Higher-order Interpolation Scheme Research Articles

Implicit large-eddy simulation of turbulent plane jet at Re = 104

In this study, implicit large-eddy simulations (ILESs) are performed for a turbulent plane jet at a Reynolds number 104 to assess the capability of ILESs in simulating turbulent transport in free shear flows. To investigate the effect of resolved turbulent scales on the self-similar behavior of the plane jet, ILESs are conducted using fifth- (ILES+P5), seventh- (ILES+P7), and ninth-order polynomial (ILES+P9) interpolation methods, as well as the third-order MUSCL method (ILES+M3). ILES+P5, ILES+P7, and ILES+P9 reproduced the results of the direct numerical simulation (DNS) for a plane jet at Re = 104. The resolution of the power spectral density of ILESs with high-order interpolation schemes, which agrees well with the DNS, increases as the order of spatial accuracy increases and resolves more than 80% of the total energy. Hence, we conclude that the spatial resolution to resolve up to the frequency region that contains approximately 80% of the turbulent energy is required to obtain good agreement with the results of DNS, which is widely accepted in LES with the subgrid-scale (SGS) model and is also valid for ILES.

A hybrid smoothed particle hydrodynamics coupled to a fictitious domain method for particulate flows and its application in a three-dimensional printing process

A hybrid smoothed particle hydrodynamics (SPH) coupled to a direct-forcing fictitious domain (FD) scheme is proposed for the simulation of particulate flows. The new method is effective in calculating the particle-fluid interaction forces and uses a kernel interpolation function to realize the momentum exchange between fluids and particles. A higher-order interpolation scheme is applied to preserve the consistency and accuracy of the kernel function. We first test our method by simulating the benchmark of flow past a cylinder at Reynolds number ranging from 20 to 200 and obtain the convergent results, including accurate lift and drag coefficients, Strouhal number, length of the recirculation bubble, and separation angle. Subsequently, we verify our new method for the typical particulate flows, including the motion of a neutrally-buoyant cylinder in Couette flow, the motion of a neutrally-buoyant cylinder in a planar Poiseuille flow, the sedimentation of a circular cylinder in a channel, and the motion of a neutrally-buoyant slender particle in Couette flow. A good agreement with the existing experimental data, with other numerical results, and with available theoretical solutions is obtained indicating the accuracy and the robustness of the new method. Subsequently, our SPH-FD is applied to simulate a slender fiber in a three-dimensional (3D) printing process, showing a good potential and advantages in simulating particulate flows of some industrial processes.

Two-Dimensional Fourier Continuation and Applications

This paper presents a method (2D-FC) for construction of bi-periodic extensions of smooth non-periodic functions defined over general two-dimensional smooth domains. The approach can be directly generalized to domains of any given dimensionality, and even to non-smooth domains, but such generalizations are not considered here. The 2D-FC extensions are produced in a two-step procedure. In the first step the one-dimensional Continuation method is applied along a discrete set of outward boundary-normal directions to produce, along such directions, continuations that vanish outside a narrow interval beyond the boundary. Thus, the first step of the algorithm produces blending-to-zero along normals for the given function values. In the second step, the extended function values are evaluated on an underlying Cartesian grid by means of an efficient, high-order boundary-normal interpolation scheme. A Continuation expansion of the given function can then be obtained by a direct application of the two-dimensional FFT algorithm. Algorithms of arbitrarily high order of accuracy can be obtained by this method. The usefulness and performance of the proposed two-dimensional Continuation method are illustrated with applications to the Poisson equation and the time-domain wave equation within a bounded domain. As part of these examples the novel Fourier Forwarding solver is introduced which, propagating plane waves as they would in free space and relying on certain boundary corrections, can solve the time-domain wave equation and other hyperbolic partial differential equations within general domains at computing costs that grow sublinearly with the size of the spatial discretization.

