On the numerical modelling of initiation of sediment transport using Smoothed Particle Hydrodynamics

Mobilization of solid particles at the interface between a porous and a free flow domain is a relevant subject in many fields of mechanical, civil and environmental engineering. One example is the initiation of sediment transport as it appears in river beds. To approximate this initiation state, various theoretical models exist. Common approaches use two-domain formulations as in [1] or one-domain formulations as in [6]. The named approaches were compared with Direct Numerical Simulations (DNS) using Smoothed Particle Hydrodynamics (SPH) in [3]. The results of these simulations showed that the theoretical models often underestimate the occurring velocities at the interface and therefore critical velocities to initialize the motion of single grains can be lower than predicted by theoretical approaches. In our numerical simulations, we study creeping flow in a free flow domain coupled to flow in a porous media applying various porous structures. To investigate velocities and shear stresses at the interface more intensively we then compare our numerical results to data from experiments that were performed on equivalent microstructures. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

薛定谔方程在非一致网格下数值模拟的界面条件

本文研究了一维线性薛定谔方程在非一致网格下数值模拟的问题.在数值模拟中,非一致网格在界面处会产生虚假反射,利用局部时间步长和界面条件的方法,成功的减小了虚假反射.改进和提高了薛定谔方程数值模拟的效率和精度.     

Convergence of a non-stiff boundary integral method for interfacial flows with surface tension

Boundary integral methods to simulate interfacial flows are very sensitive to numerical instabilities. In addition, surface tension introduces nonlinear terms with high order spatial derivatives into the interface dynamics. This makes the spatial discretization even more difficult and, at the same time, imposes a severe time step constraint for stable explicit time integration methods.

A proof of the convergence of a reformulated boundary integral method for two-density fluid interfaces with surface tension is presented. The method is based on a scheme introduced by Hou, Lowengrub and Shelley [ J. Comp. Phys. 114 (1994), pp. 312-338] to remove the high order stability constraint or stiffness. Some numerical filtering is applied carefully at certain places in the discretization to guarantee stability. The key of the proof is to identify the most singular terms of the method and to show, through energy estimates, that these terms balance one another.

The analysis is at a time continuous-space discrete level but a fully discrete case for a simple Hele-Shaw interface is also studied. The time discrete analysis shows that the high order stiffness is removed and also provides an estimate of how the CFL constraint depends on the curvature and regularity of the solution.

The robustness of the method is illustrated with several numerical examples. A numerical simulation of an unstably stratified two-density interfacial flow shows the roll-up of the interface; the computations proceed up to a time where the interface is about to pinch off and trapped bubbles of fluid are formed. The method remains stable even in the full nonlinear regime of motion. Another application of the method shows the process of drop formation in a falling single fluid.

     

A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows

This paper reports a new meshless Integrated Radial Basis Function Network (IRBFN) approach to the numerical simulation of interfacial flows in which the two-way interaction between a moving interface and the ambient viscous flow is fully investigated. When an interface between two immiscible fluids moves, not only its position and shape but also the flow variables (i.e. velocity field and pressure) change due to the presence of surface tension along the moving interface. The velocity field of the ambient flow, on the other hand, causes the interface to move and deform as a result of momentum transport between the two immiscible fluids on both sides of the interface. Numerical investigations of such a two-way interaction is reported in this paper where the level set method is used in combination with high-order projection schemes in the meshless framework of the IRBFN method. Numerical investigations on the meshless projection schemes are performed with typical benchmark incompressible viscous flow problems for verification purposes. The approach is then demonstrated with the numerical simulation of two bubbles moving, stretching and merging in an incompressible ambient fluid under the action of buoyancy force.     

