Spurious Waves Research Articles

Performance of characteristic numerical boundary conditions for mixed convective flows past a heated square cylinder using a non-Boussinesq approach

This work represents an investigation of the performance of characteristic numerical boundary conditions on the unconfined 2D mixed convective flow of air past a heated square cylinder governed by a non-Boussinesq flow model. The working fluid (air) is taken to be thermally perfect with variable thermo-physical and transport properties. The fully compressible Navier-Stokes equations are transformed using body-fitted coordinates and solved using particle velocity upwind (PVU-M+) scheme on a structured O-type grid. To investigate the performance of characteristic numerical boundary conditions, the gravity aligned upward flow past a heated square cylinder at (α = 0°, M = 0.1 and 0.2, Re = 100, Fr = 0.5 and 1.0, Pr = 0.71, γ = 1.4, and ε = 0.07 and 1.0) is considered. The overheat ratio (ε) is defined as (Tw – T∞)/T∞, where Tw and T∞ are the uniform cylinder and free stream temperatures, respectively. In order to determine the proper choice of boundary conditions to be imposed on the far boundary, algebraic characteristic boundary conditions of Euler equations (AECBC) are considered in a local 1D sense normal to the far boundary. In this work, two sets of algebraic characteristic boundary conditions are considered. In one set (BC Set I), AECBC is used over the entire far boundary. While in the other set (BC Set II), AECBC is used over a portion of far boundary, that is characterized by normal flow velocity magnitude in excess of a threshold value, and entropy preserving extrapolating conditions for pressure and density over the remaining portion of the far boundary. It is shown that BC Set I generates spurious wave reflections and numerical noise from the far boundary, whereas BC Set II eliminates these numerical artifacts and captures the flow physics far more accurately.

2-D Beam Scanning With Cylindrical-Leaky-Wave-Enhanced Phased Arrays

Directive pencil beams scannable in both elevation and azimuth are obtained through a planar phased array placed inside a Fabry–Perot cavity. The key element of the proposed approach is the exploitation of a conical element pattern (EP) with high directivity in elevation, obtained through the excitation of a dominant, weakly attenuated cylindrical TM leaky wave of azimuthal order $n = 0$ by means of a simple coaxial probe. Then, a highly reduced number $N$ of such sources are arranged to form a phased array radiating directive pencil beams. Beam-angle reconfigurability with continuous scanning both in azimuth and elevation inside a wide solid angular range is achieved by varying the array phasing and the operating frequency. An investigation on the features of truncated cylindrical leaky waves is first developed to properly characterize the EP. Then, conventional array theory is exploited to calculate the pattern of the entire array. The radiation efficiency is also evaluated accounting for the spurious surface wave related to the undesired excitation of the quasi-TEM mode. The proposed array design provides a simple and inexpensive innovative solution for obtaining a high-gain pencil beam continuously scanning in the 3-D space without suffering gain losses. In the presented implementation, the elevation angular scan, which is generally constrained by the wideband capability of the feeding system, by the requirements on the sidelobe level, and by the cutoff of the relevant leaky mode, ranges from about 21° to about 68°. Possible applications are envisaged for the next generation of wireless power transfer devices, for advanced radar and surveillance systems, earth observation, as well as for ceiling-mounted indoor localization and tracking.

Establishment of Numerical Wave Flume Based on the Second-Order Wave-Maker Theory

With growing computational power, the first-order wave-maker theory has become well established and is widely used for numerical wave flumes. However, existing numerical models based on the first-order wave-maker theory lose accuracy as nonlinear effects become prominent. Because spurious harmonic waves and primary waves have different propagation velocities, waves simulated by using the first-order wave-maker theory have an unstable wave profile. In this paper, a numerical wave flume with a piston-type wave-maker based on the second-order wave-maker theory has been established. Dynamic mesh technique was developed. The boundary treatment for irregular wave simulation was specially dealt with. Comparisons of the free-surface elevations using the first-order and second-order wave-maker theory prove that second-order wave-maker theory can generate stable wave profiles in both the spatial and time domains. Harmonic analysis and spectral analysis were used to prove the superiority of the second-order wave-maker theory from other two aspects. To simulate irregular waves, the numerical flume was improved to solve the problem of the water depth variation due to low-frequency motion of the wave board. In summary, the new numerical flume using the second-order wave-maker theory can guarantee the accuracy of waves by adding an extra motion of the wave board. The boundary treatment method can provide a reference for the improvement of nonlinear numerical flume.

