Effects of element order and interface reconstruction in FEM/volume‐of‐fluid incompressible flow simulation

In previous studies, the moment‐of‐fluid interface reconstruction method showed dramatic accuracy improvements in static and pure advection tests over existing methods, but this did not translate into an equivalent improvement in volume‐tracked multimaterial incompressible flow simulation using low‐order finite elements. In this work, the combined effects of the spatial discretization and interface reconstruction in flow simulation are examined. The mixed finite element pairs, Q₁Q₀ (with pressure stabilization) and Q₂P _{? 1} are compared. Material order‐dependent and material order‐independent first and second‐order accurate interface reconstruction methods are used. The Q₂P _{? 1} elements show significant improvements in computed flow solution accuracy for single material flows but show reduced convergence using element‐average piecewise constant density and viscosity in volume‐tracked simulations. In general, a refined Q₁Q₀ grid, with better material interface resolution, provided an accuracy similar to the Q₂P _{? 1} element grid with a comparable number of degrees of freedom. Moment‐of‐fluid shows more benefit from the higher‐order accurate flow simulation than the LVIRA, Youngs', and power diagram interface reconstruction methods, especially on unstructured grids, but does not recover the dramatic accuracy improvements it has shown in advection tests. Published 2012. This article is a US Government work and is in the public domain in the USA.     

Incompressible multiphase flow and encapsulation simulations using the moment‐of‐fluid method

A moment‐of‐fluid method is presented for computing solutions to incompressible multiphase flows in which the number of materials can be greater than two. In this work, the multimaterial moment‐of‐fluid interface representation technique is applied to simulating surface tension effects at points where three materials meet. The advection terms are solved using a directionally split cell integrated semi‐Lagrangian algorithm, and the projection method is used to evaluate the pressure gradient force term. The underlying computational grid is a dynamic block‐structured adaptive grid. The new method is applied to multiphase problems illustrating contact‐line dynamics, triple junctions, and encapsulation in order to demonstrate its capabilities. Examples are given in two‐dimensional, three‐dimensional axisymmetric (R–Z), and three‐dimensional (X–Y–Z) coordinate systems. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical errors of the volume‐of‐fluid interface tracking algorithm

One of the important limitations of the interface tracking algorithms is that they can be used only as long as the local computational grid density allows surface tracking. In a dispersed flow, where the dimensions of the particular fluid parts are comparable or smaller than the grid spacing, several numerical and reconstruction errors become considerable. In this paper the analysis of the interface tracking errors is performed for the volume‐of‐fluid method with the least squares volume of fluid interface reconstruction algorithm. A few simple two‐fluid benchmarks are proposed for the investigation of the interface tracking grid dependence. The expression based on the gradient of the volume fraction variable is introduced for the estimation of the reconstruction correctness and can be used for the activation of an adaptive mesh refinement algorithm. Copyright © 2002 John Wiley & Sons, Ltd.     

An adaptive cut‐cell method for environmental fluid mechanics

In this work we present a numerical method for solving the incompressible Navier–Stokes equations in an environmental fluid mechanics context. The method is designed for the study of environmental flows that are multiscale, incompressible, variable‐density, and within arbitrarily complex and possibly anisotropic domains. The method is new because in this context we couple the embedded‐boundary (or cut‐cell) method for complex geometry with block‐structured adaptive mesh refinement (AMR) while maintaining conservation and second‐order accuracy. The accurate simulation of variable‐density fluids necessitates special care in formulating projection methods. This variable‐density formulation is well known for incompressible flows in unit‐aspect ratio domains, without AMR, and without complex geometry, but here we carefully present a new method that addresses the intersection of these issues. The methodology is based on a second‐order‐accurate projection method with high‐order‐accurate Godunov finite‐differencing, including slope limiting and a stable differencing of the nonlinear convection terms. The finite‐volume AMR discretizations are based on two‐way flux matching at refinement boundaries to obtain a conservative method that is second‐order accurate in solution error. The control volumes are formed by the intersection of the irregular embedded boundary with Cartesian grid cells. Unlike typical discretization methods, these control volumes naturally fit within parallelizable, disjoint‐block data structures, and permit dynamic AMR coarsening and refinement as the simulation progresses. We present two‐ and three‐dimensional numerical examples to illustrate the accuracy of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical simulation of free‐surface flow using the level‐set method with global mass correction

