Fig. 4: Revisiting the KH scale, through scale-local energy fluxes.

a Probability-density-function for the normalized energy dissipation rate ε. Black dashed line is the reference single-phase case and colored dashed lines show data from multiphase simulations at different volume fractions α, i.e., 0.03 (green dashed line) 0.06 (red dashed line) 0.1 (blue dashed line) 0.5 (ocher dashed line) at constant large-scale Weber number (see Methods) \(W{e}_{{{{{{{{\mathcal{L}}}}}}}}}=42.6\) and μ_d/μ_c = 1. The stars indicate the values of epsilon computed from the DSD where ε ∝ d^−5/2 and P(ε) ∝ d^13/2N(d) and their color indicates the case (same coloring as dashed lines). The inset shows the details for the PDF at low ε. b Comparison among the different methods used to compute the Hinze scale, i.e., the original formulation d_H, and the proposed approach \({d}_{{H}_{\sigma }}\), obtained as the scale at which \({{{{{{{{\mathcal{S}}}}}}}}}_{\sigma }=0\). For d_H, we show two formulations, namely the original formulation proposed in¹⁹ (circles), and the novel interpretation \({d}_{H}^{r}\) (stars). The computation of \({d}_{H}^{r}\) uses the wavenumber-local non-linear fluxes at \({{\Pi }}({\kappa }_{{{{{{{{{\mathcal{S}}}}}}}}}_{\sigma }})\), with \({\kappa }_{{{{{{{{{\mathcal{S}}}}}}}}}_{\sigma }}=2\pi /{d}_{{H}_{\sigma }}\). The pre-factor 0.8 is used in the expression of \({d}_{H}^{r}\), to account for finite Re_λ effects. The black diagonal line shows where \({d}_{H}^{r}={d}_{{H}_{\sigma }}\), highlighting the improvement provided by the novel interpretation of the Hinze scale. Colors indicates different α (red) at \(W{e}_{{{{{{{{\mathcal{L}}}}}}}}}\) = 42.6 and μ_d/μ_c = 1; different \(W{e}_{{{{{{{{\mathcal{L}}}}}}}}}\) (green) at μ_d/μ_c = 1 and α = 0.03; different μ_d/μ_c (purple) at α = 0.1 and \(W{e}_{{{{{{{{\mathcal{L}}}}}}}}}\) = 42.6, and different μ_d/μ_c (yellow) at α = 0.03 and \(W{e}_{{{{{{{{\mathcal{L}}}}}}}}}\) = 42.6.