Framework Of Finite Volume Method Research Articles

Research on the solution of the thermo-hydro-mechanical-chemical coupling model based on the unified finite volume method framework

Heat extraction from hot dry rock (HDR) reservoirs involves complex thermal–hydraulic-mechanical-chemical (THMC) coupling processes, which pose significant challenges for efficient geothermal energy utilization. The current frameworks that combine the finite volume method (FVM) and finite element method (FEM) for THMC modeling based on the embedded discrete fracture model (EDFM) are separate, leading to complex implementations and hindering the development of efficient solutions. To address these issues, we propose an FVM-based THMC coupling model that solves fluid flow, heat transfer, and chemical reactions using the finite volume method (FVM), while mechanical deformation is addressed using the extended finite volume method (XFVM). The reaction module incorporates PHREEQC for multicomponent reactions under high-temperature and high-pressure conditions. The model is validated against experimental data for mineral reactions under high-temperature and high-pressure conditions, as well as XFEM and COMSOL simulations for displacement fields, confirming its accuracy and computational performance. Application results show that the temperature deviation between multicomponent and single-component models can reach up to 14 °C, underscoring the importance of incorporating multicomponent water–rock reactions in THMC coupling. This model accurately captures the spatiotemporal evolution of THMC processes and efficiently predicts long-term heat production in enhanced geothermal systems (EGS) reservoirs. It could significantly advance the ability to achieve more accurate and reliable predictions for the engineering development and efficient utilization of EGS reservoirs.

Dynamics of granular debris flows against slit dams based on the CFD–DEM method: effect of grain size distribution and ambient environments

AbstractEarth surface flows in nature, like debris flows and rock avalanches, have threatened people’s safety and infrastructure during past decades. Though grain size distribution (GSD) has been acknowledged as a crucial characteristic in granular material behaviour, its coupled effects associated with environments on engineering structures such as the slit dam remain unclear. To bridge the gap, this paper reveals the coupled effect of the GSD and ambient environments (i.e. slope angles and saturation conditions) on avalanche/debris flows’ impact on the slit dam using a Computational Fluid Dynamics/Discrete Element Method (CFD–DEM) model. To describe strain-dependent rheological characteristics of debris fluids, the Herschel–Bulkley–Papanastasiou model is implemented in the finite volume method framework. A power grain size distribution law is considered to quantify GSDs, in which a fractal parameter takes charge of GSD types. After model verification with experimental/theoretical results, the impact force against slit dams, granular dynamics and final deposit patterns under a series of ambient circumstances are presented. Taking advantage of the CFD–DEM method, the impact force and kinetic energy induced by fluid and solid phases are discriminated. The contribution of solid and fluid phases to both impact force and dynamics appears to be dependent on GSDs. Accordingly, compared with saturated avalanche flows (i.e. debris flows), slit dams result in higher retaining efficiency when confronted with dry avalanche flows. Regarding a narrow diameter range used in analyses, the grain diameter ratio is then enlarged up to eight to reveal the potential size effect. As for the coupled role of GSDs and slope angles, in contrast to slope angles, the influence of GSD on avalanche flow interaction with slit dams is much smaller. Additionally, provided a narrow diameter range, the effect of GSDs on impact force can be partially attributed to the change in average grain diameter. After presenting the significance of ambience and GSDs to avalanche/debris flows, a series of parametric studies around the effect of fluid grid size, particle shape and the initial porosity of granular samples are discussed, aiming to advance the understanding of their influence in the interactions between debris flows and the slit dam.

A unified consistent source term computational algorithm for the [formula omitted]-based compressible multi-fluid flow model

