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Axisymmetric diffusers find wide applications in various industrial processes, including combustion systems, pneumatic conveying, and fluidized bed reactors. Understanding and accurately modeling the behavior of two-phase flows in these devices are critical for optimizing their performance and efficiency. Various aspects of the two-phase gas-solid flow, including particle-particle and particle-wall interactions, interphase momentum transfer, and phase segregation, are investigated. Our currently developed technique, which uses a cut-cell technique, is an established new method to handle the inclined wall of an axisymmetric diffuser. The near-wall cells are treated as five faces for the new grid; one is the inclined wall. This helps treat the boundary condition at the wall in an accurate physical way. The code FORTRAN, developed in-house, was used to solve the axisymmetric diffuser using a cut-cell technique. The results of this study will provide valuable insights into the behavior of gas-solid two-phase flows in axisymmetric diffuser. A parametric study of the impact of particles diameters (100 , 200 , and 300 μ m ), the solid volume loading ratios (0.005 , 0.008 , 0.01) , and cant angle (4.5 ∘ , 7 ∘ , 9.5 ∘) effect of axisymmetric diffuser on the local skin friction, pressure, velocity, turbulent kinetic energy, and separation zone. [ABSTRACT FROM AUTHOR]

A large Courant–Friedrichs–Lewy (CFL) algorithm is presented for the explicit, finite volume solution of hyperbolic systems of conservation laws, with a focus on the shallow water equations. The Riemann problems used in the flux computation are determined using averaging kernels that extend over several computational cells. The usual CFL stability constraint is replaced with a constraint involving the kernel support size. This makes the method unconditionally stable with respect to the size of the computational cells, allowing the computational mesh to be refined locally to an arbitrary degree without altering solution stability. The practical implementation of the method is detailed for the shallow water equations with topographical source term. Computational examples report applications of the method to the linear advection, Burgers and shallow water equations. In the case of sharp bottom discontinuities, the need for improved, well‐balanced discretisations of the geometric source term is acknowledged. [ABSTRACT FROM AUTHOR]

We derive new multiple point flux approximations (MPFA) for the finite volume approximation of heterogeneous and anisotropic diffusion problems on general meshes, in dimensions 2 and 3. The resulting methods are unconditionally stable while preserving the small stencil typical of MPFA finite volumes. The key idea is to solve local variational problems with a well-designed stabilization term from which we deduce conservative flux instead of directly prescribing a flux formula and solving the usual flux continuity equations. The boundary conditions of our local variational problems are handled through additional cell-centered unknowns, leading to an overall scheme with the same number of unknowns than first-order discontinuous Galerkin methods. Convergence results follow from well-established frameworks, while numerical experiments illustrate the good behavior of the method. [ABSTRACT FROM AUTHOR]

Shock waves, characterized by abrupt changes in pressure, temperature, and density, play a significant role in various materials science processes involving fluids. These high-energy phenomena are utilized across multiple fields and applications to achieve unique material properties and facilitate advanced manufacturing techniques. Accurate simulations of these phenomena require numerical schemes that can represent shock waves without spurious oscillations and simultaneously capture acoustic waves for a wide range of wavelength scales. This work suggests a high-order discontinuous Galerkin (DG) method with a finite volume (FV) subcell limiting strategies to achieve better subcell resolution and lower numerical diffusion properties. By switching to the FV discretization on an embedded sub-cell grid, the method displays advantages with respect to both DG accuracy and FV shock-capturing ability. The FV scheme utilizes a class of high-fidelity schemes that are built upon the boundary variation diminishing (BVD) reconstruction paradigm. The method is therefore able to resolve discontinuities and multi-scale structures on the subcell level, while preserving the favorable properties of the high-order DG scheme. We have tested the present DG method up to the 6th-order accuracy for both smooth and discontinuous noise problems. [ABSTRACT FROM AUTHOR]

