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NASA NTRS1999

Compositional Analysis of Martian Soil: Synergism of APEX and MECA Experiments on MPS 2001

The APEX (ATHENA Precursor Experiment) payload for the Mars 2001 mission will analyze soil and dust with a multispectral panoramic imager and an emission spectrometer on a mast on the lander, a Moessbauer spectrometer on the lander robotic arm (RA), and APXS measurements on the Marie Curie rover. These analytical methods will provide data on elemental abundances and mineralogy. The MECA payload on the lander will apply microscopy, AFM, wet chemistry, adhesive substrates, and electrometry to determine the shape and size of particles in the soil and dust, the presence of toxic substances, and electrostatic, magnetic, and hardness qualities of particles. The two experiments will complement one another through several interactions: (1) The panoramic imager provides the geological setting in which both APEX and MECA samples are acquired, (2) The RA provides samples to MECA from the surface and subsurface and will permit APEX analytical tools access to materials below the inunediate surface, (3) Comparisons can be made between elemental analyses of the Moessbauer, IR, APXS on APEX and the wet chemistry of MECA which will define trace elements (ionic species in solution) and soil redox potential and conductivity. (4) APEX bulk compositional measurements will place MECA trace measurements in context, and similarly, MECA microscopy will provide particle size data that may correlate with compositional differences determined by the APEX instruments. Additionally, lithic fragments viewed by the NMCA microscope station should correlate with mineral/rock species inferred by APEX data, (5) If APEX instruments detect quartz for example, the scratch plates of the N4ECA microscope stage will define if a mineral of this hardness is registered during abrasion tests. This is by no means an exhaustive list of potential interactions, but it is clear that both the sheer number of analytical techniques and their complementarity should provide an analytically powerful capability for both planetary and BEDS communities.

Arvidson, R.↗

NASA NTRS1999

Compositional Analysis of Martian Soil: Synergism of APEX and MECA Experiments on MPS 2001

The APEX (ATHENA Precursor Experiment) payload for the Mars 2001 mission will analyze soil and dust with a multispectral panoramic imager and an emission spectrometer on a mast on the lander, a Moessbauer spectrometer on the lander robotic arm (RA), and APXS measurements on the Marie Curie rover. These analytical methods will provide data on elemental abundances and mineralogy. The MECA payload on the lander will apply microscopy, AFM, wet chemistry, adhesive substrates, and electrometry to determine the shape and size of particles in the soil and dust, the presence of toxic substances, and electrostatic, magnetic, and hardness qualities of particles. The two experiments will complement one another through several interactions: (1) The panoramic imager provides the geological setting in which both APEX and MECA samples are acquired, (2) The RA provides samples to MECA from the surface and subsurface and will permit APEX analytical tools access to materials below the immediate surface, (3) Comparisons can be made between elemental analyses of the Moessbauer, IR, APXS on APEX and the wet chemistry of MECA which will define trace elements (ionic species in solution) and soil redox potential and conductivity. (4) APEX bulk compositional measurements will place MECA trace measurements in context, and similarly, MECA microscopy will provide particle size data that may correlate with compositional differences determined by the APEX instruments. Additionally, lithic fragments viewed by the MECA microscope station should correlate with mineral/rock species inferred by APEX data, (5) If APEX instruments detect quartz for example, the scratch plates of the MECA microscope stage will define if a mineral of this hardness is registered during abrasion tests. This is by no means an exhaustive list of potential interactions, but it is clear that both the sheer number of analytical techniques and their complementarity should provide an analytically powerful capability for both planetary and HEDS communities.

Arvidson, R.↗

NASA NTRS2001

Advanced Methods for Aircraft Engine Thrust and Noise Benefits: Nozzle-Inlet Flow Analysis

