Disparate Length Scales Research Articles

Adaptively refined, parallelised sharp interface Cartesian grid method for three-dimensional moving boundary problems

Sharp interface Cartesian grid methods are capable of simulating complex moving boundary problems on fixed meshes while treating embedded interfaces accurately. This article further enhances the effectiveness of the sharp interface method by devising techniques for adaptive mesh resolution combined with parallel processing. These extensions enable dealing with problems involving disparate length scales encountered in many applications. A tree-based adaptive local mesh refinement scheme is developed to complement the sharp interface Cartesian grid method for efficient and optimised calculations. Detailed timing and accuracy data are presented for a variety of benchmark problems involving moving boundaries. Guidelines for selecting mesh refinement criteria for moving boundary calculations are developed. Issues associated with parallelisation of the overall framework are tackled. The capabilities of the method are demonstrated in a number of moving boundary problems, which require adequate resolution of a wide range of length scales and three-dimensional flows.

Ab initio LDA+U prediction of the tensile properties of chromia across multiple length scales

Periodic density functional theory (DFT) and DFT+U calculations are used to evaluate various mechanical properties associated with the fracture of chromia (Cr 2O 3) along the [0 0 0 1] and [ 0 1 1 ¯ ( 3 / 2 ) ( a / c ) 2 2 ] directions. The properties investigated include the tensile strength, elastic constants, and surface energies. The tensile strengths are evaluated using an ideal tensile test, which provides the theoretical tensile strength, and by fitting the calculated data to universal binding energy relationships (UBER), which permit the extrapolation of the calculated results to arbitrary length scales. The results demonstrate the ability of the UBER to yield a realistic estimate of the tensile strength of a 10-μm-thick sample of Cr 2O 3 using data obtained through calculations on nanoscopic systems. We predict that Cr 2O 3 will fracture most easily in the [ 0 1 1 ¯ ( 3 / 2 ) ( a / c ) 2 2 ] direction, with a best estimate for the tensile strength of 386 MPa for a 10 μm grain, consistent with flexural strength measurements for chromia. The grain becomes considerably stronger at the nanoscale, where we predict a tensile strength along the same direction of 32.1 GPa for 1.45 nm crystallite. The results also provide insight into the origin of the direction dependence of the mechanical properties of Cr 2O 3, with the differences in the behavior along different directions being related to the number of Cr–O bonds supporting the applied tensile load. Additionally, the results shed light on various practical aspects of modeling the mechanical properties of materials with DFT+U calculations and in using UBERs to estimate the mechanical properties of materials across disparate length scales.

Three-Dimensional Numerical Modeling of Initial Mixing of Thermal Discharges at Real-Life Configurations

A three-dimensional Reynolds-averaged Navier–Stokes computational fluid dynamics (CFD) model is developed for simulating initial mixing in the near field of thermal discharges at real-life geometrical configurations. The domain decomposition method with multilevel embedded overset grids is employed to handle the complexity of real-life diffusers as well as to efficiently account for the large disparity in length scales arising from the relative size of the ambient river reach and the typical diffuser diameter. An algebraic mixing length model with a Richardson-number correction for buoyancy effects is used for the turbulence closure. The governing equations are solved with a second-order-accurate, finite-volume, artificial compressibility method. The model is validated by applying it to simulate thermally stratified shear flows and negatively buoyant wall jet flows and the computed results are shown to be in good overall agreement with the experimental measurements. To demonstrate the potential of the numerical model as a powerful engineering simulation tool we apply it to simulate turbulent initial mixing of thermal discharges loaded from both single-port and multiport diffusers in a prismatic channel and a natural river. Comparisons of the CFD model results with those obtained by applying two widely used empirical mixing zone models show that the results are very similar in terms of both the rate of dilution and overall shape of the plumes. The CFD model further resolves the complex three-dimensional features of such flows, including the complex interplay of the ambient flow and thermal discharges as well as the interaction between each of discharges loaded from multiple ports, which are obviously not accessible by the simpler empirical models.

