Self-similar Law Research Articles

A NOVEL TRANSFORMER METHOD PRETRAINED WITH MASKED AUTOENCODERS AND FRACTAL DIMENSION FOR DIABETIC RETINOPATHY CLASSIFICATION

Diabetic retinopathy (DR) is one of the leading causes of blindness in a significant portion of the working population, and its damage on vision is irreversible. Therefore, rapid diagnosis on DR is crucial for saving the patient’s eyesight. Since Transformer shows superior performance in the field of computer vision compared with Convolutional Neural Networks (CNNs), it has been proposed and applied in computer aided diagnosis of DR. However, a large number of images should be used for training due to the lack of inductive bias in Transformers. It has been demonstrated that the retinal vessels follow self-similar fractal scaling law, and the fractal dimension of DR patients shows an evident difference from that of normal people. Based on this, the fractal dimension is introduced as a prior into Transformers to mitigate the adverse influence of lack of inductive bias on model performance. A new Transformer method pretrained with Masked Autoencoders and fractal dimension (MAEFD) is developed and proposed in this paper. The experiments on the APTOS dataset show that the classification performance for DR by the proposed MAEFD can be substantially improved. Additionally, the present model pretrained with 100,000 retinal images outperforms that pretrained with 1 million natural images in terms of DR classification performance.

Laminar Swirling Wall Jets

The problem of determining the characteristics of a laminar swirling wall jet of an incompressible fluid is considered. Numerical solutions of the Navier–Stokes equations are obtained in the stationary and non-statitonary formulations. It is shown that the jet characteristics obey a self-similar law at large distances from the jet source, as in the case of a three-dimensional laminar non-swirling jet, but in our case the jet propagates at a certain angle to the initial direction of jet blowing. With a large swirling of flow in the jet, regions of recirculation flow appear and the jet flow becomes unsteady.

Fractal thermal conductivity of unsaturated soils considering pore heterogeneity

Accurately predicting the thermal conductivity of unsaturated soils is crucial for assessing and simulating the long-term environmental impact of subterranean engineering structures. This study presents a fractal thermal conductivity model for unsaturated soils that considers the influence of heterogeneous pore topology, water content and volumetric deformation. Based on the results from the scanning electron microscopy (SEM) and mercury intrusion-extrusion cyclic porosimetry (MIECP) tests on Shanghai clay and Kaolin, we introduce the pore-throat structure to describe the soil heterogeneity and deal with the coupling of solid-pore thermal properties through the mixing law at the pore scale. Together with the self-similar fractal law, the thermal conductivity model is upscaled from the pore scale to the representative element volume (REV) scale. During the process, heterogeneity factors are introduced to describe differences between local and bulk properties, such as porosity and saturation. Thermal property tests of Shanghai clay and Kaolin with varying porosities are then conducted to validate the proposed thermal conductivity model. Additionally, nineteen published datasets on thermal conductivity are also utilized to confirm the adaptability and robustness of the model. The results show that the proposed model predicts well the thermal conductivity of unsaturated soils. Moreover, when compared to existing models, the proposed model offers superior performance in describing the thermal properties of unsaturated soils.

Laminar Swirling Wall Jets

The problem of determining the characteristics of a laminar swirling wall jet of an incompressible fluid is considered. Numerical solutions of the Navier–Stokes equations are obtained in the stationary and non-statitonary formulations. It is shown that the jet characteristics obey a self-similar law at large distances from the jet source, as in the case of a three-dimensional laminar non-swirling jet, but in our case the jet propagates at a certain angle to the initial direction of jet blowing. With a large swirling of flow in the jet, regions of recirculation flow appear and the jet flow becomes unsteady.

Spatial Probability Characteristics of Waves Generated by Polar Lows in Nordic and Barents Seas

Polar lows (PLs) are mesoscale, up to 1000 km, rather short lifetime (less than 15–30 h) cyclonic atmospheric systems formed in polar latitudes and associated with cold outbreak events. Strong winds, higher than 15 m/s, can then generate high surface waves which may pose danger to marine and coastal infrastructures. To investigate the probability of high waves generated by PLs in the Nordic and Barents Seas, analysis can be performed using available PL statistical distributions obtained from satellite passive microwave data, MODIS infrared imagery and ASCAT scatterometer data. Classical self-similar laws for wind waves development based on the extended duration concept are used to obtain first-guess estimates of significant wave height and the wavelength of waves generated by PL. All possible combinations of PL parameters (maximum wind speed, lifetime, diameter, translation velocity and direction of propagation) are considered to obtain the occurrence of waves exceeding specified levels, ranging from 2 to 15 m for significant wave height and from 100 to 500 m for wavelength. Particularly, PL-generated waves higher than 4 m occur up to 6 times a year, higher than 8 m occur up to 2–3 times a year, higher than 10 m occur up to once a year, the probability of 12 m waves is one event in several years and 15 m SWHs occur less than once in a decade. The area most affected by strong waves from PLs is the near shore zone around the Scandinavian peninsula, northward from the North Cape. The relative contribution of PLs in the formation of the waves field in the Nordic and Barents Seas is discussed.

