High-order Discretization Schemes Research Articles

Numerical study on migration and solidification behavior of molten droplets on structure surface and in debris voids during debris bed melting process

The melt migration and solidification behavior on the surface of metal structure and in the debris voids is a key phenomenon of debris bed melting, which has an important impact on the assessment of reactor pressure vessel damage and melt leakage. In this study, the moving particle semi-implicit (MPS) method with high-order discretization scheme was coupled with a surface tension model that was established based on interface reconstruction. The migration and solidification behavior of UO2–ZrO2 and Fe–Zr melts on the metal structure surface and in the debris voids were investigated with the improved MPS method. The effects of temperature, velocity, contact angle and droplet diameter on the melt migration and solidification were analyzed. The results showed that the molten UO2–ZrO2 droplet on the metal structure surface presented three stages with solidification rate decreasing, while the solidification rate of molten Fe–Zr droplet had little change. The maximum spread factor and solidification rate increased with the increase of droplet falling velocity, but the spread factor was independent of the melt type at the same velocity. The high-temperature molten droplet penetrated the voids formed by the debris with diameter of 5 mm more easily compared to the voids formed by the debris with diameter of 3 mm. The influence of contact angle on the migration of molten droplet with initial velocity was small, and the maximum difference in droplet mass fraction was about 6%. Three groups of molten droplets with different diameters penetrated the voids, and the average increments of penetration mass fraction were 2.3%, 2.7% and 5.8%, respectively, with the increase of velocity. Near the bottom of the debris bed, the molten droplet without initial velocity solidified and blocked the flow channel, but the molten droplet with initial velocity of 0.5 m/s penetrated through the debris voids. The droplet with initial velocity of 0.5 m/s had a faster solidification rate in the voids compared to the droplet without initial velocity, and the droplet solidification mass fraction was 96.27% when the initial melt temperature was 1410 K.

High resolution optimized high-order schemes for discretization of non-linear straight and mixed second derivative terms

In this paper, we propose a new set of midpoint-based high-order discretization schemes for computing straight and mixed nonlinear second derivative terms that appear in the compressible Navier-Stokes equations. Firstly, we detail a set of conventional fourth and sixth-order baseline schemes that utilize central midpoint derivatives for the calculation of second derivatives terms. To enhance the spectral properties of the baseline schemes, an optimization procedure is proposed that adjusts the order and truncation error of the midpoint derivative approximation while still constraining the same overall stencil width and scheme order. A new filter penalty term is introduced into the midpoint derivative calculation to help achieve high wavenumber accuracy and high-frequency damping in the mixed derivative discretization. Fourier analysis performed on the both straight and mixed second derivative terms show high spectral efficiency and minimal numerical viscosity with no odd-even decoupling effect. Numerical validation of the resulting optimized schemes is performed through various benchmark test cases assessing their theoretical order of accuracy and solution resolution. The results highlight that the present optimized schemes efficiently utilize the inherent viscosity of the governing equations to achieve improved simulation stability - a feature attributed to their superior spectral resolution in the high wavenumber range. The method is also tested and applied to non-uniform structured meshes in curvilinear coordinates, employing a supersonic impinging jet test case.

Convergence towards High-Speed Steady States Using High-Order Accurate Shock-Capturing Schemes

Creating time-marching unsteady governing equations for a steady state in high-speed flows is not a trivial task. Residue convergence in time cannot be achieved when using most low- and high-order spatial discretization schemes. Recently, high-order, weighted, essentially non-oscillatory schemes have been specially designed for steady-state simulations. They have been shown to be capable of achieving machine precision residues when simulating the Euler equations under canonical coordinates. In the present work, we review these schemes and show that they can also achieve machine residues when simulating the Navier–Stokes equations under generalized coordinates. This is carried out by considering three supersonic flows of perfect fluids, namely the flow upstream a cylinder, the flow over a blunt wedge, and the flow over a compression ramp.

