Impact of heterogeneously distributed solid particles in the transition zone between porous and free flow domain

We present a direct numerical simulation approach for the simulation of shallow water flow using the particle based meshfree Smoothed Particle Hydrodynamics (SPH) method. Simulations of single phase flow are done to characterize the occurring flow parameters on both macro-scale and pore-scale. More precisely, we examine initiation of motion and sediment transport as appearing at the interface between a free flow and porous flow domain under parallel flow conditions. Therefore we evaluate three theoretical models presenting analytical solutions for this coupled problem. Moreover, we discuss the influence of heterogeneities at the interface on forces on single grains by implementing and testing various microstructures into our numerical model. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenization of a two‐phase incompressible fluid in crossflow filtration through a porous medium

We study the homogenization of a slow viscous two‐phase incompressible flow in a domain consisting of a free fluid domain, a porous medium, and the interface between them. We take into account the capillary forces on the fluid‐fluid interfaces. We construct boundary layers describing the flow at the interface between the free fluid and the porous medium. We derive a macroscopic model with a viscous two‐phase fluid in the free domain, a coupled Darcy law connecting two‐phase velocities in the porous medium, and boundary conditions at the permeable interface between the free fluid domain and the porous medium.     

The oscillatory channel flow with arbitrary wall injection

In this article, we consider the laminar oscillatory flow in a low aspect ratio channel with porous walls. For small-amplitude pressure oscillations, we derive asymptotic formulations for the flow parameters using three different perturbation approaches. The undisturbed state is represented by an arbitrary mean-flow solution satisfying the Berman equation. For uniform wall injection, symmetric solutions are obtained for the temporal field from both the linearized vorticity and momentum transport equations. Asymptotic solutions that have dissimilar expressions are compared and shown to agree favourably with one another and with numerical experiments. In fact, numerical simulations of both linearly perturbed and nonlinear Navier-Stokes equations are used for validation purposes. As we insist on verifications, the absolute error associated with the total time-dependent velocities is analysed. The order of the cumulative error is established and the formulation based on the two-variable multiple-scale approach is found to be the most general and accurate. The explicit formulations help unveil interesting technical features and vortical structures associated with the oscillatory wave character. A similarity parameter is shown to exist in all formulations regardless of the mean-flow selection.     

Parallel Strategies Applied to FE Simulations of Multiphasic Problems

While the theoretical background of various porous media models is well understood, it is still a demanding task to deal with these models numerically. In this contribution, a triphasic model is presented, which is capable of describing partially saturated soils. In quasi‐static conditions, this model results in the primary variables solid displacement, pore‐liquid pressure and pore‐gas pressure. For a stable numerical implementation, Taylor‐Hood elements are required, which need quadratic ansatz functions for the displacement and linear ansatz functions for the pressure terms. Looking at numerical simulations in 2‐d, challenging finite element calculations have already been realized in combination with adaptivity in time and space [1]. Nevertheless, new strategies have to be considered for a realization of applications of the model in 3‐d in order to handle the huge amount of unknowns arising from the discretization with Taylor‐Hood elements. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase-Field Modeling of Hydraulic Fracture

Hydraulically driven fracture has gained more and more research activity in the last few years, especially due to the growing interest of the petroleum industry. Key challenge for a powerful simulation of this scenario is an effective modeling and numerical implementation of the behavior of the solid skeleton and the fluid phase, the mechanical coupling between the two phases as well as the incorporation of the fracture process. Existing models for hydraulic fracturing can be found for example in [1], where the crack path is predetermined, or in [2] who use a phase field fracture model in an elastic framework, however without incorporating the fluid flow. In this work we propose a new compact model structure for the Biot-type fluid transport in porous media at finite strains based on only two constitutive functions, that is the free energy function ψ and a dissipation potential ϕ that includes the incorporation of an additional Poiseuille-type fluid flow in cracks. This formulation is coupled to a phase field approach for fracture and is fully variational in nature, as shown in [3]. In contrast to formulations with a sharp-crack discontinuity, the proposed regularized approach has the main advantage of a straight-forward modeling of complex crack patterns including branching. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of sediment transport at the interface of a porous and a free flow domain

