Volume-integrated Average Research Articles

A multi-moment finite volume formulation for the interaction between free surface flow and moving bodies with THINC method

A numerical model is developed for fluid-rigid body interaction involving free surfaces on moving unstructured grids using the Arbitrary-Lagrangian–Eulerian (ALE) formulation. The multi-moment finite volume method is used for the spatial discretization of Navier–Stokes equations in which volume integrated average (VIA) and point values (PVs) are defined as prognostic variables and updated by integral and differential form respectively. Given VIA and PV in each cell, a higher-order reconstruction function can be built in a compact stencil which improves the numerical accuracy and robustness on the unstructured grids. More importantly, it facilitates imposing a more accurate Dirichlet–Neumann condition for the partitioned interface between the fluid and solid region. The geometrical conservation law (GCL) is also formulated to govern the spatial volume element under an arbitrary mapping. In order to capture the free surfaces of multiphase flows, the volume-of-fluid (VOF) equation is solved by THINC/QQ (THINC method with quadratic surface representation and Gaussian quadrature) scheme in varying mesh, which achieves geometrical faithful interface representation, particularly for curved surfaces in a complex and moving domain. The present model is validated by some benchmark tests in two dimensions for fluid–structure interaction problems with free surfaces. The numerical results show closer agreement with the experimental data than the reference solution of other numerical models. It convinces that the present model provides accurate and robust solutions for the simulation of the complex interaction between the free surface flows and moving bodies.

A high-fidelity solver based on hybrid numerical methods on unstructured grids for incompressible multiphase flows

A high-fidelity solver is developed on the unstructured grids to simulate incompressible multiphase flows with free surface based on volume of fluid (VOF) method. In this framework, the velocity is defined by both volume integrated average (VIA) and point value (PV) as computational variables which are updated simultaneously by volume-average/point-value multi-moment (VPM) method at each time step. Compared with our previous work, a node-based finite element (FE) method is used instead of cell-centered finite volume (FV) method for the discretization of pressure Poisson equation, which provides a new pattern that can naturally treat with the velocity and pressure coupling. It does not require Rhie–Chow interpolation or the use of a Petrov–Galerkin finite element formulation with tunable parameters to circumvent the well-known instability conditions. Moreover, we propose a novel balanced-force formulation with FE discretization to accommodate the surface tension force, which effectively eliminates the large numerical errors associated with the non-orthogonality of unstructured grids. In VOF method, the volume fraction is defined as VIA to identify the moving interface which is updated by solving advection equation through the FV formulation. The resulting model that combines hybrid discretizations, rigorously preserves the mass and momentum conservation, substantially improves the numerical accuracy and robustness of pressure field, and significantly suppresses the spurious velocity to nearly machine error even for multiphase flows with large density ratios. Various numerical examples have been presented to demonstrate the excellent capabilities of present solver which can provide high-fidelity solutions for practical applications in presence of largely deformed interface and highly complex geometry.

High-order conservative and oscillation-suppressing transport on irregular hexagonal grids

A third-order numerical scheme was developed for 2D irregular hexagonal meshes for the advection problems in this study. The scheme is based on a multi-moment constrained finite-volume method (MCV) in Cartesian coordinates and entails the introduction of a general integration method over a hexagonal cell. Unlike in the conventional finite-volume method, various discrete moments, that is, point value and volume-integrated average, are adopted as computational constraints to achieve high-order computation. The high-order spatial reconstruction can therefore be built in a local space, which considerably reduces the stencil length. The numerical scheme is tested using various idealized experiments. Compared with the existing schemes, this scheme is demonstrated to be flexible for application in irregular hexagonal meshes without increasing cost or compromising on accuracy. The general integration formulation based on a third-order polynomial helps to expand the application to arbitrary hexagons that does not require the use of centroids as computational points or Voronoi tessellation. It is also convenient to define the orthogonal wind components in the Cartesian system to directly drive the atmospheric transport.

