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All sourcesNASA NTRSDOE OSTI
NASA NTRS1989

Airborne measurements of stratospheric constituents over Antarctica in the austral spring 1987. I - Method and ozone observations

A Fourier transform spectrometer was flown aboard a DC-8 on 10 flights over Antarctica during August and September, 1987, as part of the Airborne Antarctic Ozone Experiment (AAOE). Observing the sun at infrared wavelengths, it was possible to determine the integrated column amount above the flight altitude for ozone and a number of other chemical species that are believed to be important in the perturbed chemistry of the 'ozone hole'. The paper describes the method, the observations, the data analysis procedure, and the ozone results. During the observation period, ozone developed a steep gradient near the edge of the polar vortex; deep within the vortex, the average ozone column decreased by about 1.6 percent per day during September.

Mankin, William G.↗
NASA NTRS1983

A general perturbation approach for the equations of fluid dynamics

An efficient numerical technique to produce accurate solutions to the equations of fluid dynamics is presented where the governing equations are perturbed about an approximate solution and solved by finite-difference methods on a coarsened grid. The result is a scheme which substantially reduces the number of grid points necessary to accurately resolve the flow. Applications are presented for the two-dimensional Euler equations perturbed about a solution of the transonic full potential equation. However, the concept is applicable to arbitrary equation sets, higher dimensions and for a wide variety of applications.

Chow, L. J.↗
NASA NTRS1972

Numerical method for the solution of large systems of differential equations of the boundary layer type

A numerical method for the solution of large systems of nonlinear differential equations of the boundary-layer type is described. The method is a modification of the technique for satisfying asymptotic boundary conditions. The present method employs inverse interpolation instead of the Newton method to adjust the initial conditions of the related initial-value problem. This eliminates the so-called perturbation equations. The elimination of the perturbation equations not only reduces the user's preliminary work in the application of the method, but also reduces the number of time-consuming initial-value problems to be numerically solved at each iteration. For further ease of application, the solution of the overdetermined system for the unknown initial conditions is obtained automatically by applying Golub's linear least-squares algorithm. The relative ease of application of the proposed numerical method increases directly as the order of the differential-equation system increases. Hence, the method is especially attractive for the solution of large-order systems. After the method is described, it is applied to a fifth-order problem from boundary-layer theory.

Green, M. J.↗
NASA NTRS1996

Aerodynamic Shape Optimization of Complex Aircraft Configurations via an Adjoint Formulation

This work describes the implementation of optimization techniques based on control theory for complex aircraft configurations. Here control theory is employed to derive the adjoint differential equations, the solution of which allows for a drastic reduction in computational costs over previous design methods (13, 12, 43, 38). In our earlier studies (19, 20, 22, 23, 39, 25, 40, 41, 42) it was shown that this method could be used to devise effective optimization procedures for airfoils, wings and wing-bodies subject to either analytic or arbitrary meshes. Design formulations for both potential flows and flows governed by the Euler equations have been demonstrated, showing that such methods can be devised for various governing equations (39, 25). In our most recent works (40, 42) the method was extended to treat wing-body configurations with a large number of mesh points, verifying that significant computational savings can be gained for practical design problems. In this paper the method is extended for the Euler equations to treat complete aircraft configurations via a new multiblock implementation. New elements include a multiblock-multigrid flow solver, a multiblock-multigrid adjoint solver, and a multiblock mesh perturbation scheme. Two design examples are presented in which the new method is used for the wing redesign of a transonic business jet.

Reuther, James↗
NASA NTRS1976

Pines method for circular and elliptical orbits

The Pines method for trajectory integration is derived from the equations of two-body motion. The concept of the perturbation derivative is explained and a formal definition is presented. An algorithm for applying the Pines method to numerical integration of trajectories is included.

Semar, C.↗
NASA NTRS1975

Calculation of rotational transition probabilities in molecular collisions - Application to N2 + N2

A computational method is proposed to obtain rotational transition probabilities in collisions between two diatomic molecules. The potential method of Rabitz and an exponential approximation are used to solve the semiclassical coupled equations without invoking any perturbational technique. The collision trajectory is determined in the classical modified-wave-number approximation. The method can treat systems involving strong interactions and provide probabilities for transitions even with a multiquantum jump. A simultaneous transition in the rotational states of both molecules, i.e., the rotational-rotational energy transfer, is taken into account. An application to the system N2 + N2 is presented.

