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At least 343 records · Page 19

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NASA NTRS1995

A method of solving simple harmonic oscillator Schroedinger equation

A usual step in solving totally Schrodinger equation is to try first the case when dimensionless position independent variable w is large. In this case the Harmonic Oscillator equation takes the form (d(exp 2)/dw(exp 2) - w(exp 2))F = 0, and following W.K.B. method, it gives the intermediate corresponding solution F = exp(-w(exp 2)/2), which actually satisfies exactly another equation, (d(exp 2)/dw(exp 2) + 1 - w(exp 2))F = 0. We apply a different method, useful in anharmonic oscillator equations, similar to that of Rampal and Datta, and although it is slightly more complicated however it is also more general and systematic.

Maury, Juan Carlos F.↗

NASA NTRS1995

A combined geometric approach for solving the Navier-Stokes equations on dynamic grids

A combined geometric approach for solving the Navier-Stokes equations is presented for the analysis of planar, unsteady flow about mechanisms with components in moderate relative motion. The approach emphasizes the relationships between the geometry model, grid, and flow model for the benefit of the total dynamics problem. One application is the analysis of the restart operation of a variable-geometry, high-speed inlet.

Slater, John W.↗

NASA NTRS1995

Computer program BL2D for solving two-dimensional and axisymmetric boundary layers

This report presents the formulation, validation, and user's manual for the computer program BL2D. The program is a fourth-order-accurate solution scheme for solving two-dimensional or axisymmetric boundary layers in speed regimes that range from low subsonic to hypersonic Mach numbers. A basic implementation of the transition zone and turbulence modeling is also included. The code is a result of many improvements made to the program VGBLP, which is described in NASA TM-83207 (February 1982), and can effectively supersede it. The code BL2D is designed to be modular, user-friendly, and portable to any machine with a standard fortran77 compiler. The report contains the new formulation adopted and the details of its implementation. Five validation cases are presented. A detailed user's manual with the input format description and instructions for running the code is included. Adequate information is presented in the report to enable the user to modify or customize the code for specific applications.

Iyer, Venkit↗

NASA NTRS1995

A numerical method for solving the three-dimensional parabolized Navier-Stokes equations

A numerical technique that solves the parabolized form of the Navier-Stokes equations is presented. Such a method makes it possible to obtain very detailed descriptions of the flowfield in a relatively modest CPU time. The present approach is based on a space-marching technique, uses a finite volume discretization and an upwind flux-difference splitting scheme for the evaluation of the inviscid fluxes. Second order accuracy is achieved following the guidelines of the the ENO schemes. The methodology is used to investigate three-dimensional supersonic viscous flows over symmetric corners. Primary and secondary streamwise vortical structures embedded in the boundary layer and originated by the interaction with shock waves are detected and studied. For purpose of validation, results are compared with experimental data extracted from literature. The agreement is found to be satisfactory. In conclusion, the numerical method proposed seems to be promising as it permits, at a reasonable computational expense, investigation of complex three-dimensional flowfields in great detail.

Dambrosio, Domenic↗

NASA NTRS1996

Multiple Coarse Grid Multigrid Methods for Solving Elliptic Problems

In this paper we describe some classes of multigrid methods for solving large linear systems arising in the solution by finite difference methods of certain boundary value problems involving Poisson's equation on rectangular regions. If parallel computing systems are used, then with standard multigrid methods many of the processors will be idle when one is working at the coarsest grid levels. We describe the use of Multiple Coarse Grid MultiGrid (MCGMG) methods. Here one first constructs a periodic set of equations corresponding to the given system. One then constructs a set of coarse grids such that for each grid corresponding to the grid size h there are four grids corresponding to the grid size 2*h. Multigrid operations such as restriction of residuals and interpolation of corrections are done in parallel at each grid level. For suitable choices of the multigrid operators the MCGMG method is equivalent to the Parallel Superconvergent MultiGrid (PSMG) method of Frederickson and McBryan. The convergence properties of MCGMG methods can be accurately analyzed using spectral methods.

Xiao, Shengyou↗

NASA NTRS1997

On Approximate Factorization Schemes for Solving the Full Potential Equation

An approximate factorization scheme based on the AF2 algorithm is presented for solving the three-dimensional full potential equation for the transonic flow about isolated wings. Two spatial discretization variations are presented, one using a hybrid first-order/second-order-accurate scheme and the second using a fully second-order-accurate scheme. The present algorithm utilizes a C-H grid topology to map the flow field about the wing. One version of the AF2 iteration scheme is used on the upper wing surface and another slightly modified version is used on the lower surface. These two algorithm variations are then connected at the wing leading edge using a local iteration technique. The resulting scheme has improved linear stability characteristics and improved time-like damping characteristics relative to previous implementations of the AF2 algorithm. The presentation is highlighted with a grid refinement study and a number of numerical results.