Using the criterion of one-fluid preservation, we developed a consistent algorithm for the source term to solve the γ-based compressible multi-fluid flow model with three approximate Riemann solvers, namely the Lax–Friedrichs (LxF), Kurganov, and Harten–Lax–van Leer contact (HLLC) solvers. The consistent algorithm comprises a standard Godunov solver with a high-order reconstruction and a consistent source term integration part. We prove that the present algorithm is consistent with Abgrall's criterion of the moving-material-interface property in the finite volume method framework. The cell boundary velocity for the source term discretization is found to be the same as HLLC solvers from five-equation model, but for the first time for LxF and Kurganov Riemann solvers. We develop 12 compressible multi-fluid solvers by combining four reconstruction schemes and three approximate Riemann solvers together with their consistent-source-term integration algorithms. All 12 solvers can maintain the one-fluid preservation and moving-material-interface properties numerically in several one- and two-dimensional example cases. The simulation of an underwater explosion demonstrates that a boundary-variation-diminishing reconstruction predicts an interface with width controlled within approximately three cells which is independent of the Riemann solver. The simulation of an explosion near a free surface demonstrates that the proposed model can simulate a severe compressible multi-fluid flow involving an interface across which there are large differences in density, pressure and parameters in the equation of state. In conclusion, the proposed consistent algorithm provides a unified framework for one kind of non-conservative hyperbolic system into a conservative hyperbolic system and a source term with velocity divergence, where the former can be computed by classical Godunov-type algorithms and the latter can be solved by the proposed consistent algorithm.

A new two-phase shallow water hydro-sediment-morphodynamic model based on the HLLC solver and the hybrid LTS/GMaTS approach

In this paper, we present a new computationally-efficient and high-resolution depth-averaged two-phase flow model for hydro-sediment-morphydynamic processes, featuring an advance over existing models in terms of accuracy and efficiency of numerical solution. Under the framework of finite volume method (FVM) on unstructured grids, the Harten-Lax-van Leer-Contact (HLLC) approximate Riemann solver is proposed to compute inter-cell fluxes by applying the classical upwind HLLC approach to the water-sediment mixture and the sediment phases separately, in contrast to previous two-phase flow models using centered schemes. Moreover, to improve computational efficiency, the local-time-stepping (LTS) approach is implemented, the first attempt in the field of two-phase flow modelling. After a convergence rate study, the model is tested against a series of flow-sediment-bed evolutions induced by (1) two refilling processes of dredged trenches, (2) two instantaneous dam-break flooding flows, and (3) one levee breaching process by overtopping flows. It features encouraging performance when compared to a two-phase flow model based on a centered scheme and global time bound, characterized by more accurate results and much less computational cost. The present modelling framework shows promise in practical shallow water hydro-sediment-morphodynamic modelling applications.

Effects of cell quality in grid boundary layer on the simulated flow around a square cylinder

The flow around a square cylinder is widely studied as a paradigmatic case in bluff body aerodynamics. The effects of several physical parameters of the setup, and the errors induced by turbulence models, numerical schemes and grid density have been emphasized in a huge number of studies during the past two decades. Surprisingly, the effects of the grid quality on such a class of flow has been overlooked. The lack of a shared approach and suggested best practices for high-quality grid generation among scholars and practitioners follows. The present study aims at filling this gap. The cell skewness and non-orthogonality are adopted as metrics of the grid quality. The errors induced by poor quality cells and the possible corrective measures are discussed in a Finite Volume Method framework. The effects of the cell quality on the simulated flow are systematically evaluated by a parametrical study including four different types of grid boundary layer. The obtained results are compared among them and discussed in terms of instantaneous and time-averaged flow fields, stress distribution at wall, and aerodynamic coefficients. Both the overall modelling error and the skewness-induced one are evaluated with reference to a huge number of data collected from previous studies. The local error induced by few, moderately skewed, near-wall cells upwind the cylinder propagates windward because of the convection-dominated problem, and globally affects the boundary layer separation and the vortex shedding in the wake. Skewness around the trailing edge only affects the flow to a lower extent. The skewness error on bulk aerodynamic coefficients may largely prevails on the overall modelling error, in spite of the very simple turbulence model deliberately adopted in the study. Hybrid grid boundary layer made of structured cells along the cylinder sides and unstructured ones around its edges provides results analogous to the ones obtained with a fully orthogonal grid, in spite of some clusters of few skewed cells far from the wall. Hybrid grid boundary layer is recommended as a fine balance between accuracy and flexibility in grid generation, when full orthogonal grid boundary layer is not feasible around real-world engineering applications having complicate geometries with multiple obtuse or acute edges.

