Accurate Finite Volume Method Research Articles

Application of a fourth-order accurate finite-volume method with adaptive refinement in space and time to multifluid plasma simulations

Application of a fourth-order accurate finite-volume method with adaptive refinement in space and time to multifluid plasma simulations

KFVM-WENO: A High-order Accurate Kernel-based Finite Volume Method for Compressible Hydrodynamics

This paper presents a fully multidimensional kernel-based reconstruction scheme for finite volume methods applied to systems of hyperbolic conservation laws, with a particular emphasis on the compressible Euler equations. Nonoscillatory reconstruction is achieved through an adaptive-order weighted essentially nonoscillatory (WENO) method cast into a form suited to multidimensional reconstruction. A kernel-based approach inspired by radial basis functions and Gaussian process modeling, which we call kernel-based finite volume method with WENO, is presented here. This approach allows the creation of a scheme of arbitrary order of accuracy with simply defined multidimensional stencils and substencils. Furthermore, the fully multidimensional nature of the reconstruction allows for a more straightforward extension to higher spatial dimensions and removes the need for complicated boundary conditions on intermediate quantities in modified dimension-by-dimension methods. In addition, a new simple yet effective set of reconstruction variables is introduced, which could be useful in existing schemes with little modification. The proposed scheme is applied to a suite of stringent and informative benchmark problems to demonstrate its efficacy and utility. A highly parallel multi-GPU implementation using Kokkos and the message-passing interface is also provided.

Active Flux Methods for Hyperbolic Systems Using the Method of Bicharacteristics

The Active Flux method is a third order accurate finite volume method for hyperbolic conservation laws, which is based on the use of point values as well as cell average values of the conserved quantities. The resulting method is fully discrete and has a compact stencil in space and time. An important component of Active Flux methods is the evolution formula for the update of the point values. A previously proposed exact evolution formula for acoustics is reviewed and used to construct an Active Flux method for the two-dimensional Maxwell’s equation. Furthermore, the method of bicharacteristics is discussed as a methodology for the derivation of truly multidimensional approximative evolution operators that can be used for the evolution of point values in Active Flux methods. We study accuracy and stability of the resulting methods for acoustics and compare with the Active Flux method that uses the exact evolution operator. Finally, we used the method of bicharacteristics to derive Cartesian grid Active Flux methods for the linearised and nonlinear Euler equations. Numerous test computations illustrate the performance of these new Active Flux methods.

A 4th-order accurate finite volume method for ideal classical and special relativistic MHD based on pointwise reconstructions

We present a novel implementation of a genuinely 4th-order accurate finite volume scheme for multidimensional classical and special relativistic magnetohydrodynamics (MHD) based on the constrained transport (CT) formalism. The scheme introduces several novel aspects when compared to its predecessors yielding a more efficient computational tool. Among the most relevant ones, our scheme exploits pointwise to pointwise reconstructions (rather than one-dimensional finite volume ones), employs the generic upwind constrained transport averaging and sophisticated limiting strategies that include both a discontinuity detector and an order reduction procedure. Selected numerical benchmarks demonstrate the accuracy and robustness of the method.

Simplified numerical modeling and analysis of electrolyte behavior in multiple physical fields for lithium-ion batteries

The electrolyte plays an important role in lithium-ion batteries, affecting their state and safety. However, the internal states of the electrolyte in the battery full domain are not easy to obtain directly. The electric field distribution, to which less attention has been paid, is as important as the concentration distribution, even related to battery safety. In this paper, an accurate and real-time simplified model for electrolyte is developed at the mesoscale, based on the Nernst-Planck equation and the continuum equation with some simple and reasonable assumptions. The analytical solutions of steady-state distribution for concentration and electric fields are obtained by theoretical derivation. Then, based on the assumption of internal uniformity of electrode, the numerical solution of the transient-state process is derived under full working conditions. The potential distribution and the overpotentials within electrolytes can be further calculated. Finally, the accuracy of the developed model is verified by the experiment of constant current terminal voltage and the accurate finite volume method. Besides, simulations are carried out to analyze the effects on the variables of electrolytes by varying the parameters in the model. The design suggestions of electrolyte concentration, thickness and porosity of each component in the battery are given.

