High-order Finite Volume Method Research Articles

A high-order finite volume method with improved isotherms reconstruction for the computation of multiphase flows using the Navier–Stokes–Korteweg equations

In this work we solve the Navier–Stokes–Korteweg (NSK) equations to simulate a two-phase fluid with phase change. We use these equations on a diffuse interface approach, where the properties of the fluid vary continuously across the interface that separates the different phases. The model is able to describe the behavior of both phases with the same set of equations, and it is also able to handle problems with great changes in the topology of the problem. However, high-order derivatives are present in NSK equations, which is a difficulty for the design of a numerical method to solve the problem. Here, we propose the use of a high-order Finite Volume method with Moving Least Squares approximations to handle high-order derivatives and solve the NSK equations. Moreover, a new methodology to obtain accurate equations of state is presented. In this method, we use any accurate equation of state for the pure phases. Under the saturation curve, a B-spline reconstruction fulfilling a given set of thermodynamic criteria is performed. The new EOS can be used for computations using diffuse interface modeling. Several numerical examples to show the accuracy of the new approach are presented.

Compact high order finite volume method on unstructured grids IV: Explicit multi-step reconstruction schemes on compact stencil

In the present paper, a multi-step reconstruction procedure is proposed for high order finite volume schemes on unstructured grids using compact stencil. The procedure is a recursive algorithm that can eventually provide sufficient relations for high order reconstruction in a multi-step procedure. Two key elements of this procedure are the partial inversion technique and the continuation technique. The partial inversion can be used not only to obtain lower order reconstruction based on existing reconstruction relations, but also to regularize the existing reconstruction relations to provide new relations for higher order reconstructions. The continuation technique is to extend the regularized relations on the face-neighboring cells to current cell as additional reconstruction relations. This multi-step procedure is operationally compact since in each step only the relations defined on a compact stencil are used. In the present paper, the third and fourth order finite volume schemes based on two-step quadratic and three-step cubic reconstructions are studied.

Modeling and simulations of two dimensional propagation of shear shock waves in relaxing soft solids

Soft solids, such as gelatin or soft tissue, have a shear wave speed that is smaller than the compressional wave speed. Recent observations of shear shock wave generation in the brain, which can easily reach a large Mach number regime, suggest that the cubic nonlinear behavior of soft solids could be responsible for traumatic brain injury. However, currently there are no two-dimensional (2D) models describing the propagation of linearly-polarized shear shock waves in relaxing soft solids. These models are required to model the fundamental wave propagation physics like diffraction, focusing etc., and are related to traumatic brain injuries as it can be used to model the skull/brain morphology. In this work, we present a two-dimensional system of first-order equations modeling the propagation of shear shock waves in a relaxing soft solid, and then this system is solved numerically using the piecewise parabolic method, a high-order finite volume method. The numerical solutions are validated in two parts. First, the nonlinear component, which is designed for large Mach numbers, is validated with a step-shock Riemann problem. Then relaxation mechanisms based on a generalized Maxwell body, which model the non-classical attenuation that occurs in soft tissue, are compared to analytical solutions with an error of 5%-10%. The validation of attenuation also includes dispersion due to causality which is determined by the Kramers-Kronig relations. Finally, the full numerical method, which includes nonlinearity, attenuation, and dispersion, is compared to ultrasonic measurements of shear shock wave propagation in tissue-mimicking phantoms. Two experiments were performed based on high frame-rate ultrasound imaging and tracking in a gelatin phantom 1) planar shear shock waves, and 2) focused shear shock wave propagation. The experimental and numerical waveforms closely match, e.g. the RMS amplitude error is between 12.05% and 12.27%. Moreover, the frequency-content of the temporal signal was compared for third and fifth multiples of fundamental harmonic validating the generation of odd-harmonics due to cubic nonlinearity. Furthermore, the numerical tool was able to estimate the nonlinear parameter in the phantom (β=4.4) using a grid-parameter-search. In context of traumatic brain injury, the current method can be used to study the shear shock formation in 2D-sections of human skull, and can also be used for nonlinear parameter estimation in brain.

