Turbulence is dramatic, beautiful and potentially dangerous. It happens in liquids as well as gases — think of breaking waves and raging rivers, or air streaming around a car or plane. By its very nature, turbulence is incredibly hard to describe. At the same time, a proper understanding of turbulence would have applications ranging widely across the mathematical, physical and engineering sciences.

That's why the Isaac Newton Institute for Mathematical Sciences is running a major research programme, called *Mathematical aspects of turbulence: Where do we stand?*. Here we bring you some brief introductions to the basic notions of the maths of turbulence and fluid flows and why it is so tricky. Come along for the ride, we promise it won't be too bumpy!

Why turbulence is troubling — Turbulence is everywhere. We all know what it is, but nobody understands it properly. Here's why.

Maths in a Minute: Fluid dynamics and the Euler equations — How does water, or indeed any fluid, move? The Euler equations let us look beneath the surface and mark the beginning of modern fluid dynamics.

Maths in a minute: The Navier-Stokes equations — A step up from the Euler equations are the Navier-Stokes equations. They are more general, but also present a huge challenge that could win you 1 million dollars. Find out more in this article from our library.

Maths in a minute: Computational fluid dynamics — The Navier-Stokes equations can be incredibly hard to solve, so how do you apply them in the real world? Computational fluid dynamics has the answer.

Maths in a minute: Turbulence and the Reynolds number — How does a steady and smooth flow suddenly turn turbulent? The Reynolds number helps to describe the transition.

Maths in a minute: Differential equations — All the equations mentioned so far are differential equations. If you don't know what those are, here's an introduction. Find out more in this article from our library.

How maths can make you rich and famous — The Navier-Stokes equations are at the centre of a $1 million prize offered by the Clay Mathematics Institute. Find out more in this article.

### About this package

*This collection of content forms part of our collaboration with the Isaac Newton Institute for Mathematical Sciences (INI), an international research centre and our neighbour here on the University of Cambridge's maths campus. INI attracts leading mathematical scientists from all over the world, and is open to all. Visit www.newton.ac.uk to find out more.*