Long-range Elastic Interactions Research Articles

From anti-Arrhenius to Arrhenius behavior in a dislocation-obstacle bypass: Atomistic simulations and theoretical investigation

Dislocations are the primary carriers of plasticity in metallic materials. Understanding the basic mechanisms for dislocation movement is paramount to predicting the material mechanical response. Relying on atomistic simulations, we observe a transition from non-Arrhenius to Arrhenius behavior in the rate of an edge dislocation overcoming the long-range elastic interaction with a prismatic loop in tungsten. Close to the critical resolved shear stress, the process shows a non-Arrhenius behavior at low temperatures. However, as the temperature increases, the activation entropy starts to dominate, leading to a traditional Arrhenius-like behavior. We have computed the activation entropy analytically along the minimum energy path following Schoeck’s method [1], which captures the cross-over between anti-Arrhenius and Arrhenius domains. Also, the Projected Average Force Integrator (PAFI) [2], another simulation method to compute free energies along an initial transition path, exhibits considerable concurrence with Schoeck’s formalism. We conclude that entropic effects need to be considered to understand processes involving dislocations bypassing elastic barriers close to the critical resolved shear stress. More work needs to be performed to fully understand the discrepancies between Schoeck’s and PAFI compared to molecular dynamics.

Phase-field study of the solutes-interstitial loops interaction in Fe–Cr alloys

Interstitial dislocation loops are typically formed in post-irradiated materials, such as Ferritic-Martensitic Fe–Cr alloys. These loops demonstrate solute segregation along their perimeters, effectively pinning the dislocation climb and resulting in high-density, small-sized loops that cause embrittlement. Phase-field simulations were conducted to investigate the behavior of Cr segregation in the stress field of a<100> type and various a/2<111> type interstitial dislocation loops in post-irradiated Fe–10Cr alloy. The study considers the long-range elastic interaction of solute Cr within the stress field of dislocation loops in a cubic elastic anisotropic material. The findings reveal a nonuniform stress distribution along the loop's perimeter depending on the included angle of habit plane normal n and Burgers vector b; the nonuniformity becomes more pronounced as the Burgers vector deviates from the normal direction. Furthermore, this nonuniform stress induces Cr segregation within regions experiencing tensile stress and Cr depletion in compression stress regions. Furthermore, a comprehensive analytical solution has been developed to characterize the diverse stress fields and solute segregation induced by interstitial dislocation loops with varying habit planes and Burgers vectors. The proposed analytical model effectively depicts the stress distribution of various dislocation loops, resulting in a Cr segregation profile that closely approximates those obtained through phase-field simulations. A thorough analysis of solute segregation mechanisms also elucidates the dispersed fine-scale nature and high density observed in experimental investigations of dislocation loops.

Atomically-informed dislocation dynamics model for prediction of void hardening in irradiated tungsten

Nanometric voids are one of the fundamental defects generated by neutron irradiation in fusion environment, which impede dislocation motion and cause hardening of plasma facing tungsten. Accurate prediction of voids hardening is crucial for safety evaluation of irradiated tungsten. In this paper, we develop a void-dislocation interaction model within the framework of dislocation dynamics (DD) and implement it in open-source DD software ParaDiS. We use this model to study the hardening effects of randomly-distributed voids with different sizes and number densities. The model for short-range interaction during dislocation-void contact, especially the dislocation movement on void surface, is informed and calibrated by atomistic simulations. Meanwhile, the long-range elastic interaction between dislocation and void is efficiently computed by the approach developed in our recent work (Wu et al., 2022) with Eshelby's equivalent inclusion method. Our DD simulations show that the average resistance by voids with various sizes is proportional to the 2/3 power of total number density. We also find that a modified Friedel-Kroupa-Hirsch (MFKH) model can quantify the hardening effects of nanometric voids, where the hardening coefficient depends on an effective average void diameter that accounts for the size distribution of voids. This MFKH model extends our previous work (Wu et al., 2022) that addresses the hardening of small vacancy clusters where dislocations simply cut through them, which is essentially an elastic perturbation to the dislocation dynamics. The proposed MFKH model can predict both small vacancy clusters and nanometric voids hardening effects with only difference in size dependent hardening coefficient. We further verify the MFKH model by comparing its quick predictions of the hardening effects by randomly-distributed nanometric voids with experiments on twelve neutron-irradiated tungsten samples.

