Pseudospectral Simulations Research Articles

Numerical simulations of a stochastic dynamics leading to cascades and loss of regularity: Applications to fluid turbulence and generation of fractional Gaussian fields

Motivated by the modeling of the spatial structure of the velocity field of three-dimensional turbulent flows, and the phenomenology of cascade phenomena, a linear dynamics was recently proposed that can generate high velocity gradients from a smooth-in-space forcing term. It is based on a linear partial differential equation stirred by an additive random forcing term that is δ-correlated in time. The underlying proposed deterministic mechanism corresponds to a transport in Fourier space that aims to transfer energy injected at large scales towards small scales. The key role of the random forcing is to realize these transfers in a statistically homogeneous way. Whereas at finite times and positive viscosity the solutions are smooth, a loss of regularity is observed for the statistically stationary state in the inviscid limit. We present here simulations, based on finite volume methods in the Fourier domain and a splitting method in time, which are more accurate than the pseudospectral simulations. We show that our algorithm is able to reproduce accurately the expected local and statistical structure of the predicted solutions. We conduct numerical simulations in one, two, and three spatial dimensions, and we display the solutions both in physical and Fourier space. We additionally display key statistical quantities such as second-order structure functions and power spectral densities at various viscosities. Published by the American Physical Society 2024

Parametric description of intermittent probability distribution functions in solar wind and magnetohydrodynamic turbulence

ABSTRACT In this work, we find empirical evidence that the scale-dependent statistical properties of solar wind and magnetohydrodynamic (MHD) turbulence can be described in terms of a family of parametric probability distribution functions (PDFs) known as Normal Inverse Gaussian (NIG). Understanding these PDFs is one of the most important goals in turbulence theory, as they are inherently connected to the intermittent properties of solar wind turbulence. We investigate the properties of PDFs of Elsasser increments based on a large statistical sample from solar wind observations and high-resolution numerical simulations of MHD turbulence. In order to measure the PDFs and their corresponding properties, three experiments are presented: fast and slow solar wind for experimental data and a simulation of reduced MHD (RMHD) turbulence. Conditional statistics on a 23-yr-long sample of WIND data near 1 au and high-resolution pseudo-spectral simulation of steadily driven RMHD turbulence on a $2048^3$ mesh are used to construct scale-dependent PDFs. The empirical PDFs are fitted to NIG distributions, which depend on four free parameters. Our analysis shows that NIG distributions accurately capture the evolution of the PDFs, with scale-dependent parameters, from large scales characterized by a Gaussian distribution, turning to exponential tails within the inertial range and stretched exponentials at dissipative scales. We also show that empirically-measured NIG parameters exhibit well-defined scaling properties that are similar across the three empirical data sets, which may be indicative of universal behaviour.

Differential Rotation in Convecting Spherical Shells with Non-Uniform Viscosity and Entropy Diffusivity

Contemporary three-dimensional physics-based simulations of the solar convection zone disagree with observations. They feature differential rotation substantially different from the true rotation inferred by solar helioseismology and exhibit a conveyor belt of convective “Busse” columns not found in observations. To help unravel this so-called “convection conundrum”, we use a three-dimensional pseudospectral simulation code to investigate how radially non-uniform viscosity and entropy diffusivity affect differential rotation and convective flow patterns in density-stratified rotating spherical fluid shells. We find that radial non-uniformity in fluid properties enhances polar convection, which, in turn, induces non-negligible lateral entropy gradients that lead to large deviations from differential rotation geostrophy due to thermal wind balance. We report simulations wherein this mechanism maintains differential rotation patterns very similar to the true solar profile outside the tangent cylinder, although discrepancies remain at high latitudes. This is significant because differential rotation plays a key role in sustaining solar-like cyclic dipolar dynamos.

A pseudospectral contact mechanics simulation of microindentation in two slip systems activated single crystalline material by using a homographic conformal mapping

Micromechanical behavior of two slip systems activated FCC single crystals is investigated through a pseudospectral modeling microindentation based on Dislocation Mediated Plasticity. Variational form of integral energy is evaluated by dislocations density energy, contact energy of master indenter and slave. Physical domain beneath indenter is guided to transferre into a computational domain using a homographic conformal mapping. Choosing this mapping guarantees extracting closed-form governing equations with thirty-two complex boundary conditions as remain invariant too. This is the key point that explicit integration by parts can provide requirements of imposing spectral discretization through generalized differential quadrature operators. By evaluating those operators at Guess- Lobato points, a set of algebraic equations is acquired. So, a novel framework for producing a mesh reduction scheme is handled by an efficient strong form solution. This method is fast and stable enough to calculate without emerging any aliasing. The crystal will deform with two active slip systems so that movement of contact points remain only at one direction attached to the indenter. The activity range of each slip system up to a depth of 200 microns is examined and displacement-force graphs. Good agreements are found in the results obtained from simulation compared with experiments.