High-order multi-moment finite volume method with smoothness adaptive fitting reconstruction for compressible viscous flow

We proposed a novel multi-moment finite volume method (MMFVM) to compute compressible viscous flows on arbitrary unstructured grids. Unlike the previous volume-average/point-value multi-moment (VPM) method (Xie et al. 2014 [60]), the present method, so-called VPM-Ex (VPM with extended-stencil reconstruction), formulates novel spatial reconstructions with substantially improved robustness and accuracy for both smooth and discontinuous solutions. Our efforts in this work have brought at least the following new features to the numerical formulation: 1) a more robust and accurate spatial reconstruction based on the stencil of all immediately adjacent cells that supports higher-order limiting projection; 2) two different options for both explicit and implicit reconstructions; 3) a simplified and more accurate formulation for point-value. In addition, we have proposed a smoothness-adaptive fitting (SAF) method for limiting projection to improve the solution quality for discontinuities. The resulting reconstruction method that combines VPM-Ex scheme with SAF-based limiting projection schemes is able to attain 3rd-order accuracy for smooth solution, and non-oscillatory and less-dissipative numerical results for discontinuous solution on unstructured grids of different types of elements. The numerical method has been extensively verified by various benchmark tests for scalar conservation equation, Euler equations and Navier–Stokes equations in 2 and 3 dimensions. The numerical results substantiate that the present method provides a novel numerical formulation of great practical significance to accurately resolve both continuous and discontinuous solutions.

Provably positive high-order schemes for ideal magnetohydrodynamics: analysis on general meshes

This paper proposes and analyzes arbitrarily high-order discontinuous Galerkin (DG) and finite volume methods which provably preserve the positivity of density and pressure for the ideal MHD on general meshes. Unified auxiliary theories are built for rigorously analyzing the positivity-preserving (PP) property of MHD schemes with a HLL type flux on polytopal meshes in any space dimension. The main challenges overcome here include establishing relation between the PP property and discrete divergence of magnetic field on general meshes, and estimating proper wave speeds in the HLL flux to ensure the PP property. In 1D case, we prove that the standard DG and finite volume methods with the proposed HLL flux are PP, under condition accessible by a PP limiter. For multidimensional conservative MHD system, standard DG methods with a PP limiter are not PP in general, due to the effect of unavoidable divergence-error. We construct provably PP high-order DG and finite volume schemes by proper discretization of symmetrizable MHD system, with two divergence-controlling techniques: locally divergence-free elements and a penalty term. The former leads to zero divergence within each cell, while the latter controls the divergence error across cell interfaces. Our analysis reveals that a coupling of them is important for positivity preservation, as they exactly contribute the discrete divergence-terms absent in standard DG schemes but crucial for ensuring the PP property. Numerical tests confirm the PP property and the effectiveness of proposed PP schemes. Unlike conservative MHD system, the exact smooth solutions of symmetrizable MHD system are proved to retain the positivity even if the divergence-free condition is not satisfied. Our analysis and findings further the understanding, at both discrete and continuous levels, of the relation between the PP property and the divergence-free constraint.

Experimental and numerical investigations on combined Buoyancy–Marangoni convection heat and mass transfer of power-law nanofluids in a porous composite with complex surface

In this paper, a convection and heat transfer problem of power-law fluid in a three-dimensional porous media with complex surface is studied. The Buoyancy-Marangoni convection for non-Newtonian power-law fluids in porous media is solved using a compact high-order finite volume method. For this single-phase model, the left wall is kept at high temperature and high concentration, the right wall is affected by lower temperature and lower concentration, the upper wall is a complex surface, and the remaining walls are considered to be adiabatic and impermeable. The Weierstrass–Mandelbrot function is used to simulate the shape of the upper surface. The fluid in the porous cavity is a power-law nanofluids containing copper oxide nanoparticles. The solid material of the porous medium is aluminum foam. Numerical simulations can be used to determine the power law exponent, Marangoni number, Rayleigh number, and aspect ratio on the flow, heat transfer, and mass transfer rate.