The Fluid Mechanics and Acoustics Laboratory at Hampton University (HU/FM&AL) jointly with the NASA Glenn Research Center has conducted four connected subprojects under the reporting project. Basically, the HU/FM&AL Team has been involved in joint research with the purpose of theoretical explanation of experimental facts and creation of accurate numerical simulation techniques and prediction theory for solution of current problems in propulsion systems of interest to the NAVY and NASA agencies. This work is also supported by joint research between the NASA GRC and the Institute of Mechanics at Moscow State University (IM/MSU) in Russia under a CRDF grant. The research is focused on a wide regime of problems in the propulsion field as well as in experimental testing and theoretical and numerical simulation analyses for advanced aircraft and rocket engines. The FM&AL Team uses analytical methods, numerical simulations and possible experimental tests at the Hampton University campus. The fundamental idea uniting these subprojects is to use nontraditional 3D corrugated and composite nozzle and inlet designs and additional methods for exhaust jet noise reduction without essential thrust loss and even with thrust augmentation. These subprojects are: (1) Aeroperformance and acoustics of Bluebell-shaped and Telescope-shaped designs; (2) An analysis of sharp-edged nozzle exit designs for effective fuel injection into the flow stream in air-breathing engines: triangular-round, diamond-round and other nozzles; (3) Measurement technique improvement for the HU Low Speed Wind Tunnel; a new course in the field of aerodynamics, teaching and training of HU students; experimental tests of Mobius-shaped screws: research and training; (4) Supersonic inlet shape optimization. The main outcomes during this reporting period are: (l) Publications: The AIAA Paper #00-3170 was presented at the 36th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, 17-19 June, 2000, Huntsville, AL. The AIAA Paper #01-1893 has been accepted for the AIAA/NAL-NASDA-ISAS 10th International Space Planes and Hypersonic Systems and Technologies Conference, 24-27 April 2001, Kyoto, Japan. The AIAA Paper #01 -3204 has been accepted for presentation at the 37th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, being held on 08-11 July, in Salt Lake City, UT; (2) A U.S. patent #6,082,635 was granted on July 4, 2000; (3) Grants and proposals: The H U/ FM&AL was awarded the NASA grant NAG-3-2495 in October 2000 and the laboratory is a primary U.S. research team in a joint project under the CRDF award granted to the NASA GRC and IM/MSU (Russia) in July 2000; (4) Theory and numerical simulations: Analytical theory, numerical simulation, comparison of theoretical with experimental results, and modification of theoretical approaches, models, grids, etc., have been conducted for several complicated 2D and 3D nozzle and inlet designs using NASA, ICASE, and IM/MSU codes based on full Euler and Navier-Stokes solvers: CFL3D, FLUENT, and GODUNOV, and others; (5) Experimental Tests: (a) A new course: "Advanced Aerodynamics and Aircraft Performance" presented in spring semester, 2001; training and experimental test research using the HU LSWT. (b) Small-scale M6bius-shaped screws were tested in different conditions and their application has shown essential benefits by comparison with traditional designs; (6) Installation in the FM&AL computer system: second software TECPLOT 8.0 for the UNIX SGI workstation and free TECPLOT 7.5 for the PC Dell computer, and 2D and 3D GRIDGEN (version 9) for the UNIX SGI as well as installation of two free NASA codes, 3D MAG and VULCAN; (7) Student Research Activity: Involvement of two undergraduate students as research assistants in the current research project.

Gilinsky, Mikhail↗

NASA NTRS1984

Large deflection, large amplitude vibrations and random response of symmetrically laminated rectangular plates

An analytical method is presented for determining large-deflection static bending, large-amplitude free and forced vibrations, and large-amplitude random response of a clamped, symmetrically laminated, rectangular, thin plate subjected to a uniformly distributed transverse loading. Both movable and immovable inplane boundary conditions are considered. Numerical results for bending deflections and strains, frequency ratios, mean-square center deflections and mean-square maximum strains are presented showing the parametric effects of plate length-to-width ratio, orientation of layers, and intensities of applied force for both the linear and nonlinear cases. The analytical results for large-deflection random response are verified through comparison with experimental data.

Gray, C. E., Jr.↗

NASA NTRS1982

Determination of ball bearing dynamic stiffness

The dynamic radial stiffness characteristics of rolling element bearings are currently determined by analytical methods that have not been experimentally verified. These bearing data are vital to rotating machinery design integrity because accurate critical speeds and rotor stability predictions are highly dependent on the bearing stiffness. A tester was designed capable of controlling the bearing axial preload, speed, and rotor unbalance. The rotor and support structures were constructed to permit critical speeds that are predominantly determined by a 57 mm test bearing. A curve of calculated critical speed versus stiffness was used to determine the actual bearing stiffness from the empirical data. The results of extensive testing are used to verify analytical predictions, increase confidence in existing bearing computer programs, and to serve as a data base for efforts to correct these programs.

