Implicit Finite Volume Research Articles

Degenerate Parabolic Equation with Zero-flux Boundary Condition and its Approximations

We study a degenerate parabolic-hyperbolic equation with zero-flux boundary condition. The aim of this paper is to prove convergence of numerical approximate solutions towards the unique entropy solution. We propose an implicit finite volume scheme on admissible mesh. We establish fundamental estimates and prove that the approximate solution converge towards an entropy-process solution. Contrarily to the case of Dirichlet condition, in zero-flux problem unnatural boundary regularity of the flux is required to establish that entropyprocess solution is the unique entropy solution. In the study of well-posedness of the problem, tools of nonlinear semigroup theory (stationary, mild and integral solutions) were used in order to overcome this difficulty. Indeed, in some situations including the one-dimensional setting, solutions of the stationary problem enjoy additional boundary regularity. Here, similar arguments are developed based on the new notion of integral-process solution that we introduce for this purpose.

Withdrawn: Transient modeling of natural gas in pipeline networks by two non-iterative explicit and implicit finite volume methods

This paper describes two non-iterative explicit and implicit finite volume methods (FVM) to calculate the unsteady characteristics of natural gas in distribution systems. The proposed finite volume model could be used to solve the gas network as a whole for each time step. The main advantage of this approach is that it can analyze gas distribution networks with less computational cost. In addition, contrary to the conventional iterative algorithms, it does not consist of any divergence problems (Elman et al., 1996). To achieve this goal, the explicit FVM is derived from the governing equations. Then, the linearization process is performed, and the numerical solution of the linear equations is developed by using the implicit FVM. Three classic test cases are exemplified to check the performance of the proposed methods. The results confirm that both explicit and implicit methods, even in complicated distribution networks, can predict the elements of unsteady natural gas flows. Furthermore, when the governing equations are linearized and the implicit FVM is applied, satisfactory results with less computational cost, compared to the explicit one, are produced.

Laminar convective heat transfer combining buoyancy and thermal radiation of Carreau fluid within a concentric and eccentric annulus

The present study concerns a steady-state laminar combined thermal radiation and buoyancy-driven conjugate magnetohydrodynamic (MHD) natural convective flow and heat transfer inside a fully porous medium eccentric circular annulus bounded by two infinite horizontal cylinders occupied by a Carreau fluid permeated by a uniform transverse magnetic field (B0 ) and inclined by an angle (α). The convective and thermal radiation are approximated by the Boussinesq and Roseland models, respectively. The numerical integration was performed using the implicit finite volume method. The convective terms in the momentum and energy equations were Discretized using the second-order upwind method. On the other hand, the diffusive terms were Discretized using the central difference scheme. The pressure-velocity coupling was solved using the SIMPLE approach. The findings suggest a clear trend of enhancement in the heat transfer rate when the aspect ratio and annulus eccentricity increase. This pattern is more pronounced in the negative vertical eccentricities. It is worth noting that the eccentric annulus, which has an annulus aspect ratio of 1.6 and a vertical eccentricity of −0.8, offers the best performance regarding the heat transfer mechanism inside the annulus gap. The heat transfer mechanism turns out to be strongly affected by thermal radiation, Hartmann number, and magnetic field inclination angle. The increased thermal radiation parameter and the decreased flow behavior index improve the heat transfer mechanism between the Carreau fluid and the inner cylinder in the annular gap. Furthermore, the average Nusselt number increases considerably as the induced magnetic field inclines.

