We experimentally investigate the Lagrangian dynamics of finite-sized, neutrally buoyant droplets in homogeneous isotropic turbulence. The droplet size follows a log-normal distribution whose average value decreases with increasing Reynolds number, reflecting enhanced turbulent breakup. While size-conditioned velocity and acceleration statistics show only weak finite-size dependence, temporal measures reveal clear size-dependent dynamics: larger droplets exhibit longer Lagrangian velocity integral times and an extended ballistic regime in their mean squared displacement. These findings indicate that though droplets exhibit mild deformation and internal circulation, they behave similarly to finite-size rigid particles in terms of Lagrangian dynamics. Our study opens the way to study droplet-laden turbulence and droplet-flow interactions.
This article accompanies the submission of a fluid dynamics video (entry 102248) of inertial cloud droplet clustering in isotropic turbulence for the Gallery of Fluid Motion of the 66th Annual Meeting of the American Physical Society, Division of Fluid Dynamics.
This fluid dynamics video shows the breakup of a droplet in a stationary homogeneous and isotropic turbulent flow. We consider droplets with the same density of the transporting fluid. The droplets and the fluid are numerically modelled by means of a multicompo- nent Lattice-Boltzmann method. The turbulent fluid is maintained through a large scale stirring force and the radius of stable droplets, for the parameters in our simulation, is larger than the Kolmogorov scale. Events of droplet deformation, break-up and aggregation are clearly visible from the movie. With the present database droplet…
The dynamics of droplet fragmentation in turbulence is described in the Kolmogorov-Hinze framework. Yet, a quantitative theory is lacking at higher concentrations when strong interactions between the phases and coalescence become relevant, which is common in most flows. Here, we address this issue through a fully-coupled numerical study of the droplet dynamics in a turbulent flow at high Reynolds number. By means of time-space spectral statistics, not currently accessible to experiments, we demonstrate that the characteristic scale of the process, the Hinze scale, can be precisely identified a…
The breakup of a spherical droplet in a decaying homogeneous isotropic turbulence is studied by solving the Cahn-Hilliard-Navier-Stokes equations, using the discrete unified gas kinetic scheme combined with the free-energy-based phase-field model. We focus on the combined effects of turbulence and surface tension on the breakup process by assuming that the two fluid phases have the same density and same viscosity. The key physical parameters of the system include the volume fraction (6.54%), the initial Weber number (21.7), and the initial Taylor microscale Reynolds number (58). Three distinct…
We perform direct numerical simulations (DNSs) 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 $ϕ$, we demonstrate that droplet breakup extracts kinetic energy from the larger scales while injecting energy into the smaller scales, increasingly with higher $ϕ$, with the Hinze scale dividing the two effec…
The dispersed phase in turbulence can vary from almost inviscid fluid to highly viscous fluid. By changing the viscosity of the dispersed droplet phase, we experimentally investigate how the deformability of dispersed droplets affects the global transport quantity of the turbulent emulsion. Different kinds of silicone oil are employed to result in the viscosity ratio, $ζ$, ranging from $0.53$ to $8.02$. The droplet volume fraction, $ϕ$, is varied from 0\% to 10\% with a spacing of 2\%. The global transport quantity, quantified by the normalized friction coefficient $c_{f,ϕ}/c_{f,ϕ=0}$, shows a…
We study theoretically and numerically spatial distribution and collision rate of droplets that sediment in homogeneous isotropic Navier-Stokes turbulence. It is assumed that typical turbulent accelerations of fluid particles are much smaller than gravity. This was shown to imply that the particles interact weakly with individual vortices and, as a result, form a smooth flow in most of the space. In weakly intermittent turbulence with moderate Reynolds number, rare regions where the flow breaks down can be neglected in the calculation of space averaged rate of droplet collisions. However, incr…
We study the physics of droplet breakup in a statistically stationary homogeneous and isotropic turbulent flow by means of high resolution numerical investigations based on the multicomponent lattice Boltzmann method. We verified the validity of the criterion proposed by Hinze (1955) for droplet breakup and we measured the full probability distribution function (pdf) of droplets radii at different Reynolds numbers and for different volume fraction. By means of a Lagrangian tracking we could follow individual droplets along their trajectories, define a local Weber number based on the velocity g…
Violent expiratory events, such as coughing and sneezing, are highly nontrivial examples of a two-phase mixture of liquid droplets dispersed into an unsteady turbulent airflow. Understanding the physical mechanisms determining the dispersion and evaporation process of respiratory droplets has recently become a priority given the global emergency caused by the SARS-CoV-2 infection. By means of high-resolution direct numerical simulations (DNS) of the expiratory airflow and a comprehensive Lagrangian model for the droplet dynamics, we identify the key role of turbulence on the fate of exhaled dr…
We use a stochastic model and direct numerical simulation to study the impact of turbulence on cloud droplet growth by condensation. We show that the variance of the droplet size distribution increases in time as t^{1/2}, with growth rate proportional to the large-to-small turbulent scale separation and to the turbulence integral scales but independent of the mean turbulent dissipation. Direct numerical simulations confirm this result and produce realistically broad droplet size spectra over time intervals of 20 minutes, comparable with the time of rain formation.