A conservative solver for surface-tension-driven multiphase flows on collocated unstructured grids

We developed a novel conservative model for simulating incompressible multiphase flows on collocated unstructured grids. In conventional schemes, the divergence-free condition is enforced by the numerical approximations based on the least-square approach which inherently cannot guarantee its conservative property for volume integrated average (VIA) of velocity. In this study, point values (PVs) at cell vertices are defined in addition to its VIAs for the pressure variable and a novel discretized formulation is devised for the pressure Poisson equation so as to enforce the VIA of velocity to satisfy the divergence-free condition. Considering the large density ratios in the vicinity of fluid interface, we have also put forward a non-oscillatory and less dissipative reconstruction scheme to improve the resolvability of discontinuous solutions. For the surface-tension dominated flow problems, a novel reinitialization scheme is proposed to transform the abruptly-varying volume fractions into a smooth-varying level-set function which significantly improves the solution accuracy of curvature estimation so as to suppress the spurious currents in presence of the unphysical oscillation. The resulting multiphase model that combines above numerical methods and techniques, therefore adequately assures divergence-free condition to preserve mass conservation, effectively controls the numerical oscillations and dissipations in the vicinity of moving interface and accurately resolves surface tension force with substantially suppressed spurious currents. Various numerical examples have been presented which demonstrate the excellent performance of the newly developed model to predict both fluid dynamics and interfacial deformations with high-fidelity.

A 3D Nonhydrostatic Compressible Atmospheric Dynamic Core by Multi-moment Constrained Finite Volume Method

A 3D compressible nonhydrostatic dynamic core based on a three-point multi-moment constrained finite-volume (MCV) method is developed by extending the previous 2D nonhydrostatic atmospheric dynamics to 3D on a terrain-following grid. The MCV algorithm defines two types of moments: the point-wise value (PV) and the volume-integrated average (VIA). The unknowns (PV values) are defined at the solution points within each cell and are updated through the time evolution formulations derived from the governing equations. Rigorous numerical conservation is ensured by a constraint on the VIA moment through the flux form formulation. The 3D atmospheric dynamic core reported in this paper is based on a three-point MCV method and has some advantages in comparison with other existing methods, such as uniform third-order accuracy, a compact stencil, and algorithmic simplicity. To check the performance of the 3D nonhydrostatic dynamic core, various benchmark test cases are performed. All the numerical results show that the present dynamic core is very competitive when compared to other existing advanced models, and thus lays the foundation for further developing global atmospheric models in the near future.

Multi-moment finite volume method for incompressible flows on unstructured moving grids and its application to fluid-rigid body interactions

Multi-moment finite volume method for unsteady incompressible flows on unstructured moving grids is developed under the Arbitrary Lagrangian Eulerian (ALE) framework. Two moments, volume integrated average (VIA) and point values (PV), are treated as computational variables which allow to construct quadratic or higher-order polynomials within a compact stencil. The VIAs are computed by a finite volume method (FVM) that ensures the rigorous numerical conservativeness, while the PVs are defined at cell vertices and updated efficiently by a formulation based on the governing equations in a differential form. By employing PVs at cell vertices as additional variables updated at each time step, the present multi-moment finite volume method shows great advantage when applied to the ALE framework for interactions among multiple materials, where the solution points can always coincide with the interfaces between different materials. Thus, any extra numerical step to approximate the values onto the interfaces is not necessary. We have devised two coupling schemes, i.e. an explicit weak coupling scheme and a semi-implicit strong coupling scheme, to formulate the interactions between fluid and moving solid with a wide range of mass ratios. An accurate and robust fluid-solid interaction (FSI) solver has been built by integrating the above numerical components with a radial basis function (RBF) interpolation technique for mesh movement. We in this paper present various benchmark tests to extensively verify the proposed numerical solver, which demonstrate its appealing performance in solving a large class of FSI problems.

