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DOE OSTI2021

Mesoscale Convective Systems in a Superparameterized E3SM Simulation at High Resolution

Accurately representing mesoscale convective systems (MCSs) is crucial to simulating the energy and water cycles in global climate models. Using a novel MCS identification and tracking algorithm applied to observations and model simulations, we evaluate how well the Energy Exascale Earth System Model simulates MCSs over the central US. Simulations performed by E3SM at 25 km grid spacing with (SP-E3SM) and without (E3SM) superparameterization are compared and evaluated against observations using multiple metrics highlighting important MCS characteristics. Compared to E3SM, SP-E3SM better simulates MCS number and MCS precipitation amount, diurnal cycle, propagation, and the probability distribution of precipitation rate in both spring and summer. The improvement from SP is partly contributed by improvement in simulating the large-scale environments, featuring enhanced atmospheric low-level moisture and larger moisture transport to the central US relative to E3SM. However, SP-E3SM still underestimates MCS precipitation amount, particularly in summer. This underestimation is closely related to the negative bias in MCS precipitation intensity, although the drier environments simulated during summer also contributes to underestimation of MCS frequency. Without SP, the larger bias in MCS precipitation amount is closely related to the negative bias in MCS frequency, while the substantial dry bias in the large-scale environments also contributes to underestimation of MCS intensity. Our results suggest that SP improves MCS simulation by improving modeling of the large-scale environments and convection initiation, which are both major limiting factors in E3SM even at 25 km grid spacing where deep convection is represented by a cumulus parameterization.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

Two–Dimensional Idealized Hadley Circulation Simulation for Global High Resolution Model Development

A new framework for the development of global high resolution models using a vertical- meridional, two-dimensional pole-to-pole domain is proposed. Compared with the three-dimensional global model, the two-dimensional framework can simulate cloud-scale convection at significantly reduced computational cost, data volume, and turn-around time, and accelerate the development phase. Although idealized, it allows for simulation of various clouds and scale interactions under a wide range of environmental conditions, even though zonal waves such as mid latitude storms cannot be simulated. Physics schemes can be tested and assessed over a full array of simultaneously occurring cloud regimes. The framework can serve as an intermediate step between a one-dimensional simulation and a full three-dimensional global model. Using this framework, we analyze the resolution dependencies of simulated clouds and the Hadley circulation in a range of global high resolution models by performing multiple 1000-day simulations; for some resolutions tested, for example, 2 km horizontal resolution, three-dimensional simulation is not feasible at the present time. Analysis shows that both horizontal and vertical resolution determine the properties of the Hadley circulation and clouds, and that the differences among the simulated Hadley circulations are understood as differences in scale interactions influenced by resolution.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2024

Accelerating Multiphase Simulations With Denoising Diffusion Model Driven Initializations

This study introduces a hybrid fluid simulation approach that integrates generative diffusion models with physics‐based simulations, aiming at reducing the computational costs of flow simulations while still honoring all the physical properties of interest. Pore‐scale simulations enhance our understanding of applications such as assessing hydrogen and storage efficiency in underground reservoirs. Nevertheless, they are computationally expensive and the presence of non‐unique solutions can require multiple simulations within a single geometry. To overcome the computational cost hurdle, we propose a method that couples generative diffusion models and physics‐based simulations. While training the data‐driven model, we simultaneously generate initial conditions and perform physics‐based simulations using these. This integrated approach enables us to receive real‐time feedback on a single compute node equipped with both CPUs and GPUs. By efficiently managing these processes within a single compute node, we can continuously monitor performance and halt training once the model meets the specified criteria. To test our model, we generate realizations in a real Berea sandstone fracture which shows that our technique is up to 4.4 times faster than commonly used flow simulation initializations.

