Search NASA⌕ Search

SEARCH · Search NASA

Results for “high speed imaging”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 325 records · Page 18

Ultra-Miniature Lidar Scanner for Launch Range Data Collection

The most critical component in lidar is its laser scanner, which delivers pulsed or CW laser to target with desirable field of view (FOV). Most existing lidars use a rotating or oscillating mirror for scanning, resulting in several drawbacks. A lidar scanning technology was developed that could achieve very high scanning speed, with an ultra-miniature size and much lighter weight. This technology promises at least a 10x performance improvement in these areas over existing lidar scanners. Features of the proposed ultra-miniature lidar scanner include the ability to make the entire scanner <2 mm in diameter; very high scanning speed (e.g. 5 - 20 kHz, in contrast to several hundred Hz in existing scanners); structure design to meet stringent requirements on size, weight, power, and compactness for various applications; and the scanning speed and FOV can be altered for obtaining high image resolutions of targeted areas and for diversified uses.

Geng, Jason↗

An Enduring Rapidly Moving Storm as a Guide to Saturn's Equatorial Jet's Complex Structure

Saturn has an intense and broad eastward equatorial jet with a complex three-dimensional structure mixed with time variability. The equatorial region experiences strong seasonal insolation variations enhanced by ring shadowing, and three of the six known giant planetary-scale storms have developed in it. These factors make Saturn's equator a natural laboratory to test models of jets in giant planets. Here we report on a bright equatorial atmospheric feature imaged in 2015 that moved steadily at a high speed of 450/ms not measured since 1980-1981 with other equatorial clouds moving within an ample range of velocities. Radiative transfer models show that these motions occur at three altitude levels within the upper haze and clouds. We find that the peak of the jet (latitudes 10degN to 10degS) suffers intense vertical shears reaching + 2.5/ms/km, two orders of magnitude higher than meridional shears, and temporal variability above 1 bar altitude level.

Saturn↗

Supersonic Disk Gap Band Parachute Performance in the Wake of a Viking-Type Aeroshell from Mach 2 to 2.5

Supersonic wind tunnel testing of 0.813 m diameter Disk-Gap-Band parachutes is being conducted in the NASA Glenn Research Center (GRC) 10' x 10' wind-tunnel. The tests are conducted in support of the Mars Science Laboratory Parachute Decelerator System development and qualification. Four percent of full-scale parachutes were constructed similarly to the flight-article in material and construction techniques. The parachutes are attached to a 4% scale MSL entry-vehicle to simulate the free-flight configuration. The parachutes are tested from Mach 2 to 2.5 over a Reynolds number (Re) range of 1 to 3 x 10(exp 6), representative of the MSL deployment envelope. Constrained and unconstrained test configurations are investigated to quantify the effects of parachute trim, suspension line interaction, and alignment with the capsule wake. The parachute is constrained horizontally through the vent region, to measure canopy breathing and wake interaction for fixed trim angles of 0 and 10 degrees from the velocity vector. In the unconstrained configuration the parachute is permitted to trim and cone, similar to the free-flight varying its alignment relative to the entry-vehicle wake. Test diagnostics were chosen to quantify parachute performance and to provide insight into the flow field structure. An in-line load cell provided measurement of unsteady and mean drag as a function of Mach and Re. High-speed shadowgraph video of the upstream parachute flow field was used to capture bow-shock motion and stand of distance. Particle image velocimetry of the upstream parachute flow field provides spatially and temporally resolved measurement velocity and turbulent statistics. Multiple high speed video views of targets placed in the interior of the canopy enable photo-grammetric measurement of the fabric motion in time and space from reflective. High speed video is also used to document the supersonic inflation and measure trim angle, projected area, and frequency of area oscillations.

EDL↗

Advanced communications technologies for image processing

It is essential for image analysts to have the capability to link to remote facilities as a means of accessing both data bases and high-speed processors. This can increase productivity through enhanced data access and minimization of delays. New technology is emerging to provide the high communication data rates needed in image processing. These developments include multi-user sharing of high bandwidth (60 megabits per second) Time Division Multiple Access (TDMA) satellite links, low-cost satellite ground stations, and high speed adaptive quadrature modems that allow 9600 bit per second communications over voice-grade telephone lines.

