Self-similar Behavior Research Articles

Self-similar solutions of the bed deformation

In this paper, a conclusion about the self-similar behavior of the bed surface evolution is made. It is based on the analysis of experimental and numerical studies of the bed surface evolution under the mechanical impact of liquid flow. The bottom wave has a form close to one period of a sinusoidal function with a time-varying wavelength and constant steepness. A method of constructing the automodel dependence of the bed surface on time and spatial coordinate in analytical form is proposed. It was shown that it is enough to select five bottom surfaces with given wavelengths from a series of shapes. Next, the mean values of shear stresses are calculated for them, and the rates of change of wavelengths are found. Then a degree of approximation of the wavelength dependence of its rate of change is determined, and, finally, the exact solution of the corresponding differential equation is obtained. Comparison with experimental data and numerical solutions shows that the solution error does not exceed a few percent and that computational time is reduced by 25–30 times.

Self-similarity of communities of the ABCD model

The Artificial Benchmark for Community Detection (ABCD) graph is a random graph model with community structure and power-law distribution for both degrees and community sizes. The model generates graphs similar to the well-known LFR model but it is faster and can be investigated analytically. In this paper, we show that the ABCD model exhibits some interesting self-similar behaviour, namely, the degree distribution of ground-truth communities is asymptotically the same as the degree distribution of the whole graph (appropriately normalized based on their sizes). As a result, we can not only estimate the number of edges induced by each community but also the number of self-loops and multi-edges generated during the process. Understanding these quantities is important as (a) rewiring self-loops and multi-edges to keep the graph simple is an expensive part of the algorithm, and (b) every rewiring causes the underlying configuration models to deviate slightly from uniform simple graphs on their corresponding degree sequences.

Internal gravity waves in flow past a bluff body under different levels of stratification

The flow field of a bluff body, a circular disk, that moves horizontally in a stratified environment is studied using large-eddy simulations. Five levels of stratification (body Froude numbers of ${{Fr}} = 0.5, 1, 1.5, 2$ and $5$ ) are simulated at Reynolds number of ${{Re}} = 5000$ and Prandtl number of $Pr =1$ . A higher ${{Re}} = 50\,000$ database at ${{Fr}} = 2, 10$ and $Pr =1$ is also examined for comparison. The wavelengths and amplitudes of steady lee waves are compared with a linear-theory analysis. Excellent agreement is found over the entire range of ${{Fr}}$ if an ‘equivalent body’ that includes the separation region is employed for the linear theory. For asymptotically large distances, the velocity amplitude varies theoretically as ${{Fr}}^{-1}$ but a correction owing to the dependence of the separation zone on ${{Fr}}$ is needed. The wake waves propagate in a narrow band of angles with the vertical, and have a wavelength that increases with increasing ${{Fr}}$ . The envelope of wake waves, demarcated using buoyancy variance, exhibits self-similar behaviour. The higher ${{Re}}$ results are consistent with the buoyancy effects exhibited at the lower ${{Re}}$ . The wake wave energy is larger at ${{Re}} = 50\,000$ . Nevertheless, independent of ${{Fr}}$ and ${{Re}}$ , the ratio of the wake wave potential energy to the wake turbulent energy increases to approximately 0.6–0.7 in the non-equilibrium stage showing their energetic importance besides suggesting universality in this statistic. There is a crossover of energetic dominance of lee waves at ${{Fr}} <2$ to wake-wave dominance at ${{Fr}} \approx 5$ .

Turbulent Noise Sources Reduction by Large-Eddy Breakup Devices and Riblets

An experimental study is presented on the application of large-eddy breakup (LEBU) on a flat plate as an outer layer source-targeting device to mitigate the wall pressure fluctuations and lateral coherence length scale of a turbulent boundary-layer developed at zero pressure gradient. Both represent the prominent noise sources for the trailing-edge noise radiation. When a LEBU is placed strategically at the outer part of a turbulent boundary-layer, the wall pressure spectra can establish a self-similar behavior against s′, which is a normalized separation distance between the LEBU’s trailing-edge and the targeted location for noise mitigation. It is found that s′ has to be greater than 3 to achieve an overall reduction in the wall pressure fluctuations. What appears to be the optimal LEBU configuration for the mitigation of wall pressure fluctuations, however, will not be reciprocated in the lateral coherence length. The combined effect would render the LEBU rather ineffective to reduce the frequency-integrated, overall noise radiation. Our previous study has confirmed that riblets, a near-wall source-targeting device, is more effective in the mitigation of the lateral coherence length. Under the principle of non-interference, a combination of riblets and LEBU demonstrates that reduction of the frequency-integrated, overall noise radiation can be achieved, again at s′>3.

