Flow modelling of quasi-Newtonian fluids in two-scale fibrous fabrics

Permeability is the fundamental macroscopic material property needed to quantify the flow in a fibrous medium viewed as a porous medium. Composite processing models require the permeability as input data to predict flow patterns and pressure fields. In a previous work, the expressions of macroscopic permeability were derived in a double-scale porosity medium for both Newtonian and generalized Newtonian (shear-thinning) resins. In the linear case, only a microscopic calculation on a representative volume is required, implying as many microscopic calculations as there are representative microscopic volumes in the whole fibrous structure. In the non-linear case, and even when the porous microstructure can be described by a unique representative volume, a large number of microscopic calculations must be carried out as the microscale resin viscosity depends on the macroscopic velocity, which in turn depends on the permeability that results from a microscopic calculation. An original and efficient offline-online procedure was proposed for the solution of non-linear flow problems related to generalized Newtonian fluids in porous media. In this paper, this procedure is generalized to quasi-Newtonian fluids in order to evaluate the effect of extensional viscosity on the resulting upscaled permeability. This work constitutes a natural step forward in the definition of equivalent saturated permeabilities for linear and non-linear fluids.     

Slip condition on the surface of a model fibrous porous medium

We have solved the problem of a creeping two-dimensional shear flow of a viscous incompressible fluid in a flat channel partially filled with a fibrous porous medium, which is modeled by a regular lattice of square cylinders arranged across the flow. The hydrodynamic microscopic velocity fields are determined by numerical methods. The slip coefficient at an effective boundary of the model porous medium is evaluated by macroscopic averaging.     

Permeability prediction of fibrous porous media with complex 3D architectures

We present a new three-dimensional finite element technique to solve efficiently flows in a representative porous volume with fibrous microstructures, which employs a fictitious domain method to deal with immersed microstructures and a mortar-element method to satisfy rigorously the tri-periodic boundary condition for the representative volume element. Through the extensive numerical simulations for various fiber and fabric architectures, we investigate the relationship between the permeability and fiber architectures in order to establish a reasonable approximation method in estimating the permeability of such complex 3D architectures. Specifically we discuss the applicability and the limitation of the macroscopic permeability averaging rule for those purposes, using the permeability of simple structural building blocks. Finally, we present the Kozeny constants of various microstructures for a wide range of the fiber volume fraction, which may facilitate simple permeability estimation of complex 3D porous structures using the Kozeny–Carman model.     

Two-scale homogenization of electromechanically coupled boundary value problems

The contribution addresses a direct micro-macro transition procedure for electromechanically coupled boundary value problems. The two-scale homogenization approach is implemented into a so-called FE²-method which allows for the computation of macroscopic boundary value problems in consideration of microscopic representative volume elements. The resulting formulation is applicable to the computation of linear as well as nonlinear problems. In the present paper, linear piezoelectric as well as nonlinear electrostrictive material behavior are investigated, where the constitutive equations on the microscale are derived from suitable thermodynamic potentials. The proposed direct homogenization procedure can also be applied for the computation of effective elastic, piezoelectric, dielectric, and electrostrictive material properties.     

An algorithmically consistent macroscopic tangent operator for FFT‐based computational homogenization

The present work addresses a multiscale framework for fast‐Fourier‐transform–based computational homogenization. The framework considers the scale bridging between microscopic and macroscopic scales. While the macroscopic problem is discretized with finite elements, the microscopic problems are solved by means of fast‐Fourier‐transforms (FFTs) on periodic representative volume elements (RVEs). In such multiscale scenario, the computation of the effective properties of the microstructure is crucial. While effective quantities in terms of stresses and deformations can be computed from surface integrals along the boundary of the RVE, the computation of the associated moduli is not straightforward. The key contribution of the present paper is the derivation and implementation of an algorithmically consistent macroscopic tangent operator which directly resembles the effective moduli of the microstructure. The macroscopic tangent is derived by means of the classical Lippmann‐Schwinger equation and can be computed from a simple system of linear equations. This is performed through an efficient FFT‐based approach along with a conjugate gradient solver. The viability and efficiency of the method is demonstrated for a number of two‐ and three‐dimensional boundary value problems incorporating linear and nonlinear elasticity as well as viscoelastic material response.     