A general positivity-preserving algorithm for implicit high-order finite volume schemes solving the Euler and Navier-Stokes equations

This paper presents a general positivity-preserving algorithm for implicit high-order finite volume schemes that solve compressible Euler and Navier-Stokes equations to ensure the positivity of density and internal energy (or pressure). Previous positivity-preserving algorithms are mainly based on the slope limiting or flux limiting technique, which rely on the existence of low-order positivity-preserving schemes. This dependency poses serious restrictions on extending these algorithms to temporally implicit schemes since it is difficult to know if a low-order implicit scheme is positivity-preserving. In the present paper, a new positivity-preserving algorithm is proposed in terms of the flux correction technique. And the factors of the flux correction are determined by a residual correction procedure. For a finite volume scheme that is capable of achieving a converged solution, we show that the correction factors are in the order of unity with additional high-order terms corresponding to the spatial and temporal rates of convergence. Therefore, the proposed positivity-preserving algorithm is accuracy-reserving and asymptotically consistent. The notable advantage of this method is that it does not rely on the existence of low-order positivity-preserving baseline schemes. Therefore, it can be applied to the implicit schemes solving Euler and especially Navier-Stokes equations. In the present paper, the proposed technique is applied to an implicit dual time-stepping finite volume scheme with temporal second-order and spatial high-order accuracy. The present positivity-preserving algorithm is implemented in an iterative manner to ensure that the dual time-stepping iteration will converge to the positivity-preserving solution. Another similar correction technique is also proposed to ensure that the solution remains positivity-preserving at each sub-iteration. Numerical results demonstrate that the proposed algorithm preserves positive density and internal energy in all test cases and significantly improves the robustness of the numerical schemes.

Applicability of Zero-Inertia Approximation for Overland Flow Using a Generalized Mass-Conservative Implicit Finite Volume Framework

Applicability of Zero-Inertia Approximation for Overland Flow Using a Generalized Mass-Conservative Implicit Finite Volume Framework

Thermo-Electrochemical Simulation of Large-Format Li-Ion Cells in 3D Using the Battery Modelling Toolbox (BattMo)

Computer-based modelling and simulation can be a valuable tool when developing electrochemical devices: it offers possibilities for insight into internal properties not available in experiments, it can guide the development and testing of new materials in the device, it can provide time-efficient alternatives for optimizing operating procedures, and more. In this contribution we will give an overview of the Battery Modelling Toolbox (BattMo), which is an open-source framework for continuum modelling of electrochemical-thermal systems. It has been primarily developed for the modelling and simulation of Li-ion battery cells using pseudo X-dimensional (PXD, X=2,3,4) Doyle-Fuller-Newman models, but has also successfully been applied to electrolyzers. BattMo is developed in MATLAB and has a high-performance Julia version (BattMo.jl) in development.The fundamental equations of mass, charge and energy conservation are in BattMo discretized using a low-order implicit finite volume method. These are suitable methods, since they are conservative (meaning that mass, charge, etc., are conserved), and guarantee monotone solutions (meaning that the solutions will be positive if the data is positive). These properties provide a simulation foundation that is robust and stable also on complex 3D grids -- essential features when modelling real-world systems. As with most numerical methods, the most time-consuming part is to solve the resulting linear system of equations. In BattMo we make use of state-of-the-art block-wise multigrid methods to obtain high performance.Working with BattMo may be done using a GUI, which generates a json file which completely specifies a simulation. This json file can be used directly as input to BattMo via the MATLAB/Julia prompt, or used in cloud resources. It is also possible to use a json file as a base configuration and modify the parameters manually or programmatically, for example to investigate a specific system response when changing some parameters of interest. BattMo also provides functionality for automatic gradient-based optimization and parameter estimation that can be used for maximizing or minimizing some goal quantity, and matching to experimental data. A key component for performing efficient gradient-based optimization and parameter estimation in the case of many parameters (for example when having one parameter value in each grid cell) is the use of so-called adjoints. Using adjoints allow for the gradients in the sensitivity analysis to be computed with a constant cost that is independent on the number of parameters, thus avoiding the curse of dimensionality.Much attention is spent on making the BattMo json input compatible with other simulation frameworks such as PyBamm, COMSOL and CideMod. This is to accelerate research and development by taking advantage of the strengths of different packages, as well as providing not only reproducible but also reliable science. BattMo is developed alongside BattINFO, which is a battery interface ontology based on EMMO (Elementary Multiperspective Material Ontology) that is developed in the BIG-MAP EU H2020 project. We also seek to support the BPX standard developed by the PyBamm team.In this talk we will overview the basic features of BattMo and see how the modular approach used in the software simplifies development of new models. BattMo contains many examples, and we will here present thermal simulation of 1D and 3D cells with LFP and LNMO battery chemistries in pouch cell format and investigate the effects of the tabless design in the 4680 cylindrical cell format (see the Figure below).BattMo is available at https://github.com/BattMoTeam and is developed with the support from the EU H2020 innovation programs under the grants Hydra (875527) and BIG-MAP (957189). Figure 1

