Solid-fluid Mixture Research Articles

Extended B‐spline‐based implicit material point method for saturated porous media

AbstractThe large deformation and fluidization process of a solid–fluid mixture includes significant changes to the temporal scale of the phenomena and the shape and properties of the mixed material. This paper presents an extended B‐spline (EBS)‐based implicit material point method (EBS‐MPM) for the coupled hydromechanical analysis of saturated porous media to enhance the overall versatility of MPM in addressing such diverse phenomena. The proposed method accurately represents phenomena such as high‐speed motion in both the quasi‐static and dynamic states by employing a full formulation of coupled hydromechanical modeling. The weak imposition of boundary conditions based on Nitsche's method allows representing the boundary conditions independent of the relative position of the particles and computational grid. In addition, it enables dynamic changes in the boundary domain based on the deformation. The robustness of this boundary representation is reinforced using EBS basis functions, which prevent the degradation of the condition number of the system matrices regardless of the position of the boundary domain with respect to the computational grid. Furthermore, a stabilization method based on a variational multiscale method (VMS) approach is employed to provide the flexibility in choosing arbitrary basis functions for spatial discretization, facilitating the effective construction of EBS. Numerical examples including comparisons between a full formulation and a simplified formulation are presented to demonstrate the performance of the developed method under various boundary conditions and loading states across different time scales.

Simulating glacier lake outburst floods (GLOFs) with a two-phase/layer debris flow model considering fluid-solid flow transitions

AbstractGlacier lake outburst floods (GLOFs) initiate with the rapid outburst of a glacier lake, endangering downstream populations, land, and infrastructure. The flow initiates as a mud flow; however, with the entrainment of additional solid material, the flood will often transform into a debris flow. As the run-out slope flattens, the coarse solid material deposits and the flow de-waters. The flow transforms back into a muddy, hyperconcentrated flow of fine sediments in suspension. These flow transitions change the flow composition dramatically and influence both the overall mass balance and flow rheology of the event. In this paper, we apply a two-phase/layer model to simulate flow transitions, solid–fluid phase separations, entrainment, and run-out distances of glacier lake outburst floods. A key feature of the model is the calculation of dilatant actions in the solid–fluid mixture which control flow transitions and phase separations. Given their high initial amount of fluid within the flow, GLOFs are sensitive to slope changes inducing flow transitions, which also implies changes in the flow rheology. The changes in the rheology are computed as a function of the flow composition and do not need any adaptation by ad-hoc selection of friction coefficients. This procedure allows the application of constant rheological input parameters from initiation to run-out. Our goal is to increase the prediction reliability of debris flow modeling. We highlight the problems associated with initial and boundary (entrainment) conditions. We test the new model against the well-known Lake 513 (Peru, 2010), Lake Palcacocha (Peru, 1941), and Lake Uchitel in the Aksay Valley (Kyrgyzstan) GLOF events. We show that flow transition modeling is essential when studying areas that have significant variations in slope.

IMEX_SfloW2D v2: a depth-averaged numerical flow model for volcanic gas–particle flows over complex topographies and water

Abstract. We present developments to the physical model and the open-source numerical code IMEX_SfloW2D (de' Michieli Vitturi et al., 2019). These developments consist of a generalization of the depth-averaged (shallow-water) fluid equations to describe a polydisperse fluid–solid mixture, including terms for sedimentation and entrainment, transport equations for solid particles of different sizes, transport equations for different components of the carrier phase, and an equation for temperature/energy. Of relevance for the simulation of volcanic mass flows, vaporization and entrainment of water are implemented in the new model. The model can be easily adapted to simulate a wide range of volcanic mass flows (pyroclastic avalanches, lahars, pyroclastic surges), and here we present its application to transient dilute pyroclastic density currents (PDCs). The numerical algorithm and the code have been improved to allow for simulation of sub- to supercritical regimes and to simplify the setting of initial and boundary conditions. The code is open-source. The results of synthetic numerical benchmarks demonstrate the robustness of the numerical code in simulating transcritical flows interacting with the topography. Moreover, they highlight the importance of simulating transient in comparison to steady-state flows and flows in 2D versus 1D. Finally, we demonstrate the model capabilities to simulate a complex natural case involving the propagation of PDCs over the sea surface and across topographic obstacles, through application to Krakatau volcano, showing the relevance, at a large scale, of non-linear fluid dynamic features, such as hydraulic jumps and von Kármán vortices, to flow conditions such as velocity and runout.

