Pattern formation by turbulent cascades.

Fully developed turbulence is a universal and scale-invariant chaotic state characterized by an energy cascade from large to small scales at which the cascade is eventually arrested by dissipation1-6. Here we show how to harness these seemingly structureless turbulent cascades to generate patterns. Pattern formation entails a process of wavelength selection, which can usually be traced to the linear instability of a homogeneous state7. By contrast, the mechanism we propose here is fully nonlinear. It is triggered by the non-dissipative arrest of turbulent cascades: energy piles up at an intermediate scale, which is neither the system size nor the smallest scales at which energy is usually dissipated. Using a combination of theory and large-scale simulations, we show that the tunable wavelength of these cascade-induced patterns can be set by a non-dissipative transport coefficient called odd viscosity, ubiquitous in chiral fluids ranging from bioactive to quantum systems8-12. Odd viscosity, which acts as a scale-dependent Coriolis-like force, leads to a two-dimensionalization of the flow at small scales, in contrast with rotating fluids in which a two-dimensionalization occurs at large scales4. Apart from odd viscosity fluids, we discuss how cascade-induced patterns can arise in natural systems, including atmospheric flows13-19, stellar plasma such as the solar wind20-22, or the pulverization and coagulation of objects or droplets in which mass rather than energy cascades23-25.

Localization-delocalization transition for light particles in turbulence.

Small bubbles in fluids rise to the surface due to Archimede's force. Remarkably, in turbulent flows this process is severely hindered by the presence of vortex filaments, which act as moving potential wells, dynamically trapping light particles and bubbles. Quantifying the statistical weights and roles of vortex filaments in turbulence is, however, still an outstanding experimental and computational challenge due to their small scale, fast chaotic motion, and transient nature. Here we show that, under the influence of a modulated oscillatory forcing, the collective bubble behavior switches from a dynamically localized to a delocalized state. Additionally, we find that by varying the forcing frequency and amplitude, a remarkable resonant phenomenon between light particles and small-scale vortex filaments emerges, likening particle behavior to a forced damped oscillator. We discuss how these externally actuated bubbles can be used as a type of microscopic probe to investigate the space-time statistical properties of the smallest turbulence scales, allowing to quantitatively measure physical characteristics of vortex filaments. We develop a superposition model that is in excellent agreement with the simulation data of the particle dynamics which reveals the fraction of localized/delocalized particles as well as characteristics of the potential landscape induced by vortices in turbulence. Our approach paves the way for innovative ways to accurately measure turbulent properties and to the possibility to control light particles and bubble motions in turbulence with potential applications to oceanography, medical imaging, drug/gene delivery, chemical reactions, wastewater treatment, and industrial mixing.

Coexistence of Defect Morphologies in Three-Dimensional Active Nematics.

We establish how active stress globally affects the morphology of disclination lines of a three-dimensional active nematic liquid crystal under chaotic flow. Thanks to a defect detection algorithm based on the local nematic orientation, we show that activity selects a crossover length scale in between the size of small defect loops and that of long and tangled defect lines of fractal dimension 2. This length scale crossover is consistent with the scaling of the average separation between defects as a function of activity. Moreover, on the basis of numerical simulation in a 3D periodic geometry, we show the presence of a network of regular defect loops, contractible onto the 3-torus, always coexisting with wrapping defect lines. While the length of regular defects scales linearly with the emerging active length scale, it verifies an inverse quadratic dependence for wrapping defects. The shorter the active length scale, the more the defect lines wrap around the periodic boundaries, resulting in extremely long and buckled structures.

Ductile-to-brittle transition and yielding in soft amorphous materials: perspectives and open questions.

Soft amorphous materials are viscoelastic solids ubiquitously found around us, from clays and cementitious pastes to emulsions and physical gels encountered in food or biomedical engineering. Under an external deformation, these materials undergo a noteworthy transition from a solid to a liquid state that reshapes the material microstructure. This yielding transition was the main theme of a workshop held from January 9 to 13, 2023 at the Lorentz Center in Leiden. The manuscript presented here offers a critical perspective on the subject, synthesizing insights from the various brainstorming sessions and informal discussions that unfolded during this week of vibrant exchange of ideas. The result of these exchanges takes the form of a series of open questions that represent outstanding experimental, numerical, and theoretical challenges to be tackled in the near future.

