Notebooks

Turbulence, and Fluid Mechanics in General

06 Sep 2012 00:03

The story is told of many giants of modern physics, but most plausibly of Heisenberg, that, on his death-bed, he remarked that the two great unsolved problems were reconciling quantum mechanics and general relativity, and turbulence. "Now, I'm optimistic about gravity..."

Fluid flow, it should be said, is in one sense very well understood; since the early 1800s there's been a fine, non-linear, Newtonian equation for the velocity field that seems to work, the Navier-Stokes equation. (Like Newton's law of gravitation, it should be branded on to anyone who babbles that non-linear physics is "new" or "non-Newtonian".) One of its properties is that it's invariant so long as the Reynolds number --- density*(length scale)*(velocity scale)/viscosity --- stays the same. This is why wind-tunnels work: the model in the tunnel is shorter than the original, but the mean speed is higher, so the flows are equivalent. When the Reynolds number is small, the equation is mathematically nice, the non-linearities are small, and we can solve the equation. The stream-lines --- the paths followed by small tracer particles dropped into the fluid --- form nice layers around the boundaries of the flow, which is why the flow is called laminar, and these laminæ are stable.

As you turn up the Reynolds number, the non-linearities become important, and the flow gets uglier --- it is no longer steady, but erratic (probably chaotic in the strict sense), and the nice regular stream-lines and their laminæ get snarled and then completely confused; eddies and vortices form and spin and dissolve without much obvious pattern, and the develop their own eddies in turn; odd structures with names like "von Kármán streets" appear. (Pictures make this a lot clearer; van Dyke's Album of Fluid Motion is full of handsome ones, but short on explanation.) Turbulence --- yea, "fully developed turbulence", even --- is when this decay into confusion is complete, when there are eddies and motions on all length scales, from the largest possible in the fluid on down to the so-called "dissipation scale," which is (roughly!) the minimum eddy size, as set by the mechanical properties of the fluid (its viscosity and the like). When faced with this confusion, if not well before, we give up and turn to statistics; we begin to ask questions about the statistical properties of the flow --- if you will, about all possible flows we could see under given conditions. Here we can make some nice observations, and even come up with two well-confirmed empirical laws about these statistics, and endless graphs.

So what, you may ask, is the fabled "problem of turbulence"? In essence, this: what on Earth do our statistics and our equation have to do with each other? A solution to the problem of turbulence would be, more or less, a valid derivation from the Navier-Stokes equation (and statements about the appropriate conditions) of our measured statistics. Physicists are very far from this at present. Our current closest approach stems from the work of Kolmogorov, who, by means of some statistical hypotheses about small-scale motion, was able to account for the empirical laws I mentioned. Unfortunately, no one has managed to coax the hypotheses from the Navier-Stokes equation (sound familiar?) and the hypotheses hold exactly only in the limit of infinite Reynolds number, i.e. they are not true of any actual fluid.

So what's to do? Well, all sorts of things, including more or less direct simulations of flows by cousins of cellular automata called "lattice gasses" (which is how I connect to the subject, though very vaguely). One approach uses the vorticity (the curl of the velocity field, which tells us about how the fluid swirls), since it turns out to be possible to identify some (more or less) simple objects in the flow, called vortex lines or vortex tubes, work out how they interact (there's a Hamiltonian), and then use statistical mechanics to calculate various emergent properties --- which, if you use just the right approximations, and tolerate negative temperatures (which are not impossible, and actually hotter than infinity) gives you the Kolmogorov laws. This could've been custom-tailored for my philosophical and methodological biases, which makes me suspicious, as do all the leaps in the approximation scheme used. (For the pro-vorticity case, see Chorin; reasons for caution are discussed by Frisch, pp. 189f.)

If people must find analogies for society, ecosystems, etc., from physics and engineering, turbulence is probably a better one than feedback.


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