A novel combination of mathematical modelling approaches for simulating nearly hypersonic, viscous, reacting magnetogasdynamic flows

This work focuses on the mathematical combination of magnetogasdynamics (MGD) and the Dunn and Kang chemical kinetic model with a multi-component Harten–Lax–van Leer Contact (HLLC) local Riemann solver using high-order interpolation schemes. Due to the nature of nearly hypersonic, unsteady, viscous, reacting MGD flows, the gas dissociates which is taken into account with 11 species and 26 chemical reactions along with the temperature dependent transport properties through the Wilke model. In the present work, the electromagnetic effects are considered by solving a Poisson equation simultaneously with the chemical reaction equations. The non-linear convective terms are treated numerically with third-, fifth- and seventh-order Weighted Essentially Non-Oscillatory (WENO) interpolation schemes along with second-, fourth- and sixth-order central finite difference schemes for the viscous terms. The combination of these numerical and physical models are analysed for stability and validated for benchmark problems such as (a) Hartmann flow as MGD validation test case and (b) the shock-tube problem for testing the chemical reaction model in which case the reaction rate controlling temperature suggested by Park has also been taken into account. Furthermore, a complex physical problem of an external aerodynamic flow over the cube for the Mach number varying from 1.25 to 3 have also been investigated, because a limited number of studies is available for understanding the flow behaviour of sharp cornered blunt bodies such as cubes. The knowledge of shock structures of these objects has become imperative to predict the atmospheric re-entry trajectories of de-orbiting CubeSats, space debris or fragments of launch vehicles. The main physical finding is that the shock structures are observed to be much more resolved compared to previous works, especially in the wake region. The knowledge gap is bridged by detailing the shock structures and showing the dependence of the shock standoff distance on total enthalpy of the flow.

Performance of fast fluid dynamics with a semi-Lagrangian scheme and an implicit upwind scheme in simulating indoor/outdoor airflow

Computational fluid dynamics can be time consuming for predicting indoor airflows and pollutant transport in large-scale problems or emergency management. Fast fluid dynamics (FFD) is able to accomplish efficient and accurate simulation of indoor/outdoor airflow. FFD solves the advection term of the Navier–Stokes equations either by a semi-Lagrangian (SL) scheme or an implicit upwind (IU) scheme. The SL scheme can be highly efficient, but its first-order version is not conservative and introduces significant numerical diffusion. To improve its accuracy, a high-order temporal and interpolation scheme that not only reduces dissipation and dispersion errors but also guarantees the convergence speed should be applied. Otherwise, an IU scheme instead could be used to solve the advection term. The IU scheme is conservative and introduces minor numerical diffusion, but it may increase the computation time. Therefore, this study investigated the performance of FFD with SL scheme using high-order temporal and interpolation schemes and that with IU scheme. The comparisons used experimental data of two indoor airflows and one outdoor airflow. The results showed that FFD with IU scheme was overall more accurate than FFD with SL scheme. In simulating indoor airflow, both methods were robust and the predictions were independent of time step sizes if the Courant number was less than or equal to one. In simulating the outdoor airflow, the FFD with SL scheme performed better than the FFD with IU scheme for large time step sizes. The FFD with IU scheme consumed 44%–61% computing time of the FFD with SL scheme.

On the accuracy, robustness, and performance of high order interpolation schemes for the overset method on unstructured grids

AbstractA comprehensive study on interpolation schemes used in overset grid techniques is here presented. Based on a literature review, numerous schemes are implemented, and their robustness, accuracy, and performance are assessed. Two code verification exercises are performed for this purpose: a 2D analytical solution of a laminar Poiseuille steady flow; and an intricate manufactured solution of a turbulent flow case, characteristic of a boundary layer flow combined with an unsteady separation bubble. For both cases, the influence of grid layouts, grid refinement, and time‐step is investigated. Local and global errors, convergence orders, and mass imbalance are quantified. In terms of computational performance, strong scalability, cpu timings, load imbalancing, and domain connectivity information (DCI) overhead are reported. The effect of the overset‐grid interpolation schemes on the numerical performance of the solver, that is, number of nonlinear iterations, is also scrutinized. The results show that, for a second order finite volume code, once diffusion is dominant (low Reynolds number), interpolation schemes higher than second order, for example, least squares of degree 2, are needed not to increase the total discretization errors. For convection dominated flows (high Reynolds numbers), the results suggest that second order schemes, for example, nearest cell gradient, are sufficient to prevent overset grid schemes to taint the underlying discretization errors. In terms of performance, by single‐process and parallel communication optimization, the total overset‐grid overhead (with DCI done externally to the CFD code) may be less than 4% of the total run time for second‐order schemes and 8% for third‐order ones, therefore empowering higher‐order schemes and more accurate solutions.

