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 droplet exhibit wide, non-Gaussian tails. Our study reveals that the droplet enhances the energy spectrum $E(k)$ when the wavenumber $k$ is large; this enhancement leads to dissipation reduction.
Multiplicative cascades have been used in turbulence to generate fields with multifractal statistics and long-range correlations. Examples of continuous and causal stochastic processes which generate such a random field have been carefully discussed in the literature. Here a causal lognormal stochastic process is built to represent the dynamics of pseudo-dissipation in a Lagrangian trajectory. It is introduced as the solution of a stochastic differential equation, driven by a source of noise which has sudden jumps at periodic intervals, its period being the dissipative time scale of the flow. …
Sub-Kolmogorov Intermittency and Multifractal Dissipation in Multiphase Turbulence
University of Modena and Reggio Emilia（摩德纳和雷焦艾米利亚大学）；Sorbonne Université（索邦大学）；Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Université Paris-Saclay（巴黎萨克雷大学）；Inria Saclay - Tau team（法国国家信息与自动化研究所萨克莱-Tau团队）
The leaves of many plants are superhydrophobic, a property that may have evolved to clean the leaves by encouraging water droplets to bead up and roll off. Superhydrophobic surfaces can also exhibit reduced friction and liquids flowing over such surfaces have been found to slip in apparent violations of the classical no-slip boundary condition. Here we introduce slip into a model for rolling droplets on superhydrophobic surfaces and investigate under what conditions slip might be important for the steady state motion. In particular, we examine three limiting cases where dissipation in the roll…
A droplet that impacts onto a solid substrate deforms in a complex dynamics. To extract the principal mechanisms that dominate this dynamics we deploy numerical simulations based on the phase field method. Direct comparison with experiments suggests that a dissipation local to the contact line limits the droplet spreading dynamics and its scaled maximum spreading radius $β_\mathrm{max}$. By assuming linear response through a drag force at the contact line, our simulations rationalize experimental observations for droplet impact on both smooth and rough substrates, measured through a single con…
In turbulent premixed flames, the fractal dimension of flame iso-surface is argued to be $\mathbb{D}=7/3$ for Damköhler's large-scale limit $(Da>>1)$ and $\mathbb{D}=8/3$ for Damköhler's small-scale limit $(Da\sim\mathcal{O}(1))$ based on heuristic scaling arguments. However, such scaling arguments do not consider the effect of the multifractal nature of turbulent kinetic energy dissipation on the flame surface. In this paper, we account for the effects of multifractal dissipation on the fractal dimension of low $Da$ turbulent premixed flames. We derive two corrections to the upper-limit of fr…
The off-center collision of binary bouncing droplets of equal size was studied numerically by a volume-of-fluid (VOF) method with two marker functions, which has been validated by comparing with available experimental results. A non-monotonic kinetic energy recovery with varying impact parameters was found based on the energy budget analysis. This can be explained by the prolonged entanglement time and the enhanced internal-flow-induced viscous dissipation for bouncing droplets at intermediate impact parameters, compared with those at smaller or larger impact parameters. The universality of th…
Dynamic behavior of micro water droplet condensed on a lotus leaf with two-tier roughness is studied. Under laboratory environment, the contact angle of the micro droplet on single micro papilla increases smoothly from 80 deg to 160 deg during the growth of condensed water. The best-known "self-clean" phenomenon, will be lost. A striking observation is the out-of-plane jumping relay of condensed droplets triggered by falling droplets, as well as its sustained speed obtained in continuous jumping relays, enhance the automatic removal of dropwise condensation without the help from any external f…
The rich multifractal properties of fluid turbulence illustrated by the work of Parisi and Frisch are related explicitly to Leray's weak solutions of the three-dimensional Navier-Stokes equations. Directly from this correspondence it is found that the set on which energy dissipates, $\mathbb{F}_{m}$, has a range of dimensions $\Dim=3/m$ ($1 \leq m \leq \infty$), and a corresponding range of sub-Kolmogorov dissipation inverse length scales $Lη_{m}^{-1} \leq Re^{3/(1+\Dim)}$ spanning $Re^{3/4}$ to $Re^{3}$. Correspondingly, the multifractal model scaling parameter $h$, must obey $h \geq h_{min}$…
We study a correspondence between the multifractal model of turbulence and the Navier-Stokes equations in $d$ spatial dimensions by comparing their respective dissipation length scales. In Kolmogorov's 1941 theory the key parameter $h$, which is an exponent in the Navier-Stokes invariance scaling, is fixed at $h=1/3$ but is allowed a spectrum of values in multifractal theory. Taking into account all derivatives of the Navier-Stokes equations, it is found that for this correspondence to hold the multifractal spectrum $C(h)$ must be bounded from below such that $C(h) \geq 1-3h$, which is consist…
Droplet wicking and evaporation in porous nanochannels is experimentally studied on a heated surface at temperatures ranging from 35oC to 90oC. The fabricated geometry consists of cross-connected nanochannels of height 728 nm with micropores of diameter 2 um present at every channel intersection; the pores allow water from a droplet placed on the top surface to wick into the channels. Droplet volume is also varied and a total of 16 experimental cases are conducted. Wicking characteristics such as wicked distance, capillary pressure, viscous resistance and propagation coefficient are obtained a…
