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At least 505 records · Page 28

Studying Wind Loading on CSP Collectors to Improve Performance and Reliability

Electricity generation through Concentrating Solar-thermal Power (CSP) adds the advantage of thermal energy storage and heat production. CSP collector costs constitute nearly one-third of total plant costs. Managing wind loading, especially dynamic wind loading caused by the turbulent wind flow, is a significant design challenge. Traditional collector designs rely on wind tunnel experiments and numerical simulations, which do not fully capture dynamic effects. Here, we provide an overview of NREL's activities on wind loading on CSP collectors, aiming to improve their reliability and cost-efficiency. We present insights from field measurements at operational parabolic trough and power-tower CSP plants. Over two years, we measured turbulent wind fields and resulting structural loads in a parabolic trough plant, revealing wind, turbulence, and collector field interactions. Similar measurements are ongoing at the Crescent Dunes heliostat field. These campaigns show how atmospheric turbulence, wind direction, and collector orientation affect dynamic wind loading. Upwind collector structures can generate turbulent structures, causing fluctuating loads at downstream collectors, impacting fatigue lifetime and optical efficiency. Preliminary results indicate that interior collectors experience higher turning moments than those at the edges, suggesting higher drive wear. At Crescent Dunes, we use instrumented heliostats at the field's edge and interior to study the translation of turbulent wind to dynamic structural loads. The campaign is still ongoing, and we will present first findings. Another focus of our work is tying the experimental wind loading findings to optical performance of the collectors, using simulation tools. For example, we derive torsional errors of the parabolic troughs - the angular offset from the sun position along a collector row. This error is impacted by wind-induced mirror deformations. Future work will focus on continuing the measurement efforts and combining them with computational fluid dynamic simulations. By providing detailed field measurements and validated models, our efforts aim to enhance understanding of wind loading on CSP collectors, improving their structural integrity and optical performance.

CSP↗

A new mechanism for filament disconnection at the X-point: poloidal shear in radial E × B velocity

Plasma filaments generated by turbulence in the scrape-off layer are sometimes observed in experiments and simulations to disconnect from the target plate in the vicinity of the X-point, resulting in a ‘quiescent zone’ with reduced fluctuations, possibly affecting the fluctuating part of the divertor heat loads. This phenomena is usually explained by the flux tubes squeezing, induced by flux expansion, down to scales where collisions dominate. In this work, we consider an additional mechanism spontaneously arising at the X-point: the poloidal shear in radial E × B velocity. Through a simple model, tested against 3D global turbulence simulations, herein we quantify the effectiveness of this disconnection mechanism and identify ways to control it.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Impact of plasma density/collisionality on divertor heat flux width

Both ASDEX-Upgrade (AUG) data and the generalized HD (GHD) model showed that the scrape-off width broadens as the density/collisionality increases [1, 2]. A series of BOUT++ transport simulations are performed to study the physics of the scaling characteristics of the divertor heat flux width vs density/collisionality via a plasma density scan with either fixed pressure profile or fixed temperature profile inside separatrix. Additionally, the simulations show that even in the drift dominated regime, the divertor heat flux width can be broadened due to the transition of the SOL residence time from the parallel particle flow time to the enhanced parallel conduction time as the collisionality/density increases as posited in the GHD model. In addition, the heat flux width is found to be proportional to the square root of ion mass for low collisionality while it has a weakly dependence on ion mass for high collisionality. Furthermore, our simulations show that as the density increases, the radial electric field (E r ) well shallows, which potentially weakens E r × B flow shear stabilization of turbulence at high density.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Direct Numerical Simulation of hydrogen combustion at auto-ignitive conditions: Ignition, stability and turbulent reaction-front velocity

Direct Numerical Simulations (DNS) are performed to investigate the process of spontaneous ignition of hydrogen flames at laminar, turbulent, adiabatic and non-adiabatic conditions. Mixtures of hydrogen and vitiated air at temperatures representing gas-turbine reheat combustion are considered. Adiabatic spontaneous ignition processes are investigated first, providing a quantitative characterization of stable and unstable flames. Results indicate that, in hydrogen reheat combustion, compressibility effects play a key role in flame stability and that unstable ignition and combustion are consistently encountered for reactant temperatures close to the mixture’s characteristic crossover temperature. Furthermore, it is also found that the characterization of the adiabatic processes is also valid in the presence of non-adiabaticity due to wall heat-loss. Finally, a quantitative characterization of the instantaneous fuel consumption rate within the reaction front is obtained and of its ability, at auto-ignitive conditions, to advance against the approaching turbulent flow of the reactants, for a range of different turbulence intensities, temperatures and pressure levels.