Holst, Terry L.↗

NASA NTRS1980

Fast, Conservative Algorithm for Solving the Transonic Full-Potential Equation

A fast, fully implicit approximate factorization algorithm designed to solve the conservative, transonic, full-potential equation in either two or three dimensions is described. The algorithm uses an upwind bias of the density coefficient for stability in supersonic regions. This provides an effective upwind difference of the streamwise terms for any orientation of the velocity vector (i.e., rotated differencing), thereby greatly enhancing the reliability of the present algorithm. A numerical transformation is used to establish an arbitrary body-fitted, finite-difference mesh. Computed results for both airfoils and simplified wings demonstrate substantial improvement in convergence speed for the new algorithm relative to standard successive-line over-relaxation algorithms.

Holst, Terry L.↗

NASA NTRS1998

The Space-Time Conservation Element and Solution Element Method: A New High-Resolution and Genuinely Multidimensional Paradigm for Solving Conservation Laws: The Two Dimensional Time Marching Schemes - 1

A new high resolution and genuinely multidimensional numerical method for solving conservation laws is being, developed. It was designed to avoid the limitations of the traditional methods. and was built from round zero with extensive physics considerations. Nevertheless, its foundation is mathmatically simple enough that one can build from it a coherent, robust. efficient and accurate numerical framework. Two basic beliefs that set the new method apart from the established methods are at the core of its development. The first belief is that, in order to capture physics more efficiently and realistically, the modeling, focus should be placed on the original integral form of the physical conservation laws, rather than the differential form. The latter form follows from the integral form under the additional assumption that the physical solution is smooth, an assumption that is difficult to realize numerically in a region of rapid chance. such as a boundary layer or a shock. The second belief is that, with proper modeling of the integral and differential forms themselves, the resulting, numerical solution should automatically be consistent with the properties derived front the integral and differential forms, e.g., the jump conditions across a shock and the properties of characteristics. Therefore a much simpler and more robust method can be developed by not using the above derived properties explicitly.

Chang, Sin-Chung↗

NASA NTRS1999

A Pseubo-Temporal Multi-Grid Relaxation Scheme for Solving the Parabolized Navier-Stokes Equations

A multi-grid, flux-difference-split, finite-volume code, VULCAN, is presented for solving the elliptic and parabolized form of the equations governing three-dimensional, turbulent, calorically perfect and non-equilibrium chemically reacting flows. The space marching algorithms developed to improve convergence rate and or reduce computational cost are emphasized. The algorithms presented are extensions to the class of implicit pseudo-time iterative, upwind space-marching schemes. A full approximate storage, full multi-grid scheme is also described which is used to accelerate the convergence of a Gauss-Seidel relaxation method. The multi-grid algorithm is shown to significantly improve convergence on high aspect ratio grids.

Morrison, J. H.↗

NASA NTRS1999

Solving Upwind-Biased Discretizations: Multigrid Solver Using Semicoarsening - 2

This paper studies a novel multigrid approach to the solution for a second order upwind biased discretization of the convection equation in two dimensions. This approach is based on semi-coarsening and well balanced explicit correction terms added to coarse-grid operators to maintain on coarse-grid the same cross-characteristic interaction as on the target (fine) grid. Colored relaxation schemes are used on all the levels allowing a very efficient parallel implementation. The results of the numerical tests can be summarized as follows: 1) The residual asymptotic convergence rate of the proposed V(0, 2) multigrid cycle is about 3 per cycle. This convergence rate far surpasses the theoretical limit (4/3) predicted for standard multigrid algorithms using full coarsening. The reported efficiency does not deteriorate with increasing the cycle, depth (number of levels) and/or refining the target-grid mesh spacing. 2) The full multi-grid algorithm (FMG) with two V(0, 2) cycles on the target grid and just one V(0, 2) cycle on all the coarse grids always provides an approximate solution with the algebraic error less than the discretization error. Estimates of the total work in the FMG algorithm are ranged between 18 and 30 minimal work units (depending on the target (discretizatioin). Thus, the overall efficiency of the FMG solver closely approaches (if does not achieve) the goal of the textbook multigrid efficiency. 3) A novel approach to deriving a discrete solution approximating the true continuous solution with a relative accuracy given in advance is developed. An adaptive multigrid algorithm (AMA) using comparison of the solutions on two successive target grids to estimate the accuracy of the current target-grid solution is defined. A desired relative accuracy is accepted as an input parameter. The final target grid on which this accuracy can be achieved is chosen automatically in the solution process. the actual relative accuracy of the discrete solution approximation obtained by AMA is always better than the required accuracy; the computational complexity of the AMA algorithm is (nearly) optimal (comparable with the complexity of the FMG algorithm applied to solve the problem on the optimally spaced target grid).

