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NASA NTRS2001

Method and Apparatus for Determining Position Using Global Positioning Satellites

A global positioning satellite receiver having an antenna for receiving a L1 signal from a satellite is presented. The L1 signal is processed by a preamplifier stage including a band pass filter and a low noise amplifier and output as a radio frequency (RF) signal. A mixer receives and de-spreads the RF signal in response to a pseudo-random noise code, i,e., Gold code, generated by an internal pseudo-random noise code generator. A microprocessor enters a code tracking loop, such that during the code tracking loop, it addresses the pseudorandom code generator to cause the pseudo-random code generator to sequentially output pseudo-random codes corresponding to satellite codes used to spread the L1 signal, until correlation occurs. When an output of the mixer is indicative of the occurrence of correlation between the RF signal and the generated pseudo-random codes, the microprocessor caters an operational state which slows the receiver code sequence to stay locked with the satellite cede sequence. The output of the mixer is provided to a detector which, in turn, controls certain routines of the microprocessor. The microprocessor will output pseudo range information according to an interrupt routine in response detection of correlation. The pseudo range information is to be telemetered to a ground station which determines the position of the global positioning satellite receiver.

John L Ward↗

NASA NTRS2002

Sensorimotor integration in human postural control

It is generally accepted that human bipedal upright stance is achieved by feedback mechanisms that generate an appropriate corrective torque based on body-sway motion detected primarily by visual, vestibular, and proprioceptive sensory systems. Because orientation information from the various senses is not always available (eyes closed) or accurate (compliant support surface), the postural control system must somehow adjust to maintain stance in a wide variety of environmental conditions. This is the sensorimotor integration problem that we investigated by evoking anterior-posterior (AP) body sway using pseudorandom rotation of the visual surround and/or support surface (amplitudes 0.5-8 degrees ) in both normal subjects and subjects with severe bilateral vestibular loss (VL). AP rotation of body center-of-mass (COM) was measured in response to six conditions offering different combinations of available sensory information. Stimulus-response data were analyzed using spectral analysis to compute transfer functions and coherence functions over a frequency range from 0.017 to 2.23 Hz. Stimulus-response data were quite linear for any given condition and amplitude. However, overall behavior in normal subjects was nonlinear because gain decreased and phase functions sometimes changed with increasing stimulus amplitude. "Sensory channel reweighting" could account for this nonlinear behavior with subjects showing increasing reliance on vestibular cues as stimulus amplitudes increased. VL subjects could not perform this reweighting, and their stimulus-response behavior remained quite linear. Transfer function curve fits based on a simple feedback control model provided estimates of postural stiffness, damping, and feedback time delay. There were only small changes in these parameters with increasing visual stimulus amplitude. However, stiffness increased as much as 60% with increasing support surface amplitude. To maintain postural stability and avoid resonant behavior, an increase in stiffness should be accompanied by a corresponding increase in damping. Increased damping was achieved primarily by decreasing the apparent time delay of feedback control rather than by changing the damping coefficient (i.e., corrective torque related to body-sway velocity). In normal subjects, stiffness and damping were highly correlated with body mass and moment of inertia, with stiffness always about 1/3 larger than necessary to resist the destabilizing torque due to gravity. The stiffness parameter in some VL subjects was larger compared with normal subjects, suggesting that they may use increased stiffness to help compensate for their loss. Overall results show that the simple act of standing quietly depends on a remarkably complex sensorimotor control system.

NASA Discipline Neuroscience↗

NASA NTRS2001

A vestibular phenotype for Waardenburg syndrome?

OBJECTIVE: To investigate vestibular abnormalities in subjects with Waardenburg syndrome. STUDY DESIGN: Retrospective record review. SETTING: Tertiary referral neurotology clinic. SUBJECTS: Twenty-two adult white subjects with clinical diagnosis of Waardenburg syndrome (10 type I and 12 type II). INTERVENTIONS: Evaluation for Waardenburg phenotype, history of vestibular and auditory symptoms, tests of vestibular and auditory function. MAIN OUTCOME MEASURES: Results of phenotyping, results of vestibular and auditory symptom review (history), results of vestibular and auditory function testing. RESULTS: Seventeen subjects were women, and 5 were men. Their ages ranged from 21 to 58 years (mean, 38 years). Sixteen of the 22 subjects sought treatment for vertigo, dizziness, or imbalance. For subjects with vestibular symptoms, the results of vestibuloocular tests (calorics, vestibular autorotation, and/or pseudorandom rotation) were abnormal in 77%, and the results of vestibulospinal function tests (computerized dynamic posturography, EquiTest) were abnormal in 57%, but there were no specific patterns of abnormality. Six had objective sensorineural hearing loss. Thirteen had an elevated summating/action potential (>0.40) on electrocochleography. All subjects except those with severe hearing loss (n = 3) had normal auditory brainstem response results. CONCLUSION: Patients with Waardenburg syndrome may experience primarily vestibular symptoms without hearing loss. Electrocochleography and vestibular function tests appear to be the most sensitive measures of otologic abnormalities in such patients.