Micro–macro relations for flow through random arrays of cylinders

The transverse permeability for creeping flow through unidirectional random arrays of fibers with various structures is revisited theoretically and numerically using the finite element method (FEM). The microstructure at various porosities has a strong effect on the transport properties, like permeability, of fibrous materials. We compare different microstructures (due to four random generator algorithms) as well as the effect of boundary conditions, finite size, homogeneity and isotropy of the structure on the macroscopic permeability of the fibrous medium. Permeability data for different minimal distances collapse when their minimal value is subtracted, which yields an empirical macroscopic permeability master function of porosity. Furthermore, as main result, a microstructural model is developed based on the lubrication effect in the narrow channels between neighboring fibers. The numerical experiments suggest a unique, scaling power law relationship between the permeability obtained from fluid flow simulations and the mean value of the shortest Delaunay triangulation edges (constructed using the centers of the fibers), which is identical to the averaged second nearest neighbor fiber distances. This universal lubrication relation, as valid in a wide range of porosities, accounts for the microstructure, e.g. hexagonally ordered or disordered fibrous media. It is complemented by a closure relation that relates the effective microscopic length to the packing fraction.     

Governing equations for unsaturated flow through woven fiber mats. Part 1. Isothermal flows

Correct modeling of resin flow in liquid composite molding (LCM) processes is important for accurate simulation of the mold-filling process. Recent experiments indicate that the physics of resin flow in woven (also stitched or braided) fiber mats is very different from the flow in random fiber mats. The dual length-scale porous media created by the former leads to the formation of a sink term in the equation of continuity; such an equation in combination with the Darcy's law successfully replicate the drooping inlet pressure history, and the region of partial saturation behind the flow-front, for the woven mats. In this paper, the mathematically rigorous volume averaging method is adapted to derive the averaged form of mass and momentum balance equations for unsaturated flow in LCM. The two phases used in the volume averaging method are the dense bundle of fibers called tows, and the surrounding gap present in the woven fiber mats. Averaging the mass balance equation yields a macroscopic equation of continuity which is similar to the conventional continuity equation for a single-phase flow except for a negative sink term on the right-hand side of the equation. This sink term is due to the delayed impregnation of fiber tows and is equal to the rate of liquid absorbed per unit volume. Similar averaging of the momentum balance equation is accomplished for the dual-scale porous medium. During the averaging process, the dynamic interaction of the gap flow with the tow walls is lumped together as the drag force. A representation theorem and dimensional analysis are used to replace this drag force with a linear function of an average of the relative velocity of the gap fluid with respect to the tow matrix for both the isotropic and anisotropic media. Averaging of the shear stress term of the Navier–Stokes equation gives rise to a new quantity named the interfacial kinetic effects tensor which includes the effects of liquid absorption by the tows, and the presence of slip velocity on their surface. Though the gradient of the tensor contributes a finite force in the final momentum balance equation, a scaling analysis leads to its rejection in the fibrous dual-scale porous medium if the permeability of flow through the gaps is small. For such a porous medium, the momentum equation reduces to the Darcy's law for single-phase flow.     

“Equivalent” permeability and flow in compliant porous media

Resin flow modeling for liquid composite molding processes is generally based on assumptions of rigid porous media. This is invalid for process variations utilizing compliant mold. Yet the models built on rigid porous media assumption are used with some success in analyzing such infusions.Previous work showed that for certain porous media the one dimensional flow patterns are similar to those in rigid porous media and the deformation effects can be included in a scaling factor for permeability.This note analyzes the one-dimensional linear and radial flows in porous media with generic constitutive relations between resin pressure, thickness and permeability. It shows that as long as the deformation remains moderate, the effect of deforming porous medium may be incorporated in a single scaling factor for material permeability. This scaling factor depends on material and applied injection pressure, but does not change with time, flow-front position or type of infusion (linear or radial).     

复杂多孔介质多重介质模型的表征单元体

裂缝性岩体及缝洞型碳酸盐岩储层属于复杂多孔介质,岩体内部空隙为从纳米或微米级的微孔隙到数十厘米以上的大裂缝和溶蚀孔洞,空隙尺度跨越的范围很大。对于很多实际问题,由于单重介质模型计算不准确,且又不可能对复杂空隙空间结构在微观水平上进行精确描述,因此只能应用多重连续介质方法研究复杂多孔介质中的输运问题。该文给出了多重介质模型表征单元体的定义。在所研究问题的尺度范围内,这一多重介质模型表征单元体存在,复杂多孔介质的多重连续介质理论才能够成立。还以孔隙度和渗透率作为相关状态变量进行分析,研究了单重介质模型表征单元体和多重介质模型表征单元体间的联系,给出了复杂多孔介质多重介质模型的建立方法。     

Measurement of transverse permeability using gaseous and liquid flow

Accurate measurement of transverse permeability is important for processes such as resin film infusion and vacuum-assisted resin transfer molding. In these liquid composite molding processes the out-of-plane flow is dominant and thus the transverse permeability is needed for flow prediction. This paper introduces an apparatus to measure saturated permeability for fibrous preforms using both gaseous and liquid flow. The setup creates a uniform one-dimensional flow through-the-thickness of the reinforcement by integrating a high permeability layer on the mold surfaces. A wide range of permeability as a function of fiber volume fraction can be measured in one experiment while applying a known load under a hydraulic testing machine. The system has been designed using process simulation. The measurements using the gaseous medium are comparable to the saturated fluid flow results. The measurement system can also be used to measure changes in dry fabric permeability prior to infusion due to debulking or application of binders on the fabric surface.     

