High-order Spectral Method Research Articles

Establishment and validation of a viscous-potential coupled and graphics processing unit accelerated numerical tank based on smoothed particle hydrodynamics and high-order spectral methods

The smoothed particle hydrodynamics (SPH) method is an entirely Lagrangian and truly meshless numerical tool that is inherently suitable for simulating violent viscous free-surface flows, whilst it usually renders considerable simulation time when solving large-scale engineering problems. The high-order spectral (HOS) method is an accurate and very efficient numerical tool for modeling a large-scale inviscid gravity wave field, whilst it is hard to tackle complex physical phenomena due to the dependence on the potential flow theory. This paper aims to propose a viscous-potential coupled and graphics processing unit accelerated numerical model for establishing a high-fidelity and high-efficiency numerical wave tank. In this coupling scheme, the SPH method is applied in the near-field where violent physical processes exist to accurately capture complex free-surface behaviors such as overturning and breaking, whilst the HOS method is applied in the far-field where the water undergoes gentle motion to efficiently generate a desired gravity wave. A dedicated SPH–HOS coupling interface is also presented to accurately transmit the flow information from the HOS side to the SPH side. Meanwhile, an SPH–HOS buffer region with several key techniques is also proposed to ensure a successful coupling between the two numerical methods. Different benchmarks demonstrate that the present SPH–HOS coupling method absorbs the respective advantages of the two numerical methods and shows great feasibility and reliability in solving hydrodynamic problems involving gravity wave generation, propagation, deformation, and its interaction with marine structures.

The nonlinearity of scattering waves due to interaction between focusing waves and floating production storage and offloading

The study of the full-scale wave–structure interaction is essential to our understanding of the nonlinear characteristics of offshore structures in real-sea states. This paper deals with full-scale numerical studies of the interactions between focusing waves and a fixed floating production storage and offloading (FPSO) in an efficient potential-viscous coupled method. The potential-viscous method combines in-house computational fluid dynamics code developed at Shanghai Jiao Tong University (SJTU) for naval architecture and ocean engineering (naoe) based on an open field operation and manipulation (FOAM), i.e., naoe-FOAM-SJTU solver, with high-order spectral method. The approach is verified on a model-scaled case and shows reasonably good agreement with experimental data. A phase-separation method and a dynamic mode decomposition method are utilized to extract linear and higher-order harmonic components from the scattering waves. The scattering waves around the FPSO are found to affect the higher-order harmonic components. Two kinds of scale ratios are considered to magnify the focusing wave and FPSO by 10 and 100 times to discuss the scale effect, both non-breaking and green water conditions are included to investigate the flow phenomenon. The harmonic components of scattering waves are not proportional to the scale ratio, and the scale effects influence more on higher-order harmonic components. The third- and fourth-order harmonic components of scattering waves around the FPSO in large-scale cases are obvious.

Hydrodynamic analysis and validation of the floating DeepCwind semi-submersible under 3-h irregular wave with the HOS and CFD coupling method

Recent studies, such as the Offshore Code Comparison, Collaboration, Continued, with Correlation (OC5) led by NREL, have identified significant under-prediction of low-frequency pitch and surge responses in the DeepCwind semi-submersible platform when comparing model test results with predictions from engineering tools like OpenFAST, which rely on potential flow theory assumptions for hydrodynamics. To address this mismatch, a newer experimental campaign was conducted as part of the OC6 Phase 1A study. In this work, a coupling numerical approach was employed to replicate various scenarios of the floating DeepCwind semi-submersible, including decay, regular, and 3-h irregular waves, which were explored during the OC6 Phase 1A model testing. The numerical technique involved utilizing far-field nonlinear waves simulated using the High-Order Spectral (HOS) method and applying them to a near-field Computational Fluid Dynamics (CFD) domain. This allowed for the solution of wave-structure interactions using the overset mesh capabilities of StarCCM+. The critical hydrodynamic responses of the freely floating DeepCwind semi-submersible were modeled and validated against the model test results. The results obtained from the coupling approach were then presented and compared with simulations using OpenFAST and results from other researchers. The coupling strategy demonstrated improved agreement, especially for low-frequency responses in surge and pitch motion, under the 3-h irregular wave conditions, compared to OpenFAST simulations. While the validation investigation showed overall success in capturing low-frequency and wave-frequency surge and heave motion, there was an underestimation of low-frequency pitch motion in the TCOMS coupling tool. This underprediction can be attributed, at least in part, to the effect of wave overtopping on the rear columns, which resulted in deviations from the experimental pitch motion, particularly during extreme events.

