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At least 37 records · Page 2

Nonlinear interactions in mixing layers and compressible heated round jets

The nonlinear interactions between a fundamental instability mode and both its harmonics and the changing mean flow are studied using the weakly nonlinear stability theory of Stuart and Watson, and numerical solutions of coupled nonlinear partial differential equations. The first part focuses on incompressible cold (or isothermal; constant temperature throughout) mixing layers, and for these, the first and second Landau constants are calculated as functions of wavenumber and Reynolds number. It is found that the dominant contribution to the Landau constants arises from the mean flow changes and not from the higher harmonics. In order to establish the range of validity of the weakly nonlinear theory, the weakly nonlinear and numerical solutions are compared and the limitation of each is discussed. At small amplitudes and at low-to-moderate Reynolds numbers, the two results compare well in describing the saturation of the fundamental, the distortion of the mean flow, and the initial stages of vorticity roll-up. At larger amplitudes, the interaction between the fundamental, second harmonic, and the mean flow is strongly nonlinear and the numerical solution predicts flow oscillations, whereas the weakly nonlinear theory yields saturation. In the second part, the weakly nonlinear theory is extended to heated (or nonisothermal; mean temperature distribution) subsonic round jets where quadratic and cubic nonlinear interactions are present, and the Landau constants also depend on jet temperature ratio, Mach number and azimuthal mode number. Under exponential growth and nonlinear saturation, it is found that heating and compressibility suppress the growth of instability waves, that the first azimuthal mode is the dominant instability mode, and that the weakly nonlinear solution describes the early stages of the roll-up of an axisymmetric shear layer. The receptivity of a typical jet flow to pulse type input disturbance is also studied by solving the initial value problem and then examining the behavior of the long-time solution.

Jarrah, Yousef Mohd↗

A Unified Photometric Redshift Calibration for Weak Lensing Surveys Using the Dark Energy Spectroscopic Instrument

The effective redshift distribution n(z) of galaxies is a critical component in the study of weak gravitational lensing. Here, we introduce a new method for determining n(z) for weak lensing surveys based on high-quality redshifts and neural-network-based importance weights. Additionally, we present the first unified photometric redshift calibration of the three leading stage-III weak lensing surveys, the Dark Energy Survey (DES), the Hyper Suprime-Cam (HSC) survey, and the Kilo-Degree Survey (KiDS), with state-of-the-art spectroscopic data from the Dark Energy Spectroscopic Instrument (DESI). We verify our method using a new, data-driven approach and obtain n(z) constraints with statistical uncertainties of the order of $σ_z$ ~ 0.01 and smaller. Our analysis is largely independent of previous photometric redshift calibrations and, thus, provides an important cross-check in light of recent cosmological tensions. Overall, we find excellent agreement with previously published results on the DES Y3 and HSC Y1 data sets, while there are some differences on the mean redshift with respect to the previously published KiDS-1000 results. We attribute the latter to mismatches in photometric noise properties in the COSMOS field compared to the wider KiDS self-organizing map-gold catalog. At the same time, the new n(z) estimates for KiDS do not significantly change estimates of cosmic structure growth from cosmic shear. Finally, we discuss how our method can be applied to future weak lensing calibrations with DESI data.

Lange, J. U. [American Univ., Washington, DC (Unit↗

Magnetic fields in the formation of the first stars – II. Results

ABSTRACT Beginning with cosmological initial conditions at z = 100, we simulate the effects of magnetic fields on the formation of Population III stars and compare our results with the predictions of Paper I. We use gadget-2 to follow the evolution of the system while the field is weak. We introduce a new method for treating kinematic fields by tracking the evolution of the deformation tensor. The growth rate in this stage of the simulation is lower than expected for diffuse astrophysical plasmas, which have a very low resistivity (high magnetic Prandtl number); we attribute this to the large numerical resistivity in simulations, corresponding to a magnetic Prandtl number of order unity. When the magnetic field begins to be dynamically significant in the core of the minihalo at z = 27, we map it on to a uniform grid and follow the evolution in an adaptive mesh refinement, MHD simulation in orion2. The non-linear evolution of the field in the orion2 simulation violates flux-freezing and is consistent with the theory proposed by Xu & Lazarian. The fields approach equipartition with kinetic energy at densities ∼1010–1012 cm−3. When the same calculation is carried out in orion2 with no magnetic fields, several protostars form, ranging in mass from ∼1 to 30 M⊙; with magnetic fields, only a single ∼30 M⊙ protostar forms by the end of the simulation. Magnetic fields thus suppress the formation of low-mass Pop III stars, yielding a top-heavy Pop III initial mass function and contributing to the absence of observed Pop III stars.

