Direct calculation of transition to one of two possible secondary flows in a wide spherical layer under the action of accelerated rotation of the inner sphere

Wave number selection in the presence of noise: Experimental results

“…Secondly, threshold curves were found to depend on Reinit: with increasing Reinit, the threshold values of dRe/dτ became larger. It is interesting to note that these consequences from experimental data are in agreement with numerical results, obtained earlier with fixed Refin [17] where, in the Reinit -dRe/dτ diagram, the same sequence of changes to the dominant wave number 4 →3→4 was observed with dRe/dτ increasing. It is possible that a velocity field divergence greater than 10 -6 (due to the controlled inaccuracy of the numerical calculation algorithm) in the previously mentioned numerical results may be considered as additional noise.…”

“…The first scenario is realized for very small noise amplitudes, and the wave number m=3 appears to be independent of acceleration rate. This first scenario was observed in experiments and calculations [15,17], when acceleration was begun from the state of rest or with very slow flow (Reinit = 0-10). In the second scenario, for bigger noise amplitudes, one (from 0→4 to 0→3) or two (from 0→4 to 0→3 at smaller values of dRe/dτ and from 0→3 to 0→4 at larger values) transitions were found to occur.…”

“…In the range of Re from 460 up to 500, increments of growth for linear mode λ4 (m=4) are greater than the corresponding increments for linear mode λ3 (m=3) [11,33]. So, when Re increased quasi-statically (or step by step), secondary [33] or with acceleration increasing [17], short-term conditions arise which lead to λ4 < λ3. This is a reason for selection of secondary flow with m=3, because the flow keeps a memory of its history.…”

“…When acceleration is slow and its 3 duration is longer than the linear instability e-folding timescale, a 0→1 transition takes place. In a wide gap (δ=1) inner sphere, acceleration results in a redistribution of the kinetic energy of the flow in space and time [17]. Firstly, the meridional circulation and azimuthal velocity maxima are shifted for a short time from the poles towards the equatorial plane.…”

“…It was shown in some range of initial values of Reinit at small acceleration rates that the preferred mode (m=4) is the same as for a quasi-static increase of Re, while with larger acceleration rates the wavenumber selection became essentially random [15]. It was suggested that random wave number selection, with acceleration values increasing within a restricted range of initial Re, can be explained by the influence of noise imposed in the form of deviations of the inner sphere rotational velocity from given values [17].…”