A conservative level set method for liquid-gas flows with application in liquid jet atomisation

In this paper, a methodology for modelling two-phase flows based on a conservative level set method in the framework of finite volume method is presented. The novelty of the interface capturing method used here lies on the advection of level set which is solved with a WENO scheme and corrected with a novel re-initialisation method for retaining its signed distance function character. The coupling with the volume of fluid method is done with a simple algebraic approach, and with the new algorithm the accumulated mass conservation errors remain reasonably low. The paper presents a unique coupling between the level set method and the Eulerian-Lagrangian Spray Atomisation approach for modelling spray dispersion in liquid atomisation systems. The method is shown to have good accuracy providing similar results to other numerical codes for the classical tests presented. Preliminary results are also shown for three-dimensional simulations of the primary break-up of a turbulent liquid jet obtaining results comparable to direct numerical simulations. Consequently, the coupled method can be used for simulating various two-phase flow applications offering an accurate representation of the interface dynamics.

Porous Shallow Water Modeling for Urban Floods in the Zhoushan City, China

Typhoon-induced intense rainfall and urban flooding have endangered the city of Zhoushan every year, urging efficient and accurate flooding prediction. Here, two models (the classical shallow water model that approximates complex buildings by locally refined meshes, and the porous shallow water model that adopts the concept of porosity) are developed and compared for the city of Zhoushan. Specifically, in the porous shallow water model, the building effects on flow storage and conveyance are modeled by the volumetric and edge porosities for each grid, and those on flow resistance are considered by adding extra drag in the flow momentum. Both models are developed under the framework of finite volume method using unstructured triangular grids, along with the Harten-Lax-van Leer-Contact (HLLC) approximate Riemann solver for flux computation and a flexible dry-wet treatment that guarantee model accuracy in dealing with complex flow regimes and topography. The pluvial flooding is simulated during the Super Typhoon Lekima in a 46 km2 mountain-bounded urban area, where efficient and accurate flooding prediction is challenged by local complex building geometry and mountainous topography. It is shown that the computed water depth and flow velocity of the two models agree with each other quite well. For a 2.8-day prediction, the computational cost is 120 min for the porous model using 12 cores of the Intel(R) Xeon(R) Platinum 8173M CPU @ 2.00 GHz processor, whereas it is as high as 17,154 min for the classical shallow water model. It indicates a speed-up of 143 times and sufficient pre-warning time by using the porous shallow water model, without appreciable loss in the quantitative accuracy.

A finite volume method for continuum limit equations of nonlocally interacting active chiral particles

The continuum description of active particle systems is an efficient instrument to analyze a finite size particle dynamics in the limit of a large number of particles. However, it is often the case that such equations appear as nonlinear integro-differential equations and purely analytical treatment becomes quite limited. We propose a general framework of finite volume methods (FVMs) to numerically solve partial differential equations (PDEs) of the continuum limit of nonlocally interacting chiral active particle systems confined to two dimensions. We demonstrate the performance of the method on spatially homogeneous problems, where the comparison to analytical results is available, and on general spatially inhomogeneous equations, where pattern formation is predicted by kinetic theory. We numerically investigate phase transitions of particular problems in both spatially homogeneous and inhomogeneous regimes and report the existence of different first and second order transitions.

Simulation and analysis of LPBF multi-layer single-track forming process under different particle size distributions

Laser powder bed fusion (LPBF) has the advantages of rapid and customized production and has gradually been used in aerospace, biomedicine, and other fields. However, the current LPBF technology still faces many obstacles, such as the failure to fully explore the influence of the powder layer parameters (particle size distribution, thickness of spreading layer, etc.) on the LPBF multi-layer forming process. Based on the open-source discrete element method framework Yade and the open-source finite volume method framework OpenFOAM, the simulation flow for the LPBF multi-layer single-track process was established. Among them, the considered force influence factors included gasification recoil, surface tension, Marangoni effect, viscous force, mushy zone drag force, and gravity, and the considered thermal influence factors included gasification, convection, and radiation heat dissipation. In order to analyze the influence of particle size distribution and spreading layer thickness on the LPBF multi-layer process, the forming processes of the first eight layers of powder bed with three different particle size distributions and two different spreading layer thicknesses were calculated. Regarding the surface roughness of the formed layer, it was found that when the proportion of large-size particles increased, the surface roughness of the formed layer would increase. Regarding the actual thickness of the powder layer and the depth of the solidified track, it was found that they were mainly related to the thickness of the spreading layer, but were not affected by the particle size distribution. Regarding the porosity of the formed zone, it was found that when the spreading layer thickness was smaller, the higher the proportion of small-size particles, the lower the porosity of the formed zone; when the spreading layer thickness was larger, the smaller the proportion of large-size particles, the lower the porosity of the formed zone. This paper is expected to provide support for in-depth understanding of the effects of particle size distribution and spreading layer thickness on the LPBF multi-layer forming process.

