Unstructured Finite Volume Discretization Research Articles

A constrained boundary gradient reconstruction method for unstructured finite volume discretization of the Euler equations

We propose a novel boundary gradient and solution reconstruction approach for the second-order cell-centered unstructured finite volume method for solving the inviscid Euler equations. The proposed constrained reconstruction method involving ghost cells is shown to be more accurate than several conventional boundary reconstruction methods and linear-exact in the near-boundary areas. In the present procedure, the solutions at the ghost cell centroids are given by solving constraint equations, which are established based on boundary conditions. At the same time, the solution gradients in boundary cells and the solutions in boundary surfaces are obtained altogether, while guaranteeing the compatibility of the boundary conditions with the reconstructed results. A variety of numerical test cases show that the new method effectively reduces the errors of near-boundary approximations and improves the overall numerical accuracy. Furthermore, because the constrained reconstruction is only used for the boundary cells, the extra computational cost barely changes the overall computational efficiency.

LES of H2-air jet combustion in high enthalpy supersonic crossflow

Here, we report on large eddy simulation (LES) of supersonic flow, mixing, self-ignition, and combustion in a supersonic hydrogen jet in a crossflow configuration. The configuration has been experimentally investigated at Stanford and consists of a rectilinear channel with a ramp inlet in which a hydrogen jet discharges at a 90° angle to the high enthalpy supersonic crossflow. This configuration has been extensively studied by several research groups and constitutes a good validation case for model development and physics elucidation. The LES model used is based on an unstructured finite volume discretization of the filtered mass, momentum, species, and energy equations and an explicit flow solver. In this study, we investigate the effects of the jet-to-crossflow momentum ratio, the chemical reaction mechanism, and the combustion subgrid model by comparing predictions and by comparing with experimental data including OH* chemiluminescence images and jet penetration data. In general, good agreement is found but with some departures for the smallest reaction mechanisms and some of the LES combustion models. The LES results are also used to elucidate the flow, mixing, and combustion features of this configuration.

Extending the global-direction stencil with \u201cface-area-weighted centroid\u201d to unstructured finite volume discretization from integral form

Accuracy of unstructured finite volume discretization is greatly influenced by the gradient reconstruction. For the commonly used k-exact reconstruction method, the cell centroid is always chosen as the reference point to formulate the reconstructed function. But in some practical problems, such as the boundary layer, cells in this area are always set with high aspect ratio to improve the local field resolution, and if geometric centroid is still utilized for the spatial discretization, the severe grid skewness cannot be avoided, which is adverse to the numerical performance of unstructured finite volume solver. In previous work [Kong, et al. Chin Phys B 29(10):100203, 2020], we explored a novel global-direction stencil and combined it with the face-area-weighted centroid on unstructured finite volume methods from differential form to realize the skewness reduction and a better reflection of flow anisotropy. Greatly inspired by the differential form, in this research, we demonstrate that it is also feasible to extend this novel method to the unstructured finite volume discretization from integral form on both second and third-order finite volume solver. Numerical examples governed by linear convective, Euler and Laplacian equations are utilized to examine the correctness as well as effectiveness of this extension. Compared with traditional vertex-neighbor and face-neighbor stencils based on the geometric centroid, the grid skewness is almost eliminated and computational accuracy as well as convergence rate is greatly improved by the global-direction stencil with face-area-weighted centroid. As a result, on unstructured finite volume discretization from integral form, the method also has superiorities on both computational accuracy and convergence rate.

Subgrid Models, Reaction Mechanisms, and Combustion Models in Large-Eddy Simulation of Supersonic Combustion

Here, we report on large-eddy simulation (LES) of supersonic flow, mixing, self-ignition, and combustion in a model scramjet combustor. The combustor has been experimentally investigated at DLR, German Aerospace Center Lampoldshausen and consists of a one-sided divergent channel with a wedge-shaped flame-holder: at the base of which, hydrogen is injected. This combustor has been extensively studied by many research groups and constitutes a good validation case for model development and physics elucidation. The LES model used is based on an unstructured finite volume discretization of the filtered mass, momentum, species, and energy equations, as well as an explicit flow solver. Three subgrid flow models, three reaction mechanisms, and three LES combustion models are employed to examine the sensitivity to these modeling parameters. The LES predictions are compared with experimental data for velocity, temperature, wall pressure at different cross sections, as well as schlieren and OH chemiluminesence images, showing good agreement for both first- and second-order statistics, as well as the instantaneous flow structures. Furthermore, these LES results are used to illustrate and explain the intrinsic flow, mixing, and combustion features of this combustor. The intrinsic relationship between the different modeling parameters is also discussed.