79 ASTRONOMY AND ASTROPHYSICS↗

Strong Coupling in Semiconductor Hyperbolic Metamaterials

Semiconductor-based layered hyperbolic metamaterials (HMMs) house high-wavevector volume plasmon polariton (VPP) modes in the infrared spectral range. VPP modes have successfully been exploited in the weak-coupling regime through the enhanced Purcell effect. In this paper, we experimentally demonstrate strong coupling between the VPP modes in a semiconductor HMM and the intersubband transition of epitaxially embedded quantum wells. We observe clear anticrossings in the dispersion curves for the zeroth-, first-, second-, and third-order VPP modes, resulting in upper and lower polariton branches for each mode. This demonstration sets the stage for the creation of novel infrared optoelectronic structures combining HMMs with embedded epitaxial emitter or detector structures.

77 NANOSCIENCE AND NANOTECHNOLOGY↗

Bayesian calibration of viscous anisotropic hydrodynamic simulations of heavy-ion collisions

Owing to large pressure gradients at early times, standard hydrodynamic model simulations of relativistic heavy-ion collisions do not become reliable until O(1) fm/c after the collision. To address this one often introduces a prehydrodynamic stage that models the early evolution microscopically, typically as a conformal, weakly interacting gas. In such an approach the transition from the prehydrodynamic to the hydrodynamic stage is discontinuous, introducing considerable theoretical model ambiguity. Alternatively, fluids with large anisotropic pressure gradients can be handled macroscopically using the recently developed viscous anisotropic hydrodynamics (VAH). In high-energy heavy-ion collisions VAH is applicable already at very early times, and at later times transitions smoothly into conventional second-order viscous hydrodynamics. Here we present a Bayesian calibration of the VAH model with experimental data for Pb-Pb collisions at the LHC at $\sqrt{^{S}NN}$ = 2.76 A TeV. We find that the VAH model has the unique capability of constraining the specific viscosities of the quark-gluon plasma at higher temperatures than other previously used models.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Computational Aeroacoustics by the Space-time CE/SE Method

In recent years, a new numerical methodology for conservation laws-the Space-Time Conservation Element and Solution Element Method (CE/SE), was developed by Dr. Chang of NASA Glenn Research Center and collaborators. In nature, the new method may be categorized as a finite volume method, where the conservation element (CE) is equivalent to a finite control volume (or cell) and the solution element (SE) can be understood as the cell interface. However, due to its rigorous treatment of the fluxes and geometry, it is different from the existing schemes. The CE/SE scheme features: (1) space and time treated on the same footing, the integral equations of conservation laws are solve( for with second order accuracy, (2) high resolution, low dispersion and low dissipation, (3) novel, truly multi-dimensional, simple but effective non-reflecting boundary condition, (4) effortless implementation of computation, no numerical fix or parameter choice is needed, an( (5) robust enough to cover a wide spectrum of compressible flow: from weak linear acoustic waves to strong, discontinuous waves (shocks) appropriate for linear and nonlinear aeroacoustics. Currently, the CE/SE scheme has been developed to such a stage that a 3-13 unstructured CE/SE Navier-Stokes solver is already available. However, in the present paper, as a general introduction to the CE/SE method, only the 2-D unstructured Euler CE/SE solver is chosen as a prototype and is sketched in Section 2. Then applications of the CE/SE scheme to linear, nonlinear aeroacoustics and airframe noise are depicted in Sections 3, 4, and 5 respectively to demonstrate its robustness and capability.

