Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “1D”

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 685 records · Page 38

An assessment of spectral nonoscillatory schemes

A new spectral nonoscillatory interpolation scheme is proposed that achieves spectral accuracy in smooth regions and is nonoscillatory on piecewise discontinuous data. The essential idea behind the scheme is to increase the order of an ENO scheme in proportion to the number of points, wherever possible. Numerical experiments with the new scheme on interpolation, 1D advection, and 1D gas dynamics confirm the high resolution features of the scheme. Comparisons with the results of earlier spectral nonoscillatory schemes show that the new scheme is competitive both in efficiency and accuracy.

Suresh, Ambady↗

Cross-tail current - Resonant orbits

A technique to generate self-consistent 1D current sheets is described. Groups of monoenergetic protons were followed in a modified Harris magnetic field. This sample current sheet is characterized by resonant quasi-adiabatic orbits. The magnetic moment of a quasi-adiabatic ion which is injected from outside a current sheet changes substantially during the orbit but returns to almost its initial value by the time the ion leaves. Several ion and electron groups were combined to produce a plasma sheet in which the charged particles carry the currents needed to generate the magnetic field in which the orbits were traced. An electric field also is required to maintain charge neutrality. Three distinct orbit types, one involving untrapped ions and two composed of trapped ions, were identified. Limitations associated with the use of a 1D model also were investigated; it can provide a good physical picture of an important component of the cross-tail current, but cannot adequately describe any region of the magnetotail in which the principal current sheet is separated from the plasma sheet boundary layer by a nearly isotropic outer position of the central plasma sheet.

Kaufmann, Richard L.↗

Variability of (O I) 6300-A emission near Io

Observations of (O I) 6300-A emission near Io have been obtained in 1990, 1991, and 1992 by the National Solar Observatory staff using the solar-stellar spectrography on the McMath-Pierce telescope at Kitt Peak. High-resolution spectra with a resolving power of about 1.2 x 10(exp 5) were obtained with an integration time of 10-15 min each. The viewing aperture for the observations was 5.2 arc sec x 5.2 arc sec centered on Io, with spatial resolution limited within this area by seeing conditions. Observations thus far have been reduced to obtain average brightness values over the aperture which range from approximately 200 to 1000 R for a number of different Io phase angles and Io system III longitudes. The (O I) 6300-A emission brightness exhibits an east-west asymmetry, where the average intensity for Io phase angles in the west (receding ansa) is 1.5 times brighter than in the east (approaching ansa). Similar east-west intensity ratios have also been observed for neutrals near Io and ions in the plasma torus for a number of other optical and ultraviolet emission lines which are excited by electron impact. In addition to the east-west asymmetry, the (O I) 6300-A emission brightness exhibits a strong dependence on the Io system III longitude angle, with a maximum value occurring in the range 200 deg +/- 50 deg. Earlier IUE observations of ultraviolet emission lines of O and S near Io obtained over a number of years have measured the east-west asymmetry, but the long IUE integration times of approximately 7-14 hours masked any detection of system III variability. For the (O I) 6300-A emission, the O(1D) state may be excited by electron impact of atomic oxygen and by electron impact dissociation of SO. The molecule SO may be present at the exobase or may be produced above the exobase as the dominant product of SO2 dissociation by electron impact. Preliminary assessment indicates that production of O(1D) by molecular dissociation may be more important. The (O I) 6300-A emission may therefore provide a remote signature for monitoring (1) the upward transport rates of molecular species in Io's atmosphere, (2) the relative abundance and time-variable dissociation of SO2 and/or SO at the exobase and in the corona of Io, and (3) the spatial distribution of these escaping molecular and atomic species and their ion production rates in the planetary magnetosphere.

