Finite elements in fluids: Stabilized formulations and moving boundaries and interfaces

We provide an overview of the finite element methods we developed for fluid dynamics problems. We focus on stabilized formulations and moving boundaries and interfaces. The stabilized formulations are the streamline-upwind/Petrov-Galerkin (SUPG) formulations for compressible and incompressible flows and the pressure-stabilizing/Petrov-Galerkin (PSPG) formulation for incompressible flows. These are supplemented with the discontinuity-capturing directional dissipation (DCDD) for incompressible flows and the shock-capturing terms for compressible flows. Determination of the stabilization and shock-capturing parameters used in these formulations is highlighted. Moving boundaries and interfaces include free surfaces, two-fluid interfaces, fluid-object and fluid-structure interactions, and moving mechanical components. The methods developed for this class of problems can be classified into two main categories: interface-tracking and interface-capturing techniques. The interface-tracking techniques are based on the deforming-spatial-domain/stabilized space-time (DSD/SST) formulation, where the mesh moves to track the interface. The interface-capturing techniques were developed for two-fluid flows. They are based on the stabilized formulation, over typically non-moving meshes, of both the flow equations and an advection equation. The advection equation governs the time-evolution of an interface function marking the interface location. We also describe some of the additional methods and ideas we introduced to increase the scope and accuracy of these two classes of techniques. Among them is the enhanced-discretization interface-capturing technique (EDICT), which was developed to increase the accuracy in capturing the interface. Also among them is the mixed interface-tracking/interface-capturing technique (MITICT), which was introduced for problems that involve both interfaces that can be accurately tracked with a moving-mesh method and interfaces that call for an interface-capturing technique.     

Computation of flow problems with the Mixed Interface-Tracking/Interface-Capturing Technique (MITICT)

In computation of flow problems with fluid-solid interfaces, an interface-tracking technique, where the fluid mesh moves to track the interface, would allow us to have full control of the resolution of the fluid mesh in the boundary layers. With an interface-capturing technique (or an interface locator technique in the more general case), on the other hand, independent of how accurately the interface geometry is represented, the resolution of the fluid mesh in the boundary layer will be limited by the resolution of the fluid mesh at the interface. In computation of flow problems with fluid-fluid interfaces where the interface is too complex or unsteady to track while keeping the remeshing frequency under control, interface-capturing techniques, with enhanced-discretization as needed, could be used as more flexible alternatives. Sometimes we may need to solve flow problems with both fluid-solid interfaces and complex or unsteady fluid-fluid interfaces. The Mixed Interface-Tracking/Interface-Capturing Technique (MITICT) was introduced for computation of flow problems that involve both interfaces that can be accurately tracked with a moving mesh method and interfaces that are too complex or unsteady to be tracked and therefore require an interface-capturing technique. As the interface-tracking technique, we use the Deforming-Spatial-Domain/Stabilized Space-Time (DSD/SST) formulation. The interface-capturing technique rides on this, and is based on solving over a moving mesh, in addition to the Navier-Stokes equations, the advection equation governing the time-evolution of the interface function. For the computations reported in this paper, as interface-capturing technique we are using one of the versions of the Edge-Tracked Interface Locator Technique (ETILT).     

A front-tracking method with projected interface conditions for compressible multi-fluid flows

A front-tracking method for compressible multi-fluid flows is presented, where marker points are used both for tracking fluid interfaces and also for constructing the Riemann problem on the interfaces. The Riemann problem between the two fluid phases (defined in the interface normal direction) is solved using the exact Riemann solver on the marker points. The solutions are projected onto fixed grid points and then extrapolated into the corresponding ghost-fluid regions, to be used as boundary conditions. Each fluid phase is solved separately as in the ghost-fluid method. The proposed procedures, especially the projection of the exact Riemann solutions onto the fluid grids, are designed to be simple and consistent in any spatial dimensions. Several multi-fluid problems, including the breakup of a water cylinder induced by the passage of a shock wave were computed in order to demonstrate the capability of the proposed method.     

A phase-field method for interface-tracking simulation of two-phase flows

For interface-tracking simulation of two-phase flows with a high density ratio, we propose a computational method, NS–PFM, combining Navier–Stokes (NS) equations with phase-field model (PFM) based on the free energy theory. Through the numerical simulations, it was confirmed that (1) the volume flux derived from chemical potential gradient in the Cahn–Hilliard equation of PFM plays an important role in advection and reconstruction of interface, and (2) the NS–PFM gives good predictions for the motions of immiscible, incompressible, isothermal two-phase fluid, such as air–water system, without using conventional interface-tracking techniques.     

