Ratio Of Length Scale Research Articles

Interface behavior of a thin-film bonded to a graded layer coated elastic half-plane

The interface behavior of a thin-film adhering to an elastic half-plane with an arbitrarily graded transition layer is analyzed by a linearly unified model when the interface between the thin-film and the graded layer is subjected to a mismatch strain. A series of continuous but piecewise linear functions are adopted to describe different graded variation laws of the transition layer's modulus. The fundamental solution of a graded-layer coated half-plane under a pair of normal and tangential concentrated forces is obtained first, and then the mismatch problem of a thin-film bonded to an elastic half-plane with an arbitrarily graded transition layer is further investigated. Solving the obtained integral governing equation of Cauchy singularity leads to the interfacial shear stress, the normal stress in the film, as well as the stress intensity factor near the bonded edge. It is found that the interfacial mechanics of a thin-film bonded to a graded layer is significantly affected by varying gradient law, stiffness of the film, ratio of shear modulus and length scale of the graded transition layer. The results should be helpful for the design of systems with thin-films and functional graded materials and could guide engineers to choose proper graded materials for particular applications.

Characterizing dispersivity and stagnation in porous media using NMR flow propagators

Low-field nuclear magnetic resonance (NMR) displacement probability distributions (flow propagators) are presented for water flowing through heterogeneous porous materials. Four sedimentary rocks have been chosen as example systems: Dolostone, Bentheimer sandstone, Berea sandstone, and Indiana limestone (in order of decreasing permeability). The fluid displacement is characterized by pre-asymptotic Stokes’ flow and so the probability distributions are bimodal, with peaks corresponding to stagnant fluid in dead-end pores and flowing fluid in the connected porosity. Cut-off Gaussian functions are used to fit the flowing and stagnant peaks independently. An effective dispersivity length scale Lv (also known as the mixing length scale) is estimated by fitting the portion of the probability distribution corresponding to the flowing fluid. For the relatively homogeneous Bentheimer sandstone, the ratio of effective dispersivity length scale to effective transport diameter dt is Lv/dt≈16, which is an order of magnitude larger than for randomly packed glass beads where Lv/dt≈1.8. We compare these dispersivity parameters to similar values extracted from a cumulant analysis of the entire propagator. Fitting a cut-off Gaussian avoids the usual complications of analyzing dispersion in the presence of the ubiquitous stagnant fluid, and results in a clear demonstration of the influence of long-range heterogeneities on the dispersivity for flow in real sedimentary rocks.

A Layered Description of a Premixed Flame Stabilized in Stagnating Turbulence

ABSTRACTIn the thin-flame regime of turbulent premixed combustion, instantaneous progress variable gradients are essentially fixed by propagating flamelets. The laminar flamelet internal structure thus imposes, at least to some extent, the progress variable one-point one-time statistics, i.e., the progress variable PDF . In the present study a generalized presumed probability density function (PDF) shape is considered to account for possible departures from the thin-flame limit. The parameters that characterize this PDF shape depend on the ratio of the Kolmogorov length scale to laminar flame width. The corresponding framework may be associated to a layered description of the turbulent flame brush (TFB) with the thicknesses of the different sub-layers determined from the knowledge of both the Reynolds and Karlovitz numbers, ReT and Ka. It incorporates boundary layers on both sides of the TFB, which are associated to the finite thickness of the local flamelets, i.e., small but finite values of the Karlovitz number. This description leads to a modeling proposal for premixed turbulent combustion. A tabulation is constructed based on canonical one-dimensional planar unstrained laminar premixed flame computations and the different quantities are tabulated as functions of ReT and Ka. The final closure is applied to the numerical simulations of premixed flames stabilized in stagnating turbulence.