Improving initial condition perturbations in a convection‐permitting ensemble prediction system

One of the main challenges presented by a limited‐area model ensemble prediction system (LAMEPS) concerns the limited capacity for its initial condition perturbations to correctly represent large‐scale flow uncertainties due to its limited‐size domain and deficiencies in formulating lateral boundary conditions. In addition, a mismatch between LAMEPS (initial condition) and host EPS lateral boundary perturbations can form spurious gravity waves at the boundaries. In the present work, an ensemble Jk blending method is proposed for improving representation of large‐scale uncertainties and for addressing consistent initial conditions and lateral boundary perturbations. Our approach involves employing Jk blending within a framework of three‐dimensional variation (3D‐Var) ensemble data assimilation (EDA). In such a system, small‐scale perturbations are generated from 3D‐Var EDA, while large‐scale perturbations are generated from the host ensemble via Jk blending.The ensemble Jk method is implemented in the C‐LAEF (Convection‐permitting Limited‐Area Ensemble Forecasting) system and is compared to the standard perturbed‐observation EDA approach, i.e. perturbed‐observation EDA without large‐scale constraint. The comparison shows that the ensemble Jk method gives a more skilful and reliable EPS, especially for upper‐air variables. In addition, positive effects on the surface pressure and precipitation of large‐scale perturbations are shown. Finally, the ensemble Jk method's capacity to alleviate perturbation mismatches is demonstrated.

Implementation of the Vector Vorticity Dynamical Core on Cubed Sphere for Use in the Quasi‐3‐D Multiscale Modeling Framework

AbstractThe dynamical core that predicts the three‐dimensional vorticity rather than the momentum, which is called Vector‐Vorticity Model (VVM), is implemented on a cubed sphere. Its horizontal coordinate system is not restricted to orthogonal, while the vertical coordinate is orthogonal to the horizontal surface. Accordingly, all the governing equations of the VVM, which are originally developed with Cartesian coordinates, are rewritten in terms of general curvilinear coordinates. The local coordinates on each cube surface are constructed with the gnomonic equiangular projection. Using global channel domains, the VVM on the cubed sphere has been evaluated by (1) advecting a passive tracer with a bell‐shaped initial perturbation along an east‐west latitude circle and along a north‐south meridional circle and (2) simulating the evolution of barotropic and baroclinic instabilities. The simulated results with the cubed‐sphere grids are compared to analytic solutions or those with the regular longitude‐latitude grids. The convergence with increasing spatial resolution is also quantified using standard error norms. The comparison shows that the solutions with the cubed‐sphere grids are quite reasonable for both linear and nonlinear problems when high resolutions are used. With coarse resolution, degeneracy appears in the solutions of the nonlinear problems such as spurious wave growth; however, it is effectively reduced with increased resolution. Based on the encouraging results in this study, we intend to use this model as the cloud‐resolving component in a global Quasi‐Three‐Dimensional Multiscale Modeling Framework.

