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NASA NTRS1995

Investigation of computational and spectral analysis methods for aeroacoustic wave propagation

Most computational fluid dynamics (CFD) schemes are not adequately accurate for solving aeroacoustics problems, which have wave amplitudes several orders of magnitude smaller yet with frequencies larger than the flow field variations generating the sound. Hence, a computational aeroacoustics (CAA) algorithm should have minimal dispersion and dissipation features. A dispersion relation preserving (DRP) scheme is, therefore, applied to solve the linearized Euler equations in order to simulate the propagation of three types of waves, namely: acoustic, vorticity, and entropy waves. The scheme is derived using an optimization procedure to ensure that the numerical derivatives preserve the wave number and angular frequency of the partial differential equations being discretized. Consequently, simulated waves propagate with the correct wave speeds and exhibit their appropriate properties. A set of radiation and outflow boundary conditions, compatible with the DRP scheme and derived from the asymptotic solutions of the governing equations, are also implemented. Numerical simulations are performed to test the effectiveness of the DRP scheme and its boundary conditions. The computed solutions are shown to agree favorably with the exact solutions. The major restriction appears to be that the dispersion relations can be preserved only for waves with wave lengths longer than four or five spacings. The boundary conditions are found to be transparent to the outgoing disturbances. However, when the disturbance source is placed closer to a boundary, small acoustic reflections start appearing. CAA generates enormous amounts of temporal data which needs to be reduced to understand the physical problem being simulated. Spectral analysis is one approach that helps us in extracting information which often can not be easily interpreted in the time domain. Thus, three different methods for the spectral analysis of numerically generated aeroacoustic data are studied. First, the capabilities of two traditional methods for spectral analysis, namely, the Blackman-Tukey method and periodogram method, are compared in estimating the spectra of a simple-periodic process. The periodogram is then applied to analyze transitory-deterministic processes. Finally, these two methods are compared with a more recent method, referred as the Weighted-Overlapped-Segment-Averaging (WOSA) method, in estimating the spectra of a chaotic (random-like) process. From the demonstrative case for the spectral analyses of data generated by simple-periodic process, the periodogram method is found to give a better estimate of the steep-sloped spectra than the Blackman-Tukey method. Also, for this problem, the Hanning window is found to perform better with the periodogram method than with the Blackman-Tukey method. Finally, for the spectral analysis of data generated by the chaotic process, the periodogram method does not perform well, whereas, the WOSA and Blackman-Tukey methods give equivalently good results.

Vanel, Florence O.↗

NASA NTRS1995

Investigation of dispersion-relation-preserving scheme and spectral analysis methods for acoustic waves

Important characteristics of the aeroacoustic wave propagation are mostly encoded in their dispersion relations. Hence, a computational aeroacoustic (CAA) algorithm, which reasonably preserves these relations, was investigated. It was derived using an optimization procedure to ensure, that the numerical derivatives preserved the wave number and angular frequency of the differential terms in the linearized, 2-D Euler equations. Then, simulations were performed to validate the scheme and a compatible set of discretized boundary conditions. The computational results were found to agree favorably with the exact solutions. The boundary conditions were transparent to the outgoing waves, except when the disturbance source was close to a boundary. The time-domain data generated by such CAA solutions were often intractable until their spectra was analyzed. Therefore, the relative merits of three different methods were included in the study. For simple, periodic waves, the periodogram method produced better estimates of the steep-sloped spectra than the Blackman-Tukey method. Also, for this problem, the Hanning window was more effective when used with the weighted-overlapped-segment-averaging and Blackman-Tukey methods gave better results than the periodogram method. Finally, it was demonstrated that the representation of time domain-data was significantly dependent on the particular spectral analysis method employed.

