Mathematicians derive the formulas for boundary layer turbulence 100
years after the phenomenon was first formulated

Date:
November 16, 2021
Source:
University of California - Santa Barbara
Summary:
Turbulence makes many people uneasy or downright queasy. And
it's given researchers a headache, too. Mathematicians have been
trying for a century or more to understand the turbulence that
arises when a flow interacts with a boundary, but a formulation
has proven elusive.



FULL STORY ========================================================================== Turbulence makes many people uneasy or downright queasy. And it's given researchers a headache, too. Mathematicians have been trying for a
century or more to understand the turbulence that arises when a flow
interacts with a boundary, but a formulation has proven elusive.


==========================================================================
Now an international team of mathematicians, led by UC Santa Barbara
professor Bjo"rn Birnir and the University of Oslo professor Luiza
Angheluta, has published a complete description of boundary layer
turbulence. The paper appears in Physical Review Research, and synthesizes decades of work on the topic. The theory unites empirical observations
with the Navier-Stokes equation -- the mathematical foundation of fluid dynamics -- into a mathematical formula.

This phenomenon was first described around 1920 by Hungarian physicist
Theodore von Ka'rma'n and German physicist Ludwig Prandtl, two luminaries
in fluid dynamics. "They were honing in on what's called boundary layer turbulence," said Birnir, director of the Center for Complex and Nonlinear Science. This is turbulence caused when a flow interacts with a boundary,
such as the fluid's surface, a pipe wall, the surface of the Earth and
so forth.

Prandtl figured out experimentally that he could divide the boundary
layer into four distinct regions based on proximity to the boundary. The viscous layer forms right next to the boundary, where turbulence is
damped by the thickness of the flow. Next comes a transitional buffer
region, followed by the inertial region, where turbulence is most fully developed. Finally, there is the wake, where the boundary layer flow is
least affected by the boundary, according to a formula by von Ka'rma'n.

The fluid flows quicker farther from the boundary, but its velocity
changes in a very specific manner. Its average velocity increases in the viscous and buffer layers and then transitions to a logarithmic function
in the inertial layer. This "log law," found by Prandtl and von Ka'rma'n,
has perplexed researchers, who worked to understand where it came from
and how to describe it.

The flow's variation -- or deviation from the mean velocity -- also
displayed peculiar behavior across the boundary layer. Researchers
sought to understand these two variables and derive formulas that could describe them.



==========================================================================
In the 1970s, Australian mechanical engineer Albert Alan Townsend
suggested that the shape of the mean velocity curve was influenced by
eddies attached to the boundary. If true, it could explain the odd shape
the curve takes through the different layers, as well as the physics
behind the log law, Birnir said.

Fast forward to 2010, and mathematicians at the University of Illinois
released a formal description of these attached eddies, including
formulas. The study also described how these eddies could transfer energy
away from the boundary toward the rest of the fluid. "There's a whole
hierarchy of eddies," Birnir said. The smaller eddies give energy to
the larger ones that reach all the way into the inertial layer, which
helps explain the log law.

However, there are also detached eddies, which can travel within
the fluid, and these also play an important role in boundary layer
turbulence. Birnir and his co-authors focused on deriving a formal
description of these. "What we showed in this paper is that you need to
include these detached eddies in the theory as well in order to get the
exact shape of the mean velocity curve," he said.

Their team combined all these insights to derive the mathematical
formulation of the mean velocity and variation that Prandtl and von
Ka'rma'n first wrote about some 100 years earlier. They then compared
their formulas to computer simulations and experimental data, validating
their results.

"Finally, there was a complete analytical model that explained the
system," Birnir said. With this new mathematical formulation, scientists
and engineers can adjust different parameters to predict the behavior
of a fluid.



==========================================================================
And boundary layer turbulence appears in all sorts of fields, from transportation to meteorology and beyond. "I think it's going to
have a lot of applications," Birnir remarked. For instance, a proper understanding of boundary turbulence can help make more efficient engines, reduce pollutants and minimize drag on all sorts of vehicles.

Earth's atmosphere can be modeled as a boundary flow. Despite its apparent height, the atmosphere is essentially a thin shell of moving air hugging
the planet's surface. "I think, ultimately, we will be able to use this
theory to understand both atmospheric turbulence and the jet stream,"
Birnir said. "It's going to be quite useful." The authors were surprised
to discover how important detached eddies were, especially in explaining
the turbulence transition in the buffer layer.

Studying their behavior has begun to provide insight into other types
of turbulence.

"In particular, we get insights into Lagrangian turbulence," said Birnir, referencing the theory that describes turbulent behavior in a reference
fame that moves with the flow, like a raft on a river. This contrasts
with Eulerian turbulence theory, which describes the fluid as it moves
past a fixed reference frame, like a pier on the riverbank. Attached
eddies disappear in the moving reference frame -- much like a current
seems to disappear when you're headed downstream. "But the detached
eddies are still there," Birnir said, "and they seem to play a major
role in Lagrangian turbulence." The team is currently focused on
exploring Lagrangian turbulence with these new tools, which themselves originally came from work on homogenous turbulence, where there is no
boundary. "Insights that you get in one field help you in another,"
Birnir observed.

========================================================================== Story Source: Materials provided by
University_of_California_-_Santa_Barbara. Original written by Harrison
Tasoff. Note: Content may be edited for style and length.


========================================================================== Journal Reference:
1. Bjo"rn Birnir, Luiza Angheluta, John Kaminsky, Xi Chen. Spectral
link of
the generalized Townsend-Perry constants in turbulent boundary
layers.

Physical Review Research, 2021; 3 (4) DOI: 10.1103/
PhysRevResearch.3.043054 ==========================================================================

Link to news story: https://www.sciencedaily.com/releases/2021/11/211116131724.htm

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