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NASA NTRS2009

The Principle of Energetic Consistency

A basic result in estimation theory is that the minimum variance estimate of the dynamical state, given the observations, is the conditional mean estimate. This result holds independently of the specifics of any dynamical or observation nonlinearity or stochasticity, requiring only that the probability density function of the state, conditioned on the observations, has two moments. For nonlinear dynamics that conserve a total energy, this general result implies the principle of energetic consistency: if the dynamical variables are taken to be the natural energy variables, then the sum of the total energy of the conditional mean and the trace of the conditional covariance matrix (the total variance) is constant between observations. Ensemble Kalman filtering methods are designed to approximate the evolution of the conditional mean and covariance matrix. For them the principle of energetic consistency holds independently of ensemble size, even with covariance localization. However, full Kalman filter experiments with advection dynamics have shown that a small amount of numerical dissipation can cause a large, state-dependent loss of total variance, to the detriment of filter performance. The principle of energetic consistency offers a simple way to test whether this spurious loss of variance limits ensemble filter performance in full-blown applications. The classical second-moment closure (third-moment discard) equations also satisfy the principle of energetic consistency, independently of the rank of the conditional covariance matrix. Low-rank approximation of these equations offers an energetically consistent, computationally viable alternative to ensemble filtering. Current formulations of long-window, weak-constraint, four-dimensional variational methods are designed to approximate the conditional mode rather than the conditional mean. Thus they neglect the nonlinear bias term in the second-moment closure equation for the conditional mean. The principle of energetic consistency implies that, to precisely the extent that growing modes are important in data assimilation, this term is also important.

Cohn, Stephen E.↗

NASA NTRS2005

Probabilistic Analysis of Impact of Wake Vortices on Closely-Spaced Parallel Approaches

One of the primary constraints on the capacity of the nation's air transportation system is the landing capacity of its largest airports. Many airports with closely spaced parallel runways suffer a severe runway acceptance rate when the weather conditions do not allow full utilization of these parallel runways. The present requirement for simultaneous independent landings in Instrument Meteorological Conditions, IMC, is at least 4300 feet of lateral runway spacing (as close as 3000 feet for runways with a Precision Runway Monitor). Operations in Visual Meteorological Conditions, VMC, to Closely Spaced Parallel Approaches only require a lateral runway spacing greater than 750 feet. A study by Hardy and Lewis integrated and extended earlier studies and concepts in lateral traffic separation, longitudinal station keeping, wake prediction, wake display, and the concepts of R N P into a preliminary system concept for Closely Spaced Parallel Approaches in IMC. This system allows IMC airport acceptance rates to approach those for VMC. The system concept that was developed, presented traffic and wake information on the NAVigation Display, NAV, and developed operational procedures for a mix of conventional and Runway Independent Aircraft with different approach speeds to Closely Spaced Parallel Runways. This paper first describes some improvements made on the technology needed to better predict and formulate a probabilistic representation for the time-dependent motion and spreading of the hazardous region associated with the lift-generated vortex wakes of preceding aircraft. In this way, the time at which the vortex wakes of leading aircraft intrude into the airspace of adjacent flight-corridor/runway combinations can be more reliably predicted. Such a prediction is needed because it determines restraints to be placed on in-trail separation distances; or, the allowable time intervals between aircraft executing nearly simultaneous landings or takeoffs on very closely-spaced runways. Improved estimates of wake spreading are achieved by inclusion of representations in the equations for wake spreading due to ambient turbulence and due to the long-wave instability of a vortex pair. Wake motion and spreading due to the time-averaged wind and its variations with time, are retained. The more detailed representation of wake spreading presented here permits the development of probabilistically-based uncertainty estimates for wake spreading. Measurements needed within actual aircraft wake vortices to validate and support this analysis are also described. The second part of the paper uses the improvements in the accuracy of the location of wake vortices to extend the preliminary system concept for Closely Spaced Parallel Approaches described earlier with more robust operational procedures. Additionally, improvements in longitudinal station keeping, wake display, and risk assessment methodologies are incorporated and described.

