Scaling Laws Behind Penetrative Turbulence:History and Perspectives

An unstably stratified flow entering into a stably stratified flow is referred to as penetrative convection,which is crucial to many physical processes and has been thought of as a key factor for extreme weather conditions.Past theoretical,numerical,and experimental studies on penetrative convection are reviewed,along with field studies providing insights into turbulence modeling.The physical factors that initiate penetrative convection,including internal heat sources,nonlinear constitutive relationships,centrifugal forces and other complicated factors are summarized.Cutting-edge methods for understanding transport mechanisms and statistical properties of penetrative turbulence are also documented,e.g.,the variational approach and quasilinear approach,which derive scaling laws embedded in penetrative turbulence.Exploring these scaling laws in penetrative convection can improve our understanding of large-scale geophysical and astrophysical motions.To better the model of penetrative turbulence towards a practical situation,new directions,e.g.,penetrative convection in spheres,and radiation-forced convection,are proposed.

Scaling law of resolved-scale isotropic turbulence and its application in large-eddy simulation

Eddy-damping quasinormal Markovian （EDQNM） theory is employed to calculate the resolved-scale spectrum and transfer spectrum, based on which we investigate the resolved-scale scaling law. Results show that the scaling law of the resolved-scale turbulence, which is affected by several factors, is far from that of the full-scale turbulence and should be corrected. These results are then applied to an existing subgrid model to improve its performance. A series of simulations are performed to verify the necessity of a fixed scaling law in the subgrid modeling.

Scaling laws of compressible turbulence

Spatial scaling laws of velocity kinetic energy spectra for the compressible turbulence flow and the density-weighted counterparts are formulated in terms of the wavenumber, dissipation rate, and Mach number by using a dimensional analysis. We apply the Barenblatt＇s incomplete similarity theory to both kinetic and density-weighted energy spectra. It shows that, within the initial subrange, both energy spectra approach the -5/3 and -2 power laws of the wavenumber when the Mach number tends to unity and infinity, respectively.

Empirical Scaling Laws of Neutral Beam Injection Power in HL-2A Tokamak

We present an experimental method to obtain neutral beam injection （NBI） power scaling laws with operating parameters of the NBI system on HL-2A, including the beam divergence angle, the beam power transmission efficiency, the neutralization efficiency and so on. With the empirical scaling laws, the estimating power can be obtained in every shot of experiment on time, therefore the important parameters such as the energy confinement time can be obtained precisely. The simulation results by the tokamak simulation code （TSC） show that the evolution of the plasma parameters is in good agreement with the experimental results by using the NBI power from the empirical scaling law.

The scaling laws of cabin structures subjected to internal blast loading:Experimental and numerical studies

This paper presents a combination of experimental and numerical investigations on the dynamic response of scaling cabin structures under internal blast loading.The purpose of this study is to modify the similar relationship between the scaled-down model and the prototype of the cabin structures under internal blast loading.According to the Hopkinson’s scaling law,three sets of cabin structure models with different scaling factors combined with different explosive masses were designed for the experimental study.The dynamic deformation process of the models was recorded by a three-dimensional digital imaging correlation(DIC)method and a 3D scanning technology was used to reconstruct the deformation modes of the specimen.In addition,a finite element model was developed for the modification of the scaling law.The experimental results showed that the final deflection-to-thickness ratio was increased with the increase of the model size despite of the similar trend of their deformation processes.The reason for this inconsistency was discussed based on the traditional scaling law and a modified formula considering of the effects of size and strain-rate was provided.

A scaling law of high-order harmonic generation

Using a nonperturbative quantum electrodynamics theory of high-order harmonic generation （HHG）, a scaling law of HHG is established. The scaling law states that when the atomic binding energy Eb, the wavelength ）, and the intensity I of the laser field change simultaneously to kEb, λ/k, and k3I, respectively. The characteristics of the HHG spectrum remain unchanged, while the harmonic yield is enhanced k3 times. That HHG obeys the same scaling law with above-threshold ionization is a solid proof of the fact that the two physical processes have similar physical mechanisms. The variation of integrated harmonic yields is also discussed.

On the scaling law of ramp structures in scalar turbulence

Ramp structures widely exist in scalar turbulence,such as temperature,water vapor,and carbon dioxide(CO2),which refer to the phenomenon that the physical quantity increases slowly with time and then suddenly drops.ramp structures lead to large gradients on a small scale and result in intermittency and anisotropy of turbulent flows.in this paper,wavelet analysis is used to analyze observed data from the beijing 325-m meteorological tower to extract ramp structures in temperature,water vapor,and CO2 signals.ramp structures in CO2 signals are different from those in temperature and water vapor in terms of the averaged temporal scale and normalized amplitude,and the ramp duration almost equals the cliff duration,which means ramp structures in CO2 signals are not easy to generate and different physical mechanisms may exist.in addition,both the ascending and descending part of ramp structures are linearly fitted.it is found that a scaling law exists between the slope and duration in the ascending part in the three scalar signals.the corresponding power exponents are slightly different.furthermore,the same rule exists in the descending part of ramp structures,which indicates that self-similarity may be a universal law in scalar turbulence.moreover,the maxima of selected ramp structures show the same pattern,i.e.there are ramp structures in the maximum sequence,which proves that small-scale ramp structures are superimposed on large-scale ramp structures.