Advances in Coupling Computational Fluid Dynamics and Discrete Element Method in Geotechnical Problems

In some cases, the water content in granular soil increases to the extent that it becomes saturated, which noticeably alters its responses. For example, the pore water pressure within saturated granular soil would increase rapidly under sudden external loading, which is equivalent to undrained or constant volume conditions. This reduces the effective stress in soil dramatically and may result in catastrophic failure. There have been different numerical approaches to analyse such a failure mechanism of soil to provide a deeper understanding of soil behaviour at the microscopic level. One of the most common numerical tools for such analysis is the discrete element method (DEM) due to its advantage in obtaining microscopic properties (e.g., statistics on particle contacts and fabric), reproducibility and simple feedback control. However, most DEM studies ignore the fluid phase and merely consider the solid particles while the fluid pressure is indirectly calculated by mimicking undrained condition to a constant volume condition. Note that fluid’s influence does not limit to the change of pore water pressure. For example, the external loading would induce the movement of fluid, and the fluid-solid interaction could subsequently drag the solid particles to shift within the system. In addition, the state of soil could change from solid to suspension under an excess hydraulic gradient. Therefore, the study of the fluid-solid mixture is essential as it is a typical scenario in geotechnical practice, and the simulations of saturated sand should be conducted in numerical forms in which both the solid and fluid phases can be modelled.

A semi-implicit material point method for coupled thermo-hydro-mechanical simulation of saturated porous media in large deformation

The coupled thermo-hydro-mechanical (THM) response of liquid-infiltrated porous media underpins the safe operation and maintenance of key engineering infrastructure. Challenges remain in modeling and understanding the complicated multiphysics processes of porous media subjected to THM loads and undergoing large deformation. In this study, we develop a stabilized material point method (MPM) for modeling the THM responses of large deformation problems in biphasic solid–fluid mixtures. A novel and efficient staggered solution scheme is proposed to solve the governing equations of the coupled system formulated in terms of four primary variables: solid displacement (u), liquid velocity (v), pore pressure (p), and temperature (T). The scheme solves the energy balance equation first and employs the resulting temperature to further advance the calculation of the momentum and mass balance equations using a semi-implicit fractional step method to facilitate equal-order interpolations. Both the incompressible and weakly compressible fluid are considered in the presented fractional step formulations. We also develop the axisymmetric form of the coupled MPM to increase the applicability and efficiency of the method in THM problems. The validity, stability, and robustness of the proposed method are demonstrated through three benchmark problems, including the heating of a saturated half-space, the non-isothermal consolidation of a soil column, and a three-dimensional axisymmetric problem pertaining to the thermoelastic response around a point heat source. The predictive capability of the proposed method for large deformation problems is further showcased by the simulation of the progressive failure process of a thermal-sensitive slope.

Simulation of fallpipe rock dumping utilizing a multi-phase particle-in-cell (MPPIC)-Discrete Element Method (DEM) model. Part I: Concepts and formulation

Dumping blasted rock fragments into water bodies through vertical fallpipes can be a challenging engineering activity due to the concerns of dispersion of fine contents (suspended sediments). This study aims to provide better understanding of the fallpipe rock dumping processes through numerical simulation approaches. We propose a modified multiphase particle-in-cell (MPPIC) method to simulate the fluid–solid mixture flows, and an advection–diffusion model for handling the fine contents. The modifications to the MPPIC method include: (i) changing the pure Eulerian fluid model to a hybrid Eulerian–Lagrangian model for numerical flexibility in handling large free-surface deformations, and (ii) replacing the empirical interparticle stress model with the Discrete Element Method (DEM) for numerical stability when dealing with large-size solids. The numerical model is first validated via two physical experiments: (i) wave generation by a granular collapse and (ii) sand dumping from a split hopper. The simulations generally agree well with the experimental findings, reflecting both the fluid–solid interaction and propagation of suspended sediments. Fallpipe rock dumping is then simulated with a consideration of the effects of fallpipe size on the dynamic interactions between the DEM solid particles, the fluid, and, hence, the fine contents. The results show that long narrow fallpipes perform better in terms of containing the fine contents and limiting the spreading range of the DEM solid particles on the seabed. The time-varying distribution of the fines concentration and the fluid velocity within the fallpipe are also discussed based on the simulation results.