Computation of Aeroacoustics and Fluid Flow Problems Using a Novel Dispersion Relation Preserving Scheme

A new spectrally optimized physical dispersion relation preserving scheme has been introduced to solve computational acoustics and aeroacoustics problems, accurately. The derived fourth-order accurate scheme has significant spectral resolution and physical dispersion relation preserving (DRP) nature. The scheme displays neutral stability at high CFL numbers. The developed scheme has an ability to add numerical diffusion as and when required to attenuate spurious waves present in the computed solutions. These features make the proposed scheme suitable for solving computational aeroacoustic problems. The scheme has been validated by comparing solutions of model computational acoustic problems with the available analytical solutions. Scheme has also been tested to solve the incompressible flow field around a circular cylinder executing rotary oscillations. Ability of the scheme to perform direct simulation of the computational aeroacoustic problems has been shown by computing acoustic field triggered by a laminar flow past a stationary circular cylinder. Excellent match has been observed between the present computed results and the available results in the literature which justifies applicability of the present DRP scheme to solve complex flow and aeroacoustic problems.

Applicability of Nonlinear Wavemaker Theory

Generation of high-quality waves is essential when making numerical or physically model tests. When using a wavemaker theory outside the validity area, spurious waves are generated. In order to investigate the validity of different wave generation methods, new model test results are presented where linear and nonlinear wave generation theories are tested on regular and irregular waves. A simple modification to the second-order wavemaker theory is presented, which significantly reduces the generation of spurious waves when used outside its range of applicability. For highly nonlinear regular waves, only the ad-hoc unified wave generation based on stream function wave theory was found acceptable. For irregular waves, similar conclusions are drawn, but the modified second-order wavemaker method is more relevant. This is because the ad-hoc unified generation method for irregular waves requires the wave kinematics to be calculated by a numerical model, which might be quite time-consuming. Finally, a table is presented with the range of applicability for each wavemaker method for regular and irregular waves.

Concurrent coupling of peridynamics and classical elasticity for elastodynamic problems

In this paper, a coupled peridynamic-finite element method for modeling mechanical behavior and damage growth in materials is developed. Peridynamics is a nonlocal continuum model which, in contrast to other continuum models, uses integration instead of spatial derivations in its governing equations. Utilizing integration instead of derivatives is advantageous in modeling fracture since the governing equations remain valid even after the initiation or growth of discontinuities. Although peridynamics can capture material fracture effectively, however due to its nonlocal formulation peridynamics is computationally expensive. To reduce the computational costs, we propose to couple peridynamics with finite elements and use peridynamics only in small zones where higher accuracy is needed. The main challenge in developing such a coupling method is to eliminate the artifacts introduced by the interface of the two subdomains. One of the main issues is spurious wave reflections which occurs because high frequency waves traveling from peridynamic region cannot enter the finite element zone and spuriously reflect back into the peridynamic zone. This will lead to an increase in the energy of the peridynamic zone and will drastically reduce the computational accuracy. The main feature of the proposed method is eliminating the spurious wave reflections such that the coupled method is as accurate as the pure peridynamics.

New Accurate and Efficient Method for Stiff Detonation Capturing

An alternative approach to prevent spurious behavior caused by conventional shock-capturing schemes when solving stiff detonation wave problems is introduced. In engineering research of detonation waves, conventional shock-capturing schemes usually encounter difficulties in identifying the location of the detonation front because the discontinuous solution is smeared. To overcome this excessive numerical dissipation with traditional discretized schemes used in nonreacting high-speed compressible flow, a shock-capturing scheme is introduced in which, besides the linear function constructed in the monotone upstream-centered schemes for conservation law (MUSCL) scheme, a steplike tangent of hyperbola for interface capturing (THINC) function is also employed in the reconstruction process. The final reconstruction function is determined using the boundary variation diminishing (BVD) algorithm, which reduces significantly the numerical dissipation around discontinuities. One- and two-dimensional comparative numerical tests of stiff detonation wave problems were conducted with the fifth-order weighted essentially nonoscillatory and MUSCL-THINC-BVD schemes, demonstrating that the latter scheme reproduces the correct position of detonation waves with improved resolution, whereas the former scheme, despite the higher order, produces spurious waves. Compared with other methods, which, by accepting smeared-out discontinuities profiles, require extra treatments, the current method obtains the correct but also sharp detonation front by fundamentally reducing numerical dissipation errors in shock-capturing schemes.

