Spatial Discretization Schemes Research Articles

A finite volume multi-moment method with boundary variation diminishing principle for Euler equation on three-dimensional hybrid unstructured grids

Abstract In this work, we have developed a novel numerical model for Euler equations on 3D hybrid unstructured grids including tetrahedral, hexahedral, prismatic and pyramidal elements. The model integrates the VPM (Volume integrated average and Point value based Multi-moment) spatial discretization scheme, the limiting projection with BVD (Boundary Variation Diminishing) manipulation and the Roe Riemann solver for unstructured grids. Distinguished from conventional finite volume method, both the volume-integrated average (VIA) and the point values (PV) at the cell vertices are memorized as prognostic variables and updated in time simultaneously. The VIA is computed by a finite volume formulation of flux form while the PV is point-wisely updated using the differential formulation, where the Roe solver is used to compute both conventional and differential Riemann problems. A special technique is introduced to the limiting projection that effectively suppresses both numerical oscillation and dissipation. The resulting numerical model provides remarkably improved numerical accuracy and robustness with a moderate increase in algorithmic complexity and computational cost, which makes it of practical significance for real-case applications. The numerical results of benchmark tests are presented to demonstrate the appealing solution quality of the present model in comparison with other existing methods.

Efficient spectral computation of the stationary states of rotating Bose–Einstein condensates by preconditioned nonlinear conjugate gradient methods

We propose a preconditioned nonlinear conjugate gradient method coupled with a spectral spatial discretization scheme for computing the ground states (GS) of rotating Bose–Einstein condensates (BEC), modeled by the Gross–Pitaevskii Equation (GPE). We first start by reviewing the classical gradient flow (also known as imaginary time (IMT)) method which considers the problem from the PDE standpoint, leading to numerically solve a dissipative equation. Based on this IMT equation, we analyze the forward Euler (FE), Crank–Nicolson (CN) and the classical backward Euler (BE) schemes for linear problems and recognize classical power iterations, allowing us to derive convergence rates. By considering the alternative point of view of minimization problems, we propose the preconditioned steepest descent (PSD) and conjugate gradient (PCG) methods for the GS computation of the GPE. We investigate the choice of the preconditioner, which plays a key role in the acceleration of the convergence process. The performance of the new algorithms is tested in 1D, 2D and 3D. We conclude that the PCG method outperforms all the previous methods, most particularly for 2D and 3D fast rotating BECs, while being simple to implement.

Development of Compressible Large-Eddy Simulations Combining High-Order Schemes and Wall Modeling

Compressible large-eddy simulations combining high-order methods with a wall model have been developed in order to compute wall-bounded flows at high Reynolds numbers. The high-order methods consist of low-dissipation and low-dispersion implicit finite volume schemes for spatial discretization on structured grids. In the first part, the procedure used to apply these schemes in near-wall regions is presented. It is based on a ghost cell reconstruction. Its validity is assessed by performing the large-eddy simulation of a turbulent channel flow at a friction Reynolds number of . In the second part, to consider flows at higher Reynolds numbers, a large-eddy simulation approach using a wall model is proposed. The coupling between the wall model and the high-order schemes is described. The performance of the approach is evaluated by simulating a turbulent channel flow at using meshes with grid spacings of , 200, and 300 in the streamwise direction and , 100, and 150 in the wall-normal and spanwise directions (in wall units). The effects of the choice of the point used for data exchange between the wall model and the large-eddy simulation algorithm, as well as of the positions of the ghost cells used for the coupling, are examined by performing additional computations in which these parameters vary. The results are in agreement with direct numerical simulation data. In particular, the turbulent intensities obtained in the logarithmic region of the boundary layers of the channel flow are successfully predicted.

