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.