Piecewise parabolic method for propagation of shear shock waves in relaxing soft solids: One-dimensional case.

Shear shock waves can be generated spontaneously deep within the brain during a traumatic injury. This recently observed behavior could be a primary mechanism for the generation of traumatic brain injuries. However, shear shock wave physics and its numerical modeling are relatively unstudied. Existing numerical solvers used in biomechanics are not designed for the extremely large Mach numbers (greater than 1) observed in the brain. Furthermore, soft solids, such as the brain, have a complex nonclassical viscoleastic response, which must be accurately modeled to capture the nonlinear wave behavior. Here, we develop a 1D inviscid velocity-stress-like system to model the propagation of shear shock waves in a homogeneous medium. Then a generalized Maxwell body is used to model a relaxing medium that can describe experimentally determined attenuation laws. Finally, the resulting system is solved numerically with the piecewise parabolic method, a high-order finite volume method. The nonlinear and the relaxing components of this method are validated with theoretical predictions. Comparisons between numerical solutions obtained for the proposed model and the experiments of plane shear shock wave propagation based on high frame-rate ultrasound imaging and tracking are shown to be in excellent agreement.

A very high-order finite volume method based on weighted least squares for elliptic operators on polyhedral unstructured grids

A very high-order finite volume method is proposed for the solution of the Poisson equation on unstructured grids based on the weighted least-squares method. The new method consists in an up to eight-order accurate face centered reconstruction for the integration of the diffusive fluxes. It uses a new stencil extension algorithm that maintains the local high-order near the boundaries of the computational domain.The weight function used by the scheme was optimized according to the matrix condition number, convergence order and error magnitude. From this study a new weight function is proposed for this type of reconstruction schemes which achieves the theoretical convergence order for Cartesian, triangular, polyhedral and hybrid grids. An extension for the Neumann boundary type conditions is also shown. A grid quality study, based on the non-orthogonality angle and volume ratio, proves that the scheme’s convergence order is not affected by these parameters.The efficiency criteria, defined by the required memory and solver-run time, indicate that the eight-order scheme is advantageous over the lower order ones, additionally it is demonstrated that the polyhedral grid yields more efficient solutions than the ones obtained with Cartesian or triangular grids.

High-Order Finite-Volume Method with Block-Based AMR for Magnetohydrodynamics Flows

A high-order central essentially non-oscillatory (CENO) finite volume scheme combined with a block-based adaptive mesh refinement (AMR) algorithm is proposed for the solution of the ideal magnetohydrodynamics equations. The high-order CENO finite-volume scheme is implemented with fourth-order spatial accuracy within a flexible multi-block, body-fitted, hexahedral grid framework. An important feature of the high-order adaptive approach is that it allows for anisotropic refinement, which can lead to large computational savings when anisotropic flow features such as isolated propagating fronts and/or waves, shocks, shear surfaces, and current sheets are present in the flow. This approach is designed to handle complex multi-block grid configurations, including cubed-sphere grids, where some grid blocks may have degenerate edges or corners characterized by missing neighboring blocks. A procedure for building valid high-order reconstruction stencils, even at these degenerate block edges and corners, is proposed, taking into account anisotropic resolution changes in a systematic and general way. Furthermore, a non-uniform or heterogeneous block structure is used where the ghost cells of a block containing the solution content of neighboring blocks are stored directly at the resolution of the neighbors. A generalized Lagrange multiplier divergence correction technique is applied to achieve numerically divergence-free magnetic fields while preserving high-order accuracy on the anisotropic AMR grids. Parallel implementation and local grid adaptivity are achieved by using a hierarchical block-based domain partitioning strategy in which the connectivity and refinement history of grid blocks are tracked using a flexible binary tree data structure. Physics-based refinement criteria as well as the CENO smoothness indicator are both used for directing the mesh refinement. Numerical results, including solution-driven anisotropic refinement of cubed-sphere grids, are presented to demonstrate the accuracy and efficiency of the approach.