Abstract Axisymmetric diffusers find wide applications in various industrial processes, including combustion systems, pneumatic conveying, and fluidized bed reactors. Understanding and accurately modeling the behavior of two-phase flows in these devices are critical for optimizing their performance and efficiency. Various aspects of the two-phase gas-solid flow, including particle-particle and particle-wall interactions, interphase momentum transfer, and phase segregation, are investigated. Our currently developed technique, which uses a cut-cell technique, is an established new method to handle the inclined wall of an axisymmetric diffuser. The near-wall cells are treated as five faces for the new grid; one is the inclined wall. This helps treat the boundary condition at the wall in an accurate physical way. The code FORTRAN, developed in-house, was used to solve the axisymmetric diffuser using a cut-cell technique. The results of this study will provide valuable insights into the behavior of gas-solid two-phase flows in axisymmetric diffuser. A parametric study of the impact of particles diameters ( 100 , 200 , and 300 μ m $100,\ 200,\ \text{and} \ 300\mu m$ ), the solid volume loading ratios ( 0.005 , 0.008 , 0.01 ) $(0.005,\ 0.008,\ 0.01)$ , and cant angle ( 4.5 ∘ , 7 ∘ , 9.5 ∘ ) $(4.5^{\circ},\ 7^{\circ},\ 9.5^{\circ}) $ effect of axisymmetric diffuser on the local skin friction, pressure, velocity, turbulent kinetic energy, and separation zone.

In this study, we propose a modified version of section-averaged Boussinesq equations of Winckler-Liu. The model is reformulated in conservative variables, allowing a decoupling of the Shallow-Water equations and the dispersive problem. An appropriate hybrid finite volume and finite element discretisations are performed and verified with a solitary wave solution derived for the typical case of prismatic channels with a trapezoidal cross-section. For the finite volume step we compare upwind and energy conservative numerical fluxes. The impact of this choice on the long time dynamics for Favre waves is thoroughly investigated. The proposed model and numerical approximations can accurately reproduce the main features of the wave train's free surface. The impact of the numerical dissipation introduced by upwind fluxes is discussed, emphasising the need for precaution in their applications to evaluate quantities of engineering interest such as maximum wave amplitudes. • A reformulation of Winckler and Liu's model is proposed to study wave propagation in channels with arbitrary cross-sections. • The model simplifies dispersive terms integration into existing shallow water solver, enhancing computational flexibility. • Hybrid discretization method, combining finite volume and finite element is proposed. • Two numerical flux formulations and their dissipative behaviours are investigated over long propagation times of Favre waves. • Comparing with Favre waves laboratory experiments, the numerical model captures key propagation phenomena. [ABSTRACT FROM AUTHOR]

Computationally extensive fluid flow simulations usually require numerical schemes with a high degree of numerical precision, where high wavenumber and frequency capturing capability is desired in solutions with high physical fidelity. Numerical dispersion and diffusion errors must be reduced to a minimum in order to caputre low amplitude and high frequency flow aspects, such as flow induced sound waves and minor instabilities. Such schemes also must be capable of dealing with flow discontinuities, such as shockwaves and expansion waves in cases where the flow velocity stands near or above sound velocity. In this work the Dispersion Relation Preserving (DRP) optimization is formally extended to the compact Finite Volume (FV) framework where the error is minimized along a given spectral interval. This is done by equaling the resulting algebraic compact spectral optimized Finite Difference (FD) operators with the Finite Volume formulation with the desired numeric characteristics. Numerical tests are carried over linear and nonlinear governing equations models for one and three-dimensional cases, where the travelling Gaussian wave, the transient expansion wave and the smooth to nonsmooth step wave flows are considered. The proposed compact DRP FV schemes surpasses its counterparts in some numerical tests while maintaining comparable capabilities with other more basic schemes in the remaining numerical tests. Excellent broadband content capturing capability is observed, while possessing reduced numerical instabilities generation. [ABSTRACT FROM AUTHOR]