Likens, W. C.↗

Optical Imaging of Flow Pattern and Phantom

Time-resolved optical imaging technique has been used to image the spatial distribution of small droplets and jet sprays in a highly scattering environment. The snake and ballistic components of the transmitted pulse are less scattered, and contain direct information about the sample to facilitate image formation as opposed to the diffusive components which are due to multiple collisions as a light pulse propagates through a scattering medium. In a time-gated imaging scheme, these early-arriving, image-bearing components of the incident pulse are selected by opening a gate for an ultrashort period of time and a shadowgram image is detected. Using a single shot cooled CCD camera system, the formation of water droplets is monitored as a function of time. Picosecond time-gated image of drop in scattering cells, spray droplets as a function of let speed and gas pressure, and model calcification samples consisted of calcium carbonate particles of irregular shapes ranging in size from 0. 1 to 1.5 mm affixed to a microscope slide have been measured. Formation produced by an impinging jet will be further monitored using a CCD with 1 kHz framing illuminated with pulsed light. The desired image resolution of the fuel droplets is on the 20 pm scale using early light through a highly scattering medium. A 10(exp -6)m displacement from a jet spray with a flow speed of 100 m/sec introduced by the ns grating pulse used in the imaging is negligible. Early ballistic/snake light imaging offers nondestructive and noninvasive method to observe the spatial distribution of hidden objects inside a highly scattering environment for space, biomedical, and materials applications. In this paper, the techniques we will present are time-resolved K-F transillumination imaging and time-gated scattered light imaging. With a large dynamic range and high resolution, time-gated early light imaging has the potential for improving rocket/aircraft design by determining jets shape and particle sizes. Refinements to these techniques may enable drop size measurements in the highly scattering, optically dense region of multi-element rocket injectors. These types of measurements should greatly enhance the design of stable, and higher performing rocket engines.

Galland, Pierre A.↗

NASA 9-Point LDI Code Validation Experiment

This presentation highlights the experimental work to date to obtain validation data using a 9-point lean direct injector (LDI) in support of the National Combustion Code. The LDI is designed to supply fuel lean, Jet-A and air directly into the combustor such that the liquid fuel atomizes and mixes rapidly to produce short flame zones and produce low levels of oxides of nitrogen and CO. We present NOx and CO emission results from gas sample data that support that aspect of the design concept. We describe this injector and show high speed movies of selected operating points. We present image-based species maps of OH, fuel, CH and NO obtained using planar laser induced fluorescence and chemiluminescence. We also present preliminary 2-component, axial and vertical, velocity vectors of the air flow obtained using particle image velocimetry and of the fuel drops in a combusting case. For the same combusting case, we show preliminary 3-component velocity vectors obtained using a phase Doppler anemometer. For the fueled, combusting cases especially, we found optical density is a technical concern that must be addressed, but that in general, these preliminary results are promising. All optical-based results confirm that this injector produces short flames, typically on the order of 5- to-7-mm long at typical cruise and high power engine cycle conditions.

Hicks, Yolanda R.↗

Background-Oriented Schlieren used in a hypersonic inlet test at NASA GRC

Background Oriented Schlieren (BOS) is a derivative of the classical schlieren technology, which is used to visualize density gradients, such as shock wave structures in a wind tunnel. Changes in refractive index resulting from density gradients cause light rays to bend, resulting in apparent motion of a random background pattern. The apparent motion of the pattern is determined using cross-correlation algorithms (between no-flow and with-flow image pairs) producing a schlieren-like image. One advantage of BOS is its simplified setup which enables a larger field-of-view (FOV) than traditional schlieren systems. In the present study, BOS was implemented into the Combined Cycle Engine Large-Scale Inlet Mode Transition Experiment (CCE LIMX) in the 10x10 Supersonic Wind Tunnel at NASA Glenn Research Center. The model hardware for the CCE LIMX accommodates a fully integrated turbine based combined cycle propulsion system. To date, inlet mode transition between turbine and ramjet operation has been successfully demonstrated. High-speed BOS was used to visualize the behavior of the flow structures shock waves during unsteady inlet unstarts, a phenomenon known as buzz. Transient video images of inlet buzz were recorded for both the ramjet flow path (high speed inlet) and turbine flow path (low speed inlet). To understand the stability limits of the inlet, operation was pushed to the point of unstart and buzz. BOS was implemented in order to view both inlets simultaneously, since the required FOV was beyond the capability of the current traditional schlieren system. An example of BOS data (Images 1-6) capturing inlet buzz are presented.