Itikawa, Y.↗
NASA NTRS1993

Accurate computation and continuation of homoclinic and heteroclinic orbits for singular perturbation problems

In earlier papers, Doedel and the authors have developed a numerical method and derived error estimates for the computation of branches of heteroclinic orbits for a system of autonomous ordinary differential equations in R(exp n). The idea of the method is to reduce a boundary value problem on the real line to a boundary value problem on a finite interval by using a local (linear or higher order) approximation of the stable and unstable manifolds. A practical limitation for the computation of homoclinic and heteroclinic orbits has been the difficulty in obtaining starting orbits. Typically these were obtained from a closed form solution or via a homotopy from a known solution. Here we consider extensions of our algorithm which allow us to obtain starting orbits on the continuation branch in a more systematic way as well as make the continuation algorithm more flexible. In applications, we use the continuation software package AUTO in combination with some initial value software. The examples considered include computation of homoclinic orbits in a singular perturbation problem and in a turbulent fluid boundary layer in the wall region problem.

Vaughan, William W.↗
NASA NTRS2003

Applications of Space-Filling-Curves to Cartesian Methods for CFD

The proposed paper presents a variety novel uses of Space-Filling-Curves (SFCs) for Cartesian mesh methods in 0. While these techniques will be demonstrated using non-body-fitted Cartesian meshes, most are applicable on general body-fitted meshes -both structured and unstructured. We demonstrate the use of single O(N log N) SFC-based reordering to produce single-pass (O(N)) algorithms for mesh partitioning, multigrid coarsening, and inter-mesh interpolation. The intermesh interpolation operator has many practical applications including warm starts on modified geometry, or as an inter-grid transfer operator on remeshed regions in moving-body simulations. Exploiting the compact construction of these operators, we further show that these algorithms are highly amenable to parallelization. Examples using the SFC-based mesh partitioner show nearly linear speedup to 512 CPUs even when using multigrid as a smoother. Partition statistics are presented showing that the SFC partitions are, on-average, within 10% of ideal even with only around 50,000 cells in each subdomain. The inter-mesh interpolation operator also has linear asymptotic complexity and can be used to map a solution with N unknowns to another mesh with M unknowns with O(max(M,N)) operations. This capability is demonstrated both on moving-body simulations and in mapping solutions to perturbed meshes for finite-difference-based gradient design methods.

Aftosmis, Michael J.↗
NASA NTRS1991

Aerodynamic shape optimization using sensitivity analysis on third-order Euler equations

Two major advancements of an aerodynamic optimization method with two design variables are described. The first is represnted by an improved flow prediction method that uses the third-order accurate discretization of the Euler equations. This method makes it possible to predict the flowfield of a perturbed shape which generates shocks and other large gradients without intermediate CFD analysis. The second advancement is the use of every surface grid point as a design variable in the aerodynamic shape optimization problem. The improved algorithm is demonstrated by optimizing the ramp shape of a scramjet-afterbody configuration for maximum axial thrust. It is concluded that the improved flow-field prediction method eliminates most of the flow analysis and a priori guessing of all possible shapes from which the optimum is to be selected.

Baysal, Oktay↗
NASA NTRS1997

BFS Method for Alloys Optimized and Verified for the Study of Ordered Intermetallic Material

The aerospace industry has a need for new metallic alloys that are lightweight and have high strength at elevated temperatures. The BFS (Bozzolo, Ferrante, and Smith) method is a new, computationally efficient and physically sound quantum semi-perturbative approach for describing metals and their defects. Based on a simple interpretation of the alloy formation process that identifies strain and chemical contributions to the energy of the alloy, the method provides an atom-by-atom description of an alloy. Its implementation requires little more than algebra and the solution of transcendental equations. At the NASA Lewis Research Center, we have demonstrated that BFS can investigate the properties of a large number of alloys with a minimum computational effort on low-level computers. This screening allows the selection of the best alloy candidates for a particular application and, therefore, promises large cost savings over current approaches.