Scaling law of single ion–atom impact ionization cross sections of noble gases from He to Xe at strong perturbative energies

We extend our previous work of a classical over-barrier ionization （COBI） model to calculate the single ionization cross sections of noble gases ranging from He to Xe at strong perturbative energies. The calculation results are in good agreement with extensive experimental data. The scaling law of single ion-atom impact ionization cross sections of noble gases on projectile charge q and energy E, also on target ionization energy I is drawn from the model.

Aerodynamic shape and drag scaling law of a flexible fibre in a flowing medium

The study of a flexible body immersed in a flowing medium is one of the best way to find its aerodynamic shape.This Letter revisited the problem that was first studied by Alben et al.(Nature 420,479–481,2002).To determine the aerodynamic shape of the fibre,a simpler approach is proposed.A universal drag scaling law is obtained and the universality of the Alben-Shelley-Zhang scaling law is confirmed by using dimensional analysis.A complete Maple code is provided for finding aerodynamic shape of the fibre in the flowing medium.

A modified generalized scaling law for the similitude of dynamic strain in centrifuge modeling

Soil strain is the key parameter to control the elasto-plastic deformation and even the failure processes.To overcome the defect that the strain of the model soil is always smaller than that of the prototype in Iai′s generalized scaling law(GSL),a modified scaling law was proposed based on Iai′s GSL to secure the same dynamic shear strain between the centrifuge model and the prototype by modulating the amplitude and frequency of the input motion at the base.A suite of dynamic centrifuge model tests of dry sand level ground was conducted with the same overall scaling factor(λ=200)under different centrifugal accelerations by using the technique of“modeling of models”to validate the modified GSL.The test results show that the modified GSL could achieve the same dynamic strain in model as that of the prototype,leading to better modeling for geotechnical problems where dynamic strain dominates the response or failure of soils.Finally,the applicability of the proposed scaling law and possible constraints on geometry scaling due to the capability limits of existing centrifuge shaking tables are discussed.

Studies on convergence and scaling law of Thomson backscattering spectra in strong fields

With the saddle point analysis method for the Bessel function structure and property, the convergence problem and the scaling laws of Thomson backscattering spectra are solved and studied in both cases that are for the plane wave laser field without and with applied external constant magnetic field. Some unclear points appeared in previous work are clarified. The extension of the method to a general situation for the laser field with an arbitrary polarization is discussed. We also make a simple analysis and discussion about the optimal spectra dependence of field parameters and its implication to practical applications.

SCALING LAW OF IRREGULAR EUTECTIC IN DIRECTIONAL SOLIDIFICATION

The profile features of the solidifying interface of the irregular Al-Si eutectic during directional solidification are studied by numerically solving a nonlinear coupling equation of directional solidification of this eutectic.The critical splitting point of the solidifying interface coincides with the maximum value of its position and corresponds to the marginal stability point.Argument is presented that the interlamellar spacing for local regular structure selects the critical splitting state of the α(Al)-liquid interface as its operating point,and the oper- ating point of the average interflake spacing for the irregular structure is the critical splitting state of the β(Si)-liquid interface.The scaling laws derived,which have the more general forms than the classic Jackson-Hunt scaling,gained support from the experimental results.

EXTENDED SELF-SIMILAR SCALING LAW OF MULTI-SCALE EDDY STRUCTURE IN WALL TURBULENCE

The longitudinal fluctuating velocity of a turbulent boundary layer was measured in a water channel at a moderate Reynolds number. The extended self-similar scaling law of structure function proposed by Benzi was verified. The longitudinal fluctuating velocity, in the turbulent boundary layer was decomposed into many multi-scale eddy structures by wavelet transform. The extended self-similar scaling law of structure function for each scale eddy velocity was investigated. The conclusions are I) The statistical properties of turbulence could be self-similar not only at high Reynolds number, but also at moderate and low Reynolds number, and they could be characterized by the same set of scaling exponents xi (1)(n) = n/3 and xi (2)(n) = n/3 of the fully developed regime. 2) The range of scales where the extended self-similarity valid is much larger than the inertial range and extends far deep into the dissipation range,vith the same set of scaling exponents. 3) The extended selfsimilarity is applicable not only for homogeneous turbulence, but also for shear turbulence such as turbulent boundary layers.