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.

Condition number analysis of flow fields arising from CFD simulations

Condition numbers of pressure and velocity matrices obtained from an unstructured finite volume discretization of the three dimensional Navier–Stokes equations are studied for two distinct flow configurations. The exact condition number is calculated based on singular values. The continuous estimation of condition number of the velocity matrices with respect to the time varying velocity field is achieved through the Gershgorin theoretical bounds for the eigenvalues. The focus of the present study is on the analysis of flow dynamics with respect to the degree of ill-conditioning of the linear systems. For the two benchmark problems, condition numbers of pressure and velocity matrices are presented. The pressure matrix is shown to be more ill-conditioned than velocity and requires strong preconditioning. For a 3D lid-driven cavity, the appearance and disappearance of corner vortices at higher Reynolds number is clearly reflected in the condition number variation with time. In a second application of flow past a circular cylinder, condition number variation with time during the initial phase clearly reveals the onset of vortex shedding. The condition number based on Gershgorin bounds is also used to switch the SGS and ILU preconditioners for velocity matrices when they are well conditioned. Consequently, an overall reduction in the simulation time is demonstrated.

An improved uncoupled finite volume solver for simulating unsteady shock-induced combustion

Unsteady shock-induced combustion is highly complicated due to the interactive physics-chemistry phenomena which are multi-scales in space and time. Especially, rigidity or stiffness of the Ordinary Differential Equations (ODEs) including the temporal solutions of both the chemical source terms and the flow equations may introduce significant computational cost due to the tiny time-steps of explicit methods or the complicated Jacobian matrix of implicit methods. A modified uncoupled method, which follows the idea of Strang splitting and has proved to be simple and efficient for solving the reactive flow system, is applied with using unstructured finite volume discretization and detailed reaction mechanisms. In order to further improve the efficiency of the uncoupled method, a chemical reaction detection criterion is designed to adaptively turn off chemical reaction computation. Numerical tests have shown significant efficiency improvement and satisfactory accuracy of the presented method.

Solution of fully-coupled shallow water equations and contaminant transport using a primitive-variable Riemann method

A Riemann-solver scheme, using primitive variables rather than conserved variables, is configured and tuned for the solution of the fully-coupled two-dimensional shallow water and contaminant transport equations. This scheme is based on the unstructured finite volume discretization using primitive-variable Roe-flux approximation with an entropy fix. The primitive-variable flux associated with the exact source-term balancing is well-behaved and well-balanced for both still-water and dry regions with arbitrary bed topography. Second-order accuracy is used in space and time. The present study uses a nonlinear implicit scheme based on Newton-iterative algorithm for the time integration. In order to show the accuracy of the scheme, numerical results are verified by different test cases for contaminant advection and diffusion. A scenario of contaminant transport in a complex geometry with wet and dry elements is also simulated to demonstrate that the present work can be implemented on practical applications involving flooding and contaminant transport.

Efficient CFD code implementation for the ARM-based Mont-Blanc architecture

Since 2011, the European project Mont-Blanc has been focused on enabling ARM-based technology for HPC, developing both hardware platforms and system software. The latest Mont-Blanc prototypes use system-on-chip (SoC) devices that combine a CPU and a GPU sharing a common main memory. Specific developments of parallel computing software and well-suited implementation approaches are of crucial importance for such a heterogeneous architecture in order to efficiently exploit its potential.This paper is devoted to the optimizations carried out in the TermoFluids CFD code to efficiently run it on the Mont-Blanc system. The underlying numerical method is based on an unstructured finite-volume discretization of the Navier–Stokes equations for the numerical simulation of incompressible turbulent flows. It is implemented using a portable and modular operational approach based on a minimal set of linear algebra operations. An architecture-specific heterogeneous multilevel MPI+OpenMP+OpenCL implementation of such kernels is proposed. It includes optimizations of the storage formats, dynamic load balancing between the CPU and GPU devices and hiding of communication overheads by overlapping computations and data transfers. A detailed performance study shows time reductions of up to 2.1× on the kernels’ execution with the new heterogeneous implementation, its scalability on up to 128 Mont-Blanc nodes and the energy savings (around 40%) achieved with the Mont-Blanc system versus the high-end hybrid supercomputer MinoTauro.