Hardy, Gordon H.↗

NASA NTRS2010

Sorbent, Sublimation, and Icing Modeling Methods: Experimental Validation and Application to an Integrated MTSA Subassembly Thermal Model

This paper details the validation of modeling methods for the three core components of a Metabolic heat regenerated Temperature Swing Adsorption (MTSA) subassembly, developed for use in a Portable Life Support System (PLSS). The first core component in the subassembly is a sorbent bed, used to capture and reject metabolically produced carbon dioxide (CO2). The sorbent bed performance can be augmented with a temperature swing driven by a liquid CO2 (LCO2) sublimation heat exchanger (SHX) for cooling the sorbent bed, and a condensing, icing heat exchanger (CIHX) for warming the sorbent bed. As part of the overall MTSA effort, scaled design validation test articles for each of these three components have been independently tested in laboratory conditions. Previously described modeling methodologies developed for implementation in Thermal Desktop and SINDA/FLUINT are reviewed and updated, their application in test article models outlined, and the results of those model correlations relayed. Assessment of the applicability of each modeling methodology to the challenge of simulating the response of the test articles and their extensibility to a full scale integrated subassembly model is given. The independent verified and validated modeling methods are applied to the development of a MTSA subassembly prototype model and predictions of the subassembly performance are given. These models and modeling methodologies capture simulation of several challenging and novel physical phenomena in the Thermal Desktop and SINDA/FLUINT software suite. Novel methodologies include CO2 adsorption front tracking and associated thermal response in the sorbent bed, heat transfer associated with sublimation of entrained solid CO2 in the SHX, and water mass transfer in the form of ice as low as 210 K in the CIHX.

Bower, Chad↗

DOE OSTI2022

Vapor-Phase Aggregation of Cerium Oxide Nanoparticles in a Rapidly Cooling Plasma

Local conditions, such as temperature and oxygen availability, have a pronounced effect on the formation and evolution of fallout following a nuclear explosion. While the behavior of nuclear-relevant materials such as uranium has begun to be explored under a wider range of environments, little is known about the behavior of plutonium. Here, using cerium as a surrogate, we track the vapor-phase aggregation of cerium oxide nanoparticles created in a plasma flow reactor under conditions of controlled temperature at two different oxygen fugacities. In situ optical emission spectroscopy is used to measure the variations in the spectral intensity of atomic and molecular species with temperature and oxygen content. We find that the relative rate of gas-phase oxidation of cerium is highly dependent on both temperature and local redox conditions within the flow reactor, to the extent that doubling the oxygen availability effectively doubles the amount of vapor-phase cerium monoxide at high temperatures (>2000 K). Condensed cerium oxide nanoparticles are also collected and analyzed ex situ via transmission electron microscopy and grazing-incidence small-angle X-ray scattering to determine their elemental composition, crystal structure, and size distribution. The size and morphology of the condensed nanoparticles are independent of local redox conditions, forming the same crystal type with the same size distribution regardless of oxygen availability. Postcondensation particle evolution, however, is found to be predominantly driven by temperature, with the average particle size increasing as particles cool and subsequently aggregate. These results expand our understanding of the chemical and physical behavior of refractory oxides that form during the early stages of fallout formation.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS2002

Noachian Martian Volcanics a Water Source

H2O was supplied to the Noachian atmosphere by eruptions, or in association with large impacts. Most water outgassed into an extremely cold atmosphere, and condensate deposits were inevitable. High heat flow could lead to subglacial melting only if ice thicknesses were greater than 500-1000m, which is extremely unlikely. Subareal melting and flow is contingent upon temperatures periodically exceeding 273 K, and retarding evaporative loss of the flow. In still air, evaporation into a dry atmosphere is in the free convection regime, and a stream with 2 cu m/s discharge, flowing 1 m/s could persist for hundreds of days and cover distances greater than any valley reach. The zero-wind-shear condition is considered implausible however. We investigate the possibility that evaporation rates were suppressed because the atmosphere was regionally charged with H2O as it moved over snow/ice fields. Our initial concern is precipitation from volcanic plumes. A Kilauea-style eruption on the martian surface would cover a 10km circular deposit with 10cm of H2O, if all H2O could be precipitated near the vent. The characteristics of the eruption at the vent, (vent size, temperature, H2O content, etc.) are independent of the environmental conditions. The subsequent behavior of the plume, including precipitation of ash and H2O condensate depends strongly on the environment. Hence, the proximal fate of volcanic H2O is amenable to treatment in a model. A simple bulk thermodynamic model of the rise of an H2O plume through a stably stratified CO2 atmosphere, with only adiabatic cooling, produces runaway plume rise. A more complex treatment includes the effects of latent heat release, wind shear along the plume, divergence of ash and H2O, and will yield more realistic estimates of H2O transport in eruptive plumes. Results of these simulations will be presented.

