`Mathematical and Numerical Modeling of 1-D and 2-D ConsolidationKatarina GustavssonDoctoral Dissertation Royal Institute of Technology Department of Numerical Analysis and Computing ScienceAkademisk avhandling som med tillst° av Kungl Tekniska H¨gskolan framl¨gand o a ges till offentlig granskning f¨r avl¨ggande av teknisk doktorsexamen torsdagen o a den 24 april 2003 kl 10.15 i sal Kollegiesalen, Administrationsbyggnaden, Kungl Tekniska H¨gskolan, Valhallav¨gen 79, Stockholm. o a ISBN 91-7170-419-1 TRITA-NA-9907 ISSN 0348-2952 ISRN KTH/NA/R--99/07--SE c Katarina Gustavsson, April 2003 ,AbstractWe present a mathematical model and a numerical study of a one and two dimensional consolidation processes of a flocculated suspension. The mathematical model is based on a macroscopic description of the process and the suspension is characterized by constitutive relations correlating experimental data to three material functions. The purpose here is to investigate certain mathematical properties of the model and to develop a second order accurate numerical method for this problem. In 1D we investigate and compare two models, the viscous model and the inviscid model. For cases where both models are applicable, we can show that they give similar results. However we can also show that from a numerical point of view, the viscous model is better since it gives a less restrictive condition on the time step in the numerical computations.ISBN 91-7170-419-1 · TRITA-NA-9907 · ISSN 0348-2952 · ISRN KTH/NA/R--99/07--SEiiiivContents1 Introduction 1.1 One Dimensional Gravity and Buoyancy 1.1.1 Numerical Results . . . . . . . . 1.2 Gravity and Shear Induced Separation . 1.2.1 Numerical Results . . . . . . . . 1.3 Outline . . . . . . . . . . . . . . . . . . 3 5 5 6 6 6 9 10 13 14 15 16 19 19 21 21 22 26 30 31 32 35 36 37 38 39 42 44 44Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .2 Physical Model 2.1 Consolidation of Flocculated Suspensions . . . . . . . . . 2.2 Characterization of the Suspension . . . . . . . . . . . . . 2.2.1 Consolidation Under Gravity . . . . . . . . . . . . 2.2.2 Consolidation by Shear and Gravitational Forces Model . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . Box . . . . . .3 Mathematical Models 3.1 Eulerian Two-Fluid model . . . . . . . . . . . . . . . . . . . . 3.2 Constitutive relations . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Irreversibility and Micro Structure . . . . . . . . . . . 3.2.2 Stresses: Yield Pressure and Non-Newtonian Viscosity 3.2.3 Inter-Phase Momentum Transfer: Permeability . . . . 3.3 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Indicative Froude and P´clet numbers . . . . . . . . . e 3.4 Reduced (Final) Model . . . . . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . .4 One Dimensional Consolidation Models 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Mathematical Analysis . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Well-posedness of the Linearized Viscous Model . . . . . . 4.3 Numerical Treatment of One Dimensional Consolidation Models 4.3.1 Discretization and Staggered Grid . . . . . . . . . . . . . vvi 4.3.2 Numerical Methods for Conservation Laws . . . 4.3.3 A High Resolution Method . . . . . . . . . . . . 4.3.4 Discretization of Elliptic Equation . . . . . . . . 4.3.5 Linear stability analysis . . . . . . . . . . . . . . 4.3.6 Classical Model,  = 0 . . . . . . . . . . . . . . . Numerical Results . . . . . . . . . . . . . . . . . . . . . 4.4.1 Convergence results . . . . . . . . . . . . . . . . Influence of Parameters . . . . . . . . . . . . . . . . . . 4.5.1 Comparison Between Viscous and Inviscid Model 4.5.2 Comparison to Experimental Data . . . . . . . . . . . . . . . . . Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 46 49 49 51 51 52 53 55 56 59 60 61 64 64 66 71 76 77 79 954.4 4.55 Consolidation Model in 2D 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 5.2 Mathematical Analysis . . . . . . . . . . . . . . . . 5.3 Numerical Treatment . . . . . . . . . . . . . . . . . 5.3.1 Discretization on a Staggered, Non-Uniform 5.3.2 High resolution scheme . . . . . . . . . . . 5.3.3 Discretization of the Elliptic System . . . . 5.3.4 Algorithm . . . . . . . . . . . . . . . . . . . 5.4 Convergence Study - Elliptic System . . . . . . . . 5.5 Numerical Experiments . . . . . . . . . . . . . . . 5.5.1 E. Influence of Permeability Hysteresis . . .6 Conclusions 101 6.1 Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101AcknowledgmentsviiviiiChapter 1IntroductionMathematical modeling and numerical approximation of processes involving suspensions are the main topics of this work. A suspension is a composite material consisting of two or more immiscible constituents or phases. It can for example be a fluid with small solid particles, fibers or fluid particles. Suspensions are very common in many industrial applications such as · paper manufacturing · medical applications · food industry · mineral industry · treatment of industrial and municipal waste water. Therefore, it has become an important area of research and much effort is devoted to develop models that describe the behavior of a suspension in different processes. One such process is separation, where by natural gravity or external forces, the suspension is separated into its different components. In this thesis, we focus on the dewatering of industrial or municipal waste water sludge, modeled as a suspension with agglomerates of small particles, flocs. Dewatering is a process where separation is used to reduce the water content to obtain a high level of dryness in the sludge, which is of great importance before disposal. Two different processes are studied, see Figure 1.1. In the one dimensional process, gravitational forces and buoyancy separate the particle phase from the fluid phase. The other process is a two dimensional process where, in addition to gravity, shear is used to improve and speed up the separation. The goal is to develop and study a mathematical model and perform numerical simulations of such processes with the aim to improve the understanding and, eventually, the design of separation process machinery. 34Chapter 1. Introduction0.1clear fluid0.1clear fluid0 00.10 00.1(a) Separation by gravity.(b) Separation by gravity and shear. Shear induced by movement of bottom wall.Figure 1.1. Results from numerical computation of the two different separations processes. Here, contour plots of the volume fraction of solids are displayed. Numerical results are presented in more details in sections 4.4 and 5.5.Mathematical models of a fluid with dispersed particles can be formulated in different ways depending on e.g. the application and the concentration of the suspension. One possibility is a microscopic description of the details of evolution of all particles and all their interfaces. [31, 47, 13] present results from direct simulations of rigid particles suspended in a liquid. However this is only feasible for a limited number of particles. In many practical situations, rather than considering each individual particle, a macroscopic description of the suspension is more appropriate. Then, the two phases (fluid and particle) are modeled as interacting continua, i.e. both the fluid and the solid phases are considered as continuous phases. To keep track of the different phases, a scalar volume fraction field is introduced indicating the proportion of the total volume occupied by the particle phase. To obtain a mathematical model, effective or mean equations are formulated for conservation of mass and momentum for each phase. This view of the process is considered in the present work. There is one major drawback with the macroscopic description compared to a microscopic description; The averaging process involved in obtaining the mean formulation leads to unspecified terms. In order to characterize the suspension and close the formulation, constitutive relations are needed. These are often hard to determine both from modeling as well as from experimental point of view. Here the suspension is characterized by three empirical material functions. These are fitted to experimental data obtained on waste water sludge.1.1. One Dimensional Gravity and Buoyancy Separation51.1One Dimensional Gravity and Buoyancy SeparationThe one dimensional separation process is a classical problem and has been the subject of many papers, [3, 25, 8] among others. In the classical formulation, the mathematical model reduces to one partial differential equation for the volume fraction, here referred to as the classical or inviscid model. The approach here is a slightly different model which includes a term related to the viscous forces on the suspension. This term is usually small and is neglected in the classical way of treating this problem. We choose to keep this term and solve two coupled equations, one equation for the volume fraction and one for the velocity field of the particle phase. From a numerical point of view, there is an advantage with keeping the viscous term since a less restrictive condition on the time step (CFL condition) is obtained than with the classical model. Numerical simulations as well as experimental data show a steep gradient in the volume fraction at the clear fluid interface, see Figure 1.1. The equation for the volume fraction in the classical approach is hyperbolic in the clear fluid region and parabolic otherwise. Mathematically, the regions are separated by a discontinuity. The viscous model gives a volume fraction function with very steep gradients at the interface but it is not clear whether this is a true shock or not. The fact that steep gradients are developed has to be considered in the choice of numerical method. A high resolution scheme based on second order central finite differences is developed for the viscous problem. It is constructed in such way that it also works in the inviscid limit, i.e., when the viscous terms are neglected, the classical model is obtained and the solution develops shocks.1.1.1Numerical ResultsWe present computational results for one dimensional gravity separation. The convergence of the high resolution scheme is studied showing second order accuracy and sharp gradients are well resolved. A numerical comparison between the viscous and the inviscid model shows good agreement when the viscous term is small. The less restrictive CFL conditition for the viscous model translates into faster simulation. Furthermore, our numerical results are validated by a comparison to experimental data for alumina suspensions under natural gravity. Experimentally determined concentration profiles are compared to computed profiles and the agreement is good.6Chapter 1. Introduction1.2Gravity and Shear Induced SeparationThe two dimensional process is quite different from the one dimensional process described above. The separation is irreversible and the micro structure in the suspension is very important for the dewatering. These features have to be incorporated in the mathematical model. A memory function is introduced to treat the irreversibility effects. This function is also used as an additional state variable which describes, in a simplified way, effects of structure in certain constitutive relations. The mathematical model describing the 2-D process consists of an elliptic system of equations for the fluid pressure and the two velocity components of the particle phase coupled to the volume fraction. The equation for the volume fraction is approximated numerically by a two dimensional extension of the high resolution scheme developed for the one dimensional problem. The elliptic system is discretized with second order finite differences.1.2.1Numerical ResultsNumerical experiments are performed to investigate the properties of the mathematical model and to study its sensitivity to the material functions and their influence on the solution. As it turns out the final result is sensitive to the choice of these functions and to parameters such as wall velocity. Below we show the time evolution of a typical 2d computation.1.3OutlineThe outline of this thesis is as follows. An industrial background is given and the physical process is discussed in chapter 2. In section 2.3 we present other work related to the topic. In chapter 3, the mathematical model is introduced together with the constitutive relations and their fitting to experimental data. One dimensional gravity induced separation models are discussed in chapter 4. In section 4.3.3 the high resolution method is introduced and described, and section 4.4 gives numerical results and comparison to experimental data. In chapter 5, the two dimensional gravity and shear induced process is described with a mathematical analysis of the model and the numerical treatment is presented. In section 5.5 we present numerical experiments and parametric studies of the most interesting process parameters. Chapter 6 summarizes conclusions and extensions and future work to be done on this subject.1.3. Outline70.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 00.1 0.090.050.08 0.07 0.06 0.050.2 0.10.150.04 0.03 0.020.3 0.250.01 0 00.020.040.060.080.10.020.040.060.080.1(a) t = 10s(b) t = 50s0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 00.1 0.090.050.08 0.07 0.06 0.050.2 0.10.150.04 0.03 0.020.3 0.250.01 0 00.020.040.060.080.10.020.040.060.080.1(c) t = 100s(d) t = 150s0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 00.1 0.090.050.08 0.07 0.06 0.050.2 0.10.150.04 0.03 0.020.3 0.250.01 0 00.020.040.060.080.10.020.040.060.080.1(e) t = 200s Figure 1.2. .(f) t = 250s8Chapter 1. Introduction This thesis is partially based on the following papers: 1. K. Gustavsson, Simulation of Consolidation Processes by Eulerian TwoFluid Models, Licentiate's thesis, Department of Numerical Analysis and Computing Science, Royal Institute of Technology (KTH), Stockholm, Sweden (1999). ISBN 91-7170-419-1, TRITA-NA-9907. 2. K. Gustavsson and J.Oppelstrup, A Numerical Study of the Consolidation Process of Flocculated Suspensions Using a Two-Fluid Model. Proceedings, The Third European Conference on Numerical Mathematics and Advanced Applications (ENUMATH), World Scientific (1999) 3. K. Gustavsson and J. Oppelstrup, Consolidation of concentrated suspensions - numerical simulations using a two-phase fluid model, Computing and Visualization in Science 3 (2000) 39-45 4. K. Gustavsson and J.Oppelstrup and J. Eiken, Numerical 2D models of consolidation of dense flocculated suspensions, Journal of Engineering Mathematics 41(2001) 189-201 5. K. Gustavsson and J.Oppelstrup, Consolidation of concentrated suspensions - shear and irreversible floc structure rearrangements, Computing and Visualization in Science 4 (2001) 61-66 6. K. Gustavsson and B. Sj¨green, Numerical Study of a Viscous Consolio dation Model, Proceedings of the Ninth International Conference on Hyperbolic Problems Theory, Numerics, Applications, (HYP2002), Springer (2002)Chapter 2Physical ModelContinuous separation of solids from liquids to achieve thickening of suspensions, clarification of liquids, etc., is often carried out by decanter centrifuge. One typical application is dewatering of sludge from municipal and industrial waste water. The decanter is a high speed rotational device with an outer cylindrical bowl and a screw conveyor installed inside the bowl, see Figure 2.1. The suspension is fed into the decanter through the feed pipe A, to the extreme left. &quot;Centrifugal forces&quot; cause the solids in the suspension to sediment and migrate towards the bowl wall producing a &quot;cake&quot; with high volume fraction of solids. The differential speed between the bowl and the screw conveyor transports the solids cake towards the solid outlet B, to the right. The clarified liquid is discharged from the liquid outlet C to the left. A more detailed description of the dewatering process in a decanter can be found in . The movement of the screw conveyor shears the suspension. Shear has a large influence on the separation process and makes the process more efficient:Figure 2.1. A decanter centrifuge. The suspension is fed into the decanter through the feed pipe A, right. Directly to the left of the feed pipe is the solid phase outlet B. The feed pipe empties into the bowl little bit further to the left. The clarified liquid outlet C is on the extreme left. (Picture courtesy of Alfa Laval.)910Chapter 2. Physical ModelA much higher volume fraction of the solid phase (dryer sludge) is achieved in a decanter centrifuge than in an ordinary centrifuge. The box model introduced in  is a simplified and conceptual two dimensional model for how these processes might work, see Figure 2.2. This model serves as a configuration for both experimental and mathematical modeling. The separation is driven by a gravitational force directed downwards, perpendicular to the bottom. Shear forces are induced by moving the bottom wall to the right, see Figure 2.2. Due to gravity, the particles fall to the bottom of the box where an increase in concentration is obtained. The moving bottom wall transports the solid phase to the right and a maximum concentration is obtained in the lower right corner of the box. An experimental device was constructed by Alfa Laval to verify the box model, [B Madsen REFERENCE !!]