Numerical Simulation of Free Surface Water Waves Around Wavy Hydrofoils: Prediction of Hydrodynamic Coefficients Using Machine Learning

Abstract This study uses computational methods to investigate the fluid flow characteristics around a wavy NACA 63(4)-021 hydrofoil near the water surface. It introduces notable contributions to the literature on sinusoidal edge hydrofoils by examining the behavior of this specific hydrofoil configuration close to the water surface, a previously unexplored aspect. Furthermore, the study presents a comprehensive review and analysis of various machine learning (ML) techniques applied to computational fluid dynamics (CFD) and experimental data, offering a novel approach in this field. The three-dimensional Reynolds-averaged Navier–Stokes (RANS) equations are solved using an implicit finite volume approach to simulate the turbulent flow around the hydrofoil near the free surface. The realizable k–ε turbulence model accounts for turbulent flow effects at varying submergence depths. Four machine learning models in Python are developed to predict lift and drag coefficients. Evaluation of these models on training and test datasets reveals that xgboost achieves the highest accuracy, with an impressive R2 score of 0.9775 on the test dataset. Therefore, xgboost is recommended as the optimal model for future applications in this context. The study's findings are presented for the wavy hydrofoil at three submergence depths and three angles of attack (AOA). Key results include wave profiles, magnitudes, total pressure contours around the hydrofoil and free surface, and pressure, lift, and drag coefficients. The accuracy of the numerical simulations is validated by comparing the results with available experimental data, demonstrating good agreement between the two.

Influence of External Magnetic Field on 3D Thermocapillary Convective Flow in Various Thin Annular Pools Filled with Silicon Melt

Thermocapillary convection flows can have an impact on the homogeneity of floating zone semiconductor crystals. An external magnetic field can also help to reduce this non-homogeneity. The goal of this research is to minimize thermocapillary convection in various thin annular pools filled with silicon melt. A three-dimensional (3D) numerical technique is proposed that employs an implicit finite volume formulation. The steady-state thermocapillary flow in six thin annular pools (R=0.3, 0.4, 0.5, 0.6, 0.7, and 0.8) subjected to an externally induced magnetic field was observed. Under magnetic field influence, the effects of increasing annular gap, R on the hydrothermal wave number and azimuthal pattern are obtained. The results reveal that hydrothermal waves m=14, m=11, m=8, m=6, m=4, and m=3 are observed in steady flow for R=0.3; 0.4; 0.5; 0.6; 0.7, and R=0.8, respectively. The maximum temperature occurs in the intermediate zone between the inner and outer walls when there is no magnetic field. Under a strong enough magnetic field, isothermal lines change form and become concentric circles. As the amplitude of the magnetic field (Ha) grows, the azimuthal velocity and temperature at the free surface reduce, and the asymmetric 3D flow becomes axisymmetric steady when Ha surpasses a threshold value.