Quantum dot–DNA interaction: Computational issues and preliminary insights on use of quantum dots as biosensors

In recent years, quantum dots have generated enormous interest from the life sciences community due to their (largely) untapped potential in biomedical applications; particularly in bio-labeling and sensing. While empirical work already exists on the use of quantum dots as bio-labels, their development as biosensors requires a thorough scientific understanding of their interactions with conjugated biomolecules that together ‘sense’ the molecule of interest. Some recent experiments have claimed a marked variation in the luminescence of cadmium selenide quantum dots conjugated to macromolecules linked to bacteria. The origin of this large shift in luminescence of the quantum dot (and thus by implication, the band gap) appears to be poorly understood. The knowledge of the exact nature of the interaction causing the ‘shift’ may hold the key to designing better biosensors. The objective of the present work is to address the aforementioned interaction and to that end, we have chosen a prototypical model consisting of a capped cadmium selenide quantum dot interacting with a DNA molecule. This problem is inherently multiscale due to the relatively large number of atoms, complex nature of the interactions involved in the quantum dot–DNA system and the disparate length scales present in the problem requiring a combination of methods ranging from approaches that utilize empirical molecular mechanics force fields on one hand and ab initio electronic structure (based on density functional theory) calculations on the other hand. We discuss several modeling issues that arise in the simulations of this complex problem and present some preliminary insights. Our initial results indicate a wavelength shift of roughly 19 nm in the spectrum of a 1.1 nm sized dot upon interaction with a typical DNA molecule. However, upon increase of quantum dot size, the shift decreases and thus suggests a re-examination of singular experimental data available in the literature. Our results, which are performed in vacuum rather than a solvent, may be considered as an upper-bound to the true interaction.

Elucidating the contact mechanics of aluminum silicon surfaces with Green's function molecular dynamics

We study the contact mechanics of a flat, elastic wall pressed against a rigid substrate with Green’s function molecular dynamics. The substrate’s height profiles are parametrized from atomic force microscope topography measurements of two different aluminum-silicon alloys. In both samples, roughness lives on disparate length scales, i.e., on relatively large scales defined by size and mean separation of load-bearing silicon particles and on much smaller scales associated with the roughness on top of individual particles. The major differences between the two alloys are their silicon content and the typical silicon particle geometry. These differences lead to quite different stress distributions on both mesoscale and microscale in our calculations. A common feature is that the stress distribution decays exponentially for large stresses σ and not like a Gaussian. Persson’s contact mechanics theory is generalized to the case where contact can only occur on silicon particles. This generalization predicts relatively accurate microscopic mean square stresses, however, it fails to predict accurate numbers for mean square stresses on the mesoscopic scales. Local overlap models are not accurate either, because they fail to describe the contact morphology.

A multiple-scales approach to crack-front waves

Perturbation of a steadily propagating crack with a straight edge is solved using the method of matched asymptotic expansions (MAE). This provides a simplified analysis in which the inner and outer solutions are governed by distinct mechanics. The inner solution contains the explicit perturbation and is governed by a quasi-static equation. The outer solution determines the radiation of energy away from the tip, and requires solving dynamic equations in the unperturbed configuration. The outer and inner expansions are matched via the small parameter ϵ = L/l defined by the disparate length scales: the crack perturbation length L and the outer length scale l associated with the loading. The method is illustrated for a scalar crack model and then applied to the elastodynamic mode I problem. The crack-front wave-dispersion relation is found by requiring that the energy release rate is unaltered under perturbation and dispersive properties of the crack-front wave speed are described for the first time. The example problems considered demonstrate the potential of MAE for moving-boundary-value problems with multiple scales.

Numerical solution scheme for inert, disperse, and dilute gas-particle flows

A multi-velocity formulation is proposed for the solution of an Eulerian representation of an inert, disperse, and dilute particle-phase of a gas-particle flow. Single-velocity formulations are capable of predicting regions of zero particle concentration but are problematic with crossing particle trajectories or compression waves. The multi-velocity formulation described here can account for crossing particle trajectories by splitting the particle-phase into distinct velocity families which are transported separately in the flow. Switching of the particle families at solid boundaries and due to momentum transfer with the gas-phase is conducted in a manner that enforces conservation of mass, momentum, and energy. This numerical method is combined with a parallel block-based adaptive mesh refinement algorithm that is very effective in treating problems with disparate length scales. The block-based data structure lends itself naturally to domain decomposition and thereby enables efficient and scalable implementations of the algorithm on distributed-memory multi-processor architectures. Numerical results are described to demonstrate the capabilities of the approach for predicting gas-particle flows.

