.
  • Swimming pool sloshing during an earthquake

The consequence comes from the force that is generated by the large mass of liquid moving quickly – anything standing in the way of the onrushing liquid will be subject to loads that it may not have been designed for.

The period of the slosh depends on the size and shape of the container – big containers like lakes have longer periods than small containers like coffee cups.  Things get interesting, though, when we start looking at higher mode frequencies of containers.  Consider a large cylindrical tank 20-ft in diameter, and 10-ft deep – a scaled up version of a coffee cup.  The sloshing mode everyone is familiar with in their coffee cup has one wavelength around the perimeter (i = 1), and no waves across the diameter (j = 0).  We can use some way-cool Bessel functions for this simple shape to determine the period of the primary sloshing mode, which is 4.28 seconds.  This is a long period because the tank is fairly large, and so is unlikely to interact with any structural frequencies.  

However, if we start packing more waves around the circumference of the vessel, or we start putting some waves across the diameter of the vessel (or both!), the wavelength shrinks, the speed that the wave travels shrinks, and the period of that wave mode shrinks.  Table 1 shows the resonant periods for various wave modes, and the liquid surface for each wave mode is shown at the bottom of the post.  Some of the higher modes start to get pretty wild!

Table 1:  Sloshing Periods for different wave modes in a tank that is 20-ft in radius, and 10-ft deep tank

Tank-wave-mode-periods

Interestingly, as you get to higher frequency modes, there are many modes that end up having very nearly the same period.  This becomes important if you have a forcing frequency that aligns with several modes at once.  In a reactor tank, excitation could come from gas injection into the liquid that excites the liquid surface, or a long agitator shaft that excites a surface wave at the shaft’s natural frequency, or flow periodically distorting the flexible container wall.  

In the case of jet fuel in a wing tank, sloshing in the wing tank can cause changes in the flight behavior, which recover at the same period as the sloshing, forcing feedback that eventually leads to bad things for the airplane.  The solution for airplanes is to put several restrictive baffles that break the large tank into many smaller tanks with higher frequencies that don’t interact with the plane’s flight response.  In the case of the circular tank, putting baffles in the tank can turn into a game of whack-a-mole where you might block one wave mode, only to have the energy move to a different mode with a similar period.  

If possible, sometimes the best answer is to identify that you might have a sloshing problem early in the design process, and do your best to avoid it.

Tank-wave-mode-shapes

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