Cell-centered Finite Volume Discretization Research Articles

Coupled flow and energy models with phase change in permafrost from pore- to Darcy scale: Modeling and approximation

We consider coupled models of flow and energy with phase change in thawing soils in permafrost at the pore- and Darcy scales. We develop the underlying models from first principles and define robust and conservative discretization algorithms based on cell-centered finite volume discretization on Cartesian grids and implicit time stepping for individual models. To couple the models we consider sequential approaches as well as time-lagging of some of the nonlinear properties. We also derive practical new surrogate models to obtain Darcy scale data from the pore-scale simulations. Finally, we study the sensitivity of models to the data in realistic scenarios. In particular we find that for the dynamics at the time scale of days, the nonlinear flow parameters have most significant impact on pressure, while temperature is less sensitive to data.

The cell-centered Finite-Volume self-consistent approach for heterostructures: 1D electron gas at the Si–SiO2 interface

Achieving self-consistent convergence with the conventional effective-mass approach at ultra-low temperatures (below 4.2 K) is a challenging task, which mostly lies in the discontinuities in material properties (e.g. effective-mass, electron affinity, dielectric constant). In this article, we develop a novel self-consistent approach based on cell-centered finite-volume discretization of the Sturm–Liouville form of the effective-mass Schrödinger equation and generalized Poisson’s equation (FV-SP). We apply this approach to simulate the one-dimensional electron gas formed at the Si–SiO2 interface via a top gate. We find excellent self-consistent convergence from high to extremely low (as low as 50 mK) temperatures. We further examine the solidity of FV-SP method by changing external variables such as the electrochemical potential and the accumulative top gate voltage. Our approach allows for counting electron–electron interactions. Our results demonstrate that FV-SP approach is a powerful tool to solve effective-mass Hamiltonians.

An efficient quadratic interpolation scheme for a third-order cell-centered finite-volume method on tetrahedral grids

In this paper, we propose an efficient quadratic interpolation formula utilizing solution gradients computed and stored at nodes and demonstrate its application to a third-order cell-centered finite-volume discretization on tetrahedral grids. The proposed quadratic formula is constructed based on an efficient formula of computing a projected derivative. It is efficient in that it completely eliminates the need to compute and store second derivatives of solution variables or any other quantities, which are typically required in upgrading a second-order cell-centered unstructured-grid finite-volume discretization to third-order accuracy. Moreover, a high-order flux quadrature formula, as required for third-order accuracy, can also be simplified by utilizing the efficient projected-derivative formula, resulting in a numerical flux at a face centroid plus a curvature correction not involving second derivatives of the flux. Similarly, a source term can be integrated over a cell to high-order in the form of the source term evaluated at the cell centroid plus a curvature correction, again, not requiring second derivatives of the source term. The discretization is defined as an approximation to an integral form of a conservation law but the numerical solution is defined as a point value at a cell center, leading to another feature that there is no need to compute and store geometric moments for a quadratic polynomial to preserve a cell average. Third-order accuracy and improved second-order accuracy are demonstrated and investigated for simple but illustrative test cases in three dimensions.

Assessment of a Hybrid Eulerian–Lagrangian CFD Solver for Wind Turbine Applications and Comparison with the New MEXICO Experiment

In this paper, the hybrid Lagrangian–Eulerian solver HoPFlow is presented and evaluated against wind tunnel measurements from the New MEXICO experiment. In the paper, the distinct solvers that assemble the HoPFlow solver are presented, alongside with details on their mutual coupling and interaction. The Eulerian solver, MaPFlow, solves the compressible Navier–Stokes equations under a cell-centered finite-volume discretization scheme, while the Lagrangian solver uses numerical particles that carry mass, pressure, dilatation and vorticity as flow markers in order to represent the flow-field by following their trajectories. The velocity field is calculated with the use of the decomposition theorem introduced by Helmholtz. Computational performance is enhanced by utilizing the particle mesh (PM) methodology in order to solve the Poisson equations for the scalar potential ϕ and the stream function ψ→. The hybrid solver is tested in 3-D unsteady simulations concerning the axial flow around the wind turbine (WT) model rotor tested in the New MEXICO experimental campaign. Simulation results are presented as integrated rotor loads, radial distribution of aerodynamic forces and moments and pressure distributions at various span-wise positions along the rotor blades. Comparison is made against experimental data and computational results produced by the pure Eulerian solver. A total of 5 PM nodes per chord length of the blade section at 75% have been found to be sufficient to predict the loading at the tip region of the blade with great accuracy. Discrepancies with respect to measurements, observed at the root and middle sections of the blade, are attributed to the omission of the spinner geometry in the simulations.