Scherb, F.↗

The importance of mesoscale circulations generated by subgrid-scale landscape heterogeneities in general circulation models

A mesoscale atmospheric model was used to evaluate the impact of subgrid-scale landscape discontinuities on the vertical profiles of resolved temperature, moisture, and moist static energy in the planetary boundary layer (PBL) of general circulation models (GCMs). These profiles were produced with a 3D version of the model (using a horizontal grid resolution of 7.5 km and 13 vertical layers in the PBL) by averaging horizontally the various atmospheric variables over a 180 x 180 sq km domain-about the size of the horizontal domain represented by a single grid element in a GCM. They were compared to corresponding vertical profiles produced with a 1D version of the model, which simulates the PBL, as in a GCM, over a single horizontal grid element. Differences obtained between the horizontally averaged atmospheric variables produced with the 3D simulations and the 1D simulations emphasize the impact of subgrid-scale landscape discontinuities on GCM-resolved variables. Various types of landscape discontinuities, characterized by horizontal contrasts of surface wetness and size of land patches, were simulated under various background-wind conditions. Differences of temperature, specific humidity, and moist static energy as large as 4 K, 6 g/kg, and 10 kJ/kg were obtained in some cases. These differences were not affected significantly by moderate winds but were sensitive to the spatial distribution of surface wetness. These results emphasize the need to parameterize mesoscale processes induced by landscape discontinuities in GCMss.

Lynn, Barry H.↗

A Model for Particle Microphysics, Turbulent Mixing, and Radiative Transfer in the Stratocumulus-Topped Marine Boundary Layer and Comparisons with Measurements

A detailed 1D model of the stratocumulus-topped marine boundary layer is described. The model has three coupled components: a microphysics module that resolves the size distributions of aerosols and cloud droplets, a turbulence module that treats vertical mixing between layers, and a multiple wavelength radiative transfer module that calculates radiative heating rates and cloud optical properties. The results of a 12-h model simulation reproduce reasonably well the bulk thermodynamics, microphysical properties, and radiative fluxes measured in an approx. 500-m thick, summertime marine stratocumulus cloud layer by Nicholls. However, in this case, the model predictions of turbulent fluxes between the cloud and subcloud layers exceed the measurements. Results of model simulations are also compared to measurements of a marine stratus layer made under gale conditions and with measurements of a high, thin marine stratocumulus layer. The variations in cloud properties are generally reproduced by the model, although it underpredicts the entrainment of overlying air at cloud top under gale conditions. Sensitivities of the model results are explored. The vertical profile of cloud droplet concentration is sensitive to the lower size cutoff of the droplet size distribution due to the presence of unactivated haze particles in the lower region of the modeled cloud. Increases in total droplet concentrations do not always produce less drizzle and more cloud water in the model. The radius of the mean droplet volume does not correlate consistently with drizzle, but the effective droplet radius does. The greatest impacts on cloud properties predicted by the model are produced by halving the width of the size distribution of input condensation nuclei and by omitting the effect of cloud-top radiative cooling on the condensational growth of cloud droplets. The omission of infrared scattering produces noticeable changes in cloud properties. The collection efficiencies for droplets less than 30-micrometers radius, and the value of the accommodation coefficient for condensational droplet growth, have noticeable effects on cloud properties. The divergence of the horizontal wind also has a significant effect on a 12-h model simulation of cloud structure. Conclusions drawn from the model are tentative because of the limitations of the 1D model framework. A principal simplification is that the model assumes horizontal homogeneity, and, therefore, does not resolve updrafts and downdrafts. Likely consequences of this simplification include overprediction of the growth of droplets by condensation in the upper region of the cloud, underprediction of droplet condensational growth in the lower region of the cloud, and underprediction of peak supersaturations.

Ackerman, Andrew S.↗

A Model for Particle Microphysics,Turbulent Mixing, and Radiative Transfer in the Stratocumulus-Topped Marine Boundary Layer and Comparisons with Measurements