42 ENGINEERING↗

Saturation of the kinetic ballooning instability due to the electron parallel nonlinearity

The electron parallel nonlinearity (EPN) is implemented in the gyrokinetic particle-in-cell turbulence code GEM [Y. Chen and S. E. Parker, J. Comp. Phys. 220, 839 (2007)]. Application to the Cyclone Base Case reveals a strong effect of EPN on the saturated heat transport above the kinetic ballooning mode (KBM) threshold. Evidence is provided to show that the strong effect is associated with the electron radial motion due to magnetic fluttering, which turns fine structures of the KBM eigenmode in radius into fine structures in velocity and increases the magnitude of the EPN term in the kinetic equation.

Gyrokinetic simulations↗

Presentation: Evaluating scaling effects of steam condensation physics-based model for small modular reactors safety

This study focused on assessing condensation heat transfer (CHT) models for enhancing the passive safety of small modular reactors (SMRs). It also encompassed a review of the theoretical and semi-theoretical CHT models and correlations. Condensation models such as the degradation factor, heat and mass transfer analogy (HMTA), and boundary layer models are used to study advanced nuclear reactor PCCS. The previous CHT models were developed for small tube geometries, which differ from the containment of SMR. This study examined the scaled relations of a physics-based CHT model for analyzing laminar and turbulent film condensation downward flow conditions inside scaled tube geometries.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

Progress from ASDEX Upgrade experiments in preparing the physics basis of ITER operation and DEMO scenario development

An overview of recent results obtained at the tokamak ASDEX Upgrade (AUG) is given. A work flow for predictive profile modelling of AUG discharges was established which is able to reproduce experimental H-mode plasma profiles based on engineering parameters only. In the plasma center, theoretical predictions on plasma current redistribution by a dynamo effect were confirmed experimentally. For core transport, the stabilizing effect of fast ion distributions on turbulent transport is shown to be important to explain the core isotope effect and improves the description of hollow low-Z impurity profiles. The L–H power threshold of hydrogen plasmas is not affected by small helium admixtures and it increases continuously from the deuterium to the hydrogen level when the hydrogen concentration is raised from 0 to 100%. One focus of recent campaigns was the search for a fusion relevant integrated plasma scenario without large edge localised modes (ELMs). Results from six different ELM-free confinement regimes are compared with respect to reactor relevance: ELM suppression by magnetic perturbation coils could be attributed to toroidally asymmetric turbulent fluctuations in the vicinity of the separatrix. Stable improved confinement mode plasma phases with a detached inner divertor were obtained using a feedback control of the plasma β. The enhanced D α H-mode regime was extended to higher heating power by feedback controlled radiative cooling with argon. The quasi-coherent exhaust regime was developed into an integrated scenario at high heating power and energy confinement, with a detached divertor and without large ELMs. Small ELMs close to the separatrix lead to peeling-ballooning stability and quasi continuous power exhaust. Helium beam density fluctuation measurements confirm that transport close to the separatrix is important to achieve the different ELM-free regimes. Based on separatrix plasma parameters and interchange-drift-Alfvén turbulence, an analytic model was derived that reproduces the experimentally found important operational boundaries of the density limit and between L- and H-mode confinement. Feedback control for the X-point radiator (XPR) position was established as an important element for divertor detachment control. Stable and detached ELM-free phases with H-mode confinement quality were obtained when the XPR was moved 10 cm above the X-point. Investigations of the plasma in the future flexible snow-flake divertor of AUG by means of first SOLPS-ITER simulations with drifts activated predict beneficial detachment properties and the activation of an additional strike point by the drifts.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Shear-free, inhomogeneous turbulence in a stably stratified fluid