Numerical simulation of gas-assisted injection molding using CLSVOF method

It is a typical gas-liquid two phase flow phenomenon that gas penetrates the polymer melt in gas-assisted injection molding (GAIM) process. Numerical simulation is now playing an important role in GAIM, in which the accurate simulation of moving interface is of great importance. The level set (LS) method is a popular interface tracking method, but it does not ensure naturally mass-conservation. In order to improve the mass-conservation of LS method, a coupled level-set and volume-of-fluid (CLSVOF) method with mass-correction is presented for the numerical simulations of interfacial flows in GAIM. The performance of this CLSVOF method is demonstrated by two numerical tests including the three-dimensional deformation field test and the dam break problems. Finally the CLSVOF method is employed to simulate the 3D moving interfaces in GAIM, including gas-melt interface and the melt-front interface. The influences of melt temperature and gas delay time are also analyzed detailedly. As a case study, the processes that gas penetrates the polymer melt in complex cavities are also simulated using this method, and the simulation results are in agreement with those obtained by other researchers.     

A General Cavitation Model for the Highly Nonlinear Mie-Grüneisen Equation of State

A general one-fluid cavitation model is proposed for a family of Mie-Grüneisen equations of state (EOS), which can provide a wide application of cavitation flows, such as liquid-vapour transformation and underwater explosion. An approximate Riemann problem and its approximate solver for the general cavitation model are developed. The approximate solver, which provides the interface pressure and normal velocity by an iterative method, is applied in computing the numerical flux at the phase interface for our compressible multi-medium flow simulation on Eulerian grids. Several numerical examples, including Riemann problems and underwater explosion applications, are presented to validate the cavitation model and the corresponding approximate solver.     

A Hybrid WENO Method with Modified Ghost Fluid Method for Compressible Two-Medium Flow Problems

In this paper, we develop a simplified hybrid weighted essentially non-oscillatory (WENO) method combined with the modified ghost fluid method (MGFM) [31] to simulate the compressible two-medium flow problems. The MGFM can turn the two-medium flow problems into two single-medium cases by defining the ghost fluids state in terms of the predicted the interface state, which makes the material interface “invisible”. For the single medium flow case, we adapt between the linear upwind scheme and the WENO scheme automatically by identifying the regions of the extreme points for the reconstruction polynomial as same as the hybrid WENO scheme [55]. Instead of calculating their exact locations, we only need to know the regions of the extreme points based on the zero point existence theorem, which is simpler for implementation and saves computation time. Meanwhile, it still keeps the robustness and has high efficiency. Extensive numerical results for both one and two dimensional two-medium flow problems are performed to demonstrate the good performances of the proposed method.     

A Finite Element Algorithm for Electromechanical Contact Problems

In the last hundred years a lot of work is done in describing and measuring the influence of the pressure on the resistance and wear in electrical contacts. But up to now there exists a lack of knowledge in predicting and optimizing the behavior of electrical contacts with numerical simulation tools considering the pressure dependency. The present work provides a new constitutive model for the contact interface in the case of current flow and a new friction law including electrical wear phenomena. Additionally numerical investigations are made to compare the numerical results with experimental data. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling and Numerical Simulation of Breaking Waves on Structures

This work deals with the analytical and numerical simulation of the loads on cylindrical offshore structures due to breaking waves. We investigate breaking waves impact on a cylindrical structure on the surface of the water by using a multiphase flow model of immiscible fluids based on the Volume of Fluid (VOF) method. Different numerical schemes are applied to identify the sharp interface between water and gas. The results are compared as well with the already existing experimental, analytical and numerical studies by [1-3]. The agreement shows that the analytical and numerical models are suited to describe the experimental results. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Axi symmetric 2D simulation and numerical heat transfer characteristics for the calibration furnace in a rectangular enclosure

This paper presents axi symmetric 2D numerical investigation of the spherical thermocouple calibration furnace in a rectangular enclosure. The focus is on the flow structure inside the Saturn (a hollow spherical cavity), external flow behavior due to annulus block heating and the surface temperature uniformity. Mesh sensitivity analysis is adopted to extract the mesh with minimum number of nodes but with fast convergent finite element solution. The basic strategy here is that temperature perturbation error at a single point instead of a single element contributed to the total perturbation error qualitatively remains the same. Agreement between numerical simulation results and the experiment results is good with a maximum temperature deviation 10 °C for the cavity temperature 400 °C. Finally, standard numerical temperature uncertainty due to variation in thermal conductivity is computed through the sensitivity coefficient using uncertainty analysis.     