Sediment transport involves fluid flow in two different regions. In the free flow domain, the flow is governed by the viscous Newtonian fluid; sediment only occurs as suspended particles. In the porous domain however, the flow is governed by the pore geometry of the porous skeleton consisting of sedimented grains. In nature, the interface between these two domains is not a no-slip boundary for the free flow. In this study, we quantify how sediment transport is affected by the interaction of the two different flows. We do this by comparing fluid flow in no-slip bounded flow channels to fluid flow in channels containing both a free and a porous domain. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical study of dynamics of single bubbles and bubble swarms

A systematic computational study of the dynamics of gas bubbles rising in a viscous liquid is presented. Two-dimensional simulations are carried out. Both the dynamics of single bubbles and small groups of bubbles (bubble swarms) are considered. This is a continuation of our previous studies on the two-bubble coalescence and vortex shedding [A. Smolianski, H. Haario, P. Luukka, Vortex shedding behind a rising bubble and two-bubble coalescence: a numerical approach, Appl. Math. Model. 29 (2005) 615–632]. The proposed numerical method allows us to simulate a wide range of flow regimes, accurately capturing the shape of the deforming interface of the bubble and the surface tension effect, while maintaining the mass conservation. The computed time-evolution of bubble’s position and rise velocity shows a good agreement with the available experimental data. At the same time, the results on the dynamics of bubble interface area, which are, up to our knowledge, presented for the first time, show how much the overall mass transfer would be affected by the interface deformation in the case of the bubble dissolution. Another set of experiments that are of interest for chemical engineers modelling bubbly flows concerns the bubble swarms and their behavior in different bubble-shape regimes. The ellipsoidal and spherical shape regimes are considered to represent, respectively, the coalescing and non-coalescing bubble swarms. The average rise velocities of the bubble swarms are computed and analyzed for both regimes.     

On second-order antidiffusive Lagrangian-remap schemes for multispecies kinematic flow models

This paper focuses on the numerical approximation of the solutions of multi-species kinematic flow models. These models are strongly coupled nonlinear first-order conservation laws with various applications like sedimentation of a polydisperse suspension in a viscous fluid, or traffic flow modeling. Since the eigenvalues and eigenvectors of the corresponding flux Jacobian matrix have no closed algebraic form, this is a challenging issue. A new class of simple schemes based on a Lagrangian- Eulerian decomposition (the so-called Lagrangian-remap (LR) schemes) was recently advanced in [4] for traffic flow models with nonnegative velocities, and extended to models of polydisperse sedimentation in [5]. These schemes are supported by a partial numerical analysis when one species is considered only, and turned out to be competitive in both accuracy and efficiency with several existing schemes. Since they are only first-order accurate, it is the purpose of this contribution to propose an extension to second-order accuracy using quite standard MUSCL and Runge-Kutta techniques. Numerical illustrations are proposed for both applications and involving eleven species (sedimentation) and nine species (traffic) respectively.     

On the Eulerian–Lagrangian formulation of balance equations in porous media

The article presents a general approach to modeling the transport of extensive quantities in the case of flow of multiple multicomponent fluid phases in a deformable porous medium domain under nonisothermal conditions. The models are written in a modified Eulerian–Lagrangian formulation. In this modified formulation, the material derivatives are written in terms of modified velocities. These are the velocities at which the various phase and component variables propagate in the domain, along their respective characteristic curves. It is shown that these velocities depend on the heterogeneity of various solid matrix and fluid properties. The advantage of this formulation, with respect to the usually employed Eulerian one, is that numerical dispersion, associated with the advective fluxes of extensive quantities, are eliminated. The methodology presented in the article shows how the Eulerian–Lagrangian formulation is written in terms of the relatively small number of primary variables of a transport problem. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 505–530, 1997     

用SADI方法求解方形空腔中具有高Rayleigh数的非定常自然对流问题

本文用三次样条积分计算了在方形空腔中具有高Rayleigh数Ra=107和Ra=2×107的非定常自然对流问题。二维N-S方程和能量方程是在非均匀网格中用两个交替方向的三次样条公式进行计算的。文中简要讨论了过渡流动的主要特征,所得结果与理论予估值[1,2]吻合很好。Ra=107时的稳态结果与近期文献中的结果一致。     