High resolution multi-moment finite volume method for supersonic combustion on unstructured grids

In this study, we present a novel numerical model for simulating detonation waves on unstructured grids. In contrast to the conventional finite volume method (FVM), two types of moment comprising the volume-integrated average (VIA) and the point value (PV) at the cell vertex are treated as the evolution variables for the reacting Euler equations . The VIA is computed based on a finite volume formulation of the flux form where the conventional Riemann problem is solved by the HLLC Riemann solver. The PV is updated in a point-wise manner by using the differential formulation where the Roe solver is used to compute the differential Riemann problems. In order to increase the accuracy around discontinuities, numerical oscillations and dissipations are reduced using the boundary variation diminishing algorithm. Convergence tests demonstrated that the proposed model could achieve third-order accuracy with unstructured grids for reacting Euler equations. The high resolution property of the proposed method was verified based on simulations of several detonation wave propagation problems in two and three dimensions. In particular, the current model could resolve the cellular structures with fewer degrees of freedom for the unstable oblique detonation wave problem. These fine structures may be smoothed out by the conventional FVM due to the excessive amount of numerical dissipation errors. Importantly, a simulation of stiff detonation waves showed that the proposed method could capture the correct position of the reaction front whereas the conventional FVMs produced spurious phenomena. Thus, the proposed model can obtain highly accurate solutions for detonation problems on unstructured grids, which is highly advantageous for real applications involving complex geometrical configurations .

A finite volume multi-moment method with boundary variation diminishing principle for Euler equation on three-dimensional hybrid unstructured grids

Abstract In this work, we have developed a novel numerical model for Euler equations on 3D hybrid unstructured grids including tetrahedral, hexahedral, prismatic and pyramidal elements. The model integrates the VPM (Volume integrated average and Point value based Multi-moment) spatial discretization scheme, the limiting projection with BVD (Boundary Variation Diminishing) manipulation and the Roe Riemann solver for unstructured grids. Distinguished from conventional finite volume method, both the volume-integrated average (VIA) and the point values (PV) at the cell vertices are memorized as prognostic variables and updated in time simultaneously. The VIA is computed by a finite volume formulation of flux form while the PV is point-wisely updated using the differential formulation, where the Roe solver is used to compute both conventional and differential Riemann problems. A special technique is introduced to the limiting projection that effectively suppresses both numerical oscillation and dissipation. The resulting numerical model provides remarkably improved numerical accuracy and robustness with a moderate increase in algorithmic complexity and computational cost, which makes it of practical significance for real-case applications. The numerical results of benchmark tests are presented to demonstrate the appealing solution quality of the present model in comparison with other existing methods.

A hybrid pressure–density-based Mach uniform algorithm for 2D Euler equations on unstructured grids by using multi-moment finite volume method

We propose a novel Mach-uniform numerical model for 2D Euler equations on unstructured grids by using multi-moment finite volume method. The model integrates two key components newly developed to solve compressible flows on unstructured grids with improved accuracy and robustness. A new variant of AUSM scheme, so-called AUSM+-pcp (AUSM+ with pressure-correction projection), has been devised including a pressure-correction projection to the AUSM+ flux splitting, which maintains the exact numerical conservativeness and works well for all Mach numbers. A novel 3th-order, non-oscillatory and less-dissipative reconstruction has been proposed by introducing a multi-dimensional limiting and a BVD (boundary variation diminishing) treatment to the VPM (volume integrated average (VIA) and point value (PV) based multi-moment) reconstruction. The resulting reconstruction scheme, the limited VPM-BVD formulation, is able to resolve both smooth and non-smooth solutions with high fidelity. Benchmark tests have been used to verify the present model. The numerical results substantiate the present model as an accurate and robust unstructured-grid formulation for flows of all Mach numbers.