High-order compact finite volume methods on unstructured grids with adaptive mesh refinement for solving inviscid and viscous flows

In the present paper, high-order finite volume schemes on unstructured grids developed in our previous papers are extended to solve three-dimensional inviscid and viscous flows. The high-order variational reconstruction technique in terms of compact stencil is improved to reduce local condition numbers. To further improve the efficiency of computation, the adaptive mesh refinement technique is implemented in the framework of high-order finite volume methods. Mesh refinement and coarsening criteria are chosen to be the indicators for certain flow structures. One important challenge of the adaptive mesh refinement technique on unstructured grids is the dynamic load balancing in parallel computation. To solve this problem, the open-source library p4est based on the forest of octrees is adopted. Several two- and three-dimensional test cases are computed to verify the accuracy and robustness of the proposed numerical schemes.

A reconstruction-based cell-centered high-order finite volume method for incompressible viscous flow simulation on unstructured meshes

A new high-order (> 2nd order) cell-centered finite volume method is presented for incompressible flow simulation on unstructured meshes. Artificial compressibility is employed to couple the continuity and momentum equations in a manner that allows them to be solved simultaneously. A new numerical stencil, a so-called wrapping stencil, is utilized for linear and quadratic solution reconstruction in order to achieve more accurate and robust solution reconstruction not only for the interior cells, but also for the cells on the boundary, where fewer neighboring cells typically exist. The effectiveness of the current algorithm is demonstrated by various test cases, including an analytical solution reconstruction test, Kovasznay flow simulations with various Reynolds numbers, a driven cavity flow, and flow past a square cylinder. Based on the comparison with the standard low order scheme, the proposed second and third order schemes, based on linear and quadratic solution reconstruction, show superior accuracy, which sheds light on the method's applicability in solving more challenging incompressible flow problems.

Numerical study on combined buoyancy–Marangoni convection heat and mass transfer of power-law nanofluids in a cubic cavity filled with a heterogeneous porous medium

In this paper, the combined buoyancy–Marangoni convection of non-Newtonian power-law nanofluids in a 3D heterogeneous porous cubic cavity is investigated in detail with the compact high order finite volume method. Special attentions are given to detect the effects of heterogeneity level, Marangoni number, thermal Rayleigh number, buoyancy ratio and nanoparticle volume fraction on the fluid flow as well as on rates of heat and mass transfer. It is observed that as a result of the exponential distribution of the permeability, the heat and mass transfer rates reduce as the level of heterogeneity enhances. The effect of the surface tension on the heat and mass transfer intensity becomes insignificant when the buoyancy force is strengthened. Further, the average Nusselt and Sherwood numbers increase as the Marangoni number and thermal Rayleigh number increase due to the combined effects of buoyancy and surface tension. The augmentation of the buoyancy ratio causes convection heat and mass transfer to increase for the thermal dominated flow while decrease in buoyancy ratio also augments it for the solutal dominated flow. The heat transfer (mass transfer) rate is found to increase (decrease) with increasing the nanoparticle volume fraction. Moreover, for all above studied parameters, an intensification of the flow and an increase in average Nusselt and Sherwood numbers occur with the decrease in power-law index.