Quantum annealing for microstructure equilibration with long-range elastic interactions

We demonstrate the use and benefits of quantum annealing approaches for the determination of equilibrated microstructures in shape memory alloys and other materials with long-range elastic interaction between coherent grains and their different martensite variants and phases. After a one dimensional illustration of the general approach, which requires to formulate the energy of the system in terms of an Ising Hamiltonian, we use distant dependent elastic interactions between grains to predict the variant selection for different transformation eigenstrains. The results and performance of the computations are compared to classical algorithms, demonstrating that the new approach can lead to a significant acceleration of the simulations. Beyond a discretization using simple cuboidal elements, also a direct representation of arbitrary microstructures is possible, allowing fast simulations with currently up to several thousand grains.

Thermodynamics of Light-Induced Nanoscale Polar Structures in Ferroelectric Superlattices.

We study the thermodynamics of nanoscale polar structures in PbTiO3/SrTiO3 ferroelectric superlattices induced by above-bandgap optical excitation using a phase-field model explicitly considering both structural and electronic processes. We demonstrate that the light-excited carriers provide the charge compensation of polarization bound charges and the lattice thermal energy, both of which are key to the thermodynamic stabilization of a previously observed supercrystal, a three-dimensionally periodic nanostructure, within a window of substrate strains, while different mechanical and electrical boundary conditions can stabilize a number of other nanoscale polar structures by balancing the competing short-range exchange interactions responsible for the domain wall energy and long-range electrostatic and elastic interactions. The insights into the light-induced formation and richness of nanoscale structures from this work offer theoretical guidance for exploring and manipulating the thermodynamic stability of nanoscale polar structures employing a combination of thermal, mechanical, and electrical stimuli as well as light.

Acoustic emission study on avalanche dynamics of ferroelectric switching in lead zirconate titanate ceramics

The avalanche dynamics of ferroelectric switching in lead zirconate titanate ceramics was investigated using acoustic emission. Two distinct power-law regimes for ferroelectric switching events were identified by an anomaly in the histogram of probability density, in contrast to the single power-law behavior observed in BaTiO3 single crystals. Such an anomaly is ascribed to the different minimum cut-offs of two power-law distributions. The critical energy exponents were determined to be 1.25 ± 0.10 for energies &amp;lt;103 aJ and 1.51 ± 0.14 for energies &amp;gt;103 aJ. The events in both regimes can be attributed to the depinning of domain walls from two distinct types of defects. The events in the lower energy regime are associated with domain wall junctions due to long-range electric and elastic interactions, whereas the latter is related to extrinsic defects, such as vacancies and dislocations. Moreover, for both regions, the rate of aftershocks follows the Omori power-law, indicating the same critical temporal correlations between the avalanches.

Electro-Elastic Modeling of Thermal Spin Transition in Diluted Spin-Crossover Single Crystals.

Spin-crossover solids have been studied for many years for their promising applications as optical switches and reversible high-density memories for information storage. This study reports the effect of random metal dilution on the thermal and structural properties of a spin-crossover single crystal. The analysis is performed on a 2D rectangular lattice using an electro-elastic model. The lattice is made of sites that can switch thermally between the low-spin and high-spin states, accompanied by local volume changes. The model is solved by Monte Carlo simulations, running on the spin states and atomic positions of this compressible 2D lattice. A detailed analysis of metal dilution on the magneto-structural properties allows us to address the following issues: (i) at low dilution rates, the transition is of the first order; (ii) increasing the concentration of dopant results in a decrease in cooperativity and leads to gradual transformations above a threshold concentration, while incomplete spin transitions are obtained for big dopant sizes. The effects of the metal dilution on the spatiotemporal aspects of the spin transition along the thermal transition and on the low-temperature relaxation of the photo-induced metastable high-spin states are also studied. Significant changes in the organization of the spin states are observed for the thermal transition, where the single-domain nucleation caused by the long-range elastic interactions is replaced by a multi-droplet nucleation. As to the issue of the relaxation curves: their shape transforms from a sigmoidal shape, characteristic of strong cooperative systems, into stretched exponentials for high dilution rates, which is the signature of a disordered system.

Mapping out the glassy landscape of a mesoscopic elastoplastic model.

