Abstract
We perform direct numerical simulations (DNS) of emulsions in homogeneous isotropic turbulence using a pseudopotential lattice-Boltzmann (PP-LB) method. Improving on previous literature by minimizing droplet dissolution and spurious currents, we show that the PP-LB technique is capable of long stable simulations in certain parameter regions. Varying the dispersed-phase volume fraction$\unicode[STIX]{x1D719}$, we demonstrate that droplet breakup extracts kinetic energy from the larger scales while injecting energy into the smaller scales, increasingly with higher$\unicode[STIX]{x1D719}$, with approximately the Hinze scale (Hinze,AIChE J., vol. 1 (3), 1955, pp. 289–295) separating the two effects. A generalization of the Hinze scale is proposed, which applies both to dense and dilute suspensions, including cases where there is a deviation from the$k^{-5/3}$inertial range scaling and where coalescence becomes dominant. This is done using the Weber number spectrum$We(k)$, constructed from the multiphase kinetic energy spectrum$E(k)$, which indicates the critical droplet scale at which$We\approx 1$. This scale roughly separates coalescence and breakup dynamics as it closely corresponds to the transition of the droplet size ($d$) distribution into a$d^{-10/3}$scaling (Garrettet al.,J. Phys. Oceanogr., vol. 30 (9), 2000, pp. 2163–2171; Deane & Stokes,Nature, vol. 418 (6900), 2002, p. 839). We show the need to maintain a separation of the turbulence forcing scale and domain size to prevent the formation of large connected regions of the dispersed phase. For the first time, we show that turbulent emulsions evolve into a quasi-equilibrium cycle of alternating coalescence and breakup dominated processes. Studying the system in its state-space comprising kinetic energy$E_{k}$, enstrophy$\unicode[STIX]{x1D714}^{2}$and the droplet number density$N_{d}$, we find that their dynamics resemble limit cycles with a time delay. Extreme values in the evolution of$E_{k}$are manifested in the evolution of$\unicode[STIX]{x1D714}^{2}$and$N_{d}$with a delay of${\sim}0.3{\mathcal{T}}$and${\sim}0.9{\mathcal{T}}$respectively (with${\mathcal{T}}$the large eddy timescale). Lastly, we also show that flow topology of turbulence in an emulsion is significantly more different from single-phase turbulence than previously thought. In particular, vortex compression and axial straining mechanisms increase in the droplet phase.
Highlights
An emulsion consists of a dense suspension of droplets of one fluid suspended in another fluid, and is often formed due to turbulent mixing of these two immiscible fluids
Weber number (We) perform direct numerical simulations of emulsions under homogeneous isotropic turbulence conditions performed by using the pseudopotential lattice-Boltzmann method
The process of dispersion formation is investigated for varying volume fractions of the dispersed phase and varying turbulence intensities for an emulsion with a density and viscosity ratio of 1
Summary
An emulsion consists of a dense suspension of droplets of one fluid (the dispersed phase) suspended in another fluid (the continuous phase), and is often formed due to turbulent mixing of these two immiscible fluids. Turbulent emulsions can be said to form a particular class of droplet-laden turbulent flows where there is close interplay between turbulence and the dynamics of the dispersed phase Describing these systems involves an account of the dynamics of deforming interfaces, while allowing for coalescence and breakup of droplets, resolution of a range of length and time scales of turbulent flow and the possible presence of surface active agents (surfactants) that can alter the interfacial dynamics. The dispersed phase influences turbulence by drawing turbulent kinetic energy (TKE) from the flow, which partially goes into the difference between the surface energy of parent and daughter droplets, while the rest is stored in the deformation of interfaces This reduces the effective TKE, which has consequences for the turbulence cascade and spectrum, noticeably at scales comparable to droplet sizes. Coalescing droplets in turn set finer flow structures into motion, where interfacial tension releases the energy stored in droplet deformations back as TKE into the flow at scales smaller than the droplet sizes (Dodd & Ferrante 2016)
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