Stratified Turbulence and Mixing Efficiency in a Salt Wedge Estuary

AbstractHigh-resolution observations of velocity, salinity, and turbulence quantities were collected in a salt wedge estuary to quantify the efficiency of stratified mixing in a high-energy environment. During the ebb tide, a midwater column layer of strong shear and stratification developed, exhibiting near-critical gradient Richardson numbers and turbulent kinetic energy (TKE) dissipation rates greater than 10−4 m2 s−3, based on inertial subrange spectra. Collocated estimates of scalar variance dissipation from microconductivity sensors were used to estimate buoyancy flux and the flux Richardson number Rif. The majority of the samples were outside the boundary layer, based on the ratio of Ozmidov and boundary length scales, and had a mean Rif = 0.23 ± 0.01 (dissipation flux coefficient Γ = 0.30 ± 0.02) and a median gradient Richardson number Rig = 0.25. The boundary-influenced subset of the data had decreased efficiency, with Rif = 0.17 ± 0.02 (Γ = 0.20 ± 0.03) and median Rig = 0.16. The relationship between Rif and Rig was consistent with a turbulent Prandtl number of 1. Acoustic backscatter imagery revealed coherent braids in the mixing layer during the early ebb and a transition to more homogeneous turbulence in the midebb. A temporal trend in efficiency was also visible, with higher efficiency in the early ebb and lower efficiency in the late ebb when the bottom boundary layer had greater influence on the flow. These findings show that mixing efficiency of turbulence in a continuously forced, energetic, free shear layer can be significantly greater than the broadly cited upper bound from Osborn of 0.15–0.17.

Assessment of different chemistry reduction methods based on principal component analysis: Comparison of the MG-PCA and score-PCA approaches

Two families of PCA based combustion model were recently developed: score-PCA and MG-PCA. This paper presents the first comparative study of the two aftermentioned methods. Both methods are first benchmarked on 1-D laminar flames of hydrogen–air and syngas–air, to verify the consistency of the approaches. The sensitivity to differential diffusion is carefully addressed and a rotation technique is proposed, to allow for using score-PCA with differential diffusion. Then, 2-D flame-turbulence interactions are considered, spanning a wide range of velocity and length scale ratios, to demonstrate the ability of PC-based combustion models to efficiently capture the behavior of different combustion regimes, from flamelet-like flames to distributed reaction regimes.

Removing Grit During Wastewater Treatment: CFD Analysis of HDVS Performance.

Computational Fluid Dynamics (CFD) was used to simulate the grit and sand separation effectiveness of a typical hydrodynamic vortex separator (HDVS) system. The analysis examined the influences on the separator efficiency of: flow rate, fluid viscosities, total suspended solids (TSS), and particle size and distribution. It was found that separator efficiency for a wide range of these independent variables could be consolidated into a few curves based on the particle fall velocity to separator inflow velocity ratio, Ws/Vin. Based on CFD analysis it was also determined that systems of different sizes with length scale ratios ranging from 1 to 10 performed similarly when Ws/Vin and TSS were held constant. The CFD results have also been compared to a limited range of experimental data.

Heat Transfer Analysis of Mixed Electro-osmosis Pressure-Driven Flow for Power-Law Fluids Through a Microtube

In this work, convection heat transfer for combined electro-osmotic and pressure driven flow of power-law fluid through a microtube has been analyzed. Typical results for velocity and temperature distributions, friction coefficient, and Nusselt number are illustrated for various values of key parameters such as flow behavior index, length scale ratio (ratio of Debye length to tube radius), dimensionless pressure gradient, and dimensionless Joule heating parameter. The results reveal that friction coefficient decreases with increasing dimensionless pressure gradient, and classical Poiseuille solutions can be retrieved as the dimensionless pressure gradient approaches to infinite. To increase the length scale ratio has the effect to reduce Nusselt number, while the influence of this ratio on Nusselt number diminishes as the pressure gradient increases. With the same magnitude of dimensionless Joule heating parameter, Nusselt number can be increased by increasing both the flow behavior index and dimensionless pressure gradient for surface cooling, while the opposite behavior is observed for surface heating. Also, singularities occurs in the Nusselt number variations for surface cooling as the ratio of Joule heating to wall heat flux is sufficiently large with negative sign.

Sensitivity of the two-dimensional shearless mixing layer to the initial turbulent kinetic energy and integral length scale.