Loh, Ching Y.↗

Phonon softening and slowing-down of charge density wave fluctuations in BaNi 2 As 2

BaNi 2 As 2 is a nonmagnetic analog of the iron pnictide superconductors, and exhibits an incommensurate charge density wave (IC-CDW) and a sizable elastoresistance. Here, in this Letter, phonons in BaNi 2 As 2 associated with the IC-CDW and uniform in-plane lattice distortions are investigated using high-resolution inelastic x-ray scattering. The in-plane transverse acoustic phonons reveal no softening at temperatures where the elastoresistance increases strongly, indicating the latter to be electronically driven. Systematic phonon measurements suggest the IC-CDW occurs in two stages upon cooling: Underdamped phonons first soften to zero energy well above the IC-CDW ordering temperature, then the resulting quasielastic IC-CDW fluctuations gradually slow down and coalesce into the static IC-CDW order. A possible origin for our observations is the IC-CDW in BaNi 2 As 2 being uniaxial, which provides an additional Ising degree of freedom favorable for disordered IC-CDW modulations, and accounts for the elastoresistance through a weak coupling to the lattice.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Preparation for a Measurement of Charge Asymmetry in the Bethe-Heitler Process

We have prepared a measurement of the energy asymmetry in wide- and medium-angle electron/positron pair production off protons and heavy targets. This asymmetry is caused by the interference between the first- and second-order Born diagrams and the Compton scattering diagram. It directly probes aspects of QED, as well as providing a direct measurement of the real part of the Compton amplitude. It will be conducted at the HI??S facility at Duke University, using a 60 MeV photon beam. This dissertation serves as documentation of the preparation stage of the Bethe-Heitler experiment. The major was the recommissioning of the vertical drift chambers previously used in the Q-weak experiment at the Jefferson Lab. Cosmic test runs were conducted, drift time data were collected and efficiency plateaus were measured. We made modifications to the JLAB Hall A analyzer to suit the geometry and drift characteristics of these wire chambers. The analyzer was used for the reconstruction of the trajectories of cosmic ray test runs with the results confirmed by direct measurement of trigger geometry. Spatial and angular resolution is estimated to ~300?? and 0.17° respectively. Geant 4 simulations with generated Bethe-Heitler pairs satisfying theoretical differential cross sections. It was used to check detector acceptance, optimize apparatus layout, and estimate measurable energy asymmetry. The measurable asymmetries from electron/positron pairs with polar angles around between approximately 5° and 8°, azimuthal angles differing by 180°, and energy differing by approximately 9 MeV to 15 MeV are predicted to be above 10%. The kinematics of primary vertices are reconstructed using the data from wire chambers in the simulation. The energy resolution is determined to be better than 1MeV.

Chen, Haoyu↗

Fast Solution of Fully Implicit Runge--Kutta and Discontinuous Galerkin in Time for Numerical PDEs, Part I: the Linear Setting

Fully implicit Runge--Kutta (IRK) methods have many desirable properties as time integration schemes in terms of accuracy and stability, but high-order IRK methods are not commonly used in practice with numerical PDEs due to the difficulty of solving the stage equations. This paper introduces a theoretical and algorithmic preconditioning framework for solving the systems of equations that arise from IRK methods applied to linear numerical PDEs (without algebraic constraints). Additionally, this framework also naturally applies to discontinuous Galerkin discretizations in time. Under quite general assumptions on the spatial discretization that yield stable time integration, the preconditioned operator is proven to have condition number bounded by a small, order-one constant, independent of the spatial mesh and time-step size, and with only weak dependence on number of stages/polynomial order; for example, the preconditioned operator for 10th-order Gauss IRK has condition number less than two, independent of the spatial discretization and time step. The new method can be used with arbitrary existing preconditioners for backward Euler-type time-stepping schemes and is amenable to the use of three-term recursion Krylov methods when the underlying spatial discretization is symmetric. The new method is demonstrated to be effective on various high-order finite-difference and finite element discretizations of linear parabolic and hyperbolic problems, demonstrating fast, scalable solution of up to 10th-order accuracy. The new method consistently outperforms existing block preconditioning approaches, and in several cases, the new method can achieve 4th-order accuracy using Gauss integration with roughly half the number of preconditioner applications and wallclock time as required using standard diagonally IRK methods.