Source record↗
NASA NTRS1976

Consequences of using nonlinear particle trajectories to compute spatial diffusion coefficients

The propagation of charged particles through interstellar and interplanetary space has often been described as a random process in which the particles are scattered by ambient electromagnetic turbulence. In general, this changes both the magnitude and direction of the particles' momentum. Some situations for which scattering in direction (pitch angle) is of primary interest were studied. A perturbed orbit, resonant scattering theory for pitch-angle diffusion in magnetostatic turbulence was slightly generalized and then utilized to compute the diffusion coefficient for spatial propagation parallel to the mean magnetic field, Kappa. All divergences inherent in the quasilinear formalism when the power spectrum of the fluctuation field falls off as K to the minus Q power (Q less than 2) were removed. Various methods of computing Kappa were compared and limits on the validity of the theory discussed. For Q less than 1 or 2, the various methods give roughly comparable values of Kappa, but use of perturbed orbits systematically results in a somewhat smaller Kappa than can be obtained from quasilinear theory.

Goldstein, M. L.↗
NASA NTRS1995

A Green's function method for heavy ion beam transport

The use of Green's function has played a fundamental role in transport calculations for high-charge high-energy (HZE) ions. Two recent developments have greatly advanced the practical aspects of implementation of these methods. The first was the formulation of a closed-form solution as a multiple fragmentation perturbation series. The second was the effective summation of the closed-form solution through nonperturbative techniques. The nonperturbative methods have been recently extended to an inhomogeneous, two-layer transport media to simulate the lead scattering foil present in the Lawrence Berkeley Laboratories (LBL) biomedical beam line used for cancer therapy. Such inhomogeneous codes are necessary for astronaut shielding in space. The transport codes utilize the Langley Research Center atomic and nuclear database. Transport code and database evaluation are performed by comparison with experiments performed at the LBL Bevalac facility using 670 A MeV 20Ne and 600 A MeV 56Fe ion beams. The comparison with a time-of-flight and delta E detector measurement for the 20Ne beam and the plastic nuclear track detectors for 56Fe show agreement up to 35%-40% in water and aluminium targets, respectively.

NASA Discipline Radiation Health↗
NASA NTRS1990

A parallel algorithm for nonlinear convection-diffusion equations

A parallel algorithm for the efficient solution of nonlinear time-dependent convection-diffusion equations with small parameter on the diffusion term is presented. The method is based on a physically motivated domain decomposition that is dictated by singular perturbation analysis. The analysis is used to determine regions where certain reduced equations may be solved in place of the full equation. The method is suitable for the solution of problems arising in the simulation of fluid dynamics. Experimental results for a nonlinear equation in two-dimensions are presented.

Scroggs, Jeffrey S.↗
NASA NTRS1999

High Order Approximations for Compressible Fluid Dynamics on Unstructured and Cartesian Meshes

The development of high-order accurate numerical discretization techniques for irregular domains and meshes is often cited as one of the remaining challenges facing the field of computational fluid dynamics. In structural mechanics, the advantages of high-order finite element approximation are widely recognized. This is especially true when high-order element approximation is combined with element refinement (h-p refinement). In computational fluid dynamics, high-order discretization methods are infrequently used in the computation of compressible fluid flow. The hyperbolic nature of the governing equations and the presence of solution discontinuities makes high-order accuracy difficult to achieve. Consequently, second-order accurate methods are still predominately used in industrial applications even though evidence suggests that high-order methods may offer a way to significantly improve the resolution and accuracy for these calculations. To address this important topic, a special course was jointly organized by the Applied Vehicle Technology Panel of NATO's Research and Technology Organization (RTO), the von Karman Institute for Fluid Dynamics, and the Numerical Aerospace Simulation Division at the NASA Ames Research Center. The NATO RTO sponsored course entitled "Higher Order Discretization Methods in Computational Fluid Dynamics" was held September 14-18, 1998 at the von Karman Institute for Fluid Dynamics in Belgium and September 21-25, 1998 at the NASA Ames Research Center in the United States. During this special course, lecturers from Europe and the United States gave a series of comprehensive lectures on advanced topics related to the high-order numerical discretization of partial differential equations with primary emphasis given to computational fluid dynamics (CFD). Additional consideration was given to topics in computational physics such as the high-order discretization of the Hamilton-Jacobi, Helmholtz, and elasticity equations. This volume consists of five articles prepared by the special course lecturers. These articles should be of particular relevance to those readers with an interest in numerical discretization techniques which generalize to very high-order accuracy. The articles of Professors Abgrall and Shu consider the mathematical formulation of high-order accurate finite volume schemes utilizing essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) reconstruction together with upwind flux evaluation. These formulations are particularly effective in computing numerical solutions of conservation laws containing solution discontinuities. Careful attention is given by the authors to implementational issues and techniques for improving the overall efficiency of these methods. The article of Professor Cockburn discusses the discontinuous Galerkin finite element method. This method naturally extends to high-order accuracy and has an interpretation as a finite volume method. Cockburn addresses two important issues associated with the discontinuous Galerkin method: controlling spurious extrema near solution discontinuities via "limiting" and the extension to second order advective-diffusive equations (joint work with Shu). The articles of Dr. Henderson and Professor Schwab consider the mathematical formulation and implementation of the h-p finite element methods using hierarchical basis functions and adaptive mesh refinement. These methods are particularly useful in computing high-order accurate solutions containing perturbative layers and corner singularities. Additional flexibility is obtained using a mortar FEM technique whereby nonconforming elements are interfaced together. Numerous examples are given by Henderson applying the h-p FEM method to the simulation of turbulence and turbulence transition.

