Tag: Navier-Stokes Equations

The Real Butterfly Effect
The butterfly effect — that the flapping of a butterfly’s wings in Brazil can cause a tornado in Texas — expresses the sensitivity of a chaotic system to initial conditions. In essence, because we can’t possibly track every butterfly in Brazil, we’ll never perfectly predict tornadoes in Texas, even if the equations behind our weather forecast are deterministic.
But this interpretation doesn’t fully capture the subtleties of the situation. With fluid dynamics, the small scales of a flow — like the turbulence in an individual cloud — are linked to the largest scales in the flow — for example, a hurricane. For short times, we’re actually quite good at predicting those large scales; our weather forecasts can distinguish sunny days and cloudy ones a week out. But at smaller scales, the forecast errors pile up quickly. No one can forecast that an individual cloud will form over your house three days from now. And because the small scales are linked to the larger scales, the uncertainties from the small scale cascade upward, limiting how far into the future we can reliably predict the weather.
And, unfortunately, drilling down to capture smaller and smaller scales in our models can’t fix the problem, unless our initial uncertainties are identically zero. To get around this problem, weather forecasters instead use ensemble forecasting, where they run many simulations of the weather with slightly different initial conditions. Those differences in initial conditions let the forecasters play with those initial uncertainties — how accurate is the temperature reading from that station? How reliable is the instrument reporting that humidity? How old is the satellite data coming in? Once all the forecasts are run, they can see how many predicted sunny days versus rainy ones, which ones resulted in severe weather, and so on. Often the probabilities we see in our weather app — like 30% chance of rain — depend on factors including how many of the forecasts resulted in rain.
Unfortunately, this butterfly effect permanently limits just how far into the future we can predict weather — at least until we fully understand the nature of the Navier-Stokes equations. For much more on this interesting aspect of chaos, check out this Physics Today article. (Image credit: NASA; see also T. Palmer at Physics Today)
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July 15, 2024
Abel Prize Winner Luis Caffarelli
Tomorrow mathematician Luis Caffarelli will receive the Abel Prize — one of the highest honors in mathematics — in part for his work in fluid dynamics. Caffarelli is one of the authors of a partial proof of regularity for the Navier-Stokes equations, the equations governing fluid motion. A full proof of regularity and smoothness — essentially showing that the equations never break down or blow up to infinity — is one of the open Millennium Problems. Caffarelli is the first mathematician born and educated in South America to receive the Abel Prize. Congratulations to Professor Caffarelli! (Image credit: N. Zunk/University of Texas at Austin; via Nature; submitted by Kam-Yung Soh)
May 22, 2023
An Introduction to the Reynolds Number
For those who’d like an overview of the mathematics involved in fluid dynamics, Numberphile has a lovely introduction, given by our friend Tom Crawford. The governing equations in fluid dynamics, the Navier-Stokes equations, are quite complicated, but that’s just been inspiration for scientists and mathematicians to come up with clever ways to simplify them. And, ultimately, that’s what the Reynolds number is — a way to help us judge which forces, and therefore which mathematical terms, are the most important in a given problem. (Video credit: Numberphile; submitted by COMPLETE)
December 11, 2019
Solution to a Millennium Prize Problem?
Reports emerged this weekend that Kazakh mathematician Mukhtarbay Otelbaev has published a proposed solution to the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize problems offered by the Clay Mathematics Institute. Today I want to explain some of the background of this problem, what is known about Otelbaev’s proposed solution, and what a solution would mean for fluid dynamics.
The Navier Stokes Equation
The Navier-Stokes equation is one of the governing equations of fluid dynamics and is an expression of conservation of momentum in a fluid. With the exception of a few very specific and simplified cases, there is no known general solution to equation. Instead, the equation, or a simplified model, is solved numerically using supercomputers as part of direct numerical simulation (DNS) or other forms of computational fluid dynamics (CFD). These methods allow scientists and engineers to solve the equations of fluid motion for practical problems from flow through a pipe to flow around a re-entering spacecraft.
Existence and Smoothness
Although the Navier-Stokes equation has been known for more than 150 years and can be solved numerically for many situations, some basic mathematical aspects of the equation have not yet been proven. For example, no one has proven that a general solution always exists in three-dimensions and that the energy of such a solution is bounded at all points. Colloquially, this is known as the Navier-Stokes existence and smoothness problem. The Clay Mathematics Institute has a very specific problem statement (PDF) asking for a proof (or counter-proof) of the existence and smoothness of the Navier-Stokes equation for an incompressible fluid in three-dimensions. Otelbaev contends that he has provided such a proof.
Otelbaev’s Proposed Solution
Mukhtarbay Otelbaev is an experienced mathematician with numerous papers addressing related mathematical problems. His latest paper, entitled “Existence of a strong solution to the Navier-Stokes equation,” is freely available online (PDF, in Russian, with an English abstract at the end). There is an ongoing project to translate the paper into English, and mathematicians are already evaluating the validity of this proposed solution. From what I can gather of the paper, it specifically address the Millennium Prize problem and presents Otelbaev’s proposed solution for the existence and smoothness of an incompressible fluid in three dimensions with periodic boundary conditions.
What It Means
As with any announcement of a major technical breakthrough, skepticism is warranted while experts evaluate the proposal. If the mathematical community upholds the validity of Otelbaev’s proof, he may be offered the Millennium Prize and other honors. More importantly, his solution could lead to a better understanding of the nature of the equation and the flows it describes. It is not, in itself, a general solution to the Navier-Stokes equation, but it may be a stepping stone in the path toward one. In the meantime, scientists and engineers will continue to rely on a combination of theory, experiment, and computation to progress our understanding of fluid dynamics.
For More
The story of Otelbaev’s proof and the community’s evaluation of its validity is on-going. You can follow @fyfluiddynamics and the #NavierStokes hashtag on Twitter for updates and commentary. I’d like to specially thank Catriona Stokes, Praveen C, David Sarma, and Glenn Carlson for their helpful links and observations as this story develops.
January 13, 2014
Incense in Transition
A buoyant plume of smoke rises from a stick of incense. At first the plume is smooth and laminar, but even in quiescent air, tiny perturbations can sneak into the flow, causing the periodic vortical whorls seen near the top of the photo. Were the frame even taller, we would see this transitional flow become completely chaotic and turbulent. Despite having known the governing equations for such flow for over 150 years, it remains almost impossible to predict the point where flow will transition for any practical problem, largely because the equations are so sensitive to initial conditions. In fact, some of the fundamental mathematical properties of those equations remain unproven. (Photo credit: M. Rosic)
May 31, 2013
Osborne Reynolds and Transition
How and when flow through a pipe becomes turbulent has been a conundrum for fluid mechanicians since the days of Osbourne Reynolds (~1870s):
Typically, the laminar-to-turbulence transition is studied mathematically by linearizing the Navier-Stokes equations, the governing equations of fluid dynamics, then perturbing the system. These perturbations will gradually disappear in laminar flow, but if the flow is turbulent, they’ll grow and produce chaotic motion. The transition, then, is the critical point between these two.
However, for pipe flows, this linearized approach shows that the perturbations decay for all Reynolds numbers, even though this doesn’t happen in actual experiments. In the real world, as the Reynolds number increases, small, turbulent puffs begin to split and interact, and their lifetimes increase. Eventually, these puffs carry enough turbulence to transition the flow entirely. # (submitted by David T)
July 20, 2011