The shearless mixing layer is generated from the interaction of two homogeneous isotropic turbulence (HIT) fields with different integral scales ℓ_{1} and ℓ_{2} and different turbulent kinetic energies E_{1} and E_{2}. In this study, the sensitivity of temporal evolutions of two-dimensional, incompressible shearless mixing layers to the parametric variations of ℓ_{1}/ℓ_{2} and E_{1}/E_{2} is investigated. The sensitivity methodology is based on the nonintrusive approach; using direct numerical simulation and generalized polynomial chaos expansion. The analysis is carried out at Re_{ℓ_{1}}=90 for the high-energy HIT region and different integral length scale ratios 1/4≤ℓ_{1}/ℓ_{2}≤4 and turbulent kinetic energy ratios 1≤E_{1}/E_{2}≤30. It is found that the most influential parameter on the variability of the mixing layer evolution is the turbulent kinetic energy while variations of the integral length scale show a negligible influence on the flow field variability. A significant level of anisotropy and intermittency is observed in both large and small scales. In particular, it is found that large scales have higher levels of intermittency and sensitivity to the variations of ℓ_{1}/ℓ_{2} and E_{1}/E_{2} compared to the small scales. Reconstructed response surfaces of the flow field intermittency and the turbulent penetration depth show monotonic dependence on ℓ_{1}/ℓ_{2} and E_{1}/E_{2}. The mixing layer growth rate and the mixing efficiency both show sensitive dependence on the initial condition parameters. However, the probability density function of these quantities shows relatively small solution variations in response to the variations of the initial condition parameters.

Micromechanical design of hierarchical composites using global load sharing theory

Hierarchical composites, embodied by natural materials ranging from bone to bamboo, may offer combinations of material properties inaccessible to conventional composites. Using global load sharing (GLS) theory, a well-established micromechanics model for composites, we develop accurate numerical and analytical predictions for the strength and toughness of hierarchical composites with arbitrary fiber geometries, fiber strengths, interface properties, and number of hierarchical levels, N. The model demonstrates that two key material properties at each hierarchical level—a characteristic strength and a characteristic fiber length—control the scalings of composite properties. One crucial finding is that short- and long-fiber composites behave radically differently. Long-fiber composites are significantly stronger than short-fiber composites, by a factor of 2N or more; they are also significantly tougher because their fiber breaks are bridged by smaller-scale fibers that dissipate additional energy. Indeed, an “infinite” fiber length appears to be optimal in hierarchical composites. However, at the highest level of the composite, long fibers localize on planes of pre-existing damage, and thus short fibers must be employed instead to achieve notch sensitivity and damage tolerance. We conclude by providing simple guidelines for microstructural design of hierarchical composites, including the selection of N, the fiber lengths, the ratio of length scales at successive hierarchical levels, the fiber volume fractions, and the desired properties of the smallest-scale reinforcement. Our model enables superior hierarchical composites to be designed in a rational way, without resorting either to numerical simulation or trial-and-error-based experimentation.

A most general strain gradient plate formulation for size-dependent geometrically nonlinear free vibration analysis of functionally graded shear deformable rectangular microplates

In this work, the size-dependent geometrically nonlinear free vibration of functionally graded (FG) microplates is investigated. For this purpose, with the aid of Hamilton’s principle, a nonclassical rectangular microplate model is developed based on Mindlin’s strain gradient theory, Mindlin’s plate theory and the von Karman geometric nonlinearity. For some specific values of the length scale material parameters, the simple form of size-dependent mathematical formulation based on the modified strain gradient theory (MSGT) and modified couple stress theory is obtained. The generalized differential quadrature method, numerical Galerkin scheme, periodic time differential operators and pseudo arc-length continuation method are utilized to determine the geometrically nonlinear free vibration characteristics of FG microplates with different boundary conditions. The parametric effects of thickness-to-material length scale ratio, material gradient index, length-to-thickness ratio, length-to-width ratio and boundary conditions on the nonlinear free vibration characteristics of FG microplates are studied through various numerical examples presented. It is found that a considerable difference exists between the results of various elasticity theories at small values of length scale parameter. A more precise prediction can be provided by using the size-dependent plate model based on the MSGT.

