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NASA NTRS1990

An adaptive grid algorithm for one-dimensional nonlinear equations

Richards' equation, which models the flow of liquid through unsaturated porous media, is highly nonlinear and difficult to solve. Step gradients in the field variables require the use of fine grids and small time step sizes. The numerical instabilities caused by the nonlinearities often require the use of iterative methods such as Picard or Newton interation. These difficulties result in large CPU requirements in solving Richards equation. With this in mind, adaptive and multigrid methods are investigated for use with nonlinear equations such as Richards' equation. Attention is focused on one-dimensional transient problems. To investigate the use of multigrid and adaptive grid methods, a series of problems are studied. First, a multigrid program is developed and used to solve an ordinary differential equation, demonstrating the efficiency with which low and high frequency errors are smoothed out. The multigrid algorithm and an adaptive grid algorithm is used to solve one-dimensional transient partial differential equations, such as the diffusive and convective-diffusion equations. The performance of these programs are compared to that of the Gauss-Seidel and tridiagonal methods. The adaptive and multigrid schemes outperformed the Gauss-Seidel algorithm, but were not as fast as the tridiagonal method. The adaptive grid scheme solved the problems slightly faster than the multigrid method. To solve nonlinear problems, Picard iterations are introduced into the adaptive grid and tridiagonal methods. Burgers' equation is used as a test problem for the two algorithms. Both methods obtain solutions of comparable accuracy for similar time increments. For the Burgers' equation, the adaptive grid method finds the solution approximately three times faster than the tridiagonal method. Finally, both schemes are used to solve the water content formulation of the Richards' equation. For this problem, the adaptive grid method obtains a more accurate solution in fewer work units and less computation time than required by the tridiagonal method. The performance of the adaptive grid method tends to degrade as the solution process proceeds in time, but still remains faster than the tridiagonal scheme.

Gutierrez, William E.↗

NASA NTRS2011

Optimal Padding for the Two-Dimensional Fast Fourier Transform

One-dimensional Fast Fourier Transform (FFT) operations work fastest on grids whose size is divisible by a power of two. Because of this, padding grids (that are not already sized to a power of two) so that their size is the next highest power of two can speed up operations. While this works well for one-dimensional grids, it does not work well for two-dimensional grids. For a two-dimensional grid, there are certain pad sizes that work better than others. Therefore, the need exists to generalize a strategy for determining optimal pad sizes. There are three steps in the FFT algorithm. The first is to perform a one-dimensional transform on each row in the grid. The second step is to transpose the resulting matrix. The third step is to perform a one-dimensional transform on each row in the resulting grid. Steps one and three both benefit from padding the row to the next highest power of two, but the second step needs a novel approach. An algorithm was developed that struck a balance between optimizing the grid pad size with prime factors that are small (which are optimal for one-dimensional operations), and with prime factors that are large (which are optimal for two-dimensional operations). This algorithm optimizes based on average run times, and is not fine-tuned for any specific application. It increases the amount of times that processor-requested data is found in the set-associative processor cache. Cache retrievals are 4-10 times faster than conventional memory retrievals. The tested implementation of the algorithm resulted in faster execution times on all platforms tested, but with varying sized grids. This is because various computer architectures process commands differently. The test grid was 512 512. Using a 540 540 grid on a Pentium V processor, the code ran 30 percent faster. On a PowerPC, a 256x256 grid worked best. A Core2Duo computer preferred either a 1040x1040 (15 percent faster) or a 1008x1008 (30 percent faster) grid. There are many industries that can benefit from this algorithm, including optics, image-processing, signal-processing, and engineering applications.

Dean, Bruce H.↗

NASA NTRSDate not supplied

Large-Eddy Simulation of the Variable Speed Power Turbine Cascade with Inflow Turbulence

The main goal of this is to implement a digital filtering method to generate turbulence upstream and a sub-grid model (Localized dynamic k-equation model (LDKM)) in the framework of LES which together enable reproduction of the effect of inflow turbulence on the transition seen in the VSPT experimental data at the cruise condition (incidence angle of 40o and Tu=0.5%, 5%, 10% and 15%.).