Hybrid atomistic�Ccontinuum simulations of fluid flows involving interfaces

In this work, we use an hybrid atomistic–continuum (HAC) simulation method to study transient and steady isothermal flows of Lennard-Jones fluids near interfaces. Our hybrid method is based on a domain decomposition algorithm. The flow domain is composed of two overlapping regions: an atomistic region described by molecular dynamics, and a continuum region described by a finite volume discretization of the incompressible Navier–Stokes equations. To show the interest of such an hybrid method to compute flows near fluid/solid interface, we first applied our hybrid scheme to the classical Couette flow, where the moving wall is modelled at the atomistic scale. In addition, we also studied an oscillatory shear flow. Then, to compute flows near fluid/fluid interface, we applied our method to a two-phase Couette flow (liquid/gas), where the interface is modelled at the molecular scale. We show that hybrid results can sometimes differ from those provided by analytical solutions deduced from continuum mechanics equations combined with usual boundary/interface relations. For the Couette and oscillatory shear flows, a good agreement is found between hybrid simulations and macroscopic analytical solutions, however, we noticed that the fluid in contact with the wall can be more entailed than what expected. For the liquid/gas Couette flow, the hybrid simulation exhibits an unexpected jump of the velocity in the interfacial region, corresponding to a partial slip between the two fluid phases. Those interesting results highlight the interest of using an HAC method to deal with systems for which surfaces/interfaces effects are important.     

A generalized continuous surface tension force formulation for phase-field models for multi-component immiscible fluid flows

We present a new phase-field method for modeling surface tension effects on multi-component immiscible fluid flows. Interfaces between fluids having different properties are represented as transition regions of finite thickness across which the phase-field varies continuously. At each point in the transition region, we define a force density which is proportional to the curvature of the interface times a smoothed Dirac delta function. We consider a vector valued phase-field, the velocity, and pressure fields which are governed by multi-component advective Cahn–Hilliard and modified Navier–Stokes equations. The new formulation makes it possible to model any combination of interfaces without any additional decision criteria. It is general, therefore it can be applied to any number of fluid components. We give computational results for the four component fluid flows to illustrate the properties of the method. The capabilities of the method are computationally demonstrated with phase separations via a spinodal decomposition in a four-component mixture, pressure field distribution for three stationary drops, and the dynamics of two droplets inside another drop embedded in the ambient liquid.     

Finite element methods for flow problems with moving boundaries and interfaces

Summary  This paper is an overview of the finite element methods developed by the Team for Advanced Flow Simulation and Modeling (T*AFSM) [http://www.mems.rice.edu/TAFSM/] for computation of flow problems with moving boundaries and interfaces. This class of problems include those with free surfaces, two-fluid interfaces, fluid-object and fluid-structure interactions, and moving mechanical components. The methods developed can be classified into two main categories. The interface-tracking methods are based on the Deforming-Spatial-Domain/Stabilized Space-Time (DSD/SST) formulation, where the mesh moves to track the interface, with special attention paid to reducing the frequency of remeshing. The interface-capturing methods, typically used for free-surface and two-fluid flows, are based on the stabilized formulation, over non-moving meshes, of both the flow equations and the advection equation governing the time-evolution of an interface function marking the location of the interface. In this category, when it becomes neccessary to increase the accuracy in representing the interface beyond the accuracy provided by the existing mesh resolution around the interface, the Enhanced-Discretization Interface-Capturing Technique (EDICT) can be used to to accomplish that goal. In development of these two classes of methods, we had to keep in mind the requirement that the methods need to be applicable to 3D problems with complex geometries and that the associated large-scale computations need to be carried out on parallel computing platforms. Therefore our parallel implementations of these methods are based on unstructured grids and on both the distributed and shared memory parallel computing approaches. In addition to these two main classes of methods, a number of other ideas and methods have been developed to increase the scope and accuracy of these two classes of methods. The review of all these methods in our presentation here is supplemented by a number numerical examples from parallel computation of complex, 3D flow problems.     

Decoupled Energy Stable Schemes for a Phase Field Model of Three-Phase Incompressible Viscous Fluid Flow

We develop a numerical approximation for a hydrodynamic phase field model of three immiscible, incompressible viscous fluid phases. The model is derived from a generalized Onsager principle following an energetic variational formulation and is consisted of the momentum transport equation and coupled phase transport equations. It conserves the volume of each phase and warrants the total energy dissipation in time. Its numerical approximation is given by a set of easy-to-implement, semi-discrete, linear, decoupled elliptic equations at each time step, which can be solved efficiently using fast solvers. We prove that the scheme is energy stable. Mesh refinement tests and three numerical examples of three-phase viscous fluid flows in 3D are presented to benchmark the effectiveness of the model as well as the efficiency of the numerical scheme.     