Multidisciplinary and multi-scale computational field simulations—Algorithms and applications

This paper describes the development and the validation of two computational fluid dynamics (CFD) frameworks for multidisciplinary and multi-scale simulations. The first framework is an overset grid topology with a six-degree of freedom rigid body dynamics library for both structured-grid and unstructured-grid approaches. This framework will enable the CFD flow solver to simulate the problems involving relative motions between multiple bodies such as the stage/store separation process, etc. The second framework is a hybrid scheme that integrates a direct simulation Monte-Carlo (DSMC) method for rarefied gas flows and a Navier–Stokes (NS) solver for continuum flows. The hybrid DSMC–NS numerical framework can greatly improve the computational efficiency for simulation of problems with disparate length scales or a wide range of Knudsen numbers. A density-based Navier–Stokes flow solver, HYB3D, and a pressure-based Navier–Stokes flow solver, FDNS, were employed in this study to demonstrate the capability, strength, and accuracy of these two frameworks. Numerical simulations of benchmark test cases were conducted as the preliminary evaluation for these two frameworks, and the numerical results are presented.

Pore size distribution in multiscale porous media as revealed by DDIF–NMR, mercury porosimetry and statistical image analysis

Nuclear spin diffusion in the susceptibility-contrast induced internal field (DDIF–NMR) has been recently exploited to probe pore sizes in sedimentary rock. This method has hitherto been applied to only a few samples and its ability to discern the entire spectrum of pore length scales actually present has not been fully validated against independent methods. Motivated by the need to measure the pore size distribution in vuggy carbonate rocks exhibiting structure at disparate length scales, ranging from several nanometers (matrix pores) to millimeters (vugs), we carry out a DDIF–NMR study of real and synthetic vuggy porous media. The synthetic media have controlled amounts of matrix and vuggy porosity and are made by first sintering and dissolution of known proportions of glass beads and much larger calcium carbonate particles. DDIF–NMR studies are complemented by single-point magnetic resonance imaging (SPI–MRI), which provides independent determination of the vug size distribution, mercury porosimetry and statistical image analysis (SIA) of large (2.6-cm wide), high-resolution (1.87 μm/pixel) images of thin-sections. The two-point correlation function determined in the μm-to-mm range from images is used to simulate small-angle neutron scattering, and the computed scattering intensity is extended (where appropriate) in the large- Q range using the surface fractal scaling law I ( Q ) ∝ Q D s − 6 , with D s estimated from mercury porosimetry. The extended I( Q) data are interpreted in terms of a poly-dispersed spherical pore model, allowing the determination of the pore size distribution in the nm-to-mm range. Pore size distributions obtained by statistical image analysis are found to be consistent with mercury retraction capillary pressure data. Pore size distributions from DDIF–NMR data show reasonable agreement to SIA results for samples without vuggy porosity (i.e., pore diameter < 200 μm). For other samples, significant differences are observed, underscoring the need to develop more sophisticated models for the interpretation of DDIF–NMR data.

Developing Quantitative, Multiscale Models for Microgravity Crystal Growth

Crystal growth conducted under microgravity conditions has had a profound impact on improving our understanding of melt crystal growth processes. Here, we present a brief history of microgravity crystal growth and discuss the development of appropriate models to interpret and optimize these growth experiments. The need for increased model realism and predictive capability demands new approaches for describing phenomena important at several disparate length scales. Of special importance is the ability to represent three-dimensional and transient continuum transport (flows, heat, and mass transfer), phase-change phenomena (thermodynamics and kinetics), and system design (such as furnace heat transfer during melt growth). An overview of mathematical models and numerical algorithms employed to represent such multiscale effects is presented.