The deformation and dynamics of a single droplet in isotropic turbulence is studied using a Lattice Boltzmann diffuse interface model involving exact boundary flow conditions to allow for the creation of an external turbulent flow. We focus on a small, sub-Kolmogorov droplet, whose scale is much smaller than the Kolmogorov length scale of the turbulent flow. The external flow field is obtained via pseudo-spectral simulation data describing the trajectory of a passive tracer in isotropic turbulence. In this way we combine the microscopic scale of the droplet and the macroscopic scale of the tur…
We present an extensive direct numerical simulation of statistically steady, homogeneous, isotropic turbulence in two-dimensional, binary-fluid mixtures with air-drag-induced friction by using the Cahn-Hilliard-Navier-Stokes equations. We choose parameters, e.g., the surface tension, such that we have a droplet of the minority phase moving inside a turbulent background of the majority phase. We characterize the deformation of the droplet and show that it displays multifractal dynamics. The probability distribution functions of the components of the acceleration of the center of mass of the dro…
Based on the Karman-Howarth equation in 3D incompressible fluid, a new isotropic turbulence scale evolution equation and its related theory progress. The present results indicate that the energy cascading process has remarkable similarities with the determinisitic construction rules of the isotonic oscillators in quantum mechanics.
We simulate the flow of two immiscible and incompressible fluids separated by an interface in a homogeneous turbulent shear flow at a shear Reynolds number equal to 15200. The viscosity and density of the two fluids are equal, and various surface tensions and initial droplet diameters are considered in the present study. We show that the two-phase flow reaches a statistically stationary turbulent state sustained by a non-zero mean turbulent production rate due to the presence of the mean shear. Compared to single-phase flow, we find that the resulting steady state conditions exhibit reduced Ta…
The statistics of the deformation and breakup of neutrally buoyant sub-Kolmogorov ellipsoidal drops is investigated via Lagrangian simulations of homogeneous isotropic turbulence. The mean lifetime of a drop is also studied as a function of the initial drop size and the capillary number. A vector model of drop previously introduced by Olbricht, Rallison and Leal [J. Non-Newtonian Fluid Mech. $\mathbf{10}$, 291 (1982)] is used to predict the behaviour of the above quantities analytically.
A new form of the well known Lin equation for the energy balance in isotropic turbulence is given which takes account of the filtered-partitioned transfer spectra which were used to resolve the scale paradox. It is argued that this new formulation provides a new starting point for both theoretical and numerical methods which may lead to a better understanding of the underlying energy processes in turbulence.
We conduct a careful analysis of the data provided by Krogstad & Davidson (2011) and show that their data do not support their conclusions. According to their published data, their decaying approximately homogeneous isotropic turbulent flows are, invariably, clearly different from Saffman turbulence; and very clearly marked differences exist between the far downstream turbulence behaviors generated by their conventional grid and by their multiscale cross grids.
It is shown that the correlation integrals (invariants) play an important, and at certain conditions a dominant, role in helical isotropic homogeneous turbulence both in the inertial and near-dissipation ranges. Results of direct numerical simulations and recent laboratory experiments with multiscale grids have been used for this purpose. A possibility of spontaneous breaking of reflection symmetry has been also briefly discussed in this context.
The effects of Reynolds number and Stokes number on particle-pair relative velocity (RV) were investigated systematically using a recently developed planar four-frame particle tracking technique in a novel homogeneous and isotropic turbulence chamber.
Using fluctuating hydrodynamics we investigate the effect of thermal fluctuations in the dissipation range of homogeneous, isotropic turbulence. Simulations confirm theoretical predictions that the energy spectrum is dominated by these fluctuations at length scales comparable to the Kolmogorov length. We also find that the extreme intermittency in the far-dissipation range predicted by Kraichnan is replaced by Gaussian thermal equipartition.