Flexoelectricity, incommensurate phases and the Lifshitz point

The solutions for the minimizers of the energy density f (q, p) = Aq2 + Bq4 + p2 + gA,B + describe the flexoelectric effect with a flexoelectric coupling coefficient β. The order parameters q and p can be visualized as strain and polarisation, respectively. The parameter κ denotes the ratio of intrinsic length scales for q and p.We show that the structural ground-states include 3 phases, namely the paraelastic state q = p = 0, the ferroelastic state where polarization exists inside and near twin boundaries, and the incommensurate (modulated) phases with a very rich array of structural modulations ranging from nearly pure sine waves to kink arrays and ripple states. The phases coincide in the multicritical Lifshitz point.Linear flexoelectricity is encountered only approximately inside the ferroelastic phase and near the phase boundary between the paraelastic phase and the incommensurate phase. The relationship between the polarisation and the strain gradient is highly non-linear in all other states. The spatial profiles and energy distributions are discussed in detail.

Effects of spanwise spacing on large-scale secondary flows in rough-wall turbulent boundary layers

Large-scale secondary flows can sometimes appear in turbulent boundary layers formed over rough surfaces, creating low- and high-momentum pathways along the surface (Barros & Christensen, J. Fluid Mech., vol. 748, 2014, R1). We investigate experimentally the dependence of these secondary flows on surface/flow conditions by measuring the flows over streamwise strips of roughness with systematically varied spanwise spacing. We find that the large-scale secondary flows are accentuated when the spacing of the roughness elements is roughly proportional to the boundary layer thickness ${\it\delta}$, and do not appear for cases with finer spacing. Cases with coarser spacing also generate ${\it\delta}$-scale secondary flows with tertiary flows in the spaces in between. These results show that the ratio of the spanwise length scale of roughness heterogeneity to the boundary layer thickness is a critical parameter for the occurrence of these secondary motions in turbulent boundary layers over rough walls.

Size-dependent axial dynamics of magnetically-sensitive strain gradient microbars with end attachments

The multi-parameter Sturm–Liouville eigenvalue problems, associated with the size-dependent longitudinal vibration of a finite micro-scale bar embedded in orthogonal transverse magnetic fields, are addressed in this study. Derived as a fourth-order partial differential equation, and augmented by enriched higher-order boundary conditions, the mathematical model of the micro-scale bar is predicated on the duo of the strain gradient theory of elasticity and the extended Hamilton׳s principle. The derived model is tackled with the computation scheme of the power series method. A thorough validation of the simplified form of the model is presented with benchmark results published in the literature. Presented along with the validation study is a comprehensive parametric examination of the influence of the aspect ratio, the material length scale, the mass and stiffness ratios of attachments and the magnetic field strength. Results from the analyses affirm that in the case of a lightweight mass attached to the end of the microbar, the axial resonant frequencies approach that of a microbar with a fixed–free boundary condition. However, in the case of an attachment with a heavy mass the fundamental resonant frequency of the microbar tends to zero. A Pareto analysis of the order of influence of the model׳s variables unmasks the ratio of the stiffness of the microbar and an attached elastic spring as having the most significant effect on the fundamental axial natural frequency, in the absence of the magnetic field. In the presence of the magnetic field, however, the effects of the end attachments are totally overshadowed by the influence of the magnetic field strength.

Effects of pool dimension on flame spread of aviation kerosene coating on a metal substrate

Experimental investigations are undertaken to study flame spread over aviation kerosene coating on a metal substrate with various pool dimensions. According to the experimental data, flame spread over aviation kerosene is separated into shallow and deep pool for fuel depth of 4.0mm, as well as narrow and wide pool for pool width of 12cm. The variation of flame spread velocity with pool dimension indicates that flame spread accidents on wide or deep pools are potentially more hazardous. Meanwhile, the critical liquid depth is defined as the minimum fuel depth for supporting flame propagation. For pool narrower than 16cm, the critical liquid depth declines inversely proportionally with an increase in pool width. For pool wider than 16cm, however, the critical liquid depth stabilizes at 1.2mm. At critical liquid depth, theoretical analysis verifies that surface deformation possesses a significant effect on flame extinction. The division criterion for deep and shallow pool is also validated by scaling analyses: the values of characteristic length scale ratio (ht/L) and Ra/Ma increase steeply under shallow pool conditions, but remain stable under deep pool conditions. Moreover, the variation trend of Ra/Ma confirms that Marangoni force dominates buoyancy force in driving flame spread at any fuel depths, whereas buoyant effect performs an increasingly importance as the fuel depth increases. Furthermore, the calculated Prandtl number demonstrates that the decrease of flame spread velocity for shallow pool is mainly caused by viscous shear on the metal substrate surface and secondarily caused by heat losses.

