Set Of Scaling Laws Research Articles

Protoplanet collisions: New scaling laws from smooth particle hydrodynamics simulations

One common approach for solving collisions between protoplanets in simulations of planet formation is to employ analytical scaling laws. The most widely used one was developed by Leinhardt2012 from a catalog of sim 180 N-body simulations of rubble--pile collisions. In this work, we use a new catalogue of more than 20,000 SPH simulations to test the validity and the prediction capability of Leinhardt2012 scaling laws. We find that these laws overestimate the fragmentation efficiency in the merging regime and they are not able to properly reproduce the collision outcomes in the super--catastrophic regime. In the merging regime, we also notice a significant dependence between the collision outcome, in terms of the largest remnant mass, and the relative mass of the colliding protoplanets. Here, we present a new set of scaling laws that are able to better predict the collision outcome in all regimes and it is also able to reproduce the observed dependence on the mass ratio. We compare our new scaling laws against a machine learning approach and obtain similar prediction efficiency.

Similarity design method for dynamic characteristics of rotating composite blades in thermal environment

A similarity method for rotating composite blades in thermal environment is developed for the problems of difficulty, time consumption, high cost, and risk in the dynamic test under a high-temperature environment. Within this study, a set of scaling laws are derived for thin-walled composite plates based on their dynamic models and the applicability of the scaling laws is demonstrated in the similitude design of composite blades. The present dynamic model takes into account thermal stress and centrifugal stiffening effect. Analytical calculations show that the scaled models at relatively low temperatures can well reproduce the dynamic characteristics of their corresponding prototypes at high temperatures when taking into account the scaling of temperature change and rotational speed.

A transient intrusion flow on a cooled horizontal surface

We investigate the dynamical evolution and thermal processes of a fluid on a suddenly cooled horizontal surface subjected to an isothermal or an isoflux condition. A simple scaling analysis is performed to study an intrusion flow induced by a suddenly cooled surface. Analytical results demonstrate that there exist two scenarios for the dynamical evolution and thermal processes of such an intrusion flow for Prandtl numbers smaller than unity. In a typical case, the intrusion flow could travel under different regimes such as the “conduction regime,” “accelerating convection–inertial regime,” “steady convection–inertial regime,” and “decelerating convection–viscous regime,” respectively. A set of scaling laws under different regimes have been presented based on scaling analysis. In addition, a two-dimensional numerical simulation is conducted to validate the scaling laws. The scaling laws are in agreement with our numerical results.

From stellar coronae to gyrochronology: A theoretical and observational exploration

Context.Stellar spin down is the result of a complex process involving rotation, dynamo, wind, and magnetism. Multiwavelength surveys of solar-like stars have revealed the likely existence of relationships between their rotation, X-ray luminosity, mass losses, and magnetism. They impose strong constraints on the corona and wind of cool stars.Aims.We aim to provide power-law prescriptions of the mass loss of stars, of their magnetic field, and of their base coronal density and temperature that are compatible with their observationally-constrained spin down.Methods.We link the magnetic field and the mass-loss rate from a wind torque formulation, which is in agreement with the distribution of stellar rotation periods in open clusters and the Skumanich law. Given a wind model and an expression of the X-ray luminosity from radiative losses, we constrained the coronal properties by assuming different physical scenarios linking closed loops to coronal holes.Results.We find that the magnetic field and the mass loss are involved in a one-to-one correspondence that is constrained from spin down considerations. We show that a magnetic field, depending on both the Rossby number and the stellar mass, is required to keep a consistent spin down model. The estimates of the magnetic field and the mass-loss rate obtained from our formalism are consistent with statistical studies as well as individual observations and they give new leads to constrain the magnetic field-rotation relation. The set of scaling-laws we derived can be broadly applied to cool stars from the pre-main sequence to the end of the main sequence (MS), and they allow for stellar wind modeling that is consistent with all of the observational constraints available to date.