An efficient particle tracking algorithm for large-scale parallel pseudo-spectral simulations of turbulence

Particle tracking in large-scale numerical simulations of turbulent flows presents one of the major bottlenecks in parallel performance and scaling efficiency. Here, we describe a particle tracking algorithm for large-scale parallel pseudo-spectral simulations of turbulence which scales well up to billions of tracer particles on modern high-performance computing architectures. We summarize the standard parallel methods used to solve the fluid equations in our hybrid MPI/OpenMP implementation. As the main focus, we describe the implementation of the particle tracking algorithm and document its computational performance. To address the extensive inter-process communication required by particle tracking, we introduce a task-based approach to overlap point-to-point communications with computations, thereby enabling improved resource utilization. We characterize the computational cost as a function of the number of particles tracked and compare it with the flow field computation, showing that the cost of particle tracking is very small for typical applications.

MagIC v5.10: a two-dimensional message-passing interface (MPI) distribution for pseudo-spectral magnetohydrodynamics simulations in spherical geometry

Abstract. We discuss two parallelization schemes for MagIC, an open-source, high-performance, pseudo-spectral code for the numerical solution of the magnetohydrodynamics equations in a rotating spherical shell. MagIC calculates the non-linear terms on a numerical grid in spherical coordinates, while the time step updates are performed on radial grid points with a spherical harmonic representation of the lateral directions. Several transforms are required to switch between the different representations. The established hybrid parallelization of MagIC uses message-passing interface (MPI) distribution in radius and relies on existing fast spherical transforms using OpenMP. Our new two-dimensional MPI decomposition implementation also distributes the latitudes or the azimuthal wavenumbers across the available MPI tasks and compute cores. We discuss several non-trivial algorithmic optimizations and the different data distribution layouts employed by our scheme. In particular, the two-dimensional distribution data layout yields a code that strongly scales well beyond the limit of the current one-dimensional distribution. We also show that the two-dimensional distribution implementation, although not yet fully optimized, can already be faster than the existing finely optimized hybrid parallelization when using many thousands of CPU cores. Our analysis indicates that the two-dimensional distribution variant can be further optimized to also surpass the performance of the one-dimensional distribution for a few thousand cores.

Large eddy simulations of high-magnetic Reynolds number magnetohydrodynamic turbulence for non-helical and helical initial conditions: A study of two sub-grid scale models

Pseudo-spectral large eddy simulation (LES) calculations of high-magnetic Reynolds number (Rem) incompressible magnetohydrodynamic (MHD) turbulence are carried out for two initial conditions, namely, the non-helical Orszag–Tang vortex and the strongly helical Arnold–Beltrami–Childress (ABC) flows using two eddy-viscosity-based sub-grid scale (SGS) approaches: the cross-helicity (CH) and dynamic Smagorinsky (DS) models. Validation is conducted through comparisons of 1923 LES calculations with in-house 5123 direct numerical simulations (DNS) at Reynolds number, Re=Rem=800. The results show that the CH model performs better than the DS model. The performance of the SGS models at higher Re is further evaluated by carrying out 3843 LES calculations at Re=Rem=7500. Various quantities including turbulent kinetic energy, turbulent magnetic energy, cross-helicity, helicity, vorticity structures, cosine of angle between velocity and magnetic field, cosine of angle between velocity and vorticity field, kinetic and magnetic energy spectra, and energy fluxes are analyzed to understand the capability of the two LES models in predicting the evolution of MHD turbulence. The higher Reynolds number flow shows a delay in the maximum dissipation with increased transfer of energy toward small scales, resulting in a −5/3 Kolmogorov inertial sub-range scaling. In addition, the effect of Reynolds number on the alignment between velocity and magnetic field, and the energy transfer between kinetic and magnetic energy, is studied. With the ABC flow having strong helicity and zero cross-helicity at low and high Reynolds numbers, a strong dynamo effect is also observed using the LES models, which is consistent with previous DNS.