Vanel, Florence O.↗

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

Deriving Climate Change Signal from Hyperspectral Sounders Using Spectral Fingerprinting Method

Hyperspectral observations from satellite-based sensors provide high information content for the Earth’s atmospheric temperature, water vapor and trace gas vertical profiles. We have developed a radiometrically consistent spectral fingerprinting method to derive climate change signals from Aqua AIRS/AMSU and S-NPP CrIS/ATMS data. The climate variables include temperature and water vapor profiles, cloud, trace gases, and surface skin temperature. The radiative kernels obtained via a single field of view physical retrieval algorithm under all-sky conditions. A key component to this work is a Principal Component-based Radiative Transfer Model (PCRTM). It is 4 orders of magnitude faster than a line-by-line radiative transfer model while keeping a similar accuracy (0.03 K RMS errors with close to zero bias). The PCRTM includes multiple scattering of clouds and non-thermodynamics equilibrium of CO2 in the RT calculations. Instead of quantifying the radiometric differences between AIRS/AMSU and CrIS/ATMS measurements directly using Simultaneous Nadir Overpass (SNO) or Double Difference Technique (DDT), we use the radiometric consistent fingerprinting scheme to derive two sets of space-time averaged anomalies from the Level 1 data of AIRS/AMSU and CrIS/ATMS. The derived anomalies in geophysical space will form a long-term, stable, and continuous climate data record. We can further infer the causes of any offset or drift by studying the differences between two overlapping data sets. For example, the offset in surface skin temperature anomaly time series will most likely caused by the Blackbody temperature calibration errors of the sounder instruments.

climate↗

NASA NTRS1992

Research in computational fluid dynamics and analysis of algorithms

Recently, higher-order compact schemes have seen increasing use in the DNS (Direct Numerical Simulations) of the Navier-Stokes equations. Although they do not have the spatial resolution of spectral methods, they offer significant increases in accuracy over conventional second order methods. They can be used on any smooth grid, and do not have an overly restrictive CFL dependence as compared with the O(N(exp -2)) CFL dependence observed in Chebyshev spectral methods on finite domains. In addition, they are generally more robust and less costly than spectral methods. The issue of the relative cost of higher-order schemes (accuracy weighted against physical and numerical cost) is a far more complex issue, depending ultimately on what features of the solution are sought and how accurately they must be resolved. In any event, the further development of the underlying stability theory of these schemes is important. The approach of devising suitable boundary clusters and then testing them with various stability techniques (such as finding the norm) is entirely the wrong approach when dealing with high-order methods. Very seldom are high-order boundary closures stable, making them difficult to isolate. An alternative approach is to begin with a norm which satisfies all the stability criteria for the hyperbolic system, and look for the boundary closure forms which will match the norm exactly. This method was used recently by Strand to isolate stable boundary closure schemes for the explicit central fourth- and sixth-order schemes. The norm used was an energy norm mimicking the norm for the differential equations. Further research should be devoted to BC for high order schemes in order to make sure that the results obtained are reliable. The compact fourth order and sixth order finite difference scheme had been incorporated into a code to simulate flow past circular cylinders. This code will serve as a verification of the full spectral codes. A detailed stability analysis by Carpenter (from the fluid Mechanics Division) and Gottlieb gave analytic conditions for stability as well as asymptotic stability. This had been incorporated in the code in form of stable boundary conditions. Effects of the cylinder rotations had been studied. The results differ from the known theoretical results. We are in the middle of analyzing the results. A detailed analysis of the effects of the heating of the cylinder on the shedding frequency had been studied using the above schemes. It has been found that the shedding frequency decreases when the wire was heated. Experimental work is being carried out to affirm this result.

Gottlieb, David↗

NASA NTRS2004

Discontinuous Spectral Difference Method for Conservation Laws on Unstructured Grids

A new, high-order, conservative, and efficient method for conservation laws on unstructured grids is developed. The concept of discontinuous and high-order local representations to achieve conservation and high accuracy is utilized in a manner similar to the Discontinuous Galerkin (DG) and the Spectral Volume (SV) methods, but while these methods are based on the integrated forms of the equations, the new method is based on the differential form to attain a simpler formulation and higher efficiency. A discussion on the Discontinuous Spectral Difference (SD) Method, locations of the unknowns and flux points and numerical results are also presented.

Liu, Yen↗

NASA NTRS2000

Application of the Spectral Element Method to Acoustic Radiation

This report summarizes research to develop a capability for analysis of interior noise in enclosed structures when acoustically excited by an external random source. Of particular interest was the application to the study of noise and vibration transmission in thin-walled structures as typified by aircraft fuselages. Three related topics are focused upon. The first concerns the development of a curved frame spectral element, the second shows how the spectral element method for wave propagation in folded plate structures is extended to problems involving curved segmented plates. These are of significance because by combining these curved spectral elements with previously presented flat spectral elements, the dynamic response of geometrically complex structures can be determined. The third topic shows how spectral elements, which incorporate the effect of fluid loading on the structure, are developed for analyzing acoustic radiation from dynamically loaded extended plates.