36 MATERIALS SCIENCE↗

DOE OSTI2025

Uncertainties in Simulating Flooding During Hurricane Harvey Using 2D Shallow Water Equations

Abstract Flooding is one of the most impactful weather‐related natural hazards. Numerical models that solve the two dimensional (2D) shallow water equations (SWE) represent the first‐principles approach to simulate all types of spatial flooding, such as pluvial, fluvial, and coastal flooding, and their compound dynamics. High spatial resolution (e.g., () m) is needed in 2D SWE simulations to capture flood dynamics accurately, resulting in formidable computational challenges. Thus, relatively coarser spatial resolutions are used for large‐scale simulations of flooding, which introduce uncertainties in the results. It is unclear how the uncertainty associated with the model resolution compares to the uncertainties in precipitation data sets and assumptions regarding boundary conditions when channelized flows interact with other water bodies. In this study, we compare these three sources of uncertainties in 2D SWE simulations for the 2017 Houston flooding event. Our results show that precipitation uncertainty and mesh resolution have more significant impacts on the simulated streamflow and inundation dynamics than the choice of the downstream boundary condition at the watershed outlet. We point out the viability to confine the uncertainty of coarsening mesh resolution by using the variable resolution mesh (VRM) which refines critical topographic features with far fewer grid cells. Specifically, in simulations with VRM, the simulated inundation depths over the refined region are comparable to that use the finest uniform mesh. This study contributes to understanding the challenges and pathways for applying 2D SWE models to improve the realism of flood simulations over large scales.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2024

Full Simulation of CMS for Run-3 and Phase-2

In this contribution we report the status of the CMS Geant4 simulation and the prospects for Run-3 and Phase-2. Firstly, we report about our experience during the start of Run-3 with Geant4 10.7.2, the common software package DD4hep for geometry description, and VecGeom runtime geometry library. In addition, FTFP_BERT_EMM Physics List and CMS configuration for tracking in magnetic field have been utilized. For the first time, for the Grid mass production of Monte-Carlo, this combination of components is used. Further simulation improvements are under development targeting Run-3 such as the switch to the new Geant4 11.1 in production, that provides several features important for the optimization of simulation, for example the new transportation process with built-in multiple scattering, neutron general process, custom tracking manager, G4HepEm sub-library, and others. We will present evaluation of various options, validation results, and the final choice of simulation configuration for 2023 production and beyond. The performance of the CMS full simulation for Run-2 and Run-3 will also be discussed. CMS development plan for the Phase-2 Geant4 based simulation is very ambitious, and it includes a new geometry description, physics, and simulation configurations. The progress on new detector descriptions and full simulation will be presented as well as the R&D in progress to reduce compute capacity needs.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

DOE OSTI2020

Method of information entropy for convergence assessment of molecular dynamics simulations

The lack of a reliable method to evaluate the convergence of molecular dynamics simulations has contributed to discrepancies in different areas of molecular dynamics. Here, the method of information entropy is introduced to molecular dynamics for stationarity assessment. The Shannon information entropy formalism is used to monitor the convergence of the atom motion to a steady state in a continuous spatial domain and is also used to assess the stationarity of calculated multidimensional fields such as the temperature field in a discrete spatial domain. It is demonstrated in this work that monitoring the information entropy of the atom position matrix provides a clear indicator of reaching steady state in radiation damage simulations, non-equilibrium molecular dynamics thermal conductivity computations, and simulations of Poiseuille and Couette flow in nanochannels. A main advantage of the present technique is that it is non-local and relies on fundamental quantities available in all molecular dynamics simulations. Unlike monitoring average temperature, the technique is applicable to simulations that conserve total energy such as reverse non-equilibrium molecular dynamics thermal conductivity computations and to simulations where energy dissipates through a boundary as in radiation damage simulations. The method is applied to simulations of iron using the Tersoff/ZBL splined potential, silicon using the Stillinger–Weber potential, and to Lennard–Jones fluid. Its applicability to both solids and fluids shows that the technique has potential for generalization to other areas in molecular dynamics.

74 ATOMIC AND MOLECULAR PHYSICS↗

DOE OSTI2024

Orbit-following simulations of fast-ion transport and losses due to the Alfvén eigenmode burst in the Large Helical Device

Orbit-following simulations of fast-ion transport and losses with time-dependent electromagnetic perturbations are performed to clarify the roles of Alfvén eigenmodes (AEs) and the low-frequency magnetohydrodynamic (MHD) mode observed in the kinetic-MHD hybrid simulation of AE bursts in the Large Helical Device. Fast-ion pressure profile flattening in the kinetic-MHD hybrid simulation can be reproduced by an orbit-following simulation with only the primary single AE of the time-dependent amplitude following the kinetic-MHD hybrid simulation result, while orbit-following simulations with constant AE amplitude of average level during AE burst cannot reproduce the fast-ion pressure profile flattening observed. The effects of other modes are negligible on the fast-ion pressure profile flattening. The fast-ion losses in kinetic-MHD hybrid simulation can be reproduced by an orbit-following simulation with time-dependent amplitude when the low-frequency MHD mode is considered in addition to multiple AEs. This indicates the synergetic effect of multiple AEs and the low-frequency MHD mode on fast-ion losses.