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.

A fast-running CFD formulation for unsteady ship maneuvering performance prediction

Maneuvering prediction tools are important resources for naval and commercial ship designers. They can enable designers to determine maneuvering characteristics of a design prior to construction or be useful in redesign if a ship is underperforming. Numerical methods allow for multiple hulls to be compared digitally, so a certain level of optimization can be achieved through evaluating trade-offs among designs before construction. In this paper, a novel maneuvering prediction method is presented that seeks to provide the accuracy available from today state-of-the-art numerical viscous-flow tools but with significantly reduced computational cost. The numerical formulation is based on the unsteady Reynolds-averaged Navier–Stokes (URANS) equations, and the unsteady ship generated waves are represented by the linear free-surface boundary conditions. The URANS equations are solved on an unstructured finite-volume discretization of the fluid domain around the hull and its appendages. The linearized free-surface approximation accounts for first-order wave effects while reducing the necessary extent of the computational domain and level of grid refinement required by fully nonlinear methods that employ an interface-capturing technique. The resulting formulation provides a marked improvement in computational efficiency with respect to nonlinear methods and can empower naval architects to obtain accurate results earlier in the design process.

Study on the onset and development of the three-dimensional separation around a diamond wing with incompressible flow solvers

One of the challenges in simulating aerodynamic flows is the accurate prediction of onset and progression of separation. In earlier STO research, it was found that various predictions for three-dimensional airfoils at angle of attack showed different pitch moments due to differences in the predicted location of separation areas. Therefore, a diamond shaped wing was developed within the NATO STO AVT-183 research group, to isolate the blunt leading edge separation and understand the underlying physical mechanisms. In the present article, two viscous-flow solvers dedicated to hydrodynamic applications are used to predict the flow around the diamond wing. Both codes solve the incompressible flow using unstructured grids and finite volume discretization on the same grids. The results comprise a study of with/without peniche configurations and different turbulence models ranging from isotropic or anisotropic statistical turbulence closures to hybrid LES turbulence models. Moreover, the role played by the discretization error is assessed by using anisotropic automatic grid refinement procedures based on flow-related criteria. A detailed discussion is made, highlighting the similarities and differences of the results, the influence of modeling errors and showing the potential of CFD tools to predict the type of flow under consideration.

Accuracy analysis of unstructured finite volume discretization schemes for diffusive fluxes

Numerical errors form a major source of uncertainty in CFD simulations. This source of error, which is more significant on unstructured meshes, can be alleviated by a careful choice of discretization scheme. In the context of finite volume methods, there are many suggestions for the discretization of diffusive fluxes appearing in viscous flow simulations. In this paper, a wide range of discretization schemes commonly used for diffusive flux approximation in cell-centered unstructured finite volume solvers are compared in terms of discretization and truncation errors. In addition, a novel eigenanalysis tool is proposed to relate these two forms of numerical error to each other and to interpret the error behaviors obtained by each scheme. The error comparisons are performed on unstructured meshes consisting of both isotropic and anisotropic triangles. Our results suggest that adding a solution jump term to the baseline face gradient (determined as the average of two adjacent cell gradients) provides the most accurate approximation for diffusive fluxes. Also, this term produces sufficient damping to stabilize the discretization even on highly skewed meshes.

A computational study of supersonic combustion behind a wedge-shaped flameholder

In this study, large eddy simulation (LES) has been used to examine supersonic flow, mixing, self-ignition and combustion in a model scramjet combustor and has been compared against the experimental data. The LES model is based on an unstructured finite-volume discretization, using monotonicity-preserving flux reconstruction of the filtered mass, momentum, species and energy equations. Both a two-step and a seven-step hydrogen–air mechanism are used to describe the chemical reactions. Additional comparisons are made with results from a previously presented flamelet model. The subgrid flow terms are modeled using a mixed model, whereas the subgrid turbulence–chemistry interaction terms are modeled using the partially stirred reactor model. Simulations are carried out on a scramjet model experimentally studied at Deutsches Zentrum fur Luft- und Raumfahrt consisting of a one-sided divergent channel with a wedge-shaped flame holder at the base of which hydrogen is injected. The LES predictions are compared with experimental data for velocity, temperature, wall pressure at different cross sections as well as schlieren images, showing good agreement for both first- and second-order statistics. In addition, the LES results are used to illustrate and explain the intrinsic flow, and mixing and combustion features of this combustor.

