Assumption Of Self-similarity Research Articles

Four-dimensional flow characteristics of air-entrained turbulent impinging waterjet onto quiescent water surface

This paper presents a time-resolved three-dimensional (4D) flow fields measurement of the continuous phase of a turbulent impinging jet inducing foam formation using the Lagrangian particle tracking velocimetry utilizing the Shake-The-Box algorithm. With the systems equipped with four high-speed cameras, time-series of images of fluid tracer particles were acquired. The Vortex-In-Sharp (VIC#) method was used to reconstruct the Eulerian flow fields of the particle tracks. The impinging jet was characterized as plume-like along the vertical direction with two distinct layers: developing shear and fully developed shear. The streamwise vortex structures of the continuous phase were influenced by the bubble plume motion, and the results showed high amplitude oscillations of the acceleration and deceleration near the jet source resulting in the formation of ring-like vortices, which break down as the jet moves downstream with its momentum dissipated. The flow of the continuous phase of impinging jet was self-similar both at the developed shear layer and the fully developed diffusion layer beneath the water pool and is characterized as homogeneous shear flow with anisotropy turbulence. The classical assumption of self-similarity with Gaussian profiles for continuous phase velocity is verified experimentally. We found that the results show a huge potential of blue energy harvesting from the low frequency (∼2 Hz) dissipating kinetic energy of the turbulent plume-like jet underneath the impinging water surface using triboelectric nanogenerator.

A general solution for the water entry of an arbitrary curved wedge

ABSTRACT The present study proposes a general solution for the water entry of an arbitrary curved wedge with a constant speed. The solution is based on the assumption that the impact flow is an incompressible potential flow with negligible effects of gravity and surface tension. The slamming and transition stages of the water entry of the curved wedge are addressed theoretically. For the slamming stage, pressure and hydrodynamic force models are derived from a quasi self-similar assumption based on the wetted length and a similarity solution of the water entry of linear wedges. The solution can be applied to the water entry of linear and curved wedges, where the curvature effects are absorbed into the wetted length and the derivative of the wetted length with respect to the penetration depth. For the transition stage, where a fixed separation point of flow detachment is located at the knuckle of the wedge, a hydrodynamic force model is proposed by extending the transition stage model of the water entry of linear wedges to that of curved wedges. The present solution can quickly predict the pressure distributions and hydrodynamic forces for the water entry of curved wedges. The predictions are in good agreement with the experimental and numerical results for the slamming and transition stages.

Transformational faulting in Mn2GeO4 from olivine to wadsleyite structure: Implications for physical mechanism of deep-focus earthquakes

High-pressure and temperature deformation experiments interfaced with acoustic emission (AE) monitoring have been conducted to study transformational faulting in Mn2GeO4 olivine, which transforms to the β phase, isostructural to wadsleyite. Metastable Mn2GeO4 olivine exhibits a marked embrittlement behavior at temperatures between 800 and 1100 K, emitting numerous AEs. At each temperature, brittle deformation is characterized by a two-stage process: (1) a “preparation” stage with numerous diffusedly located low-magnitude AEs and large b values (>2), and (2) a failure stage where larger-magnitude AEs form a planar distribution with b values about 1. Microstructure analysis reveals extensive kink band development in olivine grains in the recovered samples. Kink band boundaries (KBBs), with a typical thickness of ∼100 nm, are filled with a nanometric β-Mn2GeO4 “gouge”. A dense array of secondary shear localizations is often present within the kink bands, suggesting significant shear deformation therein. The combined observations suggest that faulting in metastable Mn2GeO4 olivine is a self-similar process, from grain-scale to the sample-scale. Both observed embrittlement behavior and the microstructure of metastable Mn2GeO4 olivine are essentially identical to those in Mg2GeO4 olivine we have reported previously, indicating that the physical mechanism of faulting in metastable olivine is insensitive to the specific crystallographic structure of the high-pressure phase. The low b values (about 1) observed in the faulting process in our experiments are similar to those of deep focus earthquakes in cold subduction zones. Our observed mechanism explains deep focus seismicity in cold metastable mantle wedges, provided that the self-similarity assumption holds to geological scales.