Seki, Ryohsuke↗

DOE OSTI2024

Nonlinear simulations of GAEs in NSTX-U

A set of nonlinear simulations has been performed in order to study the nonlinear evolution of unstable global Alfvén eigenmodes in the National Spherical Torus Experiment-Upgrade (NSTX-U). Results of the single toroidal mode number, n, simulations are compared with a full nonlinear simulation (all toroidal harmonics included). In single-n simulations, the conservation of two integrals of motion of a particle in a cyclotron resonance with a monochromatic wave is demonstrated, resulting in a one-dimensional evolution of the particle distribution in (E,μ,pϕ) phase-space. Nonlinear simulations (both single-n and full nonlinear) show a significant redistribution of the resonant fast ions, especially in the pitch parameter. Thus, the changes in the resonant particle's parallel and perpendicular energies can be several times larger than the total particle energy change, with only a small fraction transferring into the excitation of the mode itself. This implies that even a relatively small amplitude mode can significantly modify the beam distribution in the resonant region. For the NSTX-U case considered, the single-n simulation results are close to full nonlinear simulation only for the most unstable mode, in which case the saturation amplitudes and changes in the fast ion distribution are comparable. In contrast, peak amplitudes of subdominant modes in all-n simulations are smaller by a factor of 3–10 compared to single-n runs due to the flattening of the beam ion distribution by the fastest growing mode.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2025

Assessing the performance of MagLIF with 3D MHD simulations

The Magnetized Liner Inertial Fusion (MagLIF) experimental platform at Sandia National Laboratories has realized a number of recent improvements in electrical current and laser preheat coupling but fusion yields are below expectations, based on modeling with clean (that is, simulations with no interfacial instabilities and no mix) two-dimensional (2D) magneto-hydrodynamics (MHD) simulations, by a factor of 5 or more. However, magnetized liner implosions are known to produce helical magneto-Rayleigh–Taylor (MRT) modes and complex stagnation structures that cannot be directly modeled in 2D. This paper presents the results of 3D HYDRA MHD simulations, including helical MRT, showing that degradation from these instabilities can readily reproduce experimental yields, but it is difficult to simultaneously match the yield along with inferred fuel temperature, pressure, and burn history (assessed with x-rays), which are larger in the experiments considered here. Additional analysis methods and future experiments are proposed to help address the discrepancies. The 3D simulations also show, without substantive improvements to stability, changing the applied Bz or preheat is unlikely to increase performance of MagLIF to the same degree as clean 2D simulations. Finally, the first 3D HYDRA simulations with the Hall term show the ab initio production of helical MRT and produce a clear change in stagnation morphology compared to pre-seeded simulations without Hall. In spite of the differences, the simulations still produce comparable fusion performance but with longer, lower power, neutron yield history.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2021

Pore-scale simulation of drying in porous media using a hybrid lattice Boltzmann: pore network model