A new 3D parallel SPH scheme for free surface flows

We propose a new robust and accurate SPH scheme, able to track correctly complex three-dimensional non-hydrostatic free surface flows and, even more important, also able to compute an accurate and little oscillatory pressure field. It uses the explicit third order TVD Runge-Kutta scheme in time, following Shu and Osher [Shu C-W, Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J Comput Phys 1988;89:439-71], together with the new key idea of introducing a monotone upwind flux for the density equation, thus removing any artificial viscosity term. For the discretization of the velocity equation, the non-diffusive central flux has been used. A new flexible approach to impose the boundary conditions at solid walls is also proposed. It can handle any moving rigid body with arbitrarily irregular geometry. It does neither produce oscillations in the fluid pressure in proximity of the interfaces, nor does it have a restrictive impact on the stability condition of the explicit time stepping method, unlike the repellent boundary forces of Monaghan [Monaghan JJ. Simulating free surface flows with SPH. J Comput Phys 1994;110:399-406]. To asses the accuracy of the new SPH scheme, a 3D mesh-convergence study is performed for the strongly deforming free surface in a 3D dam-break and impact-wave test problem providing very good results.Moreover, the parallelization of the new 3D SPH scheme has been carried out using the message passing interface (MPI) standard, together with a dynamic load balancing strategy to improve the computational efficiency of the scheme. Thus, simulations involving millions of particles can be run on modern massively parallel supercomputers, obtaining a very good performance, as confirmed by a speed-up analysis. The 3D applications consist of environmental flow problems, such as dam-break flows and impact flows against a wall. The numerical solutions obtained with our new 3D SPH code have been compared with either experimental results or with other numerical reference solutions, obtaining in all cases a very satisfactory agreement.     

Specifying and controlling multi-channel web interfaces for enterprise applications

When building enterprise applications that need to be accessed through a variety of client devices, developers usually strive to implement most of the business logic device-independently while using a web browser to display the user interface. However, when those web-based front-ends shall be rendered on different devices, their differing I/O capabilities may require device-specific interaction patterns that still need to be specified and implemented efficiently. We present an approach for specifying the dialog flows in multi-channel web interfaces with very low redundancy and introduce a framework that controls web interfaces’ device-specific dialog flows according to those specifications, while keeping the enterprise application logic completely device-independent.     

A 2D unstructured multi-material Cell-Centered Arbitrary Lagrangian–Eulerian (CCALE) scheme using MOF interface reconstruction

We present an original Cell-Centered Arbitrary Lagrangian–Eulerian (CCALE) strategy using the Moment Of Fluid (MOF) interface reconstruction devoted to the numerical simulation of two-dimensional multi-material compressible fluid flows on general unstructured grids. Our methodology is assessed through several demanding two-dimensional tests and comparison with Volume Of Fluid (VOF) interface reconstruction. The corresponding numerical results provide a clear evidence of the robustness and the accuracy of this new scheme.     

基于混合驱动的柔顺性触觉接口装置

针对目前柔顺性触觉接口设备存在的容易失真的问题,提出了一种基于电机和磁流变液混合驱动的柔顺性触觉接口装置。在简单介绍了磁流变液的基础上,讨论了基于磁流变液被动驱动器的结构和原理,该驱动器采用多转子设计思路以增大力输出范围。利用磁流变液驱动器能够模拟肌体组织的黏滞性,电机能够模拟肌体组织的弹性,将驱动器与电机串联实现肌体组织的柔顺性再现,同时利用电机补偿驱动器非有益阻尼力,增强装置的保真效果。在此基础上设计了柔顺性触觉接口装置,对装置模拟自由空间、不同的变形程度柔顺性物体受力进行了介绍,分析了装置的控制方法,最后加工了触觉装置原型,开展了不同柔顺度虚拟肌体组织柔顺性再现实验,实验结果验证了所设计装置及控制方法的有效性。     

Treatment of uncertain material interfaces in compressible flows

To treat uncertain interface position is an important issue for complex applications. In this paper, we address the characterization of randomly perturbed interfaces between fluids thanks to stochastic modeling and uncertainty quantification through the 2D Euler system. The perturbed interface is modeled as a random field and represented by a Karhunen–Loève expansion. The stochastic 2D Euler system is solved applying Polynomial Chaos theory through the Intrusive Polynomial Moment Method (IPMM). This stochastic resolution method is fully explained and studied (theoretically and numerically). Stochastic Richtmyer–Meshkov unstable flows are solved and presented for several configurations of the uncertain interface (different rugosities) between the fluids. The probability density functions of the mass density of the fluid in the vicinity of the interface are computed built and compared for the different simulations: the system exhibits strong sensitivity with respect to the stochastic initially leading modes.     