. The device is sketched in Figure 2.3. It is a centrifuge with an outer cylinder that rotates with a different speed than the inner cylinder. The space between the two cylinders, the bowl, is divided into a number of chambers to obtain the effect of the walls of the screw conveyor. In the experiments, the bowl was filled with waste water, sped up, and stopped after a while. The dry solids content of the cake was determined by removing the remaining water by drying. In Figure 2.4 the concentration of the solid phase is visualized after a short time. The maximum concentration is obtained in the right lower corner where the concentrated part has climbed up along the side wall. This figure can be compare to Figure 1.2 where a numerical computation of a 2D consolidation process is displayed. The box model is also used for the numerical 2D experiments presented in chapter 5.2.1Consolidation of Flocculated SuspensionsThe waste water sludge is a two constituent fluid-particle mixture with many small solid particles dispersed in water. The suspension is dense which means that the volume fraction of particles is large, ( 10%). The density of the particles is higher than the density of the fluid: The particles will fall towards the bottom and form a layer of sediment, i.e. a separation process will take place. When certain surface active substances, known as flocculants, are added, the particles attach to each other and form larger agglomerates of particles. The agglomerates are called flocs and the suspension is said to be flocculated. Flocculating a suspension speeds up the separation process because large flocs will settle faster than small particles. Under the influence of external forces, e.g. gravitational or centrifugal forces, the flocs can form ever denser aggregates. For a solid phase volume fraction higher than the so-called gel forming fraction, the structure in the suspension changes from individual flocs to a continuous porous network filled with fluid. This network can support normal stresses and resists compression until the ap-2.1. Consolidation of Flocculated Suspensions11Figure 2.2. The original sketch of the conceptual model, , of a dewatering process of a flocculated suspension. The suspension with isolines of the concentration is pictured before and after being subjected to shear and gravity. The maximum volume fraction is found in the lower right corner. (Picture courtesy of Alfa Laval.)12Chapter 2. Physical ModelFigure 2.3. Experimental setup. The outer cylinder rotated with a different speed than the inner cylinder. The centrifuge is divided into a number of chambers.Figure 2.4. Experimental result with waste water sludge. Dry solids content in one chamber in the centrifuge sketched in Figure 2.3. The result is visualized after a short spin with wall relative movement. (Picture courtesy of Alfa Laval.)2.2. Characterization of the Suspension13plied external force exceeds a yield pressure. Once this critical force is exceeded, there will be a compression of the network resulting in fluid being released and an increase in concentration of solids. This is what we refer to as a consolidation process. A detailed discussion of flocculated suspensions can be found in e.g. .2.2Characterization of the SuspensionWe need constitutive relations to accomplish the characterization of the material as well as to form a closed set of equations. In our case experimental data is correlated to three material functions: permeability, viscosity, and yield pressure. A detailed discussion of the constitutive relations and the material functions, fitted to experimental data, is found in section 3.2. Yield Pressure The compressive response of the suspension is described by the yield pressure. When stresses in the suspension exceed the volume fraction dependent yield pressure, ps,yield (), the suspension consolidates to a higher volume fraction with a higher yield pressure. If the flocculation of the suspension is strong enough, the compression of the network is irreversible and a plastic deformation takes place. This means that even if the external compression load subsequently decreases, the suspension will remain compressed. Permeability The interaction between the solid and the fluid phase is due to drag forces on the particles or flocs when transported through the fluid. This can be described by Darcy's law and the concept of permeability from the theory of flow through porous media . Viscosity A Newtonian fluid exhibits a linear relation between the shear force,  and the velocity gradients (shear rate, )   =  .  where the viscosity  is independent of shear rate. Flocculated suspensions are non-Newtonian and viscosity decreases with increased shear rate. This kind of behavior is generally called shear thinning. Flocculated suspensions usually have very high viscosity at low shear rates and are strongly shear thinning. Viscosity also increases with increased volume fraction. A treatment of the rheological behavior of suspensions can be found in e.g.  and . The basis for this work has been that the material data needed should be furnished by techniques which can be applied routinely. Traditional pressure14Chapter 2. Physical Modelfiltration devices and piston experiments in 1D-settings are used for determining permeability and force-deformation relationships. Various types of rotating and oscillatory viscometers can be used to determine the shear behavior. For an overview of a number of techniques developed for mineral applications, see . It proposes the yield pressure and the hindered settling function - the latter related to the permeability - as important characteristics. The work gives ample illustration of the experimental difficulties and how some of these can be overcome. Usually the experimental apparatus is limited to a certain range of the parameters. In particular, the industrial sludge studied in this work has very low permeability and standard filter press devices require too long times to be practically useful. A more fundamental problem is the determination of volume fraction solids. This is easy for mineral suspensions where the density of the solids is accurately known, but we have not found any experimental technique which can measure this quantity for organic sludge. Solids weight fraction, however, is easily determined by drying, and it has been necessary to convert this to solids volume fraction by assumptions on intra-cellular water content. In what follows we assume that the models determined by 1D devices are also valid in the 2D case. For details about the parameter fitting to experimental data, see .2.2.1Consolidation Under GravityWhen no shear forces are present the sedimentation is solely driven by gravitational forces. In this gravity consolidation process, no variation occurs in the horizontal direction, and the process can be described as a 1D process in the vertical direction. If the suspension is confined to a container with an impermeable bottom, three distinct zones can be recognized during the consolidation process, see Figure 2.5. At the bottom there is the compression zone with consolidated material where the stress equals the yield pressure and a variable volume fraction of solids, . On top of that is the overburden where the stress is smaller than the critical yield stress, and the concentration is constant. This as also called the hindered settling zone. Finally, on the top there is a zone with clear liquid separated from the overburden with a sharp interface. The consolidation process will continue until the yield stress balances the gravity and buoyancy forces.2.2. Characterization of the Suspension15yclear fluid0 G0&gt; 0 0Figure 2.5. Consolidation driven solely by gravity. Distribution of the volume fraction of solids,  at some time t. Initially, the concentration was uniform,  = 0 . The compression zone ( &gt; 0 ), the overburden where the volume fraction is constant ( = 0 ) and the clear fluid zone is visualized.2.2.2Consolidation by Shear and Gravitational Forces Box ModelThe box model introduces a shearing velocity field. This process is very different compared to the one described above. There is still consolidation due to gravitational forces and the solid phase consolidates on the bottom of the box. But in addition, there are shear forces acting on the suspension by the moving bottom wall, see Figure 2.6.clear fluid0 G   &gt; 0Figure 2.6. Setup for studying a consolidation process driven by gravity and shear. Closed container with a moving bottom wall that to induce shear. Initially, the concentration in container is uniform,  = 0 .In this case, two zones can be recognized: A consolidated part on the bottom of the container and clear fluid on top of that. The concentrated part of the suspension moves with the bottom wall. As the concentration increases, the16Chapter 2. Physical Modelviscosity of the suspension will increase and the suspension behaves more like a solid and starts to climb up along the wall on the right side. There, the diverging velocity field will deform and break the suspension apart into smaller compressed flocs separated by larger channels. The fluid can flow through these channels more easily and the suspension can be further compressed. Such a change in structure of the suspension is incorporated in the mathematical model by the permeability. These structure effects are discussed in section 3.2. The break-up of the consolidated part of the suspension is a plausible contributor to the faster dewatering that has been observed in experiments where shear and gravitational forces are active.2.3Related WorkConsolidation processes subjected to gravitational forces only are well understood and have been treated both from physical, mathematical and numerical point of view. A vast number of references treat consolidation or sedimentation of concentrated suspensions. Here we will only point out a few that have been used in this work. For an extensive review of the contributions to research in sedimentation and thickening made during the 20th century, see B¨rger and Wedland, . u In 1952, Kynch, , published a paper on mathematical models for the sedimentation of a particle-fluid suspension considered as a continuum. This paper is often considered as the origin of modern sedimentation theory. Kynch's model makes a correct description of the behavior of a suspension of equally sized, small and rigid particles but it does not however work for flocculated suspensions forming compressible sediments. A more general model and mathematical theory for the sedimentation of flocculated suspensions in one dimension is presented in e.g Buscall and White, , and in Auzerais et al., . In the latter, a numerical solution to the problem is also presented. This model will be referred to as the classical 1-D model in this report. This model is also considered in Dorobantu, , where two different numerical approaches are used to produce a solution; a fixed grid two-phase method and an interface tracking method. The numerical solution is compared to experimental data published by Bergstr¨m, . o The thermodynamics of batch consolidation of suspensions is presented by B.Raniecki and J.Eiken in . B¨ rger and coworkers published a number of works concerning consolidation u of flocculated suspension see e.g. [10, 11, 13, 14, 15, 17]. In B¨rger et al., , a u phenomenological theory of consolidation of flocculated suspension is developed and a general mathematical model is derived following classical theory of continuum mechanics. This model is further analyzed in . In one space dimension it reduces to the model describing 1-D gravity consolidation mentioned above. Mathematical analysis of this model is reported in  and numerical treatment2.3. Related Work17in . In [13, 14], the theory for sedimentation of suspensions of small spheres of equal size and density is generalized to polydisperse suspensions. The multidimensional consolidation problem has also been treated by Ystr¨m, o . A mathematical model is developed, well posedness of that model is considered, and the problem is also solved numerically in two dimensions. An application to paper pulp is presented in Zahrai, , where a two phase fluid model for dewatering of an accelerating fibrous suspension is presented. The influence of different stress models for the solids phase on the dewatering is investigated.18Chapter 3Mathematical ModelsA fluid-particle system can either be considered on a meso scale where every particle or floc is explicitly kept track of or on a macro scale where the system is considered to consist of two interacting continua, see Figure 3.1. Whether to consider the fluid-particle suspension on a meso or macro scale depends on the application and whether the suspension is dilute or dense. For a dense suspension where the particles are small and the number of particles is very large, the macro scale description is more suitable. In this chapter, we discuss a mathematical macro scale model of the physical process described in chapter 2. Both the 1D-gravitational consolidation process and the process driven by gravity and shear can be described by this mathematical model. The starting point is the two-phase model, discussed in e.g Ungarish, , Drew and Passman,  and Bustos et al., . This model has been used in different applications such as dewatering of paper pulp, Zahrai, , and in fluidization processes of turbulent gas-solid suspensions, Enwald et al., . It consists of two coupled sets of conservation equations for mass and momentum and the final set of conservation equations resembles a Navier-Stokes system. This is sometimes referred to as the Eulerian/Eulerian approach, see . This is also the starting point for many of the classical 1D consolidation studies, treated in e.g Auzerais et al.,  , and B¨rger and Concha, . u The mathematical model is first formulated without any specific applications in mind. To make the model fit to a specific process or material, constitutive relations are needed. This is discussed in section 3.2.3.1Eulerian Two-Fluid modelWe are concerned with the macroscopic, averaged behavior of the suspension and a continuum approach is used to formulate a mathematical model. The 1920Parti es clFlocs: ri(t) : r t) Fluid:v (r,Chapter 3. Mathematical ModelsConti nuaV ol.frac. (r, : r t) r t) Solid:u (r,*(r, r t)Figure 3.1. Different levels of descriptiontwo phases are treated as separate interpenetrating continuous media and the presence of more than one phase in the suspension is modeled by the concept of volume fractions for each phase. This model is referred to as a two-fluid model. The general idea is to formulate the integral balance equations for a fixed control volume containing both phases and a moving interface. This yields local instantaneous equations for each phase and local instantaneous jump conditions for the interaction between the phases at the interface. In the Eulerian approach these equations are then averaged in a suitable way. Different averaging procedures and closure laws have been employed, , . The model that we present below is based on the model discussed in e.g. , also treated in , where a general phenomenological theory is developed for a consolidation process of a flocculated suspension, which reduces to the classical one dimensional model discussed in chapter 4. The following is assumed to be valid for the suspension: · The particles are small in comparison to the dimensions of the configuration containing the suspension; · The particles have the same density; · Both phases are incompressible; · No mass transfer takes place between the phases; · Surface tension between the phases is neglected.3.2. Constitutive relations21The two-fluid model conservation equations for mass and momentum look as follows,  +  · (u) = 0 t  +  · ( v) = 0 t s  Du + pf -  · T s = s g + m Dt Dv f + pf -  ·  f = f g - m Dt (3.1) (3.2) (3.3) (3.4)where the material derivatives are defined as Du = ut + u · u Dt Dv = v t + v · v. DtSubscript s denotes the solid phase and f denotes the fluid phase.   R is the volume fraction of solids and = 1 - . u, v  R2 are the velocities of the solid and fluid phase respectively. T and   R2×2 denote stress tensors and m  R2 is a phase interaction term. pf  R is the fluid pressure. s and f are densities of the two phases. These equations were postulated in  and used in  to model consolidation of suspended particles under the influence of gravity. To complete the formulation equations (3.1) - (3.4) must be supplemented by constitutive relations, boundary and initial conditions. Boundary and initial conditions are discussed in connection to the numerical treatment discussed in chapter 4 and in chapter 5. The constitutive models are discussed below.3.2Constitutive relationsConstitutive relations specify how the material behaves and how the two phases interact with each other. They are based on experience with the behavior of the material and confirmed by rheological measurements. The modeling is strongly dependent on experimental data. The experiments on industrial waste water sludge, see also section 2.2, have provided material data for the constitutive relations. Here we consider models for the stress tensors in the fluid and solid phase, T and  and the phase interaction term m.3.2.1Irreversibility and Micro StructureConsolidation is irreversible, i.e. once the flocs in the suspension are compressed they will remain compressed.22Chapter 3. Mathematical ModelsTo make the model mimic such behavior we introduce a memory function  (x, t), see , which describes the maximal  encountered since t = 0 by a material particle. It satisfies the equation D = Dt Since D = t + u ·  = - · u Dt the sign of  · u determines whether the suspension is under compression ( is increasing) or dilation ( is decreasing) along a particle path. The constitutive models should take into account the micro structure since micro structural effects are believed to be of great importance for the process. Here micro structure is incorporated in the mathematical model in a simplified way by generalizing  to be an additional state descriptor in addition to the volume fraction in the phase interaction model. This is described in details in section 3.2.3.D Dt0when    , otherwise.D Dt&gt; 0,(3.5)3.2.2Stresses: Yield Pressure and Non-Newtonian ViscosityThe solid phase is assumed to support shear forces only when velocity gradients are present, i.e., in this respect to behave like a fluid, with stress tensor T s = -ps I +  s where ps is the solid phase pressure, I is the unit tensor and  s = s (u + uT ) + s ( · u)I. We denote the viscosity of the solid phase by s and s = -2/3s similar to the Stokes hypothesis for a single phase fluid, . In the material studied the mixture viscosity of the suspension will be orders of magnitude larger than the viscosity of the fluid which we assume to be water. We will therefore neglect the fluid stress tensor,  f , and assume that the mixture viscosity equals the solid phase viscosity. This simplification will not be valid in the clear fluid limit but its influence in the compression zone, where the concentration  is large, will most likely be of little importance. With these assumptions, the solid phase stress tensor is 2  s =  (u + uT ) - ( · u)I . 