We develop a mesoscopic model to study the plastic behavior of an amorphous material under cyclic loading. The model is depinning-like and driven by a disordered thresholds dynamics that is coupled by long-range elastic interactions. We propose a simple protocol of "glass preparation" that allows us to mimic thermalization at high temperatures as well as aging at vanishing temperature. Various levels of glass stabilities (from brittle to ductile) can be achieved by tuning the aging duration. The aged glasses are then immersed into a quenched disorder landscape and serve as initial configurations for various protocols of mechanical loading by shearing. The dependence of the plastic behavior upon monotonous loading is recovered. The behavior under cyclic loading is studied for different ages and system sizes. The size and age dependence of the irreversibility transition is discussed. A thorough characterization of the disorder-landscape is achieved through the analysis of the transition graphs, which describe the plastic deformation pathways under athermal quasi-static shear. In particular, the analysis of the stability ranges of the strongly connected components of the transition graphs reveals the emergence of a phase-separation like process associated with the aging of the glass. Increasing the age and, hence, the stability of the initial glass results in a gradual break-up of the landscape of dynamically accessible stable states into three distinct regions: one region centered around the initially prepared glass phase and two additional regions characterized by well-separated ranges of positive and negative plastic strains, each of which is accessible only from the initial glass phase by passing through the stress peak in the forward and backward, respectively, shearing directions.

An efficient and minimalist scheme for continuum dislocation dynamics

Continuum dislocation dynamics methods have received considerable interests for simulating dislocation microstructures at the meso-scale, serving as potential tools for bridging the gap between micro- and macro-scale models for the plastic behaviours of crystalline materials. Recently, an exact evolution equation for the “all-dislocation” density that represents dislocation quantities over both space and dislocation-character domains has been developed by one of the present authors. The “all-dislocation” representation is superior to representations based on the Nye tensor or geometrically necessary dislocations (GND), since the statistically stored dislocation (SSD) contents will be preserved. In this paper, a numerical scheme is presented to solve the dynamics of the “all-dislocation” density efficiently, with long-range elastic interaction between dislocations accounted for via Mura's formula after singularity removal. The proposed simulation scheme is demonstrated by simulation examples in the multi-scale hierarchy, from intensive microstructures of individual dislocations including Frank-Read source and Orowan looping, to extensive microstructures of coarse-grained dislocation densities in single- and multi-slip in the face-centred cubic crystal structure.

Studying the effects of hydrogen on dislocation mobility and multiplication in nickel by phase-field method

The effects of hydrogen on dislocation mobility and operation of the Frank-Read source are evaluated using our recently developed phase-field model. This is achieved by accounting for long-range elastic hydrogen-dislocation interaction (i.e., hydrogen shielding effect) and short-range inelastic hydrogen-dislocation core interaction through atomically informed generalized stacking fault energy (GSFE, or γ-surface). The results show that, due to the short-range interaction, hydrogen impedes the dislocation motion, which is contrary to the long-range hydrogen shielding effect. The bow-out configurations of the Frank-Read source obtained by the phase-field model can agree well with that from the atomic simulations. Hydrogen can decrease the critical nucleation stress of the Frank-Read source and thus enhance the hydrogen-induced plasticity. Moreover, the hydrogen-induced dislocation core energy reduction prevails over the hydrogen-induced elastic shielding, especially when the hydrogen concentration is low. Our results suggest that the modification of the dislocation core structure due to the short-range interaction plays an important role in understanding the hydrogen-enhanced localized plasticity mechanism. Finally, quantitative influences of hydrogen on dislocation mobility and critical nucleation stress of the Frank-Read source are obtained, which can further serve upscaled dislocation dynamics simulation and dislocation-density based crystal plasticity modelling.

Collective States of Active Particles With Elastic Dipolar Interactions

Many types of animal cells exert active, contractile forces and mechanically deform their elastic substrate, to accomplish biological functions such as migration. These substrate deformations provide a mechanism in principle by which cells may sense other cells, leading to long-range mechanical inter–cell interactions and possible self-organization. Here, inspired by cell mechanobiology, we propose an active matter model comprising self-propelling particles that interact at a distance through their mutual deformations of an elastic substrate. By combining a minimal model for the motility of individual particles with a linear elastic model that accounts for substrate-mediated, inter–particle interactions, we examine emergent collective states that result from the interplay of motility and long-range elastic dipolar interactions. In particular, we show that particles self-assemble into flexible, motile chains which can cluster to form diverse larger-scale compact structures with polar order. By computing key structural and dynamical metrics, we distinguish between the collective states at weak and strong elastic interaction strength, as well as at low and high motility. We also show how these states are affected by confinement within a channel geometry–an important characteristic of the complex mechanical micro-environment inhabited by cells. Our model predictions may be generally applicable to active matter with dipolar interactions ranging from biological cells to synthetic colloids endowed with electric or magnetic dipole moments.