Self-similar bubble-front evolutions of ablative Rayleigh–Taylor instability seeded by localized perturbations

Two-dimensional numerical simulations are carried out to investigate the nonlinear bubble growth of ablative Rayleigh–Taylor instability (ARTI) seeded by localized perturbations (LPs), where the LPs are described by a Gaussian mode. It is found that the nonlinear bubble-front penetration of LP-seeded ARTI follows the self-similar scaling law αbAT∫gdt2, different from the classical case, where the self-similar behavior is not observed. It is also found that the quadratic growth coefficient αb in the LP-seeded ARTI mainly depends on the initial perturbation amplitude and initial perturbation width. When the perturbation amplitude is small, αb has a value of ∼0.03, which is not sensitive to the perturbation width. As the perturbation amplitude increases, the value of αb increases, and the phenomenon is more significant when the perturbation width is narrower. It is shown that the increase in αb is due to the spike-induced upward jet and the ablation-generated vorticity inside the bubble.

A multi-scale model for impingement drying of porous slab

Air impingement drying of a planar sheets of porous materials offers advantages of high and controllable heat and mass transfer rates. The governing heat and mass transfer phenomena involve multi-scale physical mechanisms ranging from mesoscopic pore scale to macroscopic jet scale. This work presents a pore-scale heat and mass transfer model of an unsaturated porous material based on statistically self-similar fractal scaling laws of pore structures. Furthermore, the effective transport coefficients are derived and applied to develop a cross-scale mathematical model for hot air impingement drying of a slab made of an unsaturated porous material. The flow and thermal fields of jet impingement and drying characteristics of unsaturated porous material are modeled numerically. The quantitative correlations between the mesoscopic pore structure and macroscopic heat and mass transfer properties are explored. The proposed fractal pore-scale model of porous material and cross-scale mathematical model of jet impingement show good agreement with experimental data. The effects of jet velocity, porosity as well as pore and tortuosity fractal dimensions on drying rate are discussed in detail. The computational results show that the mesoscopic pore structures indicate significant effect on the drying rate during later stage of drying. Results of this study provide useful guidance for efficient design of impingement of porous materials. • Meso-scale heat and mass transfer model of unsaturated porous material is proposed. • Multi-scale model for laminar impingement drying of porous slab is developed. • Impingement drying performance of porous slab is predicted. • Effect of mesoscopic structure on drying kinetics of porous material is explored.

Effect of particle shape on packing fraction and velocity profiles at outlet of a silo

Many studies on how the particle shape affects the discharge flow mainly focus on discharge rates and avalanche statistics. In this study, the effect of the particle shape on the packing fraction and velocities of particles in the silo discharge flow are investigated by using the discrete element method. The time-averaged packing fraction and velocity profiles through the aperture are systematically measured for superelliptical particles with different blockinesses. Increasing the particle blockiness is found to increase resistance to flow and reduce the flow rate. At an identical outlet size, larger particle blockiness leads to lower velocity and packing fraction at the outlet. The packing fraction profiles display evidently the self-similar feature that can be appropriately adjusted by fractional power law. The velocity profiles for particles with different shapes obey a uniform self-similar law that is in accord with previous experimental results, which is compatible with the hypothesis of free fall arch. To further investigate the origin of flow behaviors, the packing fraction and velocity field in the region above the orifice are computed. Based on these observations, the flow rate of superelliptical particles is calculated and in agreement with the simulated data.

Three-dimensional structure of freely-propagating flame prior to deflagration-to-detonation transition

The paper analyzes numerically the development of the freely expanding flame before the transition to detonation in three-dimensional space. The risks related to the transition to detonation in unconfined space arise at launch places and near-Earth space during rocket engine accidents. Clear and adequate models for such scenarios are in demand for the elaboration of safety measures. Here it is shown that the process of transition to detonation involves multidimensional effects related to the generation of compression waves and their amplification when interacting with the developing flame. The necessary conditions for deflagration-to-detonation transition are flame acceleration up to near-sonic speed and generation of transversal compression waves (propagating along the flame front). So, while the subsonic stage of flame acceleration can be successfully described by the quasi one-dimensional model supplemented by a known self-similar law of the flame area growth (folding factor), the model for the pre-detonation stage should take into account considered multidimensional effects. The presented results imply that there is no unambiguous relation between the transition to detonation and the folding factor.