NASA Discipline Neuroscience↗

NASA NTRS1994

Conventional and cross-correlation brain-stem auditory evoked responses in the white leghorn chick: rate manipulations

Rate-dependent changes in the chick brain-stem auditory evoked response (BAER) using conventional averaging and a cross-correlation technique were investigated. Five 15- to 19-day-old white leghorn chicks were anesthetized with Chloropent. In each chick, the left ear was acoustically stimulated. Electrical pulses of 0.1-ms duration were shaped, attenuated, and passed through a current driver to an Etymotic ER-2 which was sealed in the ear canal. Electrical activity from stainless-steel electrodes was amplified, filtered (300-3000 Hz) and digitized at 20 kHz. Click levels included 70 and 90 dB peSPL. In each animal, conventional BAERs were obtained at rates ranging from 5 to 90 Hz. BAERs were also obtained using a cross-correlation technique involving pseudorandom pulse sequences called maximum length sequences (MLSs). The minimum time between pulses, called the minimum pulse interval (MPI), ranged from 0.5 to 6 ms. Two BAERs were obtained for each condition. Dependent variables included the latency and amplitude of the cochlear microphonic (CM), wave 2 and wave 3. BAERs were observed in all chicks, for all level by rate combinations for both conventional and MLS BAERs. There was no effect of click level or rate on the latency of the CM. The latency of waves 2 and 3 increased with decreasing click level and increasing rate. CM amplitude decreased with decreasing click level, but was not influenced by click rate for the 70 dB peSPL condition. For the 90 dB peSPL click, CM amplitude was uninfluenced by click rate for conventional averaging. For MLS BAERs, CM amplitude was similar to conventional averaging for longer MPIs.(ABSTRACT TRUNCATED AT 250 WORDS).

NASA Discipline Neuroscience↗

NASA NTRS1992

System identification of perilymphatic fistula in an animal model

An acute animal model has been developed in the chinchilla for the study of perilymphatic fistulas. Micropunctures were made in three sites to simulate bony, round window, and oval window fistulas. The eye movements in response to pressure applied to the external auditory canal were recorded after micropuncture induction and in preoperative controls. The main pressure stimulus was a pseudorandom binary sequence (PRBS) that rapidly changed between plus and minus 200 mm of water. The PRBS stimulus, with its wide frequency bandwidth, produced responses clearly above the preoperative baseline in 78 percent of the runs. The response was better between 0.5 and 3.3 Hz than it was below 0.5 Hz. The direction of horizontal eye movement was toward the side of the fistula with positive pressure applied in 92 percent of the runs. Vertical eye movements were also observed. The ratio of vertical eye displacement to horizontal eye displacement depended upon the site of the micropuncture induction. Thus, such a ratio measurement may be clinically useful in the noninvasive localization of perilymphatic fistulas in humans.