Beatty, R. F.↗

NASA NTRS1985

Critical joints in large composite primary aircraft structures. Volume 2: Technology demonstration test report

A program was conducted to develop the technology for critical structural joints in composite wing structure that meets all the design requirements of a 1990 commercial transport aircraft. The results of four large composite multirow bolted joint tests are presented. The tests were conducted to demonstrate the technology for critical joints in highly loaded composite structure and to verify the analytical methods that were developed throughout the program. The test consisted of a wing skin-stringer transition specimen representing a stringer runout and skin splice on the wing lower surface at the side of the fuselage attachment. All tests were static tension tests. The composite material was Toray T-300 fiber with Ciba-Geigy 914 resin in 10 mil tape form. The splice members were metallic, using combinations of aluminum and titanium. Discussions are given of the test article, instrumentation, test setup, test procedures, and test results for each of the four specimens. Some of the analytical predictions are also included.

Bunin, Bruce L.↗

NASA NTRS1999

Strategies and Technologies for In Situ Mineralogical Investigations on Mars

Surface landers on Mars (Viking and Pathfinder) have not revealed satisfying answers to the mineralogy and lithology of the planet's surface. In part, this results from their prime directives: Viking focused on exobiology, Pathfinder focused on technology demonstration. The analytical instruments on board the landers made admirable attempts to extract the mineralogy and geology of Mars, as did countless modeling efforts after the missions. Here we suggest a framework for elucidating martian, or any other planetary geology, through an approach that defines (a) type of information required, (b) explorational strategy harmonious with acquisition of these data, (c) interpretation approach to the data, (d) compatible mission architecture, (e) instrumentation for interrogating rocks and soil. (a) Data required: The composition of a planet is ordered at scales ranging from molecules to minerals to rocks, and from geological units to provinces to planetary-scale systems. The largest ordering that in situ compositional instruments can attempt to interrogate is rock type "aggregate" information. This is what the geologist attempts to identify first. From this, mineralogy can be either directly seen or inferred. From mineralogy can be determined elemental abundances and perhaps the state of the compounds as being crystalline or amorphous. Knowledge of rock type and mineralogy is critical for elucidating geologic process. Mars landers acquired extremely valuable elemental data, but attempted to move from elements to aggregates, but this can only be done by making many assumptions and sometimes giant leaps of faith. Data we believe essential are elements, minerals, degree of ordering of compounds, and the aggregate or rock type that these materials compose. (b) Explorational strategy: A lander should function as a surrogate geologist. Of the total landscape, a geologist sees much, but gives detailed attention to an infinitesimally small amount of what is seen. To acquire samples worth detailed scrutiny, as many samples as possible need examining at a cursory or reconnaissance level. A representative, statistically-meaningful sample number cannot be overemphasized. This maxim still applies to geological exploration of our own planet of which we have abundant knowledge. Analysis of many samples mandates low-power consumption per sample. (c) Data interpretation: No single instrument can analyze the full spectrum of the x-axis. An instrument is optimized for detecting certain material characteristics and must therefore affix itself to some point on the x-axis. Any conclusions drawn about data to the left or right of the instrument's position on this axis must necessarily be derived by inference. Hence, it seems logical to include on a mission, instruments that are not closely spaced in their x-axis-position, and if only two analytical methods are used, as shown, they should start at opposite ends of the axis and work towards the center. As examples, we depict a high-resolution camera to evaluate rock type ("aggregate" state) and mineralogy, and an x-ray diffractometer-fluorescence spectrometer (XRD-XRF) to determine elements, minerals, and the degree of order of materials. (d) Mission architecture: No instrument or suite of instruments can be relied upon to always give truly unequivocal analyses. The suite of instruments should therefore permit conclusions of one instrument to be checked against those of another through closed analytical loops. These "loops" can be structured by a combination of orbital imagery, descent imagery, broad-band site viewing/analysis, and data that cover both x and y axes. For example, the detection of a basaltic-looking rock with a microscope should be checked against the elements detected, the appearance of the rock as a lava flow from descent imagery, and so forth. (e) Instrumentation: To satisfy the above criteria, it is necessary to: (i) See the rock or soil with high resolution + magnification, (ii) Examine many samples, (iii) Consume little power per analysis, (iv) Determine elemental species, (v) Determine mineralogy directly (not inferentially) and the degree of ordering of compounds, (vi) Start analyzing from both ends of the x-axis. Every geologist wants to see the hand sample first, and apply a hand lens to its surface. This has not been the starting point for missions to Mars. Thus, our technology satisfies all these criteria . This XRD-XRF-Optical instrument currently being developed, analyses rock or soil surfaces without the need for sample acquisition or preparation; this satisfies the power criterion, and enables many analyses. The device acquires direct mineralogy and determines elemental species. The embedded endoscopic camera satisfies the critical criterion of close inspection of samples; the fiber optic cable can also be used for IR, LTV, or laser sample analysis. Additional information is contained in the original (Figures).