A Two-Way Nesting Unstructured Quadrilateral Grid, Finite-Differencing, Estuarine and Coastal Ocean Model with High-Order Interpolation Schemes

The balance between the need of improving horizontal resolution in simulating local small-scale ocean processes and computational costs makes it desirable to refine model mesh locally. A three-dimensional, two-way nesting unstructured quadrilateral grid, primitive equations, finite-differencing, estuarine and coastal ocean model is developed for multi-scale modeling. Because the model grid is capable of multi-area nesting and multi-level refinement at each subdomain, the model is highly compatible with simulations involved in complex topography and studies of local small-scale ocean processes. The two-way information exchange is achieved by a virtual grid method, and its basic idea is to implement numerical integrations of variables at nesting interfaces with the support of virtual grid variables, which are interpolated or updated from actual grid variables. The model is novel for two interpolation schemes: the high-order spatial interpolation at the middle temporal level (HSIMT) parabolic interpolation scheme and HSIMT advection-equivalent interpolation scheme, and they have high-order accuracy and good consistency with the advection scheme applied to solving the tracer equations. The conservation of both volume and tracer contents is ensured via a flux correction algorithm. The two original interpolation schemes are examined in an ideal salinity advection experiment in the peak preservation skill, stability, and conservation properties. A realistic application to the Deep Waterway Project area in the Changjiang Estuary showed that the nested grid model can reproduce the hydrodynamic processes at the observed sites successfully while it failed to maintain the performance with the structured grid model in simulating the variance of salinity, for which the enforced conservation had primary responsibility. The HSIMT parabolic interpolation scheme was distinguished from other schemes for its outstanding performances in simulating salinity.

Compact Schemes for Multiscale Flows with Cell-Centered Finite Difference Method

High-order compact interpolation schemes appropriate for multiscale flows are studied within a cell-centered finite difference method (CCFDM) framework where the robustness of high-order schemes on curvilinear grids can be greatly enhanced due to the satisfaction of geometric conservation law. Two types of compact interpolations are mainly developed in this paper for shock-free flows and shock-embedded flows respectively. The present compact schemes are verified to be superior over the explicit counterparts with same orders in terms of the spectral characteristics. Regarding the shock-free flows, low-dissipation low-dispersion properties are achieved by the spectral optimization. Three optimized compact schemes (Opt4, Opt6 and Opt8) are further validated to be attractive for shock-free problems by carrying out benchmarks from computational aeroacoustics workshops and two typical turbulence cases: Tayler–Green vortex and decaying isotropic turbulence. Regarding high-speed flows in the presence of shock waves, the shock-capturing capability is realized by extending the weighting technique to the compact interpolations. The criteria to choose optimally compact nonlinear sub-stencils on a most general compact global stencil are presented. Interestingly, the explicit WENO-type schemes can be reverted within the proposed compact framework. Three nonlinear compact schemes (UI5, CI6 and CI8) on two practical stencils are analyzed and further compared with their explicit counterparts by a series of numerical experiments. The compact ones are superior to explicit ones in resolving rich flow structures as well as discontinuities.

