Self-similar Regime Research Articles

On the statistical evolution of viscous vortex-gas free shear layers

The temporal vortex-gas free shear layer is a Hamiltonian system of N-point vortices in a singly-periodic domain relevant to the (large-scale) development of plane (3D Navier–Stokes) turbulent shear layers. It has been shown (Suryanarayanan et al., 2014) that there are three distinct temporal regimes in its evolution, including an intermediate self-similar regime (RII) with a universal constant spreading rate, and a final state of relative equilibrium. Here, we study the evolution of vortex-gas free shear layers affected by viscosity as simulated by the addition of a random walk component to the motion of the vortices (Chorin, 1973). We show that while the momentum thickness of the layer, θ, scales as t1∕2 in the first and final regimes for the viscous system, it grows as t1 in the intermediate Regime II (observed here for momentum thickness Reynolds numbers Reθ from 50 to over 5000 across different simulations), with the same universal growth rate dθ∕dtΔU=0.0166±0.0002 as in the inviscid case. This suggests that the non-equilibrium universality of Regime II is robust to small perturbations of the Hamiltonian, and that this result could have general consequences for systems with long-range interactions. We further find that, even under conditions where the viscous shear stress is not negligible, the Reynolds shear stress adjusts itself in such a way that their sum is a constant in Regime II and equal to the value of the latter in the inviscid case. The implications of this observation on the dynamics of turbulent shear flows are elaborated.

Microstructure entropy guided understanding of yield strength in alloy systems characterized by grain size distribution

We introduce and adopt the concept of a phenomenological parameter, microstructure entropy, S*, to understand the yield strength in alloy systems with bimodal grain size distribution obtained from large sets of experimental data. 60 groups of bimodal grain size distribution data was obtained through the application of an ingenious concept involving severe cold deformation to obtain dislocation cell-type of martensite, followed by annealing when martensite reverts to austenite by shear or diffusion mechanism, with the aim to obtain ultrafine grain size. Six factors emerged from statistical data analysis through EBSD studies in terms of grain size and grain numbers from 60 sets of data. Microstructure entropy (S*) was obtained from the bimodal structure data in a self-similar regime. Inverse of the square root of microstructure entropy (1S*) and yield strength exhibited a linear relationship. The proposed conceptual methodology has wide acceptance for any grain size distribution to obtain microstructure entropy (S*) of a specific grain structure and predict yield strength. A generic equation similar to Hall-Petch relationship, but involving microstructure entropy is proposed to predict and understand yield strength of metallic systems characterized by grain size distribution or microstructural evolution.

Chiral Instabilities and the Onset of Chiral Turbulence in QED Plasmas.

We present a first principles study of chiral plasma instabilities and the onset of chiral turbulence in QED plasmas with strong gauge matter interaction (e^{2}N_{f}=64), far from equilibrium. By performing classical-statistical lattice simulations of the microscopic theory, we show that the generation of strong helical magnetic fields from a helicity imbalance in the fermion sector proceeds via three distinct phases. During the initial linear instability regime the helicity imbalance of the fermion sector causes an exponential growth (damping) of magnetic field modes with right- (left-) handed polarization, for which we extract the characteristic growth (damping) rates. Secondary growth of unstable modes accelerates the helicity transfer from fermions to gauge fields and ultimately leads to the emergence of a self-similar scaling regime characteristic of a decaying turbulence, where magnetic helicity is efficiently transferred to macroscopic length scales. Within this turbulent regime, the evolution of magnetic helicity spectrum can be described by an infrared power spectrum with spectral exponent κ=10.2±0.5 and dynamical scaling exponents α=1.14±0.50 and β=0.37±0.13.

Liquid bridge length scale based nondimensional groups for mapping transitions between regimes in capillary break-up experiments

Criteria to identify transitions between dynamic self-similar linear thinning regimes of liquid bridges are of utmost importance in order to accurately interpret results in capillary break-up rheometry. Currently available criteria encompass many experimental difficulties or rely on numerical approaches. Here, we introduce a different set of nondimensional groups, OhL=ηin/γρL and a=R/L, based on the experimentally relevant axial length scale of a liquid bridge L, for viscous-dominated fluids undergoing capillary break-up in air. This framework is further extended to encompass the effect of outer viscous fluids. As a result, we present a two-dimensional operating map in which the boundaries are set by fluid properties and a single geometrical parameter, related to the experimental configuration. This approach establishes guidelines to correctly interpret experimental data and identify transitions in capillary break-up experiments of liquid bridges surrounded by fluids of different viscosities.