Marshall, J. R.↗

NASA NTRS2006

Calculation of Thermally-Induced Displacements in Spherically Domed Ion Engine Grids

An analytical method for predicting the thermally-induced normal and tangential displacements of spherically domed ion optics grids under an axisymmetric thermal loading is presented. A fixed edge support that could be thermally expanded is used for this analysis. Equations for the displacements both normal and tangential to the surface of the spherical shell are derived. A simplified equation for the displacement at the center of the spherical dome is also derived. The effects of plate perforation on displacements and stresses are determined by modeling the perforated plate as an equivalent solid plate with modified, or effective, material properties. Analytical model results are compared to the results from a finite element model. For the solid shell, comparisons showed that the analytical model produces results that closely match the finite element model results. The simplified equation for the normal displacement of the spherical dome center is also found to accurately predict this displacement. For the perforated shells, the analytical solution and simplified equation produce accurate results for materials with low thermal expansion coefficients.

Soulas, George C.↗

NASA NTRS2006

Modeling of Transient Flow Mixing of Streams Injected into a Mixing Chamber

Ignition is recognized as one the critical drivers in the reliability of multiple-start rocket engines. Residual combustion products from previous engine operation can condense on valves and related structures thereby creating difficulties for subsequent starting procedures. Alternative ignition methods that require fewer valves can mitigate the valve reliability problem, but require improved understanding of the spatial and temporal propellant distribution in the pre-ignition chamber. Current design tools based mainly on one-dimensional analysis and empirical models cannot predict local details of the injection and ignition processes. The goal of this work is to evaluate the capability of the modern computational fluid dynamics (CFD) tools in predicting the transient flow mixing in pre-ignition environment by comparing the results with the experimental data. This study is a part of a program to improve analytical methods and methodologies to analyze reliability and durability of combustion devices. In the present paper we describe a series of detailed computational simulations of the unsteady mixing events as the cold propellants are first introduced into the chamber as a first step in providing this necessary environmental description. The present computational modeling represents a complement to parallel experimental simulations' and includes comparisons with experimental results from that effort. A large number of rocket engine ignition studies has been previously reported. Here we limit our discussion to the work discussed in Refs. 2, 3 and 4 which is both similar to and different from the present approach. The similarities arise from the fact that both efforts involve detailed experimental/computational simulations of the ignition problem. The differences arise from the underlying philosophy of the two endeavors. The approach in Refs. 2 to 4 is a classical ignition study in which the focus is on the response of a propellant mixture to an ignition source, with emphasis on the level of energy needed for ignition and the ensuing flame propagation issues. Our focus in the present paper is on identifying the unsteady mixing processes that provide the propellant mixture in which the ignition source is to be placed. In particular, we wish to characterize the spatial and temporal mixture distribution with a view toward identifying preferred spatial and temporal locations for the ignition source. As such, the present work is limited to cold flow (pre-ignition) conditions

Voytovych, Dmytro M.↗

NASA NTRS1983

Geomagnetism of earth's core

Instrumentation, analytical methods, and research goals for understanding the behavior and source of geophysical magnetism are reviewed. Magsat, launched in 1979, collected global magnetometer data and identified the main terrestrial magnetic fields. The data has been treated by representing the curl-free field in terms of a scalar potential which is decomposed into a truncated series of spherical harmonics. Solutions to the Laplace equation then extend the field upward or downward from the measurement level through intervening spaces with no source. Further research is necessary on the interaction between harmonics of various spatial scales. Attempts are also being made to analytically model the main field and its secular variation at the core-mantle boundary. Work is also being done on characterizing the core structure, composition, thermodynamics, energetics, and formation, as well as designing a new Magsat or a tethered satellite to be flown on the Shuttle.

