External Length Scale Research Articles

Beam-type acoustic black holes incorporating the microstructure-dependent nonlocal effect

Acoustic black holes (ABHs) are effective passive damping structures for capturing bending waves at wedge edges. However, as the wedge shape continuously reduces in size, the intrinsic length scale of the ABH's tip gradually approaches the external length scale, such as the wavelength of wave, which causes a phenomenon known as the "scale effect" to take hold of the ABH. Despite gaining considerable attention in the past two decades, the influence of intrinsic length on the energy aggregation of ABHs has not been thoroughly investigated. This study aims to explore the significant contribution of microstructure-dependent nonlocal effects on the dynamic behavior of ABHs, which holds great scientific importance. The nonlocality arises from the inherent length of the material being studied. A nonlocal elastic theory (NET) is employed to model the dynamic behavior of one-dimensional ABH beams. Geometric acoustic method is proposed to analyze the power-law beam wave number and microstructure-related reflection coefficient. Results indicate a significant impact of nonlocality on the acoustic energy of beam-type ABHs, particularly when the nonlocal intrinsic length approaches the truncation length of the wedge. Moreover, a comparison between the present study and classical theory reveals an overestimation of the reflection coefficient in conventional works for small truncation lengths. The relationship between the reflection coefficient, truncation length, and nonlocal characteristic lengths is thoroughly examined. Additionally, the influence of the damping layer and different power-law exponents m on ABH behavior is extensively studied.

Elastic solution of surface loaded layer with couple and surface stress effects

In this study, an elastic solution of an axisymmetrically surface-loaded thin layer resting on a rigid substrate is established by taking the surface stress and material microstructural effects into account. Derived solutions provide not only a means to investigate the size effects on the mechanical response but also a set of fundamental solutions essential for tackling contact problems in a micro/nano scale. In the formulation, the couple stress and surface elasticity theories are adopted to simulate the microstructured bulk layer and the surface material, respectively. A general solution of an elastic field within the bulk layer is obtained first by Hankel transform method and subsequently used together with the surface equations and boundary conditions to form a set of conditions essential for determining all unknown constants. After being fully tested with available benchmark solutions, results are used to study the role of surface and couple stresses on the load transferring mechanism to the substrate and its size-dependent characteristic for a wide range of external length scales relative to the internal length scales.

On the Svelteness as an Engineering Tool in Constructal Design: A Critical Review

The application of Constructal theory to the flow design in engineering applications connects the channels’ architecture with their freedom to morph. Assessing the evolution of the flow architecture in Constructal Design requires a core parameter. Svelteness is the best candidate, given its definition as a flow architecture’s intrinsic global geometric property. However, despite the broad applicability range of Constructal theory, research has restricted the use of Svelteness to fluid flow, focusing on using it to justify disregarding local pressure losses compared to distributed friction losses, connecting the design of the flow to its survival. This work reviews the application of Svelteness, from the intuitive perception of its meaning to its use in engineering design, namely understanding the difference between assuming the impact of Svelteness versus considering the effects of its evolution in time. This understanding allows exploring the depth and validity of applying Svelteness as a universal criterion, comparing the different methods that define it, and discussing its relevance to explaining freedom to morph in a flow. Using two types of configurations (serpentine and canopy-to-canopy), the review shows the relevance of using the configuration area for the external length scale in the presence of ramifications and a relation between the configuration area and the path followed by what flows in the absence of configurations. Finally, we discuss the establishment of Svelteness as an engineering design tool using the law of diminishing returns.

Thermoelastic responses in a nonlocal infinite solid with a circular cylindrical cavity due to a moving heat supply under the MGT model of thermal conductivity

Since the non-local continuum theory includes additional information regarding the long-range forces of molecules and atoms, the external length scale parameter may be included in the formulation of physical and engineering modeling. In this study, a new non-local thermoelastic model is presented in which the size-dependent effect has been included in the equations of motion and constitutive relations. To establish the proposed generalized model of heat conduction, the Moore-Gibson-Thompson concept was used. As an application to the presented model, a problem of thermally induced behavior in an infinite nonlocal solid with a cylindrical hole whose surrounding surface is free of traction and affected by a time-dependent moving heat supply is studied. In the transformed field of the Laplace transform, analytical formulas for thermophysical fields have been derived. To obtain the numerical results of the physical field, an approximate approach was applied to find the numerical inversion. The effect of non-localization and heat source velocity on the responses of the investigated fields is graphically illustrated and discussed in detail. The results show that Eringen’s non-local elasticity model is a promising criterion for analyzing nanostructures, taking into account the effect of small size.