CFD↗

NASA NTRS1993

Structured adaptive grid generation using algebraic methods

The accuracy of the numerical algorithm depends not only on the formal order of approximation but also on the distribution of grid points in the computational domain. Grid adaptation is a procedure which allows optimal grid redistribution as the solution progresses. It offers the prospect of accurate flow field simulations without the use of an excessively timely, computationally expensive, grid. Grid adaptive schemes are divided into two basic categories: differential and algebraic. The differential method is based on a variational approach where a function which contains a measure of grid smoothness, orthogonality and volume variation is minimized by using a variational principle. This approach provided a solid mathematical basis for the adaptive method, but the Euler-Lagrange equations must be solved in addition to the original governing equations. On the other hand, the algebraic method requires much less computational effort, but the grid may not be smooth. The algebraic techniques are based on devising an algorithm where the grid movement is governed by estimates of the local error in the numerical solution. This is achieved by requiring the points in the large error regions to attract other points and points in the low error region to repel other points. The development of a fast, efficient, and robust algebraic adaptive algorithm for structured flow simulation applications is presented. This development is accomplished in a three step process. The first step is to define an adaptive weighting mesh (distribution mesh) on the basis of the equidistribution law applied to the flow field solution. The second, and probably the most crucial step, is to redistribute grid points in the computational domain according to the aforementioned weighting mesh. The third and the last step is to reevaluate the flow property by an appropriate search/interpolate scheme at the new grid locations. The adaptive weighting mesh provides the information on the desired concentration of points to the grid redistribution scheme. The evaluation of the weighting mesh is accomplished by utilizing the weight function representing the solution variation and the equidistribution law. The selection of the weight function plays a key role in grid adaptation. A new weight function utilizing a properly weighted boolean sum of various flowfield characteristics is defined. The redistribution scheme is developed utilizing Non-Uniform Rational B-Splines (NURBS) representation. The application of NURBS representation results in a well distributed smooth grid by maintaining the fidelity of the geometry associated with boundary curves. Several algebraic methods are applied to smooth and/or nearly orthogonalize the grid lines. An elliptic solver is utilized to smooth the grid lines if there are grid crossings. Various computational examples of practical interest are presented to demonstrate the success of these methods.

Yang, Jiann-Cherng↗

NASA NTRS1974

The effect of small-scale vertical mixing of horizontal momentum in a general circulation model

Several experiments are described in which the sub-grid-scale vertical eddy viscosity in the GISS global general circulation model was varied. The results show that large viscosities suppress large-scale eddies in middle and high latitudes, but enhance the circulation in the tropical Hadley cell and increase the extent of the tropical easterlies. Comparison with observations shows that the GISS model requires eddy viscosities about 1 sq m per sec or less to give realistic results for middle and high latitudes, and eddy viscosities about 100 sq m per sec to give realistic results for low latitudes. A plausible mechanism for the implied increase in small-scale mixing in low latitudes is cumulus convection.

Stone, P. H.↗

NASA NTRS1977

Evaluation of subgrid-scale turbulence models using a fully simulated turbulent flow

An exact turbulent flow field was calculated on a three-dimensional grid with 64 points on a side. The flow simulates grid-generated turbulence from wind tunnel experiments. In this simulation, the grid spacing is small enough to include essentially all of the viscous energy dissipation, and the box is large enough to contain the largest eddy in the flow. The method is limited to low-turbulence Reynolds numbers, in our case R sub lambda = 36.6. To complete the calculation using a reasonable amount of computer time with reasonable accuracy, a third-order time-integration scheme was developed which runs at about the same speed as a simple first-order scheme. It obtains this accuracy by saving the velocity field and its first-time derivative at each time step. Fourth-order accurate space-differencing is used.