DEM-based modelling framework for spray-dried powders in ceramic tiles industry. Part I: Calibration procedure

This work describes a combined experimental-numerical study to characterize fine spray-dried powder used in the ceramic tile pressing process. A DEM-based granular assembly is endowed with a new set of scaling laws that allows for simulating reliably industrial processes using a much lower number of granules. To do it, a calibration strategy relying on three experimental setups is proposed; (i) compression test of bulk for granule stiffness, (ii) dynamic angle of repose and (iii) image analysis of the powder motion in a rotating drum for the intergranular and granular-boundary sliding and rolling friction coefficients. In order to evaluate the powder motion in a rotating drum, a robust method relying on a direct image analysis is proposed. This methodology makes it possible to quantitatively assess the frictional properties of the powder in contact with different surface materials.

Structural similitude design for a scaled composite wing box based on optimised stacking sequence

Testing appropriately-designed scaled-down structures instead of the full-size prototype structure is beneficial for quickly understanding the prototype structural behaviour in a cost-effective way. In this study, a novel approach is proposed to design scaled composite structures that can be used to predict the structural responses (e.g. static bending, modal behaviour, and compressive buckling) of the prototype. The objective is to overcome the main drawback of the conventional design method, which tends to result in low accuracy of the prototype prediction when certain variables cannot be appropriately scaled due to manufacturing constraints (e.g. ply thickness). In the present work, a set of scaling laws being independent of boundary conditions were firstly derived for plates and beams respectively based on their governing equations. The genetic algorithm (GA) was then applied to help design the stacking sequence of the scaled models, accommodating the mismatch in similarity conditions resulting from the manufacturing constraint in ply thickness. This GA-based design method was demonstrated to be effective in designing scaled plate, I-beam, and stiffened plate models, with improved accuracy in predicting the prototype structural behaviour compared with the conventional method. The application of this new design method was also extended to an A320 size wing box structure, validating its robustness for complex structures.

Buoyant flows on a vertical cone

Buoyant flows are of importance in mass and energy transformation in mountain areas, and significantly influence on local climate. The mountain is modeled by a vertical cone and three dimensional numerical simulation is carried out for Rayleigh numbers of Ra = 7.14 × 106 –2.34 × 107 in which the vertical cone is heated to the specified temperature in order to simulate buoyant flows on the mountain heated by solar radiation. Based on the comparison between numerical and experimental results, buoyant flows on the vertical cone are characterized and the mechanisms are analyzed. Further, the thickness of the thermal boundary layer, the velocity of the plume front, the vorticity, the volume and the onset of the break of the plume cap, and heat and mass transfer are measured; a set of scaling laws are obtained and verified by numerical and experimental results.

A new set of scaling relationships for DEM-CFD simulations of fluid–solid coupling problems in saturated and cohesiveless granular soils

The discrete element method (DEM) cannot effectively account for macroscale problems because of its high computational cost and time-consuming nature, especially for fluid–solid coupling problems. Therefore, a spatially scaled-down model, which should be established based on a set of scaling laws, is commonly employed to study the behaviors of the macroscale prototype. In the present study, various basic scaling relationships proposed by previous researchers for a scaled-down model are summarized first, as well as their limitations when they are used for DEM-CFD simulation of fluid–solid coupling problems in saturated and cohesiveless granular soils. Then, a new set of scaling relationships is proposed using a governing equation approach. Finally, a DEM-CFD model is established as an example analysis to study the feasibility and accuracy of the new scaling relationship present in this study for simulating fluid–solid coupling problems in saturated and cohesiveless granular soils. The results are analyzed in comparison with other scaling relationships, and the results indicate that the new set of scaling relationships is more reasonable and accurate than others.

Similitude theory for scaled friction pendulum bearings for shaking table experiments

Scaled models of friction pendulum bearings (FPBs), which require a similitude design based on a set of scaling laws, are typically employed in shaking table experiments to investigate the seismic behavior of the models. In this study, scaling laws for scaled FPBs are developed to determine the physical parameters of their models. Then, an equivalent method for the distribution of scaled FPBs in the model isolation layer is proposed for scaling cases in which fewer FPBs are arranged in the model than in the prototype. In addition to an accurate simulation of the global seismic response of the prototype isolation layer, the proposed equivalent method also aims to simulate the possible worst-case loading conditions of an individual FPB in a prototype. A model of an FPB-isolated space truss on top of four towers was scaled, manufactured and tested on a shaking table. A comparison of the seismic response of the global isolation layer and the individual FPB between the model and the prototype demonstrates the effectiveness of the developed theory.