Zent, A. P.↗

NASA NTRS2016

Development of the NASA MCAT Auxiliary Telescope for Orbital Debris Research

The National Aeronautical Space Administration has deployed the Meter Class Autonomous Telescope (MCAT) to Ascension Island with plans for it to become fully operational by summer 2016. This telescope will be providing data in support of research being conducted by the Orbital Debris Program Office at the Johnson Space Center. In addition to the main observatory, a smaller, auxiliary telescope is being deployed to the same location to augment and support observations generated by MCAT. It will provide near-simultaneous photometry and astrometry of debris objects, independent measurements of the seeing conditions, and offload low priority targets from MCAT's observing queue. Its hardware and software designs are presented here The National Aeronautical and Space Administration (NASA) has recently deployed the Meter Class Autonomous Telescope (MCAT) to Ascension Island. MCAT will provide NASA with a dedicated optical sensor for observations of orbital debris with the goal of statistically sampling the orbital and photometric characteristics of the population from low Earth to Geosynchronous orbits. Additionally, a small auxiliary telescope, co-located with MCAT, is being deployed to augment its observations by providing near-simultaneous photometry and astrometry, as well as offloading low priority targets from MCAT's observing queue. It will also serve to provide an independent measurement of the seeing conditions to help monitor the quality of the data being produced by the larger telescope. Comprised of off-the-shelf-components, the MCAT Auxiliary Telescope will have a 16-inch optical tube assembly, Sloan g'r'i'z' and Johnson/Cousins BVRI filters, and a fast tracking mount to help facilitate the tracking of objects in low Earth orbit. Tracking modes and tasking will be similar to MCAT except an emphasis will be placed on observations that provide more accurate initial orbit determination for the objects detected by MCAT. The near-simultaneous observations will also provide the opportunity for multi-filter color information of the debris objects to be obtained. Color information can further distinguish the individual objects within the population and provide insight into the reflectance properties of their surface material. The specific hardware, software, and tasking methodology of the MCAT Auxiliary Telescope is presented here..

Frith, James↗

DOE OSTI2021

Protective effects of non–catalytic proteins on endoglucanase activity at air and lignin interfaces

The manner in which added non-catalytic proteins during enzymatic hydrolysis of lignocellulosic substrates enhances hydrolysis mechanisms is not completely understood. Prior research has indicated that a reduction in the non-specific adsorption of enzymes on lignin, and deactivation of enzymes exposed to air–liquid interface provide rationale. This work investigated root causes including effects of the air–liquid interface on non-catalytic proteins, and effects of lignin on endoglucanase. Three different experimental designs and three variables (air–liquid interfacial area, the types of lignin (acid or enzymatic lignin), and the presence of non-enzymatic protein (bovine serum albumin [BSA] or soy proteins ) were used. The results showed that acid isolated lignin adsorbed almost all endoglucanase activity initially present in supernatant, independent of air interface conditions (25 or 250 ml flasks) with the presence of BSA preventing this effect. Endoglucanase lost 30%–50% of its activity due to an air–liquid interface in the presence of lignin while addition of non-enzymatic protein helped to preserve this enzyme's activity. Langmuir and Freundlich models applied to experimental data indicated that the adsorption increases with increasing temperature for both endoglucanase and BSA. Adsorption of the enzyme and protein were endothermic with an increase in entropy. Finally, these results, combined, show that hydrophobicity plays a strong role in the adsorption of both endoglucanase and BSA on lignin.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2023

Data-driven analysis and prediction of wastewater treatment plant performance: Insights and forecasting for sustainable operations