High-resolution large eddy simulations are conducted of locally forced, shear-free turbulence in the presence of an initially sharp density interface. The simulations are reminiscent of oscillating grid turbulence experiments used to isolate the effect of turbulent diffusion and entrainment from background shear. By simulating such a flow we avoid common challenges of the experiments such as secondary-flow contamination due to sidewall effects and the inevitable interaction of the stratifying agent and forcing region. To address the latter concern, we add a heating term (potential energy sink) to the governing equations in the forcing layer, thereby preventing a heat flux through the source region. This modification sets up a continuous stratification in the mixed layer that is often assumed to be negligible in experiments. Despite this difference, we are able to make meaningful comparisons in terms of the overall entrainment rate, which varies as a power law with a turbulent Richardson number. Two exponents, −2 and −1, are measured depending on the definition of the Richardson number and entrainment rate used. The definition leading to −1 is consistent with most experiments, and we argue it is the superior choice if one is able to measure the relevant quantities. We also verify the self-similar scaling of turbulence velocity and length scales in the homogeneous fluid and propose ‘inner’ and ‘outer’ scalings for the stratified cases based on a local Froude number. The detailed scaling results are useful for turbulence model validation.

54 ENVIRONMENTAL SCIENCES↗

Free-spray characteristics and spray-wall interactions of methanol on a gasoline direct injector under flash-boiling and non-flash-boiling conditions

Methanol is considered a promising alternative fuel for internal combustion engines (ICEs) due to its high-octane number, fast laminar flame speed, and elevated latent heat of vaporization, all of which support higher compression ratios and improved thermal efficiency. However, its substantial latent heat of vaporization also poses cold-start challenges, such as misfire and fuel film deposition. This study aims to investigate methanol spray morphology and spray-wall interaction using the Spray M injector from the Engine Combustion Network within a constant-pressure flow vessel. A recently developed unified numerical framework capable of modeling both flash and non-flash boiling sprays is validated against experimental liquid volume fraction data acquired via 3-D computed tomography. Here, the results reveal that flash boiling significantly alters the spray morphology, leading to smaller droplets and spray collapse due to enhanced air-entrainment-induced turbulence. Quantitative agreement between experiments and simulations confirms this behavior. Coupled 0-D equilibrium and 3-D computational fluid dynamics analyses show that flash boiling accelerates evaporation and reduces fuel residence time, while non-flash conditions maintain a persistent liquid core more susceptible to wall wetting. Wall temperature diagnostics reveal that spray collapse alters heat transfer patterns by shifting cooling effects. Mixture fraction analysis indicates that evaporation is primarily governed by shear-layer turbulence, though deviations from adiabatic equilibrium mixing emerge under low-turbulence conditions. Finally, increasing fuel, ambient, and wall temperatures reduces wall wetting and film thickness, mitigating cold-start risks. These findings enhance the understanding of methanol sprays’ behavior and support its adoption as a viable, alternative fuel for ICEs.

Engine Combustion Network↗

Direct Numerical Simulation of Transitional and Turbulent Flows Over Multi-Scale Surface Roughness—Part II: The Effect of Roughness on the Performance of a High-Pressure Turbine Blade

Abstract Turbine blades generally present surface roughness introduced in the manufacturing process or caused by in-service degradation, which can have a significant impact on aero-thermal performance. A better understanding of the fundamental physical mechanisms arising from the interaction between the roughness and the turbine flow at engine-relevant conditions can provide insights for the design of blades with improved efficiency and longer operational life. To this end, a high-fidelity numerical framework combining a well-validated solver for direct numerical simulation and a second-order accurate immersed boundary method is employed to predict roughness-induced aero-thermal effects on an LS89 high-pressure turbine (HPT) blade at engine-relevant conditions. Different amplitudes and distributions of surface roughness are investigated and a reference smooth-blade simulation under the same flow conditions is conducted for comparison. Roughness of increasing amplitude progressively shifts the blade suction side boundary layer transition upstream, producing larger values of the turbulent kinetic energy and higher total wake losses. The on-surface data-capturing capabilities of the numerical framework provide direct measurements of the heat flux and the skin friction coefficient, hence offering quantitative information between the surface topology and engineering-relevant performance parameters. This work may provide a benchmark for future numerical studies of turbomachinery flows with roughness.