A nonlinear cross-section deformable thin-walled beam finite element model with high-order interpolation of warping displacement

In this paper, a new finite element model is developed for nonlinear analysis of thin-walled beams by introducing a high-order interpolation for the warping displacement field to better simulate actual cross-section deformation. It is discovered that, due to the significant coupling between bending and membrane deformations in each wall, it is crucial for a nonlinear thin-walled beam finite element to be capable to describe complex warping displacement distribution over the cross-section. In the geometrically exact beam element model developed, the warping displacement is expressed by using a higher-order interpolation scheme, while the distortion displacements are independently described by using a number of cross-section in-plane deformation modes. Numerical results are presented to demonstrate the effectiveness of the proposed model and investigate possible impacts of interpolation schemes for the warping displacement distribution on the solution accuracy under different problem settings. The results show that with an appropriate high-order warping displacement interpolation scheme, the proposed beam model can produce very accurate results even in the case of small wall thickness, outperforming many of the existing models. It is therefore concluded that special attention is needed on the warping displacement field for a finite element model to accurately simulate the cross-section distortional deformation of thin-walled beams. • A geometrically exact cross-section deformable thin-walled beam finite element model is developed. • The distortion and warping displacements are described by distortion modes and high-order interpolation, respectively. • Numerical examples are presented for a comprehensive investigation of the present finite element model. • The importance of warping displacement field for ensuring solution accuracy is pointed out.

High-Order Conservative Positivity-Preserving DG-Interpolation for Deforming Meshes and Application to Moving Mesh DG Simulation of Radiative Transfer

Solution interpolation between deforming meshes is an important component for several applications in scientific computing, including indirect arbitrary-Lagrangian-Eulerian and rezoning moving mesh methods in numerical solution of partial differential equations. In this paper, a high-order, conservative, and positivity-preserving interpolation scheme is developed based on the discontinuous Galerkin solution of a linear time-dependent equation on deforming meshes. The scheme works for bounded but otherwise arbitrary mesh deformation from the old mesh to the new one. The cost and positivity preservation (with a linear scaling limiter) of the DG-interpolation are investigated. Numerical examples are presented to demonstrate the properties of the interpolation scheme. The DG-interpolation is applied to the rezoning moving mesh DG solution of the radiative transfer equation, an integro-differential equation modeling the conservation of photons and involving time, space, and angular variables. Numerical results obtained for examples in one and two spatial dimensions with various settings show that the resulting rezoning moving mesh DG method maintains the same convergence order as the standard DG method, is more efficient than the method with a fixed uniform mesh, and is able to preserve the positivity of the radiative intensity.

WENO interpolation for Lagrangian particles in highly compressible flow regimes

Lagrangian particles can be used either to analyze different complex flows, as massless tracers, or to model multiphase flows, as particles with mass. As particles can have arbitrary positions within the computational domain, interpolation of different flow quantities from the Eulerian grid is essential. Low-order centered interpolation schemes generally do not provide a sufficient level of accuracy for many flow configurations of interest. Thus, typically, high-order centered interpolation schemes are utilized. The current study demonstrates that for highly compressible non-reacting and reacting flow regimes, in which discontinuities in the flow field, e.g., shock waves arise, centered interpolation schemes tend to smear the shock and high order schemes also produce numerical oscillations. It is shown that this problem can be remedied by using Weighted-Essentially-Non-Oscillatory (WENO) interpolation schemes. Extensive numerical tests are performed in order to demonstrate this, comparing the performance of WENO-3 and WENO-5 interpolation schemes with a variety of centered interpolation schemes. The first test case involves specially designed steady state two-dimensional vortex flows. For a smooth flow, WENO schemes provide comparable accuracy to other high-order centered schemes. For flow fields with discontinuities, WENO-3 and WENO-5 interpolation schemes can decrease the interpolation error by more than two and three orders of magnitude, respectively, in comparison with centered schemes. Solution accuracy is further studied with a normal shock wave test. It is demonstrated that centered schemes tend to smear the shock, whereas, WENO schemes capture it in a much sharper manner. Moreover, high-order centered schemes tend to oscillate in the vicinity of the discontinuity leading to non-physical results, such as negative absolute pressure values. The same trends are retained for an unsteady three-dimensional spherical blast wave, where the shock is smeared across several grid cells, which is typical for shock-capturing flow solvers. Finally, the benefits of WENO schemes for the Lagrangian-particle tracking analysis of highly compressible reactive flows are explored by comparing various Lagrangian particle trajectories and joint probability density functions (PDFs) for a two-dimensional cellular detonation. The results indicate that high-order centered schemes lead to unphysical results, whereas WENO schemes can provide high-order interpolation that is free of severe numerical oscillations and non-physical artifacts, which is critical for the proper analysis of the flow.