A detailed 1D model of the stratocumulus-topped marine boundary layer is described. The model has three coupled components: a microphysics module that resolves the size distributions of aerosols and cloud droplets, a turbulence module that treats vertical mixing between layers, and a multiple wavelength radiative transfer module that calculates radiative heating rates and cloud optical properties. The results of a 12-h model simulation reproduce reasonably well the bulk thermodynamics, microphysical properties, and radiative fluxes measured in an approx. 500-m thick, summertime marine stratocumulus cloud layer by Nicholls. However, in this case, the model predictions of turbulent fluxes between the cloud and subcloud layers exceed the measurements. Results of model simulations are also compared to measurements of a marine stratus layer made under gate conditions and with measurements of a high, thin marine stratocumulus layer. The variations in cloud properties are generally reproduced by the model, although it underpredicts the entrainment of overlying air at cloud top under gale conditions. Sensitivities of the model results are explored. The vertical profile of cloud droplet concentration is sensitive to the lower size cutoff of the droplet size distribution due to the presence of unactivated haze particles in the lower region of the modeled cloud. Increases in total droplet concentrations do not always produce less drizzle and more cloud water in the model. The radius of the mean droplet volume does not correlate consistently with drizzle, but the effective droplet radius does. The greatest impacts on cloud properties predicted by the model are produced by halving the width of the size distribution of input condensation nuclei and by omitting the effect of cloud-top radiative cooling on the condensational growth of cloud droplets. The omission of infrared scattering produces noticeable changes in cloud properties. The collection efficiencies for droplets less than 30-micron radius, and the value of the accommodation coefficient for condensational droplet growth, have noticeable effects on cloud properties. The divergence of the horizontal wind also has a significant effect on a 12-h model simulation of cloud structure. Conclusions drawn from the model are tentative because of the limitations of the 1D model framework. A principal simplification is that the model assumes horizontal homogeneity, and, therefore, does not resolve updrafts and downdrafts. Likely consequences of this simplification include overprediction of the growth of droplets by condensation in the upper region of the cloud, underprediction of droplet condensational growth in the lower region of the cloud, and under-prediction of peak supersaturations.

Ackerman, Andrew S.↗

The Space-Time Conservation Element and Solution Element Method-A New High-Resolution and Genuinely Multidimensional Paradigm for Solving Conservation Laws: Numerical Simulation of Shock Waves and Contact Discontinuities - 2

Without resorting to special treatment for each individual test case, the 1D and 2D CE/SE shock-capturing schemes described previously (in Part I) are used to simulate flows involving phenomena such as shock waves, contact discontinuities, expansion waves and their interactions. Five 1D and six 2D problems are considered to examine the capability and robustness of these schemes. Despite their simple logical structures and low computational cost (for the 2D CE/SE shock-capturing scheme, the CPU time is about 2 micro-secs per mesh point per marching step on a Cray C90 machine), the numerical results, when compared with experimental data, exact solutions or numerical solutions by other methods, indicate that these schemes can accurately resolve shock and contact discontinuities consistently.

Wang, Xiao-Yen↗

Models of Plumes: Their Flow, Their Geometric Spreading, and Their Mixing with Interplume Flow

There are two types of plume flow models: (1) 1D models using ad hoc spreading functions, f(r); (2) MagnetoHydroDynamics (MHD) models. 1D models can be multifluid, time dependent, and incorporate very general descriptions of the energetics. They confirm empirical results that plume flow is slow relative to requirements for high speed wind. But, no published 1 D model incorporates the rapid local spreading at the base (fl(r)) which has an important effect on mass flux. The one published MHD model is isothermal, but confirms that if b=8*pi*p/absolute value(B)2<<l then the field is nearly potential below -70,000 km. Building on the MHD result, we apply a two scale approximation to calculate fl(r). We also compute the global spreading (fg(r)) out to 5.0 RSUN imposed by coronal hole geometry. Global MHD models provide a potent method of calculating fg(r). Unambiguous plume signatures have not yet been found in the solar wind. This is probably due to strong mixing of plume and interplume flows near the Sun. We describe a physical source for strong mixing due to the observed flows being unstable to shear instabilities that lead to rapid disruption.

Suess, Steven T.↗

Development of a Three-Dimensional PSE Code for Compressible Flows: Stability of Three-Dimensional Compressible Boundary Layers

A program is developed to investigate the linear stability of three-dimensional compressible boundary layer flows over bodies of revolutions. The problem is formulated as a two dimensional (2D) eigenvalue problem incorporating the meanflow variations in the normal and azimuthal directions. Normal mode solutions are sought in the whole plane rather than in a line normal to the wall as is done in the classical one dimensional (1D) stability theory. The stability characteristics of a supersonic boundary layer over a sharp cone with 50 half-angle at 2 degrees angle of attack is investigated. The 1D eigenvalue computations showed that the most amplified disturbances occur around x(sub 2) = 90 degrees and the azimuthal mode number for the most amplified disturbances range between m = -30 to -40. The frequencies of the most amplified waves are smaller in the middle region where the crossflow dominates the instability than the most amplified frequencies near the windward and leeward planes. The 2D eigenvalue computations showed that due to the variations in the azimuthal direction, the eigenmodes are clustered into isolated confined regions. For some eigenvalues, the eigenfunctions are clustered in two regions. Due to the nonparallel effect in the azimuthal direction, the eigenmodes are clustered into isolated confined regions. For some eigenvalues, the eigenfunctions are clustered in two regions. Due to the nonparallel effect in the azimuthal direction, the most amplified disturbances are shifted to 120 degrees compared to 90 degrees for the parallel theory. It is also observed that the nonparallel amplification rates are smaller than that is obtained from the parallel theory.

