Navier-Stokes equations

We are awash in a world of fluids we don’t understand. The air we breathe, the clouds overhead, the gasoline in our car engines, and the blood in our veins—these are all fluids that are liable, at any moment, to slip from a smooth and predictable flow into turbulence, with eddies of unpredictable motion-complicating existence.

For centuries, scientists have studied turbulent fluids, but describing and predicting their behavior mathematically—using a set of equations called Navier-Stokes—remains an elusive and unsolved problem, challenging physicists and super-computers alike.

Still, researchers across disciplines have been able to improve our knowledge of fluid chaos. This has led to more efficient jet engines, eerily realistic computer graphics, increasingly accurate weather forecasts, and further-flying golf balls. It’s not just humans: Animal bodies and behavior have been shaped by the physical reality of fluid dynamics—and so will those of robots in the future. Let’s give it a whirl.

54: Years George Stokes spent as Lucasian professor of mathematics at the University of Cambridge; it is the longest tenure in a job also held by Isaac Newton, Charles Babbage, and Stephen Hawking

60,000: Number of slide-rule operators required in 1922 to theoretically predict the next day’s weather using Navier-Stokes equations

9-10: Days in advance today’s meteorologists can usefully predict the weather; contemporary five-day forecasts are as accurate as one-day forecasts in 1980

1998: Release year of the first film to use computer-simulated fluids—Antz

3: Oscars won by Jos Stam, who used Navier-Stokes to pioneer advanced computer graphics

George Stokes and Claude-Louis Navier were 19th-century physicists who gave their names to the equations describing fluid motion. They were responding in part to the then-pressing problem of building sturdy railroad bridges, which were stressed by the pressure of wind and water. Each independently wrote down a mathematical description of fluid motion that engineers could use.

But the description quickly becomes complex: When flowing fluids become turbulent, “eddies” are created, which break into new eddies, from the macro scale down to the molecular level, eventually dissipating as heat due to internal friction.

Over time, mathematicians and engineers have come with simplified forms of the equation to solve problems. And today, physicists are using data from wind tunnels and other experiments to refine computer models with averages and intelligent assumptions, particularly a technique called Large-eddy simulation.

So what do these equations look like? Start with Isaac Newton’s laws, particularly his second—force equals mass times acceleration. Be sure to account for the density of the fluid, its internal and external pressure, gravity, and heat dissipation. Et voilà—fluid dynamics.

If you’re having trouble visualizing how all this works, maybe a little poetry would help?

Big whirls have little whirls
That feed on their velocity,
And little whirls have lesser whirls
And so on to viscosity.

The ditty comes to us from Lewis Fry Richardson, a British physicist known as the father of modern weather forecasting. In 1922, he published the groundbreaking book Weather Prediction by Numerical Process, which simplified equations based on Navier-Stokes—but in those pre-computer days, it took him three months of calculation to predict the next day’s weather!

“I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic.”

—British physicist Horace Lamb in 1932; a similar quote is attributed to German physicist Walter Heisenberg—the true source remains uncertain.

The Navier-Stokes equations are vital to the models used by scientists when predicting the effects of climate change. But the complexity of forecasting a system as large as the Earth’s atmosphere requires making big assumptions—for one, most models examine the world in 100-km squares, which could miss important dynamics happening at a smaller scale. Timothy Palmer, an Oxford climate scientist, argues that the world needs to invest $1 billion in an international computing center capable of modeling the climate at a scale of 1 km in order to decrease uncertainty about how humans are changing the atmosphere.

1687: First publication of Isaac Newton’s Philosophiæ Naturalis Principia Mathematica, containing the laws of motion that would begin the journey of understanding fluid dynamics.

1757: Leonhard Euler comes up with a set of equations describing the motion of fluids.

1822: Claude-Louis Navier derives what would become the Navier-Stokes equations.

1845: George Stokes derives the Navier-Stokes Equations.

1904: German physicist Ludwig Prandtl develops the concept of the boundary layer, which simplifies the Navier-Stokes equations enough to bolster early aircraft design.

1950: A team of Princeton researchers uses the ENIAC supercomputer to perform the first 24-hour weather forecast.

1980: NASA supercomputer ILLIAC-IV performs the then-largest simulations of a turbulent flow, confirming the potential of Computational Fluid Dynamics (CFD).

1999: Jos Stam publishes an influential method that uses simplified Navier-Stokes equations to create computer imagery of fluids.

2018: Supercomputer-maker Cray creates the largest CFD simulation in an attempt to understand the effect of drag on racing cyclists.

Shark skin isn’t smooth. Sharks are covered in tiny rough patches called “denticles” that, counter-intuitively, reduce their drag as they swim. This evolutionary adaptation to turbulence helped inspire engineers to put riblets on airplane wings.

People can flow, too—engineers have noticed that crowds moving through and between buildings behave a lot like flowing fluids. That means turbulence can get out of hand, most tragically when stampedes break out in massive crowds, as with the death of nearly 800 people during the Hajj pilgrimage to Mecca in 2015. Researchers have used insights from fluid mechanics to offer safety tips and even create architecture that is more conducive to safe passage of big groups. Similar insights have been applied to the problem of traffic jams.