97 MATHEMATICS AND COMPUTING↗

Dynamic Rupture Simulations of the Mw7.2 1992 Landers,California, Earthquake

An important part of the simulation technique implemented in the Recipe is the earthquake rupture characterization based on established empirical scaling relationships between source parameters and seismic moment. The scaling relationships used in the rupture characterization are derived from recorded data and are tested against dynamic and kinematic rupture models of crustal earthquakes. Currently, a revised three-stage source scaling relationship between the source rupture area and seismic moment of inland crustal earthquakes, initially proposed by Irikura and Miyake, is being used. Recent analysis of results obtained with the Recipe for the 2016 M7.1 Kumamoto earthquake performed by several researchers, including the PI, Arben Pitarka, indicate that in order to reproduce the near-fault long-period ground motions and the permanent displacements, observed during surface-rupturing earthquakes, the recipe needs modifications to the kinematic rupture parametrization in the shallow crustal weak zone. In particular, improvements are required in characterizing the slip rate function at shallow depths, above the sesimogenic zone. The main objective of this research project was the development of a dynamic rupture model for the Mw7.2 1992 Landers, California earthquake. The dynamic rupture model was used to derive kinematic rupture parameters, such as slip, slip-rate, rupture time distributions, and slip rate functions in the weak (0-4km) and seismogenic crustal zones. In addition to the knowledge gained from the dynamic rupture modeling of Chi-chi, Ridgecrest, Kumamoto and other crustal earthquakes, studied by the PI, the rupture characterization of the Landers earthquake can be helpful in updating the conventional recipe for broadband ground motion simulations of crustal earthquakes with shallow rupture. The project emphasizes the need for performing physics-based earthquake rupture modeling which ultimately will inform further improvements of rupture characterization. In particular, the modeling of rupture dynamics for recorded earthquakes, and numerical simulations for different rupture scenarios, provides physical constrains on slip evolution in time and space, used in adjustments of empirical source scaling relationships for large crustal earthquakes.

58 GEOSCIENCES↗

Prediction of active control of subsonic centrifugal compressor rotating stall

A mathematical model is developed to predict the suppression of rotating stall in a centrifugal compressor with a vaned diffuser. This model is based on the employment of a control vortical waveform generated upstream of the impeller inlet to damp weak potential disturbances that are the early stages of rotating stall. The control system is analyzed by matching the perturbation pressure in the compressor inlet and exit flow fields with a model for the unsteady behavior of the compressor. The model was effective at predicting the stalling behavior of the Purdue Low Speed Centrifugal Compressor for two distinctly different stall patterns. Predictions made for the effect of a controlled inlet vorticity wave on the stability of the compressor show that for minimum control wave magnitudes, on the order of the total inlet disturbance magnitude, significant damping of the instability can be achieved. For control waves of sufficient amplitude, the control phase angle appears to be the most important factor in maintaining a stable condition in the compressor.

Lawless, Patrick B.↗

Fracture in Phenolic Impregnated Carbon Ablator

The thermal protection materials used for spacecraft heat shields are subjected to various thermal-mechanical loads during an atmospheric reentry which can threaten the structural integrity of the system. This paper discusses the development of a novel technique to understand the failure mechanisms inside the thermal protection material, Phenolic Impregnated Carbon Ablator (PICA). PICA has successfully flown on the Stardust spacecraft and was the TPS material chosen for the Mars Science Laboratory (MSL), that will fly in 2011. Although PICA has good thermal properties, structurally, it is a weak material. To thoroughly understand failure in PICA, experiments were performed using FiberForm(Registered TradeMark) (precursor of PICA), virgin and furnace-charred PICA. Several small samples were tested inside an electron microscope to investigate the failure mechanisms. Micrographs were obtained before and after the failure in order to study crack initiation and growth. Videos were obtained to capture failure mechanisms in real time. Stress-strain data was obtained simultaneously for all the samples with the help of a data acquisition system, integrated to the mechanical stages. It was found that cracks initiated and grew in the FiberForm when a critical stress limit was reached such that the carbon fibers separated from the binder. However, both for virgin and charred PICA, crack initiation and growth occurred in the matrix (phenolic) phase. Both virgin and charred PICA showed greater strength values compared to FiberForm coupons, confirming that the presence of the porous matrix helps in absorbing the fracture energy.

