Nonlinear dynamics of waves and modulated waves in 1D thermocapillary flows. I. General presentation and periodic solutions

Abstract

We present experimental results on hydrothermal traveling-waves dynamics in long and narrow 1D channels. The onset of primary traveling-wave patterns is briefly presented for different fluid heights and for annular or bounded channels, i.e., within periodic or non-periodic boundary conditions. For periodic boundary conditions, by increasing the control parameter or changing the discrete mean-wavenumber of the waves, we produce modulated waves patterns. These patterns range from stable periodic phase-solutions, due to supercritical Eckhaus instability, to spatio-temporal defect-chaos involving traveling holes and/or counter-propagating-waves competition, i.e., traveling sources and sinks. The transition from non-linearly saturated Eckhaus modulations to transient pattern-breaks by traveling holes and spatiotemporal defects is documented. Our observations are presented in the framework of coupled complex Ginzburg-Landau equations with additional fourth and fifth order terms which account for the reflection symmetry breaking at high wave-amplitude far from onset. The second part of this paper [1] extends this study to spatially non-periodic patterns observed in both annular and bounded channel.

Study of dynamics in post-transient flows using Koopman mode decomposition

“…6(b)) slightly decreases due to the decrease in the Koopman frequencies. We note that the above methodology based on Koopman modes is already used, in a concealed form, to extract the wave numbers from experimental data on nonlinear waves in thermally-driven flows [103,104], and internal waves in stratified flows [105]. We study the the efficiency of Koopman modes in representing the flow, by computing the error of the spectralprojection models.…”

Transmission and reflection of internal wave beams

“…Previous studies have used the Hilbert transform on roll waves and hydrothermal traveling waves to demodulate the signal. 27,28 It has recently been applied to inter-nal gravity waves as a technique for separating the four wave beams emanating from an oscillating source. 29 As in that work, we use the Hilbert transform to separate upward from downward-propagating waves in a vertical time series.…”

Reflection and diffraction of internal waves analyzed with the Hilbert transform

“…In this article, we call Hilbert transform (or complex demodulation) the operation associating the complex-valued fieldŨ to the real field U. This demodulation technique has been previously used to compute local and instantaneous amplitudes, frequencies and wavenumbers [10,11,12] but, to the best of our knowledge, the present study is the first application in the context of internal gravity waves.…”