Barth, Timothy↗
NASA NTRS2023

Three-Wavelength Approach for Aerosol-Cloud Discrimination in the Sage III/ISS Aerosol Extinction Dataset

The tropical upper troposphere and lower stratosphere (UTLS) region is dominated by aerosols and clouds affecting Earth’s radiation budget and climate. Thus, satellites’ continuous monitoring and identification of these layers is crucial for quantifying their radiative impact. However, distinguishing between aerosols and clouds is challenging, especially under the perturbed UTLS conditions during post-volcanic eruptions and wildfire events. Aerosol-cloud discrimination is primarily based on their disparate wavelength-dependent scattering and absorption properties. In this study, we use aerosol extinction observations in the tropical (15°N-15°S) UTLS from June 2017 to February 2021, available from the latest generation of the Stratospheric Aerosol and Gas Experiment (SAGE) instrument-SAGE III onboard the International Space Station (ISS) to study aerosols and clouds. During this period, the SAGE III/ISS provided better coverage over the tropics at additional wavelength channels (relative to previous SAGE missions) and witnessed several volcanic and wildfire events that perturbed the tropical UTLS. We explore the advantage of having an extinction coefficient at an additional wavelength channel (1550 nm) from the SAGE III/ISS in aerosol-cloud discrimination using a method based on thresholds of two extinction coefficient ratios, R1R1 (520 nm/1020 nm) and R2R2 (1020 nm/1550 nm). This method was proposed earlier by Kent et al. [Appl. Opt. 36, 8639 (1997) [CrossRef] ] for the SAGE III-Meteor-3M but was never tested for the tropical region under volcanically perturbed conditions. We call this method the Extinction Color Ratio (ECR) method. The ECR method is applied to the SAGE III/ISS aerosol extinction data to obtain cloud-filtered aerosol extinction coefficients, cloud-top altitude, and seasonal cloud occurrence frequency during the entire study period. Cloud-filtered aerosol extinction coefficient obtained using the ECR method revealed the presence of enhanced aerosols in the UTLS following volcanic eruptions and wildfire events consistent with the Ozone Mapping and Profiler Suite (OMPS) and space-borne lidar-Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP). The cloud-top altitude obtained from the SAGE III/ISS is within 1 km of the nearly co-located observations from OMPS and CALIOP. In general, the seasonal mean cloud-top altitude from the SAGE III/ISS events peaks during the December, January, and February months, with sunset events showing higher cloud tops than the sunrise events, indicating the seasonal and diurnal variation of the tropical convection. The seasonal altitude distribution of cloud occurrence frequency obtained from the SAGE III/ISS also agrees well with CALIOP observations within 10%. We show that the ECR method is a simple approach that relies on thresholds independent of the sampling period, providing cloud-filtered aerosol extinction coefficients uniformly for climate studies irrespective of the UTLS conditions. However, since the predecessor of SAGE III did not include a 1550 nm channel, the usefulness of this approach is limited to short-term climate studies after 2017.