Mesoscopic-scale numerical investigation including the influence of scanning strategy on selective laser melting process

Selective laser melting (SLM) has become one of the most widely used laser additive manufacturing technologies today. However, the proper selection of scanning strategies is still a key concern for SLM production. In order to quantify the influences of different scanning strategies (One-direction, Two-directions, and Pre-sinter) on the SLM process, this paper calculated the spreading powder process based on the open-source discrete element method framework Yade, predicted the SLM mesoscopic molten pool dynamics behavior based on the open-source finite volume method framework OpenFOAM, and established the simulation flow of the SLM multi-layer multi-path forming process. The influences of different scanning strategies on grain orientation, pore defect, and surface roughness were analyzed and verified in comparison with experimental results. It is found that the grain orientation of the current formed layer under scanning strategies One-direction and Pre-sinter was almost the same as that of the previous formed layer, while the grain orientation of the current formed layer under scanning strategy Two-directions was significantly different from that of the previous formed layer. In addition, the porosity and surface roughness under scanning strategy Pre-sinter were lower than those under scanning strategies One-direction and Two-directions. This paper is expected to provide a basis for the selection of scanning strategies in practical SLM production.

Mesoscopic-Scale Numerical Investigation Including the Inuence of Process Parameters on LPBFMulti-Layer Multi-Path Formation

As a typical laser additive manufacturing technology, laser powder bed fusion (LPBF) has achieved demonstration applications in aerospace, biomedical and other fields. However, how to select process parameters quickly and reasonably is still the main concern of LPBF production. In order to quantitatively analyze the inuence of different process parameters (laser power, scanning speed, hatch space and layer thickness) on the LPBF process, the multi-layer and multi-path forming process of LPBF was predicted based on the open-source discrete element method framework Yade and the open-source finite volume method framework OpenFOAM. Based on the design of experiments method, a four-factor three-level orthogonal test scheme was designed, and the porosity and surface roughness data of each calculation scheme were extracted. By analyzing the orthogonal test data, it was found that as the laser power increased, the porosity decreased, and as the scanning speed, hatch space, and layer thickness increased, the porosity increased. In addition, the inuence of laser power and scanning speed on surface roughness showed a trend of decreasing first and then increasing, while the inuence of scanning distance and layer thickness on surface roughness showed a monotonous increasing trend. The order of the inuence of each process parameter on porosity was: scanning speed > laying thickness > laser power > hatch space, and the order of the inuence of each process parameter on surface roughness was: hatch space > layer thickness > laser power > scanning speed. So the porosity of the part is most sensitive to scanning speed, and the surface roughness is the most sensitive to hatch space. The above conclusions are expected to provide process control basis for actual LPBF production of the 316L stainless steel alloy.

Viscous multi-species lattice Boltzmann solver for simulating shock-wave structure

The Onsager Bhatnagar Gross Krook kinetic model proposed by Mahendra et al. [32, 33] has been extended to arrive at the viscous compressible flow equations for a multi-species diffusion problem. The multi-species model ensures (i) correct exchange coefficients, (ii) positivity, (iii) that it follows indifferentiability principle and (iv) that correct amount of entropy get added as per the Onsager’s maximum entropy production principle (MEPP). A Lattice Boltzmann method (LBM) has been developed using the derived methodology within the framework of finite volume method (FVM). This solver incorporates the physics of polyatomic gases with internal degrees of freedom with flexible Prandl number. This solver can simulate the problem with species having different specific heat ratio and molecular weights. Although the equations are derived for multi-species, only binary test cases have been considered in this paper. The solver has been validated against binary shock-wave structure problems using experimental and theoretical data. Two dimensional Richtmyer Meshkov instability test case has been compared with and without viscous effects. The diffusive (viscous and mass diffusion) effects are manifested in the reduction in amplitude and increment in width of the mushroom structure. In the framework of FVM, the proposed scheme of LBM is efficient with good resolution of solutions and is easily adaptable.

