The evolution of Hooke's law due to texture development in FCC polycrystals

Initially isotropic aggregates of crystalline grains show a texture-induced anisotropy of both their inelastic and elastic behavior when submitted to large inelastic deformations. The latter, however, is normally neglected, although experiments as well as numerical simulations clearly show a strong alteration of the elastic properties for certain materials. The main purpose of the work is to formulate a phenomenological model for the evolution of the elastic properties of cubic crystal aggregates. The effective elastic properties are determined by orientation averages of the local elasticity tensors. Arithmetic, geometric, and harmonic averages are compared. It can be shown that for cubic crystal aggregates all of these averages depend on the same irreducible fourth-order tensor, which represents the purely anisotropic portion of the effective elasticity tensor. Coupled equations for the flow rule and the evolution of the anisotropic part of the elasticity tensor are formulated. The flow rule is based on an anisotropic norm of the stress deviator defined by means of the elastic anisotropy. In the evolution equation for the anisotropic part of the elasticity tensor the direction of the rate of change depends only on the inelastic rate of deformation. The evolution equation is derived according to the theory of isotropic tensor functions. The transition from an elastically isotropic initial state to a (path-dependent) final anisotropic state is discussed for polycrystalline copper. The predictions of the model are compared with micro–macro simulations based on the Taylor–Lin model and experimental data.     

Anisotropy favoring triangulation CVD(MPFA) finite‐volume approximations

A family of flux‐continuous, control‐volume distributed multi‐point flux approximation schemes CVD (MPFA) have been developed for solving the general geometry‐permeability tensor pressure equation on structured and unstructured grids (Comput. Geo. 1998; 2 : 259–290, Comput. Geo. 2002; 6 : 433–452). The locally conservative schemes are applicable to the diagonal and full‐tensor pressure equation with generally discontinuous coefficients and remove the O(1) errors introduced by standard reservoir simulation schemes when applied to full‐tensor flow approximation. The family of flux‐continuous schemes is quantified by a quadrature parameterization. Improved numerical convergence for the family of CVD(MPFA) schemes for specified quadrature points has been observed for lower anisotropy ratios for both structured and unstructured grids in two dimensions. However, for strong full‐tensor anisotropy fields the quadrilateral schemes can induce strong spurious oscillations in the numerical solution. This paper motivates and demonstrates the benefit of using anisotropy favoring triangulation for treating such cases. Test examples involving strong full‐tensor anisotropy fields are presented in 2‐D and 3‐D, which show that the family of schemes on anisotropy favoring triangulation (prisms in 3‐D) yield well‐resolved pressure fields with little or no spurious oscillations. Copyright © 2010 John Wiley & Sons, Ltd.     

Modeling Concentration Distribution and Deformation During Convection-Enhanced Drug Delivery into Brain Tissue

Convection-enhanced drug delivery is a technique where a therapeutic agent is infused under positive pressure directly into the brain tissue. For predicting the final concentration distribution and optimizing infusion rate and catheter placement, numerical models can be of great help. However, despite advances in modeling this process, often the infused agent does not reach the targeted region prescribed in the modeling phase. In this study, patient-specific brain structure and parameters, obtained from diffusion tensor imaging (DTI), are implemented in a numerical model which describes the flow and transport in an elastic deformable matrix. To our knowledge, this is the first time that information from DTI is used in a numerical model which includes both transport of a therapeutic agent and tissue deformation. Fractional anisotropy (FA) is used to distinguish between gray and white matter and tortuosity to differentiate between inside and outside the brain tissue. One voxel in the DT-image is represented by one element of the numerical grid. The DT-images were in addition used to determine the orientation of the white matter fiber tracts and calibrate permeability and diffusion coefficients found in the literature. Values chosen for the porosity and Lamé parameters are also based on those found in the literature. Given realistic literature values, the calibration of the permeability and diffusion tensors are shown to be successful. Our result shows that preferential flow occur in direction of the white matter fiber tracts. The current model assumes linear deformation, corresponding to small porosity changes. But, because large porosity changes occur that may adversely affect drug transport, non-linear deformations should be included in the future.     