Type IIP supernova SN2016X in radio frequencies

Context. The study of radio emission from core-collapse supernovae (SNe) probes the interaction of the ejecta with the circumstellar medium (CSM) and reveals details of the mass-loss history of the progenitor. Aims. We report observations of the type IIP supernova SN 2016X during the plateau phase, at ages between 21 and 75 days, obtained with the Karl G. Jansky Very Large Array (VLA) radio observatory. Methods. We modelled the radio spectra as self-absorbed synchrotron emission, and we characterised the shockwave and the mass-loss rate of the progenitor. We also combined our results with previously reported X-ray observations to verify the energy equipartition assumption. Results. The properties of the shockwave are comparable to other type IIP supernovae. The shockwave expands according to a self-similar law R ∝ tm with m = 0.76 ± 0.08, which is notably different from a constant expansion. The corresponding shock velocities are approximately 10700–8000 km s−1 during the time of our observations. The constant mass-loss rate of the progenitor is Ṁ = (7.8 ± 0.9) × 10−7 α−8/19 (ϵB/0.1)−1 M⊙ yr−1, for an assumed wind velocity of 10 km s−1. We observe spectral steepening in the optically thin regime at the earlier epochs, and we demonstrate that it is caused by electron cooling via the inverse Compton effect. We show that the shockwave is characterised by a moderate deviation from energy equipartition by a factor of ϵe/ϵB ≈ 28, being the second type IIP supernova to show such a feature.

Decay of acoustic turbulence in two dimensions and implications for cosmological gravitational waves

Gravitational waves from a phase transition associated with the generation of the masses of elementary particles are within the reach of future space-based detectors such as LISA. A key determinant of the resulting power spectrum, not previously studied, is the lifetime of the acoustic turbulence which follows. We study decaying acoustic turbulence using numerical simulations of a relativistic fluid in two dimensions. Working in the limit of non-relativistic bulk velocities, with an ultra-relativistic equation of state, we find that the energy spectrum evolves towards a self-similar broken power law, with a high-wavenumber behaviour of $k^{-2.08 \pm 0.08}$, cut off at very high $k$ by the inverse width of the shock waves. Our model for the decay of acoustic turbulence can be extended to three dimensions using the universality of the high-$k$ power law and the evolution laws for the kinetic energy and the integral length scale. It is used to build an estimate for the gravitational wave power spectrum resulting from a collection of shock waves, as might be found in the aftermath of a strong first order phase transition in the early universe. The power spectrum has a peak wavenumber set by the initial length scale of the acoustic waves, and a new secondary scale at a lower wavenumber set by the integral scale after a Hubble time. Between these scales a distinctive new power law appears. Our results allow more accurate predictions of the gravitational wave power spectrum for a wide range of early universe phase transition scenarios.

THE CORRELATION BETWEEN FLUID FLOW AND HEAT TRANSFER OF UNSATURATED SHALE RESERVOIR BASED ON FRACTAL GEOMETRY

The dissimilar multi-scale structures of shale to conventional reservoirs make it a challenge to understand the fluid flow and heat transfer through unsaturated shale formations. In this paper, the pore structure and moisture content of shale samples are measured by low-field nuclear magnetic resonance technique and thermogravimetric differential scanning calorimetry test, respectively. A pore-scale model is accordingly developed for the immiscible two-phase fluid flow and heat conduction through unsaturated shale based on the statistically self-similar fractal scaling law of pore size distribution. The analytical expressions of effective and relative permeability, as well as effective thermal conductivity (ETC), are proposed, which indicate good agreement with experimental results. It has been shown that the capillary pressure and gas slippage play important role in multiphase flow through unsaturated shale. Both pore and tortuosity fractal dimensions show significant influence on the relative permeability for nonwetting phase (RPNW), while they indicate the marginal effect on the relative permeability for the wetting phase (RPW). The ETC decreases with the increase of pore and tortuosity fractal dimensions, and it is positively and negatively correlated with RPW and RPNW, respectively. The correlation between ETC and relative permeability is found to follow a logistic function. The present fractal model can characterize the multiscale structures of shale reservoirs and may help understand transport mechanisms of immiscible multiphase flow and heat transfer through unsaturated shale.

