Self-similar Behavior Research Articles

Photonic band gap spectra in Octonacci metamaterial quasicrystals

In this work we study theoretically the photonic band gap spectra for a one-dimensional quasicrystal made up of SiO2 (layer A) and a metamaterial (layer B) organized following the Octonacci sequence, where its nth-stage Sn is given by the inflation rule Sn=Sn−1Sn−2Sn−1 for n≥3, with initial conditions S1=A and S2=B. The metamaterial is characterized by a frequency dependent electric permittivity ε(ω) and magnetic permeability μ(ω). The polariton dispersion relation is obtained analytically by employing a theoretical calculation based on a transfer-matrix approach. A quantitative analysis of the spectra is then discussed, stressing the distribution of the allowed photonic band widths for high generations of the Octonacci structure, which depict a self-similar scaling property behavior, with a power law depending on the common in-plane wavevector kx.

Experimental characterization of three-dimensional flow vortical structures in submerged hydraulic jumps

The characteristics of three-dimensional vortical flow structures in submerged hydraulic jumps (generated downstream of a sluice gate) are analyzed in this paper. Results of a careful experimental investigation of the mean flow as well as turbulence statistics obtained with the use of Acoustic Doppler Velocimetry (ADV) and Particle Tracking Velocimetry (PTV) are presented and discussed. Experiments encompass incident Froude numbers (Fr1) of 3, 4 and 5, and submergence factors (S) ranging from 0.18 to 1.04. First, distributions in three vertical planes of values of the stream-wise velocity component and turbulent kinetic energy (TKE) are shown. With this information, the influence of Fr1 and S on turbulence statistics is assessed. For the first time to the best of our knowledge, an approximately self-similar behavior of the profiles of TKE in the fully-developed region of the submerged jump is reported herein. A “climb effect” in the TKE is identified, very much in agreement with what occurs with the velocities. Then, the analysis focuses on the zone located near the gate, aiming at explaining the physical mechanisms which are responsible for the vorticity of vertical axis in that area. A relationship between the length scale of the roller of horizontal axis (Lrsj) and that of the vortices of vertical axis (Lv) is introduced, revealing that the dynamics of the two vortices are strongly linked. Compelling evidence sustaining that vorticity of vertical axis appears as a consequence of boundary-layer separation associated with the lateral walls is also provided. Finally, the anisotropy of turbulence in the outskirts of the vertical vortices is addressed, finding that such anisotropy is a consequence rather a cause of the presence of the vortices.

Octonacci photonic crystals with negative refraction index materials

We investigate the optical transmission spectra for s-polarized (TE) and p-polarized (TM) waves in one-dimensional photonic quasicrystals on a quasiperiodic multilayer structure made up by alternate layers of SiO2 and metamaterials, organized by following the Octonacci sequence. Maxwell's equations and the transfer-matrix technique are used to derive the transmission spectra for the propagation of normally and obliquely incident optical fields. We assume Drude-Lorentz-type dispersive response for the dielectric permittivity and magnetic permeability of the metamaterials. For normally incident waves, we observe that the spectra does not have self-similar behavior or mirror symmetry and it also features the absence of optical band gap. Also for normally incident waves, we show regions of full transmittance when the incident angle θC = 0° in a particular frequency range.

A self-similar behavior for the relative viscosity of concentrated suspensions of rigid spheroids

A viscosity model for suspensions of rigid particles with predictive capability over a wide range of particle volume fraction and shear conditions is of interest to quantify the transport of suspensions in fluid flow models. We study the shear viscosity of suspensions and focus on the effect of particle aspect ratio and shear conditions on the rheological behavior of suspensions of rigid bi-axially symmetric ellipsoids (spheroids). We propose a framework that forms the basis to microscopically parameterize the evolution of the suspension microstructures and its effect on the shear viscosity of suspensions. We find that two state variables, the intrinsic viscosity in concentrated limit and the self-crowding factor, control the state of dispersion of the suspension. A combination of these two variables is shown to be invariant with the imposed shear stress (or shear rate) and depends only on the particle aspect ratio. This self-similar behavior, tested against available experimental and numerical data, allows us to derive a predictive model for the relative viscosity of concentrated suspensions of spheroids subjected to low (near zero) strain rates. At higher imposed strain rates, one needs to constrain one of the state variables independently to constrain the state of dispersion of the suspension and its shear dynamic viscosity. Alternatively, the obtained self-similar behavior provides the means to estimate the state variables from the viscosity measurements made in the laboratory, and to relate them to microstructure rearrangements and evolution occurring during deformation.