Radial basis function-differential quadrature-based physics-informed neural network for steady incompressible flows

In this work, a radial basis function differential quadrature-based physics-informed neural network (RBFDQ-PINN) is proposed to simulate steady incompressible flows. The conventional physics-informed neural network (PINN) makes use of the physical equation as a constraint to ensure that the solution satisfies the physical law and the automatic differentiation (AD) method to calculate derivatives at collocation points. Although the AD-PINN is expedient in evaluating derivatives at arbitrary points, it is time-consuming with higher-order derivatives and may lead to nonphysical solutions with sparse samples. Alternatively, the finite difference (FD) method can facilitate the calculation of derivatives, but the FD-PINN will increase the computational cost when handling random point distributions, especially with higher-order discretization schemes. To address these issues, the radial basis function differential quadrature (RBFDQ) method is incorporated into the PINN to replace the AD method for the calculation of derivatives. The RBFDQ method equips with high efficiency in the calculation of high-order derivatives as compared with the AD method and great flexibility in the distribution of mesh points as compared with the FD method. As a result, the proposed RBFDQ-PINN is not only more efficient and accurate but also applicable to irregular geometries. To demonstrate its effectiveness, the RBFDQ-PINN is tested in sample problems such as the lid-driven cavity flow, the channel flow over a backward-facing step, and the flow around a circular cylinder. Numerical results reveal that the RBFDQ-PINN achieves satisfactory accuracy without any labeled collocation points, whereas the AD-PINN struggles to solve some cases, especially for high Reynolds number flows.

Accuracy Analysis of Div-Conforming Hierarchical Higher-Order Discretization Schemes for the Magnetic Field Integral Equation

Accuracy Analysis of Div-Conforming Hierarchical Higher-Order Discretization Schemes for the Magnetic Field Integral Equation

Verification of a Free-Surface Pressure Term Extension to Represent Ships in a Nonhydrostatic Shallow-Water-Equations Solver

Abstract In recent years, increasing ship sizes and associated increasing wave loads have led to a demand for prediction tools quantifying the ship-induced loads on waterways. Depth-averaged numerical models, using a free-surface pressure term, are a prominent method to obtain the relevant design parameters. These models incorporate the wave deformation processes due to attributes of complex bathymetries, while allowing for an efficient simulation of large computational domains. The nonhydrostatic shallow-water-equations (SWE) model REEF3D::SFLOW uses a quadratic pressure approximation and high-order discretization schemes. This paper presents the implementation of a pressure term to account for the displacement of the free surface by solid moving objects. Two test cases verifying the implementation are shown based upon the analytical one-dimensional solution of the wave propagation due to surface pressure and the estimation of Havelock angles. These verification tests are the first step toward a holistic model, combining a large scale model with computational fluid dynamics (CFD) simulations near waterway banks.

Optimal Error Estimates of the Local Discontinuous Galerkin Method and High-order Time Discretization Scheme for the Swift–Hohenberg Equation

Optimal Error Estimates of the Local Discontinuous Galerkin Method and High-order Time Discretization Scheme for the Swift–Hohenberg Equation

A flexible fully nonlinear potential flow model for wave propagation over the complex topography of the Norwegian coast

Coastal wave propagation and transformation are complicated due to the significant variations of water depth and irregular coastlines, which are typically present at the Norwegian fjords. A potential flow model provides phase-resolved solutions with low demands on computational resources. Many potential flow models are developed for offshore waves and lack of numerical treatments of coastal conditions. In the presented work, several modifications are introduced to a fully nonlinear potential flow model with a σ-grid for the purpose of coastal wave modelling: Shallow water breaking criteria are included in addition to deepwater breaking algorithms to approximate breaking waves over a complete range of water depth. A new coastline algorithm is introduced to detect complex coastlines and ensure robust simulations near the coast. The algorithm is compatible with structured grid arrangement in the horizontal plane and allows for high-order discretisation schemes for the free surface boundary conditions for an accurate representation of complex free surfaces. A parallelised solver for the Laplace equation is utilised to ensure fast simulations for large domains with multi-core infrastructures.The proposed model is validated against theories and experiments for various two- and three-dimensional nonlinear wave propagation and transformation cases that represent typical coastal conditions. The simulations show a good representation of nonlinear waves, and the results compare well with experiments. Furthermore, two large-scale engineering scenarios are simulated, where the applicability of the coastline algorithm and the parallel commutation capability of the model are demonstrated.