Doyle, James F.↗

NASA NTRS1989

A pseudo-spectral collocation method applied to the problem of convective diffusive transport in fluids subject to unsteady residual accelerations

The problem of determining the sensitivity of Bridgman-Stockbarger directional solidification experiments to residual accelerations of the type associated with spacecraft in low earth orbit is analyzed numerically using a pseudo-spectral collocation method. The approach employs a novel iterative scheme combining the method of artificial compressibility and a generalized ADI method. The results emphasize the importance of the consideration of residual accelerations and careful selection of the operating conditions in order to take full advantages of the low gravity conditions.

Alexander, J. Iwan↗

NASA NTRS1990

Conforming Chebyshev spectral collocation methods for the solution of laminar flow in a constricted channel

The numerical simulation of steady planar two-dimensional, laminar flow of an incompressible fluid through an abruptly contracting channel using spectral domain decomposition methods is described. The key features of the method are the decomposition of the flow region into a number of rectangular subregions and spectral approximations which are pointwise C(1) continuous across subregion interfaces. Spectral approximations to the solution are obtained for Reynolds numbers in the range 0 to 500. The size of the salient corner vortex decreases as the Reynolds number increases from 0 to around 45. As the Reynolds number is increased further the vortex grows slowly. A vortex is detected downstream of the contraction at a Reynolds number of around 175 that continues to grow as the Reynolds number is increased further.

Karageorghis, Andreas↗

NASA NTRSDate not supplied

A Spectral Fingerprinting Method for Deriving Consistent Climate Data Records from Multiple Satellite IR Sounders

Deriving Climate Data Records (CDRs) from multiple IR sounders such AIRS on Aqua, CrIS on S-NPP, NOAA20 and JPSS-2, IASI on Metop A, B, and C requires from current operational products are challenging due to different radiative transfer models and retireavl algorithms used for processing level 2 data. We developed a Climate Fingerprinting Sounder Product (ClimFiSP) algorithm, which uses a single set of radiative kernels a robust spectral fingerprinting method to performs retrievals using spatiotemporally averaged L1 hyperspectral radiances directly. The ClimFiSP algorithm provides accurate data fusion CDR products from multiple satellite sensors. We have applied this method to both AIRS and CrIS (on SNPP and on NOAA 20) data and generated two decades climate data records for atmospheric temperature, water vapor, cloud, trace gases, and surface skin temperature. We plan to add IASI to the CDR data set in the future. The ClimFiSP is being transitioned to NASA GES DISC data center for public access.

Xu Liu↗

NASA NTRS1991

Spectral simulations of reacting turbulent flows

Spectral methods for simulating flows are reviewed, emphasizing their recent applications to reacting flow problems. Various classifications of spectral methods and their convergence properties are described and the 'spectral element' method is presented, highlighting its flexibility in dealing with complex flow geometries. The applications considered include chemical reactions in homogeneous turbulence, temporally evolving mixing layers, variable-density simulations, nonequilibrium chemistry, and spatially evolving mixing layers.

Mcmurtry, Patrick A.↗

NASA NTRS1994

Approximate Solution Methods for Spectral Radiative Transfer in High Refractive Index Layers

Some ceramic materials for high temperature applications are partially transparent for radiative transfer. The refractive indices of these materials can be substantially greater than one which influences internal radiative emission and reflections. Heat transfer behavior of single and laminated layers has been obtained in the literature by numerical solutions of the radiative transfer equations coupled with heat conduction and heating at the boundaries by convection and radiation. Two-flux and diffusion methods are investigated here to obtain approximate solutions using a simpler formulation than required for exact numerical solutions. Isotropic scattering is included. The two-flux method for a single layer yields excellent results for gray and two band spectral calculations. The diffusion method yields a good approximation for spectral behavior in laminated multiple layers if the overall optical thickness is larger than about ten. A hybrid spectral model is developed using the two-flux method in the optically thin bands, and radiative diffusion in bands that are optically thick.