3 where  denotes the mixture viscosity. (3.7) (3.6)3.2. Constitutive relations Viscosity23We assume that the mixture viscosity is a function of the volume fraction and the shear rate,  = (, ||).  The rate of strain tensor is defined as (u) =  1 (u + uT ) 2 (3.8)and the shear rate, || is a scalar formed from the invariants of the rate of  strain tensor || =  1 |(tr)2 - tr( dot · )|   2 tr = n()nn At low shear rates the flocs are assumed not to break and the viscosity is essentially constant with respect to shear rate. This is called the lower Newtonian region, 0 . At higher shear rates the flocs deform and begin to break down. The velocity gradients induce an orientation of the floc structure to form layers separated by clear fluid. This will cause the apparent viscosity to decrease and the suspension will be shear thinning. For high enough shear rate the viscosity will again reach a region with constant viscosity but at a lower level, the upper Newtonian level,  . The upper and lower regions are not well determined by available experimental data. The measurements were conducted at shear rates where the viscosity was still dependent on the shear rate. Thus 0 and  are treated as somewhat adjustable parameters for which order of magnitude estimates are available. The viscosity is also strongly increasing with the volume fraction of particles. Experimental data for the viscosity as a function of shear rate show a power law behavior and have been fit to a Carreau-Yasuda type of model, , combined with a volume fraction dependence into ||  0.05n(, ||) = Cm 1 +  C = 9.15 × 107 Pas m = 3.4 n = -0.9(3.9)Since n is close to -1 , the flow behaves almost like a Bingham fluid, which has a shear rate independent critical shear stress.24Chapter 3. Mathematical Models107106105 [Pas]104103102101=0.5 =0.3 =0.2 =0.1-2 -1 0 1 2 310 -3 100101010 shear rate [1/s]101010  (a) (, ||) as a function of || for different values of .108106104 [Pas]10210010-210-40.1 1/s 1 1/s 10 1/s 100 1/s 0.1 0.2 0.3  [-] 0.4 0.5 0.610-60 (b) (, ||) as a function of the volume fraction, , for  different values of || Figure 3.2. The viscosity is not constant but strongly shear thinning for concentrated suspensions. This means a decrease in viscosity with increasing shear rate. The viscosity also increases with .3.2. Constitutive relations Yield Pressure25The interaction between the particles is described by the solids phase pressure ps , related to the yield pressure. We saw earlier that the suspension supports normal stresses until the force exceeds a volume fraction dependent yield pressure. When the external compression load decreases, the suspension will remain compressed, and the interparticle forces vanish immediately in dilation. The irreversibility effects are modeled using the memory function  introduced in section 3.2.1, equation 5.35. The solids pressure, ps is assumed to depend on  and  as ps (,  ) = 0 ps,yield () if  &lt;  , if    . (3.10)To fit the yield pressure to experimental data we assume a power law behavior with concentration, see Figure 3.3, ps,yield () = Cn C = 5 × 106 Pa n = 4.846 (3.11)6x 1051000 9005800 700 6004ps [Pa]3p [Pa]s500 4002300 200 10010 00.10.20.3 [-]0.40.50.60.70 00.050.10.15  [-]0.20.250.3(a) Loading(b) Loading and unloadingFigure 3.3. Yield pressure as a function of  fitted to experimental data. In the right figure there is unloading at  = 0.15 (dotted line).A typical particle phase pressure - concentration relation is shown in Figure 3.3. In initial compression the pressure follows the &quot;virgin' curve given by (3.11), and when the concentration decreases from the maximal  (here = 0.15) the pressure vanishes; for a subsequent loading cycle the pressure remains zero until  reaches  , and then follows the &quot;virgin&quot; curve again.26Chapter 3. Mathematical Models3.2.3Inter-Phase Momentum Transfer: PermeabilityThe phase interaction term, m, takes into account the relative motion between the particles and the fluid and can be modeled as follows m= D 1- (v - u) + f C() (v - u), () Dt (3.12)This model, valid only for low relative velocities v - u has also been used in  and . The first term represents a drag force with a drag coefficient (), related to the permeability of the material. The second term is a virtual mass force proportional to the mass of fluid accelerated by a particle in motion. C() is called the virtual mass coefficient. The virtual mass term is an inertia force and will not be further discussed here since inertial effects are neglected based on the order of magnitude estimates in section 3.3. In  the authors split m into a dynamic and a hydrostatic part, m = md + mb . The dynamic part, md is similar to equation (3.12). The hydrostatic part is chosen such that the momentum equation for the fluid yields pf = f g in equilibrium. This is already included in our definition of the momentum equation, (3.4).  Let, k(), denote the intrinsic permeability with the dimension [m2 ] so k is a length scale representative of some pore diameter. Sludge experimental data is obtained for K() = k()/µ. K() is called the Darcy function or simply permeability function and µ is the viscosity of water. The data were fitted to a power law function of , yielding the permeability function K() = Cn C = 4.6 × 10-18 m2 /Pas n = -7.41 (3.13) PermeabilityNote the extreme variation for low values of , Figure 3.4. The model has to be modified in the limit   0 to give finite velocities. The modification should be done in such a way that the sharp limit between the clear fluid and the suspension observed in practice is kept. The modification is defined and discussed in Section 4.2.2. In a two dimensional process, the velocity field can diverge, and the solids fraction tends to decrease, hence changing the permeability. How the permeability changes, depends on details of floc sizes and structure which cannot be captured only by the volume fraction . As an example, the two structures in Figure 3.5 have the same volume fraction but different permeabilities. A heuristic model for how permeability depends on micro structure as described by  and  may be devised as follows. It has one parameter  related to3.2. Constitutive relations27105100K() [m2/Pas]10-510-1010-1510-2000.10.20.3  [-]0.40.50.6Figure 3.4. D() =k µas a function of  for the sludge experimental dataFigure 3.5. Same , different permeabilities28Chapter 3. Mathematical Modelsh*h DD+h*D+h1Figure 3.6. Conceptual model of structurefloc sizes of a network breaking up in expansion. We have no direct experimental evidence for the value of . However, the model produces qualitatively correct effects. Much more detailed theories have been put forward for yield stress models for weakly aggregated dispersions as in , using relations for fractal properties of flocs, see . Inspired by such theories, considerations of &quot;conceptual models&quot; for force balances, see , have been shown to produce results in correspondence with experiments. Other studies, such as , report only limited success in relating the observed power laws to fractal exponents. Consider the medium in Fig.3.6. The n flocs per unit volume are impermeable of size S:   nS e , 1  e  3. The exponent e may be related to the fractal properties of the flocs, see . In compression, all inter floc channels have the same width h , and the permeability is the one measured in 1D experiments. In subsequent expansion, we assume that only a fraction  of the channels widen: 1 = (1 - )n(S + h )e + n(S + h)e This expresses the relation between particle sizes and interparticle distances. 1 The resulting structure has two scales, S and S/ 3 where it is assumed that the larger aggregates are compact so the relevant fractal exponent is 3. Assuming a f -power law dependence on channel widths, with p liquid pres-3.2. Constitutive relations102950log k(,*)-5 *= *=0.1 *=0.2 *=0.3  =0.4*-10-15 00.050.10.150.2  [-]0.250.30.350.4Figure 3.7. Permeability as function of volume fraction solids. The dasheddotted line is the experimentally fitted power law function including the modification for low  for the permeability as a function of volume fraction only.sure, and D the -independent factor in the Darcy coefficient gives a volumetric flow rate Q (m/s) Q = (1 - )nh f + nhf . Dp Again f may be related to fractal exponents of the channel shapes. It follows, that with  = 1/e, the flow rate Q in initial compression when h = h is Q = (1-f ) (1 -  )f , S f -e Dp for low concentrations a power law which for many suspensions can be well fitted to data. The resulting expression for the flow rate in expansion becomes Q =  (1 - )(1/ - 1)f + S f -e Dp +1-f  ((1/ - (1 - )/ ) - 1)f .(3.14)Figure 3.7 shows ,  dependence with f = 5,  = 0.005 and the exponent e = 1. f and D were chosen to make the permeability function fit the values obtained from measurements for  &gt;  . Note the 10log-scale on the ordinate.30Chapter 3. Mathematical Models3.3Dimensional AnalysisA dimensional analysis is performed to give the orders of magnitude of the different terms in equations (3.1)-(3.4). A dimensionless formulation of the equations provides indicative parameters, insight about the relative size of different terms and their relative importance. Non-dimensional quantities for a flow in a H × H box under g gravity are introduced as x = x/H y  = y/H t = t/T u = u/U v = v/Up = p/P = / = s - fe = g/G. gA characteristic time scale is given by T = H/U where U is the free settling velocity of an individual floc at concentration 0 . U is related to the permeability of the material as U = GK0 where K0 = K(0 ) and is used as a scale factor for (). Since the consolidation process is dominated by gravity, the pressure scale is chosen to balance the gravitational forces, P = GH. Equation (3.3) together with (3.6) and (3.12) can now be written in dimensionless form (the superscript * is dropped for simplicity) Frs  1 Du 1 + p -  · s = - ps (,  ) Dt Pes Pe + s eg + () (v - u) + FrC0 C() d (v - u). dt(3.15)where C0 = C(0 ). We assume that C0  O(1). In the same way we can write equation (3.4) in dimensionless form together with equation (3.12) and under the assumption that  f can be neglected, Frf Dv + pf = f eg - (v - u) Dt () d - FrC0 C() (v - u). dt(3.16)The dimensionless P´clet numbers, Pe, Pes , and the Froude number, Fr, are e defined as GH GH U2 (3.17) Pe = Pes = Fr = Ps0 0 GH with 0 = uwall 0 /H. The stress scale factors are 0 = (0 , uwall /H) and Ps0 = ps (0 , 0 ) where uwall /H is a measure of the shear rate. Pe and Pes are P´clet numbers which relate compression and shear strength of the material to e the gravitational force. They can be viewed as a relation between the parameters related to the process and the material properties, yield pressure and shear strength. The Froude number, Fr, is the ratio of the inertia force to the gravity force.3.3. Dimensional Analysis313.3.1Indicative Froude and P´clet numbers eSince the suspension is dense, the permeability will become extremely small in areas where the concentration of the particle phase is high. In these regions the sedimentation velocity will be low and Fr will be small, typically of order 10-9 , compared to 1/Pe and 1/Pes which are of order 10-5 and 10-3 . For full details about indicative Froude and Peclet numbers, see table 3.1. To compute indicative numbers of the dimensionless variables  = 0 = 0.1 is chosen as a reference state. This is the initial volume fraction of particles in the suspension. Using (3.13) and (3.11) we obtain K0 = K(0 )  10-10 m2 /Pas and Ps0 = ps (0 , 0 )  102 Pa (3.18) (3.19)Note that K0 and Ps0 are related only to the properties of the material and not to the specific process the suspension is subjected to. From equation (3.9) the reference value of the viscosity, 0 is computed as 0 = (0 , uwall /H). This value will depend on the bottom wall speed and is given in Figure 3.2. By using the relation between the sedimentation velocity, U and the permeability: U = GK0 , the dimensionless numbers, defined in equation (3.17), can be written in the following form Pe = H G Ps0 Fr = (K0 )2 G H Pes = H 2 G 0 uwallApproximative values of Pe, Pes and Fr for different values of G and uwall , are given in Table 3.1. In all computations  = 103 kg/m3 and H = 0.1 m. The values of K0 and Ps0 are given by (3.18) and (3.19). uwall [m/s] G [m/s2 ] 103 104 105 103 104 105 1/Pes 10-2 10-3 10-4 10-2 10-3 10-4 1/Pe 10-3 10-4 10-5 10-3 10-4 10-5 Fr 10-10 10-9 10-8 10-10 10-9 10-80.010.1Table 3.1. Peclet and Froude numbers for different values of the wall speed and the G-number.32Chapter 3. Mathematical ModelsAs long as uwall and G are in the range given in Table 3.1 the inertial terms can be neglected since Fr is smaller than both 1/Pe and 1/Pes . However, if the G-number or uwall is increased further the terms will eventually be of the same order and the inertial effects have to be considered in the equations. In the application of sludge dewatering some realistic values on G and uwall are: G  15000 m/ss and uwall  0.02 m/s. Most of the numerical computations are performed with G = 10000 m/s2 and uwall ranging from 0 m/s to 0.1 m/s.3.4Reduced (Final) ModelAssuming that all dimensionless variables are of order O(1), then since Fr is much smaller than both 1/Pe and 1/Pes, all terms of O(Fr) can be discarded from the equations (3.15) and (3.16) yielding pf - 1 1 (v - u)  · s = - ps (,  ) + s eg + Pes Pe () pf = f eg - () (v - u) (3.20) (3.21)From equation (3.21) we obtain a relation between u and v as v = u - ()(pf - f eg ) If we choose () = K()/(1 - ) and introduce the reduced pressure p = p f - p 0 -  f eg · x (3.23) (3.22)where p0 is a constant, this equation is recognized as the Darcy equation, see , (1 - )(v - u) = -K()p. If equations (3.1) and (3.2) are added, we obtain one equation for the mixture from which the fluid velocity field can be eliminated by using the relation (3.22).  · (u - ()(1 - )(p) = 0. (3.24)The fluid velocity can also be eliminated from equations (3.20) and (3.21) by adding them pf - 1 1  · s = - ps (,  ) + eg + f eg Pes Pe (3.25)From equations (3.1), (3.24) and (3.25), a coupled hyperbolic-elliptic system for the unknowns , p and u is obtained. With the constitutive relation for3.4. Reduced (Final) Model33the stress tensor,  s , given by equation (3.7), it can be written as (here in dimensional form) t +  · (u) = 0  · (u - K()p) = 0 2 p -  · (, ||)(u + uT - ( · u)I)  3 = -ps (,  ) + Geg . The permeability K is given by equation (3.13), the solid phase pressure, ps is given by (3.10) and (3.11) and the viscosity by equation (3.9). (3.26) (3.27) (3.28)34Chapter 4One Dimensional Consolidation ModelsIn this chapter, mathematical features and the numerical treatment of one dimensional consolidation models is discussed. We investigate and compare two models, the viscous and the inviscid model. The purpose is to develop a numerical method for these models that can easily be generalized to the two dimensional problem. The numerical solution of the two dimensional problem is presented in chapter 5. The motivation for treating one dimensional consolidation is that the analysis is much simplified as well as the numerical treatment both when it comes to the implementation as well as shorter computational time for running test problems. There is also a considerable amount of literature on the one dimensional consolidation process. It has been treated both from experimental, mathematical and a numerical point of view in many important works, see e.g [5, 9, 25]. This chapter begins with an introduction to the mathematical models of one dimensional consolidation. Then we discuss mathematical properties of the equations and how to treat them numerically, and finally we present numerical experiments, comparisons to experimental data presented in  and numerical solutions presented in . 