A simple and robust boundary treatment for the forced Korteweg–de Vries equation

In this paper, we propose a simple and robust numerical method for the forced Korteweg–de Vries (fKdV) equation which models free surface waves of an incompressible and inviscid fluid flow over a bump. The fKdV equation is defined in an infinite domain. However, to solve the equation numerically we must truncate the infinite domain to a bounded domain by introducing an artificial boundary and imposing boundary conditions there. Due to unsuitable artificial boundary conditions, most wave propagation problems have numerical difficulties (e.g., the truncated computational domain must be large enough or the numerical simulation must be terminated before the wave approaches the artificial boundary for the quality of the numerical solution). To solve this boundary problem, we develop an absorbing non-reflecting boundary treatment which uses outward wave velocity. The basic idea of the proposing algorithm is that we first calculate an outward wave velocity from the solutions at the previous and present time steps and then we obtain a solution at the next time step on the artificial boundary by moving the solution at the present time step with the velocity. And then we update solutions at the next time step inside the domain using the calculated solution on the artificial boundary. Numerical experiments with various initial conditions for the KdV and fKdV equations are presented to illustrate the accuracy and efficiency of our method.     

Microscale Investigations of Highfrequency Wave Propagation Through Highly Porous Media

Wave propagation in highly porous materials has a well established theoretical background. Still there are parameters which require complex laboratory experimentation in order to find numerical values. This paper presents an effective method to calculate the tortuosity of aluminum foam using numerical simulations. The work flow begins with the acquisition of the foam geometry by means of a micro-CT scanner and further image segmentation and analysis. The elastodynamic wave propagation equation is solved using a velocity-stress rotated staggered finite-difference technique. The effective wave velocities are calculated and using the fluid and, aluminum effective properties, the tortuosity is determined. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convective heat transfer in a conducting fluid over a permeable stretching surface with suction and internal heat generation/absorption

We consider a convective flow in a porous medium of an incompressible viscous conducting fluid impinging on a permeable stretching surface with suction, and internal heat generation/absorption. Using a similarity transformation the governing equations of the problem are reduced to a coupled third-order nonlinear ordinary differential equations. We first examine a number of special cases for which we may obtain exact solutions. We then obtain analytical solutions (by the Homotopy Analysis Method) and numerical solutions (by a boundary value problem solver), in order to further study the behavior of the nonlinear differential equations, for various values of the physical parameters. Our numerical solutions are shown to agree with the available results in the literature. We then employ the numerical results to bring out the effects of the suction parameter, heat source/sink parameter, stretching parameter, porosity parameter, the Prandtl number and the free convection parameter on the flow and heat transfer characteristics. In the absence of suction and free convection, our findings are in agreement with the corresponding numerical results of Attia [H.A. Attia, On the effectiveness of porosity on stagnation point flow towards a stretching surface with heat generation, Comput. Mater. Sci. 38 (2007) 741-745].     

Semi-Discrete Ingham-Type Inequalities

One of the general methods in linear control theory is based on harmonic and non-harmonic Fourier series. The key of this approach is the establishment of various suitable adaptations and generalizations of the classical Parseval equality. A new and systematic approach was begun in our papers [1]-[4] in collaboration with Baiocchi. Many recent results of this kind, obtained through various Ingham-type theorems, were exposed recently in [9]. Although this work concentrated on continuous models, in connection with numerical simulations a natural question is whether these results also admit useful discrete versions. The purpose of this paper is to establish discrete versions of various Ingham-type theorems by using our approach. They imply the earlier continuous results by a simple limit process.     

An improved implementation of the McCracken and Yanosik nine point finite difference procedure

The grid orientation phenomenon present in numerical models of fluid flow in a porous media can give rise to unrealistic predictions when simulating adverse mobility displacements. McCracken and Yanosik [11] proposed a nine point finite difference scheme for approximating the solution of the continuity equations that has the potential of eliminating many of the unrealistic predictions that are observed when using five point finite difference operators. Coats and Ramesh [5] have implemented this scheme in a steamflooding model, and have noted that serious grid effects are present in the simulation of an inverted seven spot pattern. Potempa [12,13] has described a procedure which exhibits minimal grid effects for the problem described by Coats and Ramesh [5]. This paper describes modifications to the McCracken and Yanosik procedure which allow for realistic simulations of this inverted seven spot pattern under a steam drive. These modifications are based upon an approximation scheme that has been previously reported [12,13], and affect the incorporation of upstream weighting in a similator.     