In this work, a hybrid method coupling a pseudo-potential lattice Boltzmann model (LBM) and a pore network model (PNM) to simulate drying in porous media is proposed. Based on the watershed method, the porous medium is firstly decomposed into pore regions. According to the liquid–vapor phase distribution at a given time, the pore regions are further divided into four pore types, namely two-phase pores where a liquid–vapor interface exists, buffer pores next to the two-phase pores, single-liquid and single-vapor phase pores. The pseudo-potential LBM is used in the two-phase and buffer pores to simulate liquid drying and track the movement of the interfaces, while the single-phase PNM simulations are conducted in the buffer and single-phase pores to simulate vapor or liquid flow. LBM and PNM are coupled in the buffer pores through exchange of boundary information. The hybrid method is applied to simulate liquid drying in a porous medium. The whole-domain LBM simulation is considered as the reference solution to validate the hybrid method. Liquid saturation variation during the drying process and detailed phase and pressure distributions obtained by the two methods match quite well, demonstrating the accuracy of the hybrid method. For the specific case studied, the hybrid method saves more than 60% computational time compared to the whole-domain LBM simulation. In addition, the speedup of the hybrid method becomes more significant for a larger computational domain. In summary, the hybrid method developed in this work combines the accuracy of LBM and the efficiency of PNM to simulate drying in porous media at pore scale and can lead to significant reduction of computation time, thus allowing the pore-scale consideration of drying in larger porous systems.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Kinetic theory of particle-in-cell simulation plasma and the ensemble averaging technique

Abstract We derive the kinetic theory of fluctuations in physically and numerically stable particle-in-cell (PIC) simulations of electrostatic plasmas. The starting point is the single-time correlation at the start of the simulation between the statistical fluctuations of the weighted densities of macroparticle centers in the plasma particle phase-space. The fluctuations are associated with different initial conditions, typically due to the random initial conditions (in velocity space) of the macroparticles/simulation plasma, assigned according to their initial distribution of probability. The single-time correlations at all time steps and in each spatial grid cell are then determined from the Laplace–Fourier transforms of the discretized Klimontovich-like equation for the macroparticles and Maxwell’s equations for the fields, as computed by modern PIC codes. We recover the expressions for the electrostatic field and the plasma particle density fluctuation autocorrelation spectra as well as the kinetic equations describing the average evolution of PIC-simulated plasma particles, first derived by Langdon (1970b Proc. 4th Conf. Numerical Simulation of Plasmas ) using a test macroparticle approach perturbing a discretized Vlasovian plasma and then averaging the obtained physical quantity over the initial macroparticle velocity distribution. We generalize and extend these results to the modern algorithms in PIC codes using arbitrary macroparticle weights. Analytical estimates of statistical fluctuation amplitudes are derived as a function of the plasma simulation parameters, using the central limit theorem in the limit of a large number of macroparticles per cell. The theory is then used to analyze the ensemble averaging technique of PIC simulations where statistical averages are performed over ensembles of PIC simulations, modeling the same plasma physics problem but using different statistical realizations of the initial distribution functions of the macroparticles. This method is illustrated by linear Landau damping uncovering (from noise, which is usually considered numerical) the physical fluctuations driven by a single small amplitude electrostatic wave perturbing a PIC simulation plasma in equilibrium.

fluctuations correlations↗

DOE OSTI2025

Simulations of the fault current limiting operation of a long-length REBCO CORC ® superconducting cable cooled by helium gas

Conductor-on-round-core (CORC®) cables composed of rare-earth-barium-copper-oxide high-temperature superconducting (HTS) tapes are of great interest for power transmission applications due to their many advantages such as high power density, light weight, and low loss. Closed circulation loops of cryogenic helium gas can be used to cool HTS cables down to low temperatures to significantly improve their current-carrying capacity. Coupled circuit-electromagnetic-thermal finite element simulations implemented in the COMSOL Multiphysics package were developed, validated, and then used for simulating the fault current limiting (FCL) performance and the cooling processes of an 8-layer CORC® cable cooled with a flow of cryogenic helium gas. In the simulations, the temperature dependence of the electrical and thermal properties of all component materials is implemented for improved accuracy. To overcome computational challenges caused by the considerable difference in geometrical scales (i.e. few-µm-thick HTS layers versus 10 m-long HTS cable), the model is divided into two separate simulations. The first simulation is performed on the transverse cross-section of the cable to calculate the electric field, heating power and temperature rise in each component of a CORC® cable during FCL operation. The heating power calculated in the first simulation is transferred to the second model to simulate the cooling of a 10 m-long cable after the fault is cleared. The effect of the helium gas flow rate on the cooling process is also investigated to develop strategic approaches for optimizing cooling systems for HTS cables with FCL capability. The simulations indicated that a 40 ms fault with a voltage drop of 20 V m −1 along the cable can result in a temperature increase from 60 K to about 165 K inside the cable, and it takes about 500 s to cool the cable back to nearly 60 K with a flow of cold helium gas at a rate of 5 g s −1 .