A strongly coupled Eulerian Lagrangian method applied in unsteady 3D external flows around Wind Turbine rotors

This paper presents a hybrid CFD solver that couples a standard Eulerian approach that solves the compressible Navier-Stokes equations under a cell-centered finite volume discretization, with a Lagrangian one, based on particles representation which carry mass, pressure, dilatation and vorticity. The velocity field is calculated using the Helmholtz’s decomposition theorem. Computational performance is enhanced by employing the Particle Mesh (PM) technique in order to solve the Poisson equations for the scalar potential ϕ and the stream function ψ. Even though validated for 2D flows over airfoils, this specific solver is used for the first time in order reproduce the flow around a wind turbine rotor. The validation simulations concern axial flow over the wind turbine model rotor used in the New MEXICO experimental campaign. Results of the hybrid solver, presented as blade pressure distribution and axial flow velocities are compared against the ones produced by its pure Eulerian counterpart and experimental measurements. PM grids of up to 5 points per chord of the blade section at 75% radius have been used. Comparison with the standard Eulerian solver suggest that the produced blade loads are over-predicted by approximately 7% near the tip and 14% near the root. However, the calculated velocity field is much closer to the experimental measurements as compared to the one produced by the Eulerian approach, which is attributed to the reduced numerical diffusion of the Lagrangian-vorticity formulation.

A vertex-based reconstruction for cell-centered finite-volume discretization on unstructured grids

Recently, a vertex-based spatial reconstruction for unstructured cell-centered finite volume method has been proposed, showing advantages in accuracy, convergence, and efficiency. However, this method was only applied for inviscid flows. Moreover, for shock-capturing computations, the conventional cell-based limiter was the only choice to suppress numerical oscillations. In this work, the vertex-based reconstruction is extended for the solution of viscous flows, while automatically avoiding the typical decoupling problem. Moreover, a WENO-type nonlinear weighting strategy is designed and combined with the vertex-based reconstruction method, resulting in a simple and cost-efficient alternative to conventional slope limiters. In addition, the present method is combined with the recently proposed iterative near-boundary treatment, ensuring linear exactness even in the vicinity of boundaries, without sacrificing the overall computational efficiency. A series of numerical test cases involving viscous flows and shock waves provide evidence of the superior performance of the present method. The advantages of the present method are especially prominent on high aspect-ratio irregular triangular grids, which is a beneficial property for solving realistic fluid dynamic problems at high Reynolds numbers.

Algebraic Multigrid Block Preconditioning for Multi-Group Radiation Diffusion Equations

The paper focuses on developing and studying efficient block preconditioners based on classical algebraic multigrid for the large-scale sparse linear systems arising from the fully coupled and implicitly cell-centered finite volume discretization of multi-group radiation diffusion equations, whose coefficient matrices can be rearranged into the $(G+2)\times(G+2)$ block form, where $G$ is the number of energy groups. The preconditioning techniques are based on the monolithic classical algebraic multigrid method, physical-variable based coarsening two-level algorithm and two types of block Schur complement preconditioners. The classical algebraic multigrid is applied to solve the subsystems that arise in the last three block preconditioners. The coupling strength and diagonal dominance are further explored to improve performance. We use representative one-group and twenty-group linear systems from capsule implosion simulations to test the robustness, efficiency, strong and weak parallel scaling properties of the proposed methods. Numerical results demonstrate that block preconditioners lead to mesh- and problem-independent convergence, and scale well both algorithmically and in parallel.

A Conservative Semi-Lagrangian Finite Volume Method for Convection–Diffusion Problems on Unstructured Grids

A conservative semi-Lagrangian finite volume method is presented for the numerical solution of convection–diffusion problems on unstructured grids. The new method consists of combining the modified method of characteristics with a cell-centered finite volume discretization in a fractional-step manner where the convection part and the diffusion part are treated separately. The implementation of the proposed semi-Lagrangian finite volume method differs from its Eulerian counterpart in the fact that the present method is applied at each time step along the characteristic curves rather than in the time direction. To ensure conservation of mass at each time step, we adopt the adjusted advection techniques for unstructured triangular grids. The focus is on constructing efficient solvers with large stability regions and fully conservative to solve convection-dominated flow problems. We verify the performance of our semi-Lagrangian finite volume method for a class of advection–diffusion equations with known analytical solutions. We also present numerical results for a transport problem in the Mediterranean sea.