On the basic laws of anisotropic viscoelasticity

The main aim of this work is to develop a consistent formulation of the rheological behavior for different anisotropic polymer systems. The unified theory of anisotropic viscoelasticity is developed based on the symmetry principles. The Maxwell rheological equation is extended to nonsymmetric anisotropic liquids. Transitions from the most general anisotropy to particular cases of anisotropy are established. It appears that the coupled relaxation of symmetric and antisymmetric stresses is a natural phenomenon in nonsymmetric viscoelasticity. Within the concept of an internal state variable, a stress–order relation is derived for a fully nonlinear case. The order tensor dynamics is also considered. A simple method of deriving the equation of the internal rotational motion is developed for the general macroscopic anisotropy. This paper was presented at the 3rd Annual Rheology Conference, AERC 2006, April 27–29, 2006, Crete, Greece     

Dependency of Tortuosity and Permeability of Porous Media on Directional Distribution of Pore Voids

Single-phase fluid flow in porous media is usually direction dependent owing to the tortuosity associated with the internal structures of materials that exhibit inherent anisotropy. This article presents an approach to determine the tortuosity and permeability of porous materials using a structural measure quantifying the anisotropic distribution of pore voids. The approach uses a volume averaging method through which the macroscopic tortuosity tensor is related to both the average porosity and the directional distribution of pore spaces. The permeability tensor is derived from the macroscopic momentum balance equation of fluid in a porous medium and expressed as a function of the tortuosity tensor and the internal structure of the material. The analytical results generally agree with experimental data in the literature.     

Generalized Fourier analyses of the advection–diffusion equation—Part I: one‐dimensional domains

This paper presents a detailed multi‐methods comparison of the spatial errors associated with finite difference, finite element and finite volume semi‐discretizations of the scalar advection–diffusion equation. The errors are reported in terms of non‐dimensional phase and group speed, discrete diffusivity, artificial diffusivity, and grid‐induced anisotropy. It is demonstrated that Fourier analysis provides an automatic process for separating the discrete advective operator into its symmetric and skew‐symmetric components and characterizing the spectral behaviour of each operator. For each of the numerical methods considered, asymptotic truncation error and resolution estimates are presented for the limiting cases of pure advection and pure diffusion. It is demonstrated that streamline upwind Petrov–Galerkin and its control‐volume finite element analogue, the streamline upwind control‐volume method, produce both an artificial diffusivity and a concomitant phase speed adjustment in addition to the usual semi‐discrete artifacts observed in the phase speed, group speed and diffusivity. The Galerkin finite element method and its streamline upwind derivatives are shown to exhibit super‐convergent behaviour in terms of phase and group speed when a consistent mass matrix is used in the formulation. In contrast, the CVFEM method and its streamline upwind derivatives yield strictly second‐order behaviour. In Part II of this paper, we consider two‐dimensional semi‐discretizations of the advection–diffusion equation and also assess the affects of grid‐induced anisotropy observed in the non‐dimensional phase speed, and the discrete and artificial diffusivities. Although this work can only be considered a first step in a comprehensive multi‐methods analysis and comparison, it serves to identify some of the relative strengths and weaknesses of multiple numerical methods in a common analysis framework. Published in 2004 by John Wiley & Sons, Ltd.     

Incorporation of polymer diffusivity and migration into constitutive equations

Given a general one-particle constitutive equation for the stress tensor, we discuss how to incorporate the additional effects of polymer diffusivity and migration into that constitutive equation within the framework of continuum mechanics. For the example of an upper-convected Maxwell model representing the polymer contribution to the stress tensor of a dilute polymer solution, we describe i) how to modify the constitutive equation for the stress tensor to include diffusion and migration effects, ii) how to formulate a balance equation for the polymer mass density in order to describe the nonhomogeneous composition of the polymer solution resulting from migration, and iii) how to close the extended set of coupled equations by means of further constitutive equations for the migration velocity and the diffusion tensor. In order to guarantee the material objectivity for all equations, we formulate them in the body tensor formulation of continuum mechanics (and then translate them into Cartesian space). The proposed equations are compared to results of a recent kinetic-theory approach.Dedicated to Professor Arthur S. Lodge on the occasion of his 70th birthday and his retirement from the University of Wisconsin.     