High-Order Spectral Method of Density Estimation for Stochastic Differential Equation Driven by Multivariate Gaussian Random Variables

There are some previous works on designing efficient and high-order numerical methods of density estimation for stochastic partial differential equation (SPDE) driven by multivariate Gaussian random variables. They mostly focus on proposing numerical methods of density estimation for SPDE with independent random variables and rarely research density estimation for SPDE is driven by multivariate Gaussian random variables. In this paper, we propose a high-order algorithm of gPC-based density estimation where SPDE driven by multivariate Gaussian random variables. Our main techniques are (1) we build a new multivariate orthogonal basis by adopting the Gauss–Schmidt orthogonalization; (2) with the newly constructed orthogonal basis in hand, we first assume the unknown function in the SPDE has the stochastic general polynomial chaos (gPC) expansion, second implement the stochastic gPC expansion for the SPDE in the multivariate Gaussian measure space, and third we obtain and numerical calculation deterministic differential equations for the coefficients of the expansion; (3) we used high-order algorithm of gPC-based for density estimation and moment estimation. We apply the newly proposed numerical method to a known random function, stochastic 1D wave equation, and stochastic 2D Schnakenberg model, respectively. All the presented stochastic equations are driven by bivariate Gaussian random variables. The efficiency is compared with the Monte-Carlo method based on the known random function.

Nonlinear statistical characteristics of the multi-directional waves with equivalent energy

Directional distribution is believed to have a significant impact on the statistical characteristics in multi-directional sea states. In real sea states, short-crested waves are discrete not only in frequency but also in direction. For the former one, they are well explained in unidirectional mode, but for the latter one, they are not. In this paper, the kurtosis of short-crested waves with equivalent energy is first discussed. Unimodal-spectrum-multi-direction sea states and bimodal-spectrum-multi-direction sea states are simulated for a long time in a numerical wave basin based on the high-order spectral method. In the equivalent sea-swell sea state, the spatial evolution of kurtosis becomes more inhomogeneous, along with the maximum value of kurtosis being larger and the area where the maximum value occurs wider in the configuration with a crossing angle β = 40° than that with β = 0°, while little variations in swell-dominated and wind-sea-dominated states. A positive linear correlation between wavelet group steepness and kurtosis is obtained in a unimodal sea state, but not applied to a crossing sea state characterized by a bimodal spectrum. The exceedance probability of wave height and wave crest distribution at maximum kurtosis is also given. These findings can help predict the probability of extreme waves occurring, guiding the selection of ocean engineering sites to avoid extreme configurations.

Arbitrarily high-order accurate simulations of compressible rotationally constrained convection using a transfinite mapping on cubed-sphere grids

We present two major improvements over the Compressible High-ORder Unstructured Spectral difference (CHORUS) code published in Wang et al., “A compressible high-order unstructured spectral difference code for stratified convection in rotating spherical shells,” J. Comput. Phys. 290, 90–111 (2015). The new code is named CHORUS++ in this paper. Subsequently, we perform a series of efficient simulations for rotationally constrained convection (RCC) in spherical shells. The first improvement lies in the integration of the high-order spectral difference method with a boundary-conforming transfinite mapping on cubed-sphere grids, thus ensuring exact geometric representations of spherical surfaces on arbitrary sparse grids. The second improvement is on the adoption of higher-order elements (sixth-order) in CHORUS++ vs third-order elements for the original CHORUS code. CHORUS++ enables high-fidelity RCC simulations using sixth-order elements on very coarse grids. To test the accuracy and efficiency of using elements of different orders, CHORUS++ is applied to a laminar solar benchmark, which is characterized by columnar banana-shaped convective cells. By fixing the total number of solution degrees of freedom, the computational cost per time step remains unchanged. Nevertheless, using higher-order elements in CHORUS++ resolves components of the radial energy flux much better than using third-order elements. To obtain converged predictions, using sixth-order elements is 8.7 times faster than using third-order elements. This significant speedup allows global-scale fully compressible RCC simulations to reach equilibration of the energy fluxes on a small cluster of just 40 cores. In contrast, CHORUS simulations were performed by Wang et al. on supercomputers using approximately 10 000 cores. Using sixth-order elements in CHORUS++, we further carry out global-scale solar convection simulations with decreased rotational velocities. Interconnected networks of downflow lanes emerge and surround broader and weaker regions of upflow fields. A strong inward kinetic energy flux compensated by an enhanced outward enthalpy flux appears. These observations are all consistent with those published in the literature. Furthermore, CHORUS++ can be extended to magnetohydrodynamic simulations with potential applications to the hydromagnetic dynamo processes in the interiors of stars and planets.