Benton, E. R.↗

NASA NTRS1995

Determining GPS average performance metrics

Analytic and semi-analytic methods are used to show that users of the GPS constellation can expect performance variations based on their location. Specifically, performance is shown to be a function of both altitude and latitude. These results stem from the fact that the GPS constellation is itself non-uniform. For example, GPS satellites are over four times as likely to be directly over Tierra del Fuego than over Hawaii or Singapore. Inevitable performance variations due to user location occur for ground, sea, air and space GPS users. These performance variations can be studied in an average relative sense. A semi-analytic tool which symmetrically allocates GPS satellite latitude belt dwell times among longitude points is used to compute average performance metrics. These metrics include average number of GPS vehicles visible, relative average accuracies in the radial, intrack and crosstrack (or radial, north/south, east/west) directions, and relative average PDOP or GDOP. The tool can be quickly changed to incorporate various user antenna obscuration models and various GPS constellation designs. Among other applications, tool results can be used in studies to: predict locations and geometries of best/worst case performance, design GPS constellations, determine optimal user antenna location and understand performance trends among various users.

Moore, G. V.↗

NASA NTRS1996

A Comparison of Tension and Compression Creep in a Polymeric Composite and the Effects of Physical Aging on Creep Behavior

Experimental and analytical methods were used to investigate the similarities and differences of the effects of physical aging on creep compliance of IM7/K3B composite loaded in tension and compression. Two matrix dominated loading modes, shear and transverse, were investigated for two load cases, tension and compression. The tests, run over a range of sub-glass transition temperatures, provided material constants, material master curves and aging related parameters. Comparing results from the short-term data indicated that although trends in the data with respect to aging time and aging temperature are similar, differences exist due to load direction and mode. The analytical model used for predicting long-term behavior using short-term data as input worked equally as well for the tension or compression loaded cases. Comparison of the loading modes indicated that the predictive model provided more accurate long term predictions for the shear mode as compared to the transverse mode. Parametric studies showed the usefulness of the predictive model as a tool for investigating long-term performance and compliance acceleration due to temperature.

Gates, Thomas S.↗

NASA NTRS1972

A method for calculating aerodynamic heating on sounding rocket tangent ogive noses

A method is presented for calculating the aerodynamic heating and shear stresses at the wall for tangent ogive noses that are slender enough to maintain an attached nose shock through that portion of flight during which heat transfer from the boundary layer to the wall is significant. The lower entropy of the attached nose shock combined with the inclusion of the streamwise pressure gradient yields a reasonable estimate of the actual flow conditions. Both laminar and turbulent boundary layers are examined and an approximation of the effects of (up to) moderate angles-of-attack is included in the analysis. The analytical method has been programmed in FORTRAN 4 for an IBM 360/91 computer.

Wing, L. D.↗

NASA NTRS1973

A method for calculating aerodynamic heating on sounding rocket tangent ogive noses.

Wing, L. D.↗

NASA NTRS1979

Testing of containers made of glass-fiber reinforced plastic with the aid of acoustic emission analysis

Acoustic emission analysis as a quasi-nondestructive test method makes it possible to differentiate clearly, in judging the total behavior of fiber-reinforced plastic composites, between critical failure modes (in the case of unidirectional composites fiber fractures) and non-critical failure modes (delamination processes or matrix fractures). A particular advantage is that, for varying pressure demands on the composites, the emitted acoustic pulses can be analyzed with regard to their amplitude distribution. In addition, definite indications as to how the damages occurred can be obtained from the time curves of the emitted acoustic pulses as well as from the particular frequency spectrum. Distinct analogies can be drawn between the various analytical methods with respect to whether the failure modes can be classified as critical or non-critical.

