On the validity of Kolmogorov laws in bubbly flows: From bubble swarm to single bubble turbulence

Kolmogorov's 4/5 and 2/3 laws are among the few results obtained directly from the Navier–Stokes equations through the Kármán–Howarth–Monin–Hill (KHMH) framework. They describe a constant downscale energy flux and the scaling of longitudinal velocity increments in homogeneous, isotropic, statistically stationary turbulence with vanishing mean velocity gradients. Their applicability to bubble-induced turbulence (BIT) remains largely unexplored. Here we present a comparative KHMH-based assessment of both laws in three representative direct numerical simulation configurations: one homogeneous isotropic turbulence (HIT) reference case, one BIT case, and one single-bubble turbulence (SBT) case. Using one-point statistics and two-point correlations, we assess, scale by scale, how well the assumptions underlying the KHMH equation are satisfied. HIT is homogeneous, nearly isotropic, statistically stationary, and free of mean velocity gradients. BIT, generated by a swarm of 2 mm air bubbles in water, is weakly inhomogeneous but anisotropic, weakly non-stationary, and characterized by finite mean gradients. SBT, driven by a single 4 mm air bubble in water, is strongly inhomogeneous, anisotropic, non-stationary, and shear dominated. Despite these departures, HIT and BIT exhibit limited, finite-Reynolds-number-compatible 4/5-law behavior and recover the corresponding 2/3-law scaling over restricted intermediate ranges. In SBT, however, the third-order structure function departs strongly from the 4/5 law, shows a directional upscale-transfer signature over part of the scale range, and the 2/3 law is absent. Yet the coarse-grained interscale energy spectrum of Sadek and Aluie [“Extracting the spectrum of a flow by spatial filtering,” Phys. Rev. Fluids 3, 124610 (2018)] shows an approximate k−5/3 slope, showing SBT can violate these laws despite a −5/3 spectrum and that they should be tested rather than inferred from spectral slopes alone.