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.     

A simple high‐resolution advection scheme

A simple, robust, mass‐conserving numerical scheme for solving the linear advection equation is described. The scheme can estimate peak solution values accurately even in regions where spatial gradients are high. Such situations present a severe challenge to classical numerical algorithms. Attention is restricted to the case of pure advection in one and two dimensions since this is where past numerical problems have arisen. The authors' scheme is of the Godunov type and is second‐order in space and time. The required cell interface fluxes are obtained by MUSCL interpolation and the exact solution of a degenerate Riemann problem. Second‐order accuracy in time is achieved via a Runge–Kutta predictor–corrector sequence. The scheme is explicit and expressed in finite volume form for ease of implementation on a boundary‐conforming grid. Benchmark test problems in one and two dimensions are used to illustrate the high‐spatial accuracy of the method and its applicability to non‐uniform grids. Copyright © 2007 John Wiley & Sons, Ltd.     

A structured but non‐uniform Cartesian grid‐based model for the shallow water equations

In large‐scale shallow flow simulations, local high‐resolution predictions are often required in order to reduce the computational cost without losing the accuracy of the solution. This is normally achieved by solving the governing equations on grids refined only to those areas of interest. Grids with varying resolution can be generated by different approaches, e.g. nesting methods, patching algorithms and adaptive unstructured or quadtree gridding techniques. This work presents a new structured but non‐uniform Cartesian grid system as an alternative to the existing approaches to provide local high‐resolution mesh. On generating a structured but non‐uniform Cartesian grid, the whole computational domain is first discretized using a coarse background grid. Local refinement is then achieved by directly allocating a specific subdivision level to each background grid cell. The neighbour information is specified by simple mathematical relationships and no explicit storage is needed. Hence, the structured property of the uniform grid is maintained. After employing some simple interpolation formulae, the governing shallow water equations are solved using a second‐order finite volume Godunov‐type scheme in a similar way as that on a uniform grid. Copyright © 2010 John Wiley & Sons, Ltd.     

A new volume of fluid method in three dimensions—Part I: Multidimensional advection method with face‐matched flux polyhedra

A multidimensional advection scheme in 3D based on the use of face‐matched flux polyhedra to integrate the volume fraction evolution equation is proposed. The algorithm tends to reduce the formation of ‘over/undershoots’ by alleviating the over/underlapping of flux polyhedra, thus diminishing the need to use local redistribution algorithms. The accuracy and efficiency of the proposed advection algorithm, which are analyzed using different tests with prescribed velocity field, compare well with other multidimensional advection methods proposed recently. The algorithm is also applied, in combination with a Navier–Stokes solver, to reproduce the impact of a water droplet falling through air on a pool of deep water. The interfacial curvature is calculated using a height‐function technique with adaptive stencil adjustment, which provides improved accuracy in regions of low grid resolution. The comparison of the numerical results with experimental results shows a good degree of agreement. Copyright © 2008 John Wiley & Sons, Ltd.     

Adaptive Runge–Kutta discontinuous Galerkin method for complex geometry problems on Cartesian grid

A Cartesian grid method using immersed boundary technique to simulate the impact of body in fluid has become an important research topic in computational fluid dynamics because of its simplification, automation of grid generation, and accuracy of results. In the frame of Cartesian grid, one often uses finite volume method with second order accuracy or finite difference method. In this paper, an h‐adaptive Runge–Kutta discontinuous Galerkin (RKDG) method on Cartesian grid with ghost cell immersed boundary method for arbitrarily complex geometries is developed. A ghost cell immersed boundary treatment with the modification of normal velocity is presented. The method is validated versus well documented test problems involving both steady and unsteady compressible flows through complex bodies over a wide range of Mach numbers. The numerical results show that the present boundary treatment to some extent reduces the error of entropy and demonstrate the efficiency, robustness, and versatility of the proposed approach. Copyright © 2013 John Wiley & Sons, Ltd.     