Nucleation transitions in polycontextural networks toward consensus

Recently, we proposed polycontextural networks as a model of evolving systems of interacting beliefs. Here, we present an analysis of the phase transition as well as the scaling properties. The model contains interacting agents that strive for consensus, each with only subjective perception. Depending on a parameter that governs how responsive the agents are to changing their belief systems the model exhibits a phase transition that mediates between an active phase where the agents constantly change their beliefs and a frozen phase, where almost no changes appear. We observe the build-up of convention-aligned clusters only in the intermediate regime of diverging susceptibility. Here, we analyze in detail the behavior of polycontextural networks close to this transition. We provide an analytical estimate of the critical point and show that the scaling properties and the space–time structure of these clusters show self-similar behavior. Our results not only contribute to a better understanding of the emergence of consensus in systems of distributed beliefs but also show that polycontextural networks are models, motivated by social systems, where susceptibility—the sensitivity to change own beliefs—drives the growth of consensus clusters.Graphical abstract

On a Self-Similar Behavior of Logarithmic Sums

On a Self-Similar Behavior of Logarithmic Sums

Mixed bound states in the continuum: Disclosing BIC’s content via bulk normal modes

We present a nonperturbative theory of bound states in the continuum (BICs) based on the analytical theory of infinite-grating eigenmodes in the case of a planar grating waveguide, that is, 1D photonic crystal slab. We describe a mixed BIC mainly formed by a coherent mixture of two photonic-crystal normal modes in a vicinity of their avoided crossing where both of them have a nonzero coupling with a radiation-loss channel. We reveal the universal properties of the mixed BIC associated with the spatial parity symmetry breaking and self-similar spectral behavior of the bulk normal modes near Γ-point. We show that the mechanism of the mixed-BIC formation constitutes a special kind of the generic Friedrich–Wintgen mechanism. A cutoff from the radiation-loss channel and extremely high-Q narrow resonance are achieved due to the destructive interference of the two crossing normal modes. On the contrary, a conventional symmetry-protected BIC is formed mostly by a single photonic-crystal normal mode which has vanishing coupling Fourier coefficient with every available radiation-loss channel. Implementation of the mixed BICs in various photonic-crystals structures could lead to interesting applications.

Towards the understanding of convective dissolution in confined porous media: thin bead pack experiments, two-dimensional direct numerical simulations and physical models

We consider the process of convective dissolution in a homogeneous and isotropic porous medium. The flow is unstable due to the presence of a solute that induces a density difference responsible for driving the flow. The mixing dynamics is thus driven by a Rayleigh–Taylor instability at the pore scale. We investigate the flow at the scale of the pores using Hele-Shaw type experiment with bead packs, two-dimensional direct numerical simulations and physical models. Experiments and simulations have been specifically designed to mimic the same flow conditions, namely matching porosities, high Schmidt numbers and linear dependency of fluid density with solute concentration. In addition, the solid obstacles of the medium are impermeable to fluid and solute. We characterise the evolution of the flow via the mixing length, which quantifies the extension of the mixing region and grows linearly in time. The flow structure, analysed via the centreline mean wavelength, is observed to grow in agreement with theoretical predictions. Finally, we analyse the dissolution dynamics of the system, quantified through the mean scalar dissipation, and three mixing regimes are observed. Initially, the evolution is controlled by diffusion, which produces solute mixing across the initial horizontal interface. Then, when the interfacial diffusive layer is sufficiently thick, it becomes unstable, forming finger-like structures and driving the system into a convection-dominated phase. Finally, when the fingers have grown sufficiently to touch the horizontal boundaries of the domain, the mixing reduces dramatically due to the absence of fresh unmixed fluid. With the aid of simple physical models, we explain the physics of the results obtained numerically and experimentally. The solute evolution presents a self-similar behaviour, and it is controlled by different length scales in each stage of the mixing process, namely the length scale of diffusion, the pore size and the domain height.