Here this study presents a comprehensive performance and forecasting analysis of the As-Samra wastewater treatment plant (WWTP) in Jordan, with two main objectives. Firstly, a thorough evaluation of the plant's performance is conducted. The analysis involves independently assessing historical operational conditions, plant production, and their statistical correlations using various statistical techniques. The second objective focuses on developing a data-driven forecasting approach to predict the plant's production one month in advance, using multiple machine learning models. The results highlight the effectiveness of principal component analysis (PCA) in simplifying operational data, revealing distinct operational clusters, and identifying seasonal production patterns while showing correlations between operational conditions and overall power production. The support vector machine (SVM) forecasting model emerged as the top performer, showcasing the potential of a hybrid forecasting approach. The findings offer valuable perspectives for enhancing operational efficiency, refining production planning, and ultimately improving the environmental impact of the plant.

42 ENGINEERING↗

DOE OSTI2020

Anomalous Anisotropic Nanoparticle Aggregation in Cu 2 (OH) 3 Br Gels

Aerogels are of interest for their ability to uniformly incorporate nanoscale features into macroscopic assemblies, which enabled applications that require low density, high surface area, and/or bicontinuous networks. The structure of the nanoporous network is intrinsically linked to the macroscopic properties of aerogels. Hence, control of this structure is of paramount importance. Small-angle X-ray scattering (SAXS) is used here to monitor nanoparticle aggregation in situ in Cu 2 (OH) 3 Br aerogels formed via epoxide-assisted gelation. Anomalous anisotropic aggregation is observed in the absence of templating agents and is attributed to the molecular structure of the inorganic nanoparticles themselves. This is a fundamental departure from the models currently used to describe traditional inorganic sol–gel chemistry where nanoparticles are believed to undergo isotropic diffusion- and/or kinetically limited aggregation. Time-resolved SAXS indicates that Cu 2 (OH) 3 Br nanoparticles nucleate rapidly from solution to form unbranched chain-like aggregates rather than branched mass-fractal aggregates. Sizes of primary particles (~1.5 nm) and the chain-like structure of their aggregates are independent of particle concentration (gel density), while rates of particle aggregation, gelation time, and aggregate size are strongly dependent upon particle concentration, which implies that the chemistry of particle formation and the physics of particle aggregation are independent processes. Because the conditions necessary for creating anisotropic structures are not unique to Cu 2 (OH) 3 Br, these results could provide insight into the structure and gelation mechanisms of other inorganic aerogels.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

Design of Twisted Two-Dimensional Heterostructures and Performance Regulation Descriptor for Electrocatalytic Ammonia Production from Nitric Oxide

The electrocatalytic reduction of nitric oxide (NO) to ammonia (NH 3 ) represents an attractive alternative for valorizing waste NO streams (NORR). However, discovering efficient catalysts for NO-to-NH 3 conversion remains challenging. We have designed metal-intercalated twisted graphene-BN heterostructures, in which metal atoms act as electron-transfer bridges. The twisted configuration facilitates cross-interface charge transfer, redistributing electrons from the graphene–metal interface to the metal–BN interface and BN surface. This electronic modulation enables boron atom adjacent to the metal center in BN to serve as active sites, promoting strong chemisorption and enhanced activation of NO. After high-throughput screening of the stability and NO capture ability of various transition metal-intercalated twisted heterostructures, we have investigated systematically the NORR pathways across 30 candidates. The results show that the rBN-Ti-Gθ and rBN-V-Gθ heterostructures exhibit exceptional NO-to-NH 3 catalytic performance under optimized twisting conditions. Additionally, using sure independence screening and sparsifying operator (SISSO) for model training, we propose a descriptor and establish a relationship between the twist angle and catalytic activity. This study bridges the gap in applying twisted heterostructures to NORR electrocatalysis and provides new insights and strategies for designing high-performance NORR catalysts.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

Violation of energy conditions and the gravitational radius of the proton

The energy-momentum tensor (EMT) of the proton encodes fundamental information about its mass, pressure, and shear distributions. Using recent lattice QCD data for the gravitational form factors, we show that the Breit-frame Wigner EMT may be of Hawking-Ellis type IV in the proton’s core. Such EMT violates all pointwise energy conditions and lacks a causal rest frame so that the usual mechanical picture fails at short distances. We define the —a new hadronic observable—marking the scale where the EMT becomes ordinary (type I) and the classical interpretation is restored. We also derive from the averaged null energy condition nonperturbative, model-independent quantum field theory constraints on gravitational form factors.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2021