Clark, R. A.↗

NASA NTRS1977

Relationship between the kinetic energy budget and intensity of convection

Synoptic data collected over the eastern United States during the fourth Atmospheric Variability Experiment, April 24 and 25, 1975, is used to study the relationship between the kinetic energy budget and the intensity of convective activity. It is found that areas of intense convective activity are also major centers of kinetic energy activity. Energy processes increase in magnitude with an increase in convection intensity. Large generation of kinetic energy is associated with intense convection, but large quantities of energy are transported out of the area of convection. The kinetic energy budget associated with grid points having no convection differs greatly from the budgets of the three categories of convection. Weak energy processes are not associated with convection.

Fuelberg, H. E.↗

NASA NTRS1978

Analysis of dissection algorithms for vector computers

Recently two dissection algorithms (one-way and incomplete nested dissection) have been developed for solving the sparse positive definite linear systems arising from n by n grid problems. Concurrently, vector computers (such as the CDC STAR-100 and TI ASC) have been developed for large scientific applications. An analysis of the use of dissection algorithms on vector computers dictates that vectors of maximum length be utilized thereby implying little or no dissection; on the other hand, minimizing operation counts suggest that considerable dissection be performed. In this paper we discuss the resolution of this conflict by minimizing the total time required by vectorized versions of the two algorithms.

George, A.↗

NASA NTRS1993

Evapotranspiration and runoff from large land areas: Land surface hydrology for atmospheric general circulation models

A land surface hydrology parameterization for use in atmospheric GCM's is presented. The parameterization incorporates subgrid scale variability in topography, soils, soil moisture and precipitation. The framework of the model is the statistical distribution of a topography-soils index, which controls the local water balance fluxes, and is therefore taken to represent the large land area. Spatially variable water balance fluxes are integrated with respect to the topography-soils index to yield our large topography-soils distribution, and interval responses are weighted by the probability of occurrence of the interval. Grid square averaged land surface fluxes result. The model functions independently as a macroscale water balance model. Runoff ratio and evapotranspiration efficiency parameterizations are derived and are shown to depend on the spatial variability of the above mentioned properties and processes, as well as the dynamics of land surface-atmosphere interactions.

Famiglietti, J. S.↗

NASA NTRS1993

Atmospheric parameterization schemes for satellite cloud property retrieval during FIRE IFO 2

Satellite cloud retrieval algorithms generally require atmospheric temperature and humidity profiles to determine such cloud properties as pressure and height. For instance, the CO2 slicing technique called the ratio method requires the calculation of theoretical upwelling radiances both at the surface and a prescribed number (40) of atmospheric levels. This technique has been applied to data from, for example, the High Resolution Infrared Radiometer Sounder (HIRS/2, henceforth HIRS) flown aboard the NOAA series of polar orbiting satellites and the High Resolution Interferometer Sounder (HIS). In this particular study, four NOAA-11 HIRS channels in the 15-micron region are used. The ratio method may be applied to various channel combinations to estimate cloud top heights using channels in the 15-mu m region. Presently, the multispectral, multiresolution (MSMR) scheme uses 4 HIRS channel combination estimates for mid- to high-level cloud pressure retrieval and Advanced Very High Resolution Radiometer (AVHRR) data for low-level (is greater than 700 mb) cloud level retrieval. In order to determine theoretical upwelling radiances, atmospheric temperature and water vapor profiles must be provided as well as profiles of other radiatively important gas absorber constituents such as CO2, O3, and CH4. The assumed temperature and humidity profiles have a large effect on transmittance and radiance profiles, which in turn are used with HIRS data to calculate cloud pressure, and thus cloud height and temperature. For large spatial scale satellite data analysis, atmospheric parameterization schemes for cloud retrieval algorithms are usually based on a gridded product such as that provided by the European Center for Medium Range Weather Forecasting (ECMWF) or the National Meteorological Center (NMC). These global, gridded products prescribe temperature and humidity profiles for a limited number of pressure levels (up to 14) in a vertical atmospheric column. The FIRE IFO 2 experiment provides an opportunity to investigate current atmospheric profile parameterization schemes, compare satellite cloud height results using both gridded products (ECMWF) and high vertical resolution sonde data from the National Weather Service (NWS) and Cross Chain Loran Atmospheric Sounding System (CLASS), and suggest modifications in atmospheric parameterization schemes based on these results.

