Pattern Formation in Convection: The Legacy of Henry Benard and Lord Rayleigh

Pattern Formation in Convection: The Legacy of Henry Benard and Lord Rayleigh

Guenter AhlersDepartment of PhysicsUniversity of CaliforniaSanta Barbara CA USA

The Legacy of Henry Benard and Lord Rayleigh

Henri Benard 1874-1939Lord Rayleigh 1842 - 1919

Rayleigh-Benard Convection

Count Rumford (Benjamin Thompson) 1753-1814

Recognized the phenomenon of convectionin his development of an improved fireplace

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William Prout (1785-1850)

Identified a new heat transport mechanism (as opposed to radiation or conduction) and called it convection, from “Convectio”: to carry

observed a similar hexagonal pattern in soapy water cooled from the open top surface

J. Thompson, Proc. Glasg. Phil. Soc. 13, 464 (1882)

E.H. Weber, Ann. Phys. Chemie 94, 447 (1855)reported that a flow pattern consisting of an array of polygonal cells developed in a droplet of an alcohol-water solution containing a tracer, sitting in ambient air

Henri Benard (1900)

1.) he had a free upper surface2.) he found an hexagonal pattern3.) used various fluids of different viscosity4.) studied surface deformation due to convection5.) measured the characteristic length scales of the patterns6.) determined the direction of flow within the convection cells

Carried out the first systematic and quantitative study of convection in a shallow layer heated from below, and studied the associated formation of convection PATTERNS systematically and quantitatively

Ph.D. March 15 1901 College de France

William Nusselt (1882-1957)W. Nusselt, Forsch.-Arbeit auf dem Gebiete des Ing. Wesens, Heft 63,64 (1907); Gesundh.-Ing. 42,43, 477 (1915).

Heat transport by air between rigid plates. Motivated by technological issue of insulation.

N = Nusselt Numbereff = effective thermal conductivity in the presence of convection = thermal conductivity of quiescent fluid

Lord Rayleigh (John William Strutt, third Baron Rayleigh) 1842 - 1919

Fellow, Trinity College, 1866 - 1871FRS 18731879: Professor of Experimental Physics and Head of the Cavendish Laboratory (successor of Maxwell)1887 to 1905: Professor of Natural Philosophy, Royal Institution of Great Britain (successor of Tyndall)Nobel Prize 1904 “for his investigation of the density of the most important gases

and for his discovery of argon in connection with these studies”

Chancellor of Cambridge University 1908

Lord Rayleigh

Q

L T

Rayleigh carried out a stability analysis of the quiescentfluid layer. He used slip boundary conditions at top and bottom in order to permit an analytic solution

T/Tc - 1

He did not recognize that Benard’s convection actuallywas driven primarily by surface-tension gradients,as pointed out much later by J.R.A. Pearson of Cambridge “On convection cells induced by surface tension”, JFM 4 , 489 - 500 (1958).

Rayleigh’s results:





Sir Harold Jeffreys, 1891 - 1989

Fellow, St.Johns College, Cambridge 1914 to 1989Mathematics, GeophysicsPlumian Professor of AstronomyKnighted 1951

No-slip (rigid) boundary conditions

nonlinear effects: nature of the bifurcation pattern just above onset

Next milestones:

Malkus and Veronis (1958): slip (free) BCs super-critical rolls (or stripes)

nonlinear effects: nature of the bifurcation pattern just above onset

Next milestones:

Malkus and Veronis (1958): slip (free) BCs super-critical rolls (or stripes)

Schluter, Lortz, and Busse (1965): ridgid BCs super-critical rolls (or stripes)

Open circles: increasingSolid circles: decreasing T

Supercritical!

Non-Boussinesq Boussinesq Binary mixtures (diffusion driven)

Swift and Hohenberg showed that the bifurcation becomessubcritical in the presenceof additive (thermal) noise !!!

Swift and Hohenberg showed that the bifurcation becomessubcritical in the presenceof additive (thermal) noise !!!

J. Swift and P.C. Hohenberg, “Hydrodynamic fluctuations at the convective instability”, Phys. Rev. A 15, 319 (1977).

For most systems, this effect wasexpected to be observable only for < 10-6

Pierre C. Hohenberg 1934 NYU Senior Vice Provost for Research

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J. Oh and G.A., Phys. Rev. Lett. 91, 094501 (2003).

Convection in SF6 nearits critical point= 0.0000 +/- 0.0005

Fritz Busse 1936

Stable States No extremum principle

Any state inside the BusseBalloon is attainable if the phase of the pattern is pinned, e.g. by sidewalls, in an experiment.