Entropic depletion in colloidal suspensions and polymer liquids: role of nanoparticle surface topography.

We employ a hybrid Monte Carlo plus integral equation theory approach to study how dense fluids of small nanoparticles or polymer chains mediate entropic depletion interactions between topographically rough particles where all interaction potentials are hard core repulsion. The corrugated particle surfaces are composed of densely packed beads which present variable degrees of controlled topographic roughness and free volume associated with their geometric crevices. This pure entropy problem is characterized by competing ideal translational and (favorable and unfavorable) excess entropic contributions. Surface roughness generically reduces particle depletion aggregation relative to the smooth hard sphere case. However, the competition between ideal and excess packing entropy effects in the bulk, near the particle surface and in the crevices, results in a non-monotonic variation of the particle-monomer packing correlation function as a function of the two dimensionless length scale ratios that quantify the effective surface roughness. As a result, the inter-particle potential of mean force (PMF), second virial coefficient, and spinodal miscibility volume fraction vary non-monotonically with the surface bead to monomer diameter and particle core to surface bead diameter ratios. A miscibility window is predicted corresponding to an optimum degree of surface roughness that completely destroys depletion attraction resulting in a repulsive PMF. Variation of the (dense) matrix packing fraction can enhance or suppress particle miscibility depending upon the amount of surface roughness. Connecting the monomers into polymer chains destabilizes the system via enhanced contact depletion attraction, but the non-monotonic variations with surface roughness metrics persist.

Multi-scale Asymptotic Analysis of Gas Transport in Shale Matrix

Organic-rich resource shales play an important role in global natural gas production. However, many uncertainties exist in an engineering analysis of gas transport and production such as the reservoir-scale flow simulation, history-matching, and optimization. In this work, we introduce a new set of governing equations to describe the characteristic features of porous structures of the organic-rich resource shale. We apply multi-scale analysis to mass balance equations, the equation of state (for free gas), and an adsorption isotherm. Using the macroscopic model, we study gas transport in shales, consisting of nanoporous organic material (kerogen) and the inorganic material. We conclude that both gas in-place and gas production rate depend on the amount of kerogen in the shale matrix. Adsorbed-phase transport by the organic pore walls is responsible for the increase in production rate. We investigate both Henry and Langmuir adsorption as well as different values of length scale ratio and diffusion coefficients.

Effects of Free-Stream Turbulence and Reynolds Number on the Aerodynamic Characteristics of a Semicylindrical Roof

The influences of free-stream turbulence and Reynolds number on the aerodynamic characteristics of a semicylindrical roof have been investigated experimentally under uniform flow conditions. Grids were used to generate homogeneous turbulent flows with turbulence intensities varying from 5.7 to 12.2%. The diameters, D, of the semicylindrical roof models were 0.2 and 0.6 m. The Reynolds numbers based on D ranged from 6.90×104 to 8.28×105 in smooth flow and from approximately 6.06×104 to 4.79×105 in grid-generated turbulent flow. Measurements of the surface pressure indicated that, in smooth flow, the mean and RMS pressure distributions became Reynolds number independent when R>4.14×105. The introduction of turbulence has caused a premature transition of the separated shear layer from laminar to turbulent. Increasing free-stream turbulence helps to remarkably reduce the mean and fluctuating drag force at lower Reynolds number, R<1.46×105. There is also a relatively significant reduction in the mean lift force as the turbulence intensity increases when R>1.46×105. The data also suggested that the effects of turbulence essentially depend on the turbulence intensity and the ratio of turbulence length scale to cylinder diameter. The small-scale turbulence was better able to interact with the surface boundary layer before and after separation and thus dramatically increased the pressure fluctuations in both the top and wake regions.