Well posedness of fluid-solid mixture models for biofilm spread

Two phase solid-fluid mixture models are ubiquitous in biological applications. For instance, models for growth of tissues and biofilms combine time dependent and quasi-stationary boundary value problems set in domains whose boundary moves in response to variations in the mechano-chemical variables. For a model of biofilm spread, we show how to improve its stability properties by characterizing the time derivatives of relevant quasi-stationary magnitudes in terms of additional boundary value problems. We also give conditions for well posedness of time dependent submodels set in moving domains depending on the motion of the boundary. After constructing solutions for transport, diffusion and elliptic submodels for volume fractions, displacements, velocities, pressures and concentrations with the required regularity, we are able to handle the full model of biofilm spread in moving domains assuming we know the dynamics of the boundary. These techniques are general and can be applied in models with a similar structure arising in biological applications.

Study on the balance force regulation method of multi-stage pump balance drum system

Balanced drum systems are widely used in high-pressure multi-stage pump axial force balancing mechanisms. When the pump working, the fluid (solid-fluid mixture) collides with each other or rubs against the pump case, and the balance drum is affected by this situation for a long time, so wear of the balance drum gap occurs from time to time. The first is to study how the axial force of the multistage pump changes at different balance drum gap sizes in this paper. Based on the energy equation and momentum equation, and on the basis of maintaining the original balance force unchanged, a new equation of resistance pressure difference relationship is established. The corresponding relationship between balance drum gap and balance pipe orifice plate is obtained by solving this equation. The result shows that the larger the balance drum gap, the greater the balance drum balance force decreases obviously, and the multistage pump residual axial force increases in multiples. The adjustment formula of balance force can be used to obtain the radial size of the balance pipe orifice plate under different balance drum gap, when the gap increases, to keep the balance force constant, the diameter of the orifice plate should be increased gradually. After adjusting the balance force, the fluid velocity uniformity and velocity average angle of balance pipe increases, the flow pattern in the pipe becomes uniform, the gap leakage increases and the pump efficiency decreases. The pump hydraulic properties and balancing the axial force have an opposing relationship, that is, by promoting the drum balance force, the pump hydraulic properties will be reduced. Due to processing balance drum is complex, frequent replacement is time-consuming and costly, so this article provides a handy approach for enhancing the balance force of balance drum, the results of the study can provide guarantees for the balance drum system optimization and long period stable operation of the multistage pump.

Growth of Debris Flows by Soil Bed Erosion: Effects of Frictional and Hydrodynamic Shear Stresses

Debris flows are complex solid-fluid mixtures, which travel downslope under the influence of gravity. The destructive potential of a debris flow is governed by its volume and speed, both of which can be significantly enhanced by the erosion of soil bed material along the flow path. Existing friction-induced erosion theories simplify a debris flow as an equivalent fluid that induces frictional shear stress on a soil bed. Erosion occurs when the frictional stress exceeds the soil strength. Research also shows the importance of hydrodynamic stresses on the erosion of soil beds. However, the mechanism of soil bed failure under the influence of frictional and hydrodynamic shear stresses is not clear. In this extended abstract, some experimental flume test results are presented. The tests simulate the erosion of soil beds by dry sand flow and water flow to evaluate the effects of frictional and hydrodynamic shear stresses imposed on a soil bed, respectively. Findings from this research can be used to improve estimates of erosion of debris flows in mountainous regions, which will strengthen hazard assessments.