Existing energy balance models, which estimate maximum droplet spreading, insufficiently capture the droplet spreading from low to high Weber and Reynolds numbers and contact angles. This is mainly due to the simplified definition of the viscous dissipation term and incomplete modeling of the maximum spreading time. In this particular research, droplet impact on a smooth sapphire surface is studied for seven glycerol concentrations between 0% - 100%, and 294 data points are acquired using high-speed photography. Fluid properties such as density, surface tension, and viscosity are also measured…
Recent investigations on the coalescence of polymeric droplets on a solid substrate have reported strong disagreements; the heart of the issue is whether coalescence of polymeric drops is similar to that of Newtonian fluid and is independent of molecular relaxation, or whether the role of entanglement of polymeric chains leads to a transition kinetics different from that of Newtonian fluid. Via this report, we resolve the disagreements through a discussion on the effects of merging method on the dominant forces governing the coalescence process, i.e., inertia, dissipation, and relaxation. Our …
Motivated by interest in the geometry of high intensity events of turbulent flows, we examine spatial correlation functions of sets where turbulent events are particularly intense. These sets are defined using indicator functions on excursion and iso-value sets. Their geometric scaling properties are analyzed by examining possible power-law decay of their radial correlation function. We apply the analysis to enstrophy, dissipation, and velocity gradient invariants $Q$ and $R$ and their joint spatial distibutions, using data from a direct numerical simulation of isotropic turbulence at ${\rm Re…
Droplets bouncing on a vibrated liquid bath open ways to methods of manipulating droplets, creating double emulsion and performing pilot wave model experiments. In this work, we focus on the role of the droplet deformations in the vertical bouncing dynamics by neglecting the deformation of the surface of the bath. To be under this favorable conditions, low viscous oil droplet are dropped over a highly viscous oil bath. These droplets bounce vertically on the surface of the bath and exhibit many periodic trajectories and resonant modes when tuning the forcing parameters, i.e. the oscillation of…
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.
We investigate the connection between the inertial range and the dissipation range statistics of rotating turbulence through detailed simulations of a helical shell model and a multifractal analysis. In particular, by using the latter, we find an explicit relation between the (anomalous) scaling exponents of equal-time structure functions in the inertial range in terms of the generalised dimensions associated with the energy dissipation rate. This theoretical prediction is validated by detailed simulations of a helical shell model for various strengths of rotation from where the statistics of …
The energy budget and dissipation mechanisms during droplet impact on solid surfaces are studied numerically and theoretically. We find that for high impact velocities and negligible surface friction at the solid surface (i.e. free-slip), about one half of the initial kinetic energy is transformed into surface energy, independent of the impact parameters and the detailed energy loss mechanism(s). We argue that this seemingly universal rule is related to the deformation mode of the droplet and is reminiscent of pipe flow undergoing a sudden expansion, for which the head loss can be calculated b…
In many macroscopic dynamic wetting problems, it is assumed that the macroscopic interface is quasistatic, and the dissipation appears only in the region close to the contact line. When approaching the moving contact line, a microscopic mechanism is required to regularize the singularity of viscous dissipation. On the other hand, if the characteristic size of a fluidic system is reduced to a range comparable to the microscopic regularization length scale, the assumption that viscous effects are localized near the contact line is no longer justified. In the present work, such microscopic length…
This paper is devoted to the numerical study of droplet impact on solid substrates in presence of surfactants. We formulate the problem in an energetically variational framework and introduce an incompressible Cahn-Hilliard-Navier-Stokes system for the phase-field modeling of two-phase flows. Flory-Huggins potential and generalized Navier boundary condition are used to account for soluble surfactants and moving contact lines. Based on the convex splitting and pressure stabilization technique, we develop unconditionally energy stable schemes for this model. The discrete energy dissipation law f…
In this article, we study the spreading of droplets of density-matched granular suspensions on the surface of a solid. Bidispersity of the particle size distribution enriches the conclusions drawn from monodisperse experiments by highlighting key elements of the wetting dynamics. In all cases, the relation between the dynamic contact angle and the velocity of the contact line follows a similar relation as that of a simple fluid, despite the complexity introduced by the presence of particles. We extract from this relation an apparent wetting viscosity of the suspensions that differs from that m…
We study spreading of a droplet, with insoluble surfactant covering its capillary surface, on a textured substrate. In this process, the surfactant-dependent surface tension dominates the behaviors of the whole dynamics, particularly the moving contact lines. This allows us to derive the full dynamics of the droplets laid by the insoluble surfactant: (i) the moving contact lines, (ii) the evolution of the capillary surface, (iii) the surfactant dynamics on this moving surface with a boundary condition at the contact lines and (iv) the incompressible viscous fluids inside the droplet. Our deriv…