Self-similarity of heat transfer characteristics in laminar submerged and free-surface slot jet impingement

Laminar slot jet impingement of free-surface and submerged jets is investigated numerically regarding self-similar behavior in local heat transfer. Two different scaling behaviors, one for the stagnation region and one for the fully developed wall-jet region are shown, and the validity of the scaling is confirmed by fully resolved numerical simulations. Within the stagnation region, self-similarity is obtained if the relaxation process of the velocity profile is considered, which is described by the dimensionless nozzle-to-plate distance h/W·ReW as introduced in Rohlfs et al. (2014) for circular jets. Further, it is shown that the Prandtl number dependency can be well described by an analytical relationship obtained by solving the classical similarity solution of a viscous flow near stagnation point. For the wall-jet region, we first derive the scaling behavior based on the work of Watson (1964) for circular free surface jets. This analysis leads to an additional stretching of the spatial coordinate in the wall-tangential direction by the Reynolds number, e.g. x∗=x/(W·ReW).Finally, correlations for the stagnation region heat transfer are introduced for free-surface and submerged slot jets, dependent on the dimensionless nozzle-to-plate distance and constant heat flux/wall temperature, valid over a wide range of laminar flow conditions (i.e., 0.07⩽Pr≤1307,100⩽ReW⩽1000 and 4⩽h/W⩽20). Thereby, analytic Prandtl dependencies for stagnation point flows are employed and corresponding correlations for impinging slot jets are presented, valid in a very broad range (0.0005⩽Pr⩽4500).

Numerical solution of three-dimensional rectangular submerged jets with the evidence of the undisturbed region of flow

ABSTRACTThe evolution of turbulent rectangular submerged free jets has been investigated numerically with a two-dimensional (2D) approach by the present authors and, by using the large eddy simulations (LES) at several Reynolds numbers. The average numerical results confirmed the presence of the undisturbed region of flow (URF) located between the slot exit and the beginning of the potential core region (PCR) previously observed experimentally at the University of Rome “Tor Vergata” by Gori and coworkers. The 2D study of the present authors carried out under the conditions previously investigated in the literature, showed that the URF has a self-similar behavior, and proposed a new law for the evolution of the momentum. The present paper extends the LES to three-dimensional (3D) rectangular submerged free jets, in the range from Re = 5,000 to Re = 40,000, showing that the self-similar behavior of URF is also present in the 3D numerical simulations, as well as in the PCR and in the fully developed region (FDR).

Multiscale analysis of the topological invariants in the logarithmic region of turbulent channels at a friction Reynolds number of 932

The invariants of the velocity gradient tensor,$R$and$Q$, and their enstrophy and strain components are studied in the logarithmic layer of an incompressible turbulent channel flow. The velocities are filtered in the three spatial directions and the results are analysed at different scales. We show that the$R$–$Q$plane does not capture the changes undergone by the flow as the filter width increases, and that the enstrophy/enstrophy-production and strain/strain-production planes represent better choices. We also show that the conditional mean trajectories may differ significantly from the instantaneous behaviour of the flow since they are the result of an averaging process where the mean is 3–5 times smaller than the corresponding standard deviation. The orbital periods in the$R$–$Q$plane are shown to be independent of the intensity of the events, and of the same order of magnitude as those in the enstrophy/enstrophy-production and strain/strain-production planes. Our final goal is to test whether the dynamics of the flow is self-similar in the inertial range, and the answer turns out to be that it is not. The mean shear is found to be responsible for the absence of self-similarity and progressively controls the dynamics of the eddies observed as the filter width increases. However, a self-similar behaviour emerges when the calculations are repeated for the fluctuating velocity gradient tensor. Finally, the turbulent cascade in terms of vortex stretching is considered by computing the alignment of the vorticity at a given scale with the strain at a different one. These results generally support a non-negligible role of the phenomenological energy-cascade model formulated in terms of vortex stretching.