Engineering↗

LLNL FESP Theory Highlights: March 2024

Predictions of heat load widths λ q based on particle orbits alone are very pessimistic. This paper shows that pedestal peeling-ballooning (P-B) MHD turbulence broadens the stable SOL by the transport, or spreading, of fluctuation energy from the pedestal. λ q is seen to increase with Γ ε , the fluctuation energy density flux. We elucidate the fundamental physics of the spreading process. Γ ε increases with pressure fluctuation correlation length. P-B turbulence is seen to be especially effective at spreading, on account of its large effective mixing length. Spreading is shown to be a multiscale process, which is enhanced by the synergy of large and small-scale modes.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Accurate assessment of land–atmosphere coupling in climate models requires high-frequency data output

Land–atmosphere (L–A) interactions are important for understanding convective processes, climate feedbacks, the development and perpetuation of droughts, heatwaves, pluvials, and other land-centered climate anomalies. Local L–A coupling (LoCo) metrics capture relevant L–A processes, highlighting the impact of soil and vegetation states on surface flux partitioning and the impact of surface fluxes on boundary layer (BL) growth and development and the entrainment of air above the BL. A primary goal of the Climate Process Team in the Coupling Land and Atmospheric Subgrid Parameterizations (CLASP) project is parameterizing and characterizing the impact of subgrid heterogeneity in global and regional Earth system models (ESMs) to improve the connection between land and atmospheric states and processes. A critical step in achieving that aim is the incorporation of L–A metrics, especially LoCo metrics, into climate model diagnostic process streams. However, because land–atmosphere interactions span timescales of minutes (e.g., turbulent fluxes), hours (e.g., BL growth and decay), days (e.g., soil moisture memory), and seasons (e.g., variability in behavioral regimes between soil moisture and latent heat flux), with multiple processes of interest happening in different geographic regions at different times of year, there is not a single metric that captures all the modes, means, and methods of interaction between the land and the atmosphere. And while monthly means of most of the LoCo-relevant variables are routinely saved from ESM simulations, data storage constraints typically preclude routine archival of the hourly data that would enable the calculation of all LoCo metrics. Here, we outline a reasonable data request that would allow for adequate characterization of sub-daily coupling processes between the land and the atmosphere, preserving enough sub-daily output to describe, analyze, and better understand L–A coupling in modern climate models. A secondary request involves embedding calculations within the models to determine mean properties in and above the BL to further improve characterization of model behavior. Higher-frequency model output will (i) allow for more direct comparison with observational field campaigns on process-relevant timescales, (ii) enable demonstration of inter-model spread in L–A coupling processes, and (iii) aid in targeted identification of sources of deficiencies and opportunities for improvement of the models.

54 ENVIRONMENTAL SCIENCES↗

Machine Learning for Improving Surface-Layer-Flux Estimates

Abstract Flows in the atmospheric boundary layer are turbulent, characterized by a large Reynolds number, the existence of a roughness sublayer and the absence of a well-defined viscous layer. Exchanges with the surface are therefore dominated by turbulent fluxes. In numerical models for atmospheric flows, turbulent fluxes must be specified at the surface; however, surface fluxes are not known a priori and therefore must be parametrized. Atmospheric flow models, including global circulation, limited area models, and large-eddy simulation, employ Monin–Obukhov similarity theory (MOST) to parametrize surface fluxes. The MOST approach is a semi-empirical formulation that accounts for atmospheric stability effects through universal stability functions. The stability functions are determined based on limited observations using simple regression as a function of the non-dimensional stability parameter representing a ratio of distance from the surface and the Obukhov length scale (Obukhov in Trudy Inst Theor Geofiz AN SSSR 1:95–115, 1946), $$z/L$$ z / L . However, simple regression cannot capture the relationship between governing parameters and surface-layer structure under the wide range of conditions to which MOST is commonly applied. We therefore develop, train, and test two machine-learning models, an artificial neural network (ANN) and random forest (RF), to estimate surface fluxes of momentum, sensible heat, and moisture based on surface and near-surface observations. To train and test these machine-learning algorithms, we use several years of observations from the Cabauw mast in the Netherlands and from the National Oceanic and Atmospheric Administration’s Field Research Division tower in Idaho. The RF and ANN models outperform MOST. Even when we train the RF and ANN on one set of data and apply them to the second set, they provide more accurate estimates of all of the fluxes compared to MOST. Estimates of sensible heat and moisture fluxes are significantly improved, and model interpretability techniques highlight the logical physical relationships we expect in surface-layer processes.