Comparison of Node-Centered and Cell-Centered Unstructured Finite-Volume Discretizations: Inviscid Fluxes

Comparison of Node-Centered and Cell-Centered Unstructured Finite-Volume Discretizations: Inviscid Fluxes

Performance and Comparison of Cell-Centered and Node-Centered Unstructured Finite Volume Discretizations for Shallow Water Free Surface Flows

Finite volume (FV) methods for solving the two-dimensional (2D) nonlinear shallow water equations (NSWE) with source terms on unstructured, mostly triangular, meshes are known for some time now. There are mainly two basic formulations of the FV method: node-centered (NCFV) and cell-centered (CCFV). In the NCFV formulation the finite volumes, used to satisfy the integral form of the equations, are elements of the mesh dual to the computational mesh, while for the CCFV approach the finite volumes are the mesh elements themselves. For both formulations, details are given of the development and application of a second-order well-balanced Godunov-type scheme, developed for the simulation of unsteady 2D flows over arbitrary topography with wetting and drying. The popular approximate Riemann solver of Roe is utilized to compute the numerical fluxes, while second-order spatial accuracy is achieved with a MUSCL-type reconstruction technique. The Green-Gauss (G-G) formulation for gradient computations is implemented for both formulations, in order to maintain a common framework. Two different stencils for the G-G gradient computations in the CCFV formulation are implemented and tested. An edge-based limiting procedure is applied for the control of the total variation of the reconstructed field. This limiting procedure is proved to be effective for the NCFV scheme but inadequate for the CCFV approach. As such, a simple but very effective modification to the reconstruction procedure is introduced that takes into account geometrical characteristics of the computational mesh. In addition, consistent well-balanced second-order discretizations for the topography source term treatment and the wet/dry front treatment are presented for both FV formulations, ensuring absolute mass conservation, along with a stable friction term treatment. Using a controlled environment for a fair comparison, a complete assessment of both FV formulations is attempted through rigorous individual and relative performance comparisons to the approximation of analytical benchmark solutions, as well as to experimental and field data. To this end, an extensive evaluation is performed using different time dependent and steady-state test cases that incorporate topography, wetting and drying process, different types of boundary conditions as well as friction. These test cases are chosen as to compare the performance and robustness of each formulation under certain conditions and evaluate the effectiveness of the proposed modifications. Emphasis to grid convergence studies is given, with the grids used to range from regular grids to irregular ones with random perturbations of nodes. The results indicate that the quality of the mesh has a major impact on the convergence performance of the CCFV method while the NCFV method exhibits a uniform behavior on the different grid types used. The proposed correction in the computation of the reconstructed values for the CCFV formulation greatly improves the convergence behavior to the formal order of accuracy.

Comparison of Node-Centered and Cell-Centered Unstructured Finite-Volume Discretizations: Viscous Fluxes