An ISPH scheme for numerical simulation of multiphase flows with complex interfaces and high density ratios

Multiphase problems with high density ratios and complex interfaces deal with numerical instabilities and require accurate considerations for capturing the multiphase interfaces. An Incompressible Smoothed Particle Hydrodynamics (ISPH) scheme is presented to simulate such problems. In order to keep the present scheme simple and stable, well-established formulations are used for discretizing the spatial derivatives and a repulsive force is applied at the multiphase interface between particles of different fluids to maintain the interface sharpness. Special considerations are included to overcome the difficulties to model severe physical discontinuities at the interface and surface tension effects are taken into account. Different particle shifting schemes are also tested for a range of problems. Several two phase flows are investigated and the presented scheme is validated against both analytical and numerical solutions. A detailed study is also carried out on the influence of the repulsive force in an ISPH scheme showing that this simple treatment efficiently enhances the interface capturing features. The comparisons indicate that the proposed scheme is robust and capable of simulating a wide range of multiphase problems with complex interfaces including low to high ratios for density and viscosity.     

Multi-material ALE computation in inertial confinement fusion code CHIC

We present a complete Arbitrary Lagrangian Eulerian (ALE) strategy devoted to the computation of multi-material fluid flows using the volume of fluid (VOF) interface reconstruction method applied to plasma physics and inertial confinement fusion (ICF). The interface tracking is coupled to a two-temperature hydrodynamics model in the framework of a collocated Lagrangian scheme. Some numerical examples, such hot shock tube problem and ICF-like problem, provide a clear evidence of the robustness and the accuracy of the method.     

A simple implementation of the immersed interface methods for Stokes flows with singular forces

In this paper, we introduce a simple version of the immersed interface method (IIM) for Stokes flows with singular forces along an interface. The numerical method is based on applying the Taylor’s expansions along the normal direction and incorporating the solution and its normal derivative jumps into the finite difference approximations. The fluid variables are solved in a staggered grid, and a new accurate interpolating scheme for the non-smooth velocity has been developed. The numerical results show that the scheme is second-order accurate.     

A numerical method for the simulation of free surface flows with surface tension

A numerical model for the simulation of three-dimensional liquid–gas flows with free surfaces and surface tension is presented. The incompressible Navier–Stokes equations are assumed to hold in the liquid domain, while the gas pressure is assumed to be constant in each connected component of the gas domain and to follow the ideal gas law. The surface tension effects are imposed as a normal force on the interface.

An implicit splitting scheme is used to decouple the physical phenomena. Given the curvature of the liquid–gas interface, the method described in [Caboussat A, Picasso M, Rappaz J. Numerical simulation of free surface incompressible liquid flows surrounded by compressible gas. J Comput Phys 2005;203(2):626–49] is used to track the liquid domain and compute the velocity and pressure in the liquid and the pressure in the gas domain. Then the surface tension effects are added. A variational method for the computation of the curvature is presented by smoothing the characteristic function of the liquid domain and using a finite element unstructured mesh.

The model is validated and numerical results in two and three space dimensions are presented for bubbles and/or droplets flows.     

Study on a grid interface algorithm for solutions of incompressible Navier–Stokes equations

Grid interface treatment is a crucial issue in solving unsteady, three-dimensional, incompressible Navier–Stokes equations by domain decomposition methods. Recently, a mass flux based interpolation (MFBI) interface algorithm was proposed for Chimera grids [Tang HS, Jones SC, Sotiropoulos F. An overset grid method for 3D unsteady incompressible flows. J Comput Phys, 2003;191:567–600] and it has been successfully applied to a variety of flows. MFBI determines velocity and pressure at grid interfaces by mass conservation and interpolation, and it is easy to implement. Compared with the commonly used standard interpolation, which directly interpolates velocity as well as pressure, the proposed interface algorithm gives fewer solution oscillations and faster convergence rates. This paper makes a study on MFBI. Starting with discussions about grid connectivity, it is shown that MFBI is second-order accurate for mass flux across grid interface. It is also derived that the scheme provides second-order accuracy for momentum flux. In addition, another version of MFBI is presented. At last, numerical examples are presented to demonstrate that MFBI honors mass flux balance at grid interfaces and it leads to second-order accurate solutions.