Meteorology & Atmospheric Sciences↗

Cold plate with combined inclined impingement and ribbed channels

Heat transfer devices and methods for making the same that include a first enclosure having at least one inlet port; a second enclosure having a bottom plate and one or more dividing walls to establish channels, at least one internal surface of each channel having rib structures to create turbulence in a fluid flow; and a jet plate connecting the first enclosure and the second enclosure having impinging jets that convey fluid from the first enclosure to the channels, said impinging jets being set at an angular deviation from normal to cause local acceleration of fluid and to increase a local heat transfer rate.

42 ENGINEERING↗

Development of an Ultrahigh-bandwidth Phase Contrast Imaging System for detection of electron scale turbulence and Gigahertz Radio-Frequency Waves

The study of waves and turbulence is vital to the development of future reactor-grade plasma devices developed in the quest for fusion energy. These fluctuations are responsible for moving heat and particles across the magnetic field, and a predictive understanding of them is needed to achieve the density and temperature required to sustain a plasma fusion reaction. While many techniques have been developed for measuring waves and fluctuations, every measurement method has limitations. There are relatively few techniques for measuring very high frequency fluctuations, such as radio frequency waves injected to heat the plasma, unstable waves driven by suprathermal particles, or short wavelength electrostatic waves driven by electron temperature or density gradients. The present project builds upon the proven phase contrast imaging (PCI) technique to extend the response of the diagnostic by orders of magnitude in frequency and almost a factor of ten in spatial resolution. PCI provides a measurement of electron density based on small angle scattering of a CO-2 laser beam by using optical techniques to render a phase shift as an intensity change on a detector. Due to the telecommunications revolution, technological development by manufacturers has focused on components in the near -infrared, so that high- quality lasers and detectors at 1.55 µm are readily available. Shifting PCI design to a new, shorter wavelength has numerous advantages and challenges. Similar detector performance is available with room-temperature arrays with GHz bandwidth, while the detector arrays for 10.6 µm required liquid nitrogen cooling and were therefore limited to a bandwidth of about 1 MHz. Shifting to a shorter wavelength reduces the angle at which the laser beam scatters off of plasma waves, allowing more such scattered components to pass through the aperture of the vacuum vessel port and be collected by the PCI, which increases the spatial resolution. Concomitant with these benefits, various questions of performance arise. At shorter laser wave- length, the sensitivity to mirror and lens quality is increased, the contribution of the laser to the overall system noise is increased, and the sensitivity to vibrations is increased. The custom optical components at the heart of the PCI technique were required to be properly scaled for the shorter wavelength, so fabrication technologies needed to be explored. This project was designed to show that a low noise, high response PCI system at 1.55 µm could be constructed and operated, and then to quantify potential issues to allow extrapolation to a full-size production system providing physics measurements on a large plasma device. The first stage, producing the custom optical component called a Phase Plate, was successfully achieved using two methods. First, an easily reproduced masking and coating technique was able to produce good phase plates with the required parameters. Second, a nanofabrication technique was found to produce extremely high quality phase plates at a competitive price. The PCI constructed with the new phase plates and 1.55 µm laser was found to provide excellent wavelength measurements with the theoretically expected response. The sensitivity to optical surface quality was found to be in line with previous measurements at 10.6 µm. The observed signal-to-noise ratio was similar to the theoretically expected value. The effect of vibrations on PCI was studied with the first measurement of the effect of beam motion on PCI response and comparison to theory, allowing for a quantitative prediction of the effect of vibrations on a production PCI system and the requirements for improved beam stabilization. PCI is an extremely cost-effective method to provide a low noise, absolutely calibrated measurement of plasma fluctuations across a wide spatial scale. This project has shown that a 1.55 µm PCI using modern techniques and components is less expensive than the 10.6 µm PCI of a few years ago, with the largest savings in phase plate fabrication and the infrared detector array.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Impact of Forest Canopy Structure on Buoyant Plume Dynamics During Wildland Fires