Agrawal, Parul↗

Reinforcement Learning for Weakly-Coupled MDPs and an Application to Planetary Rover Control

Weakly-coupled Markov decision processes can be decomposed into subprocesses that interact only through a small set of bottleneck states. We study a hierarchical reinforcement learning algorithm designed to take advantage of this particular type of decomposability. To test our algorithm, we use a decision-making problem faced by autonomous planetary rovers. In this problem, a Mars rover must decide which activities to perform and when to traverse between science sites in order to make the best use of its limited resources. In our experiments, the hierarchical algorithm performs better than Q-learning in the early stages of learning, but unlike Q-learning it converges to a suboptimal policy. This suggests that it may be advantageous to use the hierarchical algorithm when training time is limited.

Bernstein, Daniel S.↗

Combining cosmic shear data with correlated photo-z uncertainties: constraints from DESY1 and HSC-DR1

An accurate calibration of the source redshift distribution p(z) is a key aspect in the analysis of cosmic shear data. This, one way or another, requires the use of spectroscopic or high-quality photometric samples. However, the difficulty to obtain colour-complete spectroscopic samples matching the depth of weak lensing catalogs means that the analyses of different cosmic shear datasets often use the same samples for redshift calibration. This introduces a source of statistical and systematic uncertainty that is highly correlated across different weak lensing datasets, and which must be accurately characterised and propagated in order to obtain robust cosmological constraints from their combination. In this paper we introduce a method to quantify and propagate the uncertainties on the source redshift distribution in two different surveys sharing the same calibrating sample. The method is based on an approximate analytical marginalisation of the p(z) statistical uncertainties and the correlated marginalisation of residual systematics. We apply this method to the combined analysis of cosmic shear data from the DESY1 data release and the HSC-DR1 data, using the COSMOS 30-band catalog as a common redshift calibration sample. We find that, although there is significant correlation in the uncertainties on the redshift distributions of both samples, this does not change the final constraints on cosmological parameters significantly. The same is true also for the impact of residual systematic uncertainties from the errors in the COSMOS 30-band photometric redshifts. Additionally, we show that these effects will still be negligible in Stage-IV datasets. Finally, the combination of DESY1 and HSC-DR1 allows us to constrain the "clumpiness" parameter to S 8 = ${0.768}_{-0.017}^{+0.021}$. This corresponds to a ~√(2) improvement in uncertainties with respect to either DES or HSC alone.

79 ASTRONOMY AND ASTROPHYSICS↗

The Importance of Convective Stage on Out-of-Cloud Convectively Induced Turbulence from High-Resolution Simulations

Turbulence (clear-air, mountain wave, convectively induced) is an aviation hazard that is a challenge to forecast due to the coarse resolution ultilized in operational weather models. Turbulence indices are commonly used to aid pilots in avoiding turbulence, but these indices have been designed and calibrated for midlatitude clear-air turbulence prediction (e.g., the Ellrod index). A significant limitation with current convectively induced turbulence (CIT) prediction is the lack of storm stage dependency. In this study, six high-resolution simulations of tropical oceanic and midlatitude continental convection are performed to characterize the turbulent environment near various convective types during the developing and mature stages. Second-order structure functions, a diagnostic commonly used to identify turbulence in turbulence prediction systems, are used to characterize the probability of turbulence for various convective types. Turbulence likelihood was found to be independent of region (i.e., tropical vs midlatitude) but dependent on convective stage. The probability of turbulence increased near developing convection for the majority of cases. Additional analysis of static stability and vertical wind shear, indicators of turbulence potential, showed that the convective environment near developing convection was more favorable for turbulence production than mature convection. Near developing convection, static stability decreased and vertical wind shear increased. Vertical wind shear near mature and developing convection was found to be weakly correlated to turbulence intensity in both the tropics and the midlatitudes. Finally, this study emphasizes the need for turbulence avoidance guidelines for the aviation community that are dependent on convective stage.

54 ENVIRONMENTAL SCIENCES↗

Existing and Required Modeling Capabilities for Evaluating ATM Systems and Concepts