Stress-dependence of dislocation dissociation, nucleation and annihilation in elastically anisotropic Cu

Dislocation dynamics under external stresses are an important manifestation of plastic behaviors of materials. In this study, based on the work of Shen et al. (2014), an improved microscopic phase field (MPF) model incorporating gradient energy term into the energy functional is developed to thoroughly investigate the stress dependence of whole dislocation evolution processes, including dislocation dissociation, nucleation, and annihilation in Cu with prominent elastic anisotropy. Necessary physical parameters such as elastic constants, lattice energy and gradient energy coefficients are all determined by atomistic molecular statics (MS) simulations. The residue theorem is utilized to accurately solve the anisotropic stress kernels as pre-factors characterizing the long-range elastic interaction produced by edge and screw components of mixed dislocations in anisotropic Cu. The Peierls stress of screw dislocations and stacking fault width under applied stress are evaluated by the MPF model and compared with those assessed by the semi-discrete variational Peierls-Nabarro (SVPN) model and MS simulations. The MPF model is then applied to screw dislocation dipoles and dislocation loops to examine possible dislocation evolution paths under different stress states. Through systematic simulations, we find the dislocation evolution regimes are closely related with the applied stress, and can be distinguished by three important critical stresses, i.e., critical stress, singular stress, and nucleation stress. Specifically, when the applied stress is less than the critical stress, the dislocation loops always collapse in the end, and do not depend on the final states of Shockley partials, which may exist or have annihilated. Furthermore, when the applied stress exceeds the nucleation stress, partial dislocations will spontaneously nucleate within stacking fault (SF) area, followed by a series of dislocation reactions until partial dislocations bρ1 and bρ3 are left in the end. Besides, the shear stress components perpendicular to the Escaig stress in slip plane are found to suppress the nucleation of partial dislocations in SF area.

Directing the far-from-equilibrium assembly of nanoparticles in confined liquid crystals by hydrodynamic fields.

The assembly of nematic colloids relies on long-range elastic interactions that can be manipulated through external stimuli. Confinement and the presence of a hydrodynamic field alter the defect structures and the energetic interactions between the particles. In this work, the assembly landscape of nanoparticles embedded in a nematic liquid crystal confined in a nanochannel under a pressure-driven flow is determined. The dynamics of the liquid crystal tensor alignment field is determined through a Poisson-Bracket framework, namely the Stark-Lubensky equations, coupled with the zero-Reynolds momentum equations and the liquid crystal Landau-de Gennes free energy functional. A second order semi-implicit time integration and a three-dimensional Galerkin finite element method are used to resolve flow and nematic fields under several conditions. In general, the zero Reynolds flow displaces the defects around the particles in the upstream direction and renders the surface anchoring ineffective when the flow strength dominates over the nematic elasticity. More importantly, the potential of mean force for particle assembly is non-monotonic independent of surface anchoring. Our results show that the confinement length scale determines the repulsion/attraction transition between colloids, while the flow strength modifies the static defect structure surrounding the particles and determines the magnitude of the energetic barrier for successful assembly. In the attractive regime, the particles move at different rates through the nematic until one particle eventually catches up with the other. This process occurs against or along the direction of flow depending on the flow strength. Ultimately, these results provide a template for engineering and controlling the transport and assembly of nanoparticles under far-from equilibrium conditions in anisotropic media.