Balakumar, P.↗

The Use of Vegetation For Estimating Broken-Cloud Optical Properties From Surface Measurements

It is known that because of complex three-dimensional (3D) radiative effects of broken clouds, the retrieval of cloud optical properties from upward measurements based on a one-dimensional (1D) inversion technique almost surely fails. To remove radiative effects of 3D cloud structure, we have developed a new technique that retrieves cloud optical thickness for broken clouds above green vegetation from simultaneous surface measurements in the VIS and Near Infrared (NIR) spectral regions. The theoretical basis of the method is the very different spectral behavior of cloud liquid water drops and green vegetation. For example, cloud optical properties, and hence cloud reflectivities, change little between 650 and 860 nm, while the vegetated surface albedo changes from 0.05 to 0.5 between the same two wavelengths. This spectral contrast in surface albedo suggests using ground measurements at both wavelengths not independently, but as an algebraic combination (a spectral index). For a spectral band in the NIR region, the green vegetation acts as a powerful reflector that "illuminates" horizontally inhomogeneous clouds from below. This provides the extra information needed to largely remove the 3D radiative effects, especially in the case of broken clouds; this in turn allows the retrieval of cloud optical depth using traditional 1D radiative transfer theory. This approach is similar to the so-called Green's function problem for radiative transfer where a laser beam illuminates clouds and the resulting "spot-size" of the reflected light around the beam characterizes cloud properties. We generalize Green's function theory to surf ace-cloud interaction and develop new spectral indices from which broken-cloud optical depth can be retrieved.

Marshak, A.↗

Applications of the Space-Time Conservation Element and Solution Element (CE/SE) Method to Computational Aeroacoustic Benchmark Problems

The Internal Propagation problems, Fan Noise problem, and Turbomachinery Noise problems are solved using the space-time conservation element and solution element (CE/SE) method. The problems in internal propagation problems address the propagation of sound waves through a nozzle. Both the nonlinear and linear quasi 1D Euler equations are solved. Numerical solutions are presented and compared with the analytical solution. The fan noise problem concerns the effect of the sweep angle on the acoustic field generated by the interaction of a convected gust with a cascade of 3D flat plates. A parallel version of the 3D CE/SE Euler solver is developed and employed to obtain numerical solutions for a family of swept flat plates. Numerical solutions for sweep angles of 0, 5, 10, and 15 deg are presented. The turbomachinery problems describe the interaction of a 2D vortical gust with a cascade of flat-plate airfoils with/without a downstream moving grid. The 2D nonlinear Euler Equations are solved and the converged numerical solutions are presented and compared with the corresponding analytical solution. All the comparisons demonstrate that the CE/SE method is capable of solving aeroacoustic problems with/without shock waves in a simple and efficient manner. Furthermore, the simple non-reflecting boundary condition used in the CE/SE method which is not based on the characteristic theory works very well in 1D, 2D and 3D problems.

Wang, Xiao-Yen↗

Cloud-Scale Numerical Modeling of the Arctic Boundary Layer

The research objective of this NASA grant-funded project was to determine in detail how large-scale processes. in combination with cloud-scale radiative, microphysical, and dynamical processes, govern the formation and multi-layered structure of Arctic stratus clouds. This information will be useful for developing and improving 1D (one dimensional) boundary layer models for the Arctic. Also, to quantitatively determine the effects of leads on the large-scale budgets of sensible heat, water vapor, and condensate in a variety of Arctic winter conditions. This information will be used to identify the most important lead-flux processes that require parameterization in climate models. Our approach was to use a high-resolution numerical model, the 2D (two dimensional) University of Utah Cloud Resolving Model (UU CRM), and its 1D version, the University of Utah Turbulence Closure Model (UU TCM), a boundary layer model based on third-moment turbulence closure, as well as a large-eddy simulation (LES) model originally developed by C.H. Moeng.