Analysis of mild ignition in a shock tube using a highly resolved 3D-LES and high-order shock-capturing schemes

A highly resolved three-dimensional large-eddy simulation (LES) is presented for a shock tube containing a stoichiometric hydrogen–oxygen ( $$\hbox {H}_2$$ / $$\hbox {O}_2$$ ) mixture, and the results are compared against experimental results. A parametric study is conducted to test the effects of grid resolution, numerical scheme, and initial conditions before the 3D simulations are presented in detail. An approximate Riemann solver and a high-order interpolation scheme are used to solve the conservation equations of the viscous, compressible fluid and to account for turbulence behind the reflected shock. Chemical source terms are calculated by a finite-rate model. Simultaneous results of pseudo-Schlieren, temperature, pressure, and species are presented. The ignition delay time is predicted in agreement with the experiments by the three-dimensional simulations. The mechanism of mild ignition is analysed by Lagrangian tracer particles, tracking temperature histories of material particles. We observed strongly increased temperatures in the core region away from the end wall, explaining the very early occurrence of mild ignition in this case.

High-order simplified thermal lattice Boltzmann method for incompressible thermal flows

A high-order simplified thermal lattice Boltzmann method (HSTLBM) is developed in this paper for accurate and efficient simulation of incompressible thermal flows. The derivation of HSTLBM stems from the recently developed simplified thermal lattice Boltzmann method (STLBM) and incorporates high-order interpolation algorithms, which reflects an effective combination of local second-order reconstruction and global high-order scheme. By introducing virtual streaming nodes, HSTLBM decouples the streaming distance from the mesh spacing and then correlates them through high-order interpolation scheme. Delicate parametric studies indicate that adopting 5-point Lagrange interpolation and setting the streaming distance as 0.2 times of the mesh spacing could give optimal results which balances computational accuracy, stability, and efficiency well; and third-order of global accuracy can be achieved. HSTLBM inherits various merits of STLBM, especially its nice numerical stability. As a result, HSTLBM can give accurate and stable solutions on coarser meshes for problems at high Reynolds/Rayleigh numbers. Higher efficiency and lower memory cost can thus be expected. A series of benchmark tests are provided for comprehensive evaluation of HSTLBM in modelling two- and three-dimensional problems and on uniform/non-uniform meshes.

Implementation and Validation of Semi-Implicit WENO Schemes Using OpenFOAM®

In this article, the development of high-order semi-implicit interpolation schemes for convection terms on unstructured grids is presented. It is based on weighted essentially non-oscillatory (WENO) reconstructions which can be applied to the evaluation of any field in finite volumes using its known cell-averaged values. Here, the algorithm handles convex cells in arbitrary three-dimensional meshes. The implementation is parallelized using the Message Passing Interface. All schemes are embedded in the code structure of OpenFOAM® resulting in the access to a huge open-source community and the applicability to high-level programming. Several verification cases and applications of the scalar advection equation and the incompressible Navier-Stokes equations show the improved accuracy of the WENO approach due to a mapping of the stencil to a reference space without scaling effects. An efficiency analysis indicates an increased computational effort of high-order schemes in comparison to available high-resolution methods. However, the reconstruction time can be efficiently decreased when more processors are used.

A TVD scheme for 3d unstructured grids applied to compositional reservoir simulation

Abstract In reducing the grid orientation effect for the numerical solution of partial differential equations, interpolation functions play an important role when the advective transport of the governing equations is considered. This is due to the fact that, in general, the unknowns are evaluated in the vertices of the elements and such properties must be extrapolated to inner parts of the elements. First-order schemes, such as upwind, are the easiest methods to use for performing the extrapolation of the properties. However, such methods introduce a large amount of numerical diffusion in the solution. A few higher-order interpolation schemes, on the other hand, are capable of providing solutions free of numerical diffusion, increasing the accuracy of the method and reducing the computational efforts required. In this work, we investigate the TVD interpolation scheme for three-dimensional unstructured grids in conjunction with Element-based Finite Volume Method (EbFVM) using four types of elements: hexahedron, tetrahedron, prism and pyramid.