Numerical modelling of a melting-solidification cycle of a phase-change material with complete or partial melting

A high accuracy numerical model is used to simulate an alternate melting and solidification cycle of a phase change material (PCM). We use a second order (in time and space) finite-element method with mesh adaptivity to solve a single-domain model based on the Navier-Stokes-Boussinesq equations. An enthalpy method is applied to the energy equation. A Carman-Kozeny type penalty term is introduced in the momentum equation to bring the velocity to zero inside the solid region. The mesh is dynamically adapted at each time step to accurately capture the interface between solid and liquid phases, the boundary-layer structure at the walls and the multi-cellular unsteady convection in the liquid. We consider the basic configuration of a differentially heated square cavity filled with an octadecane paraffin and use experimental and numerical results from the literature to validate our numerical system. The first study case considers the complete melting of the PCM (liquid fraction of 95%), followed by a complete solidification. For the second case, the solidification is triggered after a partial melting (liquid fraction of 50%). Both cases are analysed in detail by providing temporal evolution of the solid-liquid interface, liquid fraction, Nusselt number and accumulated heat input. Different regimes are identified during the melting-solidification process and explained using scaling correlation analysis. Practical consequences of these two operating modes are finally discussed.     

宽速域下神经网络对雷诺应力各向异性张量的预测

基于Pope修正的有效黏度假设,张量基神经网络(tensor based neural network,TBNN)构建了从雷诺平均方程湍流模型(RANS)的平均应变率张量和平均旋转率张量到高精度数值解的雷诺应力各向异性张量的映射.将高精度数值解用于TBNN的训练,从而使TBNN根据RANS求解的湍动能、湍流耗散率和速度...     

Modeling of deformation induced anisotropy in free-end torsion

The main purpose of this work is to develop a phenomenological model, which accounts for the evolution of the elastic and plastic properties of fcc polycrystals due to a crystallographic texture development and predicts the axial effects in torsion experiments. The anisotropic portion of the effective elasticity tensor is modeled by a growth law. The flow rule depends on the anisotropic part of the elasticity tensor. The normalized anisotropic part of the effective elasticity tensor is equal to the 4th-order coefficient of a tensorial Fourier expansion of the crystal orientation distribution function. Hence, the evolution of elastic and viscoplastic properties is modeled by an evolution equation for the 4th-order moment tensor of the orientation distribution function of an aggregate of cubic crystals. It is shown that the model is able to predict the plastic anisotropy that leads to the monotonic and cyclic Swift effect. The predictions are compared to those of the Taylor–Lin polycrystal model and to experimental data. In contrast to other phenomenological models proposed in the literature, the present model predicts the axial effects even if the initial state of the material is isotropic.     

Predicting Buoyant Shear Flows Using Anisotropic Dissipation Rate Models

This paper examines the modeling of two-dimensional homogeneous stratified turbulent shear flows using the Reynolds-stress and Reynolds-heat-flux equations. Several closure models have been investigated; the emphasis is placed on assessing the effect of modeling the dissipation rate tensor in the Reynolds-stress equation. Three different approaches are considered; one is an isotropic approach while the other two are anisotropic approaches. The isotropic approach is based on Kolmogorov's hypothesis and a dissipation rate equation modified to account for vortex stretching. One of the anisotropic approaches is based on an algebraic representation of the dissipation rate tensor, while another relies on solving a modeled transport equation for this tensor. In addition, within the former anisotropic approach, two different algebraic representations are examined; one is a function of the Reynolds-stress anisotropy tensor, and the other is a function of the mean velocity gradients. The performance of these closure models is evaluated against experimental and direct numerical simulation data of pure shear flows, pure buoyant flows and buoyant shear flows. Calculations have been carried out over a range of Richardson numbers (Ri) and two different Prandtl numbers (Pr); thus the effect of Pr on the development of counter-gradient heat flux in a stratified shear flow can be assessed. At low Ri, the isotropic model performs well in the predictions of stratified shear flows; however, its performance deteriorates as Ri increases. At high Ri, the transport equation model for the dissipation rate tensor gives the best result. Furthermore, the results also lend credence to the algebraic dissipation rate model based on the Reynolds stress anisotropy tensor. Finally, it is found that Pr has an effect on the development of counter-gradient heat flux. The calculations show that, under the action of shear, counter-gradient heat flux does not occur even at Ri = 1 in an air flow. This revised version was published online in July 2006 with corrections to the Cover Date.     