A self-similar model of galaxy formation and dark halo relaxation

We develop a spherical self-similar model for the formation of a galaxy through gas collapsing in an isolated self-gravitating dark matter halo. As is well known, the self-similarity assumption makes the problem eminently tractable by reducing it to a system of ordinary differential equations. We improve upon the existing literature on self-similar collapse in two ways. First, we include the effects of radiative cooling and the formation of a pseudo-disk at the center of collapse, in a parametrised manner. More importantly, we solve for the evolution of gas and dark matter simultaneously and self-consistently using a novel iterative approach. As a result, our model produces shell trajectories of both gas and dark matter that qualitatively agree with the results of full hydrodynamical simulations of self-gravitating systems. We discuss the impact of various ingredients such as the accretion rate, gas equation of state, disk radius and cooling rate amplitude on the evolution of the gas shells, although we leave the inclusion of stellar and black hole activity to future work. The self-consistent evolution of gas and dark matter allows us to study the response (or `quasi-adiabatic relaxation') of the dark matter trajectories to the presence of collapsing gas, an effect that has gained increasing importance recently in the context of precision estimates of small-scale statistics like the matter power spectrum. Our default configuration produces a relaxation relation in qualitative agreement with that seen in cosmological hydrodynamical simulations, and further allows us to easily study the impact of the model ingredients mentioned above. As an initial application, we vary one ingredient at a time and find that the accretion rate and gas equation of state have the largest impact on the relaxation relation, while the cooling amplitude plays only a minor role. Our model thus provides a convenient framework to rapidly explore the coupled nonlinear impact of multiple astrophysical processes on the mass and velocity profiles of dark matter in galactic halos, and consequently on observables such as rotation curves and gravitational lensing signals.

Subduction Interface Earthquake Rise-Time Scaling Relations

ABSTRACT The slip duration in a fault plane, also known as the rise time (Tr), is determined in finite-fault rupture models (FFRMs) through the analysis of seismic source inversions using strong ground-motion (SGM) records and teleseismic data. For subduction interface earthquakes (megathrust), models exist that provide estimates for Tr values. The finite-source rupture model database and National Earthquake Information Center databases include FFRMs that allow for the extension of source-scaling relations. Currently, Tr versus seismic moment (M0) scaling relations specifically derived for large megathrust earthquakes in the near-source region are scarce. The relationship between stress drop and M0 is not straightforward; therefore, the logarithmic distribution of stress drop among earthquakes of different magnitudes (Mw) appears to be constant or self-similar. This self-similarity refers to a symmetry of the time-dependent fields, which remain unchanged under certain scale transformations in space and time characterized by similarity exponents and a function of the scaled variable, called the scaling function. In this study, Tr scaling has been conducted using 45 FFRMs derived from large megathrust earthquakes (Mw≥7.3) obtained from the previously mentioned databases. The scaling relation derived from the FFRMs based on SGM records closely approximates log(Tr)=const+1/3log(M0), which agrees with the self-similarity assumption for earthquake ruptures. On the other hand, the scaling relation obtained from the teleseismic dataset exhibits a smaller slope, indicating that the teleseismic data may overestimate source time characteristics compared with SGM data from seismic stations located close to the source.

Impact of Galaxy Clusters on the Propagation of Ultrahigh-energy Cosmic Rays

Galaxy clusters are the largest objects in the Universe kept together by gravity. Most of their baryonic content is made of a magnetized diffuse plasma. We investigate the impact of such a magnetized environment on the propagation of ultrahigh-energy cosmic rays (UHECRs). The intracluster medium (ICM) is described according to the self-similar assumption, in which gas density and pressure profiles are fully determined by the cluster mass and redshift. The magnetic field is scaled to the thermal components of the ICM under different assumptions. We model the propagation of UHECRs in the ICM using a modified version of the Monte Carlo code SimProp, where hadronic processes and diffusion in the turbulent magnetic field are implemented. We provide a universal parameterization that approximates the UHECR fluxes escaping from the environment as a function of the most relevant quantities, such as the mass of the cluster, the position of the source with respect to the center of the cluster, and the nature of the accelerated particles. We show that galaxy clusters are an opaque environment, especially for UHECR nuclei. The role of the most massive nearby clusters in the context of the emerging UHECR astronomy is finally discussed.