We develop and analyze in this paper a momentum convection operator for variable density flows, and apply it to obtain a finite volume scheme for the Euler equations. The mesh is composed of triangular and quadrangular cells, in the two-dimensional case, and of hexahedral, tetrahedral, prismatic and pyramidal cells in three space dimensions. The approximation is staggered: the scalar variables (pressure, density and internal energy) are associated with the cells while the velocity approximation is face-centred. The derivation of the momentum convection operator extends to pyramids and prisms an already proposed construction for the other above-mentioned cells. The resulting operator takes the form of a finite volume operator, but is obtained by an algebraic process using as input the mass fluxes through the primal faces appearing in the mass balance for the definition of the velocity fluxes, with the only guideline to satisfy a discrete local kinetic energy identity. Its consistency thus deserves to be studied, and we show that this process yields a consistent convection operator in the Lax-Wendroff sense. Numerical tests confirm the expected scheme convergence, with a first-order rate on a pure shock solution. [ABSTRACT FROM AUTHOR]

This paper presents and analyzes a finite volume scheme for a fully degenerate parabolic system modeling the compressible two-phase flow, such as water-gas, problem in porous media considering the effects of temperature. The problem is written in terms of the phase formulation, i.e. the saturation of one phase, the pressure of the second phase and the temperature are primary unknowns. We use a first-order time implicit Euler scheme. The spatial discretization uses a fully coupled fully implicit cell-centered two-point flux approximation scheme where upwinding of the flux across an interface between two control volumes is performed for each fluid phase. Under physically relevant assumptions on the data, we prove the maximum principle for both saturation and temperature. We obtain a priori estimates and we show that the discrete problem is well posed. Also, we exploit various a priori estimates as well as compactness arguments to establish the convergence of the numerical scheme toward the weak solution. The open-source platform DuMu X was employed for the implementation of the developed algorithm based on the above scheme. Numerical result are presented to demonstrate the efficiency of this scheme. The test addresses the evolution in 2D of CO 2 injection into a layered aquifer. The algorithm's convergence with first order is validated by our numerical results, which support the theoretical results. To our best knowledge, this is the first convergence result of a finite volume scheme in the case of nonisothermal immiscible compressible two-phase flow in porous media. [ABSTRACT FROM AUTHOR]

This work focuses on the design of a 2D numerical scheme for the M 1 model on polygonal and conical meshes. This model is nonlinear and approximates the firsts moments of the radiative transfer equation using an entropic closure. Besides, this model admits a diffusion limit as the cross section goes to infinity. It is important for the numerical scheme to be consistent with this limit, that is to say, it should be asymptotic preserving or AP. Such a scheme already exists on polygonal meshes and the present work consists in adapting it to conical meshes. After introducing conical meshes, we explain the construction of the scheme. It is based on an analogy between the M 1 model and the Euler gas dynamics system. We also present a second order reconstruction procedure and we apply it on both polygonal and conical meshes. In the last section, some numerical test cases are given so as to compare the nodal and conical schemes. The limit scheme is studied and we observe numerically that it is consistent with the diffusion equation. [ABSTRACT FROM AUTHOR]

In this paper, we describe a new nonlinear finite-volume scheme that preserves the discrete maximum principle (DMP) for the two-dimensional sub-diffusion equation on distorted meshes. One distinguishing feature of our method is its ability to uphold the DMP for the anisotropic sub-diffusion problems, thereby ensuring the absence of spurious oscillations in numerical solutions and maintaining the physical bounds of various quantities, such as concentration, temperature, and density. Notably, our scheme offers the advantage of being applicable to distorted meshes without stringent constraints. Numerical results demonstrate that our scheme successfully preserves maximum principle on various randomly distorted meshes. [ABSTRACT FROM AUTHOR]