Effects due to aliasing on surface-wave extraction and suppression in frequency-velocity domain

Surface waves are treated as a kind of strong noise in petroleum seismic exploration. We can suppress surface waves by extracting them because of the different distribution features of surface waves and body waves in the frequency-velocity (f-v) domain. The accuracy of surface-wave extraction and suppression relies on high-resolution and high-accuracy imaging, however, while spatial aliasing will inevitably smear energy images in the f-v domain due to sparse sampling. It is crucial to analyze the effects due to spatial aliasing on surface-wave extraction and suppression. We use the reversible linear operator, high-resolution linear Radon transform, to extract and analyze the energy caused by spatial aliasing. Real-world and theoretical examples demonstrate that the retrieved surface-wave fields will be polluted by spurious body waves once the spatial aliasing energy in the f-v domain gets involved in surface-wave extraction, which will subsequently affect the quality of surface-wave suppression. We suggest that the surface-wave extraction range should be kept within Nyquist spatial sampling limits as much as possible. In addition, the results of synthetic data show that the spatial aliasing could cause mode misidentification since it is easy to be recognized as the higher modes of the surface waves in the f-v domain. In order to distinguish higher modes from spatial aliasing, we proposed using the asymptotic velocity of high modes at high frequencies and derived formulas to calculate the location of the energy due to spatial aliasing to correctly identify the high modes.

An adaptive multi-grid peridynamic method for dynamic fracture analysis

The standard way of implementing Peridynamics is a meshfree approach which uses a uniform discretization. This is inefficient when a very dense grid spacing for a localized area is required. In this paper, a radically new strategy to couple grids with different spacing is put forward. It is free of ghost forces in static cases and spurious waves in dynamic problems can be controlled and made negligible thanks to proper discretization. There is no loss of volume due to non-uniform discretization at the interface between different grids. An efficient algorithm is developed to apply the refinement adaptively. It permits to increase the resolution of the analysis only in the critical zones. The performance is investigated by solving dynamic problems, including cases of crack propagation in brittle materials. We compare the solutions of the proposed method with those of a standard peridynamic model, which employs uniform discretization, and show that the same accuracy is obtained at a much smaller computational cost.

A k · p treatment of edge states in narrow 2D topological insulators, with standard boundary conditions for the wave function and its derivative

For 2D topological insulators with strong electron–hole hybridization, such as HgTe/CdTe quantum wells, the widely used 4 × 4 k · p Hamiltonian based on the first electron and heavy hole sub-bands yields an equal number of physical and spurious solutions, for both the bulk states and the edge states. For symmetric bands and zero wave vector parallel to the sample edge, the mid-gap bulk solutions are identical to the edge solutions. In all cases, the physical edge solution is exponentially localized to the boundary and has been shown previously to satisfy standard boundary conditions for the wave function and its derivative, even in the limit of an infinite wall potential. The same treatment is now extended to the case of narrow sample widths, where for each spin direction, a gap appears in the edge state dispersions. For widths greater than 200 nm, this gap is less than half of the value reported for open boundary conditions, which are called into question because they include a spurious wave function component. The gap in the edge state dispersions is also calculated for weakly hybridized quantum wells such as InAs/GaSb/AlSb. In contrast to the strongly hybridized case, the edge states at the zone center only have pure exponential character when the bands are symmetric and when the sample has certain characteristic width values.