Numerical simulation of the filling behavior of a Non-Newtonian Fluid

A numerical model for free surface flows of non-newtonian liquids which are injected into a cavity is presented. These flows are regarded as a basic model of injection molding. Model experiments of the injection process are performed with a water-based gel. The flow equations are integrated according to the finite-volume-method. The volume of fluid method (VoF) is employed in order to describe the free surface flow of two incompressible phases, the phase interface is resolved by the method of geometric reconstruction. The Herschel-Bulkley model is used in order to describe shear-thinning behavior of the molding material and the effects of a yielding point. Different patterns of the filling flow depending on the injection parameters are evident in the experiment and the simulation. They are characterized and arranged with respect to the similarity parameters of the flow. Again, the results of the simulation are found to agree well with the experimental observations. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A numerical scheme for multi-point special boundary-value problems and application to fluid flow through porous layers

In this paper we propose a numerical scheme based on finite differences for the numerical solution of nonlinear multi-point boundary-value problems over adjacent domains. In each subdomain the solution is governed by a different equation. The solutions are required to be smooth across the interface nodes. The approach is based on using finite difference approximation of the derivatives at the interface nodes. Smoothness across the interface nodes is imposed to produce an algebraic system of nonlinear equations. A modified multi-dimensional Newton’s method is proposed for solving the nonlinear system. The accuracy of the proposed scheme is validated by examples whose exact solutions are known. The proposed scheme is applied to solve for the velocity profile of fluid flow through multilayer porous media.     

Numerical modeling of turbulence-induced interfacial instability in two-phase flow with moving interface

The present paper introduces a new interfacial marker-level set method (IMLS) which is coupled with the Reynolds averaged Navier–Stokes (RANS) equations to predict the turbulence-induced interfacial instability of two-phase flow with moving interface. The governing RANS equations for time-dependent, axisymmetric and incompressible two-phase flow are described in both phases and solved separately using the control volume approach on structured cell-centered collocated grids. The transition from one phase to another is performed through a consistent balance of kinematic and dynamic conditions on the interface separating the two phases. The topological changes of the interface are predicted by applying the level set approach. By fitting a number of interfacial markers on the intersection points of the computational grids with the interface, the interfacial stresses and consequently, the interfacial driving forces are easily estimated. Moreover, the normal interface velocity, calculated at the interfacial markers positions, can be extended to the higher dimensional level set function and used for the interface advection process. The performance of linear and non-linear two-equation k–ε turbulence models is investigated in the context of the considered two-phase flow impinging problem, where a turbulent gas jet impinging on a free liquid surface. The numerical results obtained are evaluated through the comparison with the available experimental and analytical data. The nonlinear turbulence model showed superiority in predicting the interface deformation resulting from turbulent normal stresses. However, both linear and nonlinear turbulence models showed a similar behavior in predicting the interface deformation due to turbulent tangential stresses. In general, the developed IMLS numerical method showed a remarkable capability in predicting the dynamics of the considered two-phase immiscible flow problems and therefore it can be applied to quite a number of interface stability problems.     

Nitsche-XFEM for a transport problem in two-phase incompressible flows

We consider the transport of a dissolved species in a divergence-free immiscible incompressible two-phase flow modeled by a convection diffusion equation. The so-called Henry interface condition leads to a jump condition for the concentration at the interface between the two phases. This discontinuity of the solution render the numerical solution on unfitted meshes difficult. Furthermore time discretization on moving interfaces and handling typically convection dominant situations makes the overall problem delicate. We propose a numerical method using extended finite elements and a Nitsche-type technique combined with streamline diffusion stabilization. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical simulation of liquid redistribution in permeable media involving hysteresis

In this paper, we explore new techniques for numerical simulation of liquid redistribution in permeable media involving hysteresis. We regard the hysteresis as a kind of passive fractional resistance, and integrate it into Partial Differential Equations (PDE). Also numerical methods such as the Finite Volume Method (FVM) are developed. A system of partial differential equations is derived for liquid pressure, including the effects of hysteresis and mixed media. The finite difference schemes based on the conservative law are also provided, which are well suited to the mixed media made up of different material layers. A new algorithm is deliberately designed to evaluate the total liquid volume that satisfies the conservative law exactly.