A high‐order multi‐moment constrained finite‐volume global shallow‐water model on the Yin‐Yang grid

A high‐order global shallow‐water model on a Yin‐Yang grid has been developed by using the multi‐moment constrained finite‐volume (MCV) method. Unlike the traditional finite‐volume method, more degrees of freedom (DOFs)–which are the values at the solution points within each mesh element–are defined and updated in time. The time evolution equations for these point values (PVs) are derived from a set of constraint conditions in terms of the so‐called multi‐moment quantities, such as the PV, the volume‐integrated average (VIA) and derivative. Different moments use different forms of equations which are all consistent with the shallow‐water equations, among which the VIA moment is computed from a finite‐volume formulation of flux form that guarantees rigorous numerical conservation. A fourth‐order formulation is devised with the third‐order reconstruction built over each element using the DOFs locally available. A simple and orthogonal overset grid, the Yin‐Yang grid, is used to represent the spherical geometry with quasi‐uniform grid spacing. The resulting global shallow‐water model is attractive in algorithmic simplicity and computational efficiency. The model has been validated by widely used benchmark tests. The numerical results of the present model are competitive with most existing advanced models.

Two and three dimensional multi-moment finite volume solver for incompressible Navier–Stokes equations on unstructured grids with arbitrary quadrilateral and hexahedral elements

This paper presents an extension of the robust and accurate finite volume method (FVM), so-called VPM (Volume integrated average and Point value based Multi-moment) method, to structured and unstructured grids with arbitrary quadrilateral and hexahedral mesh elements. The VPM method treats two different discretized moments of the physical fields, i.e. the volume integrated average (VIA) and the point values (PV) at the vertices of each cell, as the computational variables, which distinguishes it from conventional FVM. Given the local degrees of freedom in terms of VIA and PVs, we have properly designed the interpolation polynomials of reconstruction for quadrilateral and hexahedral mesh elements, which are then used to build a numerical formulation for incompressible viscous fluid dynamics. Numerical results of benchmark tests in both 2 and 3 dimensions are presented to verify the accuracy and robustness of the proposed method, which shows significant improvement in comparison with conventional FVM. The proposed formulation provides a practical solver that is well-balanced between numerical accuracy and algorithmic complexity.

A multi-moment finite volume method for incompressible Navier–Stokes equations on unstructured grids: Volume-average/point-value formulation

A robust and accurate finite volume method (FVM) is proposed for incompressible viscous fluid dynamics on triangular and tetrahedral unstructured grids. Differently from conventional FVM where the volume integrated average (VIA) value is the only computational variable, the present formulation treats both VIA and the point value (PV) as the computational variables which are updated separately at each time step. The VIA is computed from a finite volume scheme of flux form, and is thus numerically conservative. The PV is updated from the differential form of the governing equation that does not have to be conservative but can be solved in a very efficient way. Including PV as the additional variable enables us to make higher-order reconstructions over compact mesh stencil to improve the accuracy, and moreover, the resulting numerical model is more robust for unstructured grids.We present the numerical formulations in both two and three dimensions on triangular and tetrahedral mesh elements. Numerical results of several benchmark tests are also presented to verify the proposed numerical method as an accurate and robust solver for incompressible flows on unstructured grids.

Global shallow water models based on multi-moment constrained finite volume method and three quasi-uniform spherical grids

This is a review article to present several accurate and computationally efficient global models for shallow water equations recently developed under a general numerical framework, the multi-moment constrained finite volume (MCV) method. The multi-moment constrained finite volume method defines the unknowns (prognostic variables) as the point values at the solution points located over each mesh element. The time evolution equations to update these unknowns are derived through the constraint conditions on different moments, e.g. the point value (PV) and the volume-integrated average (VIA). Rigorous numerical conservation is guaranteed by the constraint on the VIA moment through a finite volume formulation of flux form. The resulted numerical schemes are very simple, efficient and easy to implement for both structured and unstructured grids. We have implemented the MCV method to three major spherical grids, Yin–Yang overset grid, cubed-sphere grid and geodesic icosahedral grid, which have overall quasi-uniform grid spacings and are highly popular in the community of global modeling for atmospheric and oceanic dynamics. In this paper, we present the global shallow water models based on these three spherical grids and the third-order MCV scheme. We evaluate and compare the models by widely used benchmark tests, which show the third-order convergence rate for all models, and the numerical results are competitive to other exiting models. Using MCV method as a numerical formulation is well-balanced between solution quality and computational simplicity, the proposed models provide accurate and practical bases for developing dynamic core of general circulation models on different spherical grids.

