We frequently come in contact with moving fluids like air and water, probably without realising that these mundane daily occurrences are in fact encounters with one of nature’s more profound mysteries.
Consider the smoke rising from an incense stick. For a short distance, the plume of smoke remains well-ordered with a definite, if also twisting, shape. Then the plume suddenly breaks up, contorting and swirling into multiple eddies, or whorls. This irregular, seemingly random fluid motion is turbulence.
The disordered patterns of turbulent motion rapidly mix the agarbatti’s aroma with the air, allowing you to enjoy the fragrance from across a large room just a few seconds after it is lit. Such turbulent mixing also kickstarts our mornings, when we stir milk and sugar into our tea and coffee: without turbulence, you’d have to wait for about a month to enjoy a uniformly sweetened cup. You also create turbulence with every breath you exhale: the air gushing out of your nostrils forms short-lived and complex flow patterns that become visible on a frigid winter day.
Chaotic fluctuations, sudden bursts of motion, hard-to-predict variations – these features are typical of turbulent flows. Yet they also contain persistent swirling patterns called vortices. In water streams and cloudy skies, vortices have inspired artists and imprinted themselves upon our collective visual consciousness through the work of Leonardo da Vinci and Vincent van Gogh. That turbulence has ordered patterns is a testament to its origin in the laws of mechanics: turbulent whorls don’t turn on a whim, after all, but are governed by deterministic, well-understood physical forces.
The Navier-Stokes equations
The two key physical effects that determine the state of a fluid’s motion are inertia – the tendency of a fluid to keep moving – and viscous friction, which tends to bring all motion to a halt. The strength of inertia increases with the speed of motion, the mass of the fluid, and the distances over which the flow occurs. The strength of friction is determined by the fluid’s viscosity, which is higher for honey, moderate for water, and lower for air.
When viscous effects dominate, a flow is well-ordered and predictable, and disturbances quickly dampen out. There is little mixing and the fluid tends to move as if it were composed of distinct layers, which is why it’s called laminar flow. But when inertia dominates, the flow is highly unstable. Without much friction, small disturbances don’t die out but instead grow and spread. This is what happens to a rising plume of incense smoke: tiny fluctuations in the air are amplified within the plume, causing it to become turbulent.
The balance between fluid inertia and viscosity (and other forces due to pressure differences and gravity) are precisely described by the Navier-Stokes equations, which extend Newton’s law for a rigid body (like a billiard ball) to a fluid. These equations, now about 200 years old, describe both laminar and turbulent flow. They’re compact enough to fit on a postcard and don’t look formidable – yet they are. Today, we can use powerful supercomputers to solve them to an extent to determine how some turbulent flows might behave, but even this hasn’t allowed us to crack all their mysteries.
The key difficulty is that the Navier-Stokes equations are nonlinear: they contain some terms that arise from the fluid’s inertia, which manifests in the equations as a product of the velocity with its own spatial variations. Put differently, nonlinearity allows for positive feedback, which allows small initial disturbances to amplify in time and radically change the state of the flow.
The principle of superposition doesn’t apply either. If the equations were linear, two or more different solutions – e.g. describing vortices of different sizes in a flow – would evolve independently of one another. This then would mean we could take some complex flow, break it down into simpler components, work them out, and add them all back together to get a sense of the overall flow. But nonlinearity couples all components synergistically – that is, the different vortices interact and transform each other, producing a flow whose complexity is greater than the sum of its parts.
It’s complicated
Consider the motion of air produced by a ceiling fan. The flow doesn’t just consist of one large, room-spanning vortex. Instead, the primary vortex produced by the rotation of the fan’s blades is unstable: tiny disturbances amplify and form new smaller secondary vortices. These in turn are susceptible to further instabilities and produce vortices of their own, and so on down to vortices the size of a paperclip.
From the perspective of energy conservation: kinetic energy is injected by the fan directly into the primary vortex. This energy is then handed over to the secondary vortices and so forth until we reach a scale where viscosity prevails – i.e. fluid friction dominates over inertia – and dissipates the kinetic energy as heat. This soup of interacting whorls of all possible sizes is why incense released in one corner of a room quickly makes its way to all nooks and crannies – and it’s also why working out the equations is so hard.