Image inpainting exploiting global prior refined weighted low-rank representation

Low-rank and nonlocal self-similarity are important priors for image restoration. Based on nonlocal self-similarity assumption, similar patches are grouped into a matrix that is of low-rank or approximately low-rank. Traditional algorithms model low-rank prior of local group matrices (intra-group) and ignore the global inter-group relationship, which will result in suboptimal performance without considering global consistency. Due to overlapping of patches, groups are dependent, however, conventional methods process each group independently. To make full use of the intrinsic relationship among groups, we consider all groups simultaneously and propose a global prior refined weighted low-rank representation model for image inpainting. The local-groups and the aggregation of all groups are taken into account in the proposed model. By introducing auxiliary variable, we decouple the local–global model and optimize iteratively using the alternating direction method of multipliers. The proposed global refinement model significantly improves the performance of local low-rank model. A new strategy for grouping similar patches is also proposed by dividing the neighborhood of a target patch into multiple subregions and searching for a given number of similar patches within each subregion. The proposed method is applied to various tasks (line inpainting, destriping, and depth map completion) and obtains promising performance.

Self-similar gravitational dynamics, singularities and criticality in 2D

We initiate a systematic study of continuously self-similar (CSS) gravitational dynamics in two dimensions, motivated by critical phenomena observed in higher dimensional gravitational theories. We consider CSS spacetimes admitting a homothetic Killing vector (HKV) field. For a general two-dimensional gravitational theory coupled to a dilaton field and Maxwell field, we find that the assumption of continuous self-similarity determines the form of the dilaton coupling to the curvature. Certain limits produce two important classes of models, one of which is closely related to two-dimensional target space string theory and the other being Liouville gravity. The gauge field is shown to produce a shift in the dilaton potential strength. We consider static black hole solutions and find spacetimes with uncommon asymptotic behaviour. We show the vacuum self-similar spacetimes to be special limits of the static solutions. We add matter fields consistent with self-similarity (including a certain model of semi-classical gravity) and write down the autonomous ordinary differential equations governing the gravitational dynamics. Based on the phenomenon of finite-time blow-up in ODEs, we argue that spacetime singularities are generic in our models. We present qualitatively diverse results from analytical and numerical investigations regarding matter field collapse and singularities. We find interesting hints of a Choptuik-like scaling law.

Rainy downdrafts in abyssal atmospheres

Context.Results from Juno’s Microwave Radiometer (MWR) indicate nonuniform mixing of ammonia vapor in Jupiter’s atmosphere down to tens of bars, far beneath the cloud level. Helioseismic observations suggest solar convection may require narrow, concentrated downdrafts called entropy rain to accommodate the Sun’s luminosity. Both observations suggest some mechanism of nonlocal convec-tive transport.Aims.We seek to predict the depth that a concentrated density anomaly can reach before efficiently mixing with its environment in bottomless atmospheres.Methods.We modified classic self-similar analytical models of entraining thermals to account for the compressibility of an abyssal atmosphere. We compared these models to the output of high-resolution three-dimensional fluid dynamical simulations to more accurately model the chaotic influence of turbulence.Results.We find that localized density anomalies propagate down to ~3−8 times their initial size without substantially mixing with their environment. Our analytic model accurately predicts the initial flow, but the self-similarity assumption breaks down after the flow becomes unstable at a characteristic penetration depth.Conclusions.In the context of Jupiter, our findings suggest that precipitation concentrated into localized downdrafts of size ~20 km can coherently penetrate to on the order of a hundred kilometers (tens of bars) beneath its initial vaporization level without mixing with its environment. This finding is consistent with expected convective storm length scales and Juno MWR measurements of ammonia depletion. In the context of the Sun, we find that turbulent downdrafts in abyssal atmospheres cannot maintain their coherence through the Sun’s convective layer, a potential challenge for the entropy rain hypothesis.