Low-Reynolds number mixing ventilation flows: Impact of physical and numerical diffusion on flow and dispersion

Quality assurance in computational fluid dynamics (CFD) is essential for an accurate and reliable assessment of complex indoor airflow. Two important aspects are the limitation of numerical diffusion and the appropriate choice of inlet conditions to ensure the correct amount of physical diffusion. This paper presents an assessment of the impact of both numerical and physical diffusion on the predicted flow patterns and contaminant distribution in steady Reynolds-averaged Navier–Stokes (RANS) CFD simulations of mixing ventilation at a low slot Reynolds number (Re≈2,500). The simulations are performed on five different grids and with three different spatial discretization schemes; i.e. first-order upwind (FOU), second-order upwind (SOU) and QUICK. The impact of physical diffusion is assessed by varying the inlet turbulence intensity (TI) that is often less known in practice. The analysis shows that: (1) excessive numerical and physical diffusion leads to erroneous results in terms of delayed detachment of the wall jet and locally decreased velocity gradients; (2) excessive numerical diffusion by FOU schemes leads to deviations (up to 100%) in mean velocity and concentration, even on very high-resolution grids; (3) difference between SOU and FOU on the coarsest grid is larger than difference between SOU on coarsest grid and SOU on 22 times finer grid; (4) imposing TI values from 1% to 100% at the inlet results in very different flow patterns (enhanced or delayed detachment of wall jet) and different contaminant concentrations (deviations up to 40%); (5) impact of physical diffusion on contaminant transport can markedly differ from that of numerical diffusion.

Second-order discretization in space and time for radiation-hydrodynamics

Second-order accurate discretizations for radiation-hydrodynamics are currently an area of interest in the high energy density laboratory physics and astrophysics communities. Second-order methods used to solve the hydrodynamics equations and second-order methods used to solve the radiation transport equation often differ fundamentally, making it difficult to combine them in a second-order manner. Here, we present an implicit–explicit (IMEX) method for solving the 1D equations of radiation-hydrodynamics that is second-order accurate in space and time. Our radiation-hydrodynamics model consists of the 1D Euler equations coupled with a gray radiation S2 approximation. Our RH method combines the MUSCL-Hancock method for solving the Euler equations with the TR/BDF2 time integration scheme and the linear-discontinuous Galerkin finite-element spatial discretization scheme for the S2 radiation equations. The MUSCL-Hancock method is commonly used for hydrodynamic calculations and the linear-discontinuous Galerkin scheme is the standard for the Sn equations of radiative transfer. While somewhat similar, these schemes vary fundamentally with respect to the treatment of spatial slopes. We address the challenges inherent to coupling these different numerical methods and demonstrate how these challenges can be overcome. Using the method of manufactured solutions, we show that the method is second-order accurate in space and time for both the equilibrium diffusion and streaming limit, and we show that the method is capable of computing radiative shock solutions accurately by comparing our results with semi-analytic solutions.

Finite volume schemes for the numerical simulation of tracer transport in plants

Compartmental models can be used for inverse modeling of long distance tracer transport experiments in plants. Such transport models describe axial convection and diffusion as well as exchange between compartments, and are defined by partial differential equations (PDEs). Since for inverse modeling, the forward simulation needs to be evaluated frequently, a fast PDE solver is required. Here, we compare several different finite volumes schemes up to fifth order for spatial discretization with respect to accuracy, computation time and numerical oscillations. The comparison has been performed using initial conditions with varying steepness, and periodic boundary conditions. For time discretization, standard fifth order Runge–Kutta methods are used. For smooth initial conditions, fifth order upwind schemes for spatial discretization yield the most precise and fast solutions. For higher steepness of the initial condition, higher order upwind schemes produce spurious oscillations while flux limiter schemes as well as weighted essentially non-oscillating (WENO) schemes can suppress these oscillations, at the expense of comparably slower convergence rates and higher computation times.

A fully coupled hydro-mechanical material point method for saturated dense granular materials

The stability of a dense granular assembly can be greatly reduced by a pore pressure of the interstitial fluid, and the body may fail and transit from a solid-like state to a fluid-like state. This process involves two major problems: large deformation and hydro-mechanical coupling. In this work, a three-dimensional fully coupled hydro-mechanical model using material point method (MPM) is developed. Darcy's law, considering the inertial effect, is adopted to govern the motion of interstitial water, and the conservation of momentum of the mixture is used to govern the motion of the solid, i.e., granular materials. The spatial discretization schemes for these equations are derived using the generalized integration material point method (GIMP), and the proposed coupled MPM formulation is implemented in a three-dimensional numerical code. The developed model is first quantitatively validated by comparing the simulation results of temporal evolution of spatial distribution of hydraulic pressure in a one-dimensional oedometer test with the analytical results. An experiment is designed to observe the failure of a saturated sand pile, in which the partial-saturated region is avoided by increasing the hydraulic head at the input boundary, and the kinetic energy of water is dissipated by a filtering cloth. The failure process is simulated with the MPM code. It is found that the location of the shear band in the simulation agrees with the location of the sliding surface in the experiment. The temporal evolutions of the spatial distributions of hydraulic pressure and the solid velocity distribution at a specific time are given to provide insight into the mechanism of the failure process. This work would be helpful in understanding the initiation mechanism of debris flows induced by rainfall, and sand production in gas hydrate-bearing sediments due to increasing fluid content associated with hydrate dissociation.