Titlow, James↗

NASA NTRS1999

Statistics of Cirrus Horizontal Inhomogeneity in the Southern Great Plains

Variability of cloud properties on scales smaller than that of a GCM grid is potentially important both for realistic parameterizations of microphysical processes and for the prediction of the large-scale radiative effects of clouds, have suggested that a simple model of marine stratocumulus variability, based on the assumption of Gaussian variation statistics of cloud depth, can explain the liquid water path histogram shapes observed in Landsat data. In advance of ARM SGP MMCR ice water path climatologies, we have examined aircraft ice water content statistics for cirrus clouds observed over Coffeyville, Kansas during FIRE 2. We find similar associations of histogram shape and cloud cover for these clouds, and we show that a simple modification of the model for cirrus combined with observed mean cloud depths, their standard deviations, and ambient thermodynamic conditions predicts both the histogram shape and cirrus cloud cover fairly well. This suggests that subgrid variability of cloud properties may be similar for vastly different cloud types, and that a universal parameterization of the effects of subgrid variability in GCMs as a function of only a few parameters may be a realistic goal.

Smith, S. A.↗

NASA NTRS2002

P1 Nonconforming Finite Element Method for the Solution of Radiation Transport Problems

The simulation of radiation transport in the optically thick flux-limited diffusion regime has been identified as one of the most time-consuming tasks within large simulation codes. Due to multimaterial complex geometry, the radiation transport system must often be solved on unstructured grids. In this paper, we investigate the behavior and the benefits of the unstructured P(sub 1) nonconforming finite element method, which has proven to be flexible and effective on related transport problems, in solving unsteady implicit nonlinear radiation diffusion problems using Newton and Picard linearization methods. Key words. nonconforrning finite elements, radiation transport, inexact Newton linearization, multigrid preconditioning

Kang, Kab S.↗

NASA NTRS2013

The MJO Transition from Shallow to Deep Convection in CloudSat/CALIPSO Data and GISS GCM Simulations

The relationship between convective penetration depth and tropospheric humidity is central to recent theories of the Madden-Julian oscillation (MJO). It has been suggested that general circulation models (GCMs) poorly simulate the MJO because they fail to gradually moisten the troposphere by shallow convection and simulate a slow transition to deep convection. CloudSat and Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) data are analyzed to document the variability of convection depth and its relation to water vapor during the MJO transition from shallow to deep convection and to constrain GCM cumulus parameterizations. Composites of cloud occurrence for 10MJO events show the following anticipatedMJO cloud structure: shallow and congestus clouds in advance of the peak, deep clouds near the peak, and upper-level anvils after the peak. Cirrus clouds are also frequent in advance of the peak. The Advanced Microwave Scanning Radiometer for EarthObserving System (EOS) (AMSR-E) columnwater vapor (CWV) increases by;5 mmduring the shallow- deep transition phase, consistent with the idea of moisture preconditioning. Echo-top height of clouds rooted in the boundary layer increases sharply with CWV, with large variability in depth when CWV is between;46 and 68 mm. International Satellite Cloud Climatology Project cloud classifications reproduce these climatological relationships but correctly identify congestus-dominated scenes only about half the time. A version of the Goddard Institute for Space Studies Model E2 (GISS-E2) GCM with strengthened entrainment and rain evaporation that produces MJO-like variability also reproduces the shallow-deep convection transition, including the large variability of cloud-top height at intermediate CWV values. The variability is due to small grid-scale relative humidity and lapse rate anomalies for similar values of CWV. 1.