Second-order accurate implicit finite volume method for two-dimensional modeling of PFAS transport in unsaturated porous media with variable surface tension

Per- and polyfluoroalkyl substances (PFASs) have become emerging contaminants of critical concern. Comprehensive understanding of the transport and fate of PFAS in the vadose-zone, a type of water-unsaturated porous media, is key to determination of the risks of the PFAS contamination in the subsurface and to the development of the effective remediation strategies. Accurate modeling of the PFAS transport in the unsaturated porous media is still a challenge due to the variable surface tension induced by the adsorption of PFAS to the air-water interfaces. In an effort to address this challenge, we propose a multidimensional modeling framework for the transient PFAS transport in the unsaturated porous media based on the second order accuracy finite volume method. In the modeling, the adsorption of PFAS to the solid surfaces and to the air-water interfaces is described by the two-domain sorption kinetics model, i.e., both the instantaneous and the rate limited PFAS adsorptions are taken into account. The diffusive and convective terms in the governing equations for the PFAS transport and the water flow are discretized by the central difference and the quadratic upstream interpolation for convective kinetics schemes, respectively. We investigate the effects of the convergent criteria, coupling method, and variation of the surface tension on the average and the local PFAS concentration and water content in the computational domain. We find that the convergent criteria should be chosen carefully so as to get the accurate results. The differences between the different coupling methods are affected by the boundary conditions. The variation in the surface tension due to the variation of the PFAS concentration cannot be neglected. The dimensionless parameters relevant to the properties of porous media as well as the relation between the surface tension and the PFAS concentration play an important role in transport of PFAS in the unsaturated porous media. The effects of the Péclet number, Damköhler values, and fraction of instantaneous sorption are not significant in the ranges studied in this work. These findings provide a better understanding of transport of PFAS in the vadose zone.

A three-dimensional solver for simulating detonation on curvilinear adaptive meshes

A generic solver in a parallel Cartesian adaptive mesh refinement framework is extended to simulate detonations on three-dimensional structured curvilinear meshes. A second-order accurate finite volume method is used with grid-aligned Riemann solvers for thermally perfect gas mixtures. Detailed, multi-step chemical kinetic mechanisms are employed and numerically incorporated with a splitting approach. The adaptive mesh refinement technique is applied to a mapped mesh using modified prolongation and restriction operators. The flux along the coarse-fine interface is considered in a correction procedure to ensure the conservation of the solver. The numerical accuracy, conservation and robustness of the simulations are verified and validated with suitable benchmark tests. The new solver is then used to simulate detonation problems in non-Cartesian geometries. A simulation is conducted of the three-dimensional detonation propagation in a 90-degree pipe bend. A detonation in a round tube is also simulated in a Galilean frame of reference. Both a rectangular mode and a spinning mode are observed in the simulations. In addition, the fundamental problem of detonation wave/boundary layer interaction is studied. The results show that the new solver can simulate high-speed reactive flows efficiently by the combined use of a curvilinear mapping with mesh adaptation.

Second-order accurate finite volume method for G-equation on polyhedral meshes

Second-order accurate finite volume method for G-equation on polyhedral meshes

The Cartesian Grid Active Flux Method with Adaptive Mesh Refinement

We present the first implementation of the Active Flux method on adaptively refined Cartesian grids. The Active Flux method is a third order accurate finite volume method for hyperbolic conservation laws, which is based on the use of point values as well as cell average values of the conserved quantities. The resulting method has a compact stencil in space and time and good stability properties. The method is implemented as a new solver in ForestClaw, a software for parallel adaptive mesh refinement of patch-based solvers. On each Cartesian grid patch the single grid Active Flux method can be applied. The exchange of data between grid patches is organised via ghost cells. The local stencil in space and time and the availability of the point values that are used for the reconstruction, leads to an efficient implementation. The resulting method is third order accurate, conservative and allows the use of subcycling in time.