Heterogeneous forest canopies can generate complex turbulent structures, but in the presence of a fire plume, these interactions are not fully understood. This study investigates the influence of forest canopy heterogeneity on buoyant plume dynamics resulting from surface thermal anomalies representing wildland fires, utilizing Large Eddy Simulation (LES). The Parallelized Large-Eddy Simulation Model (PALM) was employed to simulate six canopy configurations: no canopy, homogeneous canopy, external plume-edge canopy, internal plume-edge canopy, 100 m gap canopy, and 200 m gap canopy. Each configuration was analyzed with and without a static surface heat flux patch of 5000 W ∙ m -2 , resulting in a resting buoyant plume. Simulations were conducted under three crosswind speeds: 0, 5, and 10 m ∙ s -1 . Results show that canopy structure significantly modifies plume behavior, mean flow, and turbulent kinetic energy (TKE) budgets. Plume updraft speed and tilt varied with canopy configuration and crosswind speed. Horizontal pressure gradients associated with plume-atmosphere interaction were modified based on the canopy configuration, resulting in varying crosswind speed reductions at the plume region. Strong momentum absorption was observed above the canopy for the crosswind cases, with the greatest enhancement in the gap canopies. Momentum injection from below the canopy due to the heat source was also observed, resulting in plume structure modulation based on canopy configuration. TKE was found to be the largest in the gap canopy configurations. TKE budget analysis revealed that buoyant production dominated over shear production. At the center of the heat patch, the gap canopy configurations showed enhanced buoyancy within the gap. These results improve our knowledge of fire-canopy-atmosphere interactions that can inform fire models on the impacts of canopy heterogeneity on plume dynamics and ember ejections.

54 ENVIRONMENTAL SCIENCES↗

Observations of airflow around a supertall curved building and its impact on temperature and humidity in Houston's urban center

Characterization of realistically shaped skyscrapers embedded in non-uniform neighbourhoods experiencing intricate weather patterns remains inadequately investigated. Aiming to close this gap, the Center for Multiscale Applied Sensing team deployed its mobile observatory in the street canyons around the curved Wells Fargo Plaza skyscraper in downtown Houston, TX. Three deployments allowed airflow observations under different inflow wind and thermodynamic stability conditions. Doppler lidar measurements reveal that when inflow hits the curved wall of the skyscraper, perpendicular canyons experience similar vortex configurations creating two windward and two leeward circulations. Windsond measurements support that buoyancy within the deep street canyons can generate thermal updrafts as strong as 2 m s -1 which is sufficient to overturn the mechanical downwash under gentle wind conditions. Canyons experiencing venting during the daytime were observed to be more thermodynamically stable at night while thermodynamically stable canyons during the day were observed to be more thermodynamically unstable at night owing to the accumulation of heat near street-level. Fourier decomposition of the vertical velocity measurements shows that in all cases flow exhibited high Reynolds numbers and was composed of turbulent eddies of predominantly 6 and 15 min periods. Here, this observational dataset provides insights to assess wind load, pedestrian comfort, urban air mobility, and natural ventilation and may be used as a benchmark for numerical model and wind tunnel studies attempting to represent realistically complex urban conditions.

54 ENVIRONMENTAL SCIENCES↗

A learned closure method applied to phase mixing in a turbulent gradient-driven gyrokinetic system in simple geometry

We present a new method for formulating closures that learn from kinetic simulation data. We apply this method to phase mixing in a simple gyrokinetic turbulent system – temperature-gradient-driven turbulence in an unsheared slab. The closure, called the learned multi-mode (LMM) closure, is constructed by, first, extracting an optimal basis from a nonlinear kinetic simulation using singular value decomposition. Subsequent nonlinear fluid simulations are projected onto this basis and the results are used to formulate the closure. We compare the closure with other closures schemes over a broad range of the relevant two-dimensional parameter space (collisionality and gradient drive). We find that the turbulent kinetic system produces phase-mixing rates much lower than the linear expectations, which the LMM closure is capable of capturing. We also compare radial heat fluxes. A Hammett–Perkins closure, generalized to include collisional effects, is quite successful throughout the parameter space, producing ${\sim }14\,\%$ root-mean-square (r.m.s.) error. The LMM closure is also very effective: when trained at three (two) points (in a 35 point parameter grid), the LMM closure produces $8\,\%$ ( $12\,\%$ ) r.m.s. errors. The LMM procedure can be readily generalized to other closure problems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