Experimental investigation of material nonlinearity using the Rayleigh surface waves excited and detected by angle beam wedge transducers

Angle beam wedge transducers are widely used in nonlinear Rayleigh wave experiments as they can generate Rayleigh wave easily and produce high intensity nonlinear waves for detection. When such a transducer is used, the spurious harmonics (source nonlinearity) and wave diffraction may occur and will affect the measurement results, so it is essential to fully understand its acoustic nature. This paper experimentally investigates the nonlinear Rayleigh wave beam fields generated and received by angle beam wedge transducers, in which the theoretical predictions are based on the acoustic model developed previously for angle beam wedge transducers [S. Zhang, et al., Wave Motion, 67, 141–159, (2016)]. The source of the spurious harmonics is fully characterized by scrutinizing the nonlinear Rayleigh wave behavior in various materials with different driving voltages. Furthermore, it is shown that the attenuation coefficients for both fundamental and second harmonic Rayleigh waves can be extracted by comparing the measurements with the predictions when the experiments are conducted at many locations along the propagation path. A technique is developed to evaluate the material nonlinearity by making appropriate corrections for source nonlinearity, diffraction and attenuation. The nonlinear parameters of three aluminum alloy specimens – Al 2024, Al 6061 and Al 7075 – are measured, and the results indicate that the measurement results can be significantly improved using the proposed method.

Nonreflective Conditions for Perfectly Matched Layer in Computational Aeroacoustics

In computational aeroacoustics, boundary conditions such as radiation, outflow, or absorbing boundary conditions are critical issues in that they can affect the entire solution of the computation. Among these types of boundary conditions, the perfectly matched layer boundary condition, which has been widely used in computational fluid dynamics and computational aeroacoustics, is developed by augmenting the additional term in the original governing equations by an absorption function so as to stably absorb the outgoing waves. Even if the perfectly matched layer is analytically a perfectly nonreflective boundary condition, spurious waves occur at the interface, since the analysis is performed in discretized space. Hence, this study is focused on factors that affect numerical errors from perfectly matched layer to find the optimum conditions for nonreflective PML. Through a mathematical approach, a minimum width of perfectly matched layer and an optimum absorption coefficient are suggested. To validate the prediction of the analysis, numerical simulations are performed in a generalized coordinate system, as well as in a Cartesian coordinate system.

Numerical analysis of a high-order unstructured overset grid method for compressible LES of turbomachinery

Large-Eddy Simulation (LES) is recognized as a promising method for high-fidelity flow predictions in turbomachinery applications. The presented approach consists of the coupling of several instances of the same LES unstructured solver through an overset grid method. A high-order interpolation, implemented within this coupling method, is introduced and evaluated on several test cases. It is shown to be third order accurate, to preserve the accuracy of various second and third order convective schemes and to ensure the continuity of diffusive fluxes and subgrid scale tensors even in detrimental interface configurations. In this analysis, three types of spurious waves generated at the interface are identified. They are significantly reduced by the high-order interpolation at the interface. The latter having the same cost as the original lower order method, the high-order overset grid method appears as a promising alternative to be used in all the applications.

Single Precision in the Dynamical Core of a Nonhydrostatic Global Atmospheric Model: Evaluation Using a Baroclinic Wave Test Case

Abstract Reducing the computational cost of weather and climate simulations would lower electric energy consumption. From the standpoint of reducing costs, the use of reduced precision arithmetic has become an active area of research. Here the impact of using single-precision arithmetic on simulation accuracy is examined by conducting Jablonowski and Williamson’s baroclinic wave tests using the dynamical core of a global fully compressible nonhydrostatic model. The model employs a finite-volume method discretized on an icosahedral grid system and its mesh size is set to 220, 56, 14, and 3.5 km. When double-precision arithmetic is fully replaced by single-precision arithmetic, a spurious wavenumber-5 structure becomes dominant in both hemispheres, rather than the expected baroclinic wave growth only in the Northern Hemisphere. It was found that this spurious wave growth comes from errors in the calculation of gridcell geometrics. Therefore, an additional simulation was conducted using double precision for calculations that only need to be performed for model setup, including calculation of gridcell geometrics, and single precision everywhere else, meaning that all calculations performed each time step used single precision. In this case, the model successfully simulated the growth of the baroclinic wave with only small errors and a 46% reduction in runtime. These results suggest that the use of single-precision arithmetic will allow significant reduction of computational costs in next-generation weather and climate simulations using a fully compressible nonhydrostatic global model with the finite-volume method.