Self-similar aftershock rates.

In many important systems exhibiting crackling noise-an intermittent avalanchelike relaxation response with power-law and, thus, self-similar distributed event sizes-the "laws" for the rate of activity after large events are not consistent with the overall self-similar behavior expected on theoretical grounds. This is particularly true for the case of seismicity, and a satisfying solution to this paradox has remained outstanding. Here, we propose a generalized description of the aftershock rates which is both self-similar and consistent with all other known self-similar features. Comparing our theoretical predictions with high-resolution earthquake data from Southern California we find excellent agreement, providing particularly clear evidence for a unified description of aftershocks and foreshocks. This may offer an improved framework for time-dependent seismic hazard assessment and earthquake forecasting.

Investigation of the Influence of Convective Mach Number on Compressible Plane Jet Exhausting into Parallel Streams

The effects of convective Mach number on plane jets with parallel co-flow streams are investigated in the present study. Two-dimensional viscous compressible plane jet issuing to co-flowing parallel streams is numerically simulated using higher order spatial and temporal integration schemes. To remove mesh induced non-uniformities explicit tridiagonal spatial filter is applied. The mean flow field and turbulence statistics are captured from the simulated results using time-average over streamwise direction. The captured mean velocity profiles in appropriate non-dimensional form exhibit self-similar behaviour and do not change considerably with convective Mach number. The streamwise mean excess velocity profiles collapse very well with previous experimental data. The turbulent intensity profiles and Reynolds shear stress profiles do not show self-similar characteristic and increase slowly in farther downstream. The two-dimensional simulation is found capable of capturing correct jet spreading and decay.

Vorticity and mixing in Rayleigh–Taylor Boussinesq turbulence

The Rayleigh–Taylor instability induced turbulence is studied under the Boussinesq approximation focusing on vorticity and mixing. A direct numerical simulation has been carried out with an auto-adaptive multidomain Chebyshev–Fourier–Fourier numerical method. The spatial resolution is increased up to$(24\times 40)\times 940^{2}=848\,M$collocation points. The Taylor Reynolds number is$\mathit{Re}_{\unicode[STIX]{x1D706}_{zz}}\approx 142$and a short inertial range is observed. The nonlinear growth rate of the turbulent mixing layer is found to be close to$\unicode[STIX]{x1D6FC}_{b}=0.021$. Our conclusions may be summarized as follows.(i) The simulation data are in agreement with the scalings for the pressure ($k^{-7/3}$) and the vertical mass flux ($k^{-7/3}$).(ii) Mean quantities have a self-similar behaviour, but some inhomogeneity is still present. For higher-order quantities the self-similar regime is not fully achieved.(iii) In the self-similar regime, the mean dissipation rate and the enstrophy behave as$\langle \overline{\unicode[STIX]{x1D700}}\rangle \propto t$and$\langle \overline{\unicode[STIX]{x1D714}_{i}\,\unicode[STIX]{x1D714}_{i}}^{1/2}\rangle \propto t^{1/2}$, respectively.(iv) The large-scale velocity fluctuation probability density function (PDF) is Gaussian, while vorticity and dissipation PDFs show large departures from Gaussianity.(v) The pressure PDF exhibits strong departures from Gaussianity and is skewed. This is related to vortex coherent structures.(vi) The intermediate scales of the mixing are isotropic, while small scales remain anisotropic. This leaves open the possibility of a small-scale buoyancy. Velocity intermediate scales are also isotropic, while small scales remain anisotropic. Mixing and dynamics are therefore consistent.(vii) Properties and behaviours of vorticity and enstrophy are detailed. In particular, equations for these quantities are written down under the Boussinesq approximation.(viii) The concentration PDF is quasi-Gaussian. The vertical concentration gradient is both non-Gaussian and strongly skewed.