[Abridged]We present high-resolution three-dimensional direct numerical simulations of breaking waves solving the two-phase Navier-Stokes equations. We investigate the role of the Reynolds and Bond numbers on the energy, bubble and droplet statistics of strong plunging breakers, and explore the asymptotic regimes at high Reynolds and Bond numbers to be compared with laboratory breaking waves. Energetically, the breaking wave transitions from laminar to three-dimensional turbulent flow on a timescale that depends on the turbulent Reynolds number up to a limiting value of $Re_λ\sim 100$, consist…
It is shown that the distributed chaos in the simple Hamiltonian (conservative) dynamical systems, such as the Nose-Hoover oscillator and double oscillator, can mimic the distributed chaos in the isotropic homogeneous turbulence. Direct numerical simulations with the classic Toda lattice and with the nonlinear Schrödinger equation (soliton turbulence) under random initial conditions have been also discussed in this context. These properties of the distributed chaos have been related to analytical properties of the Hamiltonian systems. Decrease of the smoothness results in the power-law spectra…
Temporal decorrelations in compressible isotropic turbulence are studied using the space-time correlation theory and direct numerical simulation. A swept-wave model is developed for dilatational components while the classic random sweeping model is proposed for solenoidal components. The swept-wave model shows that the temporal decorrelations in dilatational fluctuations are dominated by two physical processes: random sweeping and wave propagation. These models are supported by the direct numerical simulation of compressible isotropic turbulence, in the sense of that all curves of normalized t…
We study the statistical properties of helicity in direct numerical simulations of fully developed homogeneous and isotropic turbulence and in a class of turbulence shell models. We consider correlation functions based on combinations of vorticity and velocity increments that are not invariant under mirror symmetry. We also study the scaling properties of high-order structure functions based on the moments of the velocity increments projected on a subset of modes with either positive or negative helicity (chirality). We show that mirror symmetry is recovered at small-scales, i.e., chiral terms…
Assuming perfect collision efficiency, we demonstrate that turbulence can initiate and sustain the rapid growth of very small water droplets in air even when these droplets are too small to cluster, and even without having to take gravity and small-scale intermittency into account. This is because the range of local Stokes numbers of identical droplets in the turbulent flow field is broad enough even when small-scale intermittency is neglected. This demonstration is given for turbulence which is one order of magnitude less intense than is typical in warm clouds but with a volume fraction which…
This article describes the dynamics of small inertial particles centrifuging out of a single vortex. It shows the importance of caustics formation in the vicinity of a single vortex: both for particle collisions and void formation. From these single-vortex studies we provide estimates of the role of caustics in high Reynolds number turbulence, and in the case of clouds, estimate how they may help in rain initiation by bridging the droplet-growth bottleneck. We briefly describe how the Basset- Boussinesq history force may be calculated by a method which does not involve huge memory costs, and p…
The streamwise and transverse velocities are measured simultaneously in grid turbulence at Reynolds numbers 100-300. We extract typical intermittency patterns, which are consistent with velocity profiles of Burgers and Lamb-Oseen vortices. The radii of these vortex tubes are estimated to be several of the Kolmogorov length.
We consider equilibrium statistics for high Reynolds number isotropic turbulence in an incompressible flow driven by steady forcing at the largest scale. Motivated by shell model observations, we develop a similarity theory for the inertial range from clearly stated assumptions. In the right variables, the theory is scaling invariant, but in traditional variables it shows anomalous scaling. We obtain the underlying probability density function, the scaling exponents, and the coefficients for the structure functions. An inertial range length scale also emerges.
Small droplets in turbulent flows can undergo highly variable deformations and orientational dynamics. For neutrally buoyant droplets smaller than the Kolmogorov scale, the dominant effects from the surrounding turbulent flow arise through Lagrangian time histories of the velocity gradient tensor. Here we study the evolution of representative droplets using a model that includes rotation and stretching effects from the surrounding fluid, and restoration effects from surface tension including a constant droplet volume constraint, while assuming that the droplets maintain an ellipsoidal shape. T…
The assumption of similarity and self-preservation, which permits an analytical determination of the energy decay in isotropic turbulence, has played an important role in the development of turbulence theory for more than half a century. Sedov (1944), who first found an ingenious way to obtain two equations from one. Nonethless, it appears that this problem has never been reinvestigated in depth subsequent to this earlier work. In the present paper, such an analysis is carried out, yielding a much more complete picture of self-preservation isotropic turbulence. Based on these exact solutions, …
This paper is the forth part of our series of work, is devoted to the analysis on the multiscales and cascade aspects of the statistical theory of isotropic turbulence based on the new Sedov-type solution. In this paper, we use the explicit map method to analyse the nonlinear dynamical behaviour for cascade in isotorpic turbulence. This deductive scale analysis is shown to provide the first visual evidence of the celebrated Richardson cascade, and reveals in partcular its multiscale character. The results also indicate that the energy cascading process has remarkable similarities with the dete…
The radial relative velocity between particles suspended in turbulent flow plays a critical role in droplet collision and growth. We present a simple and accurate approach to RV measurement in isotropic turbulence - planar 4-frame particle tracking velocimetry - using routine PIV hardware. This study demonstrates the feasibility of accurately measuring RV using routine hardware and verifies, for the first time, the path-history and inertial filtering effects on particle-pair RV at large particle separations experimentally.
