Plus Advent Calendar Door #21: Troubling turbulence

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Raging rivers and thundering waves are exciting and frightening and for mathematicians they are a massive problem. Suppose you've got a turbulent mountain stream — if you're a mathematician, you'll want to know if you can describe the flow of the water using a mathematical equation. Given a point in space (that is somewhere in the stream) and a point in time (say 5 minutes from now), you would like to know the velocity and maybe also the pressure of the water at that point in time and space.


Image: misty.

Scientists believe that turbulence is described to a reasonable level of accuracy by a very famous set of equations, known as the Navier-Stokes equations. These are partial differential equations which relate changes in velocity, changes in pressure and the viscosity of the liquid. To find the velocity and pressure of your liquid, you have to solve them.

But that's no easy feat. Exact solutions to the equations — solutions that can be written down as mathematical formulae — exist only for simplified problems that are of little or no physical interest. For most practical purposes, approximate solutions are found through computer simulations — essentially through educated guess-work — that require immense computing power.

Plus there is an additional problem that makes numerical difficulties pale into insignificance: no one knows if exact mathematical solutions even exist in all cases. And if they do exist, we still don't know if they involve oddities, such as discontinuities or infinities, that don't square with our intuition of how a liquid should behave.

It's these difficulties that have turned the understanding of the Navier-Stokes equations into one of the seven Millennium Problems posed by the Clay Mathematics Institute. Whoever proves or disproves the existence of finite and smooth solutions is set to earn a million dollars.

All this might seem strange, even scary, given that the equations are used all over the place, all the time — meteorology and aircraft design are just two examples. The fact is that, in the cases we can compute, approximate solutions do seem to give an accurate description of the motion of fluids. What we don't know is what, if anything, the model given by the Navier-Stokes equations tells us about the exact nature of fluid flow.

You can find out more about turbulence on Plus:

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