Background-oriented schlieren↗

A versatile system for processing geostationary satellite data with run-time visualization capability

To better predict global climate change, scientists are developing climate models that require interdisciplinary and collaborative efforts in their building. We are currently involved in several such projects but will briefly discuss activities in support of two such complementary projects: the Atmospheric Radiation Measurement (ARM) program of the Department of Energy and Sequoia 2000, a joint venture of the University of California, the private sector, and government agencies. Our contribution to the ARM program is to investigate the role of clouds on the top of the atmosphere and on surface radiance fields through the data analysis of surface and satellite observations and complex modeling of the interaction of radiation with clouds. One of our first ARM research activities involves the computation of the broadband shortwave surface irradiance from satellite observations. Geostationary satellite images centered over the first ARM observation site are received hourly over the Internet network and processed in real time to compute hourly and daily composite shortwave irradiance fields. The images and the results are transferred via a high-speed network to the Sequoia 2000 storage facility in Berkeley, where they are archived These satellite-derived results are compared with the surface observations to evaluate the accuracy of the satellite estimate and the spatial representation of the surface observations. In developing the software involved in calculating the surface shortwave irradiance, we have produced an environment whereby we can easily modify and monitor the data processing as required. Through the principles of modular programming, we have developed software that is easily modified as new algorithms for computation are developed or input data availability changes. In addition, the software was designed so that it could be run from an interactive, icon-driven, graphical interface, TCL-TK, developed by Sequoia 2000 participants. In this way, the data flow can be interactively assessed and altered as needed. In this environment, the intermediate data processing 'images' can be viewed, enabling the investigator to easily monitor the various data processing steps as they progress. Additionally, this environment allows the rapid testing of new processing modules and allows their effects to be visually compared with previous results.

Landsfeld, M.↗

High Speed Digital Camera Technology Review

A High Speed Digital Camera Technology Review (HSD Review) is being conducted to evaluate the state-of-the-shelf in this rapidly progressing industry. Five HSD cameras supplied by four camera manufacturers participated in a Field Test during the Space Shuttle Discovery STS-128 launch. Each camera was also subjected to Bench Tests in the ASRC Imaging Development Laboratory. Evaluation of the data from the Field and Bench Tests is underway. Representatives from the imaging communities at NASA / KSC and the Optical Systems Group are participating as reviewers. A High Speed Digital Video Camera Draft Specification was updated to address Shuttle engineering imagery requirements based on findings from this HSD Review. This draft specification will serve as the template for a High Speed Digital Video Camera Specification to be developed for the wider OSG imaging community under OSG Task OS-33.

Clements, Sandra D.↗

Experimental Investigation of Pool Boiling Heat Transfer Enhancement in Microgravity in the Presence of Electric Fields