The Sink Term to Different Kinds of Fibrous Mat During the Unsaturated Filling Process

The fibrous pre-form of resin transfer molding is a dual-scale porous medium with two distinct scales of pores, i.e., pores in intra- and inter-tow, which produce an unsaturated infiltration phenomenon during filling. A sink term representing the delayed flow rate from the inter-tow gap into the intra-tow one is introduced to establish governing equations. This study mainly analyzes the sink term by tow saturation during the microscopic flow. First, fiber-tow permeability is calculated by FLOTRAN of ANSYS, Second, periodic unit cells are built according to different structures, and the concrete expression of the sink term is indirectly obtained through the numerical simulation and date fitting of tow saturation under different pressure and viscosity conditions. Results indicate that: the FLOTRAN module can be used to calculate the permeability of fiber tow in two directions; Moreover, the filling time and infiltration process for diverse unit cells with the same volume fraction are different; under the same injection condition, different unit cells have different parameters for the sink term.     

On nonlinear evolution of convective flow in an active mushy layer

We consider the nonlinear effect of convective flow in a horizontal mushy layer during solidification. The mushy layer that we consider has a permeable mush–liquid interface and is treated as an active porous medium with variable permeability. The nonlinear partial differential equations involved in this system are conservation equations for flow momentum, mass, temperature, and concentration. Numerical solutions to the resulting weakly nonlinear equations are obtained using a fourth-order Runge–Kutta integration scheme via a shooting technique. An evolution equation of Landau type is derived in terms of linear and first-order solutions by introducing an adjoint operator. The Landau constant is calculated for both cases: constant permeability and variable permeability. The analysis reveals that there is a slow transition of the flow to a steady state with smaller amplitude for an active mushy layer.     

Barothermal Effect in a Gas-Bearing Stratum

The method of small parameter is used to derive analytical solutions to nonlinear problems on the barothermal effect in gas-bearing strata. The obtained solutions are used to perform calculations of the time–space dependences of temperature. It is demonstrated that the magnitude of the effect depends considerably on the permeability of the porous medium and on the percolation rate.     

A two-scale model for fluid flow in an unsaturated porous medium with cohesive cracks

A two-scale model is developed for fluid flow in a deforming, unsaturated and progressively fracturing porous medium. At the microscale, the flow in the cohesive crack is modelled using Darcy’s relation for fluid flow in a porous medium, taking into account changes in the permeability due to the progressive damage evolution inside the cohesive zone. From the micromechanics of the flow in the cavity, identities are derived that couple the local momentum and the mass balances to the governing equations for an unsaturated porous medium, which are assumed to hold on the macroscopic scale. The finite element equations are derived for this two-scale approach and integrated over time. By exploiting the partition-of-unity property of the finite element shape functions, the position and direction of the fractures are independent from the underlying discretization. The resulting discrete equations are nonlinear due to the cohesive crack model and the nonlinearity of the coupling terms. A consistent linearization is given for use within a Newton–Raphson iterative procedure. Finally, examples are given to show the versatility and the efficiency of the approach. The calculations indicate that the evolving cohesive cracks can have a significant influence on the fluid flow and vice versa.     

A general model for the permeability of fibrous porous media based on fluid flow simulations using the lattice Boltzmann method

Fluid flow analyses for porous media are of great importance in a wide range of industrial applications including, but not limited to, resin transfer moulding, filter analysis, transport of underground water and pollutants, and hydrocarbon recovery. Permeability is perhaps the most important property that characterizes porous media; however, its determination for different types of porous media is challenging due its complex dependence on the pore-level structure of the media. In the present work, fluid flow in three-dimensional random fibrous media is simulated using the lattice Boltzmann method. We determine the permeability of the medium using the Darcy law across a wide range of void fractions (0.08 ? ? ? 0.99) and find that the values for the permeability that we obtain are consistent with available experimental data. We use our numerical data to develop a semi-empirical constitutive model for the permeability of fibrous media as a function of their porosity and of the fibre diameter. The model, which is underpinned by the theoretical analysis of flow through cylinder arrays presented by [Gebart BR. Permeability of unidirectional reinforcements for RTM. J Compos Mater 1992; 26(8): 1100–33], gives an excellent fit to these data across the range of ?. We perform further simulations to determine the impact of the curvature and aspect ratio of the fibres on the permeability. We find that curvature has a negligible effect, and that aspect ratio is only important for fibres with aspect ratio smaller than 6:1, in which case the permeability increases with increasing aspect ratio. Finally, we calculate the permeability tensor for the fibrous media studied and confirm numerically that, for an isotropic medium, the permeability tensor reduces to a scalar value.     