The recent development of a statistical model for incompressible Navier-Stokes (NS) fluids based on inverse kinetic theory (IKT, 2004-2008) poses the problem of searching for particular realizations of the theory which may be relevant for \textit{the statistical description of turbulence} and in particular for the so-called \textit{homogeneous, isotropic}and \textit{stationary} turbulence (HIST). Here the problem is set in terms of the $1-$point velocity probability density function (PDF) which determines a complete IKT-statistical model for NS fluids. This raises the interesting question of i…
Direct numerical simulation is used to investigate the decay exponent of isotropic homogeneous turbulence over a range of Reynolds numbers sufficient to display both high and low Re number decay behavior. The initial turbulence is generated by the stirring action of the flow past many small randomly placed cubes. Stirring occurs at 1/30th of the simulation domain size so that the low-wavenumber and large scale behavior of the turbulent spectrum is generated by the fluid and is not imposed. It is shown that the decay exponent in the resulting turbulence closely matches the theoretical predictio…
We use the recently developed Macroscopic Forcing Method [Mani and Park, Physical Review Fluids, 6:054607, 2021] to compute the scale-dependent eddy diffusivity characterizing ensemble-averaged scalar and momentum transport in incompressible homogeneous isotropic turbulence. For scales larger than the energy containing eddies, eddy diffusivity is found to be constant and consistent with the Boussinesq approximation. However for small scales eddy diffusivity is found to vanish inversely proportional to the wavenumber. Behavior at all scales is reasonably captured by a non-local eddy diffusivity…
An improved understanding of turbulence is essential for the effective modelling and control of industrial and geophysical processes. Homogeneous, isotropic turbulence (HIT) is the archetypal field for developing turbulence physics theory. Based on the Schur transform, we introduce an additive decomposition of the velocity gradient tensor into a normal part (containing the eigenvalues) and a non-normal or shear-related tensor. We re-interrogate some key properties of HIT and show that the the tendency of the flow to form disc-like structures is not a property of the normal tensor; it emerges f…
Small scale characteristics of turbulence such as velocity gradients and vorticity fluctuate rapidly in magnitude and oscillate in sign. Much work exists on the characterization of magnitude variations, but far less on sign oscillations. While averages performed on large scales tend to zero because of the oscillatory character, those performed on increasingly smaller scales will vary with the averaging scale in some characteristic way. This characteristic variation at high Reynolds numbers is captured by the so-called cancellation exponent, which measures how local averages tend to cancel out …
We present the most extensive direct numerical simulations, attempted so far, of statistically steady, homogeneous, isotropic turbulence in two-dimensional fluid films with air-drag-induced friction and with polymer additives. Our study reveals that the polymers (a) reduce the total fluid energy, enstrophy, and palinstrophy, (b) modify the fluid energy spectrum both in inverse- and forward-cascade regimes, (c) reduce small-scale intermittency, (d) suppress regions of large vorticity and strain rate, and (e) stretch in strain-dominated regions. We compare our results with earlier experimental s…
We show by direct numerical simulation (DNS) that the Lagrangian cross correlation of velocity gradients in homogeneous isotropic turbulence increases at short times, whereas its auto-correlation decreases. Kinematic considerations allow to show that two invariants of the turbulent velocity field determine the short-time velocity gradient correlations. In order to get a more intuitive understanding of the dynamics for longer times, heuristic models are proposed involving the combined action of local shear and rotation. These models quantitatively reproduce the effects and disentangle the diffe…
We investigate the conditional vorticity budget of fully developed three-dimensional homogeneous isotropic turbulence with respect to coherent and incoherent flow contributions. The Coherent Vorticity Extraction based on orthogonal wavelets allows to decompose the vorticity field into coherent and incoherent contributions, of which the latter are noise-like. The impact of the vortex structures observed in fully developed turbulence on statistical balance equations is quantified considering the conditional vorticity budget. The connection between the basic structures present in the flow and the…
Using time-resolved measurements of the orientation of rigid inertial fibers in a turbulence-tank, we investigate the autocorrelation of their tumbling rate. The correlation time ($τ_d$) is well predicted by Kolmogorov inertial-range scaling based on the fiber length (L) when the fiber inertia can be neglected. For inertial fibers, we propose a simple model considering fiber inertia (measured by a tumbling Stokes number) and a viscous torque which accurately predicts both the correlation time and the variance of the tumbling rate.
We investigate the single-point probability density function of the velocity in three-dimensional stationary and decaying homogeneous isotropic turbulence. To this end we apply the statistical framework of the Lundgren-Monin-Novikov hierarchy combined with conditional averaging, identifying the quantities that determine the shape of the probability density function. In this framework the conditional averages of the rate of energy dissipation, the velocity diffusion and the pressure gradient with respect to velocity play a key role. Direct numerical simulations of the Navier-Stokes equation are…
