The chaotic flow of water cascading down a mountainside is known as turbulence. It is complex, irregular and unpredictable, but we should count our blessings that it exists. Without turbulence, we would gasp for breath, struggling to absorb oxygen, or be asphyxiated by the noxious fumes belching from motorcars, since pollutants would not be dispersed through the atmosphere.
Turbulence is everywhere. The air flowing into and out of our lungs, convective currents above hot tarmac, winds gusting along a street and large weather systems are all turbulent. Turbulence controls the aerodynamic drag on cars and aircraft, disperses pollutants and mixes the cream when we stir our coffee. Many geophysical and astrophysical flows are dominated by turbulence. The convection in the core generates and maintains the Earth’s magnetic field through turbulence.
Mathematicians, physicists and engineers are all interested in understanding turbulence. With better knowledge of turbulence we can improve the efficiency of engines, reduce the drag on automobiles, regulate the flow of blood in the heart and design better golf balls. A 10 percent reduction of drag on commercial aircraft could boost profit margins by 30 or 40 percent.
Air and water are fluids and both atmospheric and ocean flows are governed by the principles of fluid dynamics. The future course of the weather is predicted by solving the mathematical equations that govern fluids, marching the solution forward in time from weather conditions known at an initial time. This process requires powerful computers, as many millions of calculations are required. Modern weather forecast models are based firmly on the physical principles governing atmospheric flow. These principles are expressed in mathematical form as equations originally formulated by a French engineer Claude-Louis Navier and an Irish applied mathematician George Gabriel Stokes.
The Navier-Stokes equations are remarkably successful in describing a vast range of fluid behaviour. They can model laminar flow like honey dripping smoothly from a spoon or highly turbulent flow like the hot exhaust gases emanating from a jet engine. They can accurately simulate sudden transitions between smooth, regular flow and chaotic turbulence, which we can observe as a tap is opened or the plume of smoke from a cigarette accelerates upwards.
The Navier-Stokes equations are nonlinear partial differential equations. This means that they describe the way the fluid flow field changes in both space and time. Nonlinearity arises because the velocity acts upon itself. The flow field is carried along by the velocity. You could envisage the wind blowing the wind along, or a river current sweeping swirling eddies downstream. As a result, the velocity occurs in quadratic form in the equations. This makes them very difficult to solve, and numerical solution methods must be used, in a process known as computational fluid dynamics.
A major difficulty exists with the Navier-Stokes equations; it might almost be called a scandal. Although the equations are invaluable in a wide range of applications, their mathematical validity has not been proven. It is possible that, under unknown circumstances, the solution may become unrealistic, with singularities developing where the energy density becomes locally unbounded. The Clay Mathematics Institute has offered a cash prize of one million dollars to anyone who can demonstrate that the solutions of the Navier-Stokes equations remain bounded and physically realistic under all flow conditions.
Peter Lynch is emeritus professor at the school of mathematics and statistics, University College Dublin. He blogs at thatsmaths.com. His book That's Maths will be published by Gill Books next week