Analysis of concentric and eccentric contractions in biceps brachii muscles using surface electromyography signals and multifractal analysis.

Muscle contractions can be categorized into isometric, isotonic (concentric and eccentric) and isokinetic contractions. The eccentric contractions are very effective for promoting muscle hypertrophy and produce larger forces when compared to the concentric or isometric contractions. Surface electromyography signals are widely used for analyzing muscle activities. These signals are nonstationary, nonlinear and exhibit self-similar multifractal behavior. The research on surface electromyography signals using multifractal analysis is not well established for concentric and eccentric contractions. In this study, an attempt has been made to analyze the concentric and eccentric contractions associated with biceps brachii muscles using surface electromyography signals and multifractal detrended moving average algorithm. Surface electromyography signals were recorded from 20 healthy individuals while performing a single curl exercise. The preprocessed signals were divided into concentric and eccentric cycles and in turn divided into phases based on range of motion: lower (0°-90°) and upper (>90°). The segments of surface electromyography signal were subjected to multifractal detrended moving average algorithm, and multifractal features such as strength of multifractality, peak exponent value, maximum exponent and exponent index were extracted in addition to conventional linear features such as root mean square and median frequency. The results show that surface electromyography signals exhibit multifractal behavior in both concentric and eccentric cycles. The mean strength of multifractality increased by 15% in eccentric contraction compared to concentric contraction. The lowest and highest exponent index values are observed in the upper concentric and lower eccentric contractions, respectively. The multifractal features are observed to be helpful in differentiating surface electromyography signals along the range of motion as compared to root mean square and median frequency. It appears that these multifractal features extracted from the concentric and eccentric contractions can be useful in the assessment of surface electromyography signals in sports medicine and training and also in rehabilitation programs.

On the scale dependence of earthquake stress drop

We discuss the debated issue of scale dependence in earthquake source mechanics with the goal of providing supporting evidence to foster the adoption of a coherent interpretative framework. We examine the heterogeneous distribution of source and constitutive parameters during individual ruptures and their scaling with earthquake size. We discuss evidence that slip, slip-weakening distance and breakdown work scale with seismic moment and are interpreted as scale dependent parameters. We integrate our estimates of earthquake stress drop, computed through a pseudo-dynamic approach, with many others available in the literature for both point sources and finite fault models. We obtain a picture of the earthquake stress drop scaling with seismic moment over an exceptional broad range of earthquake sizes (−8 < MW < 9). Our results confirm that stress drop values are scattered over three order of magnitude and emphasize the lack of corroborating evidence that stress drop scales with seismic moment. We discuss these results in terms of scale invariance of stress drop with source dimension to analyse the interpretation of this outcome in terms of self-similarity. Geophysicists are presently unable to provide physical explanations of dynamic self-similarity relying on deterministic descriptions of micro-scale processes. We conclude that the interpretation of the self-similar behaviour of stress drop scaling is strongly model dependent. We emphasize that it relies on a geometric description of source heterogeneity through the statistical properties of initial stress or fault-surface topography, in which only the latter is constrained by observations.

Con-Patch: When a Patch Meets Its Context

Measuring the similarity between the patches in images is a fundamental building block in various tasks. Naturally, the patch size has a major impact on the matching quality and on the consequent application performance. Under the assumption that our patch database is sufficiently sampled, using large patches (e.g., 21 × 21 ) should be preferred over small ones (e.g., 7 × 7 ). However, this dense-sampling assumption is rarely true; in most cases, large patches cannot find relevant nearby examples. This phenomenon is a consequence of the curse of dimensionality, stating that the database size should grow exponentially with the patch size to ensure proper matches. This explains the favored choice of small patch size in most applications. Is there a way to keep the simplicity and work with small patches while getting some of the benefits that large patches provide? In this paper, we offer such an approach. We propose to concatenate the regular content of a conventional (small) patch with a compact representation of its (large) surroundings-its context. Therefore, with a minor increase of the dimensions (e.g., with additional ten values to the patch representation), we implicitly/softly describe the information of a large patch. The additional descriptors are computed based on a self-similarity behavior of the patch surrounding. We show that this approach achieves better matches, compared with the use of conventional-size patches, without the need to increase the database-size. Also, the effectiveness of the proposed method is tested on three distinct problems: 1) external natural image denoising; 2) depth image super-resolution; and 3) motion-compensated frame-rate up conversion.

