Energy spectra of buoyancy-driven bubbly flow in a vertical Hele-Shaw cell

TL;DR

This study uses direct numerical simulations to analyze the energy spectra of buoyancy-driven bubbly flows in a Hele-Shaw cell, revealing distinct scaling behaviors and the applicability of a simplified Navier-Stokes model.

Contribution

It provides detailed spectral analysis of bubbly Hele-Shaw flows and demonstrates that a linear drag Navier-Stokes model can approximate large-scale flow features.

Findings

Energy spectrum scales as k for low k and k^{-5} for high k.

Intermediate scaling range with k^{-3} observed around a specific wavenumber.

Linear drag Navier-Stokes model effectively captures large-scale flow behavior.

Abstract

We present direct numerical simulations (DNS) study of confined buoyancy-driven bubbly flows in a Hele-Shaw setup. We investigate the spectral properties of the flow and make comparisons with experiments. The energy spectrum obtained from the gap-averaged velocity field shows $E(k) \sim k$ for $k < k_d$, $E(k) \sim k^{-5}$ for $k > k_d$, and an intermediate scaling range with $E(k) \sim k^{-3}$ around $k \sim k_d$. We perform an energy budget analysis to understand the dominant balances and explain the observed scaling behavior. We also show that the Navier-Stokes equation with a linear drag can be used to approximate large scale flow properties of bubbly Hele-Shaw flow.