Wolitz, K.↗

NASA NTRS2011

Broadband Via-Less Microwave Crossover Using Microstrip-CPW Transitions

The front-to-back interface between microstrip and CPW (coplanar waveguide) typically requires complex fabrication or has high radiation loss. The microwave crossover typically requires a complex fabrication step. The prior art in microstrip-CPW transition requires a physical vias connection between the microstrip and CPW line on a separate layer. The via-less version of this transition was designed empirically and does not have a close form solution. The prior art of the micro wave crossover requires either additional substrate or wire bond as an air bridge to isolate two microwave lines at the crossing junction. The disadvantages are high radiation loss, no analytical solution to the problem, lengthy simulation time, and complex fabrication procedures to generate air bridges or via. The disadvantage of the prior crossover is a complex fabrication procedure, which also affects the device reliability and yield. This microstrip-CPW transition is visualized as two microstrip-slotline transitions combined in a way that the radiation from two slotlines cancels each other out. The invention is designed based on analytical methods; thus, it significantly reduces the development time. The crossover requires no extra layer to cross two microwave signals and has low radiation loss. The invention is simple to fabricate and design. It produces low radiation loss and can be designed with low insertion loss, with some tradeoff with signal isolation. The microstrip-CPW transition is used as an interface to connect between the device and the circuit outside the package. The via-less microwave crossover is used to allow two signals to cross without using an extra layer or fabrication processing step to enable this function. This design allows the solution to be determined entirely though analytical techniques. In addition, a planar via-less microwave crossover using this technique was proposed. The experimental results show that the proposed crossover at 5 GHz has a minimum isolation of 32 dB. It also has low in-band insertion loss and return loss of 1.2 dB and 18 dB, respectively, over more than 44 percent of bandwidth at room temperature. This microstrip-CPW transition requires the microstrip line to be split into two sections. Each section is connected to a microstrip quarter-wavelength openended stub. A slotline is also placed perpendicular to the microstrip section. The slot is connected to a grounded-end quarter-wavelength slotline and generates a microstrip-slotline transition. When two of these sections are placed in parallel and with the microstrip section combined at transition, a microstrip- CPW transition is formed. The slotline radiation is suppressed as two slots are excited with the electric field in an opposite direction, which cancels the radiation in far field. The invention on the crossover consists of the invented microstrip-CPW transitions combined back-to-back and a microstrip low-pass filter. One signal is crossed through to the microstrip layer, while the other signal is crossed through the CPW line located on the ground plane of the microstrip line. The microstrip low-pass filter produces a narrow line at the crossing point to enhance the system isolation. It also produces broadband response in the operating frequency band. The microstrip-CPW transition allows a microwave signal to travel from microstrip line to CPW line with low radiation loss. The crossover allows two microwave signals to cross with minimal parasitic coupling.

Stevenson, Thomas↗

NASA NTRS1972

Atmospheric pressure, density, temperature and wind variations between 50 and 200 km

Data on atmospheric pressure, density, temperature and winds between 50 and 200 km were collected from sources including Meteorological Rocket Network data, ROBIN falling sphere data, grenade release and pitot tube data, meteor winds, chemical release winds, satellite data, and others. These data were analyzed by a daily difference method and results on the distribution statistics, magnitude, and spatial structure of the irregular atmospheric variations are presented. Time structures of the irregular variations were determined by the analysis of residuals from harmonic analysis of time series data. The observed height variations of irregular winds and densities are found to be in accord with a theoretical relation between these two quantities. The latitude variations (at 50 - 60 km height) show an increasing trend with latitude. A possible explanation of the unusually large irregular wind magnitudes of the White Sands MRN data is given in terms of mountain wave generation by the Sierra Nevada range about 1000 km west of White Sands. An analytical method is developed which, based on an analogy of the irregular motion field with axisymmetric turbulence, allows measured or model correlation or structure functions to be used to evaluate the effective frequency spectra of scalar and vector quantities of a spacecraft moving at any speed and at any trajectory elevation angle.

Justus, C. G.↗

NASA NTRS1992

Flow analysis and design optimization methods for nozzle-afterbody of a hypersonic vehicle

This report summarizes the methods developed for the aerodynamic analysis and the shape optimization of the nozzle-afterbody section of a hypersonic vehicle. Initially, exhaust gases were assumed to be air. Internal-external flows around a single scramjet module were analyzed by solving the 3D Navier-Stokes equations. Then, exhaust gases were simulated by a cold mixture of Freon and Ar. Two different models were used to compute these multispecies flows as they mixed with the hypersonic airflow. Surface and off-surface properties were successfully compared with the experimental data. The Aerodynamic Design Optimization with Sensitivity analysis was then developed. Pre- and postoptimization sensitivity coefficients were derived and used in this quasi-analytical method. These coefficients were also used to predict inexpensively the flow field around a changed shape when the flow field of an unchanged shape was given. Starting with totally arbitrary initial afterbody shapes, independent computations were converged to the same optimum shape, which rendered the maximum axial thrust.

Baysal, O.↗