3536Chapter 4. One Dimensional Consolidation Models4.1IntroductionA one dimensional version of the box model confines the suspension to a closed box of height H, see Figure 2.5, with gravity acting in the negative y-direction and y  [0, H]. Our model reduces to the following coupled system of equations   + (v) = 0 t y v  p - (1 - )K() =0 y y y  4  v p = G + ps (,  ) () - 3 y y y y (4.1) (4.2) (4.3)for the volume fraction , the solid phase velocity v and the fluid pressure p. Here  is the density difference between the phases and G is the gravitational acceleration force. To obtain equation (4.3), the constitutive relation for the stress tensor  s is used as in equation (3.7). This reduces to the term 4  3 y () v yin one dimension, where  is the mixture viscosity, assumed to be a function of volume fraction only. The permeability function, K, the viscosity  and the solid phase pressure ps are derived from material properties as specified in section 3.2. In the 1D process, there is no unloading, and irreversibility is important only in the initial stage. Then  is equal to the gel forming fraction, g , so the solid supports any pressure smaller than ps,yield (g ) without deformation. This is evident in the &quot;constant concentration&quot; overburden zone explained in e.g. . At the box bottom and top there is zero normal velocity v(0) = v(H) = 0 and zero normal pressure gradient p p (0) = (H) = 0. y y At t = 0, an initial concentration profile is given, (0, y) = 0 (y). The system of equations given by (4.1)-(4.3) can now be simplified by integrating equation (4.2) yielding v p =- y () (4.4)where () = (1 - )K(). This relation can be used to eliminate the fluid pressure from equation (4.3) and we obtain an elliptic equation for the solid phase velocity field, 4  3 y () v y -  v = G + ps (, g ). () y (4.5)4.2. Mathematical Analysis Equation (4.1) and (4.5) will be referred to as the viscous model. If we multiply equation (4.5) with () we obtain 4  () 3 y () v y - v = () G +  ps (, g ) y37(4.6)The first term on the left hand side, the viscous term, is of order /H 2 . For concentrated suspensions /H 2 is usually very small. Therefore, consolidation modeling usually proceeds with the inviscid model ( = 0), see e.g [3, 17, 25], v = -()  ps (, g ) + G . y (4.7)With () = () and v according to (4.7) equation (4.1) yields ^   - t y () ^  ps (, g ) + G y =0 (4.8)We refer to equation (4.8) as the classical or the inviscid 1D model. Rewrite equation (4.8) as    - (^ ()G) -  t y y () ^ dps (, g )  d y =0 (4.9)This is a hyperbolic equation in regions where ps = 0 and parabolic elsewhere. Here we focus mainly on the viscous model. However, in the limit when   0 the classical model is obtained. Hence the two models have common features regarding the numerical treatment and some of the ideas behind the numerical scheme are partly based on analysis of the inviscid model. For a detailed description, mathematical analysis and numerical treatment of the classical model see e.g [3, 11, 25]. For cases where both the viscous and the inviscid model are applicable we show that they give similar results. However we can also show that from a numerical point of view, the viscous model is better since it gives a less restrictive condition on the time step in the calculations.4.2Mathematical AnalysisIn this section we will discuss mathematical properties of the one dimensional consolidation model. We consider the time scale of the problem and how it is related to the constitutive relations. In order to develop an efficient and accurate numerical method, it is of importance to have some estimates of the behavior of the solution to the problem. This is obtained by studying the characteristics of the inviscid model. We also consider the well-posedness of the linearized viscous model.38Chapter 4. One Dimensional Consolidation Models4.2.1Time ScalesThe time scale of the process is determined by the velocity field of the particle phase. In the viscous model, the velocity field is obtained as a solution to the elliptic equation (4.5). The solution will of course depend on the constitutive relations used to characterize the suspension, i.e. the viscosity, the permeability and the yield pressure. To obtain information about the time scales of the process and how they are affected by the constitutive relations we will look for estimates of the velocity field in terms of these functions. Assume that  is bounded but can be discontinuous and that ps ()  C. To obtain an estimate we proceed as follows. Assume that there exists a smooth solution v. Multiply equation (4.5) with v and integrate by parts. Using the Poincar´ inequality e 4 2 vy 2  2 v 2 H yields g  + (ps )y v  1 (4.10) 16  2 || + 3 H 2 min where min = min (y) and || = max |(y)|.0yH 0yHHere·is the usual L2 norm defined by f2 H= (f, f ) =0|f |2 dy.If /H 21 the estimate v  || (g  + (ps )y ). (4.11)is accurate and the particle velocity is bounded in terms of the permeability related function, . In 1D the particle velocity is direct related to the time scale of the consolidation process and a small permeability will lead to slow consolidation. 1 then If  or  becomes large, i.e. /H 2 v  C (g  + (ps )y ) min (4.12)where C = 3H 2 /16 2 , is more informative. The velocity is now bounded by 1/min . 1 the time scale When the permeability is small enough to make /H 2 is determined by the permeability. If the viscosity becomes sufficiently large it will become the limiting factor for the speed and determines the time scale of the process. The fact that the time scale of the process is influenced by the size of the viscosity is validated by numerical experiments in 1D. The results are reported in .4.2. Mathematical Analysis394.2.2CharacteristicsWe know from physical experiments that there is a sharp interface between the clear fluid phase and the overburden, see Figure 2.5. Whether this is mathematically a shock or a very sharp gradient in the solution is not clear for the viscous model. The inviscid model, given by equation (4.9), is hyperbolic in the clear fluid phase since the yield pressure is zero. This is the well known Kynch model, . In this region the structure of the solution is determined by the characteristics. To study the characteristics, look at the convective part of equation (4.9) written as a conservation law t + f ()y = 0 where f is the convective flux function and is given by f () = -G^ ().  (4.14) (4.13)The characteristics are curves, y(t) in the y - t plane determined by the equation dy df = (4.15) dt d A sharp interface between the clear fluid phase and the particle phase (between g and  = min ) is obtained or kept only if the slope of the characteristics for  = g is greater than the characteristics for  = min that is df df | &gt; | , d g d min In this case g &gt; min . (4.16)df d = - G^ () + G () .  ^ d d(4.17)So, whether the solution will develop a shock or not depends on the function  ^ related to the permeability function, () = (1 - )K(). ^ If the permeability function fitted to experimental data is used, (3.13), the flux function can be written as f exp () = -C2 (1 - )n = -C(1 - )n+2 (4.18)where C = 5 × 10-18 G and n = -7.41, see Figure ??. The -derivative of the flux function, f , can easily be computed and it is plotted in Figure 4.2.2. exp We see that f is a decreasing function of  which means that the condition for a shock, given by equation (4.16), is never fulfilled. The characteristics will diverge and the solution will not develop any shocks. The empirical Brinkman permeability model, , KB () = (2 - 3)2 2a2 9µ (3 + 4 + 3 8 - 32 ) (4.19)40Chapter 4. One Dimensional Consolidation Models-10 -10 -10 -10010202410156f()/C [-]-10 -10 -10 -10 -10 -108f ()/C [-]1010121014105161800.10.20.3 [-]0.40.50.60.710000.10.20.3 [-]0.40.50.60.7(a) Flux function, f exp ()exp (b) -derivative of f, f ()Figure 4.1. The flux function and the -derivative corresponding to equation (4.18). They are scaled with 1/C.where a is an average particle radius and µ is the viscosity of the fluid, is commonly used for the permeability of a flocculated suspension, see e.g  or . In this case we obtain a flux function as f B () = -CB 2 (1 - )KB () (4.20)where CB = 2a2 G/9µ, which is the Stoke's drag on a sphere of radius a. The low concentration behavior is consistent with the fall speed of single particles. The flux function, see Figure 4.2.2, fulfills the condition for a shock is since B B f (g ) &gt; f (min ). This means that the Brinkman permeability model will produce a sharp interface between the clear fluid phase and the overburden, as observed experimentally. Modified permeability function The flux function based on the permeability fitted to experimental data, see equation (3.13), will tend to infinity in the limit as   0. This is not physical and the permeability function has to be modified. We know from experiments that the interface between the clear fluid phase and the overburden is very sharp and the modification should be such that we obtain a characteristic structure corresponding to a solution with a sharp gradient between the clear fluid phase and the overburden. To obtain this we approximate the permeability function for small   lim with a second order polynomial. The modified permeability function is given by K() = if  &gt; lim Cn C(a0 + a1  + a2 2 ) if   lim (4.21)4.2. Mathematical Analysis4100.40.2-0.010-0.02f()/CB [-]-0.03-0.04-0.05f()/CB [-]-0.2-0.4-0.6-0.8-0.06 00.10.20.3 [-]0.40.50.60.7-1 00.10.20.3 [-]0.40.50.60.7(a) Flux function, f B ()B (b) -derivative of f, f ()Figure 4.2. The flux function and the -derivative corresponding to the Brinkman model, see equation (4.20). They are scaled with 1/CB .where C and n are given in (3.13) and lim = 0.05, see Figure 4.3. The coefficients in the polynomial are determined such that K  C 2 and are are given by1 3 a0 =n ( n2 - n + 1) lim 2 2 a1 =n-1 (-n2 + 2n) lim 1 a2 = n-2 (n2 - n) 2 lim(4.22)The modified permeability function (4.21) will change the flux function and its -derivative according to Figure 4.4(a, b). The modification produces a nonconvex flux function, see Figure 4.4 (a). For small  this implies a possibly complicated shock structure. Nevertheless, numerical experiments indicate a simple shock between the clear fluid phase and the sediment as we expect, see section 4.4.42Chapter 4. One Dimensional Consolidation Modelsx 10105-7871006k() [m2/Pas]10k() [m /Pas]-552410-103210-151-201000.10.20.3 [-]0.40.50.60.70 00.010.020.03 [-]0.040.050.060.07(a) Modified permeability function, K()(b) Close up of the modification as   0Figure 4.3. Modified permeability function. The dashed line is the modification as  tend to zero, see equation (4.21).0x 1072x 109-0.51.51 -1 0.5f()/C [-]0.02 0.04 0.06 0.08 0.1  [-] 0.12 0.14 0.16 0.18 0.2f()/C [-]-1.50-2-0.5 -2.5 -1 -3-1.5-3.5 0-2 00.020.040.060.080.1  [-]0.120.140.160.180.2(a) Flux function, f ()(b) -derivative of f, f ()Figure 4.4. The flux f () and the -derivative of f, f () when K() is used according to 4.21. Close up for small values of .4.2.3Well-posedness of the Linearized Viscous ModelWhat about non-linear model and 2D. Jacob och Heinz, Burgers new 2D model, CHECK!!!! We will here show that the linearized and localized viscous Cauchy problem is well-posed, that is the problem has a unique solution which depends continuously on data. Linearize the equations (4.1) and (4.5) around a solution 0 (t, y) and v0 (y; 0 ).4.2. Mathematical Analysis43Let the coefficients be frozen at a constant state 0 , v0 , then the equations are simplified to t + v0 y + 0 vy = 0 4 v  v0 0 vyy - = ps0 y + (G - 0 2 ) 3 0 0 (4.23) (4.24)where 0 = (0 ), 0 = (0 ) and ps0 = ps (0). The prime indicates the -derivative and ps0 = dps /d(0 ) is the elastic modulus.  and v are the perturbations from the constant state. We have also used the fact that v0 = -G0 0 . For this linear constant coefficient problem we can use Fourier transformation in the spatial variable, defined byv () = ^-^ v(y)e-iy dy and (t, ) = -(t, y)e-iy dy.Taking the Fourier transform of equations (4.23) and (4.24)gives ^ ^ t + v0 i  + 0 i^ = 0 v 4 v ^  v0 ^ ^ ^ = ps0 i  + (G - 0 2 ) - 2 0 v - 3 0 0 (4.25) (4.26)Using (4.26) to eliminate v from (4.25) we obtain an equation on the form ^ ^ ^ ^ t = P () ^ where the symbol P is ps0 0  v0 0 /2 - G0 0 ^ P () = i(v0 + 0 ) - 2 40  2 /3 + 1/0 40  2 /3 + 1/0 The linearized problem is well-posed in the sense that ^ (P ())  max 0, - 3ps0 0 40 (4.27)but it is not parabolic, because there is no  &gt; 0 such that ^ (P ())  - 2 holds for all , . From this we conclude that the linearized equation admits slightly damped dispersive wave solutions. The first term in (4.27) shows that the phase speed varies somewhat with , between v0 and v0 (2 + 0 0 /0 ). The second term, of the form  2 /( 2 + 1), is a damping term if ps0 &gt; 0, 0 &gt; 0, 0 &gt; 0, which is the case for the physical problem.44Chapter 4. One Dimensional Consolidation Modelsvj j+1/2vj+1yjyj+1Figure 4.5. The equations are discretized on a staggered grid with  defined in the cells and v on the cell boundaries.The damping is parabolic (i.e., like the equation t + ay = yy ) for low frequencies, but behaves like a sink i.e., like the equation t +ay = - for high frequencies. Note that in the limit case of low frequency, the viscosity coefficient is = ps0 0 0 , and in the limit case of high frequency, the sink strength is  = 3ps0 0 /(40 ), so that the viscosity coefficient  has little influence on the smoothing of . Damping and smoothing increase with the elastic modulus, dps /d.4.3Numerical Treatment of One Dimensional Consolidation ModelsIn the following sections, the numerical approximations of the viscous as well as the inviscid model are discussed. The aim is to develop a numerical method for the viscous model such that it also works as the viscosity goes to zero and we obtain the inviscid model. A stability analysis shows the inviscid model to have a considerably more restrictive CFL condition than the viscous model.4.3.1Discretization and Staggered GridLet time and space be discretized by choosing the time step, t, and the mesh width H y = N -1 and define the discrete mesh points yj = (j - 1)y, j = 1, 2, . . . N, yN = H tn = nt, n = 1, 2 . . . The equations are discretized on a staggered grid. Staggered means that different dependent variables are evaluated at different grid points, see figure 4.5. The choice of a staggered grid is natural in this case. Rather than pointwise approximation at grid points, we think of the domain as divided into grid cells. The volume fraction  is assigned to grid location yj + y/2 and n 1 is viewed j+ 2 as an approximation to a cell average of (y, tn ) over the cell [yj , yj+1 ]. The4.3. Numerical Treatment of One Dimensional Consolidation Models45n velocity component is assigned to yj , i.e vj  v(yj ; n ). To update n 1 , the j+ 2 fluxes into and out from the cells are computed at the cell boundaries. In our case the fluxes involves the velocity field, v, which is now defined at the cell boundaries. An advantage with a staggered grid is that we do not need any numerical boundary conditions for . They are given implicitly by the fact that the fluxes are zero at the boundaries. Standard non-staggered grid discretizations with second order central differences support an odd-even mode decoupling, see . The following standard notations for finite difference approximations on a uniform mesh will be used+ uj = uj+1 - uj - uj = uj - uj-1 + uj - uj + uj + - uj D+ u j = D- u j = D0 u j = . y y 2y4.3.2Numerical Methods for Conservation LawsThe conservation equation for the volume fraction (4.1) is non-linear and close to hyperbolic in certain regions, see Section 4.2. Hence, the solution can develop large gradients or even discontinuities. This has to be considered in the choice of a numerical method since it may lead to computational difficulties. Here follows a brief introduction to numerical methods for nonlinear hyperbolic conservation laws and a more detailed discussion is found in e.g. . Start with a conservation law on the form t + f ()y = 0. (4.28)If we let the flux be defined by f () = v(y; ), this is exactly the equation (4.1). To develop a numerical method for (4.28), we consider a finite volume discretization which is bases on the integral form of (4.28). The integral of (4.28) over the (j + 1 )-th cell, [yj , yj+1 ], see Figure 4.6, yields 2 n+1 t n+1 Fjj+1/2F nj+1tn yjj+1/2yj+1Figure 4.6. One dimensional grid with the cell average j+ 1 and the fluxes at 2 the cell boundaries yj and yj+1 .46Chapter 4. One Dimensional Consolidation Modelsd dtyj+1 yj(y, t)dy = f ((yj , t)) - f ((yj+1 , t))(4.29)Finite volume methods are closely related to finite difference methods and a finite volume method can often be interpreted as a finite difference approximation, as we will see below. If we define a piecewise constant approximation to the cell average of (y, t) in the cell [yj , yj+1 ], as j+ 1 (t) = 2 1 yyj+1(y, t)dyyj(4.30)equation (4.29) can be written in the form d 1  1 (t) = - (f ((yj+1 , t)) - f ((yj , t))). dt j+ 2 y (4.31)Let Fj represent an approximation to the value of f ((yj , t)), then a system of ordinary differential equations (ODEs) 1 d  1 (t) = - (Fj+1 - Fj ). dt j+ 2 y (4.32)is solved by an ODE method, typically a multi stage Runge-Kutta method. Equation (4.32) can be identified as a semi-discrete numerical approximation of equation (4.29). It can also be viewed as a direct finite difference approximation in space to equation (4.28) if (4.30) is thought of as a point wise approximation to (yj+ 1 ,t ). 2 To solve this, Fj has to be approximated based on the the cell averages n as (4.33) Fj = h(j- 1 (t), j+ 1 (t)) 2 2 where h is called a numerical flux function. The specific numerical method obtained depends on how h is chosen. This will be discussed below.4.3.3A High Resolution MethodA standard second order accurate method such as e.g. Lax-Wendroff,  works well as long as the computed solution is smooth enough. But, near discontinuities or large gradients it fails and oscillations are generated and the result can be very poor. The idea with a high resolution method is to construct a numerical method of second order accuracy wherever it is possible. In regions where the solutions is not behaving smoothly, it degenerates to a first order accurate scheme to prevent spurious oscillations. Many such numerical schemes are based on upwind differencing. A major disadvantage with the upwind schemes is the fact that a Riemann problem has4.3. Numerical Treatment of One Dimensional Consolidation Models47to be solved, either exact or approximately, requiring information about the eigenstructure of the problem. In our case the velocity field depends globally on  through the elliptic equation, (4.5). Therefore the concept of upwinding is not immediately meaningful and we look instead for a non-oscillatory type of scheme based on a first order central scheme modified to yield second order accuracy. This type of scheme was introduced in  where the main idea was to replace the first order piecewise constant cell average in the Lax-Friedrichs scheme with a piecewise linear approximation. A conservative, semi-discrete form of equation (4.1) on the staggered grid can be written as d 1 j+ 1 = - (Fj+1 - Fj ). (4.34) 2 dt y To simplify the notation, we write  instead of (t). To obtain a first order central differencing scheme in space, let Fj = h(j+ 1 , j- 1 , vj ) = 2 2 1 (f (j+ 1 , vj ) + f (j- 1 , vj )) 2 2 2 Q - ( 1 - j- 1 ), 2 2 j+ 2(4.35)where, according to equation (4.1), f (, v) = (y, t)v(y; ) and  = t/y. Q is called a numerical viscosity coefficient. The forward Euler difference scheme in time n+1 - n 1 d j+ 2 j+ 1 2 j+ 1  2 dt t and Q = 1, is the Lax-Friedrichs scheme. It should be noted that vj is obtained as a solution to the elliptic equation (4.5) and that vj depends on all values of  at the same time level. The discretization of the elliptic equation (4.5) is discussed in the next section. The scheme defined by the discretization (4.34) and the flux function (4.35) suffers from large numerical viscosity and is only first order accurate. In order to increase the accuracy, we introduce the piecewise linear approximation of  in the cell [yj , yj+1 ] (y)[yj ,yj+1 ] = j+ 1 + sj+ 1 2 2 (y - yj+ 1 ) 2 y , yj &lt; y &lt; yj+1 (4.36)The slope of the linear function is given by s/y, see Figure 4.7. The approximative flux function given by (4.35) is modified by using the piecewise linear approximations of  in the following fashion Fj = h(R , L , vj ) j j where R and L are approximations from the right and from the left to the j j value of  at yj .48Chapter 4. One Dimensional Consolidation Modelss j+1/2 y yj j+1/2yj+1Figure 4.7. The solution is approximated as piece wise linear in each cell. The slope is given by s/y.The approximation, R , is obtained by evaluating the linear approximation j of  in the cell [yj , yj+1 ]. From equation (4.36) R = (yj )[yj ,yj+1 ] = j+ 1 - sj+ 1 /2. j 2 2 In the same way, L is obtained by the linear approximation of  in the cell j [yj-1 , yj ] L = (yj )[yj-1 ,yj ] = j- 1 + sj- 1 /2. j 2 2 If R and L are used in (4.35) we obtain the modified numerical flux function j j Fj = h(R , L , vj ) = j j 1 (f (j+ 1 - sj+ 1 /2, vj ) + f (j- 1 + sj- 1 /2, vj ) 2 2 2 2 2 Q - (j+ 1 - sj+ 1 /2 - j- 1 - sj- 1 /2). 