Verification and Validation of openInjMoldSim, an Open-Source Solver to Model the Filling Stage of Thermoplastic Injection Molding

In the present study, the simulation of the three-dimensional (3D) non-isothermal, non-Newtonian fluid flow of polymer melts is investigated. In particular, the filling stage of thermoplastic injection molding is numerically studied with a solver implemented in the open-source computational library O p e n F O A M ® . The numerical method is based on a compressible two-phase flow model, developed following a cell-centered unstructured finite volume discretization scheme, combined with a volume-of-fluid (VOF) technique for the interface capturing. Additionally, the Cross-WLF (Williams–Landel–Ferry) model is used to characterize the rheological behavior of the polymer melts, and the modified Tait equation is used as the equation of state. To verify the numerical implementation, the code predictions are first compared with analytical solutions, for a Newtonian fluid flowing through a cylindrical channel. Subsequently, the melt filling process of a non-Newtonian fluid (Cross-WLF) in a rectangular cavity with a cylindrical insert and in a tensile test specimen are studied. The predicted melt flow front interface and fields (pressure, velocity, and temperature) contours are found to be in good agreement with the reference solutions, obtained with the proprietary software M o l d e x 3 D ® . Additionally, the computational effort, measured by the elapsed wall-time of the simulations, is analyzed for both the open-source and proprietary software, and both are found to be similar for the same level of accuracy, when the parallelization capabilities of O p e n F O A M ® are employed.

On the loss and recovery of second-order accuracy with U-MUSCL

It is shown that U-MUSCL, an unstructured-grid extension of Van Leer's κ-scheme as proposed by Burg for the edge-based discretization [AIAA Paper 2005-4999], will not be exact for linear functions and thus will not be second-order accurate with a nonzero parameter χ if the flux quadrature point is not exactly halfway between two adjacent solution points as typical in a cell-centered finite-volume discretization on general unstructured grids, and moreover that it results, contrary to expectations, in less accurate solutions with χ=1/3 than with χ=0 on irregular triangular grids. To resolve the problem while keeping the flexibility of the original κ-scheme, we propose a simple technique to guarantee the linear exactness of U-MUSCL for any parameter value on arbitrary grids. Numerical results show that the improved scheme preserves second-order accuracy on irregular grids, successfully resolves the accuracy problem on irregular grids, and produces superior solutions with χ=1/3, as expected, even on unstructured grids.

Thermodynamic consistency of cell-centered lagrangian schemes

It is well known that gas dynamics equations may develop singularities,i.e., shock waves, at a finite time. The related discontinuous solutions are thus sought under a weak form which corresponds to the integral form of the underlying conservation laws of mass, momentum and total energy. Weak solutions being not uniquely defined, the physically relevant solution is singled out by means of an extra admissibility criterion termed entropy condition. In other words, this means that the solutions of gas dynamics equations have to be consistent with the second law of thermodynamics: for smooth flows entropy has to be conserved, whereas for non smooth flows, such as shock waves, it has to be dissipated ensuring the conversion of kinetic energy into internal energy.Bearing this in mind, the purpose of the present work is to address the thermodynamic consistency of cell-centered Finite Volume discretizations dedicated to the numerical simulation of Lagrangian hydrodynamics. Firstly, we describe explicitly a general procedure to construct affordable entropy conservative numerical fluxes, extending Tadmor’s work to the Lagrangian framework. Secondly, the entropy stability of these fluxes enhanced by proper dissipation operators is investigated. Then, a multi-dimensional extension of this work is explored. Finally, these theoretical studies are assessed by various numerical experiments.

An efficient cell-centered finite-volume method with face-averaged nodal-gradients for triangular grids

In this paper, a face-averaged nodal-gradient approach is proposed as an efficient gradient method for a second-order cell-centered finite-volume discretization on triangular grids. The gradients needed in the linear reconstruction are computed in two steps: (1) compute gradients at nodes from solutions stored at cells, and (2) compute the gradient at each face by averaging the nodal gradients over the nodes forming the face. A unique feature of the proposed approach is that the same face-averaged gradient is used in linear reconstructions from both the left and right cells to the face centroid. This approach offers advantages over a conventional cell-gradient-based finite-volume method: reduced memory requirement for the gradients, smaller residual stencils, direct evaluations of viscous stresses and heat fluxes at solid boundaries, and less interpartition communication. The method is demonstrated for various inviscid and viscous problems from low to high Mach numbers on triangular grids.