Storms, Variability, and Multiple Equilibria on Hot Jupiters

Abstract Observations of hot-Jupiter atmospheres show large variations in the location of the “hot spot” and the amplitude of spectral features. Atmospheric flow simulations using the commonly employed forcing and initialization have generally produced a large, monolithic patch of nearly stationary hot area located eastward of the substellar point at high altitude. Here we perform high-resolution (T682) pseudospectral simulations that accurately capture small-scale eddies and waves inherent in hot-Jupiter atmospheres due to ageostrophy. The atmospheres contain a large number of intense storms over a wide range of scales, including the planetary scale. The latter sized storms dictate the large-scale spatial distribution and temporal variability of hot, as well as cold, regions over the planet. In addition, they exhibit quasi-periodic life cycles within multiple equilibrium states—all identifiable in the disk-integrated time series of the temperature flux.

Effect of conventionally neutral boundary layer height on turbine performance and wake mixing in offshore windfarms

In this study, we investigate the effect of conventionally neutral boundary layer height and capping inversion strength on wind farm performance. An aeroelastic multibody solver coupled with a pseudo-spectral large eddy simulation code is used to investigate the structural and aerodynamic output of wind turbines in 4 different inflow types. Farm layouts comprising of turbines in both aligned and staggered arrangements are studied. It is found that the shallower boundary layers with strong capping inversions are characterized by oscillatory structures in turbine wakes, leading to increased structural damage in downstream wind turbines due to fatigue. The power output is also observed to decrease due to poorer wake mixing.

Sub-Kolmogorov droplet dynamics in isotropic turbulence using a multiscale lattice Boltzmann scheme

The deformation and dynamics of a single droplet in isotropic turbulence is studied using a Lattice Boltzmann diffuse interface model involving exact boundary flow conditions [1] to allow for the creation of an external turbulent flow. We focus on a small, sub-Kolmogorov droplet, whose scale is much smaller than the Kolmogorov length scale of the turbulent flow. The external flow field is obtained via pseudo-spectral simulation data describing the trajectory of a passive tracer in isotropic turbulence. In this way we combine the microscopic scale of the droplet and the macroscopic scale of the turbulent flow. The results obtained from this fully resolved model are compared to previous studies on sub-Kolmogorov droplet dynamics in isotropic turbulent flows [2], where an analytical model [3], which assumes the droplet shape to be an ellipsoid at all times, is used to describe the droplet deformation. Our findings confirm, that the hybrid pseudo-spectral Lattice Boltzmann algorithm is able to study droplet deformation and breakup in a regime well beyond the ellipsoidal approximation.

A Numerical Study on the Development of Self-Similarity in a Wind Turbine Wake Using an Improved Pseudo-Spectral Large-Eddy Simulation Solver

Large-eddy simulation (LES) is performed to investigate self-similarity in a wind turbine wake flow. The turbine is represented using an actuator line model in a pseudo-spectral method-based solver. A new hybrid approach of smoothed pseudo-spectral method and finite-difference method (sPSMFDM) is proposed to alleviate the Gibbs phenomenon caused by the jump of velocity and pressure around the turbine. The LES is validated with the mean velocity and turbulence statistics obtained from wind-tunnel measurement reported in the literature. Through an appropriate choice of characteristic scales of velocity and length, self-similarity is elucidated in the normalized mean velocity and Reynolds stress profiles at various distances. The development of self-similarity is categorized into three stages based on the variation in the characteristic scales and the spanwise distribution of normalized velocity deficit. The mechanisms responsible for the transition of self-similarity stages are analyzed in detail. The findings of the flow physics obtained in this study will be useful for the modeling and fast prediction of wind turbine wake flows.

Internal electromagnetic waves, energy trapping, and energy release in simple time-domain simulations of single particle scattering.

During the decay phase of the interaction of a femto-second Gaussian pulse with a single spherical particle, the presence of quasi-periodic short oscillatory bursts of electromagnetic energy at points in the near field outside the particle have been observed in three-dimensional simulations. Analogous behavior can be very easy produced and understood in simple one-dimensional scattering calculations. In two dimensions the situation immediately becomes more complicated and interesting. Here we discuss results from two-dimensional pseudo-spectral time-domain simulations of scattering from circular, elliptical, and hexagonal particles with the real index of refraction m = 1.3. Our focus is on how energy initially trapped within a particle after interaction with an incident Gaussian pulse is released over time, and we show two kinds of events that can result in “bursts” of energy release from the particles: (i) the coalescence of counter-propagating wave-packet-like electromagnetic field structures that have maximum amplitude near the surface of the particle, and (ii) encounters of individual packets with surface regions of high curvature. The coalescence events in the circular case show the dynamical origin of a two-dimensional form of “photonic nanojet.” The two-dimensional simulations make clear the reason for quasi-periodic intermittent bursts at fixed near-field points outside the particle. Examination of field evolution shows that distinct near-surface internal field maxima, ostensibly the “source” of the emission bursts, are in fact inter-connected by caustic-like internal field structures that extend throughout the particle and have complex time evolution. The revealed intricacy of these connections suggests that understanding the origins of pulsed emissions in three dimensions, even for simple particle geometries, may be quite challenging.

