Saturday, June 22, 2024

What is turbulence? (And why are helicopters never quiet?)

Fluid mechanics is very often left out of the undergraduate physics curriculum.  This is a shame, as it's very interesting and directly relevant to many broad topics (atmospheric science, climate, plasma physics, parts of astrophysics).  Fluid mechanics is a great example of how it is possible to have comparatively simple underlying equations and absurdly complex solutions, and that's probably part of the issue.  The space of solutions can be mapped out using dimensionless ratios, and two of the most important are the Mach number (\(\mathrm{Ma} \equiv u/c_{s}\), where \(u\) is the speed of some flow or object, and \(c_{s}\) is the speed of sound) and the Reynolds number (\(\mathrm{Re} \equiv \rho u d/\mu\), where \(\rho\) is the fluid's mass density, \(d\) is some length scale, and \(\mu\) is the viscosity of the fluid). 

From Laurence Kedward, wikimedia commons

There is a nice physical interpretation of the Reynolds number.  It can be rewritten as \(\mathrm{Re} = (\rho u^{2})/(\mu u/d)\).  The numerator is the "dynamic pressure" of a fluid, the force per unit area that would be transferred to some object if a fluid of density \(\rho\) moving at speed \(u\) ran into the object and was brought to a halt.  This is in a sense the consequence of the inertia of the moving fluid, so this is sometimes called an inertial force.  The denominator, the viscosity multiplied by a velocity gradient, is the viscous shear stress (force per unit area) caused by the frictional drag of the fluid.  So, the Reynolds number is a ratio of inertial forces to viscous forces.  

When \(\mathrm{Re}\ll 1\), viscous forces dominate.  That means that viscous friction between adjacent layers of fluid tend to smooth out velocity gradients, and the velocity field \(\mathbf{u}(\mathbf{r},t) \) tends to be simple and often analytically solvable.  This regime is called laminar flow.  Since \(d\) is just some characteristic size scale, for reasonable values of density and viscosity for, say, water, microfluidic devices tend to live in the laminar regime.  

When \(\mathrm{Re}\gg 1\), frictional effects are comparatively unimportant, and the fluid "pushes" its way along.  The result is a situation where the velocity field is unstable to small perturbations, and there is a transition to turbulent flow.  The local velocity field has big, chaotic variations as a function of space and time.  While the microscopic details of \(\mathbf{u}(\mathbf{r},t)\) are often not predictable, on a statistical level we can get pretty far since mass conservation and momentum conservation can be applied to a region of space (the control volume or Eulerian approach).

Turbulent flow involves a cascade of energy flow down through eddies at length scales all the way down eventually to the mean free path of the fluid molecules.   This right here is why helicopters are never quiet.  Even if you started with a completely uniform downward flow of air below the rotor (enough of a momentum flux to support the weight of the helicopter), the air would quickly transition to turbulence, and there would be pressure fluctuations over a huge range of timescales that would translate into acoustic noise.  You might not be able to hear the turbine engine directly from a thousand feet away, but you can hear the resulting sound from the turbulent airflow.  

If you're interested in fluid mechanics, this site is fantastic, and their links page has some great stuff.

5 comments:

  1. I never really thought about why you can hear a helicopter from a distance, but in retrospect it’s pretty deep and amazing!

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  2. Anonymous7:36 PM

    To refer to the general theme at the top of your blog (rather than this particular post):

    https://www.nature.com/articles/s42254-024-00732-1

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  3. Non-paywall version of the above-linked Nature commentary: https://durham-repository.worktribe.com/output/2503794.

    I understand where the author is coming from, and when I was younger, I would probably have wholeheartedly agreed with him. But today, frankly, I find the article pejorative and elitist. Implicit throughout the exposition is the attitude that anyone interested in, or motivated by, technology and practical application is by default intellectually "dirty", "inferior", and/or "second-rate". That is a terrible and tremendously counterproductive mindset.

    It's probably (indeed, almost certainly) true that esoteric philosophical questions about the nature of reality and the universe have more sex appeal to the public than understanding how a device works. It's also true that condensed matter, in general, could do a much better job of broadly communicating such sex appeal elements of our field. But you can do the former without denigrating people who work on and/or are interested in scientific question for practical/applied reasons.

    One scientist might be interested in the non-equilibrium thermodynamics of quantum entangled systems to understand black hole entropies and the AdS/CFT correspondence. Another scientist might be interested in it so that they can design more energy-efficient quantum information technologies for biomedical data processing. At the end of the day, they are both thinking about intellectually deep questions, and we can raise one up without putting the other down.

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  4. Anon, thanks for bringing that to my attention. PPP, thanks for the non-paywalled link. I didn't read that column as negatively as you did. One issue that I haven't seen addressed in this discussion is unity of research program. In a very real sense, the particle physics community had an overarching goal that is easy to articulate: To find the ultimate building blocks of the universe and understand their interactions. After Lawrence and WWII, it was clear that this required particle accelerators and that this should be the grand strategy. In the case of condensed matter, now we can articulate a goal (we want to understand all the possible (collective) states of matter and their properties). However, CMP started much less coherently, with many people trying many different approaches to many classes of materials, and the connection to deep ideas was only revealed in a rather piecemeal fashion. That lack of cohesion was in some ways a sign of vibrancy, but definitely didn't lend itself to broad messaging.

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    Replies
    1. Thanks for your comments Doug. Looking back, maybe I initially overreacted. I just took offense with the terms “banality” and “boring” being headlines describing technology. The actual article itself is more sympathetic and balanced, though I would have chosen to blanket it in a different way.

      I do agree that lack of an easily identifiable common goal is a big roadblock that makes CMP harder to popularize.

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