In boiling high heat fluxes are possible driven by relatively small temperature differences, which make its use increasingly attractive in aerospace applications. The objective of the research is to develop ways to overcome specific problems associated with boiling in the low gravity environment by substituting the buoyancy force with the electric force to enhance bubble removal from the heated surface. Previous studies indicate that in terrestrial applications nucleate boiling heat transfer can be increased by a factor of 50, as compared to values obtained for the same system without electric fields. The goal of our research is to experimentally explore the mechanisms responsible for EHD heat transfer enhancement in boiling in low gravity conditions, by visualizing the temperature distributions in the vicinity of the heated surface and around the bubble during boiling using real-time holographic interferometry (HI) combined with high-speed cinematography. In the first phase of the project the influence of the electric field on a single bubble is investigated. Pool boiling is simulated by injecting a single bubble through a nozzle into the subcooled liquid or into the thermal boundary layer developed along the flat heater surface. Since the exact location of bubble formation is known, the optical equipment can be aligned and focused accurately, which is an essential requirement for precision measurements of bubble shape, size and deformation, as well as the visualization of temperature fields by HI. The size of the bubble and the frequency of bubble departure can be controlled by suitable selection of nozzle diameter and mass flow rate of vapor. In this approach effects due to the presence of the electric field can be separated from effects caused by the temperature gradients in the thermal boundary layer. The influence of the thermal boundary layer can be investigated after activating the heater at a later stage of the research. For the visualization experiments a test cell was developed. All four vertical walls of the test cell are transparent, and they allow transillumination with laser light for visualization experiments by HI. The bottom electrode is a copper cylinder, which is electrically grounded. The copper block is heated with a resistive heater and it is equipped with 6 thermocouples that provide reference temperatures for the measurements with HI. The top electrode is a mesh electrode. Bubbles are injected with a syringe into the test cell through the bottom electrode. The working fluids presently used in the interferometric visualization experiments, water and PF 5052, satisfy requirements regarding thermophysical, optical and electrical properties. A 30kV power supply equipped with a voltmeter allows to apply the electric field to the electrodes during the experiments. The magnitude of the applied voltage can be adjusted either manually or through the LabVIEW data acquisition and control system connected to a PC. Temperatures of the heated block are recorded using type-T thermocouples, whose output is read by a data acquisition system. Images of the bubbles are recorded with 35mm photographic and 16mm high-speed cameras, scanned and analyzed using various software packages. Visualized temperature fields HI allows the visualization of temperature fields in the vicinity of bubbles during boiling in the form of fringes. Typical visualized temperature distributions around the air bubbles injected into the thermal boundary layer in PF5052 are shown. The temperature of the heated surface is 35 C. The temperature difference for a pair of fringes is approximately 0.05 C. The heat flux applied to the bottom surface is moderate, and the fringe patterns are regular. In the image a bubble penetrating the thermal boundary layer is visible. Because of the axial symmetry of the problem, simplified reconstruction techniques can be applied to recover the temperature field. The thermal plume developing above the heated surface for more intensive heating is shown. The temperature distribution in the liquid is clearly 3D, and tomographic techniques have to be applied to recover the temperature distribution in such a physical situation. A sequence of interferometric images showing the temperature distribution around the rising bubble, recorded with a high-speed camera is shown. Again, the temperature distribution is 3D, and a more complex approach to the evaluation, the tomographic reconstruction has to be taken. Measurement of the temperature distribution from the fringe pattern temperature distributions that yield important information regarding heat transfer are determined. Two algorithms that allow the quantitative evaluation of interferometric fringe patterns and the reconstruction of temperature fields during boiling have been developed at the Heat Transfer Laboratory of the Johns Hopkins University. In the first algorithm the bubble is assumed to be axially symmetrical, which significantly reduces the computational effort for quantifying the temperature distribution around the bubble. For this purpose the thermal boundary layer around the bubble is divided into equidistant concentric shells, and the refractive index is assumed to be constant in each of the shells. Since large temperature gradients are expected in the vicinity of the bubble during boiling, the deflection of the light beam cannot be neglected in boiling experiments. Since the exit angle of the light beam is known, this allows to account for the deflections and phase shifts outside the boundary layer (in the bulk fluid and in the windows of the test cell). Three dimensional temperature distributions in the vicinity of the bubble are reconstructed using tomographic techniques. In tomography, the measurement volume is sliced into 2D planes. In the present study these planes are parallel to the heated surface. The objective is to determine the values of the field parameter of interest in form of the field function in these 2D planes. The field parameter is the change of the refractive index of the liquid in the measurement volume caused by temperature changes. By superimposing data for many 2D planes recorded at the same time instant, the 3D temperature distribution in the measurement volume is recovered.

Herman, Cila↗

Observations of C-Band Brightness Temperatures and Ocean Surface Wind Speed and Rain Rate from the Hurricane Imaging Radiometer (HIRAD) during GRIP and HS3