Transitional Shock Boundary Layer Interactions on a Compression Ramp at Mach 4

Strong laminar/transitional shock boundary layer interactions (SBLIs) have been investigated in a Mach 4 vacuum-driven wind tunnel with supporting computational fluid dynamics analysis. Such flows are extremely susceptible to large-scale separation, unsteadiness, and high surface heat flux, which can limit control authority on high-speed vehicles. The research community has heavily focused on turbulent interactions, leaving little understanding of how laminar/transitional cases behave with varying external parameters. Four compression ramp angles of 15, 18, 22, and 28° have been tested on a flat plate (unit Reynolds number ) with a range of external length scales (flat plate Reynolds number varies between and at the ramp corner). Laminar separation was observed at boundary-layer thickness Reynolds number between 2700 and 4300, with the centerline length of separation scaling mildly with Reynolds number, increasing proportionally with . Transitional reattachment is observed for strong SBLIs at higher values of . Analysis of high-speed schlieren images shows that the separation shock exhibits low-frequency unsteadiness at a Strouhal number near , consistent with turbulent SBLIs. Detailed phase analysis of schlieren data shows no evidence of upstream influence. Instead, motion of the shear layer and reattachment shock precedes motion of the separation shock foot, suggesting a downstream unsteadiness mechanism. In addition, extensive analysis of subtle features within the interaction identified a slow-moving density disturbance within the bubble that convects toward the shock foot and directly leads separation shock motion. Negligible coherence was observed between low-frequency separation shock motion and acoustic content within the separation bubble.

Pseudoelasticity in twinned α-Fe nanowires under bending

The pre-existing twin boundaries play an important role in the mechanical behaviors of metals. In the present work, we studied the bending deformation of [100]-oriented α-Fe nanowires seeded with 〈111〉/{112} twin boundaries using atomistic simulations. Bending behaviors along two different orientations were investigated. The initially flat twin boundaries become curved with a high density of 1/6 〈111〉/{112} partial dislocations under bending when the neutral plane intersects with twin plane along [01-1] direction, whereas could transform to the nonconventional {110} interfaces under bending when the neutral plane intersects with twin plane along [111] direction. The bent α-Fe nanowires could recover to the original state upon unloading, leading to a unique pseudoelasticity. The driving force for shape recovery primarily arises from the high interfacial energy, i.e. curved {112} twin boundaries or the transformed {110} interfaces. We further characterized the recoverability by the ratio η of the nanowire size (external length scale) to twin boundary spacing (internal length scale). This η-dependent recoverability matches well with the experimental data at large length scale. The present work may help understand the unique mechanical properties associated with twin boundaries in metals.

Twisting of pre-twinned α-Fe nanowires: from mild to wild avalanche dynamics

We studied the torsion behavior of α-Fe nanowires seeded with twin boundaries (TBs) using molecular dynamics simulations. Twisting the wire generates topological defects in the twin walls, namely kinks inside the twin walls for small twist angles, and junctions between kinks for large twist angles. During twisting the kink motion is jerky and uncorrelated at small twist angles. The probability density function (PDF) of jerks strength follows approximately a Gaussian distribution, indicating a mild deformation mode. The kink dynamics transforms from mild to wild at larger twist angles when complex twin patterns with a high density of junctions are generated. The collective motion of kinks now shows avalanche behavior with the energy being power-law distributed. The wildness, which measures the proportion of strain energy relaxed through such avalanches, is correlated with the junction density, and controlled by the external length scale (wire diameter) as well as an internal length scale (twin boundary spacing). Good strain-stress recoverability is achieved when unloading the wire before the formation of complex twin patterns. We correlate the evolution of twin patterns with a statistical analysis of jerk dynamics, which identifies the unique mechanical properties governed by twin boundary motion in nanowires.

Twisting of Pre-Twinned Alpha-Fe Nanowires: From Mild to Wild Avalanche Dynamics

Alpha-Fe nanowires are seeded with twin walls (TBs) with orientations perpendicular to the wire direction. Twisting the wire generates topological defects in the twin walls, namely new twin boundaries (kinks) inside the twin walls for small twist angles, and junctions between kinks for large twist angles. During twisting the kink motion is jerky and uncorrelated at small twisting angles. The probability density function (PDF) of jerk energies follows approximately a Gaussian distribution, indicating a mild deformation mode. The kink dynamics transforms from mild to wild at larger twist angles when complex twin patterns with a high density of junctions are generated. The collective motion of kinks now shows avalanche behavior with the energy being power-law distributed. The wildness, which measures the proportion of strain energy relaxed through such avalanches, is correlated with the junction density, and controlled by the external length scale (wire diameter) as well as an internal length scale (twin boundary spacing). Good strain-stress recoverability is achieved when unloading the wire before the formation of complex twin patterns. We correlate the evolution of twin patterns with a statistical analysis of jerk dynamics, which identifies the unique mechanical properties governed by twin boundary motion in nanowires.

