Rayleigh-Bénard experiment in liquid helium ; frequency locking and the onset of turbulence

2014 In a Rayleigh-Bénard experiment in liquid helium, we study the time-dependent regimes, above the onset of convection, in a parallelepipedic cell with two convective rolls of oblate cross-section 03B1 = 1.90, where 03B1 is the rolls wave number. A first frequency f1, associated to oscillating rolls, appears for a Rayleigh number R ~ 2 x 104, then for R ~ 2.7 104 a second frequency f2, much smaller, is observed. Two frequency locking regimes are present, with hysteresis, for the frequency ratios f1/f2 = 6.5 and f1/f2 = 7. The onset of turbulence germinates from the last frequency locking state for R ~ 3.4 104. This transition to turbulence is triggered by the generation of the frequencies f2/2, f2/4, ... and in real time is defined by a phase-chaotic behaviour rather than by an amplitude behaviour. LE JOURNAL DE PHYSIQUE LETTRES TOME 40, 15 AOUT 1979, Classification Physics Abstracls 67.90 47.25Q We present in this Letter some new results on a Rayleigh-Benard experiment whereby a parallelepipedic cell of liquid helium is heated from below, the liquid being at a temperature of 3.00 K under a pressure of 3 atmosphere, its Prandtl number being (J = v/x = 0.5. In such an experiment, as one increases the temperature difference AT between the top and bottom part of the fluid, the first bifurcation is associated with fluid convection. In our geometry, a parallelepipedic cell with a base 1.6 x 2.8 mm and a height 0.85 mm, we have studied the metastable state where two convective rolls, perpendicular to the largest side of the sample, are formed. Figure 1 shows the curve of marginal stability for the onset of convection. Whereas for a sample of infinite transverse dimension the roll wave number [1] is equal to ac = 3.117, our metastable state corresponds to a = 1.90 (a is a normalized wavevector, a = 2IId/~, where d is the cell height). The second bifurcation, for a small Prandtl number fluid, leads to a limit cycle, physically associated to waves propagating along the convective rolls, the so-called oscillatory [1] instability. We detect, with a local bolometer, a temperature oscillation of frequency /i ~ 550 mHz at the onset for a Rayleigh number R ~ 2 x 104, or a temperature difference AT ~ 120 mK. The third bifurcation generates a second frequency, f2, the onset of which is for R = 2.7 x 104 and its starting value f2 = 90 mHz. Then, as one keeps increasing the temperature difference, all the combination frequencies, f = mfl 1 + nf 2, m and n integer, appear as one Fourier analyses the data (Fig. 2). This is followed by two frequencylocking states (Fig. 3), for Rayleigh numbers between 3 x 104 and 3.3 x 104, the first one corresponds to fl1f2 = 6.5, and the second one to fl/f2 = 7 (Fig. 4). Finally, for R = 3.38 x 104, the onset of turbulence is observed, germinating from the last locking state, with a generation of the frequencies f 2/2, f 2/4 and