A high order weak approximation for jump-diffusions using Malliavin calculus and operator splitting

AbstractThe paper introduces a novel high order discretization scheme for expectation of jump-diffusion processes by using a Malliavin calculus approach and an operator splitting method. The test function of the target expectation is assumed to be only Lipschitz continuous in order to apply the method to financial problems. Then Kusuoka’s estimate is employed to justify the proposed discretization scheme. The algorithm with a numerical example is shown for implementation.

On the properties of high-order least-squares finite-volume schemes

High-order finite-volume schemes based on polynomial least-squares methods are an active research topic for the discretization of hyperbolic equations as they allow to obtain high-order spatial discretization schemes in arbitrary meshes. However, few studies have analyzed their performance in good-quality/near-to-uniform meshes, which are commonly used as a meshing strategy in zones where turbulent effects are important. In this paper, the theoretical numerical properties of commonly used least-squares (LSQ) k-exact high-order finite volume schemes are studied in one-dimensional and in several two-dimensional meshes (with some remarks regarding their properties in three-dimensional meshes). These results are compared to those obtained using fully-constrained polynomial reconstructions only compatible with structured meshes. The numerical properties of the schemes are investigated through the von Neumann analysis methodology applied to the one-dimensional and two-dimensional finite volume formulation, including temporal discretization errors. This analysis is also extended to non-uniform and unstructured two-dimensional meshes. At last, the schemes are tested with several numerical experiments using the linear advection, the Euler equations and the Navier-Stokes equations. The analysis of both studies yields similar conclusions regarding the numerical errors and stability of the different studied schemes showing that the high-order least-squares finite volume schemes yield stable and robust results across different uniform and non-uniform unstructured meshes. However, their performance is heavily degraded in simulations with low mesh resolution compared to schemes specially catered to structured meshes. On the other hand, the latter schemes lack stability and robustness in general structured meshes and its formulation cannot be straightforwardly extended to unstructured meshes. Moreover, this work shows that the use of weighted-LSQ can drastically improve the results of LSQ schemes in under-resolved simulations.

Two High-Order Time Discretization Schemes for Subdiffusion Problems with Nonsmooth Data

Abstract Two new high-order time discretization schemes for solving subdiffusion problems with nonsmooth data are developed based on the corrections of the existing time discretization schemes in literature. Without the corrections, the schemes have only a first order of accuracy for both smooth and nonsmooth data. After correcting some starting steps and some weights of the schemes, the optimal convergence orders O(k 3–α ) and O(k 4–α ) with 0 < α < 1 can be restored for any fixed time t for both smooth and nonsmooth data, respectively. The error estimates for these two new high-order schemes are proved by using Laplace transform method for both homogeneous and inhomogeneous problem. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Thermoacoustic instability drivers and mode transitions in a lean premixed methane-air combustor at various swirl intensities