Diskin, Boris↗

NASA NTRS2000

An Intercomparison of Lidar Ozone and Temperature Measurements From the SOLVE Mission With Predicted Model Values

Temperature profiles acquired by Goddard Space Flight Center's AROTEL lidar during the SOLVE mission onboard NASA's DC-8 are compared with predicted values from several atmospheric models (DAO, NCEP and UKMO). The variability in the differences between measured and calculated temperature fields was approximately 5 K. Retrieved temperatures within the polar vortex showed large regions that were significantly colder than predicted by the atmospheric models.

Burris, John↗

NASA NTRS2000

Arctic chemical Ozone Loss Observed by the AROTEL Instrument during the SOLVE Campaign, December 1999 - March 2000

During the winter of 1999-2000, the AROTEL instrument was deployed on the NASA DC-8 at Kiruna, Sweden for the SAGE III Ozone Loss Validation Experiment (SOLVE). Measurements of ozone, temperature and aerosols were made on 18 local science flights from December to March. Extremely low temperatures were observed throughout most of the Arctic vortex and polar stratospheric clouds were observed throughout the Arctic area during January. Significant ozone loss was measured after the sun began to rise on the vortex area in February. Ozone mixing ratios as low as 800 ppbv were observed during flights in March.

McGee, Thomas J.↗

NASA NTRS2000

Comparison of Temperature and Ozone Measured by the AROTEL Instrument on DC8 Overflights of Ny Aalesund during the SOLVE Mission

The AROTEL instrument, deployed on the NASA DC-8 at Kiruna, Sweden for the SAGE III Ozone Loss and Validation Experiment (SOLVE), flew over the NDSC station operated by the Alfred Wegner Institute at Ny Aalesund, Spitsbergen. AROTEL ozone and temperature measurements made during near overflights of Ny Aalesund are compared with sonde ozone and temperature, and lidar ozone measurements from the NDSC station. Nine of the seventeen science flights during the December through March measurement period overflew near Ny Aalesund. Agreement of AROTEL with the ground-based temperature and ozone values at altitudes from just above the aircraft to about 30 km gives strong confidence in using AROTEL temperature and ozone mixing ratio to study the mechanisms of ozone loss in the winter arctic polar region.

Hoegy, Walter R.↗

NASA NTRS2000

Towards Solving the Mixing Problem in the Decomposition of Geophysical Time Series by Independent Component Analysis

The use of the Principal Component Analysis technique for the analysis of geophysical time series has been questioned in particular for its tendency to extract components that mix several physical phenomena even when the signal is just their linear sum. We demonstrate with a data simulation experiment that the Independent Component Analysis, a recently developed technique, is able to solve this problem. This new technique requires the statistical independence of components, a stronger constraint, that uses higher-order statistics, instead of the classical decorrelation a weaker constraint, that uses only second-order statistics. Furthermore, ICA does not require additional a priori information such as the localization constraint used in Rotational Techniques.

Aires, Filipe↗

NASA NTRS2000

An Assessment of the Ozone Loss During the 1999-2000 SOLVE Campaign

Ozone observations from ozonesondes, the DIAL and AROTEL lidars aboard the DC-8, in situ ozone measurements from the ER-2 and satellite ozone measurements from POAM were used to assess ozone loss during the SOLVE 1999-2000 campaign. We compare three different methods of computing the ozone loss. The first method simply compares the time sequence of ozonesondes taken at the same station inside the vortex from December through the end of March. In the second method, ozonesondes from a variety of stations are compared using a variant on the Match technique. This method uses short (approx. 5-10 day) forward diabatic trajectories to connect various sonde launches. In the third method, the measurements are simply injected into a diabatic trajectory model and carried forward in time from December 1 to March 16. Over 60,000 individual measurements were used in the last calculation. Again, ozone loss is estimated by comparing vortex interior measurements made early in the campaign with those made later in the campaign. The diabatic nature of the second and third methods calculation presumably corrects for the normal increase in ozone within the vortex due to downward advection. The three methods agree that the largest ozone loss occurs between 400 and 460 K potential temperatures (approx. 16-20 km) with slightly over 1.5 ppmv lost over the winter period. Between 460 K and 500 K (approx. 22 km) net ozone loss is less than 0.8 ppmv. From 500K to 600K (26 km) net loss is less than 0.5 ppmv.