Experimental and theoretical similitude analysis for flexural bending of scaled-down laminated I-beams

In this paper, similitude theory is presented as a methodology on how to design a meaningful scaled-down subcomponent that emulates the strain field experienced in the full-scale component. The scaling laws for the flap-wise bending of a laminated composite I-beam with shear deformability are derived based on the similitude analysis. The set of scaling laws is treated as a criterion to design a scaled-down laminated I-beam that is partially similar to a full-scale I-beam that is representative of what is used in a utility-scale wind turbine rotor blade. The designed I-beam is manufactured and loaded using a three-point bending test while digital image correlation is used to measure the strain and displacement fields. The analytical solution is examined with the measured strain and displacement values and a finite element simulation for validation. With the derived scaling laws shown to be credible, they are then applied to the measured strain and displacement values of the scaled-down I-beam to predict the corresponding values of the full-scale I-beam. The scaled model is shown to replicate the strain field of its full-scale parent I-beam. Moreover, applicability of the scaled beam in predicting the displacement field of the full-scale I-beam using experimental data is demonstrated.

Statistical mechanics of ontology based annotations

We present a statistical mechanical theory of the process of annotating an object with terms selected from an ontology. The term selection process is formulated as an ideal lattice gas model, but in a highly structured inhomogeneous field. The model enables us to explain patterns recently observed in real-world annotation data sets, in terms of the underlying graph structure of the ontology. By relating the external field strengths to the information content of each node in the ontology graph, the statistical mechanical model also allows us to propose a number of practical metrics for assessing the quality of both the ontology, and the annotations that arise from its use. Using the statistical mechanical formalism we also study an ensemble of ontologies of differing size and complexity; an analysis not readily performed using real data alone. Focusing on regular tree ontology graphs we uncover a rich set of scaling laws describing the growth in the optimal ontology size as the number of objects being annotated increases. In doing so we provide a further possible measure for assessment of ontologies.

Nonlinear analysis of compressed elastic thin films on elastic substrates: From wrinkling to buckle-delamination

Nonlinear buckling of elastic thin films on compliant substrates is studied by modeling and simulations to reveal the roles of pre-strain, elastic modulus ratio, and interfacial properties in morphological transition from wrinkles to buckle-delamination blisters. The model integrates an interfacial cohesive zone model with the Föppl–von Kármán plate theory and Green function method within the general framework of energy minimization. A kinetics approach is developed for numerical simulations. Subject to a uniaxial pre-strain, the numerical simulations confirm the analytically predicted critical conditions for onset of wrinkling and wrinkle-induced delamination, with which a phase diagram is constructed. It is found that, with increasing pre-strain, the equilibrium configuration evolves from flat to wrinkles, to concomitant wrinkles and buckle-delamination, and to an array of parallel straight blisters. The height and width of the buckle-delamination blisters can be approximately described by a set of scaling laws with respect to the pre-strain and interfacial toughness. Subject to an equi-biaxial pre-strain, the critical conditions are determined numerically to construct a similar phase diagram for the buckling modes. Moreover, by varying the pre-strain, modulus ratio, and interfacial toughness, a rich variety of equilibrium configurations are simulated, including straight blisters, and network blisters with or without wrinkles. These results provide considerable insight into diverse surface patterns in layered material systems as a result of the mechanical interactions between the film and the substrate through their interface, which suggests potential control parameters for designing specific surface patterns.