An unstructured finite-volume level set / front tracking method for two-phase flows with large density-ratios

We extend the unstructured LEvel set / froNT tracking (LENT) method [1,2] for handling two-phase flows with strongly different densities (high-density ratios) by providing the theoretical basis for the numerical consistency between the mass and momentum conservation in the collocated Finite Volume discretization of the single-field two-phase Navier-Stokes equations. Our analysis provides the theoretical basis for the mass conservation equation introduced by Ghods and Herrmann [3] and used in [4–8]. We use a mass flux that is consistent with mass conservation in the implicit Finite Volume discretization of the two-phase momentum convection term, and solve the single-field Navier-Stokes equations with our SAAMPLE segregated solution algorithm [2]. The proposed ρLENT method recovers exact numerical stability for the two-phase momentum advection of a spherical droplet with density ratios ρ−/ρ+∈[1,104]. Numerical stability is demonstrated for in terms of the relative L∞ velocity error norm, for density-ratios in the range of [1,104], dynamic viscosity-ratios in the range of [1,104] and very strong surface tension forces, for challenging mercury/air and water/air fluid pairings. In addition, the solver performs well in cases characterized by strong interaction between two phases, i.e., oscillating droplets and rising bubbles. The proposed ρLENT method1 is applicable to any other two-phase flow simulation method that discretizes the single-field two-phase Navier-Stokes Equations using the collocated unstructured Finite Volume Method but does not solve an advection equation for the phase indicator using a flux-based approach, by adding the proposed geometrical approximation of the mass flux and the auxiliary mass conservation equation to the solution algorithm.

CFD Simulation of Low Speed Turbulent Flow Problems Using Unsteady Rans and Large Eddy Simulation Approach

The present paper focuses on the recent development of an implicit pressure-based finite volume algorithm for numerical solution of Navier Stokes equation in an inertial frame of reference for prediction of unsteady incompressible flow problems. The algorithm uses boundary-conforming, multiblock structured grid with moving boundaries, collocated variable arrangement with momentum equations resolved along cartesian directions, second order accurate spatial and temporal discretisation schemes for the convective fluxes and a pressurevelocity solution strategy. Effect of turbulence is simulated using one of the two different approaches. In the Unsteady Reynolds Averaged (URANS) approach coupled to appropriate eddy viscosity based turbulence models, the Navier Stokes (NS) equations, averaged over the whole range of turbulent length scales of the flow, are solved numerically. On the other hand, in the Large Eddy Simulation (LES) approach, the model filtered 3D NS equations are directly solved for the flow variables to resolve the large scale turbulent motions whereas the transport processes at the fine subgrid scale level only are simulated using simple algebraic turbulence model. The capabilities and limitations of both the cost-effective URANS approach and the relatively expensive but rich in physics LES approach have been demonstrated for few application problems of engineering interest.

Error estimates for a finite volume scheme for advection–diffusion equations with rough coefficients

We study the implicit upwind finite volume scheme for numerically approximating the advection–diffusion equation with a vector field in the low regularity DiPerna–Lions setting. That is, we are concerned with advecting velocity fields that are spatially Sobolev regular and data that are merely integrable. We prove that on unstructured regular meshes the rate of convergence of approximate solutions generated by the upwind scheme towards the unique solution of the continuous model is at least one. The numerical error is estimated in terms of logarithmic Kantorovich–Rubinstein distances and provides a bound on the rate of weak convergence.

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.

MHD Effect on the Thermocapillary Silicon Melt Flow in Various Annular Enclosures

AbstractTo simulate thermocapillary magnetohydrodynamic (MHD) convection in different shallow annular enclosures filled with silicon melt, a 3D numerical approach using an implicit finite volume formulation is used. Radiation is emitted upward from the annular enclosure's free top surface, the bottom one heated vertically and is subjected to an external magnetic field. The steady and unsteady thermocapillary MHD flow in four annular enclosures (R = 0.5, 0.6, 0.7, and 0.8) field is studied. The effects of varying the annular gaps, R, on the hydrothermal wave number and azimuthal pattern are obtained. The effects of the magnetic field on azimuthal velocity, temperature disturbances, and the transition from oscillatory to steady flow are also depicted. The results reveal that: hydrothermal waves m = 4, m = 6 and m = 8 are observed in steady flow for R = 0.7, R = 0.6 and R = 0.5, respectively. Oscillatory flows dominated by a hydrothermal wave m = 3 are found also when R = 0.8. The azimuthal velocity and temperature fluctuations at the free surface decrease as the magnitude of the magnetic field (Ha) increases, and the oscillatory flow would turn steady when Ha exceeds a critical value.