Characteristics of the turbulent non-turbulent interface in a spatially evolving turbulent mixing layer

The highly convoluted interface separating the turbulent and non-turbulent regions in a turbulent mixing layer is experimentally investigated using the particle image velocimetry (PIV) technique. The mixing layer was generated using a fine screen/mesh in one half of the test section of a low-speed wind tunnel. The PIV data, which were acquired with high spatial resolution in the self-similar regime of the flow, allow us to identify the turbulent/non-turbulent interface (TNTI) using a suitable threshold value of the absolute spanwise vorticity, |ωz|. The threshold values for the top and bottom interfaces of the mixing layer are found to be different, and the probability density function (PDF) of the interface position for both the interfaces is found to follow the Gaussian distribution. Interestingly, the PDF of the interface orientation reveals two clear peaks, and this is attributed to the sustained large-scale motions in a mixing layer, compared to the other free-shear flows, as is also substantiated by further analyses such as the linear stochastic estimation and the conditional analysis of the transverse velocity profile. The linear stochastic analysis also shows the presence of large vorticity structures of the order of the Taylor microscale at the mean TNTI location in a mixing layer. Furthermore, the present work reveals that, using the spanwise component of vorticity alone, we can experimentally identify and estimate the thickness of the viscous superlayer from the conditional profiles of the diffusion term and the correlation coefficient of the dissipation and the diffusion terms in the enstropy transport equation. The present value of the viscous superlayer thickness of 5η–6η (where η is the Kolmogorov length scale) compares well with the values reported in the literature for other shear flows. Although both the interfaces are found to behave like a fractal with a dimension of 1.3 in two dimensions, one can find dominant length scales of the order of the thickness of the viscous superlayer, the thickness of the TNTI and the width of the mixing layer from the pre-multiplied power spectra of the autocorrelation functions of the interface curvature, the normal velocity and the interface position, along the TNTI, respectively. In addition, we find that the TNTI characteristics do not show significant dependence on the velocity ratios and Reλ considered in the present study. Furthermore, the conditional transverse velocity profiles indicate that the entrainment characteristics for the upper and lower TNTIs may be asymmetric in nature.

Expansion of the strongly interacting superfluid Fermi gas: Symmetries and self-similar regimes

We consider an expansion of the strongly interacting superfluid Fermi gas in a vacuum, assuming absence of the trapping potential, in the so-called unitary regime (see, for instance, \cite{pitaevskii2008superfluid}) when the chemical potential $\mu \propto \hbar^2n^{2/3}/m$ where $n$ is the density of the Bose-Einstein condensate of Cooper pairs of fermionic atoms. In low temperatures, $T\to 0$, such expansion can be described in the framework of the Gross-Pitaevskii equation (GPE). Because of the chemical potential dependence on the density, $\sim n^{2/3}$, the GPE has additional symmetries, resulting in the existence of the virial theorem \cite% {vlasov1971averaged}, connecting the mean size of the gas cloud and its Hamiltonian. It leads asymptotically at $t\to\infty$ to the gas cloud expansion, linearly growing in time. We study such asymptotics, and reveal the perfect match between the quasi-classical self-similar solution and the asymptotic expansion of the non-interacting gas. This match is governed by the virial theorem, derived through utilizing the Talanov transformation \cite{talanov1970focusing}, which was first obtained for the stationary self-focusing of light in media with a cubic nonlinearity due to the Kerr effect. In the quasi-classical limit, the equations of motion coincide with 3D hydrodynamics for the perfect monoatomic gas with $\gamma=5/3$. Their self-similar solution describes, on the background of the gas expansion, the angular deformities of the gas shape in the framework of the Ermakov--Ray--Reid type system.

Buoyancy–Drag modelling of bubble and spike distances for single-shock Richtmyer–Meshkov mixing