HIRAD flew on high-altitude aircraft over Earl and Karl during NASA s GRIP (Genesis and Rapid Intensification Processes) campaign in August - September of 2010, and at the time of this writing plans to fly over Atlantic tropical cyclones in September of 2012 as part of the Hurricane and Severe Storm Sentinel (HS3) mission. HIRAD is a new C-band radiometer using a synthetic thinned array radiometer (STAR) technology to obtain cross-track resolution of approximately 3 degrees, out to approximately 60 degrees to each side of nadir. By obtaining measurements of emissions at 4, 5, 6, and 6.6 GHz, observations of ocean surface wind speed and rain rate can be retrieved. This technique has been used for many years by precursor instruments, including the Stepped Frequency Microwave Radiometer (SFMR), which has been flying on the NOAA and USAF hurricane reconnaissance aircraft for several years to obtain observations within a single footprint at nadir angle. Results from the flights during the GRIP and HS3 campaigns will be shown, including images of brightness temperatures, wind speed, and rain rate. Comparisons will be made with observations from other instruments on the campaigns, for which HIRAD observations are either directly comparable or are complementary. Features such as storm eye and eye-wall, location of storm wind and rain maxima, and indications of dynamical features such as the merging of a weaker outer wind/rain maximum with the main vortex may be seen in the data. Potential impacts on operational ocean surface wind analyses and on numerical weather forecasts will also be discussed.

Miller, Timothy L.↗

The Spectral Image Processing System (SIPS): Software for integrated analysis of AVIRIS data

The Spectral Image Processing System (SIPS) is a software package developed by the Center for the Study of Earth from Space (CSES) at the University of Colorado, Boulder, in response to a perceived need to provide integrated tools for analysis of imaging spectrometer data both spectrally and spatially. SIPS was specifically designed to deal with data from the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) and the High Resolution Imaging Spectrometer (HIRIS), but was tested with other datasets including the Geophysical and Environmental Research Imaging Spectrometer (GERIS), GEOSCAN images, and Landsat TM. SIPS was developed using the 'Interactive Data Language' (IDL). It takes advantage of high speed disk access and fast processors running under the UNIX operating system to provide rapid analysis of entire imaging spectrometer datasets. SIPS allows analysis of single or multiple imaging spectrometer data segments at full spatial and spectral resolution. It also allows visualization and interactive analysis of image cubes derived from quantitative analysis procedures such as absorption band characterization and spectral unmixing. SIPS consists of three modules: SIPS Utilities, SIPS_View, and SIPS Analysis. SIPS version 1.1 is described below.

Kruse, F. A.↗

The gravitational lens system Q0957+561 in the ultraviolet

Photometric and polarimetric observations of both images of the gravitationally lensed quasar Q0957+561 (z(sub em) = 1.41) were obtained in the UV in 1993 with the High Speed Photometer on board the Hubble Space Photometer on board the Hubble Space Telescope. The images exhibited no significant polarization in a bandpass centered on 2770 A (observer's frame); p less than or = 3.2 % (2 sigma upper limit) in each image. The ratio of the flux density in image A to that in image B in late 1993 had a constant valuee, 1.021 +/- 0.008, in four different UV bandpass between 1400 A and 3040 A observer's frame). These results are consistent with the prediction of the gravitation lens interpretation that the photometric ratio of the images measured simultaneously should be independent of frequency. Reprocessed archival spectra of the two images obtained between 1981 and 1983 by the International Ultraviolet Explorer (IUE) show that the photometric ratio of A to B varies between 0.96 and 2.0 in the Ly alpha emission line, and between 0.77 and 1.8 in the O VI lambda 1037 emission line (quasar rest frame). The photometric ratio of A to B at any single epoch is often significantly different in the two emission lines. Accepting the system as a gravitational lens implies that in the quasar the flux in the Ly alpha emsisson line can vary independently of the flux in the 0 IV emission line.

Dolan, J. F.↗

Digital video image system

Interactive recording and display device acts as very-high-speed data-input/output interface between analog (video) signals and standard digital-computer components. System can be used with various picture and memory sizes and can be controlled manually or by computer.