Self-similarity in over-tripped turbulent boundary-layer flows

The scaling universality of structure functions is studied for artificially thickened turbulent boundary-layer flows in over-tripped impacts by using hot-wire measurement datasets. The self-similarity behaviours in the inner and outer regions are examined from the viewpoint of different flow mechanisms. In the inner region, the relative ratios between structure functions for the energy-containing range of scales exhibit universality behaviour, in accordance with Townsend's attached eddy hypothesis. This universality of the energy-containing range of scales extends further away from the wall by increasing the tripping intensity. On the other hand, the impact of the external intermittency on the self-similarity of small-scale turbulence is examined through the intermittent zone in over-tripped conditions. Towards the boundary-layer edge, the structure functions exhibit a growing departure from self-similarity and analytical prediction, and it is demonstrated that the departure is primarily due to external intermittency. Moreover, based on the conditional statistics concentrated in the turbulent regimes, it is revealed that the small scales in the turbulence regime are homogenized in a self-similar behaviour, which is independent of the current tripping conditions.

On the enhancement and suppression of turbulent wall pressure by the Large Eddy BreakUp devices

An experimental study is presented on the application of Large Eddy BreakUp (LEBU) on a flat plate as an aeroacoustics noise source-targeting device to perturb the wall pressure fluctuations of a turbulent boundary layer. When interacting with a LEBU wake, the wall pressure spectra can establish a self-similar behavior against s′, which is a normalized separation distance between the LEBU's trailing edge and the targeted location for aeroacoustics noise source mitigation. It is found that s′>3 is needed to achieve an overall reduction in the wall pressure fluctuations. The fundamental mechanism by which the LEBU can reduce the wall pressure fluctuation is investigated by studying the spatio-temporal evolution of turbulent spots. When the emanated LEBU wake is interacted with the outer part of turbulent spot, the shielding effect can always inhibit the turbulent fluids ejection from the spot's leading edge. However, incursion of the high momentum fluids wall-sweeping event at the spot's trailing edge can only be effectively prevented when s′>3.

Global Solutions with Asymptotic Self-Similar Behaviour for the Cubic Wave Equation

Global Solutions with Asymptotic Self-Similar Behaviour for the Cubic Wave Equation

Evolution and breakup of a ferrofluid droplet neck through a capillary tube

The controlled generation of droplets is crucial in bioassays, chemical analysis, and inkjet printing. Despite progress in droplet formation dynamics within liquid–liquid systems, improving the controllability and flexibility of manipulating droplets is still necessary. Here, we employ a non-uniform magnetic field for non-contact control of ferrofluid droplet generation in a vertical capillary. A correlation function predicts final ferrofluid droplet volumes with an average relative error of only 6.33%. Our study demonstrates that the magnetic field actively regulates droplets' volume, deformation, and generation period. In the final stages of droplet formation, we observe a unique neck rupture mechanism influenced by the viscosity of the continuous-phase fluid, distinct from observations in air. Scaling laws fit the minimum neck and position variations, revealing self-similar neck rupture behavior for droplets at different Bond and Weber numbers. These insights into droplet dynamics have potential applications across various technological domains.

Flow characteristics in a semi-confined circular-pipe impinging jet

The impinging jet is a complex heat and mass transfer technique that involves several process variables, such as the jet Reynolds number, impingement distance, and jet configuration. In this study, the flow characteristics of a semi-confined circular-pipe impinging jet over different Reynolds numbers and impingement distances were experimentally investigated using a two-dimensional particle image velocimetry technique. The confinement was achieved by positioning a plate parallel to the impinging plate at the nozzle exit. The time-averaged velocity field exhibited a recirculation structure that gradually shifted downstream with increasing Reynolds numbers or impingement distances. Notably, at H/d = 2, this downstream shift of the structure was accompanied by an increase in the vortex intensity. Moreover, the confined plate induced alterations in the overall flow pattern within the confined region, significantly reducing the wall jet decay rate compared with both unconfined and confined radial wall jets for H/d ≥ 3. Conversely, the confinement did not affect the expansion of the wall jet. Unlike the free (unconfined) impinging jets, the semi-confined circular-pipe impinging jet did not exhibit self-similar behavior in the conventional outer-scaled coordinates, particularly concerning the turbulence intensity and Reynolds shear stress. Finally, self-similarity in the time-averaged velocity and various turbulence parameters was achieved using the parameter scale proposed in this study, thereby obtaining the corresponding scaling laws in the wall jet region. Our study results can deepen the current understanding of the flow characteristics of semi-confined circular-pipe impinging jets and are significant for optimizing the performance and efficiency of compact electronic packaging equipment.