Machine learning simulation of one-dimensional deterministic water wave propagation

Deterministic phase-resolved prediction of the evolution of surface gravity waves in water is challenging due to their complex spatio-temporal dynamics. Physics-based methods of varying complexity are available, but the conflicting objectives of numerical efficiency and accuracy impede real-time wave prediction. Data-driven methods may be able to overcome this challenge by using training data generated by complex numerical methods. This work explores the potential of a machine learning (ML) approach based on a fully convolutional encoder–decoder architecture for the efficient and accurate prediction of water waves. The high-order spectral (HOS) method forms the foundation for the generation of the training data. The HOS method is applied for different, consecutive orders of nonlinearity starting from first order up to fourth order. The JONSWAP wave energy spectrum serves as the basis for modeling the one-dimensional irregular sea states. The overall objective of this work is to evaluate whether the complex non-linear physical processes can be identified and learned by the ML approach. The trained ML flow mapper is used to perform time integration of an initial sea state. The results indicate that the proposed ML approach is able to reproduce the distinctive physical processes of the different orders of nonlinearities. It is shown that the ML approach enables fast and accurate predictions of one-dimensional waves over a time horizon that spans multiple peak periods.

Multi-Level Solution Strategies for Implicit Time Discontinuous Galerkin Discretizations

The discontinuous Galerkin method and related flavors of high-order spectral element methods provide many well-known benefits for the spatial discretization of partial differential equations such as the Navier–Stokes equations. However, practical problems of engineering relevance such as large-eddy simulation of turbulent flows over complex geometries are computationally intractable by standard explicit time integration methods, necessitating the use of implicit methods. The efficient solution of the nonlinear algebraic systems arising from implicit time integration methods applied to DG discretizations of nonlinear PDEs is challenging; standard linearization methods result in very stiff block-sparse systems with prohibitive computational and memory requirements. This paper presents a low-memory, computationally efficient, implicit solution method that combines a framework of nonlinear polynomial multigrid, adaptive explicit Runge–Kutta smoothers, implicit Jacobian-free coarse level smoothers, nonlinear Krylov subspace acceleration and adaptive time stepping using feedback control of the nonlinear solver convergence rate.

Long-term analysis of wave-induced loads using High Order Spectral Method and direct sampling of extreme wave events

An approach for running long-term analysis of the wave environment and associated wave-induced structural loads, which includes effects of nonlinearity, irregularity and short crestedness of ocean waves is presented. The suggested approach combines a novel event-sampling technique with the efficient nonlinear wave solver High Order Spectral Method (HOSM) to represent the long-term wave environment by a large, yet computationally feasible, number of short-crested wave events which are simulated in HOSM. For the present study, HOSM is enhanced with a wave-breaking model, which improves the description of steep sea states in which wave breaking plays a role.Extensive validation of HOSM and the implemented wave-breaking model is carried out through comparison with model-test results for wave-crest statistics in long- and short-crested sea states and through direct comparison with CFD simulations.The proposed methodology is demonstrated on a case study considering a 106 years long-term analysis of wave crests and wave-in-deck loads, using a simplified load model to estimate drag loads on a deck structure.