TensorBmode and stochastic Faraday mixing

This paper investigates the Faraday effect as a different source of $B$-mode polarization. The $E$-mode polarization is Faraday rotated provided a stochastic large-scale magnetic field is present prior to photon decoupling. In the first part of the paper we discuss the case where the tensor modes of the geometry are absent and we argue that the $B$ mode recently detected by the BICEP2 Collaboration cannot be explained by a large-scale magnetic field rotating, through the Faraday effect, the well-established $E$-mode polarization. In this case, the observed temperature autocorrelations would be excessively distorted by the magnetic field. In the second part of the paper the formation of Faraday rotation is treated as a stationary, random and Markovian process with the aim of generalizing a set of scaling laws originally derived in the absence of the tensor modes of the geometry. We show that the scalar, vector and tensor modes of the brightness perturbations can all be Faraday rotated even if the vector and tensor parts of the effect have been neglected, so far, by focussing the attention on the scalar aspects of the problem. The mixing between the power spectra of the $E$-mode and $B$-mode polarizations involves a unitary transformation depending nonlinearly on the Faraday rotation rate. The present approach is suitable for a general scrutiny of the polarization observables and of their frequency dependence.

Study nonlinear vibration of cross‐ply laminated plates using scale models

This article describes the establishment of necessary similarity conditions for geometrically nonlinear vibrations of thin cross‐ply laminated plates. The Von‐karman's strain‐displacement relations have been employed to model structural nonlinearity of the system. The Galerkin procedure is used to reduce the nonlinear partial differential equations to a nonlinearly second‐order ordinary differential equation. An analytical investigation based on the direct use of the governing equations of the systems is undertaken to derive the necessary scaling laws and similarity conditions. In this study, a set of scaling laws are introduced that can predict the nonlinear free vibration frequency of the prototypes by projecting the frequency of the model, accurately. The effects of distorted models with relaxations in the number of plies, stacking sequence, and different vibration amplitudes are studied. The results presented herein indicate that models with different relaxations can predict the nonlinear vibration frequency of prototypes with good accuracy. POLYM. COMPOS., 35:752–758, 2014. © 2013 Society of Plastics Engineers

Statistical symmetries and its impact on new decay modes and integral invariants of decaying turbulence

Classical decay laws of isotropic turbulence usually derived from the von Kármán–Howarth equation are essentially based on two paradigms. First, scaling symmetries of space and time, both tracing back to the Navier–Stokes equations in the limit of large Reynolds numbers (or r≫η), give rise to a temporal power-law decay for the turbulent kinetic energy and at the same time an algebraic growth of the integral length scale at an exponent that is uniquely coupled to the latter energy decay. Second, global invariants such as Birkhoff or Loitsianskii integrals determine the exponent of both power laws. We presently show that this class of decay laws may be considerably extended considering the entire set of multi-point correlation equations that admit a much wider class of symmetries. It was recently shown that these new symmetries are of paramount importance, e.g. in deriving the logarithmic law of the wall being an analytic solution of the multi-point equations. For the present case, it is particularly an additional scaling group, which we call statistical scaling group, that gives rise to two additional families of ‘canonical’ decay laws including those with an exponential characteristic for both the kinetic energy and the integral length scale. Finally, a second rather generic group admitted by all linear differential equations corresponding to the superposition principle induces an infinite set of scaling laws of rather complex form that may match rather generic initial conditions. All scaling laws are analyzed in the light of the above-mentioned integral invariants that have been further extended in the present contribution to an exponential-type invariant.

Applicability of small-scale models in prediction flutter pressure of delaminated composite beam-plates

In this study, the flutter pressure of delaminated composite beam-plates subjected to supersonic flow is investigated using similitude theory. An analytical investigation based on the direct use of the governing equations of the systems is undertaken to derive the necessary scaling laws and similarity conditions. The method presented herein has already been shown to be effective in the design of scaled down models for buckling and vibration response of laminated composite shells and panels. In this study, the flutter pressures of the prototypes are predicted by projecting the flutter pressure of the model using the derived scaling law. Based on the result of this study, for flutter of delaminated composite beam-plates, based on structural similitude, set of scaling laws can be found, which are used to develop design rules for small-scale models. For prototypes with the same material properties and stacking sequence as the prototype, distorted models with different delamination length and depth than those of the prototype can predict the behavior of the prototypes, accurately.