Parallel and fully implicit simulations of the thermal convection in the Earth’s outer core

A parallel and fully implicit method is developed for numerical simulations of the thermal convection of an incompressible fluid in a rotating spherical shell. The method is based on a pseudo-compressibility approach with dual time stepping in order to tackle with the numerical difficulty of the solenoidal condition in the incompressible Navier-Stokes equations. The numerical solution of the nonlinear governing equations is based on the method of lines, whereby the partial differential equations are transformed into ordinary differential equations by a suitable spatial discretization. A second-order cell-centered finite volume discretization based on a collocated cubed-sphere grid is used here. In order to relax the time step limit and accurately integrate the semi-discrete nonlinear ordinary differential equations in time, we employ a second-order explicit-first-step, single-diagonal-coefficient, diagonally implicit Runge–Kutta (ESDIRK) method with adaptive time stepping. The nonlinear algebraic system arising at each pseudo-time step is solved by a Newton–Krylov–Schwarz algorithm to achieve good load balance on modern supercomputers. The numerical results are in good agreement with the benchmark solutions and more accurate than those obtained with a fractional step approach in our previous research. Large-scale tests on the Tianhe-2 supercomputer indicate that the fully implicit solver can achieve good parallel scalabilities in both strong and weak senses.

A robust finite volume method for three-dimensional filling simulation of plastic injection molding

PurposeThe purpose of this paper is to develop a finite volume approach for the simulation of three-dimensional two-phase (polymer melt and air) flow in plastic injection molding which is capable of robustly handling the mesh non-orthogonality and the discontinuities in fluid properties.Design/methodology/approachThe presented numerical method is based on a cell-centered unstructured finite volume discretization with a volume-of-fluid technique for interface capturing. The over-relaxed approach is adopted to handle the non-orthogonality involved in the discretization of the face normal derivatives to enhance the robustness of the solutions on non-orthogonal meshes. A novel interpolation method for the face pressure is derived to address the numerical stability issues resulting from the density and viscosity discontinuities at the melt–air interface. Various test cases are conducted to evaluate the proposed method.FindingsThe presented method was shown to be satisfactorily accurate by comparing simulations with analytical and experimental results. Besides, the effectiveness of the proposed face pressure interpolation method was verified by numerical examples of a two-phase flow problem with various density and viscosity ratios. The proposed method was also successfully applied to the simulation of a practical filling case.Originality/valueThe proposed finite volume approach is more tolerant of non-orthogonal meshes and the discontinuities in fluid properties for two-phase flow simulation; therefore, it is valuable for engineers in engineering computations.

An energy-dissipative remedy against carbuncle: Application to hypersonic flows around blunt bodies

The carbuncle phenomenon is a critical numerical instability preventing the accurate simulation of hypersonic flows around blunted configurations. After reviewing the known remedies to handle numerically this instability (designed, all of them, for Finite Volume methods), we present and analyze in depth the only effective cure reported in literature for Residual Distribution schemes.The shock fix, which is formulated as a locally active artificial diffusive term, depends on a single controlling parameter ϵs. Extensive testing of the fix in combination with the Residual Distribution technique allows for the determination of an optimal value for ϵs, which can be related to a local Péclet-like number.After testing the performance of the fix in combination with the Residual Distribution technique it was originally designed for, the flexibility of the fix is demonstrated by coupling it with a cell-centered Finite Volume discretization: carbuncle instabilities are equally avoided in this case.Finally, a number of limitations identified in the first testing phase lead to the derivation of a more physically consistent, energy dissipative family of carbuncle fixes, whose improved capabilities are demonstrated.

