Upstream-advancing waves generated by a current over a sinusoidal bed

Abstract

Upstream-advancing waves are observed in open channel flows over a fixed sinusoidal bed with large amplitude, when the Froude number is less than the resonant value, at which stream velocity is equal to the celerity of the wave with wavelength equal to that of the bottom surface. Their wavelength is about 3-6 times as long as the bottom wavelength and the celerity is close to that obtained from potential flow theory. Therefore, the wavelength of upstream-advancing waves is determined by linear stability analyses assuming that they are induced by the Benjamin-Feir-type instability of steady flow. Here, two formulas for the wavelength with different scaling are introduced and compared with experiment. In addition, the mechanisms of upstream-advancing waves are investigated qualitatively using the forced Schr6dinger equation.

Upstream-propagating waves induced by steady current over a rippled bottom: theory and experimental observation

“…The experimental data of the flow condition on the observation of such upstream-advancing waves does not match that from the linear instability result of Yih (1976). Kyotoh & Fukushima (1997) then postulated that the observed upstream-advancing waves result from the nonlinear instability of the steady base flow and aimed to corroborate the experimental data with the Benjamin–Feir instability. Kyotoh & Fukushima (1997) analysed the modulational instability of the nonlinear steady flow near the resonance flow speed (Mei 1969).…”

“…Kyotoh & Fukushima (1997) then postulated that the observed upstream-advancing waves result from the nonlinear instability of the steady base flow and aimed to corroborate the experimental data with the Benjamin–Feir instability. Kyotoh & Fukushima (1997) analysed the modulational instability of the nonlinear steady flow near the resonance flow speed (Mei 1969). The predicted flow condition for the nonlinear instability as well as the period of upstream-propagating waves do not agree with the experimental data.…”

“…To assist in understanding the phenomenon and verify the theoretical analysis of upstream-propagating wave generation in steady flow over a rippled bottom, we conducted a series of laboratory experiments in a relatively large-scale wave flume. The experimental set-up and configurations were optimized on the basis of Kyotoh's preliminary observation of upstream-advancing waves in a rather small wave flume (Kyotoh & Fukushima 1997). While the design and initial stage of the experiments as well as some preliminary qualitative observations were reported in Fan et al (2016), the experimental set-up was improved and the experiments were recalibrated for better accuracy afterwards.…”

“…The wavenumber is calculated from its relation with the frequency and phase speed. The phase speed of the wave can be determined from the correlation coefficient of the wave elevation measured at two neighbouring wave gauges (Kyotoh & Fukushima 1997).…”

“…In a relatively small-scale flume experiment of steady flow over a bottom patch of sinusoidal ripples, Kyotoh & Fukushima (1997) observed an interesting phenomenon that a regular surface wave train of significant amplitude advancing against the incoming current is produced when the steady flow reaches the critical speed, which varies with water depth and bottom ripple wavenumber. The experimental data of the flow condition on the observation of such upstream-advancing waves does not match that from the linear instability result of Yih (1976).…”