In this paper, a finite volume (TPFA) method is employed to simulate a degenerate breast cancer model that captures the progressive mutations from a normal breast stem cell to a tumor cell. The model incorporates a degenerate parabolic equation to represent the interaction between solid tumor growth and its environment, which involves the release of degradative enzymes governed by a partial differential equation. The discrete maximum principle is verified to be satisfied by the proposed finite volume scheme, and the convergence of the existing discrete solutions to a weak solution is proven. Additionally, the development of breast cancer is demonstrated through a numerical experiment. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new positivity-preserving (new PP) technique for fifth-order finite volume unequal-sized WENO schemes when solving some extreme problems of compressible Euler equations on structured meshes. After spatial reconstruction in each time marching, a detective process is used to examine the positivity of density and pressure at some checking points in each target cell. If the negativity happens at one checking point, a new compression limiter is carried out to ensure that the modified polynomials can achieve the positivity of density and pressure in the whole target cell instead of only at some discrete points. This treatment is easily implemented, owing to a quick and simple way to overestimate the minimum and maximum values of the polynomials of density and pressure in the target cell. Some numerical experiments for classical extreme problems show that the proposed method has an advantage in computational efficiency and robustness in comparison to the classical positivity-preserving techniques (Zhang and Shu in J Comput Phys 229:8918–8934, 2010), since the CFL number of 0.6 is allowed for all tests in this paper. [ABSTRACT FROM AUTHOR]

We consider a reactive multiphase multicomponent Darcy flow in a porous medium while taking into account the effects of temperature. This flow model is coupled to an energy balance equation and ordinary and/or algebraic differential equations to model the chemical reactions. The model is discretized using a cell-centered finite volume method with implicit Euler or second order backward differential formula time discretization. Both sequential and fully coupled fully implicit strategies are considered. Two new DuMu X modules are developed based on the above schemes. Two numerical examples are presented to validate the implementation of the two methods and the capability of the code to solve CO 2 sequestration scenarios. The first test addresses a one-dimensional radial problem to study thermal effects on injectivity during CO 2 storage. For this purpose, isothermal/nonisothermal comparisons are carried out to highlight these differences, which can modify the flow. The second test is chosen to test the code to approximate solutions for a three-dimensional benchmark based on the Johansen CO 2 storage operation. Then, we compare the efficiency, performance in terms of computation time, and parallel scalability of the sequential and implicit methods. In conclusion, both modules demonstrate accuracy, numerical robustness, and the potential to solve realistic problems. With an equal time step, the sequential scheme can be faster than the implicit scheme, but the splitting can present a loss of accuracy, particularly concerning the conservation of mass of the injected CO 2 . [ABSTRACT FROM AUTHOR]

We propose in this article a monotone finite volume diffusion scheme on 3D general meshes for the radiation hydrodynamics. Primary unknowns are averaged value over the cells of the mesh. It requires the evaluation of intermediate unknowns located at the vertices of the mesh. These vertex unknowns are computed using an interpolation method. In a second step, the scheme is made monotone by combining the computed fluxes. It allows to recover monotonicity, while making the scheme nonlinear. This scheme is inserted into a radiation hydrodynamics solver and assessed on radiation shock solutions on deformed meshes. [ABSTRACT FROM AUTHOR]

We provide a numerical realization of an optimal control problem for pedestrian motion with agents that was analyzed in Herzog et al. (Appl. Math. Optim. 88(3):87, 2023). The model consists of a regularized variant of Hughes’ model for pedestrian dynamics coupled to ordinary differential equations that describe the motion of agents which are able to influence the crowd via attractive forces. We devise a finite volume scheme that preserves the box constraints that are inherent in the model and discuss some of its properties. We apply our scheme to an objective functional tailored to the case of an evacuation scenario. Finally, numerical simulations for several practically relevant geometries are performed. [ABSTRACT FROM AUTHOR]

