Modelling entrainment volume due to surface-parallel vortex interactions with an air–water interface

Abstract

We consider the entrainment volume that results from the quasi-two-dimensional interactions of rising surface-parallel vorticity with an air–water interface. Based on systematic (three-dimensional) direct numerical simulations (DNS) of the canonical problem of a rectilinear vortex pair impinging on and entraining air at the free surface, we develop a phenomenological model to predict the resulting entrainment volume in terms of four key parameters. We identify a new parameter, a circulation flux Froude number $Fr^2_\Xi =|\varGamma |W/a^2\,g$ , that predicts the dimensionless volume $\forall$ of entrained air initiated by a coherent vortical structure of circulation $\varGamma$ , effective radius $a$ , vertical rise velocity $W$ with gravity $g$ . For $Fr^2_\Xi$ below some critical value $Fr^2_{\Xi cr}$ , no air is entrained. For $Fr^2_\Xi >Fr^2_{\Xi cr}$ , the average initial entrainment $\overline {\forall }_o$ scales linearly with ( $Fr^2_\Xi -Fr^2_{\Xi cr}$ ). We also find that $\overline {\forall }_o$ is linearly dependent on circulation Weber number $We_{\varGamma }$ for a range of vortex Bond number $5 \lesssim Bo_{\varGamma } \lesssim 50$ , and parabolically dependent on circulation Reynolds $Re_{\varGamma }$ for $Re_{\varGamma }\lesssim 2580$ . Outside of these ranges, surface tension and viscosity have little effect on the initial entrainment volume. For the canonical rectilinear vortex problem, the simple model predicts $\overline {\forall }_o$ extremely well for individual coherent structures over broad ranges of $Fr^2_\Xi$ , $We_{\varGamma }$ , $Bo_{\varGamma }$ and $Re_{\varGamma }$ . We evaluate the performance of this parameterisation and phenomenological entrainment model for air entrainment due to the complex periodic vortex shedding and quasi-steady wave breaking behind a fully submerged horizontal circular cylinder. For the range of parameters we consider, the phenomenological model predicts the event-by-event dimensionless entrainment volume measured in the DNS satisfactorily for this complex application.

High-Fidelity Simulation and Novel Data Analysis of the Bubble Creation and Sound Generation Processes in Breaking Waves

“…Our bubble acoustic model and the new insights on bubble creation mechanisms can also be applied to other bubble-associated physical phenomena, for example, breaking wave and bubble generation induced by vortex/interface interaction or wave-body interaction. Hendrickson et al (2022) developed an air entrainment volume for the quasi-two-dimensional interactions of rising surface-parallel vorticity with an air-water interface and applied their model to the quasi-steady wave breaking behind a fully submerged horizontal circular cylinder. Some air cylinders (air filament or cavity) similar to what we observed in the breaking waves, can also be found in their simulations (See figures 3 and 8 in Hendrickson et al, 2022).…”

“…Hendrickson et al (2022) developed an air entrainment volume for the quasi-two-dimensional interactions of rising surface-parallel vorticity with an air-water interface and applied their model to the quasi-steady wave breaking behind a fully submerged horizontal circular cylinder. Some air cylinders (air filament or cavity) similar to what we observed in the breaking waves, can also be found in their simulations (See figures 3 and 8 in Hendrickson et al, 2022). This implies that our theory for predicting the bubble creation induced by air cylinder breakup is applicable to their simulations.…”