In this study, an in-house developed fully compressible solver utilizing high-order numerical discretization schemes is used to study the effects of swirl intensity on thermoacoustic instabilities in a lean premixed swirl stabilized combustor. Turbulent combustion modeling is achieved using large eddy simulations along with dynamically thickened flame combustion model. Obtained results reveal that there are two critical burner swirl numbers in which the combustor dynamics change drastically. Results show that the combustor is stable at low swirl numbers (S ≤ 0.5). However, as the swirl intensity increases, the combustor experiences thermoacoustic instabilities. The first critical swirl number lies within the range of 0.5–0.6, in which a transition occurs in the combustor dynamics from a state of stable operation to thermoacoustic instabilities. The second critical swirl number (S = 0.9) corresponds to the transition of different thermoacoustic instability modes. The combustor experiences limit cycle instabilities coupled with the first tangential acoustic mode of the combustor for 0.6 ≤ S < 0.9, while for S > 0.9 the instabilities are burst like and coupled with the second tangential acoustic mode of the combustor. The combustor experiences instabilities coupled with both the first and the second tangential modes, when the burner swirl number is 0.9. In order to evaluate key parameters inducing the thermoacoustic instabilities, heat release fluctuations are calculated through hydrodynamic flow features (central and side recirculation zones, coherent structures, and precessing vortex core). Investigations show that all the above features contribute almost equally in inducing heat release fluctuations in-phase with the acoustic perturbations at low swirl numbers. However, as the swirl number increases, side recirculation zone and coherent structures play the key role in producing heat release fluctuations in-phase with the acoustic perturbations, while central recirculation zone has the least effects on driving the instabilities.

A directional ghost-cell immersed boundary method for incompressible flows

This paper describes an efficient ghost-cell immersed boundary method for incompressible flow simulations. The method employs a locally directional extrapolation scheme along all discretization directions for the ghost values. Additionally, it involves fictitious discrete boundary forcing terms instead of the ghost values in the governing equations. In this way, the boundary is represented more accurately than in prior IBMs and it is possible to fulfill the divergence-free condition. When combined with high-order spatial discretization schemes, the IBM order is reduced locally near the immersed boundary in a step-wise manner. In this way, the method delivers more compact stencils and is able to deal with sharp interfaces. By handling the distance from the ghost point to the boundary carefully, the proposed method delivers lower truncation errors than standard IBM, with clean (persistent) convergence rate and enhanced stability. The parallel implementation of this approach is straightforward. Its accuracy has been checked by considering a variety of test cases, including irregular, three-dimensional, and moving boundaries. The local accuracy for the proposed method is formally second order, and is measured to be close to this value using numerical tests.

Investigation of Focusing Wave Properties in a Numerical Wave Tank with a Fully Nonlinear Potential Flow Model

Nonlinear wave interactions and superpositions among the different wave components and wave groups in a random sea sometimes produce rogue waves with extremely large wave heights that appear unexpectedly. A good understanding of the generation and evolution of such extreme wave events is of great importance for the analysis of wave forces on marine structures. A fully nonlinear potential flow (FNPF) model is proposed in the presented paper to investigate the different factors that influence the wave focusing location, focusing time and focusing wave height in a numerical wave tank. Those factors include wave steepness, spectrum bandwidth, wave generation method, focused wave spectrum, and wave spreading functions. The proposed model solves the Laplace equation together with the boundary conditions on a σ -coordinate grid using high-order discretisation schemes on a fully parallel computational framework. The model is validated against the focused wave experiments and thereafter used to obtain insights into the effects of the different factors. It is found that the wave steepness contributes to changing the location and time of focus significantly. Spectrum bandwidth and directional spreading affect the focusing wave height and profile, for example, a wider bandwidth and a wider directional spread lead to a lower focusing wave height. A Neumann boundary condition represents the nonlinearity of the wave groups better than a relaxation method for wave generation.

Assessment of artificial fluid properties for high-order accurate large-eddy simulations of shock-free compressible turbulent flows with strong temperature gradients