The Three Hundred Project: the evolution of physical baryon profiles

ABSTRACT The distribution of baryons provides a significant way to understand the formation of galaxy clusters by revealing the details of its internal structure and changes over time. In this paper, we present theoretical studies on the scaled profiles of physical properties associated with the baryonic components, including gas density, temperature, metallicity, pressure and entropy as well as stellar mass, metallicity and satellite galaxy number density in galaxy clusters from z = 4 to z = 0 by tracking their progenitors. These mass-complete simulated galaxy clusters are coming from The Three Hundred with two runs: Gizmo-SIMBA and Gadget-X. Through comparisons between the two simulations, and with observed profiles that are generally available at low redshift, we find that (1) the agreements between the two runs and observations are mostly at outer radii r ≳ 0.3r500, in line with the self-similarity assumption. While Gadget-X shows better agreements with the observed gas profiles in the central regions compared to Gizmo-SIMBA; (2) the evolution trends are generally consistent between the two simulations with slightly better consistency at outer radii. In detail, the gas density profile shows less discrepancy than the temperature and entropy profiles at high redshift. The differences in the cluster centre and gas properties imply different behaviours of the AGN models between Gadget-X and Gizmo-SIMBA, with the latter, maybe too strong for this cluster simulation. The high-redshift difference may be caused by the star formation and feedback models or hydrodynamics treatment, which requires observation constraints and understanding.

Short-Time Asymptotics for Non-Self-Similar Stochastic Volatility Models

We provide a short-time large deviation principle (LDP) for stochastic volatility models, where the volatility is expressed as a function of a Volterra process. This LDP does not require strict self-similarity assumptions on the Volterra process. For this reason, we are able to apply such an LDP to two notable examples of non-self-similar rough volatility models: models where the volatility is given as a function of a log-modulated fractional Brownian motion (Bayer, C., F. Harang, and P. Pigato. 2021. “Log-Modulated Rough Stochastic Volatility Models.” SIAM Journal on Financial Mathematics 12 (3): 1257–1284), and models where it is given as a function of a fractional Ornstein–Uhlenbeck (fOU) process (Gatheral, J., T. Jaisson, and M. Rosenbaum. 2018. “Volatility is Rough.” Quantitative Finance 18 (6): 933–949). In both cases, we derive consequences for short-maturity European option prices implied volatility surfaces and implied volatility skew. In the fOU case, we also discuss moderate deviations pricing and simulation results.

Statically Admissible Stress Fields in Conical Sand Valleys and Heaps: A Validation of Haar–von Kármán Hypothesis

The usage of the Haar–von Kármán hypothesis for circumferential stress in axisymmetric problems is still questionable. This study aims to reveal statically admissible stress fields in conical sand valleys and heaps by applying the Haar–von Kármán hypothesis to rigid-plastic material obeying the Mohr–Coulomb criterion. The original boundary-value problem reduces to a pair of ordinary differential equations by employing the self-similarity assumption in conjunction with the condition of incipient failure everywhere. The pair of governing equations are numerically integrated with the fifth-order adaptive Runge–Kutta method. The shooting method is employed to numerically convert the boundary-value problem to the initial-value problem. Moreover, the singularities that occurred in the integrations are coped with by numerical perturbation of the initial boundary conditions. A line of stress discontinuity together with the stress jump condition is imposed to link the stress solution between the rigid and plastic regions. Admissible stress fields in various axisymmetric geometries are successfully achieved whereby the load transfer mechanisms in the conical sand heaps and valleys are elucidated. The arch action that occurred in bulk solid is observed. The effect of the circumferential stress hypotheses, that is, Haar–von Kármán hypothesis and Lévy’s flow rule, on the results of rigid-plastic solution is demonstrated; therefore, the advantages and disadvantages of both the hypotheses are elucidated. This study rigorously justifies the validity of the Haar–von Kármán hypothesis, thus a guideline on determining the value of hoop stress using the Haar–von Kármán hypothesis is proposed.