2 2 2 2 (4.37)Now there remains to determine the slopes, s, in equations (4.36) and in (??). They are constructed such that the resulting numerical scheme meets the requirement for second order accuracy and guarantees that no unphysical oscillations will arise. To do this we use a non-oscillatory interpolation function, given by sj+ 1 = s(+ j+ 1 , - j- 1 ) (4.38) 2 2 2 As an interpolation function the van Albada limiter is used. It is a very common choice since it is a smooth differentiable function. s(x, y) = (x2 + )y + (y 2 + )x x2 + y 2 + 2 (4.39)Here, is a small constant to prevent division by zero when x = y = 0. The modified flux function gives a second order accurate approximation away from smooth extrema. The complete method for solving equation (4.1) is given by the numerical approximation (4.34) together with the modified flux (4.37). No boundary conditions are imposed directly on . On the boundaries y1 = 0 and yN = H f (v)y=0 = 0 f (v)y=H = 04.3. Numerical Treatment of One Dimensional Consolidation Models49since v(0) = v(H) = 0. Therefore the following conditions are imposed on the numerical fluxes F1 = 0 FN = 0. To discretize in time we use a second order Runge-Kutta method. j+ 1 = n 1 - + Fjn j+2 2(1)(2) j+ 1 22=(1) j+ 1 22- + Fj(2)2(1)(4.40)n+1 = (n 1 + j+ 1 )/2. j+ j+ 1 Here Fjn denote the numerical flux given by equation (4.37) evaluated with  and v at time level tn .4.3.4Discretization of Elliptic Equation(1)Fjn and Fj are computed from the velocity field, which in turn is computed by solving the elliptic equation (4.5). It is discretized in space on the staggered grid by second order central differences. For the coefficients and source terms,  has to be evaluated at the same grid points as v. We define a mean value of  as 1 ¯ (yj ) = j = ((yj- 1 ) + (yj+ 1 )) 2 2 2 The discrete version of (4.5) becomesn D+ ((n 1 )D- vj ) - j-2(4.41)n vj n n ¯ ¯ = F j + D+ ps (j- 1 ) 2 (n ) j(4.42)and the boundary conditions for v are v1 = vN = 0. The solution vj is obtained from a tridiagonal system by Gaussian Elimination.4.3.5Linear stability analysisA linear stability analysis is performed for the simplest central difference approximation with periodic boundaries. We start with the linearized constant coefficient differential equations (4.23) and (4.24), approximated by the second order accurate staggered semi-discrete approximation dj+ 1 (t) 2 dt = -v0 D0 j+ 1 - 0 D+ vj 2 (4.43)50Chapter 4. One Dimensional Consolidation Models 4 1 c ~ 0 D+ D- vj - vj = ps0 D+ j- 1 + (j+ 1 + j- 1 ) 2 2 2 3 0 2where c = G - 0 v0 /2 . ~ 0 Assuming periodic boundaries and length of domain 2, we can introduce the discrete Fourier transformN Nv = ^j=-Ne-iyj vj^  =j=-Ne-iyj+ 12j+ 1 2with yj = j/N and  = 0, 1, . . . N . Inserting into the difference approximation (4.44) the elliptic equation becomes - sin( y ) ^ y v ^ y ^ 160 2 )^ - v  + c cos( ) sin2 ( = 2ips0 ~ 2 3y 2 0 y 2 (4.44)^ As in the continuous case, we can express v in terms of  , and insert into ^ the transformed  equation, ^ d sin y ^ sin y/2 = -v0 i  - 0 2i v ^ dt y y The result is an equation on the form ^ d ^ ^ = Q(, y) . dt Introduce = sin( y ) 2y 2 2,^ i.e. ||  1 and (0) = 1. The Fourier symbol, Q is given by sin y c0 ~ ^ Q = -i (v0 - 4 y 0  2 + 31 0)-ps0 0  2 4 1 2 3 0  + 0.^ The stability condition is that tQ is inside the region of absolute stability ^ for the time integration method. The imaginary part of Q, corresponding to wave propagation, is estimated as follows ^ | (Q)| = 1 |~0 | c (|v0 | + 4 2 y 3 0  +1 0)1 (|v0 | + |~0 0 |) c y^ and the real part of Q is ^ - (Q) = ps0 0  2 4 1 2 3 0  + 04.4. Numerical Results51If the stability region of the time integration method encloses a rectangle -a &lt; (z) &lt; 0, -b &lt; (z) &lt; b the CFL-condition related to the convective part is t (|v0 | + |~0 0 |) &lt; b. c (4.45) y The CFL condition related to the diffusive part is then ^ t (Q)  a  or t  a 34||  N 1 4 0 + 3 0  20  2 + ps0 01 0  2=a ps0 0The right hand side has a minimum when  = 1 and  = N so t  Now, y = /N , so t  a ps0 0 0 1 4 2 l + 2 y 2 3  (4.47) a ps0 0 1 4 0 + 3 0 N 2 (4.46) where l = 0 0 . The conditions given by equation (4.45) and equation (4.47) are sufficient to guarantee linear stability. From equation (4.47) it follows that there is no parabolic time step limit as y  0. When the small scales are resolved, i.e. y &lt; l, the viscous term helps to improve the stability limit.4.3.6Classical Model,  = 0In the classical model when  = 0 the same estimate as above can be used for ^ ^ the imaginary part of Q. For the real part of Q, we have ^ - (Q) = ps0 0 0  2 which leads to the more restrictive CFL-condition t  ay 2 .  2 ps0 0 0 (4.48)We conclude that retaining the viscosity term can be motivated entirely by improved numerical stability. This is validated by numerical experiments and is reported in section 4.4.4.4Numerical ResultsIn this section we present a comparison between the viscous and inviscid models, report on a parametric study, and compare simulations to experimental data.52Chapter 4. One Dimensional Consolidation Models4.4.1Convergence resultsTo verify convergence and to estimate the order of accuracy of the numerical scheme described in section 4.3.3 we solve a test problem given by equations (4.1) and (4.5). The suspension is confined to a box of height H = 0.1 m and consolidates due to a gravitational acceleration force G = 1000 m/s2 and a density difference of  = 1000 kg/m3 . In the computations, the viscosity function fitted to experimental data is used. The viscosity is a function of the volume fraction only. It is given by equation (3.9) and modified as () = C(3.4 + Cmin ) (4.49)where C = 1.3 × 106 Pas and Cmin = 5 × 10-4 is chosen such that the velocity profile becomes smooth enough in the region between the clear fluid phase and the compression zone. For the permeability function we use the Brinkman model, see equation (4.19), (2 - 3)2 2a2 (1 - ) . (4.50) () = 9µ (3 + 4 + 8 - 32 ) The particle phase pressure is given by equations (3.10) and (3.11) with  = g = 0.1. To avoid a discontinuity in ps the function is regularized as ps (, g ) = Rw (, g )ps,yield () where Rw (, g ) = (4.51)1 tanh(w( - g )) + 1) (4.52) 2 where w determines the transition width. In the computations we present here, w = 1000. To obtain smooth coefficients and source terms in equation (4.5) we also regularize the initial data. We approximate the sharp interface between the overburden and the clear fluid phase, 1 (y, 0) = 0 (y) = (g - min ) (1 - tanh((y - y0 ))) + min . 2 (4.53)where g = 0.1, min = 0.001 and  = 100. The simulations are performed on consecutively refined meshes. Since we work with a staggered grid, the mesh is refined as y = h, h/3, h/9, h/27 and h/81 where h = H/36. H = 0.1 is the length of the domain. The time step is chosen such that  = t/y is constant, where t = 0.1 on the coarsest mesh. The numerical viscosity coefficient, Q = 0.005. Figure 4.8 shows concentration and velocity profiles at times 0, 4, 10, and 20 sec. The order of accuracy, p is estimated in the standard way as p = log f h - f h/3 f h/3 - f h/9 / log 3 (4.54)4.5. Influence of Parameters where f is a vector valued function and f the discrete L2-norm f or the max-norm fh, h,253 ·N h,1is either the discrete L1-norm |fj |,=hj=1 = hN1 2 |fj |2  ,j=1= max |fj |.jN is the number of grid points. The computed order of accuracy obtained using the second order scheme given by (4.34), (4.37) and (4.40) is presented in a Table 4.1. No of grid points N 109 325 973 Order v h,2 1.78 1.97 2.00 of convergence, v h,  1 1.76 1.76 1.97 1.97 2.00 2.00 p  2 1.64 1.95 2.00   1.45 1.92 1.99v h,1 1.81 1.98 2.00Table 4.1. Measured order of accuracy, p for the velocity field v and the volume fraction . Computations until t=4.0 s.4.5Influence of ParametersTo study how the different parameters and material functions influence the solution we start by performing a simple one dimensional study. We are interested in the influence of viscosity and permeability on the velocity field. Equation (4.5) is solved with the particle phase pressure, ps given by (4.51), the permeability, K() by equation (4.21) and the viscosity  constant. In the computations we used G = 1000 m/s2 ,  = 1000 kg/m3 and a smooth profile, given by equation 4.53 with g = 0.1, min = 0.001 and  = 100. The computations were made on grid with N = 2917 grid points. A comparison of the velocity field for  = 500, 1000 and 5000 Pas is presented in Figure 4.9. The influence on the solution is mainly close to the walls. We see in Figure 4.9 that the boundary layer becomes thicker when the viscosity increases. We can also note that the boundary layer is thicker at the top than at the bottom. The reason for this is that the thickness of the layer is proportional to K and the permeability, K, is larger at the top where  is small.54Chapter 4. One Dimensional Consolidation Models0.1 0.09 0.08 0.07 0.06 t=0 s t=4 s t=10 s t=20 s0.1 0.09 0.08 0.07 0.06 t=4 s t=10 s t=20 sy [m]y [m]0.05 0.04 0.03 0.02 0.01 0 00.05 0.04 0.03 0.02 0.01 0 -30.050.10.15 [-]0.20.250.30.35-2.5-2-1.5 v [m/s]-1-0.50 x 100.5-3(a) Volume fraction (b) Velocity vFigure 4.8. Volume fraction,  and the velocity, v at times t = 0, (for , dotted line) 4, 10, 20s. The solution is computed using the numerical method presented in section 4.3.3 and 4.3.4, 2917 grid points.0.1 0.09 0.08 0.07 0.06 =500 Pas =1000 Pas =5000 Pasy [m]0.05 0.04 0.03 0.02 0.01 0 -7-6-5-4-3 v [m/s]-2-10 x 101-5Figure 4.9. Influence of the viscosity on the velocity profile. Constant viscosity with  = 500, 1000, 5000P as.A similar comparison but with different sizes of the permeability is also performed. We let the viscosity be constant at  = 500 Pas and the permeability a function of the volume fraction, K(). The velocity profile is studied with the permeability scaled as 1 × K(), 5 × K() and 10 × K(). The result is presented in Figure 4.10. In this case the whole velocity field is affected by the change in permeability.4.5. Influence of Parameters0.1 0.09 0.08 0.07 0.06 K() m /Pas 5K() m2/Pas 10K() m2/Pas255y [m]0.05 0.04 0.03 0.02 0.01 0 -6-4 v [m/s]-2 x 100-4Figure 4.10. Influence of the permeability on the velocity profile. Constant viscosity with  = 500P as.These results are consistent with the time scale analysis performed in section 4.2.1.4.5.1Comparison Between Viscous and Inviscid ModelTo verify that the viscous term in equation (4.5) has little influence on the solution, a comparison between the numerical solutions of the viscous model, (4.1) and (4.5), and the inviscid model, (4.9), is made and presented below. The inviscid model, (4.9) is approximated numerically by the same numerical method presented for equation (4.1) i.e (4.34) and (4.37) together with (4.40). In the inviscid case the flux function is a function of  only and can be written as f () = -^ ()   ps (, g ) + G . y (4.55)In the computations we have used the same test case as in the convergence study presented in section 4.4.1, see also table 4.3. The material functions are the same except for the permeability. In this case we have used the model fitted to experimental data given by equation (4.21). The initial data is given by equation (4.53). Here, the regularizing parameter  = 1000 i.e. the initial data is not as smooth as in the convergence study.56Chapter 4. One Dimensional Consolidation Models Physical parameters G  g ms-2 kgm-3 1000 1000 1000 1000 0.10 0.10 Numerical parameters N t QH m 0.1 0.1325 3250.1 0.0010.005 0.005Viscous model Inviscid modelTable 4.2. Summary of physical and numerical parameter setting for the results presented in Figure 4.11.With the permeability fitted to experimental data, the viscosity is small and the difference between the classical and the viscous model becomes very small. The result is presented in Figure 4.11.0.1 0.09 0.08 0.07 0.06y [m]0.05 0.04 0.03 0.02 0.01 0 00.050.1 [-]0.150.20.25Figure 4.11. Comparison of solution obtained by the classical 1D model (solid line) and the solution obtained by the viscous model (dashed line) at t = 200, 2000 and 20000 s.We compared the cpu time for the two different models. The classical model requires a time step that is approximately 100 times smaller than in the viscous case, as shown by the linear stability analysis in Section 4.3.5. Even though the elliptic equation is solved every time step, the viscous solution only requires 1/50 of the total cpu time of the classical model.4.5.2Comparison to Experimental DataIn  -ray transmission was used to measure volume fraction profiles during transient settling of flocculated alumina suspensions. The objective of the study4.5. Influence of Parameters57was to investigate the effect of varying degrees of flocculation on the consolidation process. Volume fraction profiles obtained by the viscous model are compared to these experimentally measured concentration profiles. The computations and the experimental results agree quite well. A similar comparison for the classical model has been done in both  and in . In the computations we have used the same constitutive relations as in . For the permeability, we employ Brinkman's model, see equation (4.19), as in , and the flux function according to equation (4.20). The constant CB is related to the free settling velocity and has to be determined. The suspension consolidates under natural gravity, G = 9.81 m/s2 . It is observed in  that the value of CB obtained from Stoke's flow drag on a sphere is too small to describe the observed behavior. Instead, CB in  is chosen to reproduce the experimental initial settling speed. The computations we present were performed with the value from , i.e CB = 1.737 × 10-7 m/s. (4.56)The yield pressure is fitted to a power law function according to  and is given by ps () = ps0 n 0 if   g otherwise (4.57)where n = 4.3, ps0 = 80000 Pa and g = 0.15 which is the initial concentration. Of the same reason as above we chose to use the same values for n and ps0 as in . To avoid the discontinuity in the yield pressure at  = g the function is regularized using the regularizing function Rw as defined in equation (4.52) with w = 1000. Physical parameters G  g ms-2 kgm-3 9.81 3083 0.15 Numerical parameters N t QH m 0.23250.10.005Table 4.3. Summary of physical and numerical parameter setting for the results presented in Figure 4.12.In Figure 4.12 we see the numerical solution presented at 6, 11, 17, 24 and 53 days. This is compared to the experimental data presented in . We see that the agreements is very good in the compressed part of the suspension. However, the free settling rate is a little bit too slow in the beginning.580.2 0.18 0.16 0.14 0.12Chapter 4. One Dimensional Consolidation Modelsy [m]0.1 0.08 0.06 0.04 0.02 0 00.050.10.150.2  [-]0.250.30.350.4Figure 4.12. Solution at 6, 11, 17, 24 and 53 days. Solution to be compared to experimental data, BergstromThese numerical results use the viscosity function given by equation (4.49). This mimics the physics and has a computational advantage: The countervariation of viscosity and permeability decreases the variation of the local length scale.Chapter 5Consolidation Model in 2DIn this chapter, the consolidation model in two dimensions is discussed. The suspension is confined to a closed box and in addition to the gravity force, a moving bottom wall imposes a two dimensional effect of shear on the consolidation process.Hclear fluidG&gt;0 L uwallFigure 5.1. Setup for studying a consolidation process driven by gravity and shear. Closed container with a moving bottom wall that to induce shear. Initially, the concentration in container is uniform,  = 0 .We start with an introduction where the mathematical model is presented. Next, we look at the mathematical properties of this model. Guided from the numerical treatment of the one dimensional model we design a numerical solver for the two dimensional model and perform numerical experiments on a test problem. 