How the growth of ice depends on the fluid dynamics underneath.

Convective flows coupled with solidification or melting in water bodies play a major role in shaping geophysical landscapes. Particularly in relation to the global climate warming scenario, it is essential to be able to accurately quantify how water-body environments dynamically interplay with ice formation or melting process. Previous studies have revealed the complex nature of the icing process, but have often ignored one of the most remarkable particularities of water, its density anomaly, and the induced stratification layers interacting and coupling in a complex way in the presence of turbulence. By combining experiments, numerical simulations, and theoretical modeling, we investigate solidification of freshwater, properly considering phase transition, water density anomaly, and real physical properties of ice and water phases, which we show to be essential for correctly predicting the different qualitative and quantitative behaviors. We identify, with increasing thermal driving, four distinct flow-dynamics regimes, where different levels of coupling among ice front and stably and unstably stratified water layers occur. Despite the complex interaction between the ice front and fluid motions, remarkably, the average ice thickness and growth rate can be well captured with the theoretical model. It is revealed that the thermal driving has major effects on the temporal evolution of the global icing process, which can vary from a few days to a few hours in the current parameter regime. Our model can be applied to general situations where the icing dynamics occur under different thermal and geometrical conditions.

Toward learning Lattice Boltzmann collision operators.

In this work, we explore the possibility of learning from data collision operators for the Lattice Boltzmann Method using a deep learning approach. We compare a hierarchy of designs of the neural network (NN) collision operator and evaluate the performance of the resulting LBM method in reproducing time dynamics of several canonical flows. In the current study, as a first attempt to address the learning problem, the data were generated by a single relaxation time BGK operator. We demonstrate that vanilla NN architecture has very limited accuracy. On the other hand, by embedding physical properties, such as conservation laws and symmetries, it is possible to dramatically increase the accuracy by several orders of magnitude and correctly reproduce the short and long time dynamics of standard fluid flows.

Spatial population genetics with fluid flow.

The growth and evolution of microbial populations is often subjected to advection by fluid flows in spatially extended environments, with immediate consequences for questions of spatial population genetics in marine ecology, planktonic diversity and origin of life scenarios. Here, we review recent progress made in understanding this rich problem in the simplified setting of two competing genetic microbial strains subjected to fluid flows. As a pedagogical example we focus on antagonsim, i.e., two killer microorganism strains, each secreting toxins that impede the growth of their competitors (competitive exclusion), in the presence of stationary fluid flows. By solving two coupled reaction-diffusion equations that include advection by simple steady cellular flows composed of characteristic flow motifs in two dimensions (2D), we show how local flow shear and compressibility effects can interact with selective advantage to have a dramatic influence on genetic competition and fixation in spatially distributed populations. We analyze several 1D and 2D flow geometries including sources, sinks, vortices and saddles, and show how simple analytical models of the dynamics of the genetic interface can be used to shed light on the nucleation, coexistence and flow-driven instabilities of genetic drops. By exploiting an analogy with phase separation with nonconserved order parameters, we uncover how thesegeneticdrops harness fluid flows for novel evolutionary strategies, even in the presence of number fluctuations, as confirmed by agent-based simulations as well.

Self-similar properties of avalanche statistics in a simple turbulent model.

In this paper, we consider a simplified model of turbulence for large Reynolds numbers driven by a constant power energy input on large scales. In the statistical stationary regime, the behaviour of the kinetic energy is characterized by two well-defined phases: a laminar phase where the kinetic energy grows linearly for a (random) time [Formula: see text] followed by abrupt avalanche-like energy drops of sizes [Formula: see text] due to strong intermittent fluctuations of energy dissipation. We study the probability distribution [Formula: see text] and [Formula: see text] which both exhibit a quite well-defined scaling behaviour. Although [Formula: see text] and [Formula: see text] are not statistically correlated, we suggest and numerically checked that their scaling properties are related based on a simple, but non-trivial, scaling argument. We propose that the same approach can be used for other systems showing avalanche-like behaviour such as amorphous solids and seismic events. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.