Correlating micro-pillar compression behavior with bulk mechanical properties: Nanolaminated graphene-Al composite as a case study

Uniaxial compression tests were carried out on nanolaminated graphene (reduced graphene oxide, RGO)-Al composite micro-pillars of various diameters. It was found that, when the pillar diameter was an order of magnitude larger than the internal microstructural length scale (Al lamella thickness), the pillars' mechanical behavior resembled that of the corresponding bulk. When the external length scale approached the internal one, however, both mechanical strength and deformation mode of the pillars significantly deviated from their bulk behavior. These results may have implications for the dimensional design of the small scale test specimens fabricated from bulk materials with a “real” microstructure.

A scaling law for the shear-production range of second-order structure functions

Dimensional analysis suggests that the dissipation length scale ($\ell _{{\it\epsilon}}=u_{\star }^{3}/{\it\epsilon}$) is the appropriate scale for the shear-production range of the second-order streamwise structure function in neutrally stratified turbulent shear flows near solid boundaries, including smooth- and rough-wall boundary layers and shear layers above canopies (e.g. crops, forests and cities). These flows have two major characteristics in common: (i) a single velocity scale, i.e. the friction velocity ($u_{\star }$) and (ii) the presence of large eddies that scale with an external length scale much larger than the local integral length scale. No assumptions are made about the local integral scale, which is shown to be proportional to$\ell _{{\it\epsilon}}$for the scaling analysis to be consistent with Kolmogorov’s result for the inertial subrange. Here${\it\epsilon}$is the rate of dissipation of turbulent kinetic energy (TKE) that represents the rate of energy cascade in the inertial subrange. The scaling yields a log-law dependence of the second-order streamwise structure function on ($r/\ell _{{\it\epsilon}}$), where$r$is the streamwise spatial separation. This scaling law is confirmed by large-eddy simulation (LES) results in the roughness sublayer above a model canopy, where the imbalance between local production and dissipation of TKE is much greater than in the inertial layer of wall turbulence and the local integral scale is affected by two external length scales. Parameters estimated for the log-law dependence on ($r/\ell _{{\it\epsilon}}$) are in reasonable agreement with those reported for the inertial layer of wall turbulence. This leads to two important conclusions. Firstly, the validity of the$\ell _{{\it\epsilon}}$-scaling is extended to shear flows with a much greater imbalance between production and dissipation, indicating possible universality of the shear-production range in flows near solid boundaries. Secondly, from a modelling perspective,$\ell _{{\it\epsilon}}$is the appropriate scale to characterize turbulence in shear flows with multiple externally imposed length scales.

Universal power-law strengthening in metals?

The strength of most metals scales with either an internal or external length scale. Motivated by the wide applicability of this phenomenon to material type and microstructure, we develop a model which gives quantitative insight into the scaling exponent using the known universal properties of a dislocation network and the leading order stress dependence of an underlying critical stress distribution. The approach is found to be equally valid for both Hall–Petch strengthening and the smaller-is-stronger paradigm of small scale plasticity.

Scaling behaviour in impulsively started viscous flow past a finite flat plate

AbstractViscous flow past a finite flat plate which is impulsively started in the direction normal to itself is studied numerically using a high-order mixed finite-difference and semi-Lagrangian scheme. The goal is to resolve the details of the vorticity generation, and to determine the dependence of the flow on time and Reynolds number. Vorticity contours, streaklines and streamlines are presented for a range of times$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}t\in [0.0002,5]$and Reynolds numbers$\mathit{Re}\in [250,2000]$, non-dimensionalized with respect to the driving velocity and the plate length. At early times, the starting vortex is small relative to the plate length and is expected to grow as if an external length scale were absent. We identify three different types of scaling behaviours consistent with this premise. (i) At early times, solutions with different values of$\mathit{Re}$are identical up to rescaling. (ii) The solution for fixed$\mathit{Re}$satisfies a viscous similarity law in time, locally in space, as illustrated by the core vorticity maximum, the upstream boundary layer thickness, and the maximum speed, in three different regions of the flow. (iii) The vortex core trajectory and the shed circulation satisfy inviscid scaling laws for several decades in time, and are consequently essentially$\mathit{Re}$-independent at these times. In addition, the computed induced drag and tangential forces are found to follow approximate scaling laws that define their dependence on time and$\mathit{Re}$.