5960Chapter 5. Consolidation Model in 2D5.1IntroductionThe two dimensional model starts from the reduced model given by (3.26), (3.27) and (3.28). The equations written out in full are  + t p  K + x y u 2   - x 3 x  (u) + x p K - y v  + y  (v) = 0 y u v - =0 x y  u   + y y y ps p = - x x  v  +  + x x x ps p = + g - y y (5.1) (5.2)  v x x 4  3 x(5.3)4  3 yv y-2  3 yu xu y(5.4)Here u and v are the solid phase velocity field components in the x- and ydirections and p is the reduced pressure defined through (3.23), sometimes also called the excess pore pressure. The permeability function, K(), the effective solids pressure ps (,  ), and the mixture viscosity (, ||) are determined by experimental data and are  discussed in section 3.2. Given the volume fraction  and the shear rate ||, equations (5.2), (5.3) and  (5.4) form a linear elliptic system of partial differential equations for u, v and p. With the velocity field u = (u, v) given, equation (5.1) is a hyperbolic equation for the volume fraction . Boundary conditions have to be specified for the velocity components and the fluid pressure. We consider a closed box defined by (x, y)  [0, L] × [0, H] with suspension and with a moving bottom wall, see Figure 5.1. On the top of the box and on the side walls we make the assumption of no friction between the particles and the boundary and slip conditions are imposed as v v u (x, H) = 0 (0, y) = 0 (L, y) = 0 (5.5) y x x The normal component of the velocity is equal to zero on all boundaries, i.e. u(0, y) = 0 v(x, 0) = 0 u(L, y) = 0 v(x, H) = 0 (5.6)The friction between the bottom wall and the particles is large and a slip-no slip condition is imposed (depending on the size of the shear force,  = ()u/y vs.5.2. Mathematical Analysis61normal(friction)force cµ ps ()). If the friction force is large enough, the particles follows the wall with u = uwall . u (x, 0) - u(x, 0) = -uwall y (5.7)where  = cµ ps ()/(()), cµ is the friction coefficient between the bottom wall and the particles. uwall is the specified bottom velocity. The impermeable walls are consistent with the boundary condition for the reduced pressure p =0 n on all boundaries.y H u=0 vx=0 px=0 G uy=0 v=0 py=0u=0 vx=0 px=0u y +  (u-u wall ) = 0 v=0 py=0LxFigure 5.2. Physical domain and boundary conditions.In a multidimensional process, the consolidation can be irreversible, see section. Irreversibility effects are represented by the memory and structure related variable  , D when    , D &gt; 0, D Dt = Dt (5.8) Dt 0 otherwise. Here    D = +u +v (5.9) Dt t x y In this case, K = K(,  ) and ps = ps (,  ). The  -dependence is discussed in section ...5.2Mathematical AnalysisIn the two dimensional case it is not straight forward to establish a similar relation between the particle velocity and the constitutive relations as was done62Chapter 5. Consolidation Model in 2Din the one dimensional problem, see section 4.2.1. The constitutive relations give transport coefficients which are strongly dependent on the concentration. It is however still informative to consider the constant coefficient problem to get rough estimates of the influence of various parameters. With  and K constant the system of equations for u and p can be written as  · (Kp - u) = 0 u + ( + ) · u - p = f () in  (5.10) (5.11)where  = {(x, y) : 0  x  H, 0  y  H}. f () = g + ps , u = (u1 , u2 ) are the velocity components in the x- and y-directions respectively and  = -2/3. Boundary conditions are u1 (0, y) = u1 (H, y) = 0 u1 (x, 0) = uwall (x) u1,y (x, H) = 0 u2 (x, 0) = u2 (x, H) = 0 u2,x (0, y) = u2,x (H, y) = 0 Define the space L2 () as g  L2 () if |g|2 d &lt; Also define the spaces H 1 () and H 1 () as g = (g1 , g2 )  H 1 () if gi  H 1 (), i = 1, 2 where a function g is in H 1 () if the following holds (|g|2 + |g|2 )d &lt; We assume that there exists a solution p and u such that p is in H 1 () and u is in H 1 (). Also assume that f () is in L2 (). We require enough smoothness of uwall (x) that these assumptions will hold, see  and we obtain the estimate for u 4 2 f 2+ u 2 + u 2 . (5.12) u 2 i (C )2 C i which is proved below. ^ Here u is defined by u = u + u where u satisfies ·u = 0 and the bound^ ary conditions of u on . Then u satisfies homogeneous boundary conditions on . From the estimate given by equation (5.12) the velocity is bounded in terms of 1/, the bottom wall speed, uwall , which is included in u and g which is included in f .5.2. Mathematical Analysis63However, we do not obtain any dependence on the permeability in the estimates, the direct relation between u and the consolidation speed which exists in 1D vanishes in 2D. Consider the time scale of concentration change D = - · u Dt ^ The consolidation rate is related to q =  · u =  · u. From (5.10), (5.11) ^ and using u = u + u an equation for q follows, K(2 + )q - q = K · f () (5.13)Unfortunately, q is not known on the boundary so (5.13) does not lead to estimates. To prove the estimate (5.12) choose u = (y , -x ) then ·u = 0. In order for u to satisfy the boundary conditions of u we should find a  such that y (0, y) = y (H, y) = 0 y (x, 0) = uwall (x) yy (x, 1) = 0 x (x, 0) = x (x, H) = 0 xx (0, y) = xx (H, y) = 0  can be determined by e.g. transfinite interpolation  using fifth degree polynomials. ^ With u = u + u and using the fact that  · u = 0, (5.10) and (5.11) can be written as ^  · (Kp - u) = 0 ^ ^ (u + u) + ( + ) · u - p = f ()(5.14) (5.15)^ Multiply (5.14) with p, (5.15) with u and use Green's formula to obtain ^ (-K|p|2 + p · u)d = 0 (5.16)and (- i^ ^ ^ ui · (u + ui ) - ( + )( · u · u) - u · p) = ^ ^ i ^ f · ud. (5.17)Add (5.16) and (5.17) to get (K|p|2 +  i^ ^ ^i · (u + ui ) + ( + ) · u · u)d = - u ^ i · , defined by2^ f · ud.Introduce the usual L2 norm, f=|f |2 d64 then K p2Chapter 5. Consolidation Model in 2D+i^i u2^ + ( + )  · u2 f^ u +i^i uu . iIt follows that i^i u2 f^ u +i^i uu . iBy using the following inequality g we obtain  2 h 2 g 22+1 h 2  22(5.18)^i ui f^ u +u 2 . ii(5.19)^ ^ ^ Since u  H 1 e.i. u  H 1 and u = 0 on {(x, 0), 0  x  H} we can use 0 the Poincar´ inequality, , and we have e  ^ C u 22 f^ u + 2u ii2where C is a constant depending only on H. Use the inequality (5.18) again yields 4 2 ^ u 2 f 2+ u 2 . (5.20) i (C )2 C i and (5.12) follows.5.3Numerical TreatmentFinite differences on a staggered grid are used in the discretization of equations (5.1)-(5.4). Similar to the 1D problem presented in chapter 4, the conservation law for the volume fraction, (5.1), is discretized in space by a second order central difference scheme and in time a two stage Runge-Kutta method is used. The elliptic system, given by the equations (5.2)-(??), is discretized by second order central differences. The equation for the memory function, (5.9) is solved by a central second order finite difference scheme.5.3.1Discretization on a Staggered, Non-Uniform GridLet time and space be discretized by choosing a time step, t, and number of grid points, M in the x-direction and N in the y-direction.5.3. Numerical Treatment The mesh point are defined as xi yj tn = (i - 1)xi-1 , i = 1, 2, . . . M = (j - 1)yj-1 , i = 1, 2, . . . N = nt, n = 1, 2 . . .65where xi = xi+1 - xi and yj = xj+1 - xj . The length of the domain in the x-direction is given by L and in the y-direction by H and xM = L and yN = H. The discretization in space is done on a staggered grid. The value n 1 j+ 1 i+ 2 2 represents an approximation to a cell average over the grid cell [xi , xi+1 ] × [yj , yj+1 ] at time tn . n 1 j+ 1  i+2 21 xi yjxj+1 xjyj+1(x, y, tn )dxdyyj(5.21)As in the one dimensional case, one can also view n 1 j+ 1 as a point wise i+2 2approximation to (xi+ 1 , yj+ 1 , tn ) in the point (i + 1 , j + 1 ). 2 2 2 2 The velocity components, (u, v), and the pressure, p, are defined in the grid points (xi , yj ) as, see also Figure 5.3, ui,j vi,j pi,j = u(xi , yj ) = v(xi , yj ) = p(xi , yj ) i, j = 1 . . . (M, N ). yj+1 yj+1/2 yj xi x i+1/2 u,v,px i+1Figure 5.3. Grid with  and v defined in different grid points.Standard finite difference methods require a rectangular grid with equally spaced grid points. To be able to use a non-uniform grid, e.g. more grid points near a boundary, we use a grid mapping to a domain with equidistant spacing between the grid points. The numerical code for the elliptic system is implemented in such a way that it automatically transforms the equations from a physical (x, y) domain to a generalized (r, s)-domain where the equations are discretized,66Chapter 5. Consolidation Model in 2Dsee Figure 5.4 and for more details, see . The conservation equation (5.1) is discretized directly in physical space on the non-uniform grid. One reason for this is that it may not be easy to discretize the transformed equation in (r, s) in a way that conserves the correct physical quantities, .yss  yj11 00 (i,j)11 00 11 00 (i,j)x i x r rFigure 5.4. Non-uniform grid in physical space transformed to a computational space with a uniform grid.In what follows we will use the following notation for the finite difference approximations on a non-uniform mesh +x ui,j D+x ui,j = ui+1,j - ui,j -x ui,j = ui,j - ui-1.j +x ui,j -x ui,j (+x + -x )ui,j = D-x ui,j = D0x ui,j = . xi xi-1 xi + xi-1and similar for ±y and D±y defined as above but in the y-direction.5.3.2High resolution schemeTo solve equation (??) we use a second order semi discrete method with RungeKutta time stepping. This approach was introduced in section 4.3.3 for the 1D problem. The 2D extension is discussed here.5.3. Numerical Treatment67G i+1/2,j+1 j+1F i,j+1/2 j G i+1/2,j i i+1F i+1,j+1/2Figure 5.5. Definition of a cell and the numerical fluxes at the cell boundaries.Start by writing equation (5.1) on a conservation law form t + f (, u)x + g(, v)y = 0 (5.22)where f (, u) = u and g(, v) = v. Write a conservative numerical approximation on semi-discrete form as d  1 1 = -D+x Fi,j+ 1 - D+y Gi+ 1 ,j . 2 2 dt i+ 2 ,j+ 2 (5.23)Let hx denote the numerical flux function in the x-direction and hy in the y-direction. Then Fi,j+ 1 is given by 2 Fi,j+ 1 = hx (i+ 1 ,j+ 1 , i- 1 ,j+ 1 , ui,j+ 1 ) 2 2 2 2 2 2 1 (f (i+ 1 ,j+ 1 , ui,j+ 1 ) + f (i- 1 ,j+ 1 , ui,j+ 1 )) 2 2 2 2 2 2 2 Q - x (i+ 1 ,j+ 1 - i- 1 ,j+ 1 ) 2 2 2 2 2i = and Gi+ 1 ,j by 2 Gi+ 1 ,j = hy (i+ 1 ,j+ 1 , i+ 1 ,j- 1 , vi+ 1 ,j ) 2 2 2 2 2 2 1 (g(i+ 1 ,j+ 1 , vi+ 1 ,j ) + g(i+ 1 ,j- 1 , vi+ 1 ,j )) 2 2 2 2 2 2 2 Q - y (i+ 1 ,j+ 1 - i+ 1 ,j- 1 ). 2 2 2 2 2j = (5.25) (5.24)68Chapter 5. Consolidation Model in 2DHere, x = t/xi , y = t/yj and Q is the numerical viscosity coefficient. In i j the numerical approximation of the flux functions, the velocities are required at the cell boundaries. They are computed as mean values in the following fashion ui,j+ 1 = 2 vi+ 1 ,j 2 ui,j+1 + ui,j 2 vi+1,j + vi,j = 2The approximation (5.23) together with (??) and (5.25) is only first order accurate in space. To increase the accuracy, introduce a piecewise linear approximation of  in the cell given by [xi , xi+1 ] × [yj , yj+1 ] (t, x, y)[xi ,xi+1 ][yj ,yj+1 ] = i+ 1 ,j+ 1 2 2 + sx 1 ,j+ 1 i+2 2(x - xi+ 1 ) 2 xi (y - yj+ 1 ) 2 yj (5.26)+ sy 1 ,j+ 1 i+2 2where xi &lt; x &lt; xi+1 and yj &lt; y &lt; yj+1 . The numerical flux function are then evaluated using the linear reconstruction of  as (5.27) Fi,j+ 1 = hx (R 1 , L 1 , ui,j+ 1 ) i,j+ i,j+ 2 22 2and Gi+ 1 ,j = hy (R 1 ,j , L 1 ,j , vi+ 1 ,j ) i+ i+ 2 22 2(5.28)where R and L are approximations from the right and from the left of the cell boundaries. So e.g. R 1 is obtained by evaluating equation (5.26) at the cell i,j+ 2 boundary given by (xi , yj+ 1 ) 2 1 R 1 = (t, xi , yj+ 1 )[xi ,xi+1 ][yj ,yj+1 ] = i+ 1 ,j+ 1 - sx 1 j+ 1 i,j+ 2 2 2 2 2 i+ 2 2 and L 1 is obtained by evaluating the piecewise linear reconstruction in of  i,j+ 2 in the cell [xi-1 , xi ] × [yj , yj+1 ] at the cell boundary (xi-1 , yj+ 1 ) 2 1 L 1 = (t, xi-1 , yj+ 1 )[xi-1 ,xi ][yj ,yj+1 ] = i+ 1 ,j+ 1 + sx 1 j+ 1 i,j+ 2 2 2 2 2 i- 2 2 If the flux functions (5.24) and (5.25) are evaluated at R and L we obtain Fi,j+ 1 = 2 1 1 (f (i+ 1 ,j+ 1 - sx 1 ,j+ 1 , ui,j+ 1 ) 2 2 2 2 2 i+ 2 2 1 x + f (i- 1 ,j+ 1 + si- 1 ,j+ 1 , ui,j+ 1 )) 2 2 2 2 2 2 Q 1 x 1 - x (i+ 1 ,j+ 1 - si+ 1 ,j+ 1 - i- 1 ,j+ 1 - sx 1 ,j+ 1 ) 2 2 2 2 2 2 2i 2 2 i- 2 2(5.29)5.3. Numerical Treatment and Gi+ 1 ,j = 2 1 1 (f (i+ 1 ,j+ 1 - sy 1 ,j+ 1 , vi+ 1 ,j ) 2 2 2 2 2 i+ 2 2 1 y + f (i+ 1 ,j- 1 + si+ 1 ,j- 1 , vi+ 1 ,j )) 2 2 2 2 2 2 Q 1 y 1 - y (i+ 1 ,j+ 1 - si+ 1 ,j+ 1 - i+ 1 ,j- 1 - sy 1 ,j- 1 ) 2 2 2 2 2 2 2j 2 2 i+ 2 269(5.30)The slope limiters sx and sy are determined as in one dimension by sx 1 ,j+ 1 = s(+x i+ 1 ,j+ 1 , -x i- 1 ,j+ 1 ) i+ 2 2 2 22 2(5.31)and sy 1 ,j+ 1 = s(+y i+ 1 ,j+ 1 , -y i+ 1 ,j- 1 ) i+ 2 2 2 22 2(5.32)where s(x, y) is given by the van Albada limiter, (4.39). To summarize, a second order accurate numerical approximation in space of equation (5.1) is given by d  1 1 = -D+x Fi,j+ 1 - D+y Gi+ 1 ,j . 2 2 dt i+ 2 ,j+ 2 (5.33)where F and G are given by equations (5.29) and (5.30). This is a system of ordinary differential equations and it is solved by a two stage Runge-Kutta method.n i+ 1 ,j+ 1 = n 1 ,j+ 1 - x +x Fi,j+ 1 - y +y Gn 1 ,j i j i+ i+2 2 2 2 2 2(1)(2) i+ 1 ,j+ 1 2 22 2=(1) i+ 1 ,j+ 1 2 22-2(1) x +x Fi,j+ 1 i2-(1) y +y Gi+ 1 ,j j2(5.34)n+1,j+ 1 = (n 1 ,j+ 1 + i+ 1 ,j+ 1 )/2. i+ i+ 12 2(2)Here F n and GN denote the numerical fluxes evaluated with  and (u, v) at time level tn . The velocity field is obtained by solving the elliptic system of equations. Since this is rather time consuming, the velocity field is not updated between the first and second step in the Runge-Kutta scheme, i.e (u, v)n is used in both steps. This does not affect the accuracy of the solution which can be seen in Figure where a comparion is made. In Figure 5.3.2a) we present the volume fraction (U ) obtained with a velocity field that was updated between the first and second Runge-Kutta step (u ). In Figure 5.3.2b) we see the volume fraction () obtained when using the same velocity field for both steps. The difference between the two solutions can be measured as  1/2 u - h,2= xy MN|u - ij |2  iji=1 j=170Chapter 5. Consolidation Model in 2D0.09 0.08 0.07 0.060.09 0.08 0.07 0.06y [m]0.05 0.04 0.03 0.02 0.01y [m]0.01 0.02 0.03 0.04 0.05 x [m] 0.06 0.07 0.08 0.090.05 0.04 0.03 0.02 0.010.010.020.030.040.05 x [m]0.060.070.080.09(a) u (x, y, t = 100)(b) (x, y, t = 100)Figure 5.6. Concentration contours of  at t = 100s. In the left figure, the velocity field was updated between step 1 and step two in the Runge-Kutta algorithm, (5.34). In the right figure the same velocity field was used.and we find that the difference is small, u -  h,2 = 9.4501 × 10-6 . Boundary conditions are imposed on the fluxes. The normal velocities are zero at the boundaries hence,n n F1,j+ 1 = FM,j+ 1 = 02 2Gn 1 ,1 = Gn 1 ,N = 0. i+ i+2 2Discretization of Equation for  The equation for the memory and structure related function  is the linear advection equation,       +u +v = 0. (5.35) t x y Since  is partially determined by the behaviour of the volume fraction , see equation (5.8), it will contain sharp fronts. A numerical approximation of (5.35) should be such that these fronts are propagated with correct speed and that they stay sharp. A second order accurate and conservative numerical approximation is the Lax-Wendroff scheme. Written on flux formn ,n+1 1 = ,n1 ,j+ 1 - tD+x Fi,j+ 1 - tD+y Gn 1 ,j i+ i+ 1 ,j+ i+2 2 2 2 2 2(5.36)with the flux functions given byn Fi,j+ 1 =21 ,n ,n 1 ( u ) 1 1 +  i- 1 ,j+ 1 2 2 2 i,j+ 2 i+ 2 ,j+ 2 1 t 2 ,n ,n - u ) 1 ( 1 1 -  i- 1 ,j+ 1 2 2 2 xi i,j+ 2 i+ 2 ,j+ 2(5.37)5.3. Numerical Treatment and Gn 1 ,j = i+2711 v 1 (,n1 1 + ,n1 ,j- 1 ) i+ 2 2 2 i+ 2 ,j i+ 2 ,j+ 2 1 t 2 - v 1 (,n1 1 - ,n1 ,j- 1 ). i+ 2 2 2 yj i+ 2 ,j i+ 2 ,j+ 2(5.38)In order to prevent the solution from develop spurious oscillations, a small amount of additional numerical diffusion (or artificial viscosity) is added. - D+x ,n1 ,j+ 1 i-2 2is added ton Fi,j+ 1 2and - D+y ,n1 ,j- 1 i+2 2is added to Gn 1 ,j . i+ 2 At the boudaries we let the flux functions be zero,n n F1,j+ 1 = FM,j+ 1 = 02 2Gn 1 ,1 = Gn 1 ,N = 0. i+ i+2 25.3.3Discretization of the Elliptic SystemWe discretize equations (5.1)-(5.4) in space on a staggered non-uniform grid, see section 5.3.1 and Figure 5.3. The transport coefficients and source terms in (5.2), (5.3) and (5.4) are functions of the volume fraction , which is not necessarily a smooth function. To obtain numerical solutions of good convergence order some degree of smoothness of these coefficients are required. This was discussed in chapter 4 and the same Using the notation introduced in earlier the equations (5.2), (5.3) and (5.4) can be written in their discrete form as D+x (Ki- 1 ,j D-x pi,j ) + D+y (Ki,j- 1 D-y pi,j ) - D0x ui,j - D0y vi,j = 0 2 2 And for the two components of the momentum equation we obtain 4 2 D+x (i- 1 ,j D-x ui,j ) + D+y (i,j- 1 D-y ui,j ) - D0x (i,j D0y vi,j ) 2 2 3 3 +D0y (i,j D0x vi,j ) - D0x pi,j = D+x psi- 1 ,j 2 and 4 2 D+x (i- 1 ,j D-x vi,j ) + D+y (i,j- 1 D-y vi,j ) - D0y (i,j D0x ui,j ) 2 2 3 3 ¯ + D0x (i,j D0y ui,j ) - D0y pi,j = D+y psi,j- 1 + gi,j 2 (5.41) (5.40) (5.39)In the numerical approximation we need the transport coefficient evaluated at different locations in the grid, e.g. i,j = (i,j ) where  should be evaluated72Chapter 5. Consolidation Model in 2Dat (xi , yj ). The volume fraction is defined in (x1+ 1 , yj+ 1 ). This means that we 2 2 have to compute  in the grid points where it is not defined. This is done by using averages as follows 1 ¯ i- 1 ,j = (i- 1 ,j+ 1 + i- 1 ,j- 1 ) 2 2 2 2 2 2 1 ¯ i,j- 1 = (i+ 1 ,j- 1 + i- 1 ,j- 1 ) 2 2 2 2 2 2 1 ¯ i,j = (i+ 1 ,j+ 1 + i- 1 ,j+ 1 + i+ 1 ,j 1 + i- 1 ,j- 1 ). 2 2 2 2 2 2 2 2 4 Numerical approximation of a model containing convective terms as well as diffusive terms are prone to spurious oscillations. Mesh size restrictions, related to the cell Peclet number limit for a central difference approximation of a convection-diffusion equation, are necessary to avoid the oscillations, see . To explore the mesh size requirement, consider the one dimensional elliptic system with constant coefficients  2 p v =0 - y 2 y 4 2v ps p  = + g - 3 y 2 y y (5.42) (5.43)approximated numerically by second order central differences. To see how these oscillatory solutions can appear, look at the homogeneous system of difference equations for the approximate solution 4 1 (pj+1 - pj-1 ) = 0 (v - 2vj + vj-1 ) - 2 j+1 3y 2y 1 K (vj+1 - vj-1 ) + - (pj+1 - 2pj + pj-1 ) = 0 2y y 2 The trial solutions pj = P j , vj = V j , give the eigenvalue problem for 4 2 3y 2 ( - 2 + 1) 1 - 2y (2 - 1) 1 - 2y (2 - 1) K 2 y 2 ( - 2 + 1)V P=0 . 