Theoretical Aerodynamics in Today's Real World: Opportunities and Challenges

*† In spite of the CFD revolution, significant challenges still face aerodynamicists in predicting and controlling various classes of flows. These include three-dimensional separation, boundary layer transition, interaction of separation and transition such as on reentry capsules, multi-element airfoils and wings, UAV low Reynolds number flows and high angle of attack applications. Others include multi-body flows such as those occurring in store and stage separation, “stiff” combustion reacting and unsteady flows as well as plasma aerodynamics and turbulence, to name a few. In many cases, diverse multiple scales are involved and the proper identification and treatment of associated disparate length and time scales is critical in obtaining accurate prediction and effective control. The solution of such multi-scale problems can be a hurdle to effective domain decomposition, even overset, unstructured adaptive gridding and ultimately, solution accuracy. Theoretical insights can help make proper decisions on numerical pre-processing, solution, post-processing and interpretation. Opportunities for a combined theoretical, computational and experimental approach will be discussed. These will be illustrated by examples from the author’s experience. As compared to the limited “pen and paper” theoretical methods of the 50’s, illustrations will be given of the effectiveness of combined asymptotics, similitude, group invariance, approximate physics-based modeling and numerical methods for conceptual vehicle design, flow control innovation, identification of key parameters, leveraging of computational solutions, reducing the parameter space as well as providing added insight into the basic physical processes. These will be related to tradeoffs between accuracy and response speed in typical aerospace environments.

A Spatially Nonlocal Model for Polymer Desorption

In order to describe diffusion of a penetrant in a polymer entanglement network, one must incorporate nonlocal effects. Most previous models have included nonlocality in time only; however, by exploiting the disparate length scales in such systems, one can model these effects by a partial integrodifferential equation which is nonlocal in space. When considering the case of diffusion near the glass-rubber transition, a moving boundary separates the polymer into two regions, each governed by a different set of PDEs. The desorption of a semi-infinite polymer is studied using singular perturbation methods. Layers arise at the exposed surface, at the moving boundary, and initially. Analytical and phase-plane solutions are obtained for the solution, which exhibits physically realistic forms of desorption overshoot. Thus, spatially nonlocal models have the potential to replicate experimental systems, and should be considered in concert with other viscoelastic models of polymer-penetrant systems.

Development of a Continuous Model for Simulation of Turbulent Flows

The development of a continuous turbulence model that is suitable for representing both the subgrid scale stresses in large eddy simulation and the Reynolds stresses in the Reynolds averaged Navier-Stokes formulation is described. A recursion approach is used to bridge the length scale disparity from the cutoff wave number to those in the energy-containing range. The proposed model is analyzed in conjunction with direct numerical simulations of Kolmogorov flows.

Computational aeroacoustics using the B-spline collocation method

One of the major problems in computational aero-acoustics is the disparity in length scales between the flow field and the acoustic field. As a result, a mapping function is normally used to achieve a non-uniform grid distribution. In this paper, a B-spline collocation method with an arbitrary grid placement capability is proposed. This capability not only allows an optimum grid distribution but also avoids the numerical complexities associated with the mapping function. The B-spline collocation method is applied to the case of spinning co-rotating vortices. The result agrees well with the matched asymptotic solution. To cite this article: R. Widjaja et al., C. R. Mecanique 333 (2005).

Effect of Conjugate Heat Transfer on MEMS-Based Thermal Shear Stress Sensor

ABSTRACT The effect of conjugate heat transfer resulting from a microelectromechanical systems (MEMS)-based thermal shear stress is investigated. Due to the length-scale disparity and large solid–fluid thermal conductivity ratio, a two-level computation is used to examine the relevant physical mechanisms and their influences on wall shear stress. The substantial variations in transport properties between the fluid and solid phases and their interplay with regard to heat transfer and near-wall fluid flow structures are investigated. It is demonstrated that for state-of-the-art sensor design, the buoyancy effect can noticeably affect the accuracy of the shear stress measurement.

Multiscale optimization using hybrid PDE/kMC process systems with application to thin film growth

The problem of optimal time-constant and time-varying operation for transport-reaction processes is considered, when the cost functional and/or equality constraints necessitate the consideration of phenomena that occur over disparate length scales. Multiscale process models are initially developed, linking continuum conservation laws with microscopic scale simulators. Subsequently, order reduction techniques for dissipative partial-differential equations are combined with adaptive tabulation of microscopic simulation data to reduce the computational requirements of the optimization problem, which is then solved using standard search algorithms. The method is applied to a conceptual thin film deposition process to compute optimal substrate-surface temperature profiles that simultaneously maximize film-deposition-rate uniformity (macroscale objective) and minimize surface roughness (microscale objective) across the film surface for a steady-state process operation. Subsequently, optimal time-varying policies of substrate temperature and precursor inlet concentrations are computed under the assumption of quasi-steady-state process operation.

