Bubbles drive gas and chemical transfers in various industrial and geophysical contexts, in which flows are typically turbulent. As gas and chemical transfers are bubble size dependent, a robust quantification requires a prediction of bubble breakup. The most common idea, introduced independently by Kolmogorov in 1949 and Hinze in 1955, is to consider a sharp limit between breaking and non breaking bubbles, given by $\operatorname{We}_c\approx 1$, where the Weber number $\operatorname{We}$ is the ratio between inertial and capillary forces at the bubble scale. In the statistically stationary state, $\operatorname{We}_c$ sets the maximum bubble size. Yet, due to the inherent stochasticity of the flow every bubble might in reality break. In this work, we use a stochastic linear model previously developed to infer the breakup probability of bubbles in turbulence as function of both $\operatorname{We}$ and the residence time. This allows us to introduce a definition of the critical Weber number accounting for the time spent by bubbles within a turbulent region, extending the stationary description. We show that bubble breakup is a memoryless process, whose breakup rate varies exponentially with $\operatorname{We}^{-1}$. The linear model successfully reproduces experimental breakup rates from the literature. We show that the stochastic nature of bubble breakup is central when the residence time of bubbles is smaller than ten correlation times of turbulence at the bubble scale: the transition between breaking and non breaking bubbles is smooth and most bubbles can break. For large residence times, the original vision of Kolmogorov and Hinze is recovered.
Les bulles contrôlent les réactions chimiques et les transferts de gaz pour un grand nombre de situations industrielles et naturelles caractérisées, typiquement, par des écoulements turbulents. Les transferts de gaz aux interfaces et les réactions chimiques étant fonction de la taille des bulles, pour les décrire précisément il est nécessaire de connaître leur distribution de taille et en particulier de prédire leur cassure. L’idée la plus répandue, introduite indépendamment par Kolmogorov en 1949, puis Hinze en 1951, consiste à considérer une limite stricte entre bulles stables et instables, donnée par $\operatorname{We}_c \approx 1$, où le nombre de Weber, $\operatorname{We}$, est le ratio entre forces inertielles et capillaires à l’échelle de la bulle. En régime statistiquement stationnaire, $\operatorname{We}_c$ fixe une taille maximum. Or, de part le caractère intrinsèquement fluctuant de la turbulence, toutes les bulles pourraient en réalité casser. Dans ce travail, nous utilisons un modèle linéaire stochastique développé précédemment pour prédire la probabilité de cassure en fonction du $\operatorname{We}$ mais aussi du temps passé par les bulles dans l’écoulement turbulent, nommé le temps de résidence. Ceci nous permet d’introduire un $\operatorname{We}_c$ qui prend maintenant en compte le temps de résidence, étendant ainsi la vision stationnaire. Nous montrons que le processus de cassure est un processus sans mémoire, associé à des taux de cassure variant exponentiellement en $\operatorname{We}^{-1}$. Ce modèle linéaire reproduit les taux de cassure expérimentaux présents dans la littérature. Nous montrons que la nature stochastique de la cassure est centrale lorsque les temps de résidence sont inférieurs à 10 temps de corrélation de la turbulence à l’échelle de la bulle : la transition entre bulles qui cassent et qui ne cassent pas est continue et la majorité des bulles peuvent casser. En revanche, pour des temps longs, nous retrouvons la limite stricte introduite par Kolmogorov et Hinze.
Revised:
Accepted:
Published online:
Mots-clés : Bulle, turbulence, cassure, modèle stochastique
Aliénor Rivière 1 , 2 ; Stéphane Perrard 2
CC-BY 4.0
@article{CRMECA_2025__353_G1_1351_0,
author = {Ali\'enor Rivi\`ere and St\'ephane Perrard},
title = {Bubble breakup probability in turbulent flows},
journal = {Comptes Rendus. M\'ecanique},
pages = {1351--1364},
year = {2025},
publisher = {Acad\'emie des sciences, Paris},
volume = {353},
doi = {10.5802/crmeca.343},
language = {en},
}
Aliénor Rivière; Stéphane Perrard. Bubble breakup probability in turbulent flows. Comptes Rendus. Mécanique, Volume 353 (2025), pp. 1351-1364. doi: 10.5802/crmeca.343
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