ATM systems throughout the world are entering a period of major transition and change. The combination of important technological developments and of the globalization of the air transportation industry has necessitated a reexamination of some of the fundamental premises of existing Air Traffic Management (ATM) concepts. New ATM concepts have to be examined, concepts that may place more emphasis on: strategic traffic management; planning and control; partial decentralization of decision-making; and added reliance on the aircraft to carry out strategic ATM plans, with ground controllers confined primarily to a monitoring and supervisory role. 'Free Flight' is a case in point. In order to study, evaluate and validate such new concepts, the ATM community will have to rely heavily on models and computer-based tools/utilities, covering a wide range of issues and metrics related to safety, capacity and efficiency. The state of the art in such modeling support is adequate in some respects, but clearly deficient in others. It is the objective of this study to assist in: (1) assessing the strengths and weaknesses of existing fast-time models and tools for the study of ATM systems and concepts and (2) identifying and prioritizing the requirements for the development of additional modeling capabilities in the near future. A three-stage process has been followed to this purpose: 1. Through the analysis of two case studies involving future ATM system scenarios, as well as through expert assessment, modeling capabilities and supporting tools needed for testing and validating future ATM systems and concepts were identified and described. 2. Existing fast-time ATM models and support tools were reviewed and assessed with regard to the degree to which they offer the capabilities identified under Step 1. 3 . The findings of 1 and 2 were combined to draw conclusions about (1) the best capabilities currently existing, (2) the types of concept testing and validation that can be carried out reliably with such existing capabilities and (3) the currently unavailable modeling capabilities that should receive high priority for near-term research and development. It should be emphasized that the study is concerned only with the class of 'fast time' analytical and simulation models. 'Real time' models, that typically involve humans-in-the-loop, comprise another extensive class which is not addressed in this report. However, the relationship between some of the fast-time models reviewed and a few well-known real-time models is identified in several parts of this report and the potential benefits from the combined use of these two classes of models-a very important subject-are discussed in chapters 4 and 7.

Odoni, Amedeo R.↗

D retention and depth profile behavior for single crystal tungsten with high temperature neutron irradiation

Single crystalline W (tungsten) samples irradiated at 633, 963 and 1073 K by neutrons to a damage level of 0.1 dpa were exposed to a high-flux D (deuterium) plasma at 673, 873 and 973 K, respectively, in TPE (Tritium Plasma Experiment) at INL (Idaho National Laboratory). Deuterium desorption was analyzed by TDS (Thermal Desorption Spectroscopy), and D depth profiles were determined by NRA (Nuclear Reaction Analysis) at SNL (Sandia National Laboratories). HIDT (Hydrogen Isotope Diffusion and Trapping) simulation code was applied to evaluate D behavior for neutron-damaged W at higher temperature. The D retention at depths up to 3 f.tm for the neutron-damaged sample at 673 K was two orders of magnitude larger than that for undamaged tungsten, and its D desorption spectrum had a single broad stage at around 900 K. As the neutron irradiation/plasma exposure temperature increased, D retention was largely reduced, and the desorption temperature was increased to higher temperatures above 1100 K. The D depth profiles by NRA also showed D migration toward bulk by higher temperature irradiation, compared to undamaged W. The HIDT simulation indicated that the major binding energy of D was changed from 1.43 eV to 2.07 eV at higher neutron irradiation and plasma exposure temperatures, suggesting that some vacancies and small vacancy clusters would aggregate to form larger voids, or depopulation of weak traps at high D plasma exposure temperatures. It can be said that more stable trapping sites played dominant roles in the D retention at higher neutron irradiation and plasma exposure temperature. The binding energy by HIDT simulation was almost consistent with the reported value by TMAP, but the consideration of not only total D retention measured by TDS but also D depth profile by NRA led to the more accurate D behavior in neutron-damaged W.

36 MATERIALS SCIENCE↗

Midfidelity Model Verification for a Point-Absorbing Wave Energy Converter with Linear Power Takeoff: Preprint

In the preliminary design stage of a wave energy converter (WEC), fast and reliable simulation tools are required. High-fidelity numerical models are usually employed to study the wave-structure interaction, yet the computational cost is demanding. Instead, mid-fidelity models provide simulations in the order of real time. In this study, Uppsala University WEC operates in a relatively mild sea state and is modeled using WEC-Sim. The model is verified based on OpenFOAM simulations. To analyze the ability of mid-fidelity model to capture the WEC dynamics, the system is investigated separately in 1, 2 and 3 DoF. The contribution of viscous phenomena is examined. Moreover, linear and weakly non-linear solutions provided by WEC-Sim are studied. The results obtained indicate that the viscous effects in heave and surge motion can be neglected but not for the pitch. The weakly non-linear WEC-Sim solution successfully agrees with the CFD, whereas the linear solution could drive to misleading results.

CFD↗