Kruegen, Steven K.↗

High-Resolution Genuinely Multidimensional Solution of Conservation Laws by the Space-Time Conservation Element and Solution Element Method

In this overview paper, we review the basic principles of the method of space-time conservation element and solution element for solving the conservation laws in one and two spatial dimensions. The present method is developed on the basis of local and global flux conservation in a space-time domain, in which space and time are treated in a unified manner. In contrast to the modern upwind schemes, the approach here does not use the Riemann solver and the reconstruction procedure as the building blocks. The drawbacks of the upwind approach, such as the difficulty of rationally extending the 1D scalar approach to systems of equations and particularly to multiple dimensions is here contrasted with the uniformity and ease of generalization of the Conservation Element and Solution Element (CE/SE) 1D scalar schemes to systems of equations and to multiple spatial dimensions. The assured compatibility with the simplest type of unstructured meshes, and the uniquely simple nonreflecting boundary conditions of the present method are also discussed. The present approach has yielded high-resolution shocks, rarefaction waves, acoustic waves, vortices, ZND detonation waves, and shock/acoustic waves/vortices interactions. Moreover, since no directional splitting is employed, numerical resolution of two-dimensional calculations is comparable to that of the one-dimensional calculations. Some sample applications displaying the strengths and broad applicability of the CE/SE method are reviewed.

Himansu, Ananda↗

Investigation of N2O Production from 266 and 532 nm Laser Flash Photolysis of O3/N2/O2 Mixtures

Tunable diode laser absorption spectroscopy has been employed to measure the amount of N2O produced from laser flash photolysis of O3/N2/O2 mixtures at 266 and 532 nm. In the 532 nm photolysis experiments very little N2O is observed, thus allowing an upper limit yield of 7 x 10(exp -8) to be established for the process O3 + N2 yield N2O + O2, where O3 is nascent O3 that is newly formed via O(3P(sub J)) + O2 recombination (with vibrational excitation near the dissociation energy of O3). The measured upper limit yield is a factor of approx. 600 smaller than a previous literature value and is approximately a factor of 10 below the threshold for atmospheric importance. In the 266 nm photolysis experiments, significant N2O production is observed and the N2O quantum yield is found to increase linearly with pressure over the range 100 - 900 Torr in air bath gas. The source of N2O in the 266 nm photolysis experiments is believed to be the addition reaction O(1D(sub 2)) + N2 + M yields (k(sub sigma)) N2O + M, although reaction of (very short-lived) electronically excited O3 with N2 cannot be ruled out by the available data. Assuming that all observed N2O comes from the O(1D(sub 2)) + N2 + M reaction, the following expression describes the temperature dependence of k(sub sigma) (in its third-order low-pressure limit) that is consistent with the N2O yield data: k(sub sigma) = (2.8 +/- 0.1) x 10(exp -36)(T/300)(sup -(0-88+0.36)) cm(sup 6) molecule(sup -2)/s, where the uncertainties are 2(sigma) and represent precision only. The accuracy of the reported rate coefficients at the 95% confidence level is estimated to be 30 - 40% depending on the temperature. Model calculations suggest that gas phase processes initiated by ozone absorption of a UV photon represent about 1.4% of the currently estimated global source strength of atmospheric N2O. However, these processes could account for a significant fraction of the oxygen mass-independent enrichment observed in atmospheric N2O, and they appear to be the first suggested photochemical mechanism that is capable of explaining the altitude dependence of the observed mass -independent isotopic signature.

Estupinan, E. G.↗

A One Billion Year Martian Climate Model: The Importance of Seasonally Resolved Polar Caps and the Role of Wind