Fluid–fluid phase separation in a soft porous medium

Various biological and chemical processes lead to the nucleation and growth of non-wetting fluid bubbles within the pore space of a granular medium, such as the formation of gas bubbles in liquid-saturated lake-bed sediments. In sufficiently soft porous materials, the non-wetting nature of these bubbles can result in the formation of open cavities within the granular solid skeleton. Here, we consider this process through the lens of phase separation, where thermomechanics govern the separation of the non-wetting phase from a fluid–fluid–solid mixture. We construct a phase-field model informed by large-deformation poromechanics, in which two immiscible fluids interact with a poroelastic solid skeleton. Our model captures the competing effects of elasticity and fluid–fluid–solid interactions. We use a phase-field damage model to capture the mechanics of the granular solid. As a model problem, we consider an initial distribution of non-wetting fluid in the pore space that separates into multiple cavities. We use simulations and linear-stability analysis to identify the key parameters that control phase separation, the conditions that favour the formation of cavities, and the characteristic size of the resulting cavities.

Improving hydrocyclone efficiency by changing geometric parameters

Hydrocyclone is a very simple and cost-effective way to separate inhomogeneous fluid-solid mixtures. It is important to ensure adequate separation efficiency during their operation. Many researchers have worked to identify the factors that influence separation. Geometry has been identified as one of the most important influencing factors. This article summarises the results of this research in order to provide a comprehensive picture of the importance and impact of each geometric element.

Dynamic soil consolidation model using a nonlocal continuum poroelastic damage approach

AbstractWe present a novel dynamic poroelastic model for soil consolidation in an isotropic homogeneous fluid saturated porous media. A nonlocal integral‐type continuum damage formulation is applied to describe the damage evolution under dynamic excitations, in which a bilinear damage law is assumed. The governing equations are obtained by considering the conservation of momentum and mass balance for the solid–fluid mixtures, in which the fluid flow obeys Darcy's seepage law in the entire domain. The numerical solution of the fully coupled problem is achieved via a mixed FEM displacement–pressure element formulation. The mechanical equilibrium equations are evolved in time using a Newmark method and the resulting nonlinear system is solved via a Newton–Raphson method. Consistent linearization is then used to obtain the block Jacobian matrix analytically and the linear system is solved monolithically at each time step. Several numerical results are presented to study soil consolidation problems under harmonic excitations. We investigate the time‐dependence response of the skeleton displacement, pressure, and damage evolution. In addition, the examples demonstrate that the nonlocal integral‐type damage model can effectively overcome mesh dependence and yield smooth behavior.

Superposition‐based concurrent multiscale approaches for poromechanics

AbstractSuperposition‐based concurrent multiscale approaches for poromechanical problems aimed at resolving localized fluid flow and deformation of a solid skeleton including its possible fracture are developed. The main departure of the proposed formulation is that the primary unknowns in the coupling zone, the solid displacement u and pore pressure p, are discretized by possibly different numerical models. Two discretization schemes are studied: (i) superposition of two or more finite element (FE) discretizations of the Biot's u‐p model of a porous medium, and (ii) superposition of the peridynamics (PD) discretization of the fractured solid phase on the FE discretization of the Biot's u‐p model of a porous medium. In the latter case, the total stress of the fluid–solid mixture in the coupling domain is derived from the effective stress obtained from the combined PD and finite element method solutions while the pore pressure is obtained from the FE solution. The accuracy of the superposition‐based concurrent multiscale method is studied on several benchmark problems including the classical Terzaghi's one‐dimensional consolidation problem, the two‐dimensional strip footing problem, and the pressure‐ and fluid‐driven fracture propagation problem in saturated porous media. The main hydraulic fracture characteristics, including complex crack paths and fluid flow patterns, have been found to be reasonably well reproduced by the proposed method.