The Swedish philosopher Nick Bostrom imagined that in the future, humans will be able to build planet-sized super-computers. Will these colossal machines alleviate the problem of simulating the Navier-Stokes equations? You’d think the answer would be ‘yes’, but it remains ‘it’s complicated’. This is because we still won’t be able to predict how turbulent eddies evolve over long periods of time.
Glimmers of hope
A system that exhibits nonlinearity is often also a system that’s sensitive to its starting conditions. In the extreme, such sensitivity manifests as chaos – discovered as a phenomenon by Edward Lorenz with help from Helen Fetter and Margaret Hamilton, in 1961, when they were looking for a way to model atmospheric convection.
In a chaotic system, a small perturbation grows rapidly such that, in a short span of time, the perturbed flow is as different from an unperturbed one as is physically possible. Chaos thus prevents long-term predictions: minute yet inevitable errors in estimating the current state of the flow – as that of a wind, for example – eventually render long term predictions on a computer meaningless. This is why weather predictions can often ‘see’ only a week or so into the future.
Luckily short-term prediction remains feasible, and is why a significant chunk of the world’s computational and remote-sensing resources are concerned with meteorology. More accurate predictions of short-term extreme weather events, like cloudbursts, translate to more effective alerts and response systems that can save lives.
How, then, are we to approach the mercurial beast called Turbulence whose behaviour eludes prediction? Should we abandon all hopes of a simple theory to explain it, without feedback loops and endless vortices?
There are some glimmers of hope. The details of a turbulent flow remain unpredictable, but we know that the averages of some properties of a flow over time, like mean velocity in a pipe or total lift force on an aeroplane, are well-behaved (which is a mathematician’s way of saying they don’t cause one to rip out one’s hair). This is how, for example, we know climate change is real: its mechanisms display a lot of variability, but when we study a long-time average of the weather, some trends become clear.
The challenge lies in predicting the values of these quantities that scientists have measured in experiments without having to fully solve the Navier-Stokes equations.
Kolmogorov’s theory
Another glimmer of hope is the presence of order in the chaos. A celebrated example of order in turbulence is the relationship between the size and energy of a turbulent eddy. The Russian physicist Andrei Kolmogorov proposed that the ratio of the swirling velocities of any two eddies is entirely determined by their typical sizes. So the ratio of the velocities of an eddy 10 km and another 1 km wide would be the same as that for two eddies 10 metres and 1 metre wide.
This simple idea has fascinated scientists. How could this relationship hold for a variety of flows – including air churning amid clouds, water flowing through a kitchen faucet, smoke rising from a stove? Yet both experiments and computer simulations have backed up Kolmogorov’s prediction well.
(There is a caveat, however: Kolmogorov’s theory didn’t account for the fact that, once in a long while, turbulence produces some small but very strong eddies. These bursts of motion are intense enough to cause the system to deviate from Kolmogorov’s theory. Such bursty behaviour is called intermittency and remains a subject of ongoing research.)
Another striking feature of order in turbulent flows is that they often contain pockets of coherent motion. A striking example is the Great Red Spot in Jupiter’s atmosphere. The spot is really an anticyclonic storm three-times as wide as the earth, churning for at least 190 years. How can such a coherent structure arise spontaneously and then survive in the presence of so much turbulence?
Such possibilities speak to the fact that turbulence is far from a random process. Instead, it hides a deep level of organisation that we are yet to uncover.
This is why turbulence continues to attract and challenge scientists from across disciplines, while providing a reminder that profound mysteries of nature are not the sole province of massive particle colliders or giant telescopes. One could be churning right under your nose.
Siddhartha Mukherjee is a postdoctoral researcher at Université Côte d’Azur, Nice, and before that at ICTS-TIFR, and a visual artist. Jason Picardo is an assistant professor of chemical engineering at IIT Bombay, where he investigates complex fluid flows using mathematical models.
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