Simulation of 4th-order non-Gaussian random processes by higher-order spectral representation method

In this study, a new higher-order spectral representation method (HOSRM) is derived to simulate fourth-order (kurtosis-order) non-Gaussian random processes. Applying the HOSRM allows fourth-order non-Gaussian random process (symmetric non-Gaussian) sample functions to be generated accurately. As the number of terms in the cosine function used in the HOSRM increases, the simulated non-Gaussian random process sample functions can more accurately reflect the second-order (variance) and fourth-order (kurtosis) statistical properties of the objective sample functions. In particular, if the number of terms tends to infinity, then the simulated sample function characteristics will be the same as the target value. Meanwhile, a reasonable three-dimensional Hermite interpolation method is proposed to improve the simulation efficiency of stochastic processes. Finally, two examples are provided to demonstrate the accuracy and effectiveness of the proposed HOSRM for generating fourth-order non-Gaussian random processes. The results show that the statistical properties (power spectral density, trispectrum, and kurtosis) of the simulated non-Gaussian random process are consistent with those of the target random process.

On the use of a high-order spectral method and the geometric progression for the analysis of stationary bifurcation of nonlinear problems

On the use of a high-order spectral method and the geometric progression for the analysis of stationary bifurcation of nonlinear problems

On coupled envelope evolution equations in the Hamiltonian theory of nonlinear surface gravity waves

This paper presents a novel theoretical framework in the Hamiltonian theory of nonlinear surface gravity waves. The envelope of surface elevation and the velocity potential on the free water surface are introduced in the framework, which are shown to be a new pair of canonical variables. Using the two envelopes as the main unknowns, coupled envelope evolution equations (CEEEs) are derived based on a perturbation expansion. Similar to the high-order spectral method, the CEEEs can be derived up to arbitrary order in wave steepness. In contrast, they have a temporal scale as slow as the rate of change of a wave spectrum and allow for the wave fields prescribed on a computational (spatial) domain with a much larger size and with spacing longer than the characteristic wavelength at no expense of accuracy and numerical efficiency. The energy balance equation is derived based on the CEEEs. The nonlinear terms in the CEEEs are in a form of the separation of wave harmonics, due to which an individual term is shown to have clear physical meanings in terms of whether or not it is able to force free waves that obey the dispersion relation. Both the nonlinear terms that can only lead to the forcing of bound waves and those that are capable of forcing free waves are demonstrated, in the case of the latter through the analysis of the quartet and quintet resonant interactions of linear waves. The relations between the CEEEs and two other existing theoretical frameworks are established, including the theory for a train of Stokes waves up to second order in wave steepness (Fenton, ASCE J. Waterway Port Coastal Ocean Engng, vol. 111, issue 2, 1985, pp. 216–234) and a semi-analytical framework for three-dimensional weakly nonlinear surface waves with arbitrary bandwidth and large directional spreading by Li & Li (Phys. Fluids, vol. 33, issue 7, 2021, 076609).

A Study on the High-Order Spectral Model Capability to Simulate a Fully Developed Nonlinear Sea States

Modeling a nonlinear ocean wave is one of the primary concerns in ocean engineering and naval architecture to perform an accurate numerical study of wave-structure interactions. The high-order spectral (HOS) method, which can simulate nonlinear waves accurately and efficiently, was investigated to see its capability for nonlinear wave generation. An open-source (distributed under the terms of GPLv3) project named "HOS-ocean" was used in the present study. A parametric study on the "HOS-ocean" was performed with three-hour simulations of long-crested ocean waves. The considered sea conditions ranged from sea state 3 to sea state 7. One hundred simulations with fixed computational parameters but different random seeds were conducted to obtain representative results. The influences of HOS computational parameters were investigated using spectral analysis and the distribution of wave crests. The probability distributions of the wave crest were compared with the Rayleigh (first-order), Forristall (second-order), and Huang (empirical formula) distributions. The results verified that the HOS method could simulate the nonlinearity of ocean waves. A set of HOS computational parameters was suggested for the long-crested irregular wave simulation in sea states 3 to 7.