Numerical computation of fluid–solid mixture flow using the SPH–VCPM–DEM method

In this paper, a unified particle-based solver is developed to solve complex fluid–solid​ mixture flow problem involving many elastic solid bodies. Our previous Fluid Structure Interaction (FSI) solver that incorporates Smoothed Particle Hydrodynamics (SPH) for fluid modelling and Volume Compensated Particle Method (VCPM) for elastic solid modelling is extended by incorporating the Discrete Element Method (DEM) for modelling the contact force between colliding elastic solid bodies. The contact force is modelled using the non-linear Hertzian formulation. The method is firstly validated by solving several benchmark FSI problems that involve only one solid body. Then, solid–solid interaction is modelled and the collision problem involving two elastic rings and the well-known Drafting, Kissing and Tumbling (DKT) phenomenon in a fluid domain are simulated. The results agree considerably well with other benchmark numerical solutions. The collapse of Aluminium cylinder layers (in dry and wet conditions) is then simulated and it is found that the numerical result is comparable to the experimental data. Following that, the SPH–VCPM–DEM method is used to solve a more complex problem involving the dumping of solid objects through solid and perforated fall pipes for scour protection. The results show the potential of the SPH–VCPM–DEM method in simulating complex fluid–solid mixture flow.

A regularized isothermal phase-field model of two-phase solid–fluid mixture and its spatial dissipative discretization equations

Abstract The present paper is devoted to a model describing a two-phase isothermal mixture, in which one of the phases obeys solid-like (namely, elastic) rheology. A fully Eulerian description is considered. To describe the stress–strain behaviour of the solid phase the elastic energy term is added to the Helmholtz free energy. The term depends on Almansi strain tensor. In its turn, the strain tensor is defined as the solution of the corresponding evolutionary equation. Considered model belongs to the phase field family. Formally it describes two-component mixture and uses mass densities of the components as order parameters. A distinctive feature of the considered model is its preliminary regularization according to the quasi-hydrodynamic framework. The dissipativity in total energy is proved when periodic boundary conditions are imposed. A spatial dissipative semi-discrete (continuous in time and discrete in space) scheme based on staggered grids is suggested. The theoretical results remain valid in the absence of the regularization. The results of a numerical study in a 2D setting are presented.

Thermovibrational instability in a nanofluid porous medium

Thermovibrational convection in a water based metallic (Cu-water and Ag-water) or metallic oxide (Al2O3-water and TiO2-water) nanofluid saturated porous medium is addressed incorporating the Brinkman boundary correction. The nanofluid behaves more like a fluid rather than a conventional solid–fluid mixture and is studied through Khanafer–Vafai–Lightstone (KVL) model. A linear stability analysis is carried out through the Floquet theory to find the instability thresholds for convection onset in terms of critical Rayleigh number and critical wavenumber. The results corresponding to synchronous as well as subharmonic modes are obtained for the Darcy and Brinkman regimes. A rise in particle concentration is found to cause stabilizing response in general. However, it leaves an opposite effect in the Brinkman regime when the medium is filled with metallic nanofluid and is heated from above. Under this situation, the use of oblate-spheroidal particles in achieving optimal convective heat transfer enhancement is highlighted.

Robust stabilised finite element solvers for generalised Newtonian fluid flows

Various materials and solid-fluid mixtures of engineering and biomedical interest can be modelled as generalised Newtonian fluids, as their apparent viscosity depends locally on the flow field. Despite the particular features of such models, it is common practice to combine them with numerical techniques originally conceived for Newtonian fluids, which can bring several issues such as spurious pressure boundary layers, unsuitable natural boundary conditions and coupling terms spoiling the efficiency of nonlinear solvers and preconditioners. In this work, we present a finite element framework dealing with such issues while maintaining low computational cost and simple implementation. The building blocks of our algorithm are (i) an equal-order stabilisation method preserving consistency even for lowest-order discretisations, (ii) robust extrapolation of velocities in the time-dependent case to decouple the rheological law from the overall system, (iii) adaptive time step selection and (iv) a fast physics-based preconditioned Krylov subspace solver, to tackle the relevant range of discretisation parameters including highly varying viscosity. Selected numerical experiments are provided demonstrating the potential of our approach in terms of robustness, accuracy and efficiency for problems of practical interest.