Furthermore, numerical examples are conducted to imitate the following fascinating phenomena in real physics.

1. (a) The nonuniform equilibrium saturation in redistribution.

2. (b) The noncontinuity of saturation at the interface of mixed media.

3. (c) Conservation of the total liquid volume even when t is large.

The new numerical techniques in this paper are not only easy to carry out with a modest computational effect, but also effective to simulate the real porous flow in the laboratory experiments.     

Block mesh refinement for incompressible flows in curvilinear domains

A method for the solution of the Navier–Stokes equation for the prediction of flows inside domains of arbitrary shaped bounds by the use of Cartesian grids with block-refinement in space is presented. In order to avoid the complexity of the body fitted numerical grid generation procedure, we use a saw tooth method for the curvilinear geometry approximation. By using block-nested refinement, we achieved the desired geometry Cartesian approximation in order to find an accurate solution of the N–S equations. The method is applied to incompressible laminar flows and is based on a cell-centred approximation. We present the numerical simulation of the flow field for two geometries, driven cavity and stenosed tubes. The utility of the algorithm is tested by comparing the convergence characteristics and accuracy to those of the standard single grid algorithm. The Cartesian block refinement algorithm can be used in any complex curvilinear geometry simulation, to accomplish a reduction in memory requirements and the computational time effort.     

Coarsening dynamics of polymer droplets for a lubrication model with large slippage

We analyze the final stages of the dewetting process of nanoscopic thin polymer films on hydrophobized substrates using a lubrication model that captures the large slippage at the liquid-substrate interface. The final stages of this process are characterized by the slow-time coarsening dynamics of the remaining droplets. For this situation we derive a reduced system of equations from the lubrication model, using singular perturbation analysis. Such reduced models allow for an efficient numerical simulation of the coarsening process. The reduced model extends results of [2] for no-slip lubrication model. Apart from collapse and collision, we identify here some new coarsening dynamics. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Simulation of Dendritic Crystal Growth

We consider the modified Stefan-Problem with Gibbs-Thomson correction, but vanishing kinetic undercooling. In this case the interface velocity is not given by mean curvature flow, but has to be computed explicitly from the temperature gradients at the interface. A finite element method on adaptive refined multigrids is presented here. Dirichlet conditions have to be satisfied along the solid-liquid interface that in general may intersect the elements. We use an implicit level-set representation of the interface that preserves it as a sharp surface, in contrast to the phase-field method. In numerical simulations we observe dendritic patterns that show good agreement with different features of physical experiments. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vortex shedding behind a rising bubble and two-bubble coalescence: A numerical approach

In this work, we present the computational results on the wake instability in wobbling bubble regime as well as on the coalescence of two bubbles in different shape regimes. This is a continuation of our previous studies on the dynamics of a single gas bubble rising in a viscous liquid (see [A. Smolianski, H. Haario, P. Luukka, Computational Study of Bubble Dynamics, Research Report 86, Lappeenranta University of Technology, Finland]), and we use the same, finite-element/level-set/operator-splitting method that was proposed in [A. Smolianski, Numerical Modeling of Two-Fluid Interfacial Flows, Ph.D. Thesis, University of Jyväskylä, 2001]. The numerical method allows to simulate a wide range of flow regimes, accurately capturing the shape of the deforming interface of the bubble and the surface tension effect, while maintaining a good mass conservation. Due to the highly unstable and small-scale nature of the considered problems there are very few experimental investigations, but the comparison with available experimental data confirms a good accuracy of our numerical predictions. Our studies show that plausible results can be obtained with two-dimensional numerical simulations, when a single buoyant bubble or a coalescence of two bubbles is considered.