Keynote lecture. Forecasting the inundation of postfire debris flows

In the semi-arid regions of the western United States, postfire debris flows are typically runoff generated. The U.S. Geological Survey has been studying the mechanisms of postfire debris-flow initiation for multiple decades to generate operational models for forecasting the timing, location, and magnitude of postfire debris flows. Here we discuss challenges and progress for extending operational capabilities to include modeling postfire debris-flow inundation extent. Analysis of volume and impacted area scaling relationships indicated that postfire debris flows do not conform to assumptions of geometric self-similarity. We documented sensitivity of impacted areas to rainfall intensity using a candidate methodology for generating inundation hazard assessments. Our results emphasize the importance of direct measurements of debris-flow volume, inundated area, and high temporal resolution rainfall intensity.

Application of surrogate models to stability analysis and transition prediction in hypersonic flows

To increase the efficiency and robustness of stability-based transition prediction in flow simulations, simplified methods are introduced to substitute direct stability analyses for rapid disturbance growth prediction. For low-speed boundary layers, these methods are mainly established based on self-similar assumptions, which are not applicable to non-similar boundary layers in hypersonic flows. The objective of this article is to investigate the application of surrogate models to stability analysis of non-similar flows over blunt cones, focused on parameterization of boundary-layer (BL) profiles. Firstly, correlations between BL edge and profile parameters are analyzed, along with self-similar flow parameters and discrete points on BL profiles, which present four groups of BL characteristic parameters. Secondly, using these parameters as inputs, surrogate models are built for disturbance growth prediction over an MF-1 blunt cone. Results show that, surrogate models using four BL edge parameters and a BL shape factor {Ue, Te, ρe, ηe, H12} for stability analysis can achieve comparable accuracy with those using 16 discrete BL profile parameters, which are more precise than those using merely self-similar parameters or BL edge parameters. Thirdly, the established surrogate models are validated by stability analysis and transition prediction over the MF-1 blunt cone in flight experiments at the instants of t = 17 s ~ 22 s. Compared with direct linear stability analyses, the mean relative error of predicted disturbance growth rates by surrogate models is 8.0% and the maximum relative error of N factor envelopes is 6.6%, which indicates feasible applications of surrogate models to stability analysis and transition prediction of non-similar boundary layers in hypersonic flows.

Stochastic continuum-armed bandits with additive models: Minimax regrets and adaptive algorithm

We consider d-dimensional stochastic continuum-armed bandits with the expected reward function being additive β-Hölder with sparsity s for 0<β<∞ and 1≤s≤d. The rate of convergence O˜(s·T β+12β+1) for the minimax regret is established where T is the number of rounds. In particular, the minimax regret does not depend on d and is linear in s. A novel algorithm is proposed and is shown to be rate-optimal, up to a logarithmic factor of T. The problem of adaptivity is also studied. A lower bound on the cost of adaptation to the smoothness is obtained and the result implies that adaptation for free is impossible in general without further structural assumptions. We then consider adaptive additive SCAB under an additional self-similarity assumption. An adaptive procedure is constructed and is shown to simultaneously achieve the minimax regret for a range of smoothness levels.

Experimental study of turbulent bubbly jet. Part 1. Simultaneous measurement of three-dimensional velocity fields of bubbles and water

This study proposes a method for simultaneous measurements of time-resolved three-dimensional velocity fields of the dispersed and continuous phases of a turbulent bubbly jet at a low void fraction using Lagrangian particle tracking (LPT) velocimetry with the Shake-The-Box algorithm. Four high-speed cameras are used to acquire time series of images that include both bubbles and fluid tracer particles. Bubbles are firstly tracked using intensity differences between tracer particles and bubbles, then the bubble images are removed from the camera images and all tracer particles are tracked using the residual images. Subsequently, FlowFit interpolation is applied to the LPT results obtained by phase separation to investigate flow characteristics of a bubbly jet. The bubbly jet was divided into two regions along the vertical direction: jet-like and plume-like regions. Streamwise vortex structures of continuous phase were generated mainly by the rising bubbles. The Gaussian and top-hat velocity profiles matched well with the ensemble-averaged fluid and bubble velocities, respectively. The measured slip velocity in the radial direction was not constant but linearly increased. The classical assumption of self-similarity with Gaussian profiles for fluid velocity and bubble concentration is experimentally verified. The fluid volume flux and entrainment coefficient are obtained as a function of the slip velocity, void fraction, plume and bubble width based on three-dimensional measurements. We found that the classical integral theory agrees well with experiments in the plume region.