Fixation probabilities in weakly compressible fluid flows.

Competition between biological species in marine environments is affected by the motion of the surrounding fluid. An effective 2D compressibility can arise, for example, from the convergence and divergence of water masses at the depth at which passively traveling photosynthetic organisms are restricted to live. In this report, we seek to quantitatively study genetics under flow. To this end, we couple an off-lattice agent-based simulation of two populations in 1D to a weakly compressible velocity field-first a sine wave and then a shell model of turbulence. We find for both cases that even in a regime where the overall population structure is approximately unaltered, the flow can significantly diminish the effect of a selective advantage on fixation probabilities. We understand this effect in terms of the enhanced survival of organisms born at sources in the flow and the influence of Fisher genetic waves.

Stress Overshoots in Simple Yield Stress Fluids.

Soft glassy materials such as mayonnaise, wet clays, or dense microgels display a solid-to-liquid transition under external shear. Such a shear-induced transition is often associated with a nonmonotonic stress response in the form of a stress maximum referred to as "stress overshoot." This ubiquitous phenomenon is characterized by the coordinates of the maximum in terms of stress σ_{M} and strain γ_{M} that both increase as weak power laws of the applied shear rate. Here we rationalize such power-law scalings using a continuum model that predicts two different regimes in the limit of low and high applied shear rates. The corresponding exponents are directly linked to the steady-state rheology and are both associated with the nucleation and growth dynamics of a fluidized region. Our work offers a consistent framework for predicting the transient response of soft glassy materials upon startup of shear from the local flow behavior to the global rheological observables.

A lattice Boltzmann study of particle settling in a fluctuating multicomponent fluid under confinement.

We present mesoscale numerical simulations based on the coupling of the fluctuating lattice Boltzmann method for multicomponent systems with a wetted finite-size particle model. This newly coupled methodologies are used to study the motion of a spherical particle driven by a constant body force in a confined channel with a fixed square cross section. The channel is filled with a mixture of two liquids under the effect of thermal fluctuations. After some validations steps in the absence of fluctuations, we study the fluctuations in the particle's velocity at changing thermal energy, applied force, particle size, and particle wettability. The importance of fluctuations with respect to the mean settling velocity is quantitatively assessed, especially in comparison with unconfined situations. Results show that the expected effects of confinement are very well captured by the numerical simulations, wherein the confinement strongly enhances the importance of velocity fluctuations, which can be one order of magnitude larger than what expected in unconfined domains. The observed findings underscore the versatility of the proposed methodology in highlighting the effects of confinement on the motion of particles in the presence of thermal fluctuations.

Lattice Boltzmann method investigation of a reactive electro-kinetic flow in porous media: towards a phenomenological model.

A model based on the Lattice Boltzmann method is developed to study the flow of reactive electro-kinetic fluids in porous media. The momentum, concentration and electric/potential fields are simulated via the Navier-Stokes, advection-diffusion/Nernst-Planck and Poisson equations, respectively. With this model, the total density and velocity fields, the concentration of reactants and reaction products, including neutral and ionized species, the electric potential and the interaction forces between the fields can be studied, and thus we provide an insight into the interplay between chemistry, flow and the geometry of the porous medium. The results show that the conversion efficiency of the reaction can be strongly influenced by the fluid velocity, reactant concentration and by porosity of the porous medium. The fluid velocity determines how long the reactants stay in the reaction areas, the reactant concentration controls the amount of the reaction material and with different dielectric constant, the porous medium can distort the electric field differently. All these factors make the reaction conversion efficiency display a non-trivial and non-monotonic behaviour as a function of the flow and reaction parameters. To better illustrate the dependence of the reaction conversion efficiency on the control parameters, based on the input from a number of numerical investigations, we developed a phenomenological model of the reactor. This model is capable of capturing the main features of the causal relationship between the performance of the reactor and the main test parameters. Using this model, one could optimize the choice of reaction and flow parameters in order to improve the performance of the reactor and achieve higher production rates. This article is part of the theme issue 'Progress in mesoscale methods for fluid dynamics simulation'.