Extending generalized Horton laws to test embedding algorithms for topologic river networks

River networks in the landscape can be described as topologic rooted trees embedded in a three-dimensional surface. We examine the problem of embedding topologic binary rooted trees (BRTs) by investigating two space-filling embedding procedures: Top-Down, previously developed in the context of random self-similar networks (RSNs), and Bottom-Up, a new procedure developed here. We extend the concept of generalized Horton laws to interior sub catchments and create a new set of scaling laws that are used to test the embedding algorithms. We compare the two embedding strategies with respect to the scaling properties of the distribution of accumulated areas Aω and network magnitude Mω for complete order streams ω. The Bottom-Up procedure preserves the equality of distributions Aω/EAω=dMω/EMω; a feature observed in real basins. The Top-Down embedded networks fail to preserve this equality because of strong correlations of tile areas in the final tessellation. We conclude that the presence or absence of this equality is a useful test to diagnose river network models that describe the topology/geometry of natural drainage systems. We present some examples of applying the embedding algorithms to self similar trees (SSTs) and to RSNs. Finally, a technique is presented to map the resulting tiled region into a three-dimensional surface that corresponds to a landscape drained by the chosen network. Our results are a significant first step toward the goal of creating realistic embedded topologic trees, which are also required for the study of peak flow scaling in river networks in the presence of spatially variable rainfall and flood-generating processes.

Similitude requirements and scaling laws for flutter prediction of angle-ply composite plates

This study describes the establishment of the necessary similarity conditions for dynamic aeroelastic stability of laminated composite plates. An analytical investigation based on the direct use of the governing equations of the systems is undertaken to derive the necessary scaling laws and similarity conditions. The method presented herein has already been shown to be effective in the design of scaled down models for buckling and vibration response of laminated composite shells and panels. In this study, the flutter speeds of the prototypes are predicted by projecting the flutter speed of the model using the derived scaling law. The results presented herein, indicate that, for flutter of composite plates, based on structural similitude, a set of scaling laws can be found, which are used to develop design rules for small scale models. Models with different material properties, thickness, fiber orientation, Mach number, aspect ratio, and number of plies than those of the prototype can predict the flutter behavior of the prototype with good accuracy.

The velocity profile of laminar MHD flows in circular conducting pipes

We present numerical simulations without modeling of an incompressible, laminar, unidirectional circular pipe flow of an electrically conducting fluid under the influence of a uniform transverse magnetic field. Our computations are performed using a finite-volume code that uses a charge-conserving formulation [called current-conservative formulation in references (Ni et al J Comput Phys 221(1):174–204, 2007, Ni et al J Comput Phys 227(1):205–228, 2007)]. Using high resolution unstructured meshes, we consider Hartmann numbers up to 3000 and various values of the wall conductance ratio c. In the limit \({c{\ll}{\rm Ha}^{-1}}\) (insulating wall), our results are in excellent agreement with the so-called asymptotic solution (Shercliff J Fluid Mech 1:644–666, 1956). For higher values of the wall conductance ratio, a discrepancy with the asymptotic solution is observed and we exhibit regions of velocity overspeed in the Roberts layers. We characterise these overspeed regions as a function of the wall conductance ratio and the Hartmann number; a set of scaling laws is derived that is coherent with existing asymptotic analysis.

Structural Similitude for Vibration of Delaminated Composite Beam-Plates

This study describes the establishment of similarity conditions among structural systems. The main objective of this research is to study the applicability of similitude theory in establishing necessary similarity conditions for designing scaled down models for predicting the vibration behavior of delaminated composite beam-plates. The results presented herein indicate that, for vibration response of delaminated plates, based on structural similitude, a set of scaling laws can be found to establish necessary structural similitude and design rules for small scale models. For models with the same material properties and stacking sequence as prototype, distorted models with different delamination length than those of prototype can predict the behavior of the prototypes with good accuracy.