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Unformatted text preview: Physics 212: Statistical mechanics II, Fall 2006 Lecture VI Turbulence (mostly not covered in class this year) At high Reynolds number, the static solutions of viscous hydrodynamics are no longer physi- cally relevant; real systems show rapidly varying flow configurations across many length and time scales. The problem of “fully developed turbulence” for open systems (i.e., work is supplied to the system externally and dissipated by viscosity) is still not completely understood, but many simpler examples of the transition from static solutions to chaotic time-dependent solutions have been understood in the last 20-30 years. Turbulent flow is probably the most physically important example of a chaotic “steady state” in an open nonlinear system: here “steady state” means that even though the system is rapidly fluctuating, it is meaningful to define time averages for quantities like the mean fluid velocity. Now we briefly review one phenomenological law about fully developed turbulence. Weakly turbulent systems (i.e., when the Reynolds number is just above the limit for turbulent flow) have been studied in great detail with a variety of sophisticated techniques, and are connected to some major developments in chaos theory such as the notion of period doubling (cf. Berkeley physics course on nonlinear dynamics). Here we focus on statistical physics: we find a statistical scaling law in the limit of highly turbulent flows, when the intermittent variation in the velocity v λ at the smallest scales is comparable to the mean velocity u . Big whorls have little whorls That feed on their velocity, And little whorls have lesser whorls And so on to viscosity. Lewis Richardson Let’s start with an estimate of the energy dissipation due to viscosity in a system. How much energy per time per mass is dissipated? Recall that R = ρvl μ = vl ν , (1) where ν ≡ μρ- 1 is the so-called “kinematic” viscosity. In fully developed turbulence, energy is transferred from large eddies/whorls down to eddies of smaller and smaller size, until it is eventually dissipated in the smallest eddies. Since all the energy passes through large eddies, which know nothing about the viscosity, it should be possible to express the energy dissipation without using the viscosity. We define L as the typical size of a large eddy where energy is put in, and λ as the smaller length scale where viscosity becomes significant. The units of energy dissipation are energy per time per mass; to make a fluid quantity with these units, without using viscosity, we need- 3 powers of time, so we guess = ( δu ) 3 l , (2) 1 Here l is the size of an eddy and δu is the variation in the velocity for that eddy. This is an estimate of the order of magnitude of the energy dissipation based on observing eddies of size l . We can use this assumption to derive a simple picture of turbulence: assume that the above formula is correct for eddies of all different sizes, so that all the estimates of energy dissipation are consistent: by...
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