The capability of artificial fluid properties (AFP) to serve as a sub-grid scale (SGS) model and to provide stability in high-order accurate large-eddy simulations (LES) of shock-free compressible turbulent flows with strong temperature gradients is assessed. Predictive LES of such flows, which are directly relevant to applications such as engines, require an accurate description of turbulence, which typically makes them susceptible to issues of accuracy and numerical instabilities. In the present implementation, AFP schemes along with a high-order compact filter ensure robustness of otherwise non-dissipative high-order central difference spatial discretisation schemes. It is shown that a judicious choice of both filter and AFP parameters is required in order for AFP to be used as an accurate SGS model. Optimal parameter values for the AFP and the corresponding filter parameter are identified and validated in the framework of the current LES solver. First, an inviscid Taylor-Green vortex configuration is examined to evaluate the ideal combination of the filter parameter and the coefficients in the AFP framework. It is verified that AFP works as an alternative model for the unclosed subgrid-scale (SGS) dissipation terms. The AFP results are also compared to those from the dynamic Smagorinsky model (DSM), with good agreement. Second, the optimal parameter values of AFP obtained from the inviscid Taylor-Green vortex tests are validated on a series of test configurations with increasing complexity, that include an inviscid 1D advection problem involving a thermal contact discontinuity, an inviscid 2D Kelvin-Helmholtz instability test and 3D experimental unheated and heated spatially developing turbulent round jets. Robust and accurate predictions are shown with the LES solver employing the optimal model parameters in each of these tests. In particular, good agreement between the LES predictions and experimental measurements for both the flow and temperature statistics is achieved for the laboratory jets, demonstrating the accuracy of the LES/AFP simulations applied to practical turbulent configurations. The robustness of the solver in the presence of strong temperature gradients is also demonstrated by simulating a round jet case having very high temperature gradients similar to those occurring in reacting flows. In contrast, the LES/DSM simulation of this case is noted to be unstable, and failed to provide converged results.

A parallel high-order compressible flows solver with domain decomposition method in the generalized curvilinear coordinates system

PurposeThis paper aims to present the development of a highly parallel finite-difference computational fluid dynamics code in generalized curvilinear coordinates system. The objectives are to handle internal and external flows in fairly complex geometries including shock waves, compressible turbulence and heat transfer.Design/methodology/approachThe code is equipped with high-order discretization schemes to improve the computational accuracy of the solution algorithm. Besides, a new method to deal with the geometrical singularities, so-called domain decomposition method (DDM), is implemented. The DDM consists of using two different meshes communicating with each other, where the base mesh is Cartesian and the overlapped one a hollow cylinder.FindingsThe robustness of the present implemented code is appraised through several numerical test cases including a vortex advection, supersonic compressible flow over a cylinder, Poiseuille flow, turbulent channel and pipe flows. The results obtained here are in an excellent agreement when compared to the experimental data and the previous direct numerical simulation (DNS). As for the DDM strategy, it was successful as simulation time is clearly decreased and the connection between the two subdomains does not create spurious oscillations.Originality/valueIn sum, the developed solver was capable of solving, accurately and with high-precision, two- and three-dimensional compressible flows including fairly complex geometries. It is noted that the data provided by the DNS of supersonic pipe flows are not abundant in the literature and therefore will be available online for the community.

Development of a robust solver to model the flow inside the engines for high-speed propulsion

The demand for discovering new commercial routes as well as the possibility to shortening civilian long-haul flights boosted the interest of civil hypersonic vehicle designs. Among all the multiple projects started by the various nations, the European community funded project STRATOFLY aims at refining the baseline LAPCAT II-MR2.4 design for further improvements. The new aircraft would enable a flight shorter that 3 hours from Brussels to Sydney, carrying 300-passengers above the already crowed atmosphere. The wide Mach range operability, up to Mach 8, demands the use of multiple engines, leading to a highly integrated propulsion system. The current study is focused on the development of new CFD platform to estimate the performance of the combined propulsion system during the supersonic to hypersonic transition. In order to control the complex flow physics, highfidelity CFD simulations remain the fundamental tools for the preliminary investigations. On the current framework, an advanced robust compressible solver has been develop d in order to handle the different flow regimes. The new tool solves Unsteady Reynolds-Averaged Navier-Stokes (URANS) equations by employing cell-centered Finite Volume Method constructed on openFoam toolbox. Two innovative high-order discretization schemes, with different abilities, based on approximated Riemann solvers were developed for capturing the flow physics within high-speed propulsion systems. Advanced time discretization has been taken into account to increase the temporal accuracy. At the end, the whole implementation has been validated in multiple test cases, ranging from incompressible to hypersonic regimes, confirming its excellent stability, robustness and accuracy.