Coupled lattice Boltzmann finite volume method for conjugate heat transfer in porous media

ABSTRACTThis work presents a coupled lattice Boltzmann finite volume method for dealing with conjugate heat transfer problems. Lattice Boltzmann scheme is used for fluid-dynamics, while high-order finite volume method is implemented for temperature reconstruction. After a first validation with literature test cases, the method is applied to a heat exchanger with an insert made of porous medium, representative of an open-cell metal foam, innovative material largely used for its thermomechanical properties. This allows maximizing heat exchange processes with advantages in terms of efficiencies. Thus, the coupled method allows dealing with complex boundaries in multiphysics problems.

Analyses and implications of higher order finite volume methods on first-order macroscopic traffic flow models

Selection of suitable numerical scheme for solving macroscopic traffic flow models should be based on the numerical errors in various traffic parameters. Although such errors associated with state variables such as density are known to some extent, their implications to the derived parameters such as travel time and queue-length are not understood well. This paper proposes a general basis for selecting the spatial order of accuracy of a finite volume scheme for solving first-order traffic flow models. Four Measures of Accuracy that are especially relevant for traffic flow have been introduced. We illustrate that, although higher order schemes outperform the lower order ones in the quality of simulation in terms of resolution as well as computational cost, their superior performance is underestimated with the use of conventional error measures of state predictions. Localized parameters such as size and duration of the queue are more sensitive towards the order of the scheme, than domain aggregated parameters such as duration of congestion and travel time.

A NURBS-enhanced finite volume solver for steady Euler equations

In Hu and Yi (2016) [20], a non-oscillatory k-exact reconstruction method was proposed towards the high-order finite volume methods for steady Euler equations, which successfully demonstrated the high-order behavior in the simulations. However, the degeneracy of the numerical accuracy of the approximate solutions to problems with curved boundary can be observed obviously. In this paper, the issue is resolved by introducing the Non-Uniform Rational B-splines (NURBS) method, i.e., with given discrete description of the computational domain, an approximate NURBS curve is reconstructed to provide quality quadrature information along the curved boundary. The advantages of using NURBS include i). both the numerical accuracy of the approximate solutions and convergence rate of the numerical methods are improved simultaneously, and ii). the NURBS curve generation is independent of other modules of the numerical framework, which makes its application very flexible. It is also shown in the paper that by introducing more elements along the normal direction for the reconstruction patch of the boundary element, significant improvement in the convergence to steady state can be achieved. The numerical examples confirm the above features very well.

Numerical Study of the Injection of Carbon Dioxide in a Homogeneous Porous Media

This work proposes a locally conservative and less restrictive algorithm to solve the problem dealt with in [1], i.e. a two-phase flow in a homogeneous porous medium (water and CO2), with mass absorption between the fluid phases and reaction between the CO2 phase and the rock. The latter is modeled by two non-linear hyperbolic equations that represent the transport of the flowing phases for a given velocity field (equations of saturation and concentration). From the numerical point of view, we use the operator splitting technique to properly treat the time scale of each physical phenomenon and a high-order non-oscillatory central-scheme finite volume method for nonlinear hyperbolic equations proposed by [2] that was extended for a system of equations with source terms to treat the equations that govern the saturation and concentration of phases. In addition, with respect to source terms, the mass flux between fluid phases was handled using the flash methodology, whereas kinetic theory was applied for reproducing the changes in porosity and permeability that are caused by the reaction of CO2 with the rock. The same physical trends observed in [1] were obtained in our numerical results which indicate a good predictive capability. Furthermore, this method avoids the difficulties that arise when adopting small time steps enforced by CFL stability restrictions. Finally, the results obtained show that the applicability of the KT method is beyond just a single nonlinear conservation law with the absence of source terms.