Experimental characterization of permeability and fibre wetting for liquid moulding

Liquid moulding processes are unique in that resin is infused into a dry fibre preform. Appropriate wet-out of the reinforcing fibres is thus a necessity for the achievement of good composite properties. For this class of manufacturing methods, both macroscopic flow, as related to Darcy's Law and characterized by permeability, and microscopic flow, as related to fibre wet-out, are important. The current research investigates factors affecting permeability and fibre wet-out as related to liquid moulding. Specifically, it is shown that fabric permeability is dependent on the type of test fluid used. Surface tension and contact angle measurements indicate that interactions at the microscopic level between fibre and test fluid account for these differences in permeability. The investigation illustrates the competing nature of macroscopic and microscopic flow in liquid moulding.     

Connecting the macro- and microstrain responses in technical porous ceramics: modeling and experimental validations

The relation between the macroscopic and the microscopic (lattice) strain response to external uniaxial stress has been investigated for porous ceramics. Analytical and finite element modeling (FEM) have been performed and neutron diffraction data on porous sintered alumina and extruded honeycomb SiC have been used to validate the theoretical approach. By FEM simulations, it is shown that in spite of the complex pore microstructure, shear stresses are small during uniaxial compression. Analytical modeling shows that while the average microscopic stress depends on the applied macroscopic stress only through the porosity p, the average microscopic strain depends on the macroscopic stress through the pore morphology factor m, as well. Novel relationships are proposed to describe this dependence. Analytical calculations and numerical modeling perfectly agree with each other, and both show good consistency with experiments. As predicted, it has been observed that the microscopic (diffraction) Young’s modulus does not depend on the pore morphology factor, and follows the rule-of-mixtures, while the microscopic Poisson’s ratio does not even depend on porosity, but is equal to the value for the dense material property. A practical implication of these findings is that it is not possible to attach a pore morphology factor to a material, unless the processing conditions are tailored to vary p without varying m. In fact, the different values of m found for the different porosities explain why many models can be used to rationalize the experimental data. With the proposed method, the factor m can be independently evaluated by the use of macro- and micro-elastic properties of the porous body. Analogously, the macroscopic elastic properties of the dense material can be obtained by macroscopic and microscopic values measured on the correspondent porous material.     

Modelling convection in solidification processes using stabilized finite element techniques

Solidification of dendritic alloys is modelled using stabilized finite element techniques to study convection and macrosegregation driven by buoyancy and shrinkage. The adopted governing macroscopic conservation equations of momentum, energy and species transport are derived from their microscopic counterparts using the volume‐averaging method. A single domain model is considered with a fixed numerical grid and without boundary conditions applied explicitly on the freezing front. The mushy zone is modelled here as a porous medium with either an isotropic or an anisotropic permeability. The stabilized finite‐element scheme, previously developed by authors for modelling flows with phase change, is extended here to include effects of shrinkage, density changes and anisotropic permeability during solidification. The fluid flow scheme developed includes streamline‐upwind/Petrov–Galerkin (SUPG), pressure stabilizing/Petrov–Galerkin, Darcy stabilizing/Petrov–Galerkin and other stabilizing terms arising from changes in density in the mushy zone. For the energy and species equations a classical SUPG‐based finite element method is employed with minor modifications. The developed algorithms are first tested for a reference problem involving solidification of lead–tin alloy where the mushy zone is characterized by an isotropic permeability. Convergence studies are performed to validate the simulation results. Solidification of the same alloy in the absence of shrinkage is studied to observe differences in macrosegregation. Vertical solidification of a lead–tin alloy, where the mushy zone is characterized by an anisotropic permeability, is then simulated. The main aim here is to study convection and demonstrate formation of freckles and channels due to macrosegregation. The ability of stabilized finite element methods to model a wide variety of solidification problems with varying underlying phenomena in two and three dimensions is demonstrated through these examples. Copyright © 2005 John Wiley & Sons, Ltd.     

Multiphase flow in a capillary porous medium

Based on the theory of porous media, a calculation concept for the multiphase flow in a capillary porous medium will be presented. We will exclusively investigate the rise of liquids in porous bodies due to the capillarity phenomenon. The field equations used consist of the mechanical balance equations and the physical constraint conditions. The treatment of the capillary problem, based on thermomechanical investigations, yields the result that the capillarity force is a volume interaction force and depends on the free Helmholtz energy functions of the phases and the density gradient of the liquid. With respect to the aforementioned outcome, further constitutive relations for the ternary model are developed. The aim of this investigation is the numerical simulation of the behavior of liquid and gas phases in a rigid porous body at rest. Therefore, the needed weak formulations of the governing field equations, i.e. the balance equations of mass and the balance equations of momentum of the liquid and gas phases, will be given. The usefulness of the proposed theory will be demonstrated with an example.