Cavitation resulting from underwater explosions in compressible multiphase or multicomponent flows presents significant challenges due to the dynamic nature of shock–cavitation–structure interactions, as well as the complex and discontinuous nature of the involved interfaces. Achieving accurate resolution of interfaces between different phases or components, in the presence of shocks, cavitating regions, and structural interactions, is crucial for modeling such problems. Furthermore, pressure convergence in simulations involving shock–cavitation–structure interactions requires accurate algorithms. In this research paper, we employ the diffuse interface method, also known as the interface-capturing scheme, to investigate cavitation in various underwater explosion test cases near different surfaces: a free surface and a rigid surface. The simulations are conducted using the unstructured compressible Navier–Stokes (UCNS3D) finite-volume framework employing central-weighted essentially non-oscillatory (CWENO) reconstruction schemes, utilizing the five-equation diffuse interface family of methods. Quantitative comparisons are made between the performance of both models. Additionally, we examine the effects of cavitation as a secondary loading source on structures, and evaluate the ability of the CWENO schemes to accurately capture and resolve material interfaces between fluids with minimal numerical dissipation or smearing. The results are compared with existing high-order methods and experimental data, where possible, to demonstrate the robustness of the CWENO schemes in simulating cavitation bubble dynamics, as well as their limitations within the current implementation of interface capturing. [ABSTRACT FROM AUTHOR]

Summary: Hybrid algorithms are an efficient and popular choice for computing the solutions of hyperbolic conservation laws. In general, hybrid algorithms involve low‐cost high‐order accurate schemes in smooth regions and non‐oscillatory shock‐capturing schemes in the vicinity of discontinuities. Troubled‐cell indicators which measure the smoothness of the solution play a significant role in the efficiency of hybrid algorithms. This article proposes a new troubled‐cell indicator utilising the smoothness indicators of WENO schemes for hyperbolic conservation laws. The proposed troubled‐cell indicators are simple, efficient, effective, and are used to construct three new adaptive WENO algorithms of high‐order accuracy. The hybrid algorithms developed are independent of the order and type of the WENO reconstruction. For demonstration, we have considered the fifth and seventh order WENO‐Z reconstruction. The first two algorithms have comparable accuracy and resolution of the solution across discontinuities to that of the WENO‐Z scheme but at a less computational cost. The third algorithm ensures the convergence of the proposed scheme to the correct entropy solution when applied to a hyperbolic conservation law with non‐convex flux for which the WENO schemes fail. We have performed several 1D and 2D numerical experiments to demonstrate the efficiency of the proposed algorithms and their performance compared with the WENO‐Z schemes. The proposed algorithms are efficient and take 30%–75% less computational time than the WENO‐Z schemes while retaining the advantages of WENO‐Z schemes. [ABSTRACT FROM AUTHOR]

This work presents a comparison of two-dimensional numerical solutions of unsteady free surface flow. This is a simulation of the dam-break wave with different configurations using based-mesh finite volume method and meshless smoothed particle hydrodynamics (SPH). Two well-known approaches, widely used in the computational fluid dynamics (CFD). These techniques have proven their robustness in the numerical treatment of such conservation laws. The main goal is to check the ability of the SPH method and the first order finite volume HLLC solver to reproduce the numerical solutions of the 2D shallow water equations. Based on many benchmark tests, one investigates the effect of the topographic variation along the x and y directions on behavior of the numerical solutions namely at the wet-dry front. The comparison between the simulated results, the analytical solutions and the experimental measurements shows a good correlation, although the finite volume approach remains more advantageous in terms of accuracy and the CPU time.

The finite volume Laplacian can be defined in all dimensions and is a natural way to approximate the operator on a simplicial mesh. In the most general setting, its definition with orthogonal duals may require that not all volumes are positive; an example is the case corresponding to two-dimensional finite elements on a non-Delaunay triangulation. Nonetheless, in many cases two- and three-dimensional Laplacians can be shown to be negative semidefinite with a kernel consisting of constants. This work generalizes work in two dimensions that gives a geometric description of the Laplacian determinant; in particular, it relates the Laplacian determinant on a simplex in any dimension to certain volume quantities derived from the simplex geometry. [ABSTRACT FROM AUTHOR]