Computation of the self-diffusion coefficient with low-rank tensor methods: application to the simulation of a cross-diffusion system

Cross-diffusion systems arise as hydrodynamic limits of lattice multi-species interacting particle models. The objective of this work is to provide a numerical scheme for the simulation of the cross-diffusion system identified in [J. Quastel, Comm. Pure Appl. Math., 45 (1992), pp. 623–679]. To simulate this system, it is necessary to provide an approximation of the so-called self-diffusion coefficient matrix of the tagged particle process. Classical algorithms for the computation of this matrix are based on the estimation of the long-time limit of the average mean square displacement of the particle. In this work, as an alternative, we propose a novel approach for computing the self-diffusion coefficient using deterministic low-rank approximation techniques, as the minimum of a high-dimensional optimization problem. The computed self-diffusion coefficient is then used for the simulation of the cross-diffusion system using an implicit finite volume scheme.

Implicit and implicit-explicit Lagrange-projection finite volume schemes exactly well-balanced for 1D shallow water system

In this paper we consider the Lagrange-Projection technique in the framework of finite volume schemes applied to the shallow water system. We shall consider two versions of the scheme for the Lagrangian step: one fully implicit and one implicit-explicit, based on how the geometric source term is treated. First and second order well-balanced versions of the schemes are presented, in which the water at rest solutions are preserved. This allows to obtain efficient numerical schemes in low Froude number regimes, as the usual CFL restriction driven by the acoustic waves is avoided.

Combined transient 3D simulation and experimental methods to assess a slow heat-up curve used to dry a refractory concrete

Explosive spalling is a major concern when drying refractory concretes. For safety, empirically defined overestimated slow heat-up curves are used in industry. Polypropylene fibres reduce spalling; however, they are inconvenient for certain castables due to properties deterioration, such as slag penetration. There is neither a generally accepted methodology to define monolithic refractories dry-out schedules nor agreement about a mathematical model to predict spalling phenomena. This paper presents original numerical and experimental results of castable drying investigation using slow heating. Modifications in experimental methods were made to reproduce refractory lining boundary conditions. A conservative fully implicit finite volume method in three-dimensions was developed to predict pressure and moisture migration. The influence of the methods used to characterize the properties needed for the mathematical model was discussed. The combined analyses suggest that the first holding time used was not productive for drying; however, its effect on curing must be reviewed.

A Comparative Study of Entropy Generation Analysis Through Cartesian and Cylindrical Annulus Geometries

Abstract This research paper presents a comparative study dealing with entropy generation analysis in Cartesian and cylindrical annulus geometries. This numerical investigation involves heat and mass transfer coupled with forced convection and gas radiation in a semitransparent medium during the evaporation of a thin liquid film. The systems of equations are solved with interfacial and boundary conditions using an implicit finite volume method by the tridiagonal matrix algorithm method. The radiative transfer equation (RTE) was solved by the Ray tracing model. The statistical narrow band correlated-K model with the updated model parameters was applied to calculate the gas transmissivity. The main intent of this paper is to compare the entropy generation components in two different configurations. The result shows that the entropy generation related to heat and mass transfer in the channel is higher than that estimated for the cylindrical annulus. Regarding optimal design selection, the entropy generation analysis, detailed in this paper, suggests a preference for the cylindrical annulus geometry. Our results may guide designers to select from the studied configuration.

