Navier-Stokes equations: Prepare for turbulence

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]( 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. 🐦 [Tweet this!]( 🌐 [View this email on the web]( [Quartz Daily Obsession] Navier-Stokes equations February 21, 2020 “The most important unsolved problem of classical physics.” --------------------------------------------------------------- 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]( 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. 🐦 [Tweet this!]( 🌐 [View this email on the web]( By the digits [$1 million:]( Prize offered by the Clay Mathematics Institute to anyone who can formally prove the Navier-Stokes equations have a complete set of solutions [50%:]( Reduction in drag that results from adding dimples to a golf ball to reduce turbulence [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 [1 square kilometer:]( Scale to which Navier-Stokes-based climate models should be adjusted, from 100 square km today, to eliminate uncertainty [$1.1 billion:]( Cost of the supercomputers necessary to reach that resolution [10 quadrillion:]( Grid points required to accurately simulate the turbulent airflow around the wing of a 50-meter (164-foot) long aircraft flying 250 miles (402 km) per hour [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—the same number of awards as has been won by Meryl Streep giphy PREDICTING CHAOS What makes a problem unsolved? --------------------------------------------------------------- [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. This generates unpredictability, and mathematicians aren’t sure if under some conditions, the equations could [predict a fluid moving infinitely fast]( breakdown in our understanding of the world. In theory, a stone skipping across the ocean could cause the water to explode—or at least, it’s a terrifying thought experiment [from one of the world’s top mathematicians](. When it comes to applying the equations, predicting the behavior of a fluid requires modeling it on an imaginary grid that scales from a micron up to the dimensions of, for example, an aircraft. That task is still too complex for today’s supercomputers. So 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](. “[A proof] is nice to know mathematically, that you can hang your hat on it, but I think as far as I’m concerned… the numerical solutions are very much realistic because people compare them with benchmark experiments,” Stanford physicist Parviz Moin told Quartz reporter Tim Fernholz. Members only Profit with purpose? --------------------------------------------------------------- The third-largest sector in the B Corp community might surprise you. Financial services companies are increasingly seeking B Corp certification and taking on a purpose beyond the bottom line. Quartz’s Cassie Werber [reports on an industry in flux]( and the next generation of financiers. Visualized[A gif explains Navier-Stokes equations step-by-step.](Quartz) 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. Person of Interest Lewis Fry Richardson --------------------------------------------------------------- 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! Quotable “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]( true source remains uncertain. Have a friend who would enjoy our Obsession with Navier-Stokes equations? [ [Forward link to a friend](mailto:?subject=Thought you'd enjoy.&body=Read this Quartz Daily Obsession email – to the email – Giphy poll Which of these techniques have scientists not used to understand fluid turbulence? Performing interpretative danceStanding in front of a firehose Pressure-sensitive paint“Advanced guessing” Correct. You could give it a shot. Incorrect. Nope, they tried that. If your inbox doesn’t support this quiz, find the solution at bottom of email. NOAA via AP Million-dollar question Can Navier-Stokes accurately predict climate change? --------------------------------------------------------------- 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. Brief history [~1508:]( Leonardo da Vinci sketches turbulent water and muses about experiments to understand the flow of fluids. [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, a simplified form of the future Navier-Stokes equations. [1822:]( Claude-Louis Navier derives what would become the Navier-Stokes equations; during his career, he faced accusations that he [used too much math]( when building a bridge. [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 including John von Neumann use the ENIAC supercomputer to perform the first 24-hour weather forecast, [using a formula]( derived from Navier-Stokes. [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. Luc Viatour Fun fact! 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](. Watch this! A mathematician with a Navier-Stokes tattoo explains the equation --------------------------------------------------------------- Cambridge mathematician Tom Crawford spent four years studying the Navier-Stokes equations, so he got them tattooed on his side. Who else to dive deep into what makes fluid dynamics so fascinating? AP Photo take me down this 🐰 hole! 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](. Giphy poll Let’s review. Navier-Stokes equations are? [Click here to vote]( Still totally a mystery to meSomething I’m going to find a way to talk about at dinner partiesThe key to pretty much everything 💬 let's talk! In yesterday’s poll about [pole dancing]( 43% of you said that it’s not your thing, while the remaining 57% either already do pole dance, or would try it for fun or fitness. If you were wondering about [the interpretive dance that physicists did to explore Navier-Stokes]( well, it’s a real thing and you can watch it. 🤔 [What did you think of today’s email?](mailto:obsession%2Bfeedback@qz.com?cc=&subject=Thoughts%20about%20Navier-Stokes%20equations%20&body=) 💡 [What should we obsess over next?](mailto:obsession%2Bideas@qz.com?cc=&subject=Obsess%20over%20this%20next.&body=) [🎲]( [Show me a random Obsession]( Today’s email was written by [Tim Fernholz]( edited by [Annaliese Griffin]( and produced by [Tori Smith](. [facebook]( The correct answer to the quiz is Standing in front of a firehose . Enjoying the Quartz Daily Obsession? [Send this link]( to a friend! Want to advertise in the Quartz Daily Obsession? Send us an email at ads@qz.com. Not enjoying it? No worries. 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