Non-NASA Center↗

NASA NTRS1986

Dynamic posture analysis of Spacelab-1 crew members

Dynamic posture testing was conducted on the science crew of the Spacelab-1 mission on a single axis linear motion platform. Tests took place in pre- and post-flight sessions lasting approximately 20 min each. The pre-flight tests were widely spaced over the several months prior to the mission while the post-flight tests were conducted over the first, second, fourth, and sixth days after landing. Two of the crew members were also tested on the day of landing. Consistent with previous postural testing conducted on flight crews, these crew members were able to complete simple postural tasks to an acceptable level even in the first few hours after landing. Our tests were designed to induce dynamic postural responses using a variety of stimuli and from these responses, evaluate subtle changes in the postural control system which had occurred over the duration of the flight. Periodic sampling post-flight allowed us to observe the time course of readaptation to terrestrial life. Our observations of hip and shoulder position, when subjected to careful analysis, indicated modification of the postural response from pre- to post-flight and that demonstrable adjustments in the dynamic control of their postural systems were taking place in the first few days after flight. For transient stimuli where the platform on which they were asked to stand quickly moved a few centimeters fore or aft then stopped, ballistic or open loop 'programs' would closely characterize the response. During these responses the desired target position was not always achieved and of equal importance not always properly corrected some 15 seconds after the platform ceased to move. The persistent observation was that the subjects had a much stronger dependence on visual stabilization post-flight than pre-flight. This was best illustrated by a slow or only partial recovery to an upward posture after a transient base-of-support movement with eyes open. Postural responses to persistent wideband pseudorandom base-of-support translation were modeled as time invarient linear systems arrived at by Kalman adaptive filter techniques. Derived model parameters such as damping factor and fundamental frequency of the closed loop system showed significant modification between pre- and post-flight. This phenomenon is best characterized by movement of the poles toward increasing stability. While pre-flight data tended to show shoulders and hips moving in phase with each other, post-flight data showed a more disjoint behavior.(ABSTRACT TRUNCATED AT 400 WORDS).

Non-NASA Center↗

NASA NTRS2004

Examination of Surface Roughness on Light Scattering by Long Ice Columns by Use of a Two-Dimensional Finite-Difference Time-Domain Algorithm

Natural particles such as ice crystals in cirrus clouds generally are not pristine but have additional micro-roughness on their surfaces. A two-dimensional finite-difference time-domain (FDTD) program with a perfectly matched layer absorbing boundary condition is developed to calculate the effect of surface roughness on light scattering by long ice columns. When we use a spatial cell size of 1/120 incident wavelength for ice circular cylinders with size parameters of 6 and 24 at wavelengths of 0.55 and 10.8 mum, respectively, the errors in the FDTD results in the extinction, scattering, and absorption efficiencies are smaller than similar to 0.5%. The errors in the FDTD results in the asymmetry factor are smaller than similar to 0.05%. The errors in the FDTD results in the phase-matrix elements are smaller than similar to 5%. By adding a pseudorandom change as great as 10% of the radius of a cylinder, we calculate the scattering properties of randomly oriented rough-surfaced ice columns. We conclude that, although the effect of small surface roughness on light scattering is negligible, the scattering phase-matrix elements change significantly for particles with large surface roughness. The roughness on the particle surface can make the conventional phase function smooth. The most significant effect of the surface roughness is the decay of polarization of the scattered light.

Sun, W.↗

NASA NTRS1995

Cascade Optimization Strategy Maximizes Thrust for High-Speed Civil Transport Propulsion System Concept

The design of a High-Speed Civil Transport (HSCT) air-breathing propulsion system for multimission, variable-cycle operations was successfully optimized through a soft coupling of the engine performance analyzer NASA Engine Performance Program (NEPP) to a multidisciplinary optimization tool COMETBOARDS that was developed at the NASA Lewis Research Center. The design optimization of this engine was cast as a nonlinear optimization problem, with engine thrust as the merit function and the bypass ratios, r-values of fans, fuel flow, and other factors as important active design variables. Constraints were specified on factors including the maximum speed of the compressors, the positive surge margins for the compressors with specified safety factors, the discharge temperature, the pressure ratios, and the mixer extreme Mach number. Solving the problem by using the most reliable optimization algorithm available in COMETBOARDS would provide feasible optimum results only for a portion of the aircraft flight regime because of the large number of mission points (defined by altitudes, Mach numbers, flow rates, and other factors), diverse constraint types, and overall poor conditioning of the design space. Only the cascade optimization strategy of COMETBOARDS, which was devised especially for difficult multidisciplinary applications, could successfully solve a number of engine design problems for their flight regimes. Furthermore, the cascade strategy converged to the same global optimum solution even when it was initiated from different design points. Multiple optimizers in a specified sequence, pseudorandom damping, and reduction of the design space distortion via a global scaling scheme are some of the key features of the cascade strategy. HSCT engine concept, optimized solution for HSCT engine concept. A COMETBOARDS solution for an HSCT engine (Mach-2.4 mixed-flow turbofan) along with its configuration is shown. The optimum thrust is normalized with respect to NEPP results. COMETBOARDS added value in the design optimization of the HSCT engine.