Polarization of disk electrodes in high-conductivity electrolyte solutions.

We investigate the polarization of disk electrodes immersed in an electrolyte solution and subjected to a small external AC voltage over a wide range of frequencies. A mathematical model is developed based on the Debye-Falkenhagen approximation to the coupled Poisson-Nernst-Planck equations. Analytical techniques are used for predicting the spatial distribution of the electric potential and the complex impedance of the system. Scales for impedance and frequency are identified, which lead to a self-similar behavior for a range of frequencies. Experiments are conducted with gold electrodes of sizes in the range 100-350 μm immersed in a high-conductivity KCl solution over five orders of magnitude in frequency. A collapse of data on impedance magnitude and phase angle onto universal curves is observed with scalings motivated by the mathematical model. A direct comparison with the approximate analytical formula for impedance is made without any fitting parameters, and a good agreement is found for the range of frequencies where the analytical model is valid.

Accounting for the Variability of Earthquake Rates within Low-Seismicity Regions: Application to the 2022 Aotearoa New Zealand National Seismic Hazard Model

ABSTRACT The distribution of earthquakes in time and space is seldom stationary, which could hinder a robust statistical analysis, particularly in low-seismicity regions with limited data. This work investigates the performance of stationary Poisson and spatially precise forecasts, such as smoothed seismicity models (SSMs), in terms of the available training data. Catalog bootstrap experiments are conducted to: (1) identify the number of training data necessary for SSMs to perform spatially better than the least-informative Uniform Rate Zone (URZ) models; and (2) describe the rate temporal variability accounting for the overdispersion and nonstationarity of seismicity. Formally, the strict-stationarity assumption used in traditional forecasts is relaxed into local and incremental stationarity (i.e., a catalog is only stationary in the vicinity of a given time point t) along with self-similar behavior described by a power law. The results reveal rate dispersion up to 10 times higher than predicted by Poisson models and highlight the impact of nonstationarity in assuming a constant mean rate within training-forecast intervals. The temporal rate variability is translated into a reduction of spatial precision by means of URZ models. First, counting processes are devised to capture rate distributions, considering the rate as a random variable. Second, we devise a data-driven method based on geodetic strain rate to spatially delimit the precision of URZs, assuming that strain/stress rate is related to the timescales of earthquake interactions. Finally, rate distributions are inferred from the available data within each URZ. We provide forecasts for the New Zealand National Seismic Hazard Model update, which can exhibit rates up to ten times higher in low-seismicity regions compared with SSMs. This study highlights the need to consider nonstationarity in seismicity models and underscores the importance of appropriate statistical descriptions of rate variability in probabilistic seismic hazard analysis.

DETECTION AND CHARACTERIZATION OF CUSP SINGULARITIES

Studies on nonlinear analysis of system dynamics have increased in recent years. Since most systems that exist in nature have complex dynamics and therefore exhibit nonlinear behavior; there are various methods and theories developed in this context. Self-similar functions are mathematical functions exhibiting self-similar and scale-invariant behaviors performing fractal structures. Singularities are the basis for producing the self-similar behavior of these functions. Singularity analysis is mainly carried out by using wavelet transform (WT). The cusp singularities are non-oscillating singularities which are characterized by their singularity strength. However, the representation and behavior of this type of singularity differs depending on the sign of the exponent, known as the singularity strength. The cusp singularity functions with negative exponent show irregular behavior progressively different than positively valued functions since the value of the function is undefined at that particular singular point. It is commonly accepted that the singularity strength is studied as Hölder exponent of the cusp function, but by definition, the value of this exponent cannot take negative values. We present a new method to estimate the singularity strength of cusp singularities with negative exponent. The developed method is based on analyzing and redistributing the amplitudes of a cusp function with negative exponent by taking the WT. The redistribution of amplitudes over time is achieved by applying curve fitting process to frequency values of the analyzing function.