Stable Cell-Centered Finite Volume Discretization for Biot Equations

In this paper we discuss a new discretization for the Biot equations. The discretization treats the coupled system of deformation and flow directly, as opposed to combining discretizations for the two separate subproblems. The coupled discretization has the following key properties, the combination of which is novel: (1) The variables for the pressure and displacement are co-located and are as sparse as possible (e.g., one displacement vector and one scalar pressure per cell center). (2) With locally computable restrictions on grid types, the discretization is stable with respect to the limits of incompressible fluid and small time-steps. (3) No artificial stabilization term has been introduced. Furthermore, due to the finite volume structure embedded in the discretization, explicit local expressions for both momentum-balancing forces and mass-conservative fluid fluxes are available. We prove stability of the proposed method with respect to all relevant limits. Together with consistency, this proves convergence of the method. Finally, we give numerical examples verifying both the analysis and the convergence of the method.

Convergence of a Cell-Centered Finite Volume Discretization for Linear Elasticity

We show convergence of a cell-centered finite volume discretization for linear elasticity. The discretization, termed the MPSA method, was recently proposed in the context of geological applications, where cell-centered variables are often preferred. Our analysis utilizes a hybrid variational formulation, which has previously been used to analyze finite volume discretizations for the scalar diffusion equation. The current analysis deviates significantly from the previous in three respects. First, additional stabilization leads to a more complex saddle-point problem. Second, a discrete Korn's inequality has to be established for the global discretization. Finally, robustness with respect to the Poisson ratio is analyzed. The stability and convergence results presented herein provide the first rigorous justification of the applicability of cell-centered finite volume methods to problems in linear elasticity.

Multislope MUSCL method for general unstructured meshes

The multislope concept has been recently introduced in the literature to deal with MUSCL reconstructions on triangular and tetrahedral unstructured meshes in the finite volume cell-centered context. Dedicated scalar slopes are used to compute the interpolations on each face of a given element, in opposition to the monoslope methods in which a unique limited gradient is used. The multislope approach reveals less expensive and potentially more accurate than the classical gradient techniques. Besides, it may also help the robustness when dealing with hyperbolic systems involving complex solutions, with large discontinuities and high density ratios. However some important limitations on the mesh topology still have to be overcome with the initial multislope formalism. In this paper, a generalized multislope MUSCL method is introduced for cell-centered finite volume discretizations. The method is freed from constraints on the mesh topology, thereby operating on completely general unstructured meshes. Moreover optimal second-order accuracy is reached at the faces centroids. The scheme can be written with nonnegative coefficients, which makes it L∞-stable. Special attention has also been paid to equip the reconstruction procedure with well-adapted dedicated limiters, potentially CFL-dependent. Numerical tests are provided to prove the ability of the method to deal with completely general meshes, while exhibiting second-order accuracy.

A higher order Finite Volume resolution method for a system related to the inviscid primitive equations in a complex domain

We construct the cell-centered Finite Volume discretization of the two-dimensional inviscid primitive equations in a domain with topography. To compute the numerical fluxes, the so-called Upwind Scheme (US) and the Central-Upwind Scheme (CUS) are introduced. For the time discretization, we use the classical fourth order Runge---Kutta method. We verify, with our numerical simulations, that the US (or CUS) is a robust first (or second) order scheme, regardless of the shape or size of the topography and without any mesh refinement near the topography.

Advanced Workflows for Fluid Transfer in Faulted Basins

The traditional 3D basin modeling workflow is made of the following steps: construction of present day basin architecture, reconstruction of the structural evolution through time, together with fluid flow simulation and heat transfers. In this case, the forward simulation is limited to basin architecture, mainly controlled by erosion, sedimentation and vertical compaction. The tectonic deformation is limited to vertical slip along faults. Fault properties are modeled as vertical shear zones along which rock permeability is adjusted to enhance fluid flow or prevent flow to escape.For basins having experienced a more complex tectonic history, this approach is over-simplified. It fails in understanding and representing fluid flow paths due to structural evolution of the basin. This impacts overpressure build-up, and petroleum resources location.Over the past years, a new 3D basin forward code has been developed in IFP Energies nouvelles that is based on a cell centered finite volume discretization which preserves mass on an unstructured grid and describes the various changes in geometry and topology of a basin through time. At the same time, 3D restoration tools based on geomechanical principles of strain minimization were made available that offer a structural scenario at a discrete number of deformation stages of the basin.In this paper, we present workflows integrating these different innovative tools on complex faulted basin architectures where complex means moderate lateral as well as vertical deformation coupled with dynamic fault property modeling.Two synthetic case studies inspired by real basins have been used to illustrate how to apply the workflow, where the difficulties in the workflows are, and what the added value is compared with previous basin modeling approaches.