Source record↗

NASA NTRS2007

Influence of Electrotactile Tongue Feedback on Controlling Upright Stance during Rotational and/or Translational Sway-referencing with Galvanic Vestibular Stimulation

Integration of multi-sensory inputs to detect tilts relative to gravity is critical for sensorimotor control of upright orientation. Displaying body orientation using electrotactile feedback to the tongue has been developed by Bach-y-Rita and colleagues as a sensory aid to maintain upright stance with impaired vestibular feedback. MacDougall et al. (2006) recently demonstrated that unpredictably varying Galvanic vestibular stimulation (GVS) significantly increased anterior-posterior (AP) sway during rotational sway referencing with eyes closed. The purpose of this study was to assess the influence of electrotactile feedback on postural control performance with pseudorandom binaural bipolar GVS. Postural equilibrium was measured with a computerized hydraulic platform in 10 healthy adults (6M, 4F, 24-65 y). Tactile feedback (TF) of pitch and roll body orientation was derived from a two-axis linear accelerometer mounted on a torso belt and displayed on a 144-point electrotactile array held against the anterior dorsal tongue (BrainPort, Wicab, Inc., Middleton, WI). Subjects were trained to use TF by voluntarily swaying to draw figures on their tongue, both with and without GVS. Subjects were required to keep the intraoral display in their mouths on all trials, including those that did not provide TF. Subjects performed 24 randomized trials (20 s duration with eyes closed) including four support surface conditions (fixed, rotational sway-referenced, translating the support surface proportional to AP sway, and combined rotational-translational sway-referencing), each repeated twice with and without GVS, and with combined GVS and TF. Postural performance was assessed using deviations from upright (peak-to-peak and RMS sway) and convergence toward stability limits (time and distance to base of support boundaries). Postural stability was impaired with GVS in all platform conditions, with larger decrements in performance during trials with rotation sway-referencing. Electrotactile feedback improved performance with GVS toward non-GVS levels, again with the greatest improvement during trials with rotation sway-referencing. These results demonstrate the effectiveness of tongue electrotactile feedback in providing sensory substitution to maintain postural stability with distorted vestibular input.

Wood, Scott J.↗

NASA NTRS2010

Motion Perception and Manual Control Performance During Passive Tilt and Translation Following Space Flight

This joint ESA-NASA study is examining changes in motion perception following Space Shuttle flights and the operational implications of post-flight tilt-translation ambiguity for manual control performance. Vibrotactile feedback of tilt orientation is also being evaluated as a countermeasure to improve performance during a closed-loop nulling task. METHODS. Data has been collected on 5 astronaut subjects during 3 preflight sessions and during the first 8 days after Shuttle landings. Variable radius centrifugation (216 deg/s) combined with body translation (12-22 cm, peak-to-peak) is utilized to elicit roll-tilt perception (equivalent to 20 deg, peak-to-peak). A forward-backward moving sled (24-390 cm, peak-to-peak) with or without chair tilting in pitch is utilized to elicit pitch tilt perception (equivalent to 20 deg, peak-to-peak). These combinations are elicited at 0.15, 0.3, and 0.6 Hz for evaluating the effect of motion frequency on tilt-translation ambiguity. In both devices, a closed-loop nulling task is also performed during pseudorandom motion with and without vibrotactile feedback of tilt. All tests are performed in complete darkness. PRELIMINARY RESULTS. Data collection is currently ongoing. Results to date suggest there is a trend for translation motion perception to be increased at the low and medium frequencies on landing day compared to pre-flight. Manual control performance is improved with vibrotactile feedback. DISCUSSION. The results of this study indicate that post-flight recovery of motion perception and manual control performance is complete within 8 days following short-duration space missions. Vibrotactile feedback of tilt improves manual control performance both before and after flight.