Influence of Random Pinning on the Crystallization Process in Suspensions of Hard Spheres

We discuss crystal formation in supersaturated suspensions of monodisperse hard spheres with a concentration of hard spheres randomly pinned in space and time. The pinning procedure introduces an external length scale and an external time scale that restrict the accessible number of configureurations and ultimately the number of pathways leading to crystallization. We observe a significant drop in the nucleation rate density at a characteristic pinning concentration that can be directly related to the structure of the critical nucleus and the dynamics of its formation in the unpinned system.

Turbulence: Large-scale sweeping and the emergence of small-scale Kolmogorov spectra

The dynamics of fully developed hydrodynamic turbulence still is a basically unsolved theoretical problem, due to the strong-coupling long-range nonlinearities in the Navier-Stokes equations. The present analysis focuses on the small-scale fluctuations in a turbulent boundary layer with one external length scale y(o). After taking a (2+1)D spatiotemporal spectral transform of the fluctuating vorticity fields, care is taken of large-scale sweeping which arises as a collective zero mode from the nonlinear flow terms. The "unswept" small-scale nonlinearities are then shown to be asymptotically locally isotropic (i.e., for wave numbers k→∞) by internal consistency, which allows to close the nonlinear hierarchy. The Navier-Stokes equations (without external forcing) are integrated to give the spectral response of the fluctuating small-scale velocity fields on the presence of a locally isotropic blob of turbulence while it is being swept around over an arbitrary steady state mean velocity profile, using viscous boundary conditions at y=0. Averaging the response spectrum over all possible orientational configurations and sweep velocities results in a novel self-consistency integral for the 4D energy spectrum function. The distribution of turbulence sweep velocities is modeled by means of Lévy-type densities, having an algebraic tail with power p>1. The generic case (which includes Von Kármán's logarithmic mean velocity profile) is found to correspond to 1<p<3. Asymptotic analysis of the self-consistency integral leads to a differential equation which fixes the scaling exponent λ of the unswept frequency Δ and admits a nonempty, integrable and positive definite Airy-type frequency spectrum E(ı)(k,Δ/k(λ))∼k(μ) with so-called "normal" Kolmogorov scaling, that is, μ=-7/3 and λ=2/3. Anomalous scaling is possible for one special mean profile.

Length Scales Interaction in Nonlocal Plastic Strain Localization of Bars of Varying Section

In computational mechanics, strain softening generates ill-posed boundary value problems, which cannot be solved without being regularized, e.g., through the introduction of an internal length scale. This paper investigates how the internal length scale that is introduced by regularization may interact with the external length scale arising from boundary conditions in the particular case of a strain-softening bar of varying cross section and a nonlocal averaging regularization. The interaction of internal and external length scales is examined using an analytical closed-form solution for overnonlocal softening plasticity that derives from a Fredholm equation of the second kind. In the absence of external length (bars of constant section), the analysis shows that the overnonlocal averaging confines and smoothly distributes plastic strain into a localized band. The localization width, plastic strain distribution inside the band, and load-displacement response are controlled by the internal length of the averaging function and the overnonlocal weighting factor. In the presence of external length (bar of varying section), the analytical solution shows that the localization width is controlled by the interaction of external and internal length scales. This interaction is significant when the external and internal lengths are of comparable magnitude, and decreases when the external length becomes large compared to the internal length. The bandwidth is found to depend on the internal and external lengths and stress level while strain localizes, and to relate only to the internal length when the bar collapses.

Elemental T and Y Shapes of Tree Networks of Ducts with Various Cross-Sectional Shapes

This paper reports optimal bifurcation shapes T and Y in turbulent regime of tree-shaped flows. Unlike earlier studies of T and Y constructs, here the effect of pressure losses at the junction is taken into account, and the wall roughness and duct cross-sectional shapes are free to vary. The optimal ratio of duct cross-sectional areas as a generalization of Murray's law, the optimal ratio of duct lengths, and the optimal angle between the branches of the Y are presented. These optimal geometrical features are reported as functions of the flow direction splitting flow versus merging flow, wall roughness, duct cross-sectional shape, and svelteness. The svelteness, Sv, is a global property defined as the external length scale of the flow construct divided by the internal length scale. It is shown that the effect of junction pressure losses on the optimized architecture can be neglected when Sv 3/2 is greater than approximately 10 4 . Two dimension- less terms are introduced and shown to be useful for the optimization of flow networks. DOI: 10.1061/ASCE0733-94292009135:2132 CE Database subject headings: Cross sections; Bifurcations; Walls; Roughness; Hydraulic networks.