Dislocation interactions during low-temperature plasticity of olivine and their impact on the evolution of lithospheric strength

The strength of the lithosphere is typically modelled based on constitutive equations for steady-state flow. However, strain hardening may cause significant evolution of strength in the colder load-bearing portion of the lithosphere. Recent rheological data from low-temperature deformation experiments on olivine suggest that strain hardening occurs due to the presence of temperature-independent back stresses generated by long-range elastic interactions among dislocations. These interpretations provided the basis for a flow law that incorporates hardening by the development of back stress. Here, we test this dislocation-interaction hypothesis by examining the microstructures of olivine samples deformed plastically at room temperature either in a deformation-DIA apparatus at differential stresses of ≤4.3GPa or in a nanoindenter at applied contact stresses of ≥10.2GPa. High-angular resolution electron backscatter diffraction maps reveal the presence of geometrically necessary dislocations with densities commonly above 1014m−2 and intragranular heterogeneities in residual stress on the order of 1 GPa in both sets of samples. Scanning transmission electron micrographs reveal straight dislocations aligned in slip bands and interacting with dislocations of other types that act as obstacles. The resulting accumulations of dislocations in their slip planes, and associated stress heterogeneities, are consistent with strain hardening resulting from long-range back-stresses acting among dislocations and thereby support the form of the flow law for low-temperature plasticity. Based on these observations, we predict that back stresses among dislocations will impart significant mechanical anisotropy to deformed lithosphere by enhancing or reducing the effective stress. Therefore, strain history, with associated microstructural and micromechanical evolution, is an important consideration for models of lithospheric strength. The microstructural observations also provide new criteria for identifying the operation of back-stress induced strain hardening in natural samples and therefore provide a means to test the applicability of the flow law for low-temperature plasticity.

Theoretical framework for predicting solute concentrations and solute-induced stresses in finite volumes with arbitrary elastic fields

A theoretical model for computing the interstitial solute concentration and the interstitial solute-induced stress field in a three-dimensional finite medium with any arbitrary elastic fields was developed. This model can be directly incorporated into two-dimensional or three-dimensional discrete dislocation dynamics simulations, continuum dislocation dynamics simulations, or crystal plasticity simulations. Using this model, it is shown that a nano-hydride can form in the tensile region below a dissociated edge dislocation at hydrogen concentration as low as χ0=5×10−5, and its formation induces a localized hydrogen elastic shielding effect that leads to a lower stacking fault width for the edge dislocation. Additionally, the model also predicts the segregation of hydrogen at Σ109(13 7 0)/33.4∘ symmetric tilt grain boundary dislocations. This segregation strongly alters the magnitude of the shear stresses at the grain boundary, which can subsequently alter dislocation-grain boundary interactions and dislocation slip transmissions across the grain boundary. Moreover, the model also predicts that the hydrogen concentration at a mode-I central crack tip increases with increasing external loading, higher intrinsic hydrogen concentration, and/or larger crack lengths. Finally, linearized approximate closed-form solutions for the solute concentration and the interstitial solute-induced stress field were also developed. These approximate solutions can effectively reduce the computation cost to assess the concentration and stress field in the presence of solutes. These approximate solutions are also shown to be a good approximation when the positions of interest are several nanometers away (i.e. long-ranged elastic interactions) from stress singularities (e.g. dislocation core and crack tip), for low solute concentrations, and/or at high temperatures.

Quantifying the dynamics of dislocation kinks in iron and tungsten through atomistic simulations

When high-Peierls-barrier materials such as iron (Fe) and tungsten (W) are deformed, dislocation kinks can be easily activated. The subsequent kink dynamics may dictate the dislocation mobility and the material's overall performance under certain conditions. In this work, taking the thermal-induced kink diffusion along ½<111> screw dislocation lines as an example, the kink dynamics in b.c.c. iron and tungsten are quantified through atomistic simulations. Results show that in both Fe and W, the kink dynamics, including its diffusion coefficient (Dkink) and dissipation parameter (γkink), are sensitive to the dimension (noted as L) of a simulation cell size along the dislocation line direction: the larger L, the higher Dkink, and the smaller γkink. A scaling law for describing the three-stage L-dependent kink dynamics is extracted from a series of computational analysis of the kink diffusion along dislocations with L ranging from tens to hundreds of nanometers. It is found that, if a converged Dkink is desired from atomistic simulations, the minimum L needs to be at least hundreds of nanometers. This is beyond the reach of an atomic-level model using a modest computational resource. To explain the L-dependent kink dynamics, we calculate the kink-induced local stress fields using two different atomistic stress formula, i.e., a widely-used Virial and a recently developed mechanical stress formula. Results suggest: (i) the L-dependent kink dynamics is caused by the long-range elastic interaction between the kink and its periodic images; and (ii) the Virial stress formula underestimates such interactions. This work lays the continuum description of kink-controlled dislocation dynamics on an atomistic foundation. It will also support the development of multiscale methods for addressing the coupled dynamics between the motion of a μm-long dislocation line and the atomic-level kink diffusion along the line itself in b.c.c. metals or other high-Peierls-barrier materials under deformation.