A Novel Highly Accurate Finite-Element Family

A novel Nth order finite element for interior acoustics and structural dynamics is presented, with N arbitrarily large. The element is based upon a three-dimensional extension of the Coons patch technique, which combines high-order Lagrange and Hermite interpolation schemes. Numerical applications are presented, which include the evaluation of the natural frequencies and modes of vibration of (1) air inside a cavity (interior acoustics) and (2) finite-thickness beams and plates (structural dynamics). The numerical results presented are assessed through a comparison with analytical and numerical results. They show that the proposed methodology is highly accurate. The main advantages however are (1) its flexibility in obtaining different level of accuracy (p-convergence) simply by increasing the number of nodes, as one would do for h-convergence, (2) the applicability to arbitrarily complex configurations, and (3) the ability to treat beam- and shell-like structures as three-dimensional small-thickness elements.

On the Spurious Mode Generation Induced by Spectral-Like Optimized Interpolation Schemes Used in Computational Acoustics

AbstractThe present work constitutes a fraction of a more extensive study that is devoted to numerical methods in acoustics. More precisely, we address here the interpolation process, which is more and more frequently used in Computational Acoustics–whether it is for enabling multi-stage hybrid calculations, or for easing the proper handling of complex configurations via advanced techniques such as Chimera grids or Immersed Boundary Conditions. In that regard, we focus on high-order interpolation schemes, so as to analyze their intrinsic features and to assess their effective accuracy. Taking advantage of specific insights that had been previously achieved by the present authors regarding standard high-order interpolation schemes (of centered nature), we here focus on their so-calledspectral-like optimizedcounterparts (of both centered and noncentered nature). The latter spectral-like optimized schemes are analyzed thoroughly thanks to dedicated theoretical developments, which allow highlighting better what their strengths and weaknesses are. Among others, the various ways such interpolation schemes can degrade acoustic signals they are applied to are carefully investigated from a theoretical point-of-view. Besides that, specific criteria that could help in optimizing interpolation schemes better are provided, along with generic rules about how to minimize the signal degradation induced by existing interpolation schemes, in practice.

파랑 중 선박의 자유도 운동해석을 위한 중첩격자 기반의 수치해법

This paper introduces a computational method for analysis of the 6-DOF motions of a ship in waves using an overset grid technique which consists of inner and outer domains for representing body motions and numerical wave tank, respectively. High order interpolation scheme is employed to increase numerical accuracy over the interface where physical values, such as velocities and pressure, interact between the inner and outer domains. The numerical schemes and algorithm are addressed in the present paper. An application to motion of KCS container carrier in head waves is presented, and the comparison of responses on heave and pitch motions shows good agreement with those of model tests.

High-order interpolation schemes for shear instability simulations

Purpose The purpose of this paper is to evaluate the performance of a numerical method for the solution to shallow-water equations on a staggered grid, in simulations for shear instabilities at two convective Froude numbers.

Compilation and validation of bathymetric data for the South China Sea with an emphasis on shallow region

Bathymetric data with a resolution of 5 min called gbscs5 have been compiled for a shelf region shallower than 200 m in the South China Sea by referring to paper charts, electronic charts, and trackline sounding data. Through a comparison between the new dataset and existing datasets (etopo5, etopo1, gebco08), characteristics of each dataset, including the existence of artificial features derived from erroneous sounding data or a conversion error of depth units, were documented for each dataset. It was found that small-scale bathymetric features observed along the shelf edge in gebco08 bathymetry are bounded by survey lines and were likely to be a spurious figure generated when applying a high-order interpolation scheme. Tidal simulations conducted with changing bathymetries have shown that the newly compiled dataset will produce smaller deviation from observed values than the existing bathymetries. While all four datasets analyzed in this study were found to be suitable for estimating overall tidal features of the South China Sea, a care must be taken when applying the tidal model results to a particular coastal region.