A study of the regularized lid-driven cavity’s progression to chaos

Computational simulations of a two-dimensional incompressible regularized lid-driven cavity flow were performed and analyzed to identify the dynamic behavior of the flow through multiple bifurcations which ultimately result in Eulerian chaotic flow. Semi-implicit, pseudo-spectral numerical simulations were performed at Reynolds numbers from 1000 to 25,000. Poincaré maps were used to identify transitions as the Reynolds number increased from stable-laminar flow to periodic flow, periodic flow to quasi-periodic flow, quasi-periodic flow to chaotic flow, and a sudden, brief return from chaotic flow to periodic flow. The first critical Reynolds number, near 10,250, is found in agreement with existing literature. An additional bifurcation is observed near a Reynolds number of 15,500. A power spectrum analysis, in which the novel concepts of frequency shredding and power capacity are introduced, was performed with the conclusion that no further bifurcations occurred at Reynolds numbers above 15,500 even though Eulerian chaos was not formally observed until Reynolds numbers above 18,000. While qualitative changes in the fluid flow’s attractor were apparent from trajectories in the phase space, mechanisms by which such changes can occur were apparent from power spectra of flow time histories. The novel power spectrum analysis may also serve as a new approach for characterizing multi-scale nonlinear dynamical systems.

Comparison of dealiasing schemes in large-eddy simulation of neutrally stratified atmospheric flows

Abstract. Aliasing errors arise in the multiplication of partial sums, such as those encountered when numerically solving the Navier–Stokes equations, and can be detrimental to the accuracy of a numerical solution. In this work, a performance and cost analysis is proposed for widely used dealiasing schemes in large-eddy simulation, focusing on a neutrally stratified, pressure-driven atmospheric boundary-layer flow. Specifically, the exact 3∕2 rule, the Fourier truncation method, and a high-order Fourier smoothing method are intercompared. Tests are performed within a newly developed mixed pseudo-spectral finite differences large-eddy simulation code, parallelized using a two-dimensional pencil decomposition. A series of simulations are performed at varying resolution, and key flow statistics are intercompared among the considered runs and dealiasing schemes. The three dealiasing methods compare well in terms of first- and second-order statistics for the considered cases, with modest local departures that decrease as the grid stencil is reduced. Computed velocity spectra using the 3∕2 rule and the FS method are in good agreement, whereas the FT method yields a spurious energy redistribution across wavenumbers that compromises both the energy-containing and inertial sublayer trends. The main advantage of the FS and FT methods when compared to the 3∕2 rule is a notable reduction in computational cost, with larger savings as the resolution is increased (15 % for a resolution of 1283, up to a theoretical 30 % for a resolution of 20483).

A numerical tool for the study of the hydrodynamic recovery of the Lattice Boltzmann Method

We investigate the hydrodynamic recovery of Lattice Boltzmann Method (LBM) by analyzing exact balance relations for energy and enstrophy derived from averaging the equations of motion on sub-volumes of different sizes. In the context of 2D isotropic homogeneous turbulence, we first validate this approach on decaying turbulence by comparing the hydrodynamic recovery of an ensemble of LBM simulations against the one of an ensemble of Pseudo-Spectral (PS) simulations. We then conduct a benchmark of LBM simulations of forced turbulence with increasing Reynolds number by varying the input relaxation times of LBM. This approach can be extended to the study of implicit subgrid-scale (SGS) models, thus offering a promising route to quantify the implicit SGS models implied by existing stabilization techniques within the LBM framework.