A new numerical method that couples the incompressible Navier–Stokes equations with the global mass correction level‐set method for simulating fluid problems with free surfaces and interfaces is presented in this paper. The finite volume method is used to discretize Navier–Stokes equations with the two‐step projection method on a staggered Cartesian grid. The free‐surface flow problem is solved on a fixed grid in which the free surface is captured by the zero level set. Mass conservation is improved significantly by applying a global mass correction scheme, in a novel combination with third‐order essentially non‐oscillatory schemes and a five stage Runge–Kutta method, to accomplish advection and re‐distancing of the level‐set function. The coupled solver is applied to simulate interface change and flow field in four benchmark test cases: (1) shear flow; (2) dam break; (3) travelling and reflection of solitary wave and (4) solitary wave over a submerged object. The computational results are in excellent agreement with theoretical predictions, experimental data and previous numerical simulations using a RANS‐VOF method. The simulations reveal some interesting free‐surface phenomena such as the free‐surface vortices, air entrapment and wave deformation over a submerged object. Copyright © 2009 John Wiley & Sons, Ltd.     

A three-dimensional fictitious domain method for incompressible fluid flow problems

A new Galerkin finite element method for the solution of the Navier–Stokes equations in enclosures containing internal parts which may be moving is presented. Dubbed the virtual finite element method, it is based upon optimization techniques and belongs to the class of fictitious domain methods. Only one volumetric mesh representing the enclosure without its internal parts needs to be generated. These are rather discretized using control points on which kinematic constraints are enforced and introduced into the mathematical formulation by means of Lagrange multipliers. Consequently, the meshing of the computational domain is much easier than with classical finite element approaches. First, the methodology will be presented in detail. It will then be validated in the case of the two-dimensional Couette cylinder problem for which an analytical solution is available. Finally, the three-dimensional fluid flow inside a mechanically agitated vessel will be investigated. The accuracy of the numerical results will be assessed through a comparison with experimental data and results obtained with a standard finite element method. © 1997 John Wiley & Sons, Ltd.     

Modelling of two‐dimensional laminar flow using finite element method

In this paper, a Galerkin weighted residual finite element numerical solution method, with velocity material time derivative discretisation, is applied to solve for a classical fluid mechanics system of partial differential equations modelling two‐dimensional stationary incompressible Newtonian fluid flow. Classical examples of driven cavity laminar flow and laminar flow past a cylinder are presented. Numerical results are compared with data found in the literature. Copyright © 1999 John Wiley & Sons, Ltd.     

黏性不可压缩流体流动前沿的数值模拟

提出了模拟注射成型中黏性、不可压缩流体流动前沿的新方法. 将Hele-Shaw流动应用于非 等温条件下的黏性、不可压缩流体，建立了流动分析模型，用充填因子的输运方程描述流动 前沿. 应用高阶Taylor展开式计算每一时间步长的充填因子，用Galerkin方法导出了计算 充填因子各阶导数的递推公式. 给出了时间增量的选取方法，证明了它的稳定性. 针对Han 设计的试验模具，用相同的材料及工艺条件模拟充填过程，比较了传统方法和该方法的模 拟结果与实验结果的差异. 算例分析表明，该方法可以有效地提高注射成型中流动前沿的 模拟精度和计算效率.     

A second‐order time accurate finite element method for quasi‐incompressible viscous flows

A finite element method for quasi‐incompressible viscous flows is presented. An equation for pressure is derived from a second‐order time accurate Taylor–Galerkin procedure that combines the mass and the momentum conservation laws. At each time step, once the pressure has been determined, the velocity field is computed solving discretized equations obtained from another second‐order time accurate scheme and a least‐squares minimization of spatial momentum residuals. The terms that stabilize the finite element method (controlling wiggles and circumventing the Babuska–Brezzi condition) arise naturally from the process, rather than being introduced a priori in the variational formulation. A comparison between the present second‐order accurate method and our previous first‐order accurate formulation is shown. The method is also demonstrated in the computation of the leaky‐lid driven cavity flow and in the simulation of a crossflow past a circular cylinder. In both cases, good agreement with previously published experimental and computational results has been obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

Vortex‐in‐cell method combined with a boundary element method for incompressible viscous flow analysis