A ternary phase bi-scale FE-model for diffusion-driven dendritic alloy solidification processes

It is of high interest to describe alloy solidification processes with numerical simulations. In order to predict the material behavior as precisely as possible, a ternary phase, bi-scale numerical model will be presented. This paper is based on a coupled thermo-mechanical, two-phase, two-scale finite element model developed by Moj et al. [2], where the theory of porous media (TPM) [1] has been used. Finite plasticity extended by secondary power-law creep is utilized to describe the solid phase and linear visco-elasticity with Darcy's law of permeability for the liquid phase, respectively. Here, the microscopic, temperature-driven phase transition approach is replaced by the diffusion-driven 0D model according to Wang and Beckermann [3]. The decisive material properties during solidification are captured by phenomenological formulations for dendritic growth and solute diffusion processes. A columnar as well as an equiaxial solidification example will be shown to demonstrate the principal performance of the presented model. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Uncertainty Quantification for Two-Phase Flow in Heterogeneous Porous Media

The simulation of flow and transport in porous media such as aquifers often involve dealing with complex heterogeneities. They are characterized by varying hydrogeological properties which differ strongly from the adjacent medium and often lead to significant changes of the flow behavior. However detailed information about the location of such heterogeneities is not always known. The deterministic models thus need to be extended stochastically to quantify uncertainties. As mathematical model we use the capillarity-free fractional flow formulation for two immiscible and incompressible fluid phases in a two-dimensional and partitioned domain. To cope with the randomly located heterogeneity interfaces we employ a stochastic Galerkin (SG) method [4]. The physical space of this system then is modelled by a central upwind finite volume scheme [5] in combination with mixed finite elements [7]. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On some oblique derivative problems arising in the fluid flow in porous media. A theoretical and numerical approach

In this paper we consider some free boundary problems related to the fluid flow in a porous medium. By applying a method due to Baiocchi [1] these problems are reduced to nonlinear problems on a fixed domain. The main difficulty here lies in the fact that such problems are not variational because of jump discontinuities in the direction of the oblique derivative in the boundary condition. We give a uniqueness result and by a constructive method we establish at the same time an existence result and a new algorithm for the numerical solution of the original free boundary problem. Some numerical results are given.     

An equivalent pipe network model for free surface flow in porous media

Free surface flow analysis in porous media is challenging in many practical applications with strong non-linearity. An equivalent pipe network model is proposed for the simulation and evaluation of free surface flow in porous media. On the basis of representative elementary volume with homogeneous pore-scale patterns, the pore space of the homogeneous isotropic porous media is conceptualized as a collection of capillary tubes. According to Hagen-Poiseulle's law and flux equivalence principle, equivalent hydraulic parameters and unified governing formulations for the pipe network model are deduced. The two-dimensional free surface flow problem is reduced to a one-dimensional problem of pipe networks and a one-dimensional procedure based on the finite element method is then developed by introducing a continuous penalized Heaviside function. The proposed equivalent pipe network model is verified with results from numerical solutions and laboratory-measured data available in the literature, and good agreements are obtained. The proposed equivalent pipe network model is shown to be effective in analyzing the free surface flow in porous media. The numerical results also indicate that the proposed equivalent pipe network model has weak sensitivity of the mesh size and penalty parameters.     

Finite element methods with numerical quadrature for elliptic problems with smooth interfaces

The purpose of this paper is to study the effect of the numerical quadrature on the finite element approximation to the exact solution of elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with classical finite element methods [Z. Chen, J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math. 79 (1998) 175-202]. We derive error estimates in finite element method with quadrature for elliptic interface problems in a two-dimensional convex polygonal domain. Optimal order error estimates in L² and H¹ norms are shown to hold even if the regularity of the solution is low on the whole domain. Finally, numerical experiment for two dimensional test problem is presented in support of our theoretical findings.