Siegel, R.↗

NASA NTRS2016

Development of a Perfectly Matched Layer Technique for a Discontinuous-Galerkin Spectral-Element Method

The perfectly matched layer (PML) technique is developed in the context of a high- order spectral-element Discontinuous-Galerkin (DG) method. The technique is applied to a range of test cases and is shown to be superior compared to other approaches, such as those based on using characteristic boundary conditions and sponge layers, for treating the inflow and outflow boundaries of computational domains. In general, the PML technique improves the quality of the numerical results for simulations of practical flow configurations, but it also exhibits some instabilities for large perturbations. A preliminary analysis that attempts to understand the source of these instabilities is discussed.

Spectral-Element↗

NASA NTRS1987

Relaxation schemes for spectral multigrid methods

The effectiveness of relaxation schemes for solving the systems of algebraic equations which arise from spectral discretizations of elliptic equations is examined. Iterative methods are an attractive alternative to direct methods because Fourier transform techniques enable the discrete matrix-vector products to be computed almost as efficiently as for corresponding but sparse finite difference discretizations. Preconditioning is found to be essential for acceptable rates of convergence. Preconditioners based on second-order finite difference methods are used. A comparison is made of the performance of different relaxation methods on model problems with a variety of conditions specified around the boundary. The investigations show that iterations based on incomplete LU decompositions provide the most efficient methods for solving these algebraic systems.

Phillips, Timothy N.↗

NASA NTRS1987

Spectral multigrid methods for the solution of homogeneous turbulence problems

New three-dimensional spectral multigrid algorithms are analyzed and implemented to solve the variable coefficient Helmholtz equation. Periodicity is assumed in all three directions which leads to a Fourier collocation representation. Convergence rates are theoretically predicted and confirmed through numerical tests. Residual averaging results in a spectral radius of 0.2 for the variable coefficient Poisson equation. In general, non-stationary Richardson must be used for the Helmholtz equation. The algorithms developed are applied to the large-eddy simulation of incompressible isotropic turbulence.

Erlebacher, G.↗

NASA NTRS1988

Spectral multigrid methods for the solution of homogeneous turbulence problems

Erlebacher, G.↗

NASA NTRS2005

Martian Radiative Transfer Modeling Using the Optimal Spectral Sampling Method

The large volume of existing and planned infrared observations of Mars have prompted the development of a new martian radiative transfer model that could be used in the retrievals of atmospheric and surface properties. The model is based on the Optimal Spectral Sampling (OSS) method [1]. The method is a fast and accurate monochromatic technique applicable to a wide range of remote sensing platforms (from microwave to UV) and was originally developed for the real-time processing of infrared and microwave data acquired by instruments aboard the satellites forming part of the next-generation global weather satellite system NPOESS (National Polarorbiting Operational Satellite System) [2]. As part of our on-going research related to the radiative properties of the martian polar caps, we have begun the development of a martian OSS model with the goal of using it to perform self-consistent atmospheric corrections necessary to retrieve caps emissivity from the Thermal Emission Spectrometer (TES) spectra. While the caps will provide the initial focus area for applying the new model, it is hoped that the model will be of interest to the wider Mars remote sensing community.

Eluszkiewicz, J.↗

NASA NTRS2009

Evolutionary Computing Methods for Spectral Retrieval

A methodology for processing spectral images to retrieve information on underlying physical, chemical, and/or biological phenomena is based on evolutionary and related computational methods implemented in software. In a typical case, the solution (the information that one seeks to retrieve) consists of parameters of a mathematical model that represents one or more of the phenomena of interest. The methodology was developed for the initial purpose of retrieving the desired information from spectral image data acquired by remote-sensing instruments aimed at planets (including the Earth). Examples of information desired in such applications include trace gas concentrations, temperature profiles, surface types, day/night fractions, cloud/aerosol fractions, seasons, and viewing angles. The methodology is also potentially useful for retrieving information on chemical and/or biological hazards in terrestrial settings. In this methodology, one utilizes an iterative process that minimizes a fitness function indicative of the degree of dissimilarity between observed and synthetic spectral and angular data. The evolutionary computing methods that lie at the heart of this process yield a population of solutions (sets of the desired parameters) within an accuracy represented by a fitness-function value specified by the user. The evolutionary computing methods (ECM) used in this methodology are Genetic Algorithms and Simulated Annealing, both of which are well-established optimization techniques and have also been described in previous NASA Tech Briefs articles. These are embedded in a conceptual framework, represented in the architecture of the implementing software, that enables automatic retrieval of spectral and angular data and analysis of the retrieved solutions for uniqueness.

Terrile, Richard↗