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2021

Selecting durable building envelope systems with machine learning assisted hygrothermal simulations database

Hygrothermal simulations provide insight into the energy performance and moisture durability of building envelope components under dynamic conditions. The inputs required for hygrothermal simulations are extensive, and carrying out simulations and analyses requires expert knowledge. An expert system, the Building Science Advisor (BSA), has been developed to predict the performance and select the energy-efficient and durable building envelope systems for different climates. The BSA consists of decision rules based on expert opinions and thousands of parametric simulation results for selected wall systems. The number of potential wall systems results in millions, too many to simulate all of them. We present how machine learning can help predict durability data, such as mold growth, while minimizing the number of simulations needed to run. The simulation results are used for training and validation of machine learning tools for predicting wall durability. We tested Artificial Neural Network (ANN) and Gradient Boosted Decision Trees (GBDT) for their applicability and model accuracy. Models developed with both methods showed adequate prediction performance (root mean square error of 0.195 and 0.209, respectively). Finally, we introduce how the information supports guidance for envelope design via an easy-to-use web-based tool that does not require the end-user to run hygrothermal simulations.

Salonvaara, Mikael↗

DOE OSTI2024

Validation of new and existing methods for time-domain simulations of turbulence and loads

We seek to obtain a second-by-second match between the simulated and measured structural loads of a utility-scale wind turbine. To obtain the one-to-one load simulations, we start with the furthest upstream component of the modeling chain: the turbulent inflow. We consider new and existing methods to generate constrained-turbulence flow fields. The new method is based on large-eddy simulations (LES) and machine learning (ML). The existing methods include Kaimal-based TurbSim and the superstatistical wind field model. The inflow measurements used to constrain these simulations are obtained with a nacelle-mounted scanning lidar. We compare the flow fields for the different inflow simulation approaches and validate their associated load predictions against measurements collected in the Rotor Aero-dynamics, Aeroelastics, and Wake (RAAW) field campaign. We find that the rotor-position control developed for this study is key in enabling the time match between measurements and simulations. When this control approach is used, the load simulation performance tracks with the inflow simulation fidelity, with LES+ML yielding errors ≤ 4% for the damage-equivalent loads of flapwise bending moment, and tower fore-aft bending moments.

17 WIND ENERGY↗

DOE OSTI2022

Hydrodynamic simulations of electron-capture supernovae: progenitor and dimension dependence

ABSTRACT We present neutrino-transport hydrodynamic simulations of electron-capture supernovae (ECSNe) in flash with new two-dimensional (2D) collapsing progenitor models. These progenitor models feature the 2D modelling of oxygen-flame propagation until the onset of core collapse. We perform axisymmetric simulations with six progenitor models that, at the time of collapse, span a range of propagating flame front radii. For comparison, we also perform a simulation with the same set-up using the canonical, spherically symmetrical progenitor model n8.8. We found that the variations in the progenitor models inherited from simulations of stellar evolution and flame propagation do not significantly alter the global properties of the neutrino-driven ECSN explosion, such as the explosion energy (∼1.36–1.48 × 1050 erg) and the mass (∼0.017–0.018 M⊙) and composition of the ejecta. Due to aspherical perturbations induced by the 2D flame, the ejecta contains a small amount (≲1.8 × 10−3 M⊙) of low-Ye (0.35 < Ye < 0.4) component. The baryonic mass of the protoneutron star is ∼1.34 M⊙ (∼1.357 M⊙) with the new (n8.8) progenitor models when simulations end at ∼400 ms and the discrepancy is due to updated weak-interaction rates in the progenitor evolutionary simulations. Our results reflect the nature of ECSN progenitors containing a strongly degenerate oxygen–neon–magnesium (ONeMg) core and suggest a standardized ECSN explosion initialized by ONeMg core collapse. Moreover, we carry out a rudimentary three-dimensional simulation and find that the explosion properties are fairly compatible with the 2D counterpart. Our paper facilitates a more thorough understanding of ECSN explosions following the ONeMg core collapse, though more three-dimensional simulations are still needed.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2025

