Now an international team of mathematicians, led by UC Santa Barbara professor Björn 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 Kármán 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 Kármán.
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 Kármán, 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.
Journal Reference:
Björn Birnir, Luiza Angheluta, John Kaminsky, et al. Spectral link of the generalized Townsend-Perry constants in turbulent boundary layers [open], Physical Review Research (DOI: 10.1103/PhysRevResearch.3.043054)
(Score: 1, Interesting) by Anonymous Coward on Wednesday November 17 2021, @11:49PM
I mean, just WOW!
I've watched some serious analysts work on this nonlinear problem, covering pages (and later screenfuls) with math. One friend worked on it for a couple of years straight, but the math was so crazy that he had to smoke pot for it to make sense.
If true, congrats to the team.
(Score: 1, Interesting) by Anonymous Coward on Wednesday November 17 2021, @11:50PM (1 child)
I'm too far removed from being able to talk-the-talk on this stuff. I'm going to have to pester my nonlinear fluids guys to understand whether this is as big a deal as the PR makes it sound, or whether they just found their own fit parameters.
(Score: 1, Interesting) by Anonymous Coward on Thursday November 18 2021, @07:01AM
It may be a big deal. I'm not a boundary layer specialist, so I don't already know the models involved, and I can't give a direct answer.
The fact that they provide a formula for all moments of the fluctuations makes this a very useful result. Put differently, they predict the full probability density function for fluctuations (which is NOT Gaussian). Put simply, this sort of result is needed to predict extreme events, i.e. maximum load in a wind-farm, maximum wind-speed in a storm, etc.
But I'd need more time to digest it before I can say (1) how original it is and (2) whether it can be used for any other flow configurations of interest (for instance a pipe flow). The fact that it showed up in Phys Rev Research means that it may not be that novel (PRL or even nature would have accepted it otherwise, although they'd need a special type of shameless buzzword spouting corresponding author to get pure turbulence papers in those two journals).
(Score: 4, Informative) by khallow on Thursday November 18 2021, @12:13AM (9 children)
The basic Navier-Stokes equation [wikipedia.org] (NS equations) is notoriously difficult to work with. So much effort is spent to study special cases where some parts of the equation or the environment are suppressed (often by setting terms equal to zero). Here, we see that with the four regions (though I'm not sure what they're doing with the second).
The first is a restriction of the system to two dimensions plus viscosity (the thin layer at the boundary). The transitional region is a mystery to me, but it has to glue together the boundary behavior to the eddy environment of the third layer - I'd have to study the paper to see what they did there. The third is a low viscosity fluid (they might even zero out viscosity altogether). And the fourth is a low viscosity fluid with mostly laminar fluid flow (no turbulence) with large scale eddies.
This is a typical treatment of the NS equations. It's common to break up the dynamics of a fluid into regions with different behavior and stitch them together at the boundaries between regions. This is an example of a more general principle: come up with complex equations that explain everything in your model. Then restrict your attention to particular model behavior and cut out the parts of the equations that don't apply to this sort of behavior.
Finally, while this is heralded as an accurate model of NS fluid flow along a boundary, it's not necessarily the only way. There may be some other similarly or more accurate approach that breaks up the regions differently. It may depend on the fluid as to what approach is more accurate - for example, a slime fluid (think snot) might be better modeled with a different breakdown.
(Score: 0) by Anonymous Coward on Thursday November 18 2021, @01:06AM (8 children)
And to think you were doing so good, right up to the last line. Here's a little Hallmark advice for you (from a birthday card):
Don't kiss your honey,
when your nose is runny.
You may think it's funny,
but it's snot!
(Score: 1) by khallow on Thursday November 18 2021, @01:19AM (7 children)
(Score: -1, Redundant) by Anonymous Coward on Thursday November 18 2021, @01:30AM (1 child)
Does this explain why the ropes of jism arc prior to hitting chins?
(Score: 2) by hendrikboom on Friday November 19 2021, @03:31AM
gravity's rainbow?
(Score: 0) by Anonymous Coward on Thursday November 18 2021, @01:30AM (1 child)
Just invert the ketchup bottle (30 deg from vertical is a good starting point) and then slap it on the lower side (not the bottom of the bottle, the SIDE). The acceleration moves the ketchup to one side of the bottle making a path for air to get in, then the ketchup flows easily.
(Score: 1, Informative) by Anonymous Coward on Thursday November 18 2021, @06:21AM
The sideways tap to the ketchup bottle promotes a Rayleigh–Taylor instability in the air-to-ketchup interface which enables the ketchup to flow in a civilized fashion.
--
...and that's almost everything I remember about fluid dynamics.
(Score: 2, Interesting) by pTamok on Thursday November 18 2021, @07:56AM (2 children)
Don't think so: ketchup is the canonical example of thixotropy [wikipedia.org](time-dependent shear thinning), which is a subset of non-Newtonian. AFAIK, slime is not thixotropic. If you think of slime like the stuff dumped on people in game shows which can include oobleck [wikipedia.org], it can actually be anti-thixotropic (rheopectic - stiffens when shear is applied).
...
Turns out, I was wrong. So I learned something. This paper [ Micro- and macrorheology of mucus: Samuel K. Lai, Ying-Ying Wang, Denis Wirtz, and Justin Hanes4,*Adv Drug Deliv Rev. 2009 Feb 27; 61(2): 86–100. doi: 10.1016/j.addr.2008.09.012 [nih.gov] ] describes mucus as:
(Score: 0) by Anonymous Coward on Friday November 19 2021, @12:44AM (1 child)
The ketchup problem has been solved...by putting it in a plastic bottle with a valve in the cap. Squeeze the bottle and out it squirts. The valve has some preload so it's OK to store cap down, nothing leaks out in the fridge. The label is even stuck on "upside down", it's meant to be stored cap down.
Now if you are still using an old school skinny glass ketchup bottle, the slap on the side works very nicely. It may still come out un-evenly, a little bit with each light slap, but it doesn't do the: nada, nada, nada, blort! that happens if you whap the bottom or try to shake it out.
(Score: 1) by pTamok on Friday November 19 2021, @08:17AM
The plastic 'squeezy' bottle brings its own problems: when partly empty and stored valve lowermost, when you take it out of the fridge the air above the ketchup warms up and expands, forcing ketchup out as it stands on the table waiting to be used.
The 'trick' to the old style glass bottles with metal caps was, once there was some space above the ketchup, to give the bottle a good shake and get the ketchup runny before pouring. The downside was if you forgot to check the cap was firmly closed before shaking, as you got a kitchen and yourself covered in ketchup - something you only do rarely (you do it again once the pain of cleaning up has faded from your memory).