Clouds Aerosols↗
NASA NTRS1987

The characteristic Rossby frequency

The characteristic Rossby frequency is defined for a fixed zonal wavenumber perturbation as the variational integral of the Rayleigh-Ritz method. It is a measure of the time scale of the disturbance. For a disturbance which locally has the shape of an eigenfunction but is not global in extent, the characteristic Rossby frequency is very close to the true eigenvalue, and additionally remains unchanged under linear inviscid dynamics. Results are presented for the shallow water equations, both with and without a mean zonal wind. The characteristic Rossby frequency of a wavenumber 1 perturbation having the shape of the second symmetric Rossby mode but confined to the Northern Hemisphere is close to the corresponding Rossby frequency. This finding is helpful in understanding the behavior of the observed wavenumber 1 pattern of January 1979, which propagated westward with nearly the pure Rossby frequency but was discernible only in the Northern Hemisphere (as discussed by Daley and Williamson).

Straus, David M.↗
NASA NTRS1995

Contact and pressure balance structures in two-fluid cosmic-ray hydrodynamics

The role of cosmic-ray-modified contact discontinuities and pressure balance structures in two-fluid cosmic-ray hydrodynamics in one Cartesian space dimension are investigated by means of analytic and numerical solution examples, as well as by weakly nonlinear asymptotics. The fundamental wave modes of the two-fluid cosmic-ray hydrodynamic equations in the long-wavelength limit consist of the backward and forward propagating cosmic-ray-modified sound waves, with sound speed dependent on both the cosmic-ray and thermal gas pressures; the contact discontinuity; and a pressure balance mode in which the sum ofthe cosmic ray and thermal gas pressure perturbations is zero. The pressure balance mode, like the contact discontinuity is advected with the background flow. The interaction of the pressure balance mode with the contact discontinuity is investigated by means of the method of multiple scales. The thermal gas and cosmic-ray pressure perturbations satisfy a linear diffusion equation, and entropy perturbations arising from nonisentropic initial conditions for the thermal gas are frozen into the fluid. The contact discontinuity and pressure balance eigenmodes both admit nonzero perturbations in the thermal gas, whereas the cosmic-ray-modified sound waves are isentropic. The total entropy perturbation is shared between the contact discontinuity and pressure balance eigenmodes, and examples are given in which there is a transfer of entropy between the two modes. In particular, N-wave type density disturbances are obtained which arise as a result of the entropy transfer between the two modes. A weakly nonlinear geometric optics perturbation expansion is used to study the long timescale evolution of the short-wavelength entropy wave and the thermal gas sound waves in a slowly varying, large-scale background flow. The weakly nonlinear geometric optics expansion is also used to generalize previous studies of squeezing instability for short-wavelength sound waves in the two fluid model, by including a weakly nonlinear wave steepening term that leads to shock formation, as well as the effect of long time and space dependence of the background flow. Implications of cosmic-ray-modified pressure balance structures and contact discontinuities in models of the interaction of traveling interplanetary shocks and compression and rarefraction waves with the solar wind termination shock are briefly discussed.

Webb, G. M.↗
NASA NTRS1993

A comparison of two multi-variable integrator windup protection schemes

Two methods are examined for limit and integrator wind-up protection for multi-input, multi-output linear controllers subject to actuator constraints. The methods begin with an existing linear controller that satisfies the specifications for the nominal, small perturbation, linear model of the plant. The controllers are formulated to include an additional contribution to the state derivative calculations. The first method to be examined is the multi-variable version of the single-input, single-output, high gain, Conventional Anti-Windup (CAW) scheme. Except for the actuator limits, the CAW scheme is linear. The second scheme to be examined, denoted the Modified Anti-Windup (MAW) scheme, uses a scalar to modify the magnitude of the controller output vector while maintaining the vector direction. The calculation of the scalar modifier is a nonlinear function of the controller outputs and the actuator limits. In both cases the constrained actuator is tracked. These two integrator windup protection methods are demonstrated on a turbofan engine control system with five measurements, four control variables, and four actuators. The closed-loop responses of the two schemes are compared and contrasted during limit operation. The issue of maintaining the direction of the controller output vector using the Modified Anti-Windup scheme is discussed and the advantages and disadvantages of both of the IWP methods are presented.

Mattern, Duane↗