A New View of Silicon Electrochemistry

The salient features of the current burst model for the Si–electrolyte interface is presented in a short overview. In the current burst model, charge transfer at the Si–electrolyte interface is localized in space and time and intrinsically stochastic; all phenomena encountered in Si electrochemistry then are manifestations of the nucleation probabilities for current bursts, which depend on the system parameter in an unambiguous way. One of the unique features of the model is the existence of an internal time constant (equal to the average duration of a current pulse) which ultimately is the reason for the observed oscillations of the current in time and in space, the latter case describing pores. For pores, the intrinsic time constant is expressed as an intrinsic length scale which is determined by the interaction of current bursts within the same cycle, i.e. by the size of synchronized domains. The pore morphology is the result of the interaction of the intrinsic length scale with external length scales, e.g. the space charge region depth; smooth macropores are obtained when the relevant length scales are similar. The existence of an intrinsic time constant and length scale will be demonstrated by resonance phenomena which were observed for the first time in pore growth.

Internal degrees of freedom in a thermodynamical model for intracell biological transport

The mechanism for intracell transport based on the 1D motion of certain proteins along the cytoskeleton microtubular fibres, substantiated by recent experimental biological ressearch, is analysed from the perspective of a non-equilibrium thermodynamical model which incorporates an internal protein degree of freedom. The role of resonances between internal and external length scales in the mechanism functioning is succintly analysed from the perspective of the cell-biological potentialities of the model.

Propagative Sine-Gordon solitons in the spatially forced Kelvin-Helmholtz instability

The spatio-temporal evolution of the vortex sheet separating two finite-depth layers of immiscible fluids is examined in the vicinity of threshold when spatially periodic forcing is imposed at the horizontal boundaries. As a result of the Galilean invariance of the problem, the interface deformation is shown to satisfy a coupled system of evolution equations involving not only the usual “short-wave” at the critical wavenumber but also a shallow-water “long-wave” associated with the mean elevation of the interface. The weakly nonlinear model is further studied in the Boussinesq approximation where it reduces to a forced Klein-Gordon equation. Thus, the secondary Benjamin-Feir instability of nonlinear Stokes wavetrains is analysed in the absence of forcing. When spatial forcing is reintroduced, the competition between the imposed external length scale and the natural length scale of the interface is shown analytically to give rise to one-dimensional propagating Sine-Gordon phase solitons. Numerical simulations of the Klein-Gordon evolution model fully confirm this prediction and also lead to the determination of the range of stability of phase solitons.

Turbulence, critical fluctuations, and intermittency

An explicit analogy is developed between the long-range fluctuations near a critical point and the long-range fluctuations in $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ space in a turbulent fluid in the limit of large Reynolds number. Mathematically the analogy is between the probability functional of the spins on a lattice and the probability functional for the Fourier-transformed velocity fluctuations. The latter is given by the stationary solution of an exact Fokker-Planck equation originally derived by Edwards. This functional, which is not explicitly known, plays the role of a Hamiltonian for stationary turbulence. The limit of zero viscosity is equivalent to the temperature approaching the critical temperature from above. The order parameter is the vorticity-vorticity-correlation function in $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ space, which in three dimensions is the same as the energy spectrum $E(k)$. The mechanism of energy dissipation constrains the critical exponent $\ensuremath{\gamma}$ to satisfy $\ensuremath{\gamma}=1$. The fluid is stirred at an external length scale ${k}_{0}^{\ensuremath{-}1}$. If effects due to the nonzero value of ${k}_{0}$ are ignored, the scaling properties of the full probability functional can be explicitly calculated, and lead to the original Kolmogorov theory. This can be characterized by the critical exponents $\ensuremath{\nu}=\frac{3}{4}$, $\ensuremath{\eta}=\frac{2}{3}$, indicating that the equivalent Hamiltonian is of long range. For ${k}_{0}\ensuremath{\ne}0$, the range of fluctuations in $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$, space still diverges for zero viscosity. This limit no longer leads to a probability functional with the scale symmetry of a fixed point. We examine the cascade process under the assumption that $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$-space correlation functions go as some power of $r$. The original Kolmogorov theory is recovered by neglecting fluctuations in the cascade process. When these fluctuations are included, correlation functions of different order do not scale in the same way. This is in accord with phenomenological theories of intermittency and with experiment. An effective critical exponent ${\ensuremath{\eta}}^{\ensuremath{'}}=\frac{2}{3}+\ensuremath{\zeta}$ is still defined by the energy spectrum $E(k)$. The parameter $E(k)$ is one of a family of intermittency exponents which are intrinsic properties of the Navier-Stokes equations. Their calculation from the dynamical equations remains an objective for future work.