An implicit high-order radial basis function-based differential quadrature-finite volume method on unstructured grids to simulate incompressible flows with heat transfer

A high-order implicit radial basis function-based differential quadrature-finite volume (IRBFDQ-FV) method is presented in this work to efficiently simulate incompressible flows with heat transfer on unstructured mesh. The velocity and temperature fields are solved by locally using the lattice Boltzmann flux solver and the high-order finite volume method. Specifically, the proposed highly accurate finite volume method utilizes a high-order Taylor polynomial to approximate the solution within every control cell. Spatial derivatives are the corresponding coefficients in the polynomial, and they are approximated by the meshless radial basis function-based differential quadrature (RBFDQ) method. The diffusive and convective fluxes at each cell interface are simultaneously evaluated through local reconstruction of lattice Boltzmann solution using D2Q9 lattice velocity model. To efficiently calculate the solution with high-order accuracy, an implicit time-marching method incorporating the lower-upper symmetric Gauss-Seidel (LU-SGS) and the explicit first stage, singly-diagonally implicit Runge-Kutta (ESDIRK) approaches is devised. The proposed method is comprehensively validated by a series of numerical experiments containing both steady-state and time-dependent heat transfer problems with/without curved boundaries at a wide variety of Rayleigh numbers and Grashof numbers. The obtained results demonstrate a high degree of accuracy and reliability of the proposed method for complex flows on unstructured mesh. In comparison with the classical second-order method, the proposed high-order method has better computational efficiency when comparable results are achieved.

A high-order finite volume method solving viscous incompressible flows using general pressure equation

A high-order accurate finite volume method on Cartesian meshes has been developed to solve viscous incompressible flows using the general pressure equation (GPE). The GPE is a pressure evolution equation that replaces the divergence-free constraint equation when computing incompressible flows. The high-order spatial accuracy of the current method is deduced from the high-order flow reconstruction model using the conserved volume integrals, and the high-order accurate computations of the flux integrals. The Roe’s flux difference scheme was adapted to compute the inviscid fluxes. The grid convergence tests using the solution of Taylor-Green decaying vortices showed that the current method is spatially fifth order accurate for both velocity and pressure. The problem of doubly periodic shear layers and the lid-driven cavity flow were computed to demonstrate the capability of the current method in transient and steady state computations.

High‐order accurate conjugate heat transfer solutions with a finite volume method in anisotropic meshes with application in polymer processing

AbstractThe computational modeling has become an indispensable tool to support the engineering design and control of many industrial processes. In polymer processing applications, the simulation of conjugate heat transfer phenomena is critical as rigorous temperature control is necessary to ensure that the produced parts meet the required specifications. In this article, a high‐order accurate finite volume method is proposed to improve the numerical accuracy and the computational efficiency of conjugate heat transfer simulations with application in polymer processing. Anisotropic meshes are investigated to significantly reduce the number of unknowns in convection‐dominated problems where the higher temperature variations occur perpendicularly to curved boundaries and interfaces. The reconstruction for off‐site data method based on polynomial reconstructions is employed to fulfill the prescribed boundary and interface conditions solely using polygonal meshes to avoid the limitations of curved mesh approaches. A code verification benchmark based on manufactured solutions proves that the proposed method provides a fourth‐order of convergence and is computationally more cost‐effective than the classical second‐order of convergence. Moreover, meshes with higher aspect ratios improve the calculation efficiency but suffer from a small accuracy penalty due to conditioning deterioration. Comparison with the classical finite volume discretization techniques, widely implemented in commercial and open‐source software packages, is also provided. The applicability and performance of the proposed method are further supported with a practical case study for the sheet extrusion cooling stage.

A solver for simulating shock-induced combustion on curvilinear adaptive meshes

A generic solver in a structured Cartesian adaptive mesh refinement framework is extended to simulate unsteady shock-induced combustion problems on a structured curvilinear mesh. A second-order accurate finite volume method is used with a grid-aligned Riemann solver for inviscid thermally perfect gas mixtures. To solve these reactive problems, detailed chemical kinetic mechanisms are employed with a splitting approach. The prolongation and restriction operators are modified to implement the adaptive mesh refinement algorithm on a mapped mesh. The developed solver is verified with several benchmark tests and is then used to simulate unsteady shock-induced combustion. The results show that the computed stand-off distance of waves and oscillation frequencies of mass fraction of products observed at the stagnation point are in good agreement with the results from experiments.