DelGenio, Anthony G.↗

NASA NTRS2018

Computational Fluid Dynamics Analysis of the Stall Characteristics of a Wing Designed Based on Prandtl's Mininum Induced Drag

Stall characteristics of a wing designed based on Prandtls minimum induced drag solution using the bending moment as the design constraint is presented. Flow field is resolved using the Reynold-Averaged Navier Stokes solver OVERFLOW, version 2.2l, with fully turbulent flow approximation. The turbulence was resolved using the Spalart-Allmaras turbulence model with rotational and curvature correction. Grid independence study show acceptable grid resolution is achieved. The stall angle-of-attack was predicted at 17.25. CFD analysis show that large separations begins inboard, near the symmetry plane, and exists at stall. However, the separated region remain localized and the flow at the wing tip remain attached.

drag↗

NASA NTRS2015

Development of a Grid-Independent Geos-Chem Chemical Transport Model (v9-02) as an Atmospheric Chemistry Module for Earth System Models

The GEOS-Chem global chemical transport model (CTM), used by a large atmospheric chemistry research community, has been re-engineered to also serve as an atmospheric chemistry module for Earth system models (ESMs). This was done using an Earth System Modeling Framework (ESMF) interface that operates independently of the GEOSChem scientific code, permitting the exact same GEOSChem code to be used as an ESM module or as a standalone CTM. In this manner, the continual stream of updates contributed by the CTM user community is automatically passed on to the ESM module, which remains state of science and referenced to the latest version of the standard GEOS-Chem CTM. A major step in this re-engineering was to make GEOS-Chem grid independent, i.e., capable of using any geophysical grid specified at run time. GEOS-Chem data sockets were also created for communication between modules and with external ESM code. The grid-independent, ESMF-compatible GEOS-Chem is now the standard version of the GEOS-Chem CTM. It has been implemented as an atmospheric chemistry module into the NASA GEOS- 5 ESM. The coupled GEOS-5-GEOS-Chem system was tested for scalability and performance with a tropospheric oxidant-aerosol simulation (120 coupled species, 66 transported tracers) using 48-240 cores and message-passing interface (MPI) distributed-memory parallelization. Numerical experiments demonstrate that the GEOS-Chem chemistry module scales efficiently for the number of cores tested, with no degradation as the number of cores increases. Although inclusion of atmospheric chemistry in ESMs is computationally expensive, the excellent scalability of the chemistry module means that the relative cost goes down with increasing number of cores in a massively parallel environment.

CTM↗

NASA NTRS2004

Three-Dimensional High-Order Spectral Volume Method for Solving Maxwell's Equations on Unstructured Grids

A three-dimensional, high-order, conservative, and efficient discontinuous spectral volume (SV) method for the solutions of Maxwell's equations on unstructured grids is presented. The concept of discontinuous 2nd high-order loca1 representations to achieve conservation and high accuracy is utilized in a manner similar to the Discontinuous Galerkin (DG) method, but instead of using a Galerkin finite-element formulation, the SV method is based on a finite-volume approach to attain a simpler formulation. Conventional unstructured finite-volume methods require data reconstruction based on the least-squares formulation using neighboring cell data. Since each unknown employs a different stencil, one must repeat the least-squares inversion for every cell at each time step, or to store the inversion coefficients. In a high-order, three-dimensional computation, the former would involve impractically large CPU time, while for the latter the memory requirement becomes prohibitive. In the SV method, one starts with a relatively coarse grid of triangles or tetrahedra, called spectral volumes (SVs), and partition each SV into a number of structured subcells, called control volumes (CVs), that support a polynomial expansion of a desired degree of precision. The unknowns are cell averages over CVs. If all the SVs are partitioned in a geometrically similar manner, the reconstruction becomes universal as a weighted sum of unknowns, and only a few universal coefficients need to be stored for the surface integrals over CV faces. Since the solution is discontinuous across the SV boundaries, a Riemann solver is thus necessary to maintain conservation. In the paper, multi-parameter and symmetric SV partitions, up to quartic for triangle and cubic for tetrahedron, are first presented. The corresponding weight coefficients for CV face integrals in terms of CV cell averages for each partition are analytically determined. These discretization formulas are then applied to the integral form of the Maxwell equations. All numerical procedures for outer boundary, material interface, zonal interface, and interior SV face are unified with a single characteristic formulation. The load balancing in a massive parallel computing environment is therefore easier to achieve. A parameter is introduced in the Riemann solver to control the strength of the smoothing term. Important aspects of the data structure and its effects to communication and the optimum use of cache memory are discussed. Results will be presented for plane TE and TM waves incident on a perfectly conducting cylinder for up to fifth order of accuracy, and a plane wave incident on a perfectly conducting sphere for up to fourth order of accuracy. Comparisons are made with exact solutions for these cases.