A variational minimization formulation for hydraulically induced fracturing in elastic-plastic solids

A variational modeling framework for hydraulically induced fracturing of elastic-plastic solids is developed in the present work. The developed variational structure provides a global minimization problem. While fracture propagation is modeled by means of a phase-field approach to fracture, plastic effects are taken into account by using a Drucker–Prager-type yield-criterion function. This yield-criterion function governs the plastic evolution of the fluid-solid mixture. Fluid storage and transport are described by a Darcy–Biot-type formulation. Thereby the fluid storage is decomposed into a contribution due to the elastic deformations and one due to the plastic deformations. A local return mapping scheme is used for the update of the plastic quantities. The global minimization structure demands a $$H(\mathrm{div})$$ -conforming finite-element formulation. Furthermore this is combined with an enhanced-assumed-strain formulation in order to overcome locking phenomena arising from the plastic deformations. The robustness and capabilities of the presented framework will be shown in a sequence of numerical examples.

Finite Element Implementation of Biphasic-Fluid Structure Interactions in febio.

In biomechanics, solid-fluid mixtures have commonly been used to model the response of hydrated biological tissues. In cartilage mechanics, this type of mixture, where the fluid and solid constituents are both assumed to be intrinsically incompressible, is often called a biphasic material. Various physiological processes involve the interaction of a viscous fluid with a porous-hydrated tissue, as encountered in synovial joint lubrication, cardiovascular mechanics, and respiratory mechanics. The objective of this study was to implement a finite element solver in the open-source software febio that models dynamic interactions between a viscous fluid and a biphasic domain, accommodating finite deformations of both domains as well as fluid exchanges between them. For compatibility with our recent implementation of solvers for computational fluid dynamics (CFD) and fluid-structure interactions (FSI), where the fluid is slightly compressible, this study employs a novel hybrid biphasic formulation where the porous skeleton is intrinsically incompressible but the fluid is also slightly compressible. The resulting biphasic-FSI (BFSI) implementation is verified against published analytical and numerical benchmark problems, as well as novel analytical solutions derived for the purposes of this study. An illustration of this BFSI solver is presented for two-dimensional (2D) airflow through a simulated face mask under five cycles of breathing, showing that masks significantly reduce air dispersion compared to the no-mask control analysis. In addition, we model three-dimensional (3D) blood flow in a bifurcated carotid artery assuming porous arterial walls and verify that mass is conserved across all fluid-permeable boundaries. The successful formulation and implementation of this BFSI solver offers enhanced multiphysics modeling capabilities that are accessible via an open-source software platform.

Quantifying the transition of impact mechanisms of geophysical flows against flexible barrier

Flexible barriers have been increasingly used worldwide for mitigating natural hazards involving various geophysical flows. The analysis and design of flexible barriers need to distinguish two major impact mechanisms, runup and pile-up, which remain poorly understood due to complicated interactions between the impacting flow and the deformable barrier. In this study, a unified computational approach based on coupled computational fluid dynamics and discrete element method (CFD-DEM) is employed to examine the impact mechanisms of a wide spectrum of geophysical flows against a deformable and permeable flexible barrier. We consider geophysical flows including rock avalanche, debris avalanche, debris flow, debris flood and mud flow, and a flexible barrier consisting of a barrier net, cables and brakes. The signatures of runup and pile-up mechanisms are thoroughly analyzed, in terms of flow features and barrier responses. The effects of Froude numbers (Fr = 0.46–7.40), solid volume concentrations (0.1–1), and fluid rheologies (Newtonian and non-Newtonian fluids) on the impact mechanism transitions are systematically examined. Two nondimensional indices, the static-peak load ratio of the barrier and the momentum reduction ratio of the flow, are proposed for identifying the transition of impact mechanism from pile-up to runup. The transition is found to occur with either an increase in Fr or solid volume concentration of the flow. Increased viscosity of the fluid in a solid-fluid mixture may result in a transition occurring at a higher Fr. This study helps gain insights into the two impact mechanisms and their transitions and may provide a useful reference for the future design of flexible barriers in mitigating hazardous geophysical flows.