Enhancing thermal mixing of supercritical water through a confined co-flowing planar jet

Previous studies have reported that the thermal mixing of supercritical water (SCW) would be inhibited by the density gradient in jet flow. The confined co-flowing planar jet which has one central inlet and two outer inlets is expected to enhance thermal mixing through stronger turbulent entrainment induced by double mixing layers. Direct numerical simulations (DNS) of planar jet of supercritical water (653–843 K, 25 MPa) are performed. The effects of the density ratio ρr (1.1, 3, 6) between jet and ambient fluids, the Reynolds number based on the density, velocity, diameter, and viscosity of central inlet Rein=ρinUinDin/μin (1000–4000), and the buoyancy on thermal mixing properties are investigated. We find that increasing ρr results in the decay of turbulence near the double mixing layers and the attenuation of thermal mixing. The self-similar behavior for co-flowing planar jet of supercritical water can be more likely to achieve for the mean field than for the turbulence field. While increasing Rein results in the amplification of turbulence production in the far-field region due to the vortex stretching mechanism induced by larger velocity gradient, the enhancement of thermal mixing is insignificant. The gravity wave along the normal direction leads to density stratification and inhibition of turbulent mixing near the mixing layers when Rein less than 2000. The gravity effect can be neglected when Rein greater than 2000 due to the increasing turbulence production. Finally, we find that the enhancement of thermal mixing can be achieved by increasing the turbulent intensity of outer inlets.

On self-similar patterns in coupled parabolic systems as non-equilibrium steady states.

We consider reaction-diffusion systems and other related dissipative systems on unbounded domains with the aim of showing that self-similarity, besides the well-known exact self-similar solutions, can also occur asymptotically in two different forms. For this, we study systems on the unbounded real line that have the property that their restriction to a finite domain has a Lyapunov function (and a gradient structure). In this situation, the system may reach local equilibrium on a rather fast time scale, but on unbounded domains with an infinite amount of mass or energy, it leads to a persistent mass or energy flow for all times; hence, in general, no true equilibrium is reached globally. In suitably rescaled variables, however, the solutions to the transformed system converge to so-called non-equilibrium steady states that correspond to asymptotically self-similar behavior in the original system.

Analysis of the n-Term Klein-Gordon Equations in Cantor Sets

The effectiveness of the local fractional reduced differential transformation method (LFRDTM) for the approximation of the solution related to the extended n-term local fractional Klein-Gordon equation is the main aim of this paper in which fractional complex transform and local fractional derivative have been employed to analyze the n-term Klein-Gordon equations, and Cantor sets. The proposed method, along with the existence of the solutions demonstrated through some examples, provides a powerful mathematical means in solving fractional linear differential equations. Considering these points, the paper also provides an accurate and effective method to solve complex physical systems that display fractal or self-similar behavior across various scales. In conclusion, the fractional complex transform with the local fractional differential transform method has been proven to be a robust and flexible approach towards obtaining effective approximate solutions of local fractional partial differential equations.

Self-similar evolution of wave turbulence in Gross-Pitaevskii system.

We study the universal nonstationary evolution of wave turbulence (WT) in Bose-Einstein condensates (BECs). Their temporal evolution can exhibit different kinds of self-similar behavior corresponding to a large-time asymptotic of the system or to a finite-time blowup. We identify self-similar regimes in BECs by numerically simulating the forced and unforced Gross-Pitaevskii equation(GPE) and the associated wave kinetic equation(WKE) for the direct and inverse cascades, respectively. In both the GPE and the WKE simulations for the direct cascade, we observe the first-kind self-similarity that is fully determined by energy conservation. For the inverse cascade evolution, we verify the existence of a self-similar evolution of the second kind describing self-accelerating dynamics of the spectrum leading to blowup at the zero mode (condensate) at a finite time. We believe that the universal self-similar spectra found in the present paper are as important and relevant for understanding the BEC turbulence in past and future experiments as the commonly studied stationary Kolmogorov-Zakharov (KZ) spectra.