New algorithm improves fluid interface dynamics calculations

Scientists at the Lawrence Berkeley National Laboratory devised a new framework that can more accurately resolve the Navier-Stokes equations, a series of mathematical statements used to predict how fluids will flow, according to a study published Friday in Science Advances.

The Navier-Stokes equations are based on the application of Newton’s second law to the motion of fluids, along with the assumption that the stress in fluid is the combination of a diffusing viscous term and a pressure term. Currently, they are used in a vast array of different fields, such as special effects for movies, industrial research, and engineering, lab officials explained.

However, as they noted in a statement, some computational methods used to solve these complex mathematical problems are unable to accurately resolve intricate fluid dynamics occurring beside moving boundaries or surfaces, or how tiny structures influence the motion of those surfaces and the surrounding environment. Hence, the need to improve those computational methods.

Research makes it easier to determine speed, pressure of fluids

Enter Robert Saye, a research fellow in the Berkeley Lab mathematics group. He has come up with a new formula for the Navier-Stokes equations that makes them earlier to calculate using numerical computation methods. Saye’s algorithms are capable of capturing both tiny features near evolving interfaces and the influence those structures have on distant dynamics.

“These algorithms can accurately resolve the intricate structures near the surfaces attached to the fluid motion,” he explained. “As a result, you can learn all sorts of interesting things about how the motion of the interface affects the global dynamics, which ultimately allows you to design better materials or optimize geometry for better efficiency.”

“For example, in a glass of champagne, the motion of the little gas bubbles depends crucially on boundary layers surrounding the bubbles,” Saye added. “These boundary layers need to be accurately resolved, otherwise you won’t see the slight zig-zag pattern that real bubbles take as they float to the top of the glass. This particular phenomena is important in bubble aeration, a process used widely in industry to oxygenate liquids and transport materials in liquid chambers.”

His work will make it easier for researchers to use the Navier-Stokes equations to determine how quickly a fluid is moving in its environment, the amount of pressure it is under and what forces it exerts on its surroundings, the lab noted. Saye’s algorithm will also enable experts to more easily and accurately gain new insight into how each of these traits influence one another.

What is it about this method that makes it better?

As the lab explains, researchers have in the past attempted to come up with several different ways to simplify Navier-Stokes equations and their solutions, including one method in which liquids (and sometimes gases) are modeled as incompressible. Most of these methods are so-called low-order methods, Saye said, while his new technique is a high-order one.

“High-order methods are in some sense more accurate,” he explained. “One interpretation is that, for fixed computing resources, a high-order method results in more digits of accuracy compared to a low-order method. On the other hand, it is often the case that you only need a handful of digits of accuracy in your simulation. In this case, a high-order method requires less computing power, sometimes significantly less.”

Also, low-order methods for fluid interface dynamics tend to introduce something known as a numerical boundary layer into the calculated results. These can lead to imperfections, limiting a scientist’s ability to closely examine and analyze the fluid dynamics next to the interface. When intricate dynamics are involved, things move very quickly or small features are included in the interface, high-order methods are needed, Saye said.

“I wanted to make these numerical algorithms significantly more accurate. When I thought about it that way, I realized that I needed a whole new technique to solve the equations,” he said. To do so, he applied gauge methods to the equations. “Gauge methods are about the freedom one has in choosing variables in the equations. So I essentially used these ideas to rewrite the Navier-Stokes equations in a way that is more amenable to developing very accurate simulation algorithms.”

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Image credit: Berkeley Mathematics Lab

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