Clement, Gilles↗

NASA NTRS2006

NASA Tech Briefs, November 2006

Topics include: Simulator for Testing Spacecraft Separation Devices; Apparatus for Hot Impact Testing of Material Specimens; Instrument for Aircraft-Icing and Cloud-Physics Measurements; Advances in Measurement of Skin Friction in Airflow; Improved Apparatus for Testing Monoball Bearings; High-Speed Laser Scanner Maps a Surface in Three Dimensions; Electro-Optical Imaging Fourier-Transform Spectrometer; Infrared Instrument for Detecting Hydrogen Fires; Modified Coaxial Probe Feeds for Layered Antennas; Detecting Negative Obstacles by Use of Radar; Cryogenic Pound Circuits for Cryogenic Sapphire Oscillators; PixelLearn; New Software for Predicting Charging of Spacecraft; Conversion Between Osculating and Mean Orbital Elements; Generating a 2D Representation of a Complex Data Structure; Making Activated Carbon by Wet Pressurized Pyrolysis; Composite Solid Electrolyte Containing Li+- Conducting Fibers; Electrically Conductive Anodized Aluminum Surfaces; Rapid-Chill Cryogenic Coaxial Direct-Acting Solenoid Valve; Variable-Tension-Cord Suspension/Vibration- Isolation System; Techniques for Connecting Superconducting Thin Films; Versatile Friction Stir Welding/Friction Plug Welding System; Thermal Spore Exposure Vessels; Enumerating Spore-Forming Bacteria Airborne with Particles; Miniature Oxidizer Ionizer for a Fuel Cell; Miniature Ion-Array Spectrometer; Promoted-Combustion Chamber with Induction Heating Coil; Miniature Ion-Mobility Spectrometer; Mixed-Salt/Ester Electrolytes for Low-Temperature Li+ Cells; Miniature Free-Space Electrostatic Ion Thrusters; Miniature Bipolar Electrostatic Ion Thruster; Holographic Plossl Retroreflectors; Miniature Electrostatic Ion Thruster With Magnet; Using Apex To Construct CPM-GOMS Models; Sequence Detection for PPM Optical Communication With ISI; Algorithm for Rapid Searching Among Star-Catalog Entries; Expectation-Based Control of Noise and Chaos; Radio Heating of Lunar Soil to Release Gases; Using Electrostriction to Manipulate Ullage in Microgravity; Equations for Scoring Rules When Data Are Missing; Insulating Material for Next-Generation Spacecraft; and Pseudorandom Switching for Adding Radar to the AFF Sensor.

Source record↗

NASA NTRS2005

Coherent Laser Instrument Would Measure Range and Velocity

A proposed instrument would project a narrow laser beam that would be frequency-modulated with a pseudorandom noise (PN) code for simultaneous measurement of range and velocity along the beam. The instrument performs these functions in a low mass, power, and volume package using a novel combination of established techniques. Originally intended as a low resource- footprint guidance sensor for descent and landing of small spacecraft onto Mars or small bodies (e.g., asteroids), the basic instrument concept also lends itself well to a similar application guiding aircraft (especially, small unmanned aircraft), and to such other applications as ranging of topographical features and measuring velocities of airborne light-scattering particles as wind indicators. Several key features of the instrument s design contribute to its favorable performance and resource-consumption characteristics. A laser beam is intrinsically much narrower (for the same exit aperture telescope or antenna) than a radar beam, eliminating the need to correct for the effect of sloping terrain over the beam width, as is the case with radar. Furthermore, the use of continuous-wave (CW), erbium-doped fiber lasers with excellent spectral purity (narrow line width) permits greater velocity resolution, while reducing the laser s power requirement compared to a more typical pulsed solid-state laser. The use of CW also takes proper advantage of the increased sensitivity of coherent detection, necessary in the first place for direct measurement of velocity using the Doppler effect. However, measuring range with a CW beam requires modulation to "tag" portions of it for time-of-flight determination; typically, the modulation consists of a PN code. A novel element of the instrument s design is the use of frequency modulation (FM) to accomplish both the PN-modulation and the Doppler-bias frequency shift necessary for signed velocity measurements. This permits the use of a single low-power waveguide electrooptic phase modulator, while simultaneously mitigating the effects of speckle as a noise source in the coherent detection.

Chang, Daniel↗

NASA NTRS2011

A Low Cost Remote Sensing System Using PC and Stereo Equipment

A system using a personal computer, speaker, and a microphone is used to detect objects, and make crude measurements using a carrier modulated by a pseudorandom noise (PN) code. This system can be constructed using a personal computer and audio equipment commonly found in the laboratory or at home, or more sophisticated equipment that can be purchased at reasonable cost. We demonstrate its value as an instructional tool for teaching concepts of remote sensing and digital signal processing.