The Buoyancy–Drag model is a simple model, based on ordinary differential equations, for estimating the growth of a turbulent mixing zone at an interface between fluids of different density due to Richtmyer–Meshkov and Rayleigh–Taylor instabilities. The early stages of the mixing process are very dependent on the initial conditions and modifications to the Buoyancy–Drag model are needed to obtain correct results. In a recent paper, Youngs & Thornber (2019), the results of three-dimensional turbulent mixing simulations were used to construct the modifications required to represent the evolution of the overall width of the mixing zone due to single-shock Richtmyer–Meshkov mixing evolving from narrowband initial random perturbations. The present paper extends this analysis to give separate equations for the bubble and spike distances (the depths to which the mixing zone penetrates the dense and light fluids). The data analysis depends on novel integral definitions of the bubble and spike distances which vary smoothly with time. Results are presented for two pre-shock density ratios, ρ1∕ρ2=3and20. New insights are given for the variation of asymmetry of the mixing zone with time. At early time, values of the spike-to-bubble distance are very high. The asymmetry greatly reduces as mixing proceeds towards a self-similar state. For the overall (integral) mixing width, W, the Buoyancy–Drag model gives satisfactory results at both density ratios using the same parameters. However, for the bubble and spike distances the behaviour is very different at the two density ratios. The method used to analyse the data provides a new way of estimating the self-similar growth exponent θ (W∼tθ). The values obtained are approximate because of the difficulty in running the three dimensional simulations far enough into the self-similar regime. Estimates of θ are consistent with the theoretical value of 1/3 given by the model of Elbaz & Shvarts (2018). The corrected form of the Buoyancy–Drag model gives accurate fits to the data for W,hbandhsover the whole time range with θ =1/3 for ρ1∕ρ2=3 and θ=0.35 for ρ1∕ρ2=20.

Dynamics of Subsiding Shells in Actively Growing Clouds with Vertical Updrafts

Abstract The dynamics of a subsiding shell at the edges of actively growing shallow cumulus clouds with updrafts is analyzed using direct numerical simulation. The actively growing clouds have a fixed in-cloud buoyancy and velocity. Turbulent mixing and evaporative cooling at the cloud edges generate a subsiding shell that grows with time. A self-similar regime is observed for first- and second-order moments when normalized with respective maximum values. Internal scales derived from integral properties of the flow problem are identified. A self-similarity analysis using these scales reveals that contrary to classical self-similar flows, the turbulent kinetic energy budget terms and velocity moments scale according to the buoyancy and not with the mean velocity. The shell thickness is observed to increase linearly with time. The buoyancy scale remains time invariant and is set by the initial cloud–environment thermodynamics. The shell accelerates ballistically with a magnitude set by the saturation value of the buoyancy of the cloud–environment mixture. In this regime, the shell is buoyancy driven and independent of the in-cloud velocity. Relations are obtained for predicting the shell thickness and minimum velocities by linking the internal scales with external flow parameters. The values thus calculated are consistent with the thickness and velocities observed in typical shallow cumulus clouds. The entrainment coefficient is a function of the initial state of the cloud and the environment, and is shown to be on the same order of magnitude as fractional entrainment rates calculated for large-scale models.

Spectral, statistical, and vertex functions in scalar quantum field theory far from equilibrium

We compute the far-from-equilibrium dynamics of relativistic scalar quantum fields in $3+1$ space-time dimensions starting from over-occupied initial conditions. We determine universal scaling exponents and functions for two-point correlators and the four-vertex in a self-similar regime in time and space or momenta. The scaling form of the momentum-dependent four-vertex exhibits a dramatic fall-off toward low momenta. Comparing spectral functions (commutators) and statistical correlations (anticommutators) of field operators allows us to detect strong violations of the fluctuation-dissipation relation in this non-perturbative infrared regime. Based on a self-consistent expansion in the number of field components to next-to-leading order, a wide range of interaction strengths is analyzed and compared to weak-coupling estimates in effective kinetic theory and classical-statistical field theory.

On transition to self-similar acceleration of spherically expanding flames with cellular instabilities

The evolution of the self-acceleration of spherically expanding flames subjected to cellular instabilities was experimentally studied with H2/O2/N2 mixtures over a wide range of Lewis numbers, flame thicknesses, and density ratios. Results show that the flames manifest a self-similar acceleration behavior, and the acceleration exponent, α, continuously increases and follows a power-law relation with the normalized Peclet number beyond a transition acceleration stage, which is different from the usually suggested self-similar propagation characterized by a constant α after the short transition acceleration period. The self-similar acceleration with reduced growth rate is found to be the results of the continuous extension of cellular spectrum on the long-wavelength side but the saturation of cellular spectrum on the short-wavelength side. The critical Peclet numbers associated with onset (Pecr) and transition (Pect) of self-acceleration have a similar non-linear relationship with the Marksetin number, caused by the non-equal differential-thermal effects on the development of Darrieus–Landau (DL) cells for sub-unity (<1) and super-unity (>1) Lewis numbers. Based on the experimental Pecr and Pect, a regime diagram is presented for the laminar premixed hydrogen flames with cellular instabilities, which consists smooth propagation, transition acceleration, and self-similar acceleration regimes. In addition, an empirical power-law correlation for the α is suggested to describe the whole self-acceleration process. This correlation is able to predict the present α as well as the experimental results of the previous work. The measured α in the range of present interpellation is smaller than 1.5, the suggested value for the flame self-turbulization. However, this value could possibly be attained at an extremely large Peclet number (or flame radius) based on the extrapolation of the present correlation.