Neely, P. L.↗

Mars Balloon Flight Test Results

This paper describes a set of four Earth atmosphere flight test experiments on prototype helium superpressure balloons designed for Mars. Three of the experiments explored the problem of aerial deployment and inflation, using the cold, low density environment of the Earth's stratosphere at an altitude of 30-32 km as a proxy for the Martian atmosphere. Auxiliary carrier balloons were used in three of these test flights to lift the Mars balloon prototype and its supporting system from the ground to the stratosphere where the experiment was conducted. In each case, deployment and helium inflation was initiated after starting a parachute descent of the payload at 5 Pa dynamic pressure, thereby mimicking the conditions expected at Mars after atmospheric entry and high speed parachute deceleration. Upward and downward looking video cameras provided real time images from the flights, with additional data provided by onboard temperature, pressure and GPS sensors. One test of a 660 cc pumpkin balloon was highly successful, achieving deployment, inflation and separation of the balloon from the flight train at the end of inflation; however, some damage was incurred on the balloon during this process. Two flight tests of 12 m diameter spherical Mylar balloons were not successful, although some lessons were learned based on the failure analyses. The final flight experiment consisted of a ground-launched 12 m diameter spherical Mylar balloon that ascended to the designed 30.3 km altitude and successfully floated for 9.5 hours through full noontime daylight and into darkness, after which the telemetry system ran out of electrical power and tracking was lost. The altitude excursions for this last flight were +/-75 m peak to peak, indicating that the balloon was essentially leak free and functioning correctly. This provides substantial confidence that this balloon design will fly for days or weeks at Mars if it can be deployed and inflated without damage.

deployment↗

Optical data processing study

An onboard coherent optical data processing system was designed to handle large information content of imagery at high speeds, reduce redundancy in picture transmission, and detect specific image features. Theoretically derived Fourier transform characteristics for simple but representative two dimensional images served as a basis for predicting expected features of actual target images. Fourier transformation and spatial filtering of coherent optical images was accomplished theoretically and in the laboratory. The effect of various parameters such as optical aperature, incidence angles, the transparency assumption, the thin lens approximation on resolution, and performance of the system are predicted and tested.

Pinson, L. J.↗

Video transmission on ATM networks

The broadband integrated services digital network (B-ISDN) is expected to provide high-speed and flexible multimedia applications. Multimedia includes data, graphics, image, voice, and video. Asynchronous transfer mode (ATM) is the adopted transport techniques for B-ISDN and has the potential for providing a more efficient and integrated environment for multimedia. It is believed that most broadband applications will make heavy use of visual information. The prospect of wide spread use of image and video communication has led to interest in coding algorithms for reducing bandwidth requirements and improving image quality. The major results of a study on the bridging of network transmission performance and video coding are: Using two representative video sequences, several video source models are developed. The fitness of these models are validated through the use of statistical tests and network queuing performance. A dual leaky bucket algorithm is proposed as an effective network policing function. The concept of the dual leaky bucket algorithm can be applied to a prioritized coding approach to achieve transmission efficiency. A mapping of the performance/control parameters at the network level into equivalent parameters at the video coding level is developed. Based on that, a complete set of principles for the design of video codecs for network transmission is proposed.

Chen, Yun-Chung↗

Fundamental Studies on Two-Phase Gas-Liquid Flows Through Packed Beds in Microgravity

In the typical operation of a packed-bed reactor, gas and liquid flow simultaneously through a fixed bed of solid particles. Depending on the application, the particles can be of various shapes and sizes and provide for intimate contact and high rates of transport between the phases needed to sustain chemical or biological reactions. The packing may also serve as either a catalyst or as a support for growing biological material. NASA has flown two of these packed-bed systems in a microgravity environment with limited or no success. The goal of this research is to develop models (with scale-up capability) needed for the design of the physicochemical equipment to carry out these unit operations in microgravity. New insight will also lead to improvements in normal gravity operations. Our initial experiment was flown using an existing KC-135 two-phase flow rig with a modified test section. The test section is a clear polycarbonate rectangular column with a depth of 2.54 cm, a width of 5.08 cm, and 60 cm long. The column was randomly packed with spherical glass beads by slowly dropping the beads into the bed. Even though care was taken in handling the column after it was filled with packing, the alternating high and low gravity cycles with each parabola created a slightly tighter packed bed than is typically reported for this type. By the usual method of comparing the weight difference of a completely dry column versus a column filled with water, the void fraction was found to be .345 for both sizes of beads used. Five flush mounted differential pressure transducers are spaced at even intervals with the first location 4 cm from the inlet port and the subsequent pressure transducers spaced at 13 cm intervals along the column. Differential pressure data was acquired at 1000 Hz to adequately observe pulse formation and characteristics. Visual images of the flow were recorded using a high-speed SVHS system at 500 frames per second. Over 250 different test conditions were evaluated along with a companion set of tests in normal gravity. The flow rates, fluid properties and packing properties were selected to provide a range of several orders-of-magnitude for the important dimensionless parameters. Additional information is included in the original extended abstract.

Balakotaiah, Vemuri↗