A pressure‐based unstructured grid method for all‐speed flows

An all‐speed algorithm based on the SIMPLE pressure‐correction scheme and the ‘retarded‐density’ approach has been formulated and implemented within an unstructured grid, finite volume (FV) scheme for both incompressible and compressible flows, the latter involving interaction of shock waves. The collocated storage arrangement for all variables is adopted, and the checkerboard oscillations are eliminated by using a pressure‐weighted interpolation method, similar to that of Rhie and Chow [Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA Journal 1983; 21 : 1525]. The solution accuracy is greatly enhanced when a higher‐order convection scheme combined with adaptive mesh refinement (AMR) are used. Copyright © 2000 John Wiley & Sons, Ltd.     

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.     

A discontinuous Galerkin‐based sharp‐interface method to simulate three‐dimensional compressible two‐phase flow

A numerical method for the simulation of compressible two‐phase flows is presented in this paper. The sharp‐interface approach consists of several components: a discontinuous Galerkin solver for compressible fluid flow, a level‐set tracking algorithm to follow the movement of the interface and a coupling of both by a ghost‐fluid approach with use of a local Riemann solver at the interface. There are several novel techniques used: the discontinuous Galerkin scheme allows locally a subcell resolution to enhance the interface resolution and an interior finite volume Total Variation Diminishing (TVD) approximation at the interface. The level‐set equation is solved by the same discontinuous Galerkin scheme. To obtain a very good approximation of the interface curvature, the accuracy of the level‐set field is improved and smoothed by an additional P_NP_M‐reconstruction. The capabilities of the method for the simulation of compressible two‐phase flow are demonstrated for a droplet at equilibrium, an oscillating ellipsoidal droplet, and a shock‐droplet interaction problem at Mach 3. Copyright © 2015 John Wiley & Sons, Ltd.     

The intersection marker method for 3D interface tracking of deformable surfaces in finite volumes

Currently, the majority of computational fluid dynamics (CFD) codes use the finite volume method to spatially discretise the computational domain, sometimes as an array of cubic control volumes. The Finite volume method works well with single‐phase flow simulations, but two‐phase flow simulations are more challenging because of the need to track the surface interface traversing and deforming within the 3D grid. Surface area and volume fraction details of each interface cell must be accurately accounted for, in order to calculate for the momentum exchange and rates of heat and mass transfer across the interface. To attain a higher accuracy in two‐phase flow CFD calculations, the intersection marker (ISM) method is developed. The ISM method is a hybrid Lagrangian–Eulerian front‐tracking algorithm that can model an arbitrary 3D surface within an array of cubic control volumes. The ISM method has a cell‐by‐cell remeshing capability that is volume conservative and is suitable for the tracking of complex interface deformation in transient two‐phase CFD simulations. Copyright © 2015 John Wiley & Sons, Ltd.     

A fast nested multi‐grid viscous flow solver for adaptive Cartesian/Quad grids

A nested multi‐grid solution algorithm has been developed for an adaptive Cartesian/Quad grid viscous flow solver. Body‐fitted adaptive Quad (quadrilateral) grids are generated around solid bodies through ‘surface extrusion’. The Quad grids are then overlapped with an adaptive Cartesian grid. Quadtree data structures are employed to record both the Quad and Cartesian grids. The Cartesian grid is generated through recursive sub‐division of a single root, whereas the Quad grids start from multiple roots—a forest of Quadtrees, representing the coarsest possible Quad grids. Cell‐cutting is performed at the Cartesian/Quad grid interface to merge the Cartesian and Quad grids into a single unstructured grid with arbitrary cell topologies (i.e., arbitrary polygons). Because of the hierarchical nature of the data structure, many levels of coarse grids have already been built in. The coarsening of the unstructured grid is based on the Quadtree data structure through reverse tree traversal. Issues arising from grid coarsening are discussed and solutions are developed. The flow solver is based on a cell‐centered finite volume discretization, Roe's flux splitting, a least‐squares linear reconstruction, and a differentiable limiter developed by Venkatakrishnan in a modified form. A local time stepping scheme is used to handle very small cut cells produced in cell‐cutting. Several cycling strategies, such as the saw‐tooth, W‐ and V‐cycles, have been studies. The V‐cycle has been found to be the most efficient. In general, the multi‐grid solution algorithm has been shown to greatly speed up convergence to steady state—by one to two orders. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical simulation of 3D free surface flows,with multiple incompressible immiscible phases. Applications to impulse waves