Lid-Driven Square Cavity Flow: A Benchmark Solution With an 8192 × 8192 Grid

Abstract The incompressible steady-state fluid flow inside a lid-driven square cavity was simulated using the mass conservation and Navier–Stokes equations. This system of equations is solved for Reynolds numbers of up to 10,000 to the accuracy of the computational machine round-off error. The computational model used was the second-order accurate finite volume (FV) method. A stable solution is obtained using the iterative multigrid methodology with 8192 × 8192 volumes, while degree-10 interpolation and Richardson extrapolation were used to reduce the discretization error. The solution vector comprised five entries of velocities, pressure, and location. For comparison purposes, 65 different variables of interest were chosen, such as velocity profile, its extremum values and location, and extremum values and location of the stream function. The discretization error for each variable of interest was estimated using two types of estimators and their apparent order of accuracy. The variations of the 11 selected variables are shown across 38 Reynolds number values between 0.0001 and 10,000. In this study, we provide a more accurate determination of the Reynolds number value at which the upper secondary vortex appears. The results of this study were compared with those of several other studies in the literature. The current solution methodology was observed to produce the most accurate solution till date for a wide range of Reynolds numbers.

Three-dimensional numerical simulation of ferrofluid magnetoconvection in cubical enclosure heated with an inner spherical hot block

In this study, three-dimensional computational analysis is performed to investigate the magnetoconvection of ferrofluid ([Formula: see text]-water) within a cubical enclosure heated by an inner spherical hot block. The ferrofluid, considered as a working fluid, is modeled as a single-phase fluid. The inner spherical block is put at high temperature while all the remaining walls of the enclosure are exposed to low temperature. Two radii values ([Formula: see text] and [Formula: see text]) of the inner hot sphere are examined. Governing equations with corresponding boundary conditions are solved numerically applying a second-order accurate finite volume method on a staggered grid system, using an accelerated multigrid model. Simulations are carried out based on various flow-governing parameters such as Rayleigh number [Formula: see text], Hartmann number [Formula: see text] and ferrofluid nanoparticle volume fraction [Formula: see text]. The effects of the pertinent parameters in the performance of the system are also studied. The flow and thermal fields, the local and surface-averaged Nusselt numbers on the sphere and the enclosure for both configurations are detailed. The flow remains steady and laminar for all Rayleigh numbers regardless of the sphere radius. Obviously, heat transfer rate improves with [Formula: see text] augmentation and minimizes with Ha decrease. At the highest Ra and lowest Ha, higher inner sphere radius shows significantly better heat transfer rate (more than [Formula: see text]). Useful correlations are presented to quantify the surface-averaged heat transfer rate through the cubical enclosure.

A new auxiliary volume-based gradient algorithm for triangular and tetrahedral meshes

To have a second-order accurate centroid-based finite volume method, the gradients at the cell-centroids need to be evaluated with at least first-order accuracy. This again requires the cell face-center solution values to be interpolated with second-order accuracy when using Green-Gauss methods. It is not always possible to obtain the second-order accuracy in face interpolations with simple averaging or linear interpolation by Traditional Green-Gauss methods when the meshes are irregular in nature. In this paper, a simple and novel way to calculate first-order accurate gradients on any tri and tetra type meshes is proposed. The new method, referred to as the Circle Green-Gauss (CGG) reconstruction, is based on constructing an auxiliary volume around the cell of interest from the neighboring cells. The solution values at the faces of the constructed auxiliary volume of the cell can then be easily estimated to second-order accuracy, thereby making the gradients first-order accurate. Qualitative comparison of the proposed method with Traditional Green-Gauss (TGG) reconstruction and the Least-Squares (LS) reconstruction (used in most commercial solvers) is shown on different mesh topologies that are irregular in nature. The stability and applicability of CGG are discussed in detail through representative test problems including the method of manufactured solutions that demonstrate that the gradients are first-order accurate (so that the overall method is second-order accurate) on triangular and tetrahedral meshes.