A Versatile Simulated Data Transport Layer for in Situ Workflows Performance Evaluation

In situ processing does not only allow scientific applications to face the explosion in data volume and velocity but also to address the time constraints of many simulation-analysis workflows by providing scientists with early insights about their applications at runtime. Multiple frameworks implement the concept of a data transport layer (DTL) to enable such in situ workflows. These tools are very versatile, directly or indirectly access the data generated on the same node, another node of the same compute cluster, or a completely distinct node, and allow data publishers and subscribers to run on the same computing resources or not. This versatility puts on researchers the onus of taking key decisions related to resource allocation and how to transport data to ensure the most efficient execution of their in situ workflows. However, domain scientists and workflow practitioners lack the appropriate tools to assess the respective performance of particular design and deployment options. In this paper we introduce a versatile simulated DTL designed to provide researchers with insights on the respective performance of different execution scenarios of in situ workflows. This open-source, standalone library builds on the SimGrid toolkit and can be linked to any SimGrid-based simulator. It facilitates the evaluation of the performance behavior, at scale, of different data transport configurations and the study of the effects of resource allocation strategies. We demonstrate the scalability, versatility, and accuracy of this simulated DTL by reproducing the execution of two synthetic benchmarks and of a real-world in situ workflow composed of an MPI application and a parallel data analysis. Results of simulations run on a single core show that the proposed library can simulate the interactions of tens of thousands of simulated processes deployed on two interconnected commodity clusters in a few seconds, and the execution by a thousand simulated processes of an in situ workflow in less than three minutes.

Suter, Fred [ORNL] (ORCID:0000000319021955)↗

DOE OSTI2023

High-Performance Computing Based EMT Simulation of Large PV or Hybrid PV Plants

Faults in the transmission grid have led to reduced power generation from power electronics resources that are typically not connected to the faulted transmission line. In many of the cases, partial loss of power is observed within the power electronics resources like large photovoltaic (PV) power plants. This phenomena is not captured in existing simulation models and/or simulators. High-fidelity switched system electromagnetic transient (EMT) dynamic models of PV power plants can improve the fidelity of models available for accurate analysis of the impact on PV plants during simulation of faults. However, these models are extremely computationally expensive and take a long time to simulate. Long simulation times limit the ability to use these models as larger regions are studied in EMT simulations with more power electronics resources. In this paper, numerical simulation algorithms are combined with high-performance computing techniques and applied to the high-fidelity switched system EMT model of PV plants. Using these techniques, a speed-up of up to 58x is obtained, while preserving the accuracy of the simulation at greater than 98%.

Debnath, Suman↗

DOE OSTI2022

Characterization of the Finite Element Computational Fluid Dynamics Capabilities in the Multiphysics Object Oriented Simulation Environment

We report the multiphysics object-oriented simulation environment (moose) is a code package that couples a variety of physics modules, allowing for highly accessible multiphysics simulations. The physics modules include a finite element Navier–Stokes (N–S) module that is designed to solve laminar fluid dynamics problems. The usage of this module in multiple recent studies coupled with the growing interest in moose for usage in nonlight water reactor safety studies by the Nuclear Regulatory Commission (NRC) prompted the authors to investigate the computational fluid dynamics capabilities of moose. A two-dimensional laminar flow past a circular cylinder scenario is simulated in the moose framework to investigate the effectiveness of the N–S module. Simulations assumed an unsteady laminar flow with a Reynolds number of 200. To verify the results from moose, similar simulations were conducted using the well-utilized simulation of turbulent flow in arbitrary regions—computational continuum mechanics C++ (star-ccm + ) finite volume code. Results from both codes are also compared to some results from literature. Velocity and pressure profiles of both transient simulations were compared. The numerical and input errors in moose are also visualized with contour plots to qualitatively understand the evolution of the errors across time and space. The comparisons between moose and star-ccm + showed nearly perfect agreement between the codes for velocity and pressure, especially after the development of the vortex street in later time-steps. The force coefficients showed excellent agreement after the development of the vortex street, but demonstrated notable discrepancies prior to the vortex street development, which is likely due to how each code simulated the approach to the vortex street in earlier time-steps.

97 MATHEMATICS AND COMPUTING↗