Estimating the Fractal Dimensions of Vascular Networks and Other Branching Structures: Some Words of Caution

Branching patterns are ubiquitous in nature; consequently, over the years many researchers have tried to characterize the complexity of their structures. Due to their hierarchical nature and resemblance to fractal trees, they are often thought to have fractal properties; however, their non-homogeneity (i.e., lack of strict self-similarity) is often ignored. In this paper we review and examine the use of the box-counting and sandbox methods to estimate the fractal dimensions of branching structures. We highlight the fact that these methods rely on an assumption of self-similarity that is not present in branching structures due to their non-homogeneous nature. Looking at the local slopes of the log–log plots used by these methods reveals the problems caused by the non-homogeneity. Finally, we examine the role of the canopies (endpoints or limit points) of branching structures in the estimation of their fractal dimensions.

Smooth Transonic Flows Around Cones

&lt;p style='text-indent:20px;'&gt;We consider a conical body facing a supersonic stream of air at a uniform velocity. When the opening angle of the obstacle cone is small, the conical shock wave is attached to the vertex. Under the assumption of self-similarity for irrotational motions, the Euler system is transformed into the nonlinear ODE system. We reformulate the problem in a non-dimensional form and analyze the corresponding ODE system. The initial data is given on the obstacle cone and the solution is integrated until the Rankine-Hugoniot condition is satisfied on the shock cone. By applying the fundamental theory of ODE systems and technical estimates, we construct supersonic solutions and also show that no matter how small the opening angle is, a smooth transonic solution always exists as long as the speed of the incoming flow is suitably chosen for this given angle.&lt;/p&gt;

Deep Autoencoder for Hyperspectral Unmixing via Global-Local Smoothing

Hyperspectral unmixing is to decompose the mixed pixels into pure spectral signatures (endmembers) and their proportions (abundances). Recently, deep learning-based methods have been applied to enhance the representation ability of unmixing models by extracting joint spatial–spectral characteristics of the hyperspectral data. However, most deep learning based-unmixing methods usually conduct global smoothing by convolutions on the whole hyperspectral imagery, which may ignore the variations within the imagery and result in oversmoothing. In this article, we propose a deep network for hyperspectral unmixing based on a new global–local smoothing autoencoder (GLA). GLA is an unsupervised model, which aims at exploring the local homogeneity and the global self-similarity of hyperspectral imagery. The proposed GLA network mainly includes two modules: a Local Continuous conditional random field Smoothing (LCS) module and a global recurrent smoothing (GRS) module. In LCS, we propose a conditional random field-based smoothing strategy to describe the joint spatial–spectral information within a local homogeneity region, which also reduces the risk of abundance maps boundary blurry. In GRS, we follow the self-similarity assumption for hyperspectral imagery and develop a recurrent neural network structure to exploit potential long-distance dependency relationships among pixels. The GLA is compared with several state-of-the-art unmixing methods on both real and synthetic data, and the abundance estimation results indicate that our method is promising. We will publish the code of GLA if this article has the honor to be accepted.

Critical behavior for impact fragmentation of spherical solid bodies sensitive to strain rate

We consider the impact fragmentation of two spherical solid bodies sensitive to the strain rate in a three-dimensional setting. We use both dimensional analysis and numerical simulations by the smoothed-particle hydrodynamics (SPH) method to shed light on this problem. The key point of the work is the assumption of complete self-similarity of the problem under consideration with respect to the effective strain rate parameter , which is verified by numerical simulations. As a result, we consider the two cases corresponding to high-velocity and low-velocity loading. The size of the system may be characterized by the total number of SPH particles N tot approximating each sphere. It is shown that for the finite system the critical velocity of fragmentation at high-velocity loading exceeds that at low-velocity loading, i.e. V c∞ > V c0. With an unlimited increase in the system size, these velocities become the same. It is shown that V c∞ and V c0 depend on the system size in a quadratic manner, i.e. where ν is a correlation length exponent.