Efficient approach toward the application of the Godunov method to hydraulic transients

Abstract The proposed study investigated the applicability of the finite volume method (FVM) based on the Godunov scheme to transient water hammer with shock front simulation, in which intermediate fluxes were computed using either first-order or second-order Riemann solvers. Finite volume (FV) schemes are known to conserve mass and momentum and produce the efficient and accurate realization of shock waves. The second-order solution of the Godunov scheme requires an efficient slope or a flux limiter for error minimization and time optimization. The study examined a range of limiters and found that the MINMOD limiter is the best for modeling water hammer in terms of computational time and accuracy. The first- and second-order FVMs were compared with the method of characteristics (MOCs) and experimental water hammer measurements available in the literature. Both the FV methods accurately predicted the numerical and experimental results. Parallelization of the second-order FVM reduced the computational time similar to that of first-order. Thus, the study presented a faster and more accurate FVM which is comparable to that of MOC in terms of computational time and precision, therefore it is a good substitute for the MOC. The proposed study also investigated the implementation of a more complex convolution-based unsteady friction model in the FVM to capture real pressure dissipation. The comparison with experimental data proved that the first-order FV scheme with the convolution integral method is highly accurate for computing unsteady friction for sudden valve closures.

A monotonicity-preserving higher-order accurate finite-volume method for Kapila’s two-fluid flow model

In preparation of the study of liquefied natural gas (LNG) sloshing in ships and vehicles, we model and numerically analyze compressible two-fluid flow. We consider a five-equation two-fluid flow model, assuming velocity and pressure continuity across two-fluid interfaces, with a separate equation to track the interfaces. The system of partial differential equations is hyperbolic and quasi-conservative. It is discretized in space with a tailor-made third-order accurate finite-volume method, employing an HLLC approximate Riemann solver. The third-order accuracy is obtained through spatial reconstruction with a limiter function, for which a novel formulation is presented. The non-homogeneous term is handled in a way consistent with the HLLC treatment of the convection operator. We study the one-dimensional case of a liquid column impacting onto a gas pocket entrapped at a solid wall. It mimics the impact of a breaking wave in an LNG containment system, where a gas pocket is entrapped at the tank wall below the wave crest. Furthermore, the impact of a shock wave on a gas bubble containing the heavy gas R22, immersed in air, is simulated in two dimensions and compared with experimental results. The numerical scheme is shown to be higher-order accurate in space and capable of capturing the important characteristics of compressible two-fluid flow.

Shock wave reflection from a short orifice open to vacuum

We study axisymmetric time-dependent problem of reflection of a shock wave running into the wall with a short orifice and associated rarefied gas flow into vacuum through a pipe. The main characteristics under consideration are the mass flow rate through the pipe and its transition to the constant value, the influence of the orifice presence, pipe relative length and Knudsen number on the reflected shock wave velocity. The analysis is based on the numerical solution of the S-model kinetic equation using the second-order accurate finite volume method, implemented into the “Nesvetay-3D” software package.

Buoyancy-induced effects on large-scale motions in differentially heated vertical channel flows studied in direct numerical simulations

Direct numerical simulations of turbulent convection in a differentially heated vertical channel flow with Prandtl number Pr=0.71 are conducted with a fourth-order accurate finite volume method for a bulk Reynolds number Reb=4328 and three Grashof numbers Gr ∈ {0, 6.4 · 105, 9.5 · 105}. The discussion of instantaneous flow field snapshots, first- and second-order moments, budget equations, power density spectra and quadrant analyses shows that the turbulent velocity fluctuations are attenuated in the aiding flow and enhanced in the opposing flow. In contrast, temperature fluctuations are attenuated in the opposing flow and enhanced in the aiding flow. The analyses further reveal that the low-momentum flow structures in the aiding flow are warmer than the high-momentum flow structures and vice versa in the opposing flow. Due to their different temperatures, buoyancy accelerates and decelerates the flow structures differently, which leads to reduced and increased pressure and shear fluctuations in the aiding and opposing flow. Thus, the redistribution of turbulent velocity fluctuations is lower in the aiding flow and higher in the opposing flow. The Reynolds shear stresses are consequently decreased in the former and increased in the latter, influencing the production of streamwise velocity fluctuations accordingly. In summary, the discussed physical mechanisms underline the indirect effect of buoyancy on the turbulent velocity fluctuations.