Non‐linear flux‐splitting schemes with imposed discrete maximum principle for elliptic equations with highly anisotropic coefficients

Families of flux‐continuous, locally conservative, finite‐volume schemes have been developed for solving the general tensor pressure equation of petroleum reservoir simulation on structured and unstructured grids. The schemes are applicable to diagonal and full tensor pressure equation with generally discontinuous coefficients and remove the O(1) errors introduced by standard reservoir simulation schemes when applied to full tensor flow approximation. The family of flux‐continuous schemes is quantified by a quadrature parameterization. Improved convergence using the quadrature parameterization has been established for the family of flux‐continuous schemes. When applied to strongly anisotropic full‐tensor permeability fields the schemes can fail to satisfy a maximum principle (as with other FEM and finite‐volume methods) and result in spurious oscillations in the numerical pressure solution. This paper presents new non‐linear flux‐splitting techniques that are designed to compute solutions that are free of spurious oscillations. Results are presented for a series of test‐cases with strong full‐tensor anisotropy ratios. In all cases the non‐linear flux‐splitting methods yield pressure solutions that are free of spurious oscillations. Copyright © 2010 John Wiley & Sons, Ltd.     

Viscosity,first and second normal-stress coefficients,and molecular stretching in concentrated solutions and melts

A numerical method to solve the diffusion equation for the encapsulated FENE-dumbbell model is presented. Viscosity, first and second normal-stress coefficients, and molecular stretching for steady state shear flow are calculated. From the graphs of these quantities one can see the effect of anisotropy of Stokes' resistance and the Brownian force on rheological properties. The accuracy of the approximate method used by Bird and DeAguiar is assessed.     

Fully discrete arbitrary-order schemes for a model parabolic equation

A fully discrete methodology is investigated from which two-level, explicit, arbitrary-order, conservative numerical schemes for a model parabolic equation can be derived. To illustrate this, fully discrete three-, five-, seven- and nine-point conservative numerical schemes are presented, revealing that a higher-order scheme has a better stability condition. A method from which high-order numerical schemes for a scalar advection-diffusion equation can be developed is discussed. This method is based on high-order schemes of both the advection and diffusion equations.     

Molecular model and flow calculation: I. The numerical solutions to multibead-rod models in homogeneous flows

The Galerkin method is used to solve the diffusion equation of the distribution function in configurational space for a multibead-rod model, and the dimensionless components of the extra stress tensor are then calculated by means of the expression of ensemble average. The material functions for steady-state shear flow and uniaxial flow and the mechanical properties of rigid-rodlike molecule suspensions in superposed flows are obtained numerically. The results indicate that it is promising to employ the multibead-rod models without the constitutive equation in numerical simulations of flows of suspensions. The project supported by the National Natural Science Fundation of China.     

Application of streamwise diffusion to time-dependent free convection of liquid metals

A numerical analysis is given for the application of streamwise diffusion to high-intensity flows with marginal spatial resolution. Terms are added to the momentum equation which are similar to those used in the Petrov-Galerkin, Taylor-Galerkin and balancing tensor diffusivity methods. Values for the streamwise viscosity are obtained from mesh refinement studies. An illustration is given for the time-dependent free convection of a liquid metal in a cavity with differentially heated sided walls. The spatial problem is solved with the Galerkin finite element method and the time integration is performed with the backward Euler method. Solution quality and computation time are compared for results with and without added streamwise diffusion. For the cases presented, streamwise diffusion eliminates spurious oscillations and saves computation time without compromising the solution.     