Wave-number selection processes occur when the phase can slip at some pointIn the pattern

In the 1970’s and thereafter the Busse Balloon became the “playground” of experimentalistsand theorists with an interest in nonlinear physics and pattern formation

Four uniquely nonlinear issues:

1.) Wavenumber selection

2.) Spatio-temporal chaos (STC)

3.) Coarsening

4.) Localized structures (Pulses)

Wave-number selection by curved rolls:Curved rolls normally induce mean flow.For the rotationally symmetric target pattern mean flow can not occur.Thus the mean flow must be balanced precisely by a pressure gradient.This condition leads to a unique wave number unrelated to any extremum principle.

L. Koschmieder and S. Pallas, Int. J. Heat Mass Trans. 17, 991 (1974);M. Cross, Phys. Rev. A 27, 490 (1983); P. Manneville and J.M. Piquemal, Phys. Rev. A 28, 1774 (1983);M.C. Cross and A.C. Newell, Physica D 10, 299 (1984);J. Buell and I. Catton, Phys. Fluids 29, 1 (1986)A.C. Newell, T. Passot, and M. Souli, Phys. Rev. Lett. 64, 2378 (1990); J. Fluid Mech. 220, 187 (1990).

J.R. Royer, P. O'Neill, N. Becker, and G. A., Phys. Rev. E 70 , 036313 (2004).

Cross-Newell equationfor the phase of the pattern[M.C. Cross and A.C. Newell, Physica D 10, 299 (1984)].

Photo credit: Ekaterina G. Tribelskaya

Wave-number selection by a domain wall.The wall can be diplaced, thereby changing the wave number of the rolls parallel to it. This yields a wave number that differs from the one selected by the target patterns.M.C. Cross and G. Tessauro, Phil. Mag. A 56, 703 (1987).

Spatio-temporal chaos



Experiment: S. Morris, E. Bodenschatz, D.S. Cannell, and G.A., Phys. Rev. Lett. 71, 2026 (1993).

Bi-stability (straight rolls and Spiral-Defect-Chaos)Dynamics driven by competition between different wavenumber-selection mechanisms and mean flow[Chiam, M. R. Paul, M. C. Cross, and H. S. Greenside, Phys. Rev. E67, 056206 (2003)].

Q

d T





Prandtl = 0.9 CO2 Omega = 17Movies by N. Becker

Spatio-temporal chaos at onset above a supercritical bifurcation

Coarsening in a non-potential system

Much numerical work (Vinals, Cross, Paul, and others)starting from random small-amplitude initial conditions.Amplitude evolves fast to a highly disordered pattern.Pattern evolves from extreme disorder, perhaps to stripesor possibly to a ‘glassy” state retaining some disorder.What are the time scales and the dominant mechanismsof this evolution?

Experiments are difficult because random initial conditionsare difficult to maintain. Instead:

Quench from spiral-defect chaos down to a small wherestraight rolls are known to be the only statisticallystationary state.



N. Becker and G.A., unpublished.

Roll curvature in the bulk heals quickly;disclinations disappear.

Dislocations become scarce soon after.

Domain walls seemto dominate the late pattern evolution.

Some dislocations are lingering at late times.

0.5 to 0.05



Localized structures (Pulses, dissipative solitons)

convect. with rotation(Ecke, Knobloch et al.)




Binary mixture convection in an annulusExperiments: E. Moses, J. Fineberg, and V. Steinberg, Phys. Rev. 35, 2757 (1987);R. Heinrichs, G.A., and D.S. Cannell, Phys. Rev. A 35, 2761 (1987);Numerous subsequent papers by Kolodner and others. Theory: O. Thual and S. Fauve, Europhys. Lett. 49, 749 (1988);and numerous papers thereafter.






Binary mixture convectionExperiment:K. Lerman, D.S. Cannell, and G. A.,

Phys. Rev. E 53, R2041 (1996).Theory: I. Mercader, M. Net, and E. Knobloch, Phys. Rev. E 51, 339 (1995).

Binary mixture convectionExperiment:K. Lerman, E. Bodenschatz, D.S. Cannell, and G. A,

Phys. Rev. Lett. 70, 3572 (1993).



Count Rumford Henri Benard Lord Rayleigh William Nusselt Sir Harold

PierreHohenberg

Mike Cross Alan Newell

Fritz Busse

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Pattern Formation in Convection: The Legacy of Henry Benard and Lord Rayleigh

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