In this paper, new third-order finite volume unequal-sized weighted essentially non-oscillatory (US-WENO) Lagrangian schemes are designed to solve Euler equations in two and three dimensions. The spatial reconstruction procedures are implemented by using a convex combination of a quadratic polynomial with several linear polynomials specified on unequal-sized stencils, so the new US-WENO Lagrangian schemes can achieve the designed third-order accuracy and maintain an essentially non-oscillatory property near strong discontinuities in multi-dimensions. Unlike the traditional WENO reconstruction procedures specified on unstructured meshes, the linear weights of these new two-dimensional and three-dimensional US-WENO spatial reconstructions can be selected as any positive numbers as long as their summation equals one and they are not related to the local mesh topology or the location of quadrature points. Moreover, the linear weights do not have to be recalculated even if the grid moves with the fluid, avoiding the appearance of negative linear weights, thus improving computation efficiency and robustness in multi-dimensional Lagrangian numerical simulations. Finally, extensive benchmark numerical cases are employed to display the excellent capability of the presented US-WENO Lagrangian schemes. [ABSTRACT FROM AUTHOR]

This research study performs a numerical assessment of the inward solidification of phase change material (PCM) integrated with metal foam (MF) encapsulated in a horizontal cylindrical geometry. The cylinder is not at melting temperature at the beginning, and its outer surface is subjected to convection boundary conditions. The 1D phase change model is resolved by applying the temperature transforming technique using an in‐house computer program written in C++. Four different MF materials, aluminum, copper, nickel, and stainless steel, with various porosities, are used to examine the thermal behavior of the storage system. It is found that the use of copper as a MF with higher effective thermal conductivity remarkably enhances the elapsed time for complete solidification. Additionally, for a porosity value of 0.92, the time for total solidification of copper is diminished by 94%, while the complete solidification time for Stainless Steel is decreased by 65% in comparison to the situation where no MF is used. [ABSTRACT FROM AUTHOR]

Fluid flow in deformable fractures is important to many applications including hydraulic stimulation. Fracture flow simulations typically rely on the Poiseuille flow model which presumes laminar quasi‐steady‐state flow conditions with negligible inertia. The high flow rates involved in industrial applications bring these assumptions into question. The GG22 flow model is a new method to capture inertial effects and turbulent flow phenomena with a moderate increase in computational complexity. Here we develop and verify the first hydro‐mechanically coupled finite element – finite volume model for fracture flow which uses the GG22 model. We use the model to examine fracture flow between oscillating elastic plates and show that inertia may elicit phase‐shifts in the fluid response, larger fluid pressures and rock mass stresses, and induce wave‐like behaviour even in a quasi‐static rock mass. We then apply the model to hydraulic stimulation of a cemented fracture in both the viscous and toughness dominant propagation regimes and examine the influence of inertia and turbulence compared to Poiseuille flow. We demonstrate that inertial effects due to the variation of aperture are negligible in two‐dimensional KGD‐like fractures with constant injection rates. If the flow rates are large enough to invoke turbulent flow behaviour, then significantly different fracture propagation behaviour is observed. The modelling established in this article will serve as the basis to examine the role of inertia and turbulence in axisymmetrical radial fractures and dynamic stimulation with pressure pulsing where we expect the role of inertia to be more significant. [ABSTRACT FROM AUTHOR]

A problem with 1-D water hammer modelling is in the application of accurate unsteady friction. Moreover, investigating the time response of fluid dynamics and unsteady turbulence structures during the water hammer is not possible with a 1-D model. This review article provides a summary of 1-D modelling using the recent finite volume approach and the discussion extends to a quasi-2-D model and historical developments as well as recent advancements in 3-D CFD simulations of water hammer. The eddy viscosity model is excellent in capturing pressure profiles but it is computationally intensive and requires more computational time. This article reviews 3-D CFD simulations with sliding mesh, an immersed solid approach, and dynamic mesh approaches for modelling valve closures. Despite prediction accuracy, a huge computational time and high computer resources are required to execute 3-D flow simulations with advanced valve modelling techniques. Experimental validation shows that a 3-D CFD simulation with a flow rate reduction curve as a boundary condition predicted accurate pressure variation results. Finally, a brief overview of the transient flow turbulence structures for a rapidly accelerated and decelerated pipe flow using DNS (Direct numerical simulation) data sets is presented. Overall, this paper summarises past developments and future scope in the field of water hammer modelling using CFD. [ABSTRACT FROM AUTHOR]