Conservation Properties of Iterative Methods for Implicit Discretizations of Conservation Laws

Conservation properties of iterative methods applied to implicit finite volume discretizations of nonlinear conservation laws are analyzed. It is shown that any consistent multistep or Runge-Kutta method is globally conservative. Further, it is shown that Newton’s method, Krylov subspace methods and pseudo-time iterations are globally conservative while the Jacobi and Gauss-Seidel methods are not in general. If pseudo-time iterations using an explicit Runge-Kutta method are applied to a locally conservative discretization, then the resulting scheme is also locally conservative. However, the corresponding numerical flux can be inconsistent with the conservation law. We prove an extension of the Lax-Wendroff theorem, which reveals that numerical solutions based on these methods converge to weak solutions of a modified conservation law where the flux function is multiplied by a particular constant. This constant depends on the choice of Runge-Kutta method but is independent of both the conservation law and the discretization. Consistency is maintained by ensuring that this constant equals unity and a strategy for achieving this is presented. Simulations show that this strategy improves the convergence rate of the pseudo-time iterations. Experiments with GMRES suggest that it also suffers from inconsistency but that this is automatically accounted for after some finite number of iterations. Similar experiments with coarse grid corrections based on agglomeration indicate no inconsistency.

Study on Numerical Algorithm of the N-S Equation for Multi-Body Flows around Irregular Disintegrations in Near Space

There has been a concern that the accurate numerical simulation of multi-body flow, which is caused by the multiple disintegrations of expired spacecraft re-entering into the near space, has a critical bottleneck impact on the falling area of the disintegrated debris. To solve this problem, an O-type grid topology method has been designed for the multi-body flow field of irregular debris formed by multiple disintegrations in near space, and a finite-volume implicit numerical scheme has been constructed for the Navier-Stokes equations to solve the aerodynamic interference characteristics of irregular multi-body flow, and further the N-S equation numerical algorithm has been established for the irregular multi-body flows in near space. The reliability of the method has been verified by the comparison of the present computation and the experiment of the low-density wind tunnel for the two-body flow of sphere, cylinder and square scripts. The objects of this study are from the multiple disintegrations of the Tiangong-1 spacecraft during uncontrolled re-entry into the atmosphere, including propelling cylinders and low-temperature lock cabinets. A series of simulations of multi-body flow mechanisms around different combinations have been carried out with varied shapes and spacing. As a result, it is found that when the distance of irregular debris (e.g., two propelling cylinders) in the near space is in the range of Δy < 3D or Δx < D, there is an obvious multi-body interference between debris, and the flow characteristics are obviously changed. When the distance between the debris in near space reaches a certain level, the influence of mutual interference can be ignored. For example, when the y-direction distance between multiple bodies is greater than 3D, the flow interference tends to be small and can be ignored, and we can regard them as two separate pieces to be carried out by the numerical prediction of flight track and falling area in engineering application. The results provide a practical design criterion for the integrated simulation platform which is used to simulate the multi-physics complex aerodynamics of space vehicles from the free-molecule flow of the outer space to the near-ground continuum flow.

Computationally inexpensive semi-analytical thermal model to predict melt-pool dimensions for a single-track in Selective Laser Melting

Selective laser melting (SLM) has recently gained momentum as a state-of-the-art manufacturing process capable of building highly customized products. Localized heat input and short interaction time of laser beam intensify thermal gradient and deteriorate the part's quality due to thermal stresses and uncontrolled distortions. An algorithm is proposed that comprises of analytical solution for volumetric moving heat source combined with numerically formulated conduction and convection cooling to estimate the temperature distribution and melt-pool geometry during the SLM process. Fully implicit finite volume equations are framed with appropriate part boundary conditions for the two-dimensional domain and iteratively solved by Alternating Direction Implicit (ADI) method. The proposed methodology has shown the potential to develop a deep understanding of this dynamic process and deliver results at low computational costs. Simulations for single-track melting have been studied by changing the process parameters, and phase change from powder to solid is well exhibited by adopting temperature-dependent material properties. The proposed model has been validated by experimental data from the literature for two alloys - Inconel 718 and SS316L.