Statistical Properties of Turbulence in the Presence of a Smart Small-Scale Control.

By means of high-resolution numerical simulations, we compare the statistical properties of homogeneous and isotropic turbulence to those of the Navier-Stokes equation where small-scale vortex filaments are strongly depleted, thanks to a nonlinear extra viscosity acting preferentially on high vorticity regions. We show that the presence of such smart small-scale drag can strongly reduce intermittency and non-Gaussian fluctuations. Our results pave the way towards a deeper understanding on the fundamental role of degrees of freedom in turbulence as well as on the impact of (pseudo)coherent structures on the statistical small-scale properties. Our work can be seen as a first attempt to develop smart-Lagrangian forcing or drag mechanisms to control turbulence.

Plasma-induced catalysis: towards a numerical approach.

A lattice Boltzmann (LB) model is developed, validated and used to study simplified plasma/flow problems in complex geometries. This approach solves a combined set of equations, namely the Navier-Stokes equations for the momentum field, the advection-diffusion and the Nernst-Planck equations for electrokinetic and the Poisson equation for the electric field. This model allows us to study the dynamical interaction of the fluid/plasma density, velocity, concentration and electric field. In this work, we discuss several test cases for our numerical model and use it to study a simplified plasma fluid flowing and reacting inside a packed bed reactor. Inside the packed bed, electric breakdown reactions take place due to the electric field, making neutral species ionize. The presence of the packed beads can help enhance the reaction efficiency by locally increasing the electric field, and the size of packed beads and the pressure drop of the packed bed do influence the outflux. Hence trade-offs exist between reaction efficiency and packing porosity, the size of packing beads and the pressure drop of the packed bed. Our model may be used as a guidance to achieve higher reaction efficiencies by optimizing the relevant parameters. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

A multi-component lattice Boltzmann approach to study the causality of plastic events.

Using a multi-component lattice Boltzmann (LB) model, we perform fluid kinetic simulations of confined and concentrated emulsions. The system presents the phenomenology of soft-glassy materials, including a Herschel-Bulkley rheology, yield stress, ageing and long relaxation time scales. Shearing the emulsion in a Couette cell below the yield stress results in plastic topological re-arrangement events which follow established empirical seismic statistical scaling laws, making this system a good candidate to study the physics of earthquakes. One characteristic of this model is the tendency for events to occur in avalanche clusters, with larger events, triggering subsequent re-arrangements. While seismologists have developed statistical tools to study correlations between events, a process to confirm causality remains elusive. We present here, a modification to our LB model, involving small, fast vibrations applied to individual droplets, effectively a macroscopic forcing, which results in the arrest of the topological plastic re-arrangements. This technique provides an excellent tool for identifying causality in plastic event clusters by examining the evolution of the dynamics after 'stopping' an event, and then checking which subsequent events disappear. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

Unified Theoretical and Experimental View on Transient Shear Banding.

Dense emulsions, colloidal gels, microgels, and foams all display a solidlike behavior at rest characterized by a yield stress, above which the material flows like a liquid. Such a fluidization transition often consists of long-lasting transient flows that involve shear-banded velocity profiles. The characteristic time for full fluidization τ_{f} has been reported to decay as a power law of the shear rate γ[over ˙] and of the shear stress σ with respective exponents α and ß. Strikingly, the ratio of these exponents was empirically observed to coincide with the exponent of the Herschel-Bulkley law that describes the steady-state flow behavior of these complex fluids. Here we introduce a continuum model, based on the minimization of a "free energy," that captures quantitatively all the salient features associated with such transient shear banding. More generally, our results provide a unified theoretical framework for describing the yielding transition and the steady-state flow properties of yield stress fluids.

Statistical properties of thermally expandable particles in soft-turbulence Rayleigh-Bénard convection.