A Problem on a Viscous Layer Deformation

The axisymmetric motion of a viscous incompressible fluid in a layer bounded by a solid plane and the free surface parallel to it is considered. There are three regimes of motion in the problem: stabilization to rest, collapse in a finite time, and an intermediate self-similar regime in which the viscous layer expands unlimitedly in an infinite time.

Monte Carlo Simulations of Topological Properties in Two-Phase Polycrystalline Materials for Several Diffusion Mechanism

Numerical simulations by means of the Monte Carlo Potts model have been provided to simulate grain structures in two-phase polycrystalline materials. The topological features in the simulated microstructure analyzed for different diffusion mechanisms over a broad range of volume fractions for both phases. The topological properties include the average number of sides, grain topology distribution and the topological size relation function. It is found that the average number of sides depends proportionally on the volume fraction. It increases as the volumes fraction increases and vice versa. Moreover, it is shown that the grain topology distribution in the self-similar growth regime can be described by time unchanged function of the relative grain size. Additionally, topological size function in the simulated microstructure can be evaluated by a quadratic function.

Rogue waves, self-similar statistics, and self-similar intermediate asymptotics

We advance a statistical theory of extreme event emergence in random nonlinear wave systems with self-similar intermediate asymptotics. We show, within the framework of a generic (1 + 1)D nonlinear Schrodinger equation with linear gain, that extreme events and even rogue waves in weakly nonlinear, statistical open systems emerge as parabolic-shape giant fluctuations in the self-similar asymptotic propagation regime. We analytically demonstrate the self-similar structure of the non-Gaussian statistics of emergent rogue waves and validate our results with numerical simulations. Our results shed new light on generic statistical features of extreme events in nonlinear open systems with self-similar intermediate asymptotics.

Numerical modeling of the Kolmogorov flow in a viscous and inviscid media

In this paper we consider the problem proposed by A.N. Kolmogorov to study the reasons of turbulence. We consider the action of the external periodic field along one of the coordinates on a viscous and inviscid conductive media in the two-dimensional case. We propose a numerical study of this problem based on the solution of the systems of the Euler and mainly the Navier-Stokes equations. The calculations show that under certain conditions periodic vortex structures may appear in the liquid, similar to the “parquet” mode in the Kolmogorov task, with the subsequent development of the self-similar regime.

Plasma optics in the context of high intensity lasers

The use of plasmas provides a way to overcome the low damage threshold of classical solid-state based optical materials, which is the main limitation encountered in producing and manipulating intense and energetic laser pulses. Plasmas can directly amplify or alter the characteristics of ultra-short laser pulses via the three-wave coupling equations for parametric processes. The strong-coupling regime of Brillouin scattering (sc-SBS) is of particular interest: recent progress in this domain is presented here. This includes the role of the global phase in the spatio-temporal evolution of the three-wave coupled equations for backscattering that allows a description of the coupling dynamics and the various stages of amplification from the initial growth to the so-called self-similar regime. The understanding of the phase evolution allows control of the directionality of the energy transfer via the phase relation between the pulses. A scheme that exploits this coupling in order to use the plasma as a wave plate is also suggested.

Optimal perturbations for controlling the growth of a Rayleigh–Taylor instability