Liu, Yen↗

NASA NTRSDate not supplied

Wall-modeled LES of Flow Over a Gaussian Bump

We perform wall-modeled large eddy simulations (WMLES) of turbulent flow over a Gaussian-shaped bump geometry, to assess its performance in the acceleration and separation regions of the flow. An unstructured finite-volume solver is used along with an equilibrium wall model. The Mach number of the oncoming flow selected for the simulation is 0.176, and two Reynolds numbers (Re) are simulated that are about 10000 and 36000 based on boundary layer thickness upstream of the bump. Spanwise periodic results for the lower Re case are compared with the available DNS data, while the full three-dimensional simulation results for the higher Re case are compared to available experimental data. Sensitivity of WMLES results are assessed for a number of factors including grid resolution, wall model exchange location, type of wall model, and unstructured grid topology. The WMLES results for the lower Re case agree well with available DNS data in terms of the wall pressure variation, velocity and turbulent stress profile comparisons. The skin-friction predictions show a reduced tendency to separate. This appears to be accentuated for the higher Re case, for the medium resolution grid used in this study, where WMLES does not predict any flow separation in the mean, which is in contrast to the large separation bubble observed in the experiments.

turbulence↗

NASA NTRS2010

Community-Based Services that Facilitate Interoperability and Intercomparison of Precipitation Datasets from Multiple Sources

Over the past 12 years, large volumes of precipitation data have been generated from space-based observatories (e.g., TRMM), merging of data products (e.g., gridded 3B42), models (e.g., GMAO), climatologies (e.g., Chang SSM/I derived rain indices), field campaigns, and ground-based measuring stations. The science research, applications, and education communities have greatly benefited from the unrestricted availability of these data from the Goddard Earth Sciences Data and Information Services Center (GES DISC) and, in particular, the services tailored toward precipitation data access and usability. In addition, tools and services that are responsive to the expressed evolving needs of the precipitation data user communities have been developed at the Precipitation Data and Information Services Center (PDISC) (http://disc.gsfc.nasa.gov/precipitation or google NASA PDISC), located at the GES DISC, to provide users with quick data exploration and access capabilities. In recent years, data management and access services have become increasingly sophisticated, such that they now afford researchers, particularly those interested in multi-data set science analysis and/or data validation, the ability to homogenize data sets, in order to apply multi-variant, comparison, and evaluation functions. Included in these services is the ability to capture data quality and data provenance. These interoperability services can be directly applied to future data sets, such as those from the Global Precipitation Measurement (GPM) mission. This presentation describes the data sets and services at the PDISC that are currently used by precipitation science and applications researchers, and which will be enhanced in preparation for GPM and associated multi-sensor data research. Specifically, the GES-DISC Interactive Online Visualization ANd aNalysis Infrastructure (Giovanni) will be illustrated. Giovanni enables scientific exploration of Earth science data without researchers having to perform the complicated data access and match-up processes. In addition, PDISC tool and service capabilities being adapted for GPM data will be described, including the Google-like Mirador data search and access engine; semantic technology to help manage large amounts of multi-sensor data and their relationships; data access through various Web services (e.g., OPeNDAP, GDS, WMS, WCS); conversion to various formats (e.g., netCDF, HDF, KML (for Google Earth)); visualization and analysis of Level 2 data profiles and maps; parameter and spatial subsetting; time and temporal aggregation; regridding; data version control and provenance; continuous archive verification; and expertise in data-related standards and interoperability. The goal of providing these services is to further the progress towards a common framework by which data analysis/validation can be more easily accomplished.

Liu, Zhong↗