Modelled land use and land cover change emissions – a spatio-temporal comparison of different approaches

Abstract. Quantifying the net carbon flux from land use and land cover changes (fLULCC) is critical for understanding the global carbon cycle and, hence, to support climate change mitigation. However, large-scale fLULCC is not directly measurable and has to be inferred from models instead, such as semi-empirical bookkeeping models and process-based dynamic global vegetation models (DGVMs). By definition, fLULCC estimates are not directly comparable between these two different model types. As an important example, DGVM-based fLULCC in the annual global carbon budgets is estimated under transient environmental forcing and includes the so-called loss of additional sink capacity (LASC). The LASC results from the impact of environmental changes on land carbon storage potential of managed land compared to potential vegetation and accumulates over time, which is not captured in bookkeeping models. The fLULCC from transient DGVM simulations, thus, strongly depends on the timing of land use and land cover changes mainly because LASC accumulation is cut off at the end of the simulated period. To estimate the LASC, the fLULCC from pre-industrial DGVM simulations, which is independent of changing environmental conditions, can be used. Additionally, DGVMs using constant present-day environmental forcing enable an approximation of bookkeeping estimates. Here, we analyse these three DGVM-derived fLULCC estimations (under transient, pre-industrial, and present-day forcing) for 12 models within 18 regions and quantify their differences as well as climate- and CO2-induced components and compare them to bookkeeping estimates. Averaged across the models, we find a global fLULCC (under transient conditions) of 2.0±0.6 PgC yr−1 for 2009–2018, of which ∼40 % are attributable to the LASC (0.8±0.3 PgC yr−1). From 1850 onward, the fLULCC accumulated to 189±56 PgC with 40±15 PgC from the LASC. Around 1960, the accumulating nature of the LASC causes global transient fLULCC estimates to exceed estimates under present-day conditions, despite generally increased carbon stocks in the latter. Regional hotspots of high cumulative and annual LASC values are found in the USA, China, Brazil, equatorial Africa, and Southeast Asia, mainly due to deforestation for cropland. Distinct negative LASC estimates in Europe (early reforestation) and from 2000 onward in the Ukraine (recultivation of post-Soviet abandoned agricultural land), indicate that fLULCC estimates in these regions are lower in transient DGVM compared to bookkeeping approaches. Our study unravels the strong dependence of fLULCC estimates on the time a certain land use and land cover change event happened to occur and on the chosen time period for the forcing of environmental conditions in the underlying simulations. We argue for an approach that provides an accounting of the fLULCC that is more robust against these choices, for example by estimating a mean DGVM ensemble fLULCC and LASC for a defined reference period and homogeneous environmental changes (CO2 only).

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS1976

Sound radiation from randomly vibrating beams of finite circular cross section

The radiation of sound from vibrating cylindrical beams is analyzed based on the frequency of the beam vibrations and the physical characteristics of the beam and its surroundings. A statistical analysis of random beam vibrations allows this result to be independent of the boundary conditions at the ends of the beam. The acoustic power radiated by the beam can be determined from a knowledge of the frequency band vibration data without a knowledge of the individual modal vibration amplitudes. A practical example of the usefulness of this technique is provided by the application of the theoretical calculations to the prediction of the octave band acoustic power output of the picking sticks of an automatic textile loom. Calculations are made of the expected octave band sound pressure levels based on measured acceleration data. These theoretical levels are subsequently compared with actual sound pressure level measurements of loom noise.

Sutterlin, M. W.↗

NASA NTRS1981

Data acquisition and analysis of range-finding systems for spacing construction

For space missions of future, completely autonomous robotic machines will be required to free astronauts from routine chores of equipment maintenance, servicing of faulty systems, etc. and to extend human capabilities in hazardous environments full of cosmic and other harmful radiations. In places of high radiation and uncontrollable ambient illuminations, T.V. camera based vision systems cannot work effectively. However, a vision system utilizing directly measured range information with a time of flight laser rangefinder, can successfully operate under these environments. Such a system will be independent of proper illumination conditions and the interfering effects of intense radiation of all kinds will be eliminated by the tuned input of the laser instrument. Processing the range data according to certain decision, stochastic estimation and heuristic schemes, the laser based vision system will recognize known objects and thus provide sufficient information to the robot's control system which can develop strategies for various objectives.