A discrete adjoint-based method is employed to control multi-mode Rayleigh–Taylor (RT) instabilities via strategic manipulation of the initial interfacial perturbations. We seek to find to what extent mixing and growth can be enhanced at late stages of the instability and which modes are targeted to achieve this. Three objective functions are defined to quantify RT mixing and growth: (i) variance of mole fraction, (ii) a kinetic energy norm based on the vertical velocity and (iii) variations of mole fraction with respect to the unperturbed initial state. The sensitivity of these objective functions to individual amplitudes of the initial perturbations are studied at various stages of the RT instability. The most sensitive wavenumber during the early stages of the instability closely matches the most unstable wavenumber predicted by linear stability theory. It is also shown that randomly changing the initial perturbations has little effect at early stages, but results in large variations in both RT growth and its sensitivity at later times. The sensitivity obtained from the adjoint solution is employed in gradient-based optimization to both suppress and enhance the objective functions. The adjoint-based optimization framework was capable of completely suppressing mixing by shifting all of the interfacial perturbation energy to the highest modes such that diffusion dominates. The optimal initial perturbations for enhancing the objective functions were found to have a broadband spectrum resulting in non-trivial coupling between modes and depends on the particular objective function being optimized. The objective functions were increased by as much as a factor of nine in the self-similar late-stage growth regime compared to an interface with a uniform distribution of modes, corresponding to a 32% increase in the bubble growth parameter and 54% increase in the mixing width. It was also found that the interfacial perturbations optimized at early stages of the instability are unable to predict enhanced mixing at later times, and thus optimizing late-time multi-mode RT instabilities requires late-time sensitivity information. Finally, it was found that the optimized distribution of interfacial perturbations obtained from two-dimensional simulations was capable of enhancing the objective functions in three-dimensional flows. As much as 51% and 99% enhancement in the bubble growth parameter and mixing width, respectively, was achieved, even greater than what was reached in two dimensions.

Kinetics of coarsening have dramatic effects on the microstructure: Self-similarity breakdown induced by viscosity contrast

The viscous coarsening of a phase separated mixture is studied and the effects of the viscosity contrast between the phases are investigated. From an analysis of the microstructure, it appears that for moderate departure from the perfectly symmetric regime the self-similar bicontinuous regime is robust. However, the connectivity of one phase decreases when its volume fraction decreases or when it is becoming less viscous than the complementary phase. Eventually self-similarity breakdown is observed and characterized.

Restoring universality to the pinch-off of a bubble

The pinch-off of a bubble is an example of the formation of a singularity, exhibiting a characteristic separation of length and time scales. Because of this scale separation, one expects universal dynamics that collapse into self-similar behavior determined by the relative importance of viscous, inertial, and capillary forces. Surprisingly, however, the pinch-off of a bubble in a large tank of viscous liquid is known to be nonuniversal. Here, we show that the pinch-off dynamics of a bubble confined in a capillary tube undergo a sequence of two distinct self-similar regimes, even though the entire evolution is controlled by a balance between viscous and capillary forces. We demonstrate that the early-time self-similar regime restores universality to bubble pinch-off by erasing the system's memory of the initial conditions. Our findings have important implications for bubble/drop generation in microfluidic devices, with applications in inkjet printing, medical imaging, and synthesis of particulate materials.

Joule-Level High-Efficiency Energy Transfer to Subpicosecond Laser Pulses by a Plasma-Based Amplifier

Laser plasma amplification of sub-picosecond pulses above the Joule level is demonstrated, a major milestone for this scheme to become a solution for the next-generation of ultra-high intensity lasers. By exploring over 6 orders of magnitude the influence of the incident seed intensity on Brillouin laser amplification, we reveal the importance of a minimum intensity to ensure an early onset of the self-similar regime, and a large energy transfer with a very high efficiency, up to 20%. Evidence of energy losses of the seed by spontaneous backward Raman is found at high amplification. The first three-dimensional envelope simulations of the sub-picosecond amplification were performed, supplemented by one-dimensional PIC simulations. Comparisons with the experimental results demonstrate the capability of quantitative predictions on the transferred energy. The global behavior of the amplification process, is reproduced, including the evolution of the spatial profile of the amplified seed.

The transient force profile of low-speed droplet impact: measurements and model

The transient force exerted by a low-speed liquid droplet impinging onto a flat rigid surface is investigated experimentally. The measurements employ a high-sensitivity piezo-electric sensor, along with a high-speed camera, and cover four decades in droplet Reynolds number and greater than two decades in Weber number. Across these ranges, the peak of individual force profiles span from 3 mN to over 1300 mN. Once normalised, the force–time profiles support the existence of an inertially dominated self-similar regime. Within this regime, previous numerical and theoretical studies predict a$\sqrt{t}$dependence of impact normal force during the initial pre-peak rise. While our measurements confirm this finding, they also indicate that, after the peak force the profiles exhibit an exponential decay. This long-time decay law suggests treatment of the momentum transport from the droplet using a lumped model. An observed linear dependence between the force and force decay rate supports this approach. The reason for the efficacy of treating this system via a lumped model apparently connects to the physics right at the surface that limit the rate of momentum transport from the droplet to the surface. This is explored by estimating the momentum transfer by solely using the deforming droplet shape, but under the condition of negligible momentum gradients within the droplet. The short- and long-time solutions are combined and the resulting model equation is shown to accurately cover the entire force–time profile.