A Robust and Practical Multi-Moment Finite Volume Model for Computational Fluid Dynamics

A robust and practical CFD code has been developed. The numerical framework, so-called VSIAM3 (Volume/Surface Integrated Average based Multi-Moment Method) makes use of two kinds of integrated moments of physical field, i.e. the volume integrated average (VIA) and the surface integrated average (SIA), which are treated as the computational variables and separately updated in time. VSIAM3 formulation is essentially different from conventional finite volume method and provides a convenient and robust framework to accommodate many existing numerical techniques for simulating various complex flows. In this paper, we will present the underlying idea of VSIAM3 and the extensions to make it applicable to various practical problems. Efforts toward high computational performance on hard wares with distributed memory and GPGPU will be also reported.

A global shallow‐water model on an icosahedral–hexagonal grid by a multi‐moment constrained finite‐volume scheme

AbstractA third‐order global shallow‐water model was developed on an icosahedral–hexagonal grid with the application of the multi‐moment constrained finite‐volume (MCV) method. Using the hexagonal grid, the sphere is divided into quasi‐uniform elements and free of polar problems. The MCV model defines seven degrees of freedom (DOFs) as the prognostic variables which are the point values at the six vertices and the centre for each hexagonal element to construct a third‐order scheme. The time evolution equations to update the DOFs are derived through the constraint conditions on different moments, i.e. the point value and the volume‐integrated average (VIA) moments. Rigorous conservation is guaranteed by the constraint on the VIA through a flux form. The MCV formulation is very simple and easy to implement. We evaluated the model with benchmark tests, and the competitive results reveal the proposed model to be an accurate and practical framework for developing general circulation models.

A Multimoment Constrained Finite-Volume Model for Nonhydrostatic Atmospheric Dynamics

Abstract The two-dimensional nonhydrostatic compressible dynamical core for the atmosphere has been developed by using a new nodal-type high-order conservative method, the so-called multimoment constrained finite-volume (MCV) method. Different from the conventional finite-volume method, the predicted variables (unknowns) in an MCV scheme are the values at the solution points distributed within each mesh cell. The time evolution equations to update the unknown point values are derived from a set of constraint conditions based on the multimoment concept, where the constraint on the volume-integrated average (VIA) for each mesh cell is cast into a flux form and thus guarantees rigorously the numerical conservation. Two important features make the MCV method particularly attractive as an accurate and practical numerical framework for atmospheric and oceanic modeling. 1) The predicted variables are the nodal values at the solution points that can be flexibly located within a mesh cell (equidistant solution points are used in the present model). It is computationally efficient and provides great convenience in dealing with complex geometry and source terms. 2) High-order and physically consistent formulations can be built by choosing proper constraints in view of not only numerical accuracy and efficiency but also underlying physics. In this paper the authors present a dynamical core that uses the third- and the fourth-order MCV schemes. They have verified the numerical outputs of both schemes by widely used standard benchmark tests and obtained competitive results. The present numerical core provides a promising and practical framework for further development of nonhydrostatic compressible atmospheric models.