A numerical method for the solution to the density‐dependent incompressible Navier–Stokes equations modeling the flow of N immiscible incompressible liquid phases with a free surface is proposed. It allows to model the flow of an arbitrary number of liquid phases together with an additional vacuum phase separated with a free surface. It is based on a volume‐of‐fluid approach involving N indicator functions (one per phase, identified by its density) that guarantees mass conservation within each phase. An additional indicator function for the whole liquid domain allows to treat boundary conditions at the interface between the liquid domain and a vacuum. The system of partial differential equations is solved by implicit operator splitting at each time step: first, transport equations are solved by a forward characteristics method on a fine Cartesian grid to predict the new location of each liquid phase; second, a generalized Stokes problem with a density‐dependent viscosity is solved with a FEM on a coarser mesh of the liquid domain. A novel algorithm ensuring the maximum principle and limiting the numerical diffusion for the transport of the N phases is validated on benchmark flows. Then, we focus on a novel application and compare the numerical and physical simulations of impulse waves, that is, waves generated at the free surface of a water basin initially at rest after the impact of a denser phase. A particularly useful application in hydraulic engineering is to predict the effects of a landslide‐generated impulse wave in a reservoir. Copyright © 2014 John Wiley & Sons, Ltd.     

Discretization of transport equations on 2D Cartesian unstructured grids using data from remote cells for the convection terms

This paper presents a new finite volume discretization methodology for the solution of transport equations on locally refined or unstructured Cartesian meshes. The implementation of the cell‐face values of the dependent variables enables the employment of data from remote cells and thus the use of higher‐order differencing schemes. It also results in simple and flux‐conservative multiple‐scale stencils for the discretization of the governing equations. The latter are finally cast into a generalized form that does not depend on the local mesh structure. The performance of the numerical model is demonstrated on some classical 2D problems using various gridding techniques and a bounded second‐order upwind scheme. A stable and efficient behaviour of the algorithm is observed in all test cases. The results indicate that the combination in the present model of both local grid refinement and second‐order discretization can produce substantially more accurate solutions than each of the above techniques alone, for the same computational effort. The method is also applicable to turbulent flows and can be easily extended to three‐dimensions. Copyright © 2003 John Wiley & Sons, Ltd.     

Comparison of adjoint‐based and feature‐based grid adaptation for functional outputs

Adjoint‐based and feature‐based grid adaptive strategies are compared for their robustness and effectiveness in improving the accuracy of functional outputs such as lift and drag coefficients. The output‐based adjoint approach strives to improve the adjoint error estimates that relate the local residual errors to the global error in an output function via adjoint variables as weight functions. A conservative adaptive indicator that takes into account the residual errors in both the primal (flow) and dual (adjoint) solutions is implemented for the adjoint approach. The physics‐based feature approach strives to identify and resolve significant features of the flow to improve functional accuracy. Adaptive indicators that represent expansions and compressions in the flow direction and gradients normal to the flow direction are implemented for the feature approach. The adaptive approaches are compared for functional outputs of three‐dimensional arbitrary Mach number flow simulations on mixed‐element unstructured meshes. Grid adaptation is performed with h‐refinement and results are presented for inviscid, laminar and turbulent flows. Copyright © 2006 John Wiley & Sons, Ltd.     

A multigrid adaptive mesh refinement strategy for 3D aerodynamic design

Presently, improving the accuracy and reducing computational costs are still two major CFD objectives often considered incompatible. This paper proposes to solve this dilemma by developing an adaptive mesh refinement method in order to integrate the 3D Euler and Navier–Stokes equations on structured meshes, where a local multigrid method is used to accelerate convergence for steady compressible flows. The time integration method is a LU‐SGS method (AIAA J 1988; 26: 1025–1026) associated with a spatial Jameson‐type scheme (Numerical solutions of the Euler equations by finite volume methods using Runge–Kutta time‐stepping schemes. AIAA Paper, 81‐1259, 1981). Computations of turbulent flows are handled by the standard k–ω model of Wilcox (AIAA J 1994; 32: 247–255). A coarse grid correction, based on composite residuals, has been devised in order to enforce the coupling between the different grid levels and to accelerate the convergence. The efficiency of the method is evaluated on standard 2D and 3D aerodynamic configurations. Copyright © 2004 John Wiley & Sons, Ltd.     