Symplectic spatial integration schemes for systems of balance equations

A method to generate geometric pseudo-spectral spatial discretization schemes for hyperbolic or parabolic partial differential equations is presented. It applies to the spatial discretization of systems of conservation laws with boundary energy flows and/or distributed source terms. The symplecticity of the proposed spatial discretization schemes is defined with respect to the natural power pairing (form) used to define the port-Hamiltonian formulation for the considered systems of balance equations. The method is applied to the resistive diffusion model, a parabolic equation describing the plasma dynamics in tokamaks. A symplectic Galerkin scheme with Bessel conjugated bases is derived from the usual Galerkin method, using the proposed method. Besides the spectral and energetic properties expected from the symplecticity of the method, it is shown that more accurate approximation of eigenfunctions and reduced numerical oscillations result from this choice of conjugated approximation bases. Finally, the obtained numerical results are validated against experimental data from the tokamak Tore Supra facility.

Application of the Path Tubes Method to the Navier-Stokes Equations

This work deals with an extension of the Path Tubes method for the solution of the time-dependent Navier-Stokes equations for an incompressible Newtonian fluid. Departing from a physically intuitive methodology based on the theoretical basis of the mechanics of continuous media, a robust numerical technique is obtained. This version of the Path Tubes method draws on a semi-Lagrangian time-discretization that employs the Reynolds’ transport theorem, and a localization approach, to establish an implicit semi-Lagrangian algorithm that allows the use of classical schemes for spatial discretization, such as central-difference formulas, without the need to use upwind techniques, or high-order corrections for time derivatives. Some of the extensive numerical tests are shown herein, in particular for Reynolds’ numbers typical of advection dominated flows. The tests show the method is accurate, even for coarse grids.

ARES: A Parallel Discrete Ordinates Transport Code for Radiation Shielding Applications and Reactor Physics Analysis

ARES is a multidimensional parallel discrete ordinates particle transport code with arbitrary order anisotropic scattering. It can be applied to a wide variety of radiation shielding calculations and reactor physics analysis. ARES uses state-of-the-art solution methods to obtain accurate solutions to the linear Boltzmann transport equation. A multigroup discretization is applied in energy. The code allows multiple spatial discretization schemes and solution methodologies. ARES currently provides diamond difference with or without linear-zero flux fixup, theta weighted, directional theta weighted, exponential directional weighted, and linear discontinuous finite element spatial differencing schemes. Discrete ordinates differencing in angle and spherical harmonics expansion of the scattering source are adopted. First collision source method is used to eliminate or mitigate the ray effects. Traditional source iteration and Krylov iterative method preconditioned with diffusion synthetic acceleration are applied to solve the linear system of equations. ARES uses the Koch-Baker-Alcouffe parallel sweep algorithm to obtain high parallel efficiency. Verification and validation for the ARES transport code system have been done by lots of benchmarks. In this paper, ARES solutions to the HBR-2 benchmark and C5G7 benchmarks are in excellent agreement with published results. Numerical results are presented which demonstrate the accuracy and efficiency of these methods.