An analytical study of wave propagation in a peridynamic bar with nonuniform discretization

In this paper, we use an analytical approach to study the propagation of a plane wave and its spurious reflection in a peridynamic bar using two different methods. In the first method, a coupled peridynamic–finite element approach is used in which peridynamic formulation is used in one part of the domain and finite element is used in the other part. In the second method, peridynamic formulation is used in the entire domain but the bar is discretized by two grids of different sizes. In both cases, the size of the grid of each zone does not change and the two grids share one node with each other. The incident wave travels from the finer grid toward the coarser grid. For the case when peridynamics is used on the entire domain, the size of the peridynamics horizon changes based on the size of the gird. For both cases, we investigate the impact of the relative size of the girds on the amplitude and energy of the transmitted and reflected waves. Our analytical and numerical results show that more spurious reflections occur when the size mismatch between the two grids is larger. In both cases, the issue of spurious wave reflection becomes more severe when the peridynamic horizon size increases. For the case of coupled peridynamic–finite element, even when the size of the two grids are the same, spurious wave reflection occurs which is due to the change in the formulation from a nonlocal to a local continuum. The spurious reflection reduces when the wavelength of the incident wave is large compared with the coarse grid.

Passing waves from atomistic to continuum

Progress in the development of coupled atomistic–continuum methods for simulations of critical dynamic material behavior has been hampered by a spurious wave reflection problem at the atomistic–continuum interface. This problem is mainly caused by the difference in material descriptions between the atomistic and continuum models, which results in a mismatch in phonon dispersion relations. In this work, we introduce a new method based on atomistic dynamics of lattice coupled with a concurrent atomistic–continuum method to enable a full phonon representation in the continuum description. This permits the passage of short-wavelength, high-frequency phonon waves from the atomistic to continuum regions. The benchmark examples presented in this work demonstrate that the new scheme enables the passage of all allowable phonons through the atomistic–continuum interface; it also preserves the wave coherency and energy conservation after phonons transport across multiple atomistic–continuum interfaces. This work is the first step towards developing a concurrent atomistic–continuum simulation tool for non-equilibrium phonon-mediated thermal transport in materials with microstructural complexity.

Eliminating corner effects in square lattice simulation

Using an alternative source decomposition, we propose new exact boundary conditions on numerical boundary of a square lattice for out-of-plane motion over the whole space. A set of recurrence relations are found for the resulting kernel functions, hence allow their efficient and accurate evaluation with a system of ordinary differential equations. Stability of the boundary conditions is proved rigorously. Numerical results illustrate effective suppression for spurious wave reflection, and elimination of corner effects. This approach may be extended to other lattice structures and in higher dimensions.

Nonreflecting outlet boundary conditions for incompressible flows using SPH

In this paper we implement a simple strategy, based on Jin and Braza’s method, to deal with nonreflecting outlet boundary conditions for incompressible Navier-Stokes flows using the method of smoothed particle hydrodynamics (SPH). The outflow boundary conditions are implemented using an outflow zone downstream of the outlet, where particles are moved using an outgoing wave equation for the velocity field so that feedback noise from the outlet boundary is greatly reduced. For unidirectional flow across the outlet, this condition reduces to Orlanski’s wave equation. The performance of the method is demonstrated through several two-dimensional test problems, including unsteady, plane Poiseuille flow, flow between two inclined plates, the Kelvin–Helmholtz instability in a channel, and flow in a constricted conduit, and in three-dimensions for turbulent flow in a 90° section of a curved square pipe. The results show that spurious waves incident from the outlet are effectively absorbed and that steady-state laminar flows can be maintained for much longer times compared to periodic boundary conditions. In addition, time-dependent anisotropies in the flow, like fluid recirculations, are convected across the outlet in a very stable and accurate manner.