Nonlithographic Hierarchical Patterning of Semiconducting Nanoparticles via Polymer/Polymer Phase Separation

We demonstrate a new and versatile nonlithographic strategy for the lateral patterning of semiconducting quantum dots (QDs) in thin polymer films via phase separation between two immiscible polymer components, one of which is attached as a stabilizing layer to the QD surface. Simple spin-casting of blend solutions containing block copolymer-stabilized cadmium sulfide (CdS) QDs with an external polystyrene (PS) brush layer (PS−CdS) and poly(methyl methacrylate) (PMMA) homopolymer results in solvent evaporation-induced organization of QDs on two disparate length scales: polymer/polymer spinodal decomposition gives rise to tunable microscopic lateral patterning of PS−CdS features (with mesoscopic feature heights), while simultaneous steric interactions between the polymer brush layers on neighboring nanoparticles give rise to a nanoscale liquidlike distribution of QDs within the PS−CdS domains. The correlation length and morphology of the lateral patterns are easily controlled via the spin-casting rotation speed and blend composition, respectively. Subsequent selective removal of the PMMA component allows the production of various photoluminescent PS−CdS features with structural hierarchy on glass substrates, including cellular and wirelike networks and arrays of spatially correlated islands.

Prediction of radiation from finite-length fluid-loaded periodic structures subject to broadband excitation using local-global homogenization

Local-Global Homogenization (LGH) is a method to predict directly the smooth global response of periodic fluid-loaded structures in a self-contained manner. Many fluid-loaded structures, such as fuselages and hulls, have periodically spaced braces, ribs, or attachments. Structural motion, acoustic radiation, and the interior sound field are of interest. Calculating the motion of such fluid-loaded structures is difficult because of their complexity and the disparity of length scales. Periodic discontinuities cause the structural response to occur in a broad spectrum of spatial wavenumbers, and to exhibit stop-band and pass-band behavior. The broad spatial wavenumber spectrum contains both radiating and non-radiating components. The global low-wavenumber part of the response is most efficiently coupled to the acoustic field, since low wavenumbers correspond to supersonic phase speeds. In the LGH reformulation, an infinite order operator that embodies both the structural modes and the evanescent component of the fluid loading governs the equivalent smooth global problem. Numerical implementation is demonstrated for the structural response and acoustic radiation from a finite-length fluid-loaded plate with periodic impedance discontinuities subject to broadband excitation. Calculations show good agreement with the exact solution and substantially improved computational efficiency. Radiation characteristics are discussed for single frequency and broadband structural motions.

OPTIMAL OPERATION OF THIN FILM GROWTH WITH MULTISCALE PROCESS OBJECTIVES

The issue of optimal time-varying operation for transport-reaction processes is considered, when the cost functional and/or equality constraints necessitate the consideration of phenomena that occur over disparate length scales. Multiscale process models are initially developed, linking continuum conservation laws with microscopic scale simulators. Subsequently, order reduction techniques for dissipative partial-differential equations are combined with adaptive tabulation methods for microscopic simulators to reduce the computational requirements of the optimization problem, which is then solved using standard search algorithms. The method is demonstrated on a thin film deposition process, where optimal surface temperature profiles and inlet switching times that simultaneously maximize thickness uniformity and minimize surface roughness across the film surface are computed.

A least square extrapolation method for thea posteriori error estimate of the incompressible Navier Stokes problem

A posteriori error estimators are fundamental tools for providing confidence in the numerical computation of PDEs. To date, the main theories of a posteriori estimators have been developed largely in the finite element framework, for either linear elliptic operators or non-linear PDEs in the absence of disparate length scales. On the other hand, there is a strong interest in using grid refinement combined with Richardson extrapolation to produce CFD solutions with improved accuracy and, therefore, a posteriori error estimates. But in practice, the effective order of a numerical method often depends on space location and is not uniform, rendering the Richardson extrapolation method unreliable. We have recently introduced (Garbey, 13th International Conference on Domain Decomposition, Barcelona, 2002; 379–386; Garbey and Shyy, J. Comput. Phys. 2003; 186:1–23) a new method which estimates the order of convergence of a computation as the solution of a least square minimization problem on the residual. This method, called least square extrapolation, introduces a framework facilitating multi-level extrapolation, improves accuracy and provides a posteriori error estimate. This method can accommodate different grid arrangements. The goal of this paper is to investigate the power and limits of this method via incompressible Navier Stokes flow computations. Copyright © 2005 John Wiley & Sons, Ltd.