Self-similar roll-up of a vortex sheet driven by a shear flow: Hyperbolic double spiral

In this paper, we consider the roll-up of an infinite vortex sheet and investigate its self-similar behavior. We address the question of whether the unsteady double spiral produced by the curvature singularity in finite time exhibits self-similar behavior. We find a self-similar solution of the double-spiral vortex sheet, which in fact, is a hyperbolic spiral. The radius of the spiral asymptotically grows with time and is proportional to the inverse of the angle from the spiral center. The curvature singularity plays the role of triggering spiral formation, but the source of vorticity for forming the spiral is the initial vorticity of the sheet. We show analytically that the self-similar solution satisfies the Birkhoff-Rott equation asymptotically. Numerical validation is also given by applying the blob-regularization model to the vortex sheet with a periodic perturbation. We examine various asymptotic relations among primitive variables for the spiral turns and find agreement of numerical results of the inner turns of the vortex sheet with the analytic solution. Our study clarifies contrasting results on the existence of the self-similar double-spiral of a large structure in the previous studies. Our solution also suggests the possibility of bifurcation of the self-similar solution of the double-spiral as the sheet strength varies.

Forecasting stock market returns over multiple time horizons

In this paper, we seek to demonstrate the predictability of stock market returns and explain the nature of this return predictability. To this end, we introduce investors with different investment horizons into the news-driven, analytic, agent-based market model developed in Gusev et al. [Algo. Finance, 2015, 4, 5–51]. This heterogeneous framework enables us to capture dynamics at multiple timescales, expanding the model’s applications and improving precision. We study the heterogeneous model theoretically and empirically to highlight essential mechanisms underlying certain market behaviours, such as transitions between bull and bear markets and the self-similar behaviour of price changes. Most importantly, we apply this model to show that the stock market is nearly efficient on intraday timescales, adjusting quickly to incoming news, but becomes inefficient on longer timescales, where news may have a long-lasting nonlinear impact on dynamics, attributable to a feedback mechanism acting over these horizons. Then, using the model, we design algorithmic strategies that utilize news flow, quantified and measured, as the only input to trade on market return forecasts over multiple horizons, from days to months. The backtested results suggest that the return is predictable to the extent that successful trading strategies can be constructed to harness this predictability.

Numerical evidence of an undisturbed region of flow in a turbulent rectangular submerged free jet

ABSTRACTThe evolution of turbulent rectangular submerged free jets is described in the literature by the presence of two regions of flow: the potential core region (PCR) and the fully developed region (FDR). However, experiments carried out in the last decade showed that a third region of flow is present, the undisturbed region of flow (URF), so-called in the average visualization, or the negligible disturbances flow (NDF) plus the small disturbances flow (SDF), so-called in the instant visualization. The URF is located between the slot exit and the beginning of the PCR. The main characteristics of URF, and NDF, are that velocity and turbulence profiles remain almost equal to those measured on the slot exit, and the height of the jet remains equal to the slot one. In the SDF the jet height undergoes small variations, i.e., contractions or expansions, but without formation of the vortex. To date, no numerical evidence of the presence of URF has been given by the literature. The present study, which concerns a two-dimensional jet, presents Large Eddy Simulations (LES), carried out at four Reynolds numbers, which are able to predict and characterize URF. The present numerical results are compared to previous theoretical approaches and confirm the presence of URF, between the slot exit and the PCR. Moreover, URF has a self-similar behavior and a new law for the evolution of the momentum is proposed.

Viscosity effect on the flow patterns in T-type micromixers

Flow patterns and mixing of liquids with different viscosities in T-type micromixers are numerically investigated on the Reynolds number range from 1 to 250. The viscosity ratio of the mixing media varied from 1 to 2; its effect on the flow structure and the mixing is studied. The dependences of the mixing efficiency and the pressure difference in the channel on the viscosity ratio and the Reynolds number are obtained. It is shown that the viscosity ratio has a considerable effect on the flow structure before and after transition from the symmetric to the asymmetric flow pattern. The self-similar behavior of the asymmetric flow pattern is established.