Implementation and Comparison of High-Resolution Spatial Discretization Schemes for Solving Two-Fluid Seven-Equation Two-Pressure Model

As compared to the two-fluid single-pressure model, the two-fluid seven-equation two-pressure model has been proved to be unconditionally well-posed in all situations, thus existing with a wide range of industrial applications. The classical 1st-order upwind scheme is widely used in existing nuclear system analysis codes such as RELAP5, CATHARE, and TRACE. However, the 1st-order upwind scheme possesses issues of serious numerical diffusion and high truncation error, thus giving rise to the challenge of accurately modeling many nuclear thermal-hydraulics problems such as long term transients. In this paper, a semi-implicit algorithm based on the finite volume method with staggered grids is developed to solve such advanced well-posed two-pressure model. To overcome the challenge from 1st-order upwind scheme, eight high-resolution total variation diminishing (TVD) schemes are implemented in such algorithm to improve spatial accuracy. Then the semi-implicit algorithm with high-resolution TVD schemes is validated on the water faucet test. The numerical results show that the high-resolution semi-implicit algorithm is robust in solving the two-pressure two-fluid two-phase flow model; Superbee scheme and Koren scheme give two highest levels of accuracy while Minmod scheme is the worst one among the eight TVD schemes.

Evaluating the impact of grid cell properties in spatial discretization of groundwater model for a tropical karst catchment in Rote Island, Indonesia

Abstract To assess the effect of three grid cell properties (size, mean slope of the surface and distance between centre of grid and observation well) on groundwater models' performances, a tropical karst catchment characterized by monsoonal season in Rote Island, Indonesia was selected. Here, MODFLOW was used to develop models with five different spatial discretization schemes: 10 × 10 m, 20 × 20 m, 30 × 30 m, 40 × 40 m and 50 × 50 m. Using parameter estimation method, hydraulic conductivity and specific yield values over a selection of pilot points were estimated. The trends of the performances were calculated at each observation well in order to recommend the most appropriate location for observation well placement in terms of topographical characteristic. It is confirmed that the deterioration of model performance is mainly controlled by the increase of distance between well and centre of the cell, and the mean slope of the surface. Results reveal that model performance increases substantially for areas of low slope (<3%) and medium slope (3–10%) for a smaller grid cell size. Therefore, to improve model performance, it is recommended that the observations wells are placed in areas of low and medium slopes.

A fully-implicit high-order system thermal-hydraulics model for advanced non-LWR safety analyses

An advanced system analysis tool is being developed for advanced reactor safety analysis. This paper describes the underlying physics and numerical models used in the code, including the governing equations, the stabilization schemes, the high-order spatial and temporal discretization schemes, and the Jacobian Free Newton Krylov solution method. The effects of the spatial and temporal discretization schemes are investigated. Additionally, a series of verification test problems are presented to confirm the high-order schemes. It is demonstrated that the developed system thermal-hydraulics model can be strictly verified with the theoretical convergence rates, and that it performs very well for a wide range of flow problems with high accuracy, efficiency, and minimal numerical diffusions.

Numerical simulation of two phase flow in wellbores by means of drift flux model and pressure based method

This paper presents a drift flux correlation and a pressure based method to overcome convergence problems of two phase flow simulation in wellbores. The drift flux correlation is flow pattern independent so that results discontinuities at flow pattern transitions are eliminated. Using the experiment data of large diameter annuli, the performance of drift flux correlation has been testified. The proposed pressure based method combines iteration and traversal to ensure convergence of the algorithm in which the staggered grid finite volume method and the fully implicit backward Euler method are adopted for the spatial discretization and time integration schemes. Test case comparison with Roe solver has proved the good performance of proposed pressure based method. Moreover, a full scale experiment is conducted to demonstrate the successful implementation of the pressure based method and drift flux correlation.

Analyzing slow drying in a porous medium placed adjacent to laminar airflow using a pore-network model

ABSTRACTA novel unified implicit computational framework and a unique pore-to-cell meshing method is used to predict, by coupling the inside-drying outside-flow processes, the slow drying of a pore-network-represented porous medium placed adjacent to a laminar flow of air in a slit. The effects of numerical algorithms, spatial and temporal discretization schemes, and meshing methods in minimizing computational effort are studied. The numerical solutions of the outer and inner processes are tested against several benchmark studies. The external air velocity affects drying during the initial stages. The microstructure of the pore network (P-N is found to have a strong influence on drying.