Experimental researches for scaling laws of low-order structure function in boundary layer

Scaling laws are addressed by analysing moments of velocity increments which obtained by Particle-image Velocimetry(PIV)system in the boundary layer of a flat plate.In the paper,we measure the moments of increments of upstream velocity(u'),longitudinal velocity(v')and ponderance of vorticity(dv'/dx)at Reθ=2167 in different wall distance and verify the anomaly of the scaling exponents of high-order structure functions with the increasing order of the moments,discuss the scaling of non-integer moments of order between+2 and-1.The difference of scaling exponents of low-order structure functions between the experimental data and Kolmogorov's,SL's(She & Leveque)prediction increases as the moment order decreases toward-1,which shows that the anomaly is manifested in low-oeder moments as well.However,for same order structure functions,the scaling exponents of v' and dv'/dx are not changeable in different wall distance.

A refinement of scaling laws in wall turbulence

As a universal conclusion of turbulent scale, scaling laws are important to the research on statistic turbulence. We measured two-dimensional instantaneous velocity field in turbulent boundary layers of flat plate with the momentum thickness Reynolds number Reθ=2 167. Scaling laws have different forms in different wall distance and scale. We proposed an expected scaling law and compared it with the She-Leveque (SL) scaling law based on the wavelet analysis and traditional statistical methods. Results show that the closer to the wall, the more the expected scaling law approached to the SL scaling law.

EFFECTS OF POISSON’S RATIO ON SCALING LAW IN HERTZIAN FRACTURE

In this paper the Auerbach＇s scaling law of Hertzian fracture induced by a spherical indenter pressing on a brittle solid is studied. In the analysis, the singular integral equation method is used to analyze the fracture behavior of the Hertzian contact problem. The results show that the Auerbach＇s constant sensitively depends on the Poisson＇s ratio, and the effective Auerbach＇s domain is also determined for a given value of the Poisson＇s ratio.

Scaling Law of s-Wave Valence Proton Distributions

Inthe framework of the single-particle potential model, the root-meamsquare radii of s-wave valence proton distributions for P and S isotopes have been systematically calculated. The scaling law of s-wave valence proton distributions is obtained by analyzing the relation between radii and separation energies. Necessary condition for the occurrence of 2sl/2-state proton halos is deduced from the scaling law. Quantitative condition derived from this work can to some extent serve as reference for proton halo search.

Comparison of Coarse Graining DEM Models Based on Exact Scaling Laws

The simulation of a large number of particles requires unacceptable computational time that is the most criticalproblem existing in the industrial application of the DEM. Coarse graining is a promising approach to facilitatethe application of DEM to industrial problems. While the current coarse graining framework is often developedin an ad-hoc manner, leading to different formulations and different solution accuracy and efficiency. Therefore,in this paper, existing coarse graining techniques have been carefully analysed by the exact scaling law which canprovide the theory basis for the upscaling method. A proper scaling rule for the size of particles and samples as wellas interaction laws have been proposed. The scaling rule is applied to a series simulations of biaxial compressiontests with different scale factors to investigate the precision of the coarse graining system. The error between theoriginal system and the coarse system shows a growing tendency as the scale factor increases. It can be concludedthat the precision of the coarse graining system is accepted when applying scaling rules based on the exact scalinglaws.

Scaling laws governing the elastic properties of 3D graphenes

In this study,we comprehensively investigated the scaling law for elastic properties of three-dimensional honeycomb-like graphenes(3D graphenes)using hybrid neural network potential-based molecular dynamics simulations and theoretical analyses.The elastic constants were obtained as functions of honeycomb hole size,denoted by the graphene wall length L.All five independent elastic constants in the large-L limit are proportional to L^(-1).The associated coefficients are combinations of elastic constants of two-dimensional graphene.High-order terms including L^(-2)and L^(-3)emerge for finite L values.They have three origins,the distorted areas close to the joint lines of 3D graphenes,the variation in solid angles between graphene plates,and the bending distortion of graphene plates.Significantly,the chirality becomes essential with decreasing L because the joint line structures are different between the armchair and zigzag-type 3D graphenes.Our findings provide insights into the elastic properties of graphene-based superstructures and can be used for further studies on graphene-based materials.

Underwater explosion in centrifuge partⅠ:Validation and calibration of scaling laws

Centrifuge is a promising tool for underwater explosion(UNDEX) research as both Mach and Froude similitudes could be satisfied with hyper-gravity in models, which would result in similarities in both shock wave and bubble oscillation. Scaling laws for UNDEX in centrifuge have been proposed based on dimensional analysis. Two dimensionless numbers, i.e.,π_3 and π_4, are used to characterize shock wave and bubble oscillation, respectively. To validate scaling laws, 17 UNDEX tests are designed by varying accelerations or explosive weights and positions in centrifuge. The tests are classified into different groups to validate scaling laws as well as calibrate coefficients in empirical formulae for both shock wave and bubble oscillation. The results show that changes of gravity acceleration or hydrodynamic pressure almost has no influence on shock wave peak pressure and time constant as long as π3 is constant. The dimensionless bubble period and maximum radius agreed with each other when π4 is constant. Based on the research, an example is exhibited to suggest method for the computation of initial loading conditions for a submerged obstacle subjected to UNDEX.