4D variational data assimilation for locally nested models: Complementary theoretical aspects and application to a 2D shallow water model

We consider the application of a four‐dimensional variational data assimilation method to a numerical model, which employs local mesh refinement to improve its solution. We focus on structured meshes where a high‐resolution grid is embedded in a coarser resolution one, which covers the entire domain. The formulation of the nested variational data assimilation algorithm was derived in a preliminary work (Int. J. Numer. Meth. Fluids 2008; under review). We are interested here in complementary theoretical aspects. We present first a model for the multi‐grid background error covariance matrix. Then, we propose a variant of our algorithms based on the addition of control variables in the inter‐grid transfers in order to allow for a reduction of the errors linked to the interactions between the grids. These formulations are illustrated and discussed in the test case experiment of a 2D shallow water model. Copyright © 2010 John Wiley & Sons, Ltd.     

A pressure correction‐volume of fluid method for simulation of two‐phase flows

A pressure correction method coupled with the volume of fluid (VOF) method is developed to simulate two‐phase flows. A volume fraction function is introduced in the VOF method and is governed by an advection equation. A modified monotone upwind scheme for a conservation law (modified MUSCL) is used to solve the solution of the advection equation. To keep the initial sharpness of an interface, a slope modification scheme is introduced. The continuum surface tension (CST) model is used to calculate the surface tension force. Three schemes, central‐upwind, Parker–Youngs, and mixed schemes, are introduced to compute the interface normal vector and the gradient of the volume fraction function. Moreover, a height function technique is applied to compute the local curvature of the interface. Several basic test problems are performed to check the order of accuracy of the present numerical schemes for computing the interface normal vector and the gradient of the volume fraction function. Three physical problems, two‐dimensional broken dam problem, static drop, and spurious currents, and three‐dimensional rising bubble, are performed to demonstrate the efficiency and accuracy of the pressure correction method. Copyright © 2010 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.     

Multi‐size‐mesh multi‐time‐step algorithm for noise computation on curvilinear meshes

Aeroacoustic problems are often multi‐scale and a zonal refinement technique is thus desirable to reduce computational effort while preserving low dissipation and low dispersion errors from the numerical scheme. For that purpose, the multi‐size‐mesh multi‐time‐step algorithm of Tam and Kurbatskii [AIAA Journal, 2000, 38 (8), p. 1331–1339] allows changes by a factor of two between adjacent blocks, accompanied by a doubling in the time step. This local time stepping avoids wasting calculation time, which would result from imposing a unique time step dictated by the smallest grid size for explicit time marching. In the present study, the multi‐size‐mesh multi‐time‐step method is extended to general curvilinear grids by using a suitable coordinate transformation and by performing the necessary interpolations directly in the physical space due to multidimensional interpolations combining order constraints and optimization in the wave number space. A particular attention is paid to the properties of the Adams–Bashforth schemes used for time marching. The optimization of the coefficients by minimizing an error in the wave number space rather than satisfying a formal order is shown to be inefficient for Adams–Bashforth schemes. The accuracy of the extended multi‐size‐mesh multi‐time‐step algorithm is first demonstrated for acoustic propagation on a sinusoidal grid and for a computation of laminar trailing edge noise. In the latter test‐case, the mesh doubling is close to the airfoil and the vortical structures are crossing the doubling interface without affecting the quality of the radiated field. The applicability of the algorithm in three dimensions is eventually demonstrated by computing tonal noise from a moderate Reynolds number flow over an airfoil. Copyright © 2013 John Wiley & Sons, Ltd.     

A projection scheme for incompressible multiphase flow using adaptive Eulerian grid: 3D validation

A three‐dimensional finite element method for incompressible multiphase flows with capillary interfaces is developed based on a (formally) second‐order projection scheme. The discretization is on a fixed (Eulerian) reference grid with an edge‐based local h‐refinement in the neighbourhood of the interfaces. The fluid phases are identified and advected using the level‐set function. The reference grid is then temporarily reconnected around the interface to maintain optimal interpolations accounting for the singularities of the primary variables. Using a time splitting procedure, the convection substep is integrated with an explicit scheme. The remaining generalized Stokes problem is solved by means of a pressure‐stabilized projection. This method is simple and efficient, as demonstrated by a wide range of difficult free‐surface validation problems, considered in the paper. Copyright © 2005 John Wiley & Sons, Ltd.