Shen, C. N.↗

NASA NTRS1981

The preferred-mode coherent structure in the near field of an axisymmetric jet with and without excitation

Visualization and phase-locked hot-wire measurements have been used to investigate the preferred-mode coherent structure in the near field of an axisymmetric jet for different Reynolds numbers and for the two limiting asymptotic states of the exit boundary layer: laminar and fully developed turbulent. Distributions of phase-averaged longitudinal and lateral velocities, coherent vorticity, coherent and incoherent turbulence Reynolds stresses, and coherent strain rate and production have been obtained over the spatial extent of the coherent structure for a Strouhal number of 0.3. It is found that these properties are independent of the initial conditions, but weakly dependent on the Reynolds number up to about 100,000. The naturally occurring coherent structure in the unexcited jet, educed via an optimized conditional sampling scheme, shows no clear dependence on the jet Reynolds number up to 800,000.

Hussain, A. K. M. F.↗

NASA NTRS1985

Long-time asymptotics of a system for plasma diffusion

A system of parabolic nonlinear equations that describe the diffusion of a fully collisional plasma across a strong magnetic field is discussed. It is demonstrated that the solution to this system tends to a time asymptotic state which is of space-time separable form, theta(t)f(x). Furthermore, f(x) is independent of the initial conditions and theta(t) depends slightly on the initial conditions. The rate of decay of the temporal part is governed by a nonlinear eigenvalue problem. Since the equations are considered in a bounded domain we are able to analyze the effect of boundary conditions on the evolution of the system. Additional effects on radiation, heating, and particle injection can also be accounted for. Essential differences between the behavior of a fully-coupled system and a scalar equation are observed.

Rosenau, P.↗

NASA NTRS1985

Activated recombinative desorption: A potential component in mechanisms of spacecraft glow

The concept of activated recombination of atomic species on surfaces can explain the production of vibrationally and translationally excited desorbed molecular species. Equilibrium statistical mechanics predicts that the molecular quantum state distributions of desorbing molecules is a function of surface temperature only when the adsorption probability is unity and independent of initial collision conditions. In most cases, the adsorption probability is dependent upon initial conditions such as collision energy or internal quantum state distribution of impinging molecules. From detailed balance, such dynamical behavior is reflected in the internal quantum state distribution of the desorbing molecule. This concept, activated recombinative desorption, may offer a common thread in proposed mechanisms of spacecraft glow. Using molecular beam techniques and equipment available at Los Alamos, which includes a high translational energy 0-atom beam source, mass spectrometric detection of desorbed species, chemiluminescence/laser induced fluorescence detection of electronic and vibrationally excited reaction products, and Auger detection of surface adsorbed reaction products, a fundamental study of the gas surface chemistry underlying the glow process is proposed.

Cross, J. B.↗

NASA NTRS1986

The nonlinear development of Gortler vortices in growing boundary layers

The development of Gortler vortices in boundary layers over curved walls in the nonlinear regime is investigated. The growth of the boundary layer makes a parallel flow analysis impossible except in the high wavenumber regime so in general the instability equations must be integrated numerically. Here the spanwise dependence of the basic flow is described using Fourier series expansion whilst the normal and streamwise variations are taken into account using finite differences. The calculations suggest that a given disturbance imposed at some position along the wall will eventually reach a local equilibrium state essentially independent of the initial conditions. In fact, the equilibrium state reached is qualitatively similar to the large amplitude high wave-number solution described asymptotically by Hall (1982). In general, it is found that the nonlinear interactions are dominated by a mean field type of interaction between the mean flow and the fundamental. Thus, even though higher harmonics of the fundamental are necessarily generated, most of the disturbance energy is confined to the mean flow correction and the fundamental. A major result of the calculations is finding that the downstream velocity field develops a strongly inflection character as the flow moves downstream. The latter result suggests that the major effect of Gortler vortices on boundary layers of practical importance might be to make them highly receptive to rapidly growing Rayleigh modes of instability.

Hall, Philip↗