On wetting characteristics of droplet on a spherical particle in film boiling regime

This study reports droplet-particle interaction of size ratio less than unity in the film boiling regime on a highly thermally conductive spherical particle surface. Specifically, the effects of impact Weber number (We) of subcooled state droplets comprising water (We=3.9–103.6) and isopropyl alcohol (IPA) (We=8.6–194.6) were studied using high speed imaging technique in the particle temperature range of 250–350°C. In general, non-wetting interaction behaviour was observed with two distinct outcomes – rebound and complete disintegration demarcated by a critical Weber number range instead of a single threshold value. Extent of surface wetting was characterised by the maximum droplet spread diameter parameter which was found to scale with impact Weber number in a power law form which agrees with the theoretical scaling argument. Additionally, an energy balance model was developed to compute this parameter which provided good agreement with the experimental measurements in the lower Weber number regime, however, higher deviations were noted near the transition regime. Also quantified from experiments was the droplet-particle contact time which exhibited a power law dependency on Weber number in the rebound regime, however, was noted to be almost independent of Weber number in the disintegration regime. Particle surface wettability was characterised by the experimentally measured dynamic contact angles which were found to vary in the range of 120–160o in low Weber number regime manifesting the hydrophobic nature of particle surface in film boiling regime. Also, all the parameters such as contact line velocity, particle temperature and droplet size apparently had relatively insignificant influence on the variation of dynamic contact angle. Temporal variation of non-dimensional spreading parameter exhibited a self-similar behaviour wherein all data collapsed on a single power law profile. It was further shown that the behaviour could also be described by a recovery type exponential profile through suitable non-dimensionalization and both profiles can be utilized to produce a spreading kinetics.

Self-similarity of the large-scale motions in turbulent pipe flow

Townsend’s attached eddy hypothesis assumes the existence of a set of energetic and geometrically self-similar eddies in the logarithmic layer in wall-bounded turbulent flows, which can be scaled with their distance to the wall. To examine the possible self-similarity of the energetic eddies in fully developed turbulent pipe flow, we performed stereo particle image velocimetry measurements together with a proper orthogonal decomposition analysis. For two Reynolds numbers, $Re_{{\it\tau}}=1330$ and 2460, the resulting modes/eddies were shown to exhibit self-similar behaviour for eddies with wall-normal length scales spanning a decade. This single length scale provides a complete description of the cross-sectional shape of the self-similar eddies.

The signature of initial production mechanisms in isotropic turbulence decay

In the present work the quantification of the time-lasting effects of production mechanisms in homogeneous isotropic turbulence decay is addressed. The analysis is developed through the use of theoretical tools as well as numerical calculations based on the eddy damped quasinormal Markovian (EDQNM) model. In both cases a modified Lin equation is used, which accounts for production mechanisms as proposed by Meldi, Lejemble, and Sagaut [“On the emergence of non-classical decay regimes in multiscale/fractal generated isotropic turbulence,” J. Fluid Mech. 756, 816–843 (2014)]. The approaches used show that an exponential decay law can be observed if the intensity of the forcing is strong enough to drive the turbulence dynamics, before a power-law decay is eventually attained. The EDQNM numerical results indicate that the exponential regime can persist for long evolution times, longer than the observation time in grid turbulence experiments. A rigorous investigation of the self-similar behavior of the pressure spectrum has been performed by a comprehensive comparison of EDQNM data with direct numerical simulation (DNS)/experiments in the literature. While DNS and free decay EDQNM simulations suggest the need of a very high Reλ threshold in order to observe a clear −7/3 slope of the pressure inertial range, experimental data and forced EDQNM calculations indicate a significantly lower value. This observation suggests that the time-lasting effects of production mechanisms, which cannot be excluded in experiments, play a role in the lack of general agreement with classical numerical approaches. These results reinforce the urge to evolve the numerical simulation state of the art towards the prediction of realistic physical states.