0If  is real and positive, non-oscillatory solutions are guaranteed. This yields the following restriction on the step size y 4 y   3 K (5.44) The term K, the square root of the &quot;apparent permeability&quot;, is the length scale related to the viscous boundary layer velocity profile. For the material5.3. Numerical Treatment73studied the boundary layer if of typical size 0.1 - 1mm and the size of the computational domain is of order 0.1 m. To fulfill the condition on the step size given by (5.44) very small cells are required and this is too costly. A commonly used remedy for this problem is to use an upwind difference for the convective terms in (5.42) and (5.43). In that case the solution is never oscillatory but on the other hand it thickens the boundary layer significantly for large y due to numerical viscosity. Another recipe to avoid oscillations is to artificially increase the amount of viscosity. This can be done locally, that is in a cell with step size y the total viscosity should be large enough to fulfill the restriction given by (5.44),  = max( ^ 3y 2 , ). 16K (5.45)We deal with this by using a non-uniform grid. With more grid points close to the bottom where the &quot;apparent permeability&quot; is small, the length scales can be resolved and the condition (5.44) can be fulfilled. However, if this is not sufficient we increase the amount of viscosity locally, wherever it is needed. For a given x and y i,j = max c ^2 3yj 3x2 i , c , i,j ) 16Ki,j 16Ki,j(5.46)where c is a constants to be chosen. So, instead of (5.40) and (5.41) we solve 4 2 D+x (^i- 1 ,j D-x ui,j ) + D+y (^i,j- 1 D-y ui,j ) - D0x (i,j D0y vi,j )  2  2 3 3 +D0y (i,j D0x vi,j ) - D0x pi,j = D+x psi- 1 ,j 2 and 4 2  2  D+x (^i- 1 ,j D-x vi,j ) + D+y (^i,j- 1 D-y vi,j ) - D0y (i,j D0x ui,j ) 2 3 3 ¯ + D0x (i,j D0y ui,j ) - D0y pi,j = D+y psi,j- 1 + gi,j 2(5.47)(5.48)with  given by equation (5.46). ^ This approach can also be applied to the permeability function instead of the viscosity. Then, equation (5.39) is solved with a modified permeability function. Implementation of Boundary Conditions The slip conditions on the velocity and the zero normal derivative condition for the pressure are implemented with second order skew differences. For example,74Chapter 5. Consolidation Model in 2Dthe pressure boundary condition on the top, (x, y = H), and on the bottom, (x, y = 0), of the box is implemented as follow: -3pi,1 + 4pi,2 - p(i, 3) p (x, 0) = y x1 + x2 p 3pi,N y - 4pi,N y-1 + p(i, N y - 2) (x, H) = y xN y-1 + xN y-2x 10-3(5.49) (5.50)1.210.80.60.40.20 00.010.020.030.040.050.060.070.080.090.1Figure 5.7. Bottom velocit. No slip boundary condition, where uw all goes to zero towards the corners.Pressure Non-Unicity Problem The equations (5.2), (5.3), (5.4) and the boundary conditions depend on p only through its derivatives. If p(y) is a solution, then for any constant C the function pC = p(y) + C is also a solution. This means that the algebraic system to be solved for an consistent numerical approximation is singular. To overcome this problem the algebraic system is modified such that is becomes non-singular in a way that is described below. The algebraic system obtained after the discretization described in section 5.3.3 is a linear system of equation for the 3M N unknowns gathered in the vectors w = (u, v), p  RMN . L11 L21LL12 L22w p=f 0(5.51)represents the discrete second order derivaThe submatrix L11  R tive operators applied to u and v. The submatrix L12  R2MN ×MN represents2MN ×2MN5.3. Numerical Treatment75the gradient and L21  RMN ×2MN the divergence operator. L22  RMN ×MN is the discrete Laplace operator. Finally, f  R2MN is given by the right hand side of (??). The singularity of L implies that L11 L21 L12 L22 0 p0 =0 (5.52)where p0 is the vector spanning the right null space of L22 . To obtain a unique pressure solution an additional condition or constraint on the system is required. There is also a single vector n such that nT L22 = 0. If L22 is symmetric, n = p0 . If we multiply the last equation in (5.51) from the left with nT we get nT L21 w = 0 (5.53)where we have used nT L22 = 0. Equation (5.53) represents the condition that has to be satisfied in order to have any solutions to (5.51). Depending on the implementation of boundary conditions and due to round off errors, equation (5.53) is in general not satisfied exactly and there are no solutions to (5.51). In that case, instead of solving L21 w + L22 p = 0 the following constrained minimization problem is solved min L22 p + L21 v 2 . mT p=0 Where the constraint mT p = 0 is added in order to get a unique solution. m could be arbitrary as long as mT p0 is non-zero, and we will take m = p0 in the theorem below. That a unique solution exists to the minimization problem is shown by the following theorem: Theorem 5.1. Assume that dim(null(L22)) = 1 and nT L22 = 0. Then the solution of min L22 p - b 2 s.t. pT p = 0 is unique and given by 0 L22 pT 0 n 0 p  = b 0 (5.54)So, to handle the pressure non-unicity problem, the system (5.51), is replaced by the augmented system      L11 L12 0 w f L21 L22 n  p  =  0  (5.55) 0 nT 0  0^ L76Chapter 5. Consolidation Model in 2D^ where L  R3MN +1×3MN +1 . The null vector is computed with the shifted inverse power method according to the following algorithm nk+1 = (LT - I)-1 nk , 22 here  is a small positive number. Solving the Algebraic System The algebraic system produced by the discretization described above is very sparse and a direct method for solving sparse systems is used to solve the linear system. This is currently the limiting factor for the spatial resolution. nk+1 = nk+1 nk+1 (5.56)25.3.4AlgorithmIn this section we present a summary of the numerical algorithm used to solve the full problem. The hyperbolic equation and the elliptic system are coupled through the velocity field (u, v) and the volume fraction . Initially the volume fraction  and the memory function  is known, either from initial conditions or the previous time step. If we are at time step tn the algorithm can be described as follows: i) Evaluate the material functions: (n ), ps (n , n ) and K(n , n ). ii) Comstruct the discrete elliptic operator, L, corresponding to equations ()(), with coefficients given by (n ) and K(n , n ) and the righ hand side vector with F = F (ps (n , n )) L11 L21LL12 L22w pn=F 0(5.57)iii) Compute the left null space vector of L22 . For a general case, the transpose of the null vector nT is determined by the relation nT L22 = 0. Finding the left eigenvector corresponding to the zero eigenvalue of L22 can efficiently be done by the shifted inverse power method, see algorithm 5.56. Only a few iterations are needed in order for the algorithm to converge. iv) Solve an extended linear system of equations of size (3M N +1)×(3M N +1). This is given by the discretization of the elliptic system and the minimization process discussed in section 5.3.3,  n   w F (ps (n , n ))  0 L((n ), K(n , n ))  p  =  0 -5.4. Convergence Study - Elliptic System77L is a sparse matrix and a direct method for sparse systems is used to solve Lw = F . v) Compute n+1 by using the modified Lax-Friedrichs scheme applied to t + (u(x, y; n ))x + (v(x, y; n ))y = 0 and the 2-stage Runge-Kutta scheme to advance in time. The velocity field is not updated between the two stages. vi) Solve a linear advection equation for n+1  + u(x, y; n ) + v(x, y; n ) = 0 t x y by using the Lax-Wendroff scheme. vii) This complete one time step. Repeat i)-v) to advance in time.5.4Convergence Study - Elliptic SystemIn this section we present results from runs on four consecutively refined meshes. This is done to test the spatial convergence of the numerical treatment of the elliptic system. For the simulations we have used a regularly subdivided mesh on [0, 0.1] × [0, 0.1] with mesh width h. We start with a mesh where h = 0.01 and then we reduce the mesh size to h/2, h/4 and h/8. On each mesh, the elliptic system of equations given by (5.2)-(5.4) with boundary conditions as in section 5.1 is solved by the numerical approximation described in section 5.3.3. The boundary condition for u on the bottom is a non-slip condition. To avoid inconsistencies in the velocity boundary conditions in the lower corners of the box, the velocity goes linearly to zero at the vertical walls. The viscosity, the permeability and the yield pressure are functions of the volume fraction, . The volume fraction is here given as a smooth function of y, 1 (x, y, 0) = 0 (x, y) = (g - min ) (1 - tanh((y - y0 ))) + min . 2 where g = 0.1, min = 0.001 and the regularizing parameter,  = 100. The viscosity function scaled by a factor of 500 in order to satisfy the step size restriction (5.44) on the coarsest mesh. The difference in L2-norm of the velocity components and the fluid pressure computed on the different meshes and the order of convergence is given in Tables 5.1, 5.2 and 5.3. We define the discrete L2-norm, · h,2 , of the grid function u  RN ×N as  1/2 uh,2= h2 NN|uij |2 i=1 j=178Chapter 5. Consolidation Model in 2Dand the convergence rate, q is in this case defined as uh - uh/2 uh/2 - uh/4q = log/ log 2where the subindex on u denotes the spatial resolution. The observed order for the velocity components, see Tables 5.1 and 5.2, is slightly lower than two because of the error introduced by the pressure correction discussed in section 5.3.3.huh - uh/2h,2uh -uh/2 h,2 uh/2 -uh/4 h,2Observed order, q0.01 0.005 0.00251.2093 × 10-5 3.5049 × 10-6 9.7552 × 10-7 0.2898 0.2783 1.79 1.85Table 5.1. Order of accuracy for the velocity component u.hv h - v h/2h,2v h -v h/2 h,2 v h/2 -v h/4 h,2Conv rate, q0.01 0.005 0.00251.2207 × 10-5 3.3884 × 10-6 9.3263 × 10-7 0.2776 0.2752 1.85 1.86Table 5.2. Order of accuracy for the velocity component v.The order of accuracy for the fluid pressure is .....5.5. Numerical Experiments79hph - ph/2h,2ph -ph/2 h,2 ph/2 -ph/4 h,2Conv rate, q0.01 0.005 0.0025358.06 0.3351 120.00 0.2948 35.37 1.76 1.56Table 5.3. Order of accuracy for the fluid pressure p.5.5Numerical ExperimentsNumerical simulations are performed to verify, and validate the numerical and the mathematical model. The following sections present the results of parameter and model assumptions variations. These simulations are intended to exemplify the influence on the solution from using different material functions and parameters. Different viscosity models are tried out and the effect of irreversibility in the permeability model is studied. One important parameter for the consolidation process is the bottom wall speed and simulations are carried out with different wall speeds and wall boundary conditions. We know from physical experiences that a major increase in the consolidation speed is one result of applied shear. Unfortunately the amount of physical experiments are very limited for these type of processes and the results presented here are validated qualitatively rather than quantitatively. For the simulations in this section, the suspension is confined to a H × L box, gravitational acceleration is G and the bottom wall moves with velocity uwall . The material functions, discussed in section 3.2, are used in various forms. For simplicity, they are repeated here. The viscosity function is modified according to (, ||) = 9.15 × 107 3.4 1 +  ||  0.05-0.9+ CminPas(5.58)where Cmin is chosen such that the viscosity is sufficiently large in the clear fluid phase (  0) and satisfies the restriction given by equation (5.44). In a reference case we want the viscosity to be a function of volume fraction only. Then (5.59) () = 1.3 × 106 (3.4 + Cmin ) Pas This corresponds to a constant shear rate of approximately 6s-1 in the shear thinning model given by equation (5.58).80Chapter 5. Consolidation Model in 2DThe particle phase pressure is given by equations (3.10) and (3.11). To avoid a discontinuity in ps the function is regularized as ps (,  ) = Rw (,  )ps,yield () where Rw (,  ) = (5.60)1 tanh(w( -  )) + 1) (5.61) 2 where w determines the transition width. Two different permeability models are tested. Firstly the model fitted to experimental data according to K() = 4.6 × 10-18 -7.41 C(a0 + a1  + a2 2 ) if  &gt; 0.05 if   0.05. (5.62)The coefficients in the polynomial are determined by equation (4.22). Secondly, simulations with the permeability hysteresis function is carried out. This model is derived in section 3.2.3, see equation (3.14). Default values for physical and numerical parameters are given below and details of the numerical simulations are given in Table 5.4. The case with all defaults is the reference case. Results from the numerical experiments are presented by 2D plots showing contours of the volume fraction and mass flow arrows, f s = u = (u, v), for different times. A quick view of the result is provided by the development of mean concentration in the &quot;cake&quot;, the quantity of direct engineering interest in dewatering. We quantify this by defining Ac to be the area for which  is larger than a threshold value, here taken to be c = 0.05. Then mean = 1 Ac dAc .Ac(5.63)The integrals are computed by first order formulas and exhibit jumps (see e.g. Figure 5.15.) when the threshold iso-line crosses the grid points. This is especially noticeable for uwall = 0 when iso-lines are horizontal. In all the simulations the initial data are: (x, y, 0) = 0 (y) = 0.001 if y  0.065 0.1 otherwise.5.5. Numerical Experiments RUN A. Reference Solution (Defaults) · H = L = 0.1 m81· G = 10000 m/s2·  = 1000 kg/m3· Cmin = 2 × 10-4· M = N = 41 (x = 0.0025, y varies, smallest at bottom- to avoid the use of artificial viscosity)· t = 0.05 s for stability reasons· Q = 0.05, for stability reasons· c = 0, no artificial viscosity· w = 100, see equation (5.61)· Particle phase pressure, ps = ps (,  ), see equation (5.60)· No shear thinning,  = (), see equation (5.59)· No permeability hysteresis, K = K(), see equation (5.62)· No-slip wall boundary condition.82Chapter 5. Consolidation Model in 2DRUN B. Influence of Viscosity Model RUN C. Influence of Wall Speed RUN D. Influence of Wall Boundary Conditions RUN E. Influence of Permeability Hysteresis RUN F. Influence of Shear Thinning and Permeability HysteresisRUNuwallVisc. model  ()Perm. model K 100 × K()Yield press.model model , ps ps (,  )Result pres. inA * B B1 B2 C C1 C2 C3 D0.001Fig. 5.85.9 Fig. 5.105.14 = (, ||)   = 100 × ()  = (, ||)  0 0.005 0.01 slip/ no slip 0.01 100 × K(,  ) slip/ no slip/  = (, ||)  100 × K(,  )  = (, ||) Fig. 5.155.16Fig. 5.175.19 Fig. 5.20E E1 E2 F *Fig. 5.215.23Table 5.4. Summary of the model setting in the numerical experiments presented in sections A-F. *) A grid refinement of this case is presented.A. Reference Solution In this section, numerical results from running the reference case is presented. Figure 5.8 shows a time study of the process. The wall velocity is small and in the very beginning it behaves essentially like a 1D process, see Figure 5.8 (a).5.5. Numerical Experiments830.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.183 0.01 0.22 0 0 0.02 0.04 0.06 0.256 0.08 0.10.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0 0.0446 0.2230.0367 0.1830.267 0.312 0.02 0.04 0.06 0.08 0.1(a) Volume fraction contours after t = 10 s(b) Volume fraction contours after t = 50 s0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.0505 0.02 0.01 0 0 0.252 0.303 0.3530.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.24 0.01 0 0 0.299 0.359 0.02 0.04 0.06 0.08 0.1 0.05990.020.040.060.080.1(c) Volume fraction contours after t = 250 s(d) Volume fraction contours after t = 500 sFigure 5.8. RUN A. Time evolution of the volume fraction (contours are marked with the actual value). The arrows represent the mass flow. The bottom wall velocity is uwall = 0.001 m/s, K = 100K(), ps = ps (,  ) and  = ().However, very soon the 2D effects of the wall movement can clearly be seen. The movement of the wall sets up a large rotating velocity field and a cake with high concentration is produced in the lower right corner, see Figure 5.8 (b).84Chapter 5. Consolidation Model in 2DWhen the volume fraction increases, the viscosity increases and the viscous forces become large enough to &quot;drag&quot; the cake up along the right wall. Then it follows the velocity field back into the middle of the box, making the region with high concentration grow, see Figures 5.8 (c) and (d). A grid refinement for this case is performed. The solution is computed on a 61 × 61 mesh and shows qualitatively good agreement with the default case, (41 × 41), see Figure 5.9. The position of the interface and the concentrations contours are captured rather well by the 41 × 41 case. However as expected, the interface is sharper and due to smaller cells the high concentration cake in the corner is smaller in the in the 61 × 61 case.0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 00.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 00.020.040.060.080.10.020.040.060.080.1(a) Volume fraction contours, 41 × 41(b) Volume fraction contours, 61 × 61Figure 5.9. RUN A. Grid refinement. Contour plots of volume fraction at t = 250 s.B. Influence of Viscosity Model In this section numerical experiments with the shear thinning model, (5.58) is carried out. In Figure 5.10 we show a sequence of solutions at four different times. As in the reference case the suspension consolidates fast in the beginning. If we compare Figure 5.10 to Figure 5.8 the difference is obvious. A large cake is formed in the lower right corner. The volume fraction in the cake is large and it follows the motion almost like a solid body. Close to the bottom in Figure 5.8 (c) and (d) the suspension starts to slip and a thin fluid layer is observed.5.5. Numerical Experiments850.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0 0.195 0.274 0.02 0.04 0.06 0.08 0.1 0.0392 0.1560.1 0.09 0.08 0.07 0.06 0.05 0.0733 0.04 0.22 0.03 0.02 0.01 0.293 0 0 0.02 0.04 0.06 0.08 0.513 0.1(a) t = 10s(b) t = 50s0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.524 0 0 0.02 0.04 0.06 0.08 0.1 0.375 0.0749 0.30.