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 consider the impingement of a droplet onto a wall with high impact speed. To model this process we favour a diffuse-interface concept. Precisely, we suggest a compressible Navier--Stokes--Allen--Cahn model. Basic properties of the model are discussed. To cope with the fluid-wall interaction, we derive thermodynamically consistent boundary conditions that account for dynamic contact angles. We briefly discuss an discontinuous Galerkin scheme which approximates the energy dissipation of the system exactly and illustrate the results with a series of numerical simulations. Currently, these simu…
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…
We study theoretically the self-propulsion dynamics of a small droplet on general curved surfaces by a variational approach. A new reduced model is derived based on careful computations for the capillary energy and the viscous dissipation in the system. The model describes quantitatively the spontaneous motion of a liquid droplet on general surfaces. In particular, it recovers previous models for droplet motion on the outside surface of a cone. In this case, we derive a scaling law of the displacement $s\sim t^{1/3}$ of a droplet with respect to time $t$ by asymptotic analysis. Theoretical res…
The multifractal theory of turbulence is used to investigate the energy cascade in the Northwestern Atlantic ocean. The statistics of singularity exponents of velocity gradients computed from in situ measurements are used to show that the anomalous scaling of the velocity structure functions at depths between 50 ad 500 m has a linear dependence on the exponent characterizing the strongest velocity gradient, with a slope that decreases with depth. Since the distribution of exponents is asymmetric about the mode at all depths, we use an infinitely divisible asymmetric model of the energy cascade…
We compare experiments and direct numerical simulations to evaluate the accuracy of the Stokes-drag model, which is used widely in studies of inertial particles in turbulence. We focus on statistics at the dissipation scale and on extreme values of relative particle velocities for moderately inertial particles (St < 1). The probability distributions of relative velocities in the simulations were qualitatively similar to those in the experiments. The agreement improved with increasing Stokes number and decreasing relative velocity. Simulations underestimated the probability of extreme events, w…
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…
The probability density functions (PDFs) for energy dissipation rates, created from time-series data of grid turbulence in a wind tunnel, are analyzed in a high precision by the theoretical formulae for PDFs within multifractal PDF theory which is constructed under the assumption that there are two main elements constituting fully developed turbulence, i.e., coherent and incoherent elements. The tail part of PDF, representing intermittent coherent motion, is determined by Tsallis-type PDF for singularity exponents essentially with one parameter with the help of new scaling relation whose valid…
In this paper, we report experimental results on the shape and motion of a mercury droplet, placed in a horizontally rotating cylinder in the rpm range 8-93, so that the Reynolds number of the drop 2500<Re<26000 and its capillary number 0.0002<Ca<0.0023. When contact angle variations can be neglected at low speeds (Re<8150), the velocity of the drop is much lower than that predicted by the Ho. Young Kim's [6] relation. This observed discrepancy is overcome by modifying Kim's relation by substituting the dissipation estimated from a boundary layer near the solid surface instead of bulk dissipat…
We develop a stochastic model for the velocity gradients dynamics along a Lagrangian trajectory. Comparing with different attempts proposed in the literature, the present model, at the cost of introducing a free parameter known in turbulence phenomenology as the intermittency coefficient, gives a realistic picture of velocity gradient statistics at any Reynolds number. To achieve this level of accuracy, we use as a first modelling step a regularized self-stretching term in the framework of the Recent Fluid Deformation (RFD) approximation that was shown to give a realistic picture of small scal…
This paper investigates a variational approach to viscous flows with contact line dynamics based on energy-dissipation modeling. The corresponding model is reduced to a thin-film equation and its variational structure is also constructed and discussed. Feasibility of this modeling approach is shown by constructing a numerical scheme in 1D and by computing numerical solutions for the problem of gravity driven droplets. Some implications of the contact line model are highlighted in this setting.
Multiphase shear flows often show banded structures that affect the global behavior of complex fluids e.g. in microdevices. Here we investigate numerically the banding of emulsions, i.e. the formation of regions of high and low volume fraction, alternated in the vorticity direction and aligned with the flow (shear bands). These bands are associated with a decrease of the effective viscosity of the system. To understand the mechanism of banding experimentally observed we have performed interface resolved simulations of the two-fluid system. The experiments were perfomed starting with a random d…
Reduction of Triadic Interactions Suppresses Intermittency and Anomalous Dissipation in Turbulence
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（国际理论科学中心，塔塔基础研究所）；Department of Mechanical Engineering, Johns Hopkins University（约翰斯·霍普金斯大学机械工程系）；Complex Fluids and Flows Unit, Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学复杂流体与流动单元）
We study both experimentally and theoretically the descending motion due to gravity of a fluid drop surrounded by another immiscible fluid in a confined space between two parallel plates, i.e., in the Hele-Shaw cell. As a result, we show a new scaling regime of a nonlinear drag friction in viscous liquid that replaces the well-known Stokes' drag friction through a clear collapse of experimental data thanks to the scaling law. In the novel regime, the dissipation in the liquid thin film formed between the drop and cell walls governs the dynamics. The crossover of this scaling regime to another …
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…