Large Eddy Simulation of Gasoline Sprays in a Lagrangian–Eulerian Framework Using the High-Order Spectral Element Method

Abstract Predicting the spray evolution using simulations requires accurate modeling of the turbulent gas-phase flow field. In this study, the high-order spectral-element method (SEM), implemented in the code Nek5000, was used to provide highly resolved solutions to the turbulent flow field. Spray modeling capabilities were implemented into the Nek5000 code. The spray is modeled in a Lagrangian–Eulerian (LE) framework, where the liquid is represented by discrete parcels of droplets. The method for coupling liquid and gas in the context of SEM is described, which allows for very fine meshes to be used without affecting the stability of the solution. Large-eddy simulations (LES) of the eight-hole ECN Spray G gasoline injector were conducted. Numerical results are compared against experimental data for liquid penetration, droplet size and gas velocity. The morphology of the multiplume spray is compared against experimental data. The effect of different spray injection inputs is analyzed. It was found that using a plume direction of 33 deg and an injection cone angle of 30 deg produced the best results overall. This work shows the applicability of SEM for spray modeling applications, where use of a high-order flow solver can help us understand the multiplume spray aerodynamics and how it leads to plume collapse under certain conditions. Results also highlight the need for tuning spray input parameters in the LE framework, even when high-fidelity gas flow solutions are possible.

High Order Spectral Analysis of Ferroresonance Phenomena in Electric Power Systems

This study presents a comprehensive investigation of ferroresonance, a dangerous electrical phenomenon that poses significant financial risks. Using real electrical transmission line parameters, we simulated synthetic ferroresonance scenarios on a model and analyzed the resulting data using high-order spectral analysis methods, including the Wigner-Ville method, Welch method, frequency-power analysis, and spectral methods. Our analysis revealed changes in frequency and power before and after ferroresonance, with third and fourth-order cumulants being calculated. We confirmed the accuracy of the power transmission line’s base frequency and power before the ferroresonance event and determined the frequency and power values before and after ferroresonance with frequency-power analysis. Our cumulative analysis results showed symmetrical results that are consistent with the properties of ferroresonance. Additionally, we found that the Wigner Ville method’s high-resolution results were significantly more effective than conventional methods. Our study’s findings provide valuable insights into ferroresonance’s behavior and may inform the development of more effective prevention and mitigation strategies.

High-order optimal mode decomposition analysis of the ground effect on flow past two tandem inclined plates

Two-dimensional flow past two tandem near-ground plates with inclination angles of 25° at the Reynolds number of 150 is numerically simulated via the high-order spectral element method. Plate-to-ground gap is varied from G = 0.2L to 1.6L with intervals of 0.2L at two representative inter-plate spacings (i.e., X = 2.5L and 6L). The ground effect on the fluid force, power spectral density, asymmetric gap flow, and wake structure of plates is systematically evaluated. Then, the high-order optimal mode decomposition (HOOMD) method is proposed to synchronously analyze the velocity and pressure fields. The results show that the fluid force and flow structure are closely dependent on G. The presence of the ground inhibits vortex shedding when G < 0.6L; as the gap increases from 0.6 L to 1.4 L, the fluctuating forces are continuously enhanced until the ground effect basically disappears at G > 1.4L. The ground effect exacerbates the asymmetry of the vortex structure near the upper and lower parts of the inclined plates, consequently changing the fluid force. The downstream plate is more sensitive to the ground effect because of impingement from the upward-biased jet flow generated in the narrow gap between the upstream plate and ground. The HOOMD method well captures the spatial morphology and temporal evolution features of different dominant modes at the transition or vortex shedding flow regime. Mode analysis affords a correspondence between the coherent vortex structure and fluid force of plates. Furthermore, the ground effect can simultaneously change the global mode energy and local pressure mode shape, subsequently influencing the fluid force. However, the global mode energy plays the determinant role in the variation of the fluid force of plates with the plate-to-ground distance herein.

Numerical investigation on nonlinear evolution behavior and water particle velocity of wave crests for narrow-band wave field with Gaussian spectrum

The nonlinear evolution of initial narrow-band Gaussian spectrum wave field, as well as the water particle velocity of wave crests for Freak waves are investigated by phase-resolved numerical model of high-order spectral (HOS) method. The long-time evolution behavior (up to 2000 wave periods) are considered under the effects of initialized parameters, including initial spectral width, wave steepness and BFI. The effects of initial BFI on occurrence probability of Freak waves indicate various distinctive stages. Moreover, the evolution characteristics of spectral width, local wave steepness, BFI and kurtosis are analyzed. And near-periodic oscillating pattern of spectral width after its long-time evolution is obtained and effected by initial spectral width and wave steepness with different features. Besides, the positive (or negative) linear correlations between wave height extreme value and local wave steepness & kurtosis (or BFI evolution value) are effected by initial spectral width primarily. At last, the behavior of water particle velocity (including the extreme value) of Freak wave crest is discussed. And the nonlinear correlation between wave crest height and water particle velocity of wave crest is obtained quantitatively.