Campbell, Joel F.↗

NASA NTRS2013

Time Shifted PN Codes for CW Lidar, Radar, and Sonar

A continuous wave Light Detection and Ranging (CW LiDAR) system utilizes two or more laser frequencies and time or range shifted pseudorandom noise (PN) codes to discriminate between the laser frequencies. The performance of these codes can be improved by subtracting out the bias before processing. The CW LiDAR system may be mounted to an artificial satellite orbiting the earth, and the relative strength of the return signal for each frequency can be utilized to determine the concentration of selected gases or other substances in the atmosphere.

Campbell, Joel F.↗

NASA NTRS2015

Modulated Sine Waves for Differential Absorption Measurements Using a CW Laser System

Campbell, Joel F.↗

NASA NTRSDate not supplied

Transition Induced by an Egg-Crate Roughness on a Flat Plate in Supersonic Flow

Measurements were made in the wake of a roughness patch with a hot wire to determine mode shapes and frequencies of the dominant instabilities. The egg-crate roughness pattern is an analytic function described by a sinusoidal equation, similar to arrays of discrete elements that are positioned in a spanwise and streamwise grid, which were considered in previous computations. This is an intermediate configuration toward understanding the underlying physics of pseudorandom distributed roughness. At a freestream unit Reynolds number of10.8×106/m, the boundary layer did not transition within the measurement region. The largest amplitude boundary layer instability was centered near 53 kHz and appeared to be similar to an antisymmetric mode centered behind the region downstream of all of the peaks in the egg-crate roughness. This mode was similar to a mode found to have the highest N factor in previous computational studies. At a freestream unit Reynolds number of12.6×106/m, the boundary layer transitioned to turbulence. The dominant instability was centered near 74 kHz and grew linearly before saturating and breaking down. The mode shape of the 74-kHz mode was again an antisymmetric mode centered behind all of the peaks in the egg-crate roughness.

Amanda Chou↗

NASA NTRS2020

Adaptive Stress Testing: Finding Likely Failure Events with Reinforcement Learning

Finding the most likely path to a set of failure states is important to the analysis of safety-critical systems that operate over a sequence of time steps, such as aircraft collision avoidance systems and autonomous cars. In many applications such as autonomous driving, failures cannot be completely eliminated due to the complex stochastic environment in which the system operates.As a result, safety validation is not only concerned about whether a failure can occur, but also discovering which failures are most likely to occur. This article presents adaptive stress testing (AST), a framework for finding the most likely path to a failure event in simulation. We consider a general black box setting for partially observable and continuous-valued systems operating in an environment with stochastic disturbances. We formulate the problem as a Markov decision process and use reinforcement learning to optimize it. The approach is simulation-based and does not require internal knowledge of the system, making it suitable for black-box testing of large systems. We present different formulations depending on whether the state is fully observable or partially observable. In the latter case, we present a modified Monte Carlo tree search algorithm that only requires access to the pseudorandom number generator of the simulator to overcome partial observability. We also present an extension of the framework, called differential adaptive stress testing (DAST), that can find failures that occur in one system but not in another. This type of differential analysis is useful in applications such as regression testing, where we are concerned with finding areas of relative weakness compared to a baseline. We demonstrate the effectiveness of the approach on an aircraft collision avoidance application, where a prototype aircraft collision avoidance system is stress tested to find the most likely scenarios of near mid-air collision.

Verification and Validation↗

NASA NTRSDate not supplied

Instabilities in the Wake of Roughness on a Flat Plate in a Quiet Supersonic Tunnel

Roughness-induced transition is an unavoidable reality in practical high-speed vehicles. Typical prediction of transition due to roughness include algebraic correlations and, more recently, semi-empirical methods. In the NASA Langley Research Center Supersonic Low Disturbance Tunnel, a Mach 3.5 quiet tunnel, several transition experiments have been performed in the past decade to better understand the mechanisms by which small roughness causes transition in a supersonic boundary layer. The study started first with isolated roughness elements of different planforms and shapes and progressed to increasingly more complicated geometries before arriving at a pseudorandom roughness, defined by an analytic function. Concurrent computational efforts progressed with these studies as well, starting with the use of linear stability theory and progressing to the use of harmonic linearized Navier-Stokes to predict the growth of boundary layer stabilities.

Amanda Chou↗