The dynamics of inertial particles in Rayleigh-Bénard convection, where both particles and fluid exhibit thermal expansion, is studied using direct numerical simulations (DNS) in the soft-turbulence regime. We consider the effect of particles with a thermal expansion coefficient larger than that of the fluid, causing particles to become lighter than the fluid near the hot bottom plate and heavier than the fluid near the cold top plate. Because of the opposite directions of the net Archimedes' force on particles and fluid, particles deposited at the plate now experience a relative force towards the bulk. The characteristic time for this motion towards the bulk to happen, quantified as the time particles spend inside the thermal boundary layers (BLs) at the plates, is shown to depend on the thermal response time, [Formula: see text], and the thermal expansion coefficient of particles relative to that of the fluid, [Formula: see text]. In particular, the residence time is constant for small thermal response times, [Formula: see text], and increasing with [Formula: see text] for larger thermal response times, [Formula: see text]. Also, the thermal BL residence time is increasing with decreasing K. A one-dimensional (1D) model is developed, where particles experience thermal inertia and their motion is purely dependent on the buoyancy force. Although the values do not match one-to-one, this highly simplified 1D model does predict a regime of a constant thermal BL residence time for smaller thermal response times and a regime of increasing residence time with [Formula: see text] for larger response times, thus explaining the trends in the DNS data well.

Preferential Rotation of Chiral Dipoles in Isotropic Turbulence.

We introduce a new particle shape which shows preferential rotation in three dimensional homogeneous isotropic turbulence. We call these particles chiral dipoles because they consist of a rod with two helices of opposite handedness, one at each end. 3D printing is used to fabricate these particles with a length in the inertial range and their rotations are tracked in a turbulent flow between oscillating grids. High aspect ratio chiral dipoles preferentially align with their long axis along the extensional eigenvectors of the strain rate tensor, and the helical ends respond to the extensional strain rate with a mean spinning rate that is nonzero. We use Stokesian dynamics simulations of chiral dipoles in pure strain flow to quantify the dependence of spinning on particle shape. Based on the known response to pure strain, we build a model that gives the spinning rate of small chiral dipoles using velocity gradients along Lagrangian trajectories from high resolution direct numerical simulations. The statistics of chiral dipole spinning determined with this model show surprisingly good agreement with the measured spinning of much larger chiral dipoles in the experiments.

Symmetry Breaking Drift of Particles Settling in Homogeneous Shear Turbulence.

We investigate the influence of shear on the gravitational settling of heavy inertial particles in homogeneous shear turbulence (HST). In addition to the well-known enhanced settling velocity, observed for heavy inertial particles in homogeneous isotropic turbulence (HIT), a horizontal drift velocity is also observed in the shearing direction due to the presence of a nonzero mean vorticity (introducing symmetry breaking due to the mean shear). This drift velocity is due to the combination of shear, gravity, and turbulence, and all three of these elements are needed for this effect to occur. We extend the mechanism responsible for the enhanced settling velocity in HIT to the case of HST. Two separate regimes are observed, characterized by positive or negative drift velocity, depending on the particle settling velocity.

Turbulence on a Fractal Fourier Set.

A novel investigation of the nature of intermittency in incompressible, homogeneous, and isotropic turbulence is performed by a numerical study of the Navier-Stokes equations constrained on a fractal Fourier set. The robustness of the energy transfer and of the vortex stretching mechanisms is tested by changing the fractal dimension D from the original three dimensional case to a strongly decimated system with D=2.5, where only about 3% of the Fourier modes interact. This is a unique methodology to probe the statistical properties of the turbulent energy cascade, without breaking any of the original symmetries of the equations. While the direct energy cascade persists, deviations from the Kolmogorov scaling are observed in the kinetic energy spectra. A model in terms of a correction with a linear dependency on the codimension of the fractal set E(k)∼k(-5/3+3-D) explains the results. At small scales, the intermittency of the vorticity field is observed to be quasisingular as a function of the fractal mode reduction, leading to an almost Gaussian statistics already at D∼2.98. These effects must be connected to a genuine modification in the triad-to-triad nonlinear energy transfer mechanism.