Numerical study of dispersing pollutant clouds in a built-up environment

In this paper, we study numerically the dispersion of a passive scalar released from an instantaneous point source in a built-up (urban) environment using a Reynolds-averaged Navier–Stokes method. A nonlinear k–? turbulence model [Speziale, C.G., 1987. On nonlinear k–l and k–? models of turbulence. J. Fluid Mech., 178, 459–475] was used for the closure of the mean momentum equations. A tensor diffusivity model [Yoshizawa, A., 1985. Statistical analysis of the anisotropy of scalar diffusion in turbulent shear flows. Phys. Fluids, 28, 3226–3231] was used for closure of the scalar transport equations. The concentration variance was also calculated from its transport equation, for which new values of Yoshizawa’s closure coefficients are used, in order to account for the instantaneous tracer release and the complex geometry. A new dissipation length-scale model, required for the modelling of the dissipation rate of concentration variance, is also proposed. The numerical results for the flow, the pollutant concentration and the concentration variance, are compared with experimental data. This data was obtained from a water-channel simulation of a full-scale field experiment of tracer dispersion through a large array of building-like obstacles known as the Mock Urban Setting Trial (MUST).     

Correcting Anisotropic Gradient Transport of k

The gradient transport model for k is extended to classes of turbulent flows for which the gradient transport hypothesis is relevant but the anisotropy of the Reynolds stress, to which the eddy diffusivity is proportional, is large and variable. In highly anisotropic turbulence the standard isotropic model used in engineering practice is fundamentally wrong and the conventional anisotropic approximation inadequate. The work is motivated by the important observations that the eddy diffusivity coefficient for a standard gradient transport model for various transported quantities is a factor of 3–10 times larger in highly anisotropic turbulence than that used in standard engineering models. While the conventional anisotropic eddy diffusivity approximation appears adequate for material conserved scalars it is inadequate for k. The problem is solved by addressing the anisotropy of the turbulent transport of k at the level of the underlying third order tensor. It is shown that, unlike the traditional transport models for k, the orientation of the anisotropy with respect to the direction of the gradient plays a crucial role not accounted for in conventional models used in engineering calculations. The new anisotropic eddy diffusivity tensor is quadratic in the anisotropy (the traditional model is linear in the anisotropy). It is shown that the new more rigorous anisotropic eddy diffusivity varies 300% more than the standard model comparing the isotropic limit to the value for the two-dimensional limit. The two-dimensional limit is important in strongly stably stratified flows, in pressure gradient or shock driven flows and in rotating flows. Using the simple shear and the homogeneous non-equilibrium Rayleigh Taylor turbulence the new anisotropic diffusivity tensor is validated in inhomogeneous Rayleigh Taylor turbulence at early and late times.     

The Fast Exact Closure for Jeffery’s equation with diffusion

Jeffery’s equation with diffusion is widely used to predict the motion of concentrated fiber suspensions in flows with low Reynold’s numbers. Unfortunately, the evaluation of the fiber orientation distribution can require excessive computation, which is often avoided by solving the related second order moment tensor equation. This approach requires a ‘closure’ that approximates the distribution function’s fourth order moment tensor from its second order moment tensor. This paper presents the Fast Exact Closure (FEC) which uses conversion tensors to obtain a pair of related ordinary differential equations; avoiding approximations of the higher order moment tensors altogether. The FEC is exact in that when there are no fiber interactions, it exactly solves Jeffery’s equation. Numerical examples for dense fiber suspensions are provided with both a Folgar–Tucker (1984) [3] diffusion term and the recent anisotropic rotary diffusion term proposed by Phelps and Tucker (2009) [9]. Computations demonstrate that the FEC exhibits improved accuracy with computational speeds equivalent to or better than existing closure approximations.     

Towards a constitutive law for the unsteady contact stress in granular media

In order to build a unified modelling for granular media by means of Eulerian averaged equations, it is necessary to study two contributions in the effective Cauchy stress tensor: the first one concerns solid and fluid matter, including contact and collisions between grains; the second one focuses on the random movements of grains and fluid, similar to Reynolds stress for turbulent flows. It is shown that the point of view of piecewise continuous media already used for two phase flows allows one to derive a constitutive equation for the first contribution, under the form of a partial differential equation. Similarly as for the Reynolds stress in turbulent flows, this equation cannot be written only in terms of averaged quantities without adequate approximations. The structure of the closed equation is discussed with respect to irreversible thermodynamics, and in connection with some already existing models. It is emphasised that numerical simulations by the discrete elements method can be used in order to validate these approximations, through numerical experiments statistically considered. Finally an extension of this approach to the second contribution of the effective Cauchy stress tensor is briefly discussed, showing how the modelling of both contributions have to be coupled.