A New Formulation of Coupling and Sliding Motions of Grain Boundaries Based on Dislocation Structure

A continuum model of the two dimensional low angle grain boundary motion and the dislocation structure evolution on the grain boundaries has been developed in [L. Zhang and Y. Xiang, J. Mech. Phys. Solids, 117 (2018), pp. 157--178]. The model is based on the motion and reaction of the constituent dislocations of the grain boundaries. The long-range elastic interaction between dislocations is included in the continuum model, and it maintains a stable dislocation structure described by Frank's formula for grain boundaries. In this paper, we develop a new continuum model for the coupling and sliding motions of grain boundaries that avoids the time-consuming calculation of the long-range elastic interaction. In this model, the long-range elastic interaction is replaced by a constraint of Frank's formula. The constrained evolution problem in our new continuum model is further solved by using the projection method. Effects of the coupling and sliding motions in our new continuum model and relationship with the classical motion by curvature model are discussed. The continuum model is validated by comparisons with discrete dislocation dynamics model and the early continuum model [L. Zhang and Y. Xiang, J. Mech. Phys. Solids, 117 (2018), pp. 157--178] in which the long-range dislocation interaction is explicitly calculated.

Friedel Oscillations and Helium Bubble Ordering in Molybdenum

Helium ions implanted into metals can evolve into ordered bubbles isomorphic to the host lattice. Long-range elastic interaction is generally believed to drive the formation of bubble superlattice, but little is known about the thermodynamics at the very initial stage. Our first-principles calculations demonstrate that in molybdenum, Friedel oscillations induced by individual helium generate both potential barriers and wells for helium clustering at short He-He distances. Such repulsion and attraction at high concentration provide a thermodynamic diving force to assist lining up randomly distributed He atoms into ordered bubbles. Friedel oscillations might have general impact on solute-solute interactions in alloys.

Emergence of Cracklike Behavior of Frictional Rupture: The Origin of Stress Drops

The process of frictional rupture, i.e. the failure of frictional systems, abounds in the technological and natural world around us, ranging from squealing car brake pads to earthquakes along geological faults. A general framework for understanding and interpreting frictional rupture commonly involves an analogy to ordinary crack propagation, with far-reaching implications for various disciplines, from engineering tribology to geophysics. An important feature of the analogy to cracks is the existence of a reduction in the stress-bearing capacity of the ruptured interface, i.e. of a drop from the applied stress, realized far ahead of a propagating rupture, to the residual stress left behind it. Yet, how and under what conditions such finite and well-defined stress drops emerge from basic physics are not well understood. Here we show that for rapid rupture a stress drop is directly related to wave radiation from the frictional interface to the bodies surrounding it and to long-range bulk elastodynamics, and not exclusively to the physics of the contact interface. Furthermore, we show that the emergence of a stress drop is a transient effect, affected by the wave travel time in finite systems and by the decay of long-range elastic interactions. Finally, we supplement our results for rapid rupture with predictions for slow rupture. All of the theoretical predictions are supported by available experimental data and by extensive computations. Our findings elucidate the origin of stress drops in frictional rupture, i.e. they offer a comprehensive and fundamental understanding of why, how and to what extent frictional rupture might be viewed as an ordinary fracture process.

Soft Nucleation of an Elastic Crease.

Creasing instability is ubiquitous in soft solids; however, its inception remains enigmatic as it cannot be captured by the standard linearization techniques. It also does not fit the conventional picture of a barrier-crossing nucleation, and instead carries some features of a second order phase transition. Here we show that despite its fundamentally nonlinear nature, creasing has its origin in marginal stability which is, however, obscured by the dominance of long-range elastic interactions. We argue that despite its supercritical (soft) character, creasing bifurcation can be identified by the condition that the (generalized) driving force acting on an incipient stress singularity degenerates. The analytic instability criterion, obtained in this way, shows an excellent agreement with both physical experiments and direct numerical simulations.