Implementation of spectral difference method on overset grids for compressible inviscid flows

This work investigates the use of a high-order spectral difference (SD) method on overset grids for the solution of the Euler equations. A spectral difference method of up to fourth order for hexahedral elements is employed as the spatial discretization scheme and the explicit four-stage Runge–Kutta method is utilized for time integration. In order to preserve the high-order accuracy throughout the whole computation domain, an interpolation method for overlapping interfaces is developed. To validate this interpolation method and the accuracy of the overall algorithm, the inviscid convection of a vortex and subsonic flow over a wing are calculated. Then the flow over a hovering rotor is simulated by the high-order SD method on overset grids. The numerical results demonstrated the capability and accuracy of the present algorithm on overset grids.

Verification of a low Mach variable-density Navier-Stokes solver for turbulent combustion

We describe the low Mach variable-density Navier-Stokes numerical iterative solution procedure implemented in the finite-volume unstructured T-FlowS code. As the test cases we use a number of analytic manufactured solutions and Rayleigh-Taylor instability problem from the literature for algorithm verification purposes. The tests show that the code is second-order accurate in agreement with the spatial discretization scheme. We outline the recent combustion ADEF model implemented in the program.

Complex reaction network generation for Steady State Isotopic Transient Kinetic Analysis: Fischer-Tropsch Synthesis

A versatile modeling strategy for Steady State Isotopic Transient Kinetic Analysis (SSITKA) data, acquired in a plug flow reactor, is elaborated with particular attention to complex reaction networks and Fischer-Tropsch Synthesis as example. A spatial discretization scheme optimizing accuracy and CPU time is developed. For low switch time constants, the van Leer and van Albada flux limiter functions used in conjunction with the DASPK solver yields the lowest CPU time for the integration of the resulting ordinary differential equations. For larger switch time constants, conventional central differencing can be applied. A network generation methodology is implemented accounting for the isotopic labeling. It reduces the exponential dependence of the number of considered species on the carbon number to a quadratic dependence. For a reaction network allowing a maximum chain length of 5 carbon atoms a gain in CPU time up to a factor of 10 can be achieved.

On the implementation of moment transport equations in OpenFOAM: Boundedness and realizability

In this contribution some crucial numerical aspects concerning the implementation of quadrature-based moment methods (QBMM) into the open-source Computational Fluid Dynamics (CFD) code OpenFOAM are discussed. As well-known QBMM are based on the simple idea of solving a kinetic master equation, not in terms of the underlying number density function (NDF), but in terms of the moments of the NDF itself, via moment transport equations. These numerical aspects are in fact very often overlooked, resulting in implementations that do not satisfy the properties of boundedness and realizability for the moments of the NDF. Boundedness is an important property (i.e., moments of the NDF have to be bounded between some minimal and maximum values), that in turn depends on the initial and boundary conditions. Boundedness can be guaranteed by using a consistent approach with respect to the constraints imposed on the transport variables, such as dispersed phase volume fraction. Realizability is instead related to the existence of an underlying NDF that corresponds to a specific moment set. It is well-known that time and spatial discretization schemes can corrupt a moment set unless they are specifically designed to preserve realizability. One popular choice is to use ad-hoc pseudo high-order schemes to ensure realizable moment set is obtained. In this work, moment transport equations are implemented in the CFD code OpenFOAM using a similar form proposed by Weller (2002) for preserving the boundedness of the volume fraction, together with the numerical schemes of Vikas et al. (2011) for ensuring moment realizability. The effectiveness of such implementation is illustrated on two simple examples, taken from our work on the simulation of fluid-fluid multiphase systems.

New analytical solutions to the two-phase water faucet problem

The one-dimensional water faucet problem is one of the classical benchmark problems originally proposed by Ransom to study the two-fluid two-phase flow model. With simplifications, such as massless gas phase and no wall and interfacial frictions, analytical solutions had been previously obtained for the transient liquid velocity and void fraction distribution. The water faucet problem and its analytical solutions have been widely used for code assessment, benchmark, and numerical verification. In this work, we present a new set of analytical solutions to the water faucet problem at the steady-state condition, with the gas-phase density’s effect on pressure distribution considered. This new set of analytical solutions is used in a rigorous numerical verification process from which the anticipated second-order spatial accuracy is achieved for a second-order spatial discretization scheme. On the contrary, the same anticipated order of accuracy could not be obtained using the Ransom solutions as the reference. In addition, extended Ransom transient solutions for the gas-phase velocity and pressure are derived with the assumption of decoupled liquid and gas pressures. Numerical benchmark on the extended Ransom solutions is also presented.