Self-similarity of density and velocity profiles in a 2D hopper flow of elliptical particles: Discrete element simulation

Two-dimensional discharge flow of elliptical particles with different aspect ratios from a silo is investigated by using discrete element simulation. The measured flow rate of the particles is well fitted via Beverloo's law. The flow rate shows a monotonic decrease with increasing aspect ratio. The density and velocity profiles of particles at the outlet were systematically measured. At an identical outlet size, the speed of particles through the aperture decreases with increasing aspect ratio. The velocity profiles in all cases follow the same definite self-similar law and are in good agreement with the reported experimental results, thereby showing compatibility with the free fall arch postulation. The density profiles show obvious self-similar feature and can be fitted conveniently by using a fractional power law. Based on these observations and the results of the previous work, the flow rate for elliptical particles is calculated and is in agreement with the simulated results.

Flame structure, turbulent burning velocity and its unified scaling for lean syngas/air turbulent expanding flames

A systematic experimental study of lean premixed syngas/air turbulent expanding flames has been conducted under a wide range of turbulence intensities (0–3.54 m/s), initial pressures (0.5–5 bar), and hydrogen volumetric fractions up to 80% (20%, 50% and 80%). Flame structure and turbulent flame propagation dynamics were investigated. Results show that the flame becomes more refined and wrinkled with the increasing of both turbulence intensity and initial pressure, leading to a larger flame area and the associated turbulent burning velocity (ST). With hydrogen fraction increased, ST is also enhanced significantly, which is mainly due to the promotion of laminar burning velocity (SL) and diffusional-thermal instability. ST/SL is nearly kept constant with hydrogen fraction, which is a trade-off between strengthened diffusional-thermal instability and weakened turbulence stretch. A unified scaling of ST is obtained, indicating that turbulent Reynolds number (ReT) is a practical method to correlate ST when Lewis number is close to unity. Furthermore, at least in the interpretation domain, ST of spherical flames continually increases as the flame expands, which has been referred as flame acceleration phenomenon. It appears that only effective turbulence intensity itself is not able to reflect acceleration phenomenon completely. Turbulent expanding flames follow a self-similar propagation law and the quantitative ST dependence with flame expanding is ST∼R0.5 approximately.

Numerical simulation and self-similarity of the mean mass transfer in turbulent round jets

The paper investigates the mean mass transfer/passive scalar spreading in turbulent submerged round jets. Two regions of flow are present in the jet evolution: the Near-Field Region (NFR) and the Fully Developed Region (FDR). This group of research investigates from some years the mean evolution of turbulent rectangular jets with the new physical finding that two sub-regions (not a single one) are present in the NFR. The first region of the two is the newly discovered Undisturbed Region of Flow (URF), while the second one is the known Potential Core Region (PCR). In a recent paper we showed that the flow evolution of turbulent round jets, as far as momentum spreading is concerned, is self-similar also in the NFR. Literature shows that mass transfer spreading is self-similar only in FDR. The present paper presents new mean mass transfer results of the numerical Large Eddy Simulation (LES) in turbulent round jets. Four Reynolds numbers, from 2492 to 19,988, and two laminar Schmidt numbers, 1 and 10, are investigated. The first novel result of this paper is that mass transfer is self-similar in the NFR. The second result is that two new analytical models describe the passive scalar spreading in the URF and PCR. The third result is that two new self-similar laws describe the passive scalar spreading in the FDR. The fourth result states that the well-known power-law relationship, between passive scalar and axial momentum in the FDR, holds regardless of the modeling of turbulent viscosity and turbulent Schmidt number.

Length scales in turbulent free shear flows

ABSTRACTWe address the important point of the proportionality between the longitudinal integral lengthscale (L) and the characteristic mean flow width (δ) using experimental data of an axisymmetric wake and a turbulent planar jet. This is a fundamental hypothesis when deriving the self-similar scaling laws in free shear flow. We show that L/δ is indeed constant, at least in a range of streamwise distances between 15 and 50 times the characteristic inlet dimension. We revisit turbulence closure models such as the Prandtl mixing length and the eddy viscosity in the light of the non-equilibrium dissipation scaling. We show that the mixing length model does not comply with the scalings stemming from the non-equilibrium version of the theory even if it does comply with the theory’s equilibrium version. Similarly, the eddy viscosity model holds in the case of the non-equilibrium version of the theory provided that the eddy viscosity is constant everywhere. We conclude by comparing the results of the different models with each other and with experimental data and with an improved model (following Townsend) that corrects for the eddy viscosity by considering the intermittency of the flow.