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.449 0.02 0.524 0.01 0 0 0.3 0.07490.020.040.060.080.1(c) t = 250s(d) t = 500sFigure 5.10. RUN B1. Time evolution of the consolidation process with uwall = 0.001 m/s. The figure shows contour plots of the volume fraction and the mass flow (arrows). Here, K = 100K(), ps = ps (,  ) and  = (, ||). If we compare the obtained viscosity in the shear thinning case, see Figure 5.12 (b) to the viscosity in the reference case, see Figure 5.11 (b), it is much larger in the shear thinning case. This explains the more solid like behavior of the compressed part of the suspension. We can also note that in contrast to the one dimensional gravity consolidation86Chapter 5. Consolidation Model in 2D0.1 0.09 0.080.045 0.04 0.0350.070.030.060.0250.050.020.04 0.03 0.0505 0.02 0.01 0 0 0.252 0.303 0.3530.005 0.015 0.016.67e+03 3.88e+04 2.59e+045.16e+040.020.040.060.080.10 00.020.040.060.080.1(a) Volume fraction contours,  = ()(b) Viscosity,  = ()Figure 5.11. RUN A. Contour plot of the volume fraction, (a), and the corresponding viscosity, (b), at t = 250 s. In this process uwall = 0.001 m/s, K = 100K(), ps = ps (,  ) and  = ().0.1 0.09 0.080.045 0.04 0.035 1.02e+060.070.030.060.0250.05 0.04 0.03 0.02 0.01 0.3750.07490.020.30.015 2.05e+06 0.01 0.0057.16e+06 3.07e+068.18e+060.524 0 0 0.02 0.04 0.06 0.08 0.10 0.04 0.05 0.06 0.07 0.08 0.09 0.1(a) Volume fraction contours,  = (, ) (b) Viscosity,  = (, ) Figure 5.12. RUN B1.Volume fraction contours and mass flow, (a), and the corresponding viscosity, (b), at t = 250 s. In this process uwall = 0.001 m/s,  K = 100K(), ps = ps (,  ) and  = (, ||), a shear thinning viscosity model.5.5. Numerical Experiments870.1 0.09 0.080.045 0.04 0.035 2.52e+060.070.030.06 0.050.07520.0254.99e+060.301 0.04 0.376 0.03 0.02 0.01 0 00.02 0.015 0.01 0.005 0 0.04 7.46e+060.526 0.02 0.04 0.06 0.08 0.11.73e+07 0.05 0.06 0.07 0.08 0.09 0.1(a) Volume fraction contours,  = 100 × ()(b) Viscosity,  = 100 × ()Figure 5.13. RUN B2. Volume fraction contours and mass flow, (a), and the corresponding viscosity, (b), at t = 250 s. In this process uwall = 0.001 m/s, K = 100K(), ps = ps (,  ) and  = (). The viscosity is scaled with scale = 100.where there is no shearing motion, the viscosity influences the solution every where in this case. To understand whether the observed difference between the shear thinning case and the reference case depends only on the &quot;over all size&quot; of the viscosity or if shear thinning has an effect, we do a computation where the reference case viscosity is scaled by a factor of 100 to be approximately of the same order as the shear thinning case. The result after t = 250 s is displayed in Figure 5.13. The solution shows resemblance to the shear thinning case but there is no thin fluid layer at the bottom i this case. This can be seen more clearly in the viscosity plots, Figure 5.12 (b) and 5.13 (b). The viscosity is similar except in the corner of the box where it is lower for the shear thinning case. This is due to the high shear rate at the bottom. The suspension can locally become more compressed contributing to a higher mean concentration, see Figure 5.14, and it continues to grow with a larger rate than for the non shear thinning viscosity models. In Figure 5.14 the mean concentration is computed for the different cases. It is obvious that the viscosity has an influence on the solution. The mean concentration decreases as the viscosity increases. But in the shear thinning case (dotted line) a higher mean concentration is obtained.88Chapter 5. Consolidation Model in 2D0.30.25mean [-]0.20.150.10.05050100 t [s]150200250Figure 5.14. RUN B. The mean concentration as a function of time using different viscosity models.5.5. Numerical Experiments C. Influence of Wall Speed89To study the influence of different wall speeds on the process i computations with uwall = 0 (1D), 0.001 0.005 and 0.01 m/s are performed. The comparison is made using the mean concentration as a function of time, see Figure 5.15.0.40.350.30.25mean[-]0.20.150.10.05uwall=0 uwall=0.001 u =0.005 wall u =0.01wall0 050100 t [s]150200250Figure 5.15. RUN C1,B1,C2,C3. The mean concentration as a function of time for four different wall velocities. In these computations K = 100K(), ps = ps (,  ) and  = (, ||). In the beginning of the process there is not much difference between the cases. They all behave like the 1D case. After some time, the mean concentration for the largest wall speed, uwall = 0.01 m/s, becomes steady at the lowest mean concentration. As the wall speed decreases the mean concentration increases. The highest mean concentration is obtained when uwall = 0.005 m/s. However the difference is small.90Chapter 5. Consolidation Model in 2D0.10.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.0749 0.30.090.080.070.060.050.040.030.0560.020.28 0.3360.02 0.010.3750.010.5240 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 00.020.040.060.080.1(a) uwall = 0(b) uwall = 0.0010.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.489 0.01 0.489 0 0 0.02 0.04 0.06 0.08 0.1 0.28 0.06990.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.415 0 0 0.02 0.04 0.06 0.08 0.1 0.296 0.3550.0592(c) uwall = 0.005(d) uwall = 0.01Figure 5.16. RUN C1,B1,C2,C3. Contour plots of the volume fraction and mass flow (arrows) at t = 250 s for different bottom wall speeds. In these computations K = 100K(), ps = ps (,  ) and  = (, ||). 5.5. Numerical Experiments D. Influence of Wall Boundary Condition91We saw in previous section that the wall speed influenced the process. Now we want to study how th actual boundary conditions at the bottom wall affects the process.A computation is made with the same parameters as in the previous case except for the boundary condition. Here we use the slip/no-slip condition described in section 5.1, equation (5.7). Parameters in this model is chosen such that a reasonable wall speed is obtained. The friction coefficient was, cµ = 2, the steepness parameter  = 0.01 and uwall = 0.01 m/s.In Figures 5.17 and 5.18 we present the solution at four different times. We also show a plot of the obtained wall velocity. We see a parabolic profile that flattens out and decreases for larger times. At t = 250, see Figure 5.18 (d), a peak has appeared in the velocity profile in the region where the volume fraction is small. In this region the viscosity is also small, and the peak probably appear due to bad resolution of the boundary layer.92Chapter 5. Consolidation Model in 2D0.1 0.09 0.08 0.07 0.062 3 x 10-42.5u(x,0) [m/s]0.05 0.04 0.03 0.02 0.0333 0.1671.510.50.01 0 00.2 0.233 0.02 0.04 0.06 0.08 0.10 0 0.01 0.02 0.03 0.04 0.05 x [m] 0.06 0.07 0.08 0.09 0.1(a) t = 10 s(b) t = 10 s, solid phase velocity along the bottom wall0.1 0.09 0.08 0.07 0.062 3 x 10-42.5u(x,0) [m/s]0.05 0.04 0.03 0.02 0.01 0.274 0 0 0.329 0.02 0.04 0.06 0.08 0.1 0.0548 0.2191.510.50 00.010.020.030.040.05 x [m]0.060.070.080.090.1(c) t = 50 s(d) t = 50 s, solid phase velocity along the bottom wallFigure 5.17. RUN D. Contour plots of the volume fraction and the mass flow (arrows) at t = 10 s, (a) and at t = 50 s, (c). The boundary condition on the bottom wall is given by the slip-no slip condition, see equation 5.7. In the figures on the right hand side, (b) and (d), we show the actual velocity along the bottom wall. In these computations K = 100K(), ps = ps (,  ) and  = (, ||). The velocity at the bottom is of order 2.5 × 10-4 which is 1/4 of the lowest velocity in the previous case. In this case the bottom wall has moved approximately 1/2 of its own length after 250 s, see Figure ?? (b). This is comparable5.5. Numerical Experiments93to the situation in Figure 5.10 (b) where uwall = 0.001 m/s and the solution is presented at t = 50 s. The solutions appears similar but the suspension is more compressed in this case. In Figure 5.19, the mean concentration is plotted as a function of time and compared to the previous case. The curve in the slip/no slip case (solid line) follows the uwall = 0.001 m/s-curve (dashed line) in the beginning of the process but after some time the process with the slip/no slip condition, slows down. This could be an effect of the decreasing bottom velocity.94Chapter 5. Consolidation Model in 2D0.1 0.09 0.08 0.07 0.062 3 x 10-42.5u(x,0) [m/s]0.05 0.04 0.0689 0.03 0.02 0.01 0 0 0.2761.510.50.345 0.415 0.02 0.04 0.06 0.08 0.10 0 0.01 0.02 0.03 0.04 0.05 x [m] 0.06 0.07 0.08 0.09 0.1(a) t = 100 s(b) t = 100 s, solid phase velocity along the bottom wall0.1 0.09 0.08 0.07 0.062 3 x 10-42.5u(x,0) [m/s]0.05 0.04 0.03 0.294 0.02 0.01 0.515 0 0 0.02 0.04 0.06 0.08 0.1 0.368 0.07361.510.50 00.010.020.030.040.05 x [m]0.060.070.080.090.1(c) t = 250 s(d) t = 250 s, solid phase velocity along the bottom wallFigure 5.18. RUN D. Contour plots of the volume fraction and mass flow (arrows) at t = 100 s, (a) and at t = 250 s, (c). The boundary condition on the bottom wall is given by the slip/no slip condition, see equation 5.7. In the figures on the right hand side, (b) and (d), we show the actual velocity along the bottom wall. In these computations K = 100K(), ps = ps (,  ) and  = (, ||). 5.5. Numerical Experiments950.30.25mean[-]0.20.150.10.05050100 t [s]150200250Figure 5.19. RUN D,B1. The mean concentration as a function of time comparing slip/no slip case with the uwall = 0.001 case..5.5.1E. Influence of Permeability HysteresisTo study the effects of permeability hysteresis on the consolidation process results obtained with this model is compared to the reference case. In both cases the wall speed was uwall = 0.01 m/s. The solutions are presented at t = 250 s in Figure 5.20 (a) and (b). We clearly see a difference between the two solutions with a completely different cake shape and a higher maximum value of volume fraction in the irreversible case. As in the case with large viscosity the consolidated part of the suspension behaves solid like and because of the wall movement it climbs up the right wall. But when it reaches the diverging velocity field, close to the clear fluid interface, it starts to break up into smaller compressed flocs separated by large channels. The permeability increases rapidly due to hysteresis effects and a further compression of the suspension is possible. A result of this is an increase in overall consolidation speed as shown in Figure 5.20 (c) where the mean concentration is shown as function of time. We can observe a higher mean volume fraction for the irreversible permeability model (solid line).96Chapter 5. Consolidation Model in 2D0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.0466 0.03 0.02 0.01 0 0 0.327 0.02 0.04 0.06 0.08 0.1 0.1870.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.233 0.02 0.01 0.389 0 0 0.02 0.04 0.06 0.08 0.1 0.278 0.333 0.0555(a) Contour plots at t = 250 s. K(), uwall = 0.01 m/s(b) Contour plots at t = 250 s. K(,  ), uwall = 0.01 m/s0.40.350.3mean [-]0.250.20.150.1K(, ) K()*0.05 050100 t [s]150200250(c) mean as function of time. uwall = 0.01 m/s Figure 5.20. RUN E1,E2. The upper right and left figures show conotur plots of the volume fraction and mass flow (arrows) at t = 250 s comparing two permeability models. In this process uwall = 0.01 m/s, ps = ps (,  ) and  = (). In (a) K = K() and in (b) K = K(,  ). In (c), the mean concentration as function of time is presented for the two different permeability models.5.5. Numerical Experiments F. Influence of Shear Thinning and Permeability Hysteresis97In this last test case, a run is performed with the shear thinning viscosity model and the irreversible permeability function. The boundary conditions at the bottom wall are the slip/no slip conditions investigated in the previous section.0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.184 0.01 0.257 0 0 0.02 0.04 0.06 0.08 0.1 0.0377 0.1470.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.0723 0.02 0.01 0 0 0.288 0.36 0.02 0.04 0.06 0.08 0.504 0.1(a) t = 100 s(b) t = 50 s, bottom velocity0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.295 0.02 0.01 0 00.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.516 0.02 0.04 0.06 0.08 0.1 0 0 0.02 0.04 0.06 0.08 0.294 0.367 0.441 0.514 0.1 0.07330.07370.369(c) t = 100 s(d) t = 200 s, bottom velocityFigure 5.21. RUN F. Time evolution of the consolidation process. Here, K = 100K(,  ), ps = ps (,  ) and  = (, ||). The boundary condition on the  bottom wall is given by the slip-no slip condition, see equation 5.7. The figure shows contour plots of the volume fraction and the mass flow (arrows).98Chapter 5. Consolidation Model in 2DAgain, the result is displayed at four different times , see Figure 5.21: We were only able to compute a solution to t = 200 s. After this, the computation became unstable. Like we saw in the previous case, a velocity peak in the velocity profile at the bottom started grow and eventually the computations became unstable. A remedy for this would probably be more grid points and smaller time steps. But this is for ... But if the solution is compared and the velocity at the bottom wall was approximately 0.0005 m/s.0.30.25mean [-]0.20.150.10.052D 1D020406080100 t [s]120140160180200Figure 5.22. RUN F. The mean concentration as a function of time, comparing this case (solid line) to the 1D case (dashed line).We also performed a grid refinement of this case and computed the solution on a grid with 61 × 61 grid points. In the 41 × 41-case the contours are more smeared out.5.5. Numerical Experiments990.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 00.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 00.020.040.060.080.10.020.040.060.080.1(a) Mesh: 41 × 41(b) Mesh: 61 × 61Figure 5.23. RUN F. Grid refinement. In (a) we use the default mesh with 41 × 41 grid points. In (b) the mesh had 61 × 61 grid points. Here, K =  100K(,  ), ps = ps (,  ) and  = (, ||). The boundary condition on the bottom wall is given by the slip-no slip condition, see equation ??. The figure shows contour plots of the volume fraction and the volume solid flux (arrows).Conclusions of Numerical Experiments We have seen that the choice of viscosity model is important for the process and that 2D effects of shearing motions are more pronounced. Also, a larger maximum value of concentration as well as mean value is obtained with a shear thinning model, at least for low wall speeds. The permeability hysteresis model contributes to a major effect on the consolidation process and produces results which are at least qualitatively correct. A faster dewatering is obtained both for the shear thinning case as for the reference case.100Chapter 6Conclusions· A numerical model for consolidation of dense suspensions was developed with constitutive relations fitted to measurements. Specifically a new heuristic model for how the permeability function depends on micro structure was developed. The simulation results in 2D are plausible. The effects of shear on the process are significant when the irreversibility model is employed. However, the model deviates from reality in a number of ways. The material should resist not only pressure but also some shear in equilibrium. The plastic material models used for geo-mechanics are interesting. The shape of the plastic yield surface for such materials make shear forces decrease the normal forces necessary for compression, and shear will speed up consolidation by that effect too. The sensitivity of the results to the viscosity model, Gustavsson and Oppelstrup 2000b, also suggests that a more detailed study needs to be performed to fully understand the effects of shear and viscosity. Direct numerical simulations of fluid-particle suspensions can be a helpful tool in order to study these effects on the particle scale. 2D model: One effect of shear thinning is a smaller effective viscosity close to the bottom due to high shear rates. This leads to a higher mean volume fraction than with a non-shear thinning model even though the &quot;over-all-viscosity&quot; is of the same size.6.1Further WorkWe have seen that the result is very dependent on the different material fuctions and the chioce of parameters. This motivates further work in that area. Finally, we will also briefly discuss a particle simulation model which we have used to simulate a number of deforming particles in a fluid. The idea with these type of simulations is to investigate how microscopical properties influence the macroscopical behavior of the suspension. The results we present are very 1011023Chapter 6. Conclusions2.521.510.50 -1-0.500.51Figure 6.1. The inital configuration.preliminary but we would like to point out that these kind of simulations can be very powerful in developing new type of constitutive relations for multi phase fluids. ** We have also performed some very preliminary two-dimensional simulations with deformable particles settling in a suspending fluid under the influence of gravity. Particle simulations can be used to develop accurate bulk models. In this specific case we solve Navier-Stokes equations for the fluid. We have performed simulations with 30 deformable particles suspended in a fluid. They settle under the influence of gravity.6.1. Further Work103250 ts1000 ts 3332.52.52.52221.51.51.51110.50.50.50 -1-0.500.510 -1-0.500.510 -1-0.500.51(a)(b)(c)2000 ts 3 33000 ts3 at 0, 250, 500, 1000, 2000, 3000 ts.2.52.52.52221.51.51.5y1110.50.50.500 -1-0.500.510 -100.10.20.3-0.500.510.40.50.60.7(d)(e)(f)Figure 6.2. Concentration contours of  at t = 100s using two different permeability models. The volume solids flux is shown as arrows. In a) k() and in b) k(,  ).104Chapter 6. Conclusions at 0, 250, 500, 1000, 2000, 3000 ts.32.521.5y10.5000.10.20.30.40.50.60.7Figure 6.3. The mean concentration as a function of time for two different permeability models. G = 10000m/s2 and uwall = 0.01m/s. Solid line: 1D case with consolidation due to gravity only.332.52.5221.51.5110.50.50 -1-0.500.510 -1-0.500.51(a)(b)Figure 6.4. Concentration contours of  at t = 100s using two different permeability models. The volume solids flux is shown as arrows. In a) k() and in b) k(,  ).Bibliography G. 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