An improved Serre model: Efficient simulation and comparative evaluation

• A new Serre model with improved dispersion characteristics is considered. • A careful development of the new model allows an efficient numerical implementation. • A splitting scheme based on high order finite volume and finite difference is used. • Numerical experiments illustrate the superiority of the proposed model. The so-called Serre or Green and Naghdi equations are a well-known set of fully nonlinear and weakly dispersive equations that describe the propagation of long surface waves in shallow water. In order to extend its range of application to intermediate water depths, some modifications have been proposed in the literature. In this work, we analyze a new Serre model with improved linear dispersion characteristics. This new Serre system, herein denoted by Serre α, β , presents additional terms of dispersive origin, thus extending its applicability to more general depth to wavelength ratios. A careful development of the Serre α, β model allows a straightforward and efficient numerical implementation. This model is suitable for numerical integration by a splitting strategy which requires the solution of a hyperbolic problem and a dispersive problem. The hyperbolic part is discretized using a high-order finite volume method. For the dispersive part standard finite differences are used. A set of numerical experiments are conducted to validate the Serre α, β model and to test the robustness of our numerical scheme. Theoretical solutions and benchmark experimental data are used. Moreover, comparisons against the classical Serre equations and against another well established Serre model with improved dispersion characteristics are also made.

Analysis of entropy generation in combined buoyancy-Marangoni convection of power-law nanofluids in 3D heterogeneous porous media

In this paper, entropy generation due to the combined buoyancy-Marangoni convection of non-Newtonian power-law nanofluids in a 3D heterogeneous porous cubic cavity is investigated in detail with the compact high order finite volume method. The fluid in the cavity is a CMC-based nanofluids containing CuO nanoparticles, and the numerical study is conducted for a wide range of heterogeneity level ranging from 0 to 1.5, Marangoni number from 0 to 1000, thermal Rayleigh number from 104 to 106, buoyancy ratio from −2.0 to −0.1, nanoparticle volume fraction from 0 to 0.1 and power-law index from 0.76 to 1.0 to identify the irreversibility characteristics. The results show that the total entropy generation is considerably affected by the heterogeneity in permeability; it increases when the level of heterogeneity increases. Marangoni number becomes a more effective parameter on total entropy generation for lower values of thermal Rayleigh numbers. For thermal dominated flow, the increase in the buoyancy ratio provokes various irreversibilities to enhance; on the other hand, for solutal dominated flow, the rise of the buoyancy ratio declines the entropy generation. Moreover, it is also shown that the enhancement of the nanoparticle volume fraction reduces the total entropy generation. Apart from that, our numerical tests show the flow field and entropy generation are influenced appreciably by the presence of the power-law index.

A Novel Multi-Dimensional Limiter for High-Order Finite Volume Methods on Unstructured Grids

AbstractThis paper proposes a novel distance derivative weighted ENO (DDWENO) limiter based on fixed reconstruction stencil and applies it to the second- and highorder finite volume method on unstructured grids. We choose the standard deviation coefficients of the flow variables as the smooth indicators by using the k-exact reconstruction method, and obtain the limited derivatives of the flow variables by weighting all derivatives of each cell according to smoothness. Furthermore, an additional weighting coefficient related to distance is introduced to emphasize the contribution of the central cell in smooth regions. The developed limiter, combining the advantages of the slope limiters and WENO-type limiters, can achieve the similar effect of WENO schemes in the fixed stencil with high computational efficiency. The numerical cases demonstrate that the DDWENO limiter can preserve the numerical accuracy in smooth regions, and capture the shock waves clearly and steeply as well.

A family of HLL-type solvers for the generalized Riemann problem

The generalized Riemann problem (GRP) is the initial value problem for a conservation law with piecewise smooth, but discontinuous initial data. We provide a new method for solving the GRP approximately, that can be used as a building block for high order finite volume or discontinuous Galerkin methods. Our new GRP solvers use the approximate states and wave speeds obtained through a HLL-type Riemann solver and use this information to build an approximation of the state in the GRP of any order. What is new about this approach compared to most previous solvers is that we no longer need to solve a classical Riemann problem exactly. We give a detailed explanation of this strategy for HLL and HLLC solvers for the Euler equations, as well as for the HLLD solver for MHD equations. We demonstrate the performance of the solvers from this new family of GRP solvers for a broad range of test problems.