Fourth order transport model on Yin-Yang grid by multi-moment constrained finite volume scheme

A fourth order transport model is proposed for global computation with the application of multi-moment constrained finite volume (MCV) scheme and Yin-Yang overset grid. Using multi-moment concept, local degrees of freedom (DOFs) are point-wisely defined within each mesh element to build a cubic spatial reconstruction. The updating formulations for local DOFs are derived by adopting multi moments as constraint conditions, including volume-integrated average (VIA), point value (PV) and first order derivative value (DV). Using Yin-Yang grid eliminates the polar singularities and results in a quasi-uniform mesh over the whole globe. Each component of Yin-Yang grid is a part of the LAT-LON grid, an orthogonal structured grid, where the MCV formulations designed for Cartesian grid can be applied straightforwardly to develop the high order numerical schemes. Proposed MCV model is checked by widely used benchmark tests. The numerical results show that the present model has fourth order accuracy and is competitive to most existing ones.

An Adaptive Multimoment Global Model on a Cubed Sphere

Abstract An adaptive global shallow-water model is proposed on cubed-sphere grid using the multimoment finite volume scheme and the Berger–Oliger adaptive mesh refinement (AMR) algorithm that was originally designed for a Cartesian grid. On each patch of the cubed-sphere grid, the curvilinear coordinates are constructed in a way that the Berger–Oliger algorithm can be applied directly. Moreover, an algorithm to transfer data across neighboring patches is proposed to establish a practical integrated framework for global AMR computation on the cubed-sphere grid. The multimoment finite volume scheme is adopted as the fluid solver and is essentially beneficial to the implementation of AMR on the cubed-sphere grid. The multimoment interpolation based on both volume-integrated average (VIA) and point value (PV) provides the compact reconstruction that makes the present scheme very attractive not only in dealing with the artificial boundaries between different patches but also in the coarse–fine interpolations required in the AMR computations. The single-cell-based reconstruction avoids involving more than two nesting levels during interpolations. The reconstruction profile of constrained interpolation profile–conservative semi-Lagrangian scheme with third-order polynomial function (CIP-CSL3) is adopted where the slope parameter provides a flexible and convenient switching to get the desired numerical properties, such as high-order (fourth order) accuracy, monotonicity, and positive definiteness. Numerical experiments with typical benchmark tests for both advection equation and shallow-water equations are carried out to evaluate the proposed model. The numerical errors and the corresponding CPU times of numerical experiments on uniform and adaptive meshes verify the performance of the proposed model. Compared to the uniformly refined grid, the AMR technique is able to achieve the similar numerical accuracy with less computational cost.

Global advection transport model on hexagonal-pentagonal geodesic grid by multi-moment scheme

A 3rd order numerical model is developed for global advection transport computation. The multi-moment finite volume method has been implemented to the hexagonal-pentagonal geodesic grid for spherical geometry. Two kinds of moments, i.e. point value and volume-integrated average, are used as the constraint conditions to derive the time evolution equations to update the computational variables, which are in the present model the values defined at the specified points over each mesh element. The numerical model has rigorous numerical conservation and 3rd order accuracy, and provides an accurate and practical formulation for advection calculation in the hexagonal-pentagonal type geodesic grid.

A multi-moment transport model on cubed-sphere grid

A conservative, single-cell-based semi-Lagrangian transport model is proposed in this paper. Using multi-moment concept, an additional moment, i.e. volume-integrated average (VIA), is treated as the model variable besides the point value (PV) updated in the traditional semi-Lagrangian schemes. A quadratic interpolation function is constructed based on local degrees of freedom defined within each single cell. The PV moment is advanced by the semi-Lagrangian formulation, whereas the VIA moment is updated by a finite volume formulation to rigorously ensure the numerical conservation. The numerical fluxes are computed from the PV moments defined along the boundary edges of the control volume. The scheme is extended to the spherical geometry through the application of the cubed-sphere grid that eliminates the polar singularity in the conventional longitude/latitude coordinates by using the quasi-uniform grid spacing covering the whole sphere. The single-cell-based scheme is well suited for the treatment of the connections between different patches. A simple quasi-monotone limiter to the PV moment is applied to suppress non-physical oscillations. The proposed scheme has been validated via representative benchmark tests and the performance is competitive to other existing transport schemes.