Schoeberl, M. R.↗

NASA NTRS2001

Lidar Temperature Measurements During the SOLVE Campaign and the Absence of PSCs from Regions of Very Cold Air

NASA Goddard Space Flight Center's Airborne Raman Ozone, Temperature and Aerosol Lidar (AROTEL) measured extremely cold temperatures during all three deployments (December 1-16, 1999, January 14-29, 2000 and February 27-March 15, 2000) of the Sage III Ozone Loss and Validation Experiment (SOLVE). Temperatures were significantly below values observed in previous years with large regions regularly below 191 K and frequent temperature retrievals yielding values at or below 187 K. Temperatures well below the saturation point of type I polar stratospheric clouds (PSCs) were regularly encountered but their presence was not well correlated with PSCs observed by the NASA Langley Research Center's Aerosol Lidar co-located with AROTEL. Temperature measurements by meteorological sondes launched within areas traversed by the DC-8 showed minimum temperatures consistent in time and vertical extent with those derived from AROTEL data. Calculations to establish whether PSCs could exist at measured AROTEL temperatures and observed mixing ratios of nitric acid and water vapor showed large regions favorable to PSC formation. On several occasions measured AROTEL temperatures up to 10 K below the NAT saturation temperature were insufficient to produce PSCs even though measured values of nitric acid and water were sufficient for their formation.

Burris, John↗

NASA NTRS2001

Validation of Temperature Measurements from the Airborne Raman Ozone Temperature and Aerosol Lidar During SOLVE

The Airborne Raman Ozone, Temperature and Aerosol Lidar (AROTEL) participated in the recent Sage III Ozone Loss and Validation Experiment (SOLVE) by providing profiles of aerosols, polar stratospheric clouds (PSCs), ozone and temperature with high vertical and horizontal resolution. Temperatures were derived from just above the aircraft to approximately 60 kilometers geometric altitude with a reported vertical resolution of between 0.5 and 1.5 km. The horizontal footprint varied from 4 to 70 km. This paper explores the measurement uncertainties associated with the temperature retrievals and makes comparisons with independent, coincident, measurements of temperature. Measurement uncertainties range from 0.1 K to approximately 4 K depending on altitude and integration time. Comparisons between AROTEL and balloon sonde temperatures retrieved under clear sky conditions using both Rayleigh and Raman scattered data showed AROTEL approximately 1 K colder than sonde values. Comparisons between AROTEL and the Meteorological Measurement System (MMS) on NASA's ER-2 show AROTEL being from 2-3 K colder for altitudes ranging from 14 to 18 km. Temperature comparisons between AROTEL and the United Kingdom Meteorological Office's model showed differences of approximately 1 K below approximately 25 km and a very strong cold bias of approximately 12 K at altitudes between 30 and 35 km.

Burris, John↗

NASA NTRS2001

A Space-Time Conservation Element and Solution Element Method for Solving the Two- and Three-Dimensional Unsteady Euler Equations Using Quadrilateral and Hexahedral Meshes

In this paper, we report a version of the Space-Time Conservation Element and Solution Element (CE/SE) Method in which the 2D and 3D unsteady Euler equations are simulated using structured or unstructured quadrilateral and hexahedral meshes, respectively. In the present method, mesh values of flow variables and their spatial derivatives are treated as independent unknowns to be solved for. At each mesh point, the value of a flow variable is obtained by imposing a flux conservation condition. On the other hand, the spatial derivatives are evaluated using a finite-difference/weighted-average procedure. Note that the present extension retains many key advantages of the original CE/SE method which uses triangular and tetrahedral meshes, respectively, for its 2D and 3D applications. These advantages include efficient parallel computing ease of implementing non-reflecting boundary conditions, high-fidelity resolution of shocks and waves, and a genuinely multidimensional formulation without using a dimensional-splitting approach. In particular, because Riemann solvers, the cornerstones of the Godunov-type upwind schemes, are not needed to capture shocks, the computational logic of the present method is considerably simpler. To demonstrate the capability of the present method, numerical results are presented for several benchmark problems including oblique shock reflection, supersonic flow over a wedge, and a 3D detonation flow.

Zhang, Zeng-Chan↗