Abstract Natural convection in cavity with built-in plates has been widely used in fluid mechanics and heat transfer engineering, such as heat exchangers, electronics, solar collector, growing crystals, etc. However, the natural convection flow physics and heat transfer characteristics in cavity with plate arrays remain to be investigated further. In this study, numerical analysis and computational fluid dynamics simulation are conducted for natural convection in cavity with plate arrays by finite volume method. Key influence factors impacting heat transfer are analyzed, including the line dip-angle of plate arrays θ, rotary angle of plates to the array φ, with relationships comparison between the Nusselt number and Rayleigh number in different built-in plate arrays. Moreover, for illustrative application example, the natural convection in solar greenhouse with PV arrays roof is investigated, with inside temperature distribution comparison under different design parameters. The preliminary results indicate that plates spacing could remarkably impacts the number of streamlines between ”late’arrays. The dip-angle of arrays θ leads to different shape flow in the cavity, and the rotary angle of plates φ notably affects the fluid flow direction in the vicinity areas of plates, especially when φ = 90°. Solar greenhouse case study indicates that increasing array group numbers can improve air temperature distribution uniformity in the cavity, which is favorable for built environment accurate control and operation. This work can provide modeling method support and reference for natural heat convection applications. [ABSTRACT FROM AUTHOR]

In this work, we develop numerical methods for the solution of blood flow and coagulation on dynamic adaptive moving meshes. We consider the blood flow as a flow of incompressible Newtonian fluid governed by the Navier–Stokes equations. The blood coagulation is introduced through the additional Darcy term, with a permeability coefficient dependent on reactions. To this end, we introduce moving mesh collocated finite‐volume methods for the Navier–Stokes equations, advection–diffusion equations, and a method for the stiff cascade of reactions. A monolithic nonlinear system is solved to advance the solution in time. The finite volume method for the Navier–Stokes equations features collocated arrangement of pressure and velocity unknowns and a coupled momentum and mass flux. The method is conservative and inf‐sup stable despite the saddle point nature of the system. It is verified on a series of analytical problems and applied to the blood flow problem in the deforming domain of the right ventricle, reconstructed from a time series of computed tomography scans. At last, we demonstrate the ability to model the coagulation process in deforming microfluidic capillaries. [ABSTRACT FROM AUTHOR]

We consider a 3D-2D-1D mixed-dimensional diffusive model in a fractured porous medium coupling the 1D model along the centerline skeleton of a tubular network, the 2D model on a network of planar fractures and the 3D model in the surrounding matrix domain. The transmission conditions are based on a potential continuity assumption at matrix fracture interfaces, and on Robin type conditions at the resolved interfaces between the tubular network and the matrix and fracture network domains. The discretization of this mixed-dimensional model is formulated in the gradient discretization framework (Droniou, J., Eymard, R. & Herbin, R. (2016) Gradient schemes: generic tools for the numerical analysis of diffusion equations. ESAIM Math. Model. Numer. Anal., 50, 749–781), which covers a large class of conforming and nonconforming schemes and provides stability and error estimates based on general coercivity, consistency and limit-conformity properties. As an example of discretization fitting this framework, the mixed-dimensional version of the vertex approximate gradient (VAG) scheme is developed. It is designed to allow nonconforming meshes at the interface between the 1D and 3D-2D domains, to provide a conservative formulation with local flux expressions and to be asymptotic preserving in the limit of high transfer coefficients. Numerical experiments are provided on analytical solutions for simplified geometries, which confirm the theoretical results. Using its equivalent finite volume formulation, the VAG discretization is extended to a drying mixed-dimensional model coupling the Richards equation in a fractured porous medium and the convection diffusion of the vapor molar fraction along the 1D domain. It is applied to simulate the drying process between an operating tunnel and a radioactive waste storage rock with explicit representation of the fractures in the excavated damaged zone. [ABSTRACT FROM AUTHOR]