A second-order cell-centered Lagrangian scheme with a HLLC Riemann solver of elastic and plastic waves for two-dimensional elastic-plastic flows

In this paper, we propose a fast and efficient second-order cell-centered Lagrangian scheme for 2D elastic-plastic flows with the hypo-elastic constitutive model and von Mises' yielding condition. First, we develop a novel HLLC-type Riemann solver with elastic and plastic waves (HLLCEP) for 2D elastic-plastic flows. Then, we present a two-directional momentum conservative method to determine the moving speed of grid vertexes. Moreover, we introduce a special symmetry-preserving second-order reconstruction method for scalars, vectors or tensors in order to keep the good symmetric property of the proposed second-order scheme. Finally, a second-order cell-centered Lagrangian scheme, based on the developed Riemann solver (HLLCEP) and the finite volume method framework, is constructed. A number of numerical tests have been carried out, and the numerical results show that the proposed scheme reaches the second-order accuracy for problems with smooth solutions, and is essentially non-oscillatory, and appears to be convergent and symmetric. Moreover, the current HLLCEP Riemann solver is more efficient than the FRRSE solver developed in [11].

A Two-Layer Hydrostatic-Reconstruction Method for High-Resolution Solving of the Two-Layer Shallow-Water Equations over Uneven Bed Topography

In this study, attention is focused on numerically solving the two-dimensional, two-layer nonlinear shallow-water equations (2LSWEs) over uneven bed topography. A two-layer hydrostatic-reconstruction method (2LHRM) is proposed for face value reconstructions. A numerical model is then developed based on the 2LHRM in the framework of finite-volume methods using the slope-limited centered (SLIC) approximate Riemann solver for flux evaluations. The validations against various benchmark tests show that the 2LHRM by working with the SLIC scheme predicts robust solutions around large gradients and is able to simulate lake-at-rest solutions without using any ad-hoc techniques.

Trough OpenMP Platform for Reducing Computational Time Cost in Underwater Landslide Simulation on Inclined Bottom

Simulation of underwater landslide becomes important, since underwater landslide phenomena is very dangerous in real life. One of the enormous disasters caused by this phenomena can be a Tsunami. Computer simulation of underwater landslide can reduce cost of time and money from conventional simulation (using laboratory). However, to obtain high resolution of computer simulation, large discrete points should be computed. In this paper, the numerical simulation of underwater landslide using two-layers shallow water equations (SWE) and OpenMP platform is elaborated. Here, the finite volume method framework using upwinding dispersive correction hydrostatic reconstruction (UDCHR) scheme is used. The results of numerical simulation is in a good agreement with the numerical simulation using Nasa-Vof2d numerical scheme. In parallel performance, speedup and efficiency of this numerical simulation are observed 2.8 times and 76% respectively at t=0.8 s final time simulation.

Shapes of an Air Taylor Bubble in Stagnant Liquids Influenced by Different Surface Tensions

The aim of this work is to propose an empirical model for predicting shapes of a Taylor bubble, which is a part of slug flows, under different values of the surface tension in stagnant liquids by employing numerical simulations. The k - Ɛ turbulence model was used in the framework of finite volume method for simulating flow fields in a unit of slug flow and also the pressure distribution on a Taylor bubble surface. Assuming that an air pressure distribution inside the Taylor bubble must be uniform, a grid search method was exploited to find an appropriate shape of a Taylor bubble for six values of surface tension. It was found that the shape of a Taylor bubble would be blunter if the surface tension was increased. This was because the surface tension affected the Froude number, controlling the flow around a Taylor bubble. The simulation results were also compared with the Taylor bubble shape, created by the Dumitrescu-and-Taylor model and former studies in order to ensure that they were consistent. Finally, the empirical model was presented from the simulation results.