Integration of a deep fluid equation with a free surface

We show that the Euler equations describing the unsteady potential flow of a two-dimensional deep fluid with a free surface in the absence of gravity and surface tension can be integrated exactly under a special choice of boundary conditions at infinity. We assume that the fluid surface at infinity is unperturbed, while the velocity increase is proportional to distance and inversely proportional to time. This means that the fluid is compressed according to a self-similar law. We consider perturbations of a self-similarly compressible fluid and show that their evolution can be accurately described analytically after a conformal map of the fluid surface to the lower half-plane and the introduction of two arbitrary functions analytic in this half-plane. If one of these functions is equal to zero, then the solution can be written explicitly. In the general case, the solution appears to be a rapidly converging series whose terms can be calculated using recurrence relations.

Simulation of primordial black hole formation using pseudo-spectral methods

In this work we have used for the first time pseudo-spectral methods to perform numerical simulations of spherically symmetric black hole formations on a Friedman-Robertson-Walker universe. With these methods, the differential equations describing the gravitational collapse are partially solved algebraically. With our publicly available code we then independently check, and confirm, previous numerical estimations of the thresholds to form primordial black holes. By using an excision technique and analytical estimations of accretion rates, we were also able to estimate the black holes mass even in the case of large deviations from the threshold. There, we confirm, with an explicit example, that the estimation of the black hole mass via the self-similar scaling law is only accurate up to $O(15\%)$, for the largest allowed mass.

A buoyancy–shear–drag-based turbulence model for Rayleigh–Taylor, reshocked Richtmyer–Meshkov, and Kelvin–Helmholtz mixing

A phenomenological turbulence model for Rayleigh–Taylor, reshocked Richtmyer–Meshkov, and Kelvin–Helmholtz instability-induced mixing is developed using a general buoyancy–shear–drag model. Analytic solutions to the simplified model equations corresponding to each instability are derived separately, which are then used to calibrate the model coefficients to predict specific values of the mixing layer growth parameters and exponents. The buoyancy–shear–drag equations for the bubble and spike mixing layer widths are then solved numerically, and a turbulent diffusivity (or viscosity) is constructed dimensionally as the product of the mixing layer width, h, and its time-derivative, dh/dt. Surrogate turbulent fields are then constructed by multiplying a presumed, approximate self-similar spatial profile by appropriate functions of h and dh/dt. Using several simplifying approximations, the turbulent diffusion equations satisfied by the mean mass fraction and mean shear velocity are solved analytically in a reference frame moving with the mean advection velocity, and it is shown that the mean fields evolve in space and time as expected. The explicit modeling and solution of turbulent transport equations (e.g., for the turbulent kinetic energy and its dissipation rate or a lengthscale) are not required. By using separate equations for the bubble and spike evolution, the model naturally captures the asymmetry induced by an increasing density contrast between the heavy and light fluids. The model also includes molecular dissipation and diffusion, and therefore, can describe transition to fully-developed turbulence if the initial turbulent diffusivity/viscosity is much smaller than the molecular diffusivity/viscosity. This model is applied to constant-acceleration Rayleigh–Taylor, impulsively reshocked Richtmyer–Meshkov, and variable-density Kelvin–Helmholtz mixing layers to demonstrate its utility. It is shown that the numerical solutions of the model calibrated using specific values of the instability growth parameters and exponents: (1) produces mixing layer widths in agreement with the expected self-similar growth power laws; (2) gives spatiotemporal profiles of turbulent fields that are expected and consistent with previous results; (3) predicts the expected power-law growths and decays of the spatially-integrated turbulent fields, and; (4) gives spatiotemporal profiles of the mean fields that are expected and consistent with previous results. This relatively simple model can be implemented numerically, and is expected to be useful for fundamental as well as astrophysical and inertial confinement fusion applications.

Relationship between cellular morphology and self-acceleration in lean hydrogen-air expanding flames

Experiments and quantitative evaluation are conducted upon the cellular morphology and self-acceleration in lean H2-air expanding flames, trying to interpret the inherent relationship between them. Results confirm the existence of two distinct self-acceleration stages (transition and saturation). The transition between them happens when the averaged cell radius decreases to a minimum stable value. Self-similar laws exist both in the transition stage and the saturation stage, with the fractal excess of transition stage much greater than saturation stage. In the saturation stage, fractal excess slowly increases with pressure but is always smaller than 0.33. Basing on evolution profiles of cell radius, the physical interpretation of greater self-acceleration exponent in the transition stage is quantitatively confirmed by correlating the decreasing rate of cell wavelength with flame radius. Finally, the inconsistency of fractal excess in the saturation stage is interpreted from the perspective of the discrete wavelength spectrum in cellular flames.