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

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.