An efficient methodology for the simulation of nonlinear irregular waves in computational fluid dynamics solvers based on the high order spectral method with an application with OpenFOAM

An efficient methodology for simulating nonlinear irregular waves in a Computational Fluid Dynamics (CFD) solver is proposed. The High Order Spectral (HOS) method is used to generate nonlinear irregular waves in an open ocean and a numerical wave tank. The inverse Fast Fourier Transforms (FFTs) and multi-dimensional interpolation from the HOS simulation results are used for the efficient reconstruction of nonlinear waves in the CFD solver. The proposed procedure is published as an open-source project called Grid2Grid, which is developed to interface with a generic CFD solver. It provides the function Application Programming Interface (API), which can communicate with different programming languages. Extreme wave events were used to validate the proposed procedure. The predicted wave breaking events are reported both in OpenFOAM CFD and HOS simulations, and the wave elevations of CFD during simulations show good agreement with experiments and with HOS simulation.

Space-time statistics of extreme ocean waves in crossing sea states

The study of extreme ocean waves has gained considerable interest in recent years, due to their importance for offshore design and navigation safety, and several theoretical approaches have been developed for their statistical description. However, in the case of crossing seas, where two or more wave systems of different characteristics are present, a full understanding of the main physical mechanisms responsible for the occurrence of very high individual waves is still lacking. As a consequence, the prediction of extremes in such conditions currently relies on integrated parameters of the total sea state, such as the spectral wave steepness. In this study, to gain further insight into the role of the crossing wind sea and swell wave systems in producing extreme individual waves, we investigate realistic sea states during typhoon Kong-rey (2018) using an ensemble of numerical simulations obtained from a phase-resolving wave model based on the High-Order Spectral (HOS) method. The reliability of the numerical fields is assessed for the first time with stereo wave measurements of the sea surface elevation field collected from an offshore platform in the area of interest. We show that, in specific conditions, space-time extreme crest heights in crossing seas can be larger than in unimodal seas due to second-order bound wave interactions between the wind sea and the swell. To improve existing prediction capabilities, we propose a novel formulation for the wave steepness in crossing seas, which includes nonlinear effects up to the second order and accounts for the spectral parameters of the interacting wave systems.

Numerical study of wall shear stress fluctuations and near-wall structures in a square duct at low Reynolds numbers

Direct numerical simulations (DNS) of turbulent flow through a square duct at two different low Reynolds numbers is performed using OpenFOAM to study the instantaneous wall shear stress fluctuations and near-wall structures. We first benchmarked our simulation results against other DNS studies which used higher-order spectral methods, thereby confirming the accuracy of our results. The mean profile of streamwise wall shear stress is well studied in the literature, however its higher order statistics has not been yet explored. There is also a lack of information on the spanwise component of wall shear stress and its higher-order statistics. Here, we provide an extensive study on streamwise and spanwise wall shear stresses, and their higher-order statistics. The intensities of both components of wall shear stress fluctuations are close to zero near the corner, and monotonically increase while moving towards the duct center. This is in contrast with their mean profiles which do not exhibit similar monotonic trends. However, the third-order moment, i.e. skewness of the streamwise wall shear stress again shows a non-monotonic behavior from the corner to the center. The profiles of the mean as well as the higher-order statistics of wall shear stress follow a better scaling between the two Reynolds number studied, when the spanwise location is normalized using inner scaling. Streamwise velocity correlations near the wall show that the flow structures in a square duct are similar to that of a channel flow near the center, whereas there is a cross-talk of the flow structures between the neighboring walls near the corner. An investigation into the effect of corner on the probability of back flow events is made at a Reynolds number lower than previously reported. The probability follows a non-monotonic behavior as we move from the corner towards the center, where it reaches a minimum at around 50 wall units from the corner. Similar to the wall shear stress profiles, the behavior of the near-wall flow structures as well as the spanwise variation of the probability of backflow events also show a qualitatively similar trend for the two Reynolds numbers, when the spanwise location is normalized using inner scaling.