Numerical simulation of the geometrical-optics reduction of CE2 and comparisons to quasilinear dynamics

Zonal flows have been observed to appear spontaneously from turbulence in a number of physical settings. A complete theory for their behavior is still lacking. Recently, a number of studies have investigated the dynamics of zonal flows using quasilinear (QL) theories and the statistical framework of a second-order cumulant expansion (CE2). A geometrical-optics (GO) reduction of CE2, derived under an assumption of separation of scales between the fluctuations and the zonal flow, is studied here numerically. The reduced model, CE2-GO, has a similar phase-space mathematical structure to the traditional wave-kinetic equation, but that wave-kinetic equation has been shown to fail to preserve enstrophy conservation and to exhibit an ultraviolet catastrophe. CE2-GO, in contrast, preserves nonlinear conservation of both energy and enstrophy. We show here how to retain these conservation properties in a pseudospectral simulation of CE2-GO. We then present nonlinear simulations of CE2-GO and compare with direct simulations of quasilinear (QL) dynamics. We find that CE2-GO retains some similarities to QL. The partitioning of energy that resides in the zonal flow is in good quantitative agreement between CE2-GO and QL. On the other hand, the length scale of the zonal flow does not follow the same qualitative trend in the two models. Overall, these simulations indicate that CE2-GO provides a simpler and more tractable statistical paradigm than CE2, but CE2-GO is missing important physics.

An integrated pseudo-spectral simulation of high-speed discharging at an electromagnetic forming conveying a conductive driver sheet

An integrated computational methodology is implemented based on a pseudo-spectral solution of governing differential equations in a high-speed discharging of the flat spiral coils, which is used in the electromagnetic-forming systems. Both conductive driver sheet and coil are divided into several peripheral segments. Also, eddy currents induced by each segment are computed as well as the fields generated by each wire of the coil corresponding to defined local source points. The model will be established based on the Chebyshev polynomials to calculate spatial and temporal distributions of the magnetic fields. The Houbolt marching method is applied for the time discretization of the Maxwell equations. By adopting a backward process for implicit boundary conditions and summing up overall components of the magnetic fields at the lower surface, it leads to obtaining an integrated response to the problem. A multiple regression analysis is implemented based on the least square error norm to remove additional equations generated in the spectral discretization. An exponential filtering is used to control aliasing phenomenon due to collocation treatment of the mathematical terms. Validation is carried out by comparisons of the results with other works related in the literature.

The Effect of Aliased and De-aliased Formulation for DNS Analysis in Plane Poiseuille Flow

In this research, the performance of various aliased and de-aliased schemes was studied in pseudo-spectral (PS) simulation. The PS method was used to perform aliased and de-aliased direct numerical simulation of a turbulent plane Poiseuille channel flow. The effects of aliasing errors in Navier–Stokes equations were examined for both common skew-symmetric and rotational forms. Due to boundary conditions, Fourier series in periodic directions and Chebyshev expansion in normal direction were considered. A third-order backward difference scheme was used for temporal approximation method, and different types of “2/3 rule” de-aliasing procedures in various directions were examined. The results for aliased and de-aliased formulation showed that generally, the skew-symmetric form gives the most accurate results. But de-aliased rotational form was more robust and presented fewer errors. The amount of aliasing errors for the ordinary rotation form is about two times larger than for De-aliased XYZ rotational scheme and common skew-symmetric form. From a practical point of view, the choice between the skew-symmetric form and de-aliased rotational method with the same accuracy reduces to the last one if final decision rests on the amount of computational time.

Multiscale aeroelastic simulations of large wind farms in the atmospheric boundary layer

In large wind farms, the turbulence induced by each turbine results in high overall turbulence levels that can be detrimental for downstream wind turbine components. In the current study, we scrutinize structural loads and dynamics, and their correlation to turbulent flow structures by conducting aeroelastic simulations in wind farms. To this end, a pseudospectral large-eddy simulation solver is coupled with a multibody dynamics module in a multiscale framework. The multirate approach leads us naturally to the development of an aeroelastic actuator sector model that represents the wind turbine forces on the flow. This makes it computationally feasible to simulate long time horizons of the two-way coupled aeroelastic system. Hence, it allows us to look at the interaction of the turbine structure with the turbulent boundary layer and the wakes of multiple turbine arrays, and to get estimates of damage equivalent loads and structural loading statistics, as longer time series are available. Results are shown for two typical wind farm layouts, i.e. aligned and staggered, for above-rated flow regimes.

A convergent Born series for solving the inhomogeneous Helmholtz equation in arbitrarily large media

We present a fast method for numerically solving the inhomogeneous Helmholtz equation. Our iterative method is based on the Born series, which we modified to achieve convergence for scattering media of arbitrary size and scattering strength. Compared to pseudospectral time-domain simulations, our modified Born approach is two orders of magnitude faster and nine orders of magnitude more accurate in benchmark tests in 1, 2, and 3-dimensional systems.