Computational Fluid Dynamics Simulations of Nonlinear Sloshing in a Rotating Rectangular Tank Using the Level Set Method

In this paper, numerical simulations of nonlinear sloshing in rectangular tanks are presented. Model implementations in the open source software reef3d are tested, and the results are compared with experimental data from three different conditions. The interface location is compared for both linear and nonlinear sloshing. The nonlinear sloshing is simulated in both two-dimensional (2D) and three-dimensional (3D). Video images from the SPHERIC project are compared with simulations for the interface. A condition with lateral wave impacts in sloshing, with a frequency close to the natural frequency of the first mode, can be found in this case. The numerical model is solving the Reynolds-averaged Navier–Stokes (RANS) equations with the k–ω turbulence model. The level set method is used to capture the interface. Higher order discretization schemes are implemented to handle time-evolution and convective fluxes. A ghost cell method is used to account for solid boundaries and parallel computations. It is found that the limiting factor for the eddy-viscosity has significant influence in the nonlinear sloshing cases. As the sloshing becomes more violent, the increased strain at the gas–liquid interface overproduces turbulence energy with unrealistically high damping of the motion. Three-dimensional simulations show slightly better comparison than 2D. Due to nonlinearities and small damping, the time to reach steady-state may take several cycles. The last case shows promising results for the global motion. As expected, the breakup of the liquid surface makes it difficult to resolve each phase. But overall, the numerical model predicts the sloshing motion reasonably well.

Non-normal stability of embedded boundary methods through pseudospectra

We present non-normal linear stability of embedded boundary (EB) methods employing pseudospectra and resolvent norms. We analyze the stability of the linear wave equation in a domain with an embedded boundary in one and two dimensions. In particular, we characterize the stability in terms of the normalized distance of the EB to the nearest ghost node (α). An important objective is that the CFL condition remains unaffected by the EB which is taken to be that of the underlying grid (as if the EB is absent or aligned with the regular grid). We consider various spatial and temporal discretization methods including both central and upwind-biased schemes. Stability is guaranteed when α≤αmax where αmax ranges between 0.5 and 0.77 depending on the discretization scheme. The stability analysis is also briefly examined for the one-dimensional advection–diffusion equation. Sharper limits on the sufficient conditions for stability are obtained based on the pseudospectral radius (the Kreiss constant) than the restrictive limits based on the usual singular value decomposition analysis. A resulting sufficient condition for stability is that the intersection of the EB with the surface normal passing through a ghost node (x→g) must lie within the ghost cell (0.5Δx distance from x→g). This condition leads us to propose a simple and robust reclassification scheme for the ghost cells (dubbed “hybrid ghost cells”) to ensure Lax stability of the discrete system. This has been tested successfully for both low and high order discretization schemes with transient growth of at most O(1). Moreover, we present a fourth order EB reconstruction scheme taking into account both accuracy and stability.

High-resolution method for evolving complex interface networks

In this paper we describe a high-resolution transport formulation of the regional level-set approach for an improved prediction of the evolution of complex interface networks. The novelty of this method is twofold: (i) construction of local level sets and reconstruction of a global level set, (ii) local transport of the interface network by employing high-order spatial discretization schemes for improved representation of complex topologies. Various numerical test cases of multi-region flow problems, including triple-point advection, single vortex flow, mean curvature flow, normal driven flow, dry foam dynamics and shock–bubble interaction show that the method is accurate and suitable for a wide range of complex interface-network evolutions. Its overall computational cost is comparable to the Semi-Lagrangian regional level-set method while the prediction accuracy is significantly improved. The approach thus offers a viable alternative to previous interface-network level-set method.