Cell-centered and node-centered approaches have been compared for unstructured finite-volume discretization of inviscid fluxes. The grids range from regular grids to irregular grids, including mixed-element grids and grids with random perturbations of nodes. Accuracy, complexity, and convergence rates of defect-correction iterations are studied for eight nominally second-order accurate schemes: two node-centered schemes with weighted and unweighted least-squares (LSQ) methods for gradient reconstruction and six cell-centered schemes two node-averaging with and without clipping and four schemes that employ different stencils for LSQ gradient reconstruction. The cell-centered nearest-neighbor (CC-NN) scheme has the lowest complexity; a version of the scheme that involves smart augmentation of the LSQ stencil (CC-SA) has only marginal complexity increase. All other schemes have larger complexity; complexity of node-centered (NC) schemes are somewhat lower than complexity of cell-centered node-averaging (CC-NA) and full-augmentation (CC-FA) schemes. On highly anisotropic grids typical of those encountered in grid adaptation, discretization errors of five of the six cell-centered schemes converge with second order on all tested grids; the CC-NA scheme with clipping degrades solution accuracy to first order. The NC schemes converge with second order on regular and/or triangular grids and with first order on perturbed quadrilaterals and mixed-element grids. All schemes may produce large relative errors in gradient reconstruction on grids with perturbed nodes. Defect-correction iterations for schemes employing weighted least-square gradient reconstruction diverge on perturbed stretched grids. Overall, the CC-NN and CC-SA schemes offer the best options of the lowest complexity and secondorder discretization errors. On anisotropic grids over a curved body typical of turbulent flow simulations, the discretization errors converge with second order and are small for the CC-NN, CC-SA, and CC-FA schemes on all grids and for NC schemes on triangular grids; the discretization errors of the CC-NA scheme without clipping do not converge on irregular grids. Accurate gradient reconstruction can be achieved by introducing a local approximate mapping; without approximate mapping, only the NC scheme with weighted LSQ method provides accurate gradients. Defect correction iterations for the CC-NA scheme without clipping diverge; for the NC scheme with weighted LSQ method, the iterations either diverge or converge very slowly. The best option in curved geometries is the CC-SA scheme that offers low complexity, second-order discretization errors, and fast convergence.

A primitive-variable Riemann method for solution of the shallow water equations with wetting and drying

A Riemann flux that uses primitive variables rather than conserved variables is developed for the shallow water equations with nonuniform bathymetry. This primitive-variable flux is both conservative and well behaved at zero depth. The unstructured finite-volume discretization used is suitable for highly nonuniform grids that provide resolution of complex geometries and localized flow structures. A source-term discretization is derived for nonuniform bottom that balances the discrete flux integral both for still water and in dry regions. This primitive-variable formulation is uniformly valid in wet and dry regions with embedded wetting and drying fronts. A fully nonlinear implicit scheme and both nonlinear and time-linearized explicit schemes are developed for the time integration. The implicit scheme is solved by a parallel Newton-iterative algorithm with numerically computed flux Jacobians. A concise treatment of characteristic-variable boundary conditions with source terms is also given. Computed results obtained for the one-dimensional dam break on wet and dry beds and for normal-mode oscillations in a circular parabolic basin are in very close agreement with the analytical solutions. Other results for a forced breaking wave with friction interacting with a sloped bottom demonstrate a complex wave motion with wetting, drying and multiple interacting wave fronts. Finally, a highly nonuniform, coastline-conforming unstructured grid is used to demonstrate an unsteady simulation that models an artificial coastal flooding due to a forced wave entering the Gulf of Mexico.

Stable Boundary Treatment for the Wave Equation on Second-Order Form

A stable and accurate boundary treatment is derived for the second-order wave equation. The domain is discretized using narrow-diagonal summation by parts operators and the boundary conditions are imposed using a penalty method, leading to fully explicit time integration. This discretization yields a stable and efficient scheme. The analysis is verified by numerical simulations in one-dimension using high-order finite difference discretizations, and in three-dimensions using an unstructured finite volume discretization.

Unstructured finite volume discretization of two‐dimensional depth‐averaged shallow water equations with porosity

AbstractThis paper deals with the numerical discretization of two‐dimensional depth‐averaged models with porosity. The equations solved by these models are similar to the classic shallow water equations, but include additional terms to account for the effect of small‐scale impervious obstructions which are not resolved by the numerical mesh because their size is smaller or similar to the average mesh size. These small‐scale obstructions diminish the available storage volume on a given region, reduce the effective cross section for the water to flow, and increase the head losses due to additional drag forces and turbulence. In shallow water models with porosity these effects are modelled introducing an effective porosity parameter in the mass and momentum conservation equations, and including an additional drag source term in the momentum equations. This paper presents and compares two different numerical discretizations for the two‐dimensional shallow water equations with porosity, both of them are high‐order schemes. The numerical schemes proposed are well‐balanced, in the sense that they preserve naturally the exact hydrostatic solution without the need of high‐order corrections in the source terms. At the same time they are able to deal accurately with regions of zero porosity, where the water cannot flow. Several numerical test cases are used in order to verify the properties of the discretization schemes proposed. Copyright © 2009 John Wiley & Sons, Ltd.