Earthen embankments are subjected to increasing threats because of climate change inducing sequences of severe drought periods followed by floods, possibly leading to overtopping of the structures. Consequently, the water saturation of the dike can vary significantly both in space and time, and the resulting groundwater flow can affect the free-surface flow in case of overtopping. Conversely, the free-surface flow can modify the pore water content, which controls erosion and slope instabilities. In this paper, a combined approach to such situations is presented, in which the degree of saturation and the flow through the embankment are simulated by solving the two-dimensional Richards equation on an unstructured mesh with an implicit finite volume scheme that is coupled to the system of shallow-water equations solved in one dimension using an explicit finite-volume scheme. The coupled model is validated on several situations of flows through and over earthen embankments with different constitutive materials. [ABSTRACT FROM AUTHOR]

Finite volume, weighted essentially non-oscillatory (WENO) schemes require the computation of a smoothness indicator. This can be expensive, especially in multiple space dimensions. We consider the use of the simple smoothness indicator σ S = 1 N S - 1 ∑ j ( u ¯ j - u ¯ m) 2 , where N S is the number of mesh elements in the stencil, u ¯ j is the local function average over mesh element j, and index m gives the target element. Reconstructions utilizing standard WENO weighting fail with this smoothness indicator. We develop a modification of WENO-Z weighting that gives a reliable and accurate reconstruction of adaptive order, which we denote as SWENOZ-AO. We prove that it attains the order of accuracy of the large stencil polynomial approximation when the solution is smooth, and drops to the order of the small stencil polynomial approximations when there is a jump discontinuity in the solution. Numerical examples in one and two space dimensions on general meshes verify the approximation properties of the reconstruction. They also show it to be about 10 times faster in two space dimensions than reconstructions using the classic smoothness indicator. The new reconstruction is applied to define finite volume schemes to approximate the solution of hyperbolic conservation laws. Numerical tests show results of the same quality as standard WENO schemes using the classic smoothness indicator, but with an overall speedup in the computation time of about 3.5–5 times in 2D tests. Moreover, the computational efficiency (CPU time versus error) is noticeably improved. [ABSTRACT FROM AUTHOR]

Summary: This paper is dedicated to a numerical method based on a random choice as proposed in Glimm's scheme. It is applied to the problem of advection of a scalar quantity. The numerical scheme proposed here relies on a fractional step approach for which: the first step is performed using any classical finite‐volume scheme, and the second step is a cell‐wise update. This second step is a projection based on a random choice. The resulting scheme possesses a very low level of numerical diffusion. In order to assess the capabilities of this approach, several test cases have been investigated including convergence studies with respect to the mesh‐size. The algorithm performs very well on one‐dimensional and multi‐dimensional problems. This algorithm is very easy to implement even for multi‐processor computations. [ABSTRACT FROM AUTHOR]

This paper outlines the methodology and results for a two-species finite volume scalar computational drug transport model developed for simulating the mass transport of Poly(lactic-co-glycolic acid (PLGA)) from a half-embedded single strut implanted in a coronary arterial vessel wall. The mathematical drug transport model incorporates the convection-diffusion equation in scalar form (dimensionless) with a two-species (free-drug and bound-drug) mass transport setup, including reversible equilibrium reaction source terms for the free and bound-drug states to account for the pharmaco-kinetic reactions in the arterial wall. The relative reaction rates of the added source terms control the interconversion of the drug between the free and bound-drug states. The model is solved by a 2D finite-volume method for discretizing and solving the free and bound drug transport equations with anisotropic vascular drug diffusivities. This model is an improvement over previously developed models using the finite-difference and finite element method. A dimensionless characteristic scaling pre-analysis was conducted a priori to evaluate the significance of implementing the reaction source terms in the transport equations. This paper reports the findings of an investigation of the interstitial flow profile into the arterial wall and the free and bound drug diffusion profiles with a parametric study of varying the polymer drug concentration (low and high), tortuosity, porosity, and Peclet and DamKöhler numbers over the course of 400 h (16.67 days). The results also reveal how a single species drug delivery model that neglects both a reversible binding reaction source term and the porosity and tortuosity of the arterial wall cannot accurately predict the distribution of both the free and bound drug.