Model experimental study of scale invariant wetting behaviors in Cassie-Baxter and Wenzel regimes.

In this work, we discuss quantitatively two basic relations describing the wetting behavior of microtopographically patterned substrates. Each of them contains scale invariant topographical parameters that can be easily expressed onto substrates decorated with specifically designed micropillars. The first relation discussed in this paper describes the contact angle hysteresis of water droplets in the Cassie-Baxter regime. It is shown that the energy at the origin of the hysteresis, that has to be overcome for moving the triple line, can be invariantly expressed for hexagonal pillars by varying the pillars width and interpillar distance. Identical contact angle hystereses are thus measured on substrates expressing this scale invariance for pillar widths and interpillar distances ranging from 4 to 128 μm. The second relation we discuss concerns the faceting of droplets spreading on microtopographically patterned substrates. It is shown in this case that the condition for pinning of the triple line can be fulfilled by simultaneously varying the height of the pillars and the interpillar distance, leading to faceted droplets of similar morphologies. The invariance of these two wetting phenomena resulting from the simultaneous and homothetic variation of topographical parameters is demonstrated for a wide range of pattern dimensions. Our results show that either of those two wetting behaviors can be simply achieved by the proper choice of a dimensionless ratio of topographical length scales.

On grid-generated turbulence in the near- and far field regions

AbstractUsing a conventional bi-planar turbulence-generating grid, we confirm the recent findings (Valente &amp; Vassilicos,Phys. Rev. Lett., vol. 108, 2012, art. 214503) that show there is a turbulence decay region close to the generating grid that departs from the ‘classical’ turbulence decay (Comte-Bellot &amp; Corrsin,J. Fluid Mech., vol. 25, 1966, pp. 657–682). In this ‘near-field’ region, the turbulence energy decays more rapidly than in the far-field and it exhibits unusual scaling properties. Based on the velocity decay laws, we show that for our conventional grid, the near-field extends from$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}x/M \sim 6$to$x/M \sim 12$where$x$is the downstream distance from the grid and$M$is the mesh size. This corresponds to$1.1 \le x/x* \le 2.3$where$x*$is the wake interaction length scale (Mazellier &amp; Vassilicos,Phys. Fluids, vol. 22, 2010, art. 075101). However, other statistics (velocity derivatives and length-scale ratios) indicate that the extent of the initial period depends on the grid mesh Reynolds number,$R_M$, extending further for higher values of$R_M$. In the near-field the turbulence approaches isotropy both at the large and small scales but there still is inhomogeneity in the derivative statistics. The derivative skewness also departs from values observed at comparable Reynolds numbers in the far-field decay region, and in other turbulent flows at comparable Reynolds numbers. Two values of$R_M$were studied:$42 \times 10^3$and$76 \times 10^3$.

A Generalized Critical Heat Flux Correlation for Submerged and Free Surface Jet Impingement Boiling

A critical heat flux (CHF) correlation is developed for jet impingement boiling of a single round jet on a flat circular surface. The correlation is valid for submerged jets as well as for free surface jets with Reynolds numbers (Re) between 4000 and 60,000. Data for the correlation are obtained from an extensive experimental study of submerged jet impingement boiling performed by the authors with water at subatmospheric pressures and with FC-72 at atmospheric pressure. Additional experimental data from a free surface jet study are also incorporated to include the effect of variation in surface diameter relative to a fixed nozzle diameter, additional working fluids (water and R-113 both at atmospheric pressure), and jet configuration. The range of parameters considered include Re from 0 (pool boiling) to 60,000, jet diameter to capillary length scale ratios (dj/Lc) ranging from 0.44 to 5.50, surface diameter to capillary length scale ratios (ds/Lc) ranging from 4.47 to 38.42, and liquid-to-vapor density ratios from 119 to 8502. The proposed correlation is built on the framework of a forced convective CHF model. Using this correlation, 95% of the experimental CHF jet impingement data can be predicted within ±22% error. The corresponding average absolute error and the maximum absolute error are 8% and 36%, respectively, over the range of parameters considered.