Wind deflation and deposition are powerful agents of surface change in the present Mars climate regime. Recent studies indicate that, while the distribution of regions of potential deflation (or erosion) and deposition is remarkably insensitive to changes in orbital parameters (obliquity, timing of perihelion passage, etc.), rates of aeolian surface modification may be highly sensitive to these parameters even if the atmospheric mass remains constant. But previous work suggested the atmospheric mass is likely to be sensitive to obliquity, especially if a significant mass of carbon dioxide can be stored in the regolith or deposited in the form of massive polar caps. Deflation and erosion are highly sensitive to surface pressure, so feedback between orbit variations and surface pressure can greatly enhance the sensitivity of aeolian modification rates to orbital parameters. We used statistics derived from a 1 Gyr orbital integration of the spin axis of Mars, coupled with 3D general circulation models (GCMs) at a variety of orbital conditions and pressures, to explore this feedback. We also employed a seasonally resolved 1D energy balance model to illuminate the gross characteristics of the longterm atmospheric evolution, wind erosion and deposition over one billion years. We find that seasonal polar cycles have a critical influence on the ability for the regolith to release CO2 at high obliquities, and find that the atmospheric CO2 actually decreases at high obliquities due to the cooling effect of polar deposits at latitudes where seasonal caps form. At low obliquity, the formation of massive, permanent polar caps depends critically on the values of the frost albedo, A(sub frost), and frost emissivity, E(sub frost). Using our 1D model with values of A(sub frost) = 0.67 and E(sub frost) = 0.55, matched to the NASA Ames GCM results, we find that permanent caps only form at low obliquities (< 10 degrees). Thus, contrary to expectations, the Martian atmospheric pressure is remarkable static over time, and decreases both at high and low obliquity. Also, from our one billion year orbital model, we present new results on the fraction of time Mars is expected to experience periods of high and low obliquity. Finally, using GCM runs at a variety of pressures, we examine the likely role of wind erosion under an early more massive Martian atmosphere.

J C Armstrong↗

Robust and Simple Non-Reflecting Boundary Conditions for the Euler Equations: A New Approach Based on the Space-Time CE/SE Method

This paper reports on a significant advance in the area of non-reflecting boundary conditions (NRBCs) for unsteady flow computations. As a part of the development of the space-time conservation element and solution element (CE/SE) method, sets of NRBCs for 1D Euler problems are developed without using any characteristics-based techniques. These conditions are much simpler than those commonly reported in the literature, yet so robust that they are applicable to subsonic, transonic and supersonic flows even in the presence of discontinuities. In addition, the straightforward multidimensional extensions of the present 1D NRBCs have been shown numerically to be equally simple and robust. The paper details the theoretical underpinning of these NRBCs, and explains their unique robustness and accuracy in terms of the conservation of space-time fluxes. Some numerical results for an extended Sod's shock-tube problem, illustrating the effectiveness of the present NRBCs are included, together with an associated simple Fortran computer program. As a preliminary to the present development, a review of the basic CE/SE schemes is also included.

Chang, Sin-Chung↗

Evidence for Three-Dimensional Radiative Effects in MODIS Cloud Optical Depths Retrieved at Back Scattering View Angles

This study addresses the question whether 1D radiative transfer theory describes well the angular distribution of shortwave cloud reflection. The statistical analysis of a large set of MODIS observations indicates that in oblique backward scattering directions, cloud reflection is stronger than 1D theory would predict. After considering a variety of possible causes, the paper concludes that the most likely reason for the increase lies in 3D radiative interactions. The results' main implication is that cloud optical depths retrieved at back scattering view angles larger than about 50 degrees tend to be overestimated and should be used only with great caution.

Varnai, Tamas↗

Advanced Applications of Adifor 3.0 for Efficient Calculation of First-and Second-Order CFD Sensitivity Derivatives

This final report will document the accomplishments of the work of this project. 1) The incremental-iterative (II) form of the reverse-mode (adjoint) method for computing first-order (FO) aerodynamic sensitivity derivatives (SDs) has been successfully implemented and tested in a 2D CFD code (called ANSERS) using the reverse-mode capability of ADIFOR 3.0. These preceding results compared very well with similar SDS computed via a black-box (BB) application of the reverse-mode capability of ADIFOR 3.0, and also with similar SDs calculated via the method of finite differences. 2) Second-order (SO) SDs have been implemented in the 2D ASNWERS code using the very efficient strategy that was originally proposed (but not previously tested) of Reference 3, Appendix A. Furthermore, these SO SOs have been validated for accuracy and computational efficiency. 3) Studies were conducted in Quasi-1D and 2D concerning the smoothness (or lack of smoothness) of the FO and SO SD's for flows with shock waves. The phenomenon is documented in the publications of this study (listed subsequently), however, the specific numerical mechanism which is responsible for this unsmoothness phenomenon was not discovered. 4) The FO and SO derivatives for Quasi-1D and 2D flows were applied to predict aerodynamic design uncertainties, and were also applied in robust design optimization studies.

Taylor, Arthur C., III↗