In this study, an immersed boundary vortex‐in‐cell (VIC) method for simulating the incompressible flow external to two‐dimensional and three‐dimensional bodies is presented. The vorticity transport equation, which is the governing equation of the VIC method, is represented in a Lagrangian form and solved by the vortex blob representation of the flow field. In the present scheme, the treatment of convection and diffusion is based on the classical fractional step algorithm. The rotational component of the velocity is obtained by solving Poisson's equation using an FFT method on a regular Cartesian grid, and the solenoidal component is determined from solving an integral equation using the panel method for the convection term, and the diffusion term is implemented by a particle strength exchange scheme. Both the no‐slip and no‐through flow conditions associated with the surface boundary condition are satisfied by diffusing vortex sheet and distributing singularities on the body, respectively. The present method is distinguished from other methods by the use of the panel method for the enforcement of the no‐through flow condition. The panel method completes making use of the immersed boundary nature inherent in the VIC method and can be also adopted for the calculation of the pressure field. The overall process is parallelized using message passing interface to manage the extensive computational load in the three‐dimensional flow simulations. Copyright © 2011 John Wiley & Sons, Ltd.     

A numerical study of projection algorithms in the finite element simulation of three‐dimensional viscous incompressible flow

In this work a comparative study of two versions of the projection algorithm used either for time integration or as an iterative method to solve the three‐dimensional incompressible Navier–Stokes equations is presented. It is also shown that these projection algorithms combined with the finite element method are particularly suited for the treatment of outflow boundary conditions in the context of external flows. This assertion is illustrated by means of some numerical examples in which five types of boundary conditions are compared. The scheme is applied to simulate the flow past a cylinder clamped on two fixed parallel solid walls. Comparison with experimental data available for this problem shows good agreement of the velocity and pressure fields, both computed with continuous piecewise linear elements. Copyright © 1999 John Wiley & Sons, Ltd.     

Adaptive subdivision piecewise linear interface calculation (ASPLIC) for 2D multi‐material hydrodynamic simulation codes

In this work, we propose an adaptive subdivision piecewise linear interface calculation (PLIC) for 2D multimaterial hydrodynamic simulation codes. Classical volume‐of‐fluid PLIC technique uses one line segment and one given normal to separate two materials. Unfortunately, these paradigms are not sufficient when filaments occur, leading to the creation of flotsam and jetsam. We propose to detect such situations and to split the computational mixed cell into reconstruction subzones. Within these subzones, one computes a so‐called subgradient using an incomplete stencil of neighbors, and the material is distributed in these subzones. Given subzone volume fraction and the subgradient, one computes one line segment using classical PLIC method, leading to a modified PLIC method for subscale material entity. The subdivision procedure relies on a splitting point, which is chosen as a specific information about the relative location of the filament in the cell, leading to an adaptive subdivision for PLIC reconstructions. Numerical tests are carried out in a 2D Lagrange + Remap multimaterial hydrodynamics Eulerian code. Static and dynamic filaments and fragments are simulated in advection or stretched in vortex‐like motion. The full hydrodynamics equations are solved on a more realistic test (shock‐bubble impact). Results show that our approach supplements classical PLIC method for situations when filaments and fragments occur. Copyright © 2014 John Wiley & Sons, Ltd.     

Fringe element reconstruction for front tracking for three‐dimensional incompressible flow analysis

Fringe element reconstruction technique for tracking the free surface in three‐dimensional incompressible flow analysis was developed. The flow field was calculated by the mixed formulation based on a four‐node tetrahedral element with a bubble function at the centroid (P1+/P1). Since an Eulerian approach was employed in this study, the flow front interface was advected by the flow through a fixed mesh. For accurate modelling of interfacial movement, a fringe element reconstruction method developed can provide not only an accurate treatment of material discontinuity but also surface tension across the interface. The effect of surface tension was modelled by imposing tensile stress directly on the constructed surface elements at the flow front interface. To verify the numerical approach developed, the developed algorithm was applied to two examples whose solutions are available in references. Good agreement was obtained between the simulation results and these solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

New stabilized finite element method for time‐dependent incompressible flow problems

A new stabilized finite element method is considered for the time‐dependent Stokes problem, based on the lowest‐order P₁?P₀ and Q₁?P₀ elements that do not satisfy the discrete inf–sup condition. The new stabilized method is characterized by the features that it does not require approximation of the pressure derivatives, specification of mesh‐dependent parameters and edge‐based data structures, always leads to symmetric linear systems and hence can be applied to existing codes with a little additional effort. The stability of the method is derived under some regularity assumptions. Error estimates for the approximate velocity and pressure are obtained by applying the technique of the Galerkin finite element method. Some numerical results are also given, which show that the new stabilized method is highly efficient for the time‐dependent Stokes problem. Copyright © 2009 John Wiley & Sons, Ltd.     