Implicit finite volume simulation of 2D shallow water flows in flexible meshes

In this work, an implicit method for solving 2D hyperbolic systems of equations is presented, focusing on the application to the 2D shallow water equations. It is based on the first order Roe’s scheme, in the framework of finite volume methods. A conservative linearization is done for the flux terms, leading to a non-structured matrix for unstructured meshes thus requiring iterative methods for solving the system. The validation is done by comparing numerical and exact solutions in both unsteady and steady cases. In order to test the applicability of the implicit scheme to real world situations, a laboratory scale tsunami simulation is carried out and compared to the experimental data. The implicit schemes have the advantage of the unconditional stability, but a quality loss in the transient solution can appear for high CFL numbers. The properties of the scheme are well suited for the simulation of unsteady shallow water flows over irregular topography using all kind of meshes.

Second-order accurate genuine BGK schemes for the ultra-relativistic flow simulations

This paper presents second-order accurate genuine BGK schemes in the framework of finite volume method for the ultra-relativistic flows. Different from the existing kinetic flux-vector splitting (KFVS) or BGK-type schemes for the ultra-relativistic Euler equations, the present schemes are derived from the analytical solution of the Anderson–Witting model, which is given for the first time and includes the “genuine” particle collisions in the gas transport process. The proposed schemes for the ultra-relativistic viscous flows are also developed and two examples of ultra-relativistic viscous flow are designed. Several 1D and 2D numerical experiments are conducted to demonstrate that the proposed schemes not only are accurate and stable in simulating ultra-relativistic inviscid and viscous flows, but also have higher resolution at the contact discontinuity than the KFVS or BGK-type schemes.

High-order finite-volume solutions of the steady-state advection–diffusion equation with nonlinear Robin boundary conditions

We propose high-order finite-volume schemes for numerically solving the steady-state advection–diffusion equation with nonlinear Robin boundary conditions. Although the original motivation comes from a mathematical model of blood clotting, the nonlinear boundary conditions may also apply to other scientific problems. The main contribution of this work is a generic algorithm for generating third-order, fourth-order, and even higher-order explicit ghost-filling formulas to enforce nonlinear Robin boundary conditions in multiple dimensions. Under the framework of finite volume methods, this appears to be the first algorithm of its kind. Numerical experiments on boundary value problems show that the proposed fourth-order formula can be much more accurate and efficient than a simple second-order formula. Furthermore, the proposed ghost-filling formulas may also be useful for solving other partial differential equations.

An integrated model coupling open-channel flow, turbidity current and flow exchanges between main river and tributaries in Xiaolangdi Reservoir, China

The ever growing importance of sustainable management of reservoir sedimentation has promoted the development and applications of turbidity current models. However, there are few effective and practical models in literature for turbidity currents in a reservoir where the impounded area involves both the main river and its many tributaries. An integrated numerical model coupling open-channel flow, turbidity current and flow exchanges between main river and tributaries is proposed, which can simulate the complex flow and sediment transport in a reservoir where these three physical processes coexist. The model consists of two sets of governing equations for the open-channel flow and turbidity current, which are based on the modified St. Venant equations by taking into account the effect of lateral flow exchanges. These two sets of equations are solved in the finite volume method framework and the solutions are executed in an alternating calculation mode. Different methods are respectively proposed to calculate the discharge of flow exchange caused by free surface gradient and turbidity current intrusion. For the surface-gradient driven flow exchange, a storage cell method, which re-defines the relationship between water level and representative cross-sectional area, is used to update the water level at confluence. For the turbidity current intrusion, a discharge formula is proposed based on the analysis of the energy and momentum transformation in the intruding turbid water body. This formula differs from previous ones in that the effect of tributary bed slope is considered. Two events of water-sediment regulation conducted in the Xiaolangdi Reservoir in 2004 and 2006 were simulated to test the ability of this model. The predicted reservoir drawdown process, the turbidity current evolution and the sediment venting efficiency were in close agreement with the measurements. The necessity to couple the flow exchanges was demonstrated by comparing the performance of the proposed model with different intrusion discharge formulas and the common one-dimensional model neglecting the main river and tributary interaction.