Topological approach for the measurement of mixing state quality in a vertical rectangular channel

•Betti number ratio are introduced to quantify the degree of bubble overlap.•Evolution mechanism of flow pattern in rectangular channel is elucidated.•Modified 0-1 test is used to detect chaos in the rectangular channel.•Bubbly-mixed flow and chaotic nature are found in the rectangular channel. The phenomenon of gas-liquid two-phase mixing inside rectangular channels has been encountered in metallurgical and chemical industries. However, a large number of traditional measurement methods are unable to reliably characterize the degree of bubbles overlap in this sort of channel. In this work, the mixing state quality of the gas-liquid two-phase inside rectangular channels are accurately characterized with the help of the topological invariants (i.e., Betti numbers). Specifically, the indicator Betti numbers ratio is introduced to quantitatively describe the degree of bubbles overlap and the complexity of Betti numbers ratio time series of the gas-liquid mixing process is quantified by the permutation entropy and the modified 0-1 test for chaos for the first time. Experimental results show that the evolution of the flow pattern inside the rectangular channels is mainly caused by bubbles collision, coalescence and circling during the gas-liquid two-phase mixing process. The inexplicable mixing states were observed other than the identifiable flow pattern inside the rectangular channels. Betti numbers ratio as a novel topological indicator can be successfully used to distinguish quantitatively the various inexplicable mixing state with the same Betti number. In the context of considering the bubbles overlap, the presented working condition with the greatest Betti numbers ratio owns the lowest complexity and chaoticity.