A PLIC volume tracking method for the simulation of two‐fluid flows

Numerical methodologies for computer simulations of two‐fluid flows are presented. These methodologies fall into the category of volume tracking methods with piecewise‐linear interface calculation (PLIC). The scope of this work is limited to laminar flows of immiscible, non‐reacting, incompressible Newtonian fluids, without phase change, in planar two‐dimensional geometries. The following novel or enhanced procedures are proposed: a parallelogram scheme for multidimensional advection of the volume‐fraction field; a circle‐fit technique for the orientation of the interface segments and the calculation of curvature; a novel contact angle treatment; and a staggered formulation for volumetric body forces that can accurately balance pressure forces in the vicinity of the interface. In addition, surface‐tension‐derived and hydrostatic‐derived pressure adjustments are introduced as a means of accurately calculating pressure forces in cells that contain the interface, so as to minimize the non‐physical flows that afflict many available volume tracking methods. The proposed method is validated using four test problems that involve simulations of pure advection, a static drop, an oscillating bubble, and a static meniscus. Copyright © 2006 John Wiley & Sons, Ltd.     

A Hermite finite element method for incompressible fluid flow

We describe some Hermite stream function and velocity finite elements and a divergence‐free finite element method for the computation of incompressible flow. Divergence‐free velocity bases defined on (but not limited to) rectangles are presented, which produce pointwise divergence‐free flow fields (∇· u _h≡0). The discrete velocity satisfies a flow equation that does not involve pressure. The pressure can be recovered as a function of the velocity if needed. The method is formulated in primitive variables and applied to the stationary lid‐driven cavity and backward‐facing step test problems. Copyright © 2009 John Wiley & Sons, Ltd.     

Spurious velocities in the steady flow of an incompressible fluid subjected to external forces

We show that a non-physical velocity may appear in the numerical computation of the flow of an incompressible fluid subjected to external forces. A distorted mesh and the use of a numerical method which does not rigorously ensure the incompressibility condition turn out to be responsible for this phenomenon. We illustrate it with numerical examples and we propose a projection method which improves the results. © 1997 John Wiley & Sons, Ltd.     

Finite‐element simulation of incompressible fluid flow in an elastic vessel

Finite‐element simulation was performed to predict the incompressible Navier–Stokes flow in a domain, partly bounded by an elastic vessel, which is allowed to vary with time. Besides satisfying the physical conservation laws, both surface and the volume conservation laws are satisfied at the discrete level for ensuring the balance between physical and geometrical variables. Several problems which are amenable to analytical solutions were tested for validating the method. The simulated results are observed to agree favourably with analytical solutions. Having verified the applicability of the finite‐element code to problems involving moving grids, we consider an incompressible fluid flow bounded by rigid and elastic vessel walls. Our emphasis was placed on the validation of the formulation developed within the moving‐grid framework. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical simulation of flow‐induced noise using LES/SAS and Lighthill's acoustic analogy

We present a finite element (FE) formulation of Lighthill's acoustic analogy for the hybrid computation of noise generated by turbulent flows. In the present approach, the flow field is computed using large eddy simulation and scale adaptive simulation turbulence models. The acoustic propagation is obtained by solving the variational formulation of Lighthill's acoustic analogy with the FE method. In order to preserve the acoustic energy, we compute the inhomogeneous part of Lighthill's wave equation by applying the FE formulation on the fine flow grid. The resulting acoustic nodal loads are then conservatively interpolated to the coarser acoustic grid. Subsequently, the radiated acoustic field can be solved in both time and frequency domains. In the latter case, an enhanced perfectly matched layer technique is employed, allowing one to truncate the computational domain in the acoustic near field, without compromising the numerical solution. Our hybrid approach is validated by comparing the numerical results of the acoustic field induced by a corotating vortex pair with the corresponding analytical solution. To demonstrate the applicability of our scheme, we present full 3D numerical results for the computed acoustic field generated by the turbulent flow around square cylinder geometries. The sound pressure levels obtained compare well with measured values. Copyright © 2009 John Wiley & Sons, Ltd.