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The spectral method for distinguishing crustal from core-source magnetic fields has been re-examined, modified, and applied to both a comprehensive geomagnetic field model and an altitude normalized magnetic map of Mars. Additional information is contained in the original extended abstract.
The effect of a nonuniform mean flow on the normal modes; the inflow/outflow nonreflecting boundary conditions; and the sound power are studied. The normal modes in an annular duct are computed using a spectral method in combination with a shooting method. The swirl causes force imbalance which couples the acoustic and vortical modes. The acoustic modes are distinguished from the vortical modes by their large pressure and small vorticity content. The mean swirl also produces a Doppler shift in frequency. This results in more counter-spinning modes cut-on at a given frequency than modes spinning with the swirl. Nonreflecting boundary conditions are formulated using the normal mode solutions. The inflow/outflow boundary conditions are implemented in a linearized Euler scheme and validated by computing the propagation of acoustic and vortical waves in a duct for a variety of swirling mean flows. Numerical results show that the evolution of the vortical disturbances is sensitive to the inflow conditions and the details of the wake excitations. All three components of the wake velocity must be considered to correctly compute the wake evolution and the blade upwash. For high frequencies, the acoustic-vortical mode coupling is weak and a conservation equation for the acoustic energy can be derived. Sound power calculations show significant mean flow swirl effects, but mode interference effects are small.
The transition in confined rotating flows is a topical problem with many industrial and fundamental applications. The purpose of this study is to investigate the Taylor-Couette flow in a finite-length cavity with counter-rotating walls, for two aspect ratios L=5 or L=6. Two complex regimes of wavy vortex and spirals are emphasized for the first time via direct numerical simulation, by using a three-dimensional spectral method. The spatio-temporal behavior of the solutions is analyzed and compared to the few data actually available. c2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.
The implementation and performance of a parallel spatial direct numerical simulation (PSDNS) approach on the Intel iPSC/860 hypercube and IBM SP1 and SP2 parallel computers is documented. Spatially evolving disturbances associated with the laminar-to-turbulent transition in boundary-layer flows are computed with the PSDNS code. The feasibility of using the PSDNS to perform transition studies on these computers is examined. The results indicate that PSDNS approach can effectively be parallelized on a distributed-memory parallel machine by remapping the distributed data structure during the course of the calculation. Scalability information is provided to estimate computational costs to match the actual costs relative to changes in the number of grid points. By increasing the number of processors, slower than linear speedups are achieved with optimized (machine-dependent library) routines. This slower than linear speedup results because the computational cost is dominated by FFT routine, which yields less than ideal speedups. By using appropriate compile options and optimized library routines on the SP1, the serial code achieves 52-56 M ops on a single node of the SP1 (45 percent of theoretical peak performance). The actual performance of the PSDNS code on the SP1 is evaluated with a "real world" simulation that consists of 1.7 million grid points. One time step of this simulation is calculated on eight nodes of the SP1 in the same time as required by a Cray Y/MP supercomputer. For the same simulation, 32-nodes of the SP1 and SP2 are required to reach the performance of a Cray C-90. A 32 node SP1 (SP2) configuration is 2.9 (4.6) times faster than a Cray Y/MP for this simulation, while the hypercube is roughly 2 times slower than the Y/MP for this application. KEY WORDS: Spatial direct numerical simulations; incompressible viscous flows; spectral methods; finite differences; parallel computing.
We have calculated admittance spectra using the spatio-spectral method [14] for Venus by moving the central location of the spectrum over a 1 grid, create 360x180 admittance spectra. We invert the observed admittance using top-loading (TL), hot spot (HS), and bottom loading (BL) models, resulting in elastic, crustal, and lithospheric thickness estimates (Te, Zc, and Zl) [0]. The result is a global map for interpreting subsurface structure. Estimated values of Te and Zc concur with previous TL local admittance results, but BL estimates indicate larger values than previously suspected.
The spectral method for distinguishing crustal from core-source magnetic fields has been re-examined, modified and applied to both a comprehensive geomagnetic field model and an altitude normalized magnetic map of Mars. These observational spectra are fairly fitted by theoretical forms expected from certain elementary classes of magnetic sources. For Earth we found fields from a core of radius 3512 plus or minus 64 km, in accord with the 3480 km seismologic radius, and a crust represented by a shell of random dipolar sources at radius 6367 plus or minus 14 km, just beneath the 6371.0 km mean radius. For Mars we found only a field from a crust represented in same way, but 46 plus or minus 10 km below the planetary mean radius of 3389.5 km, and with sources about 9.6 plus or minus 3.2 times stronger than Earth's. It is remarkable that the same simple theoretical form should fairly fit crustal magnetic spectra for both worlds and return crustal-source depth estimates of plausible magnitude. Evidently, the idea of an ensemble of compact, quasi-independent, magnetized regions within these planetary crusts has some merit. Yet such estimates, at best a kind of average, depend upon both the observational spectrum fitted and the physical basis of the theoretical spectrum.
Under certain conditions, such as in thin liquid films or microgravity, surface tension variations along a free surface can induce convection. Convection onset due to surface tension variation is important to many terrestrial technological processes in addition to microgravity materials processing applications. Examples include coating, drying crystallization, solidification, liquid surface contamination, and containerless processing. In double-diffusive and multicomponent systems, the spatial variations of surface tension are associated with two or more stratifying agencies, respectively. For example, both temperature and species (concentration) gradients are associated with convection in the solidification of binary alloys or salt ponds. The direction of the two (or more) gradients has a profound effect on the nature of the flow at or slightly beyond the onset of convection. Our recent work at the NASA Lewis Research Center focused on characterizing surface-tension-induced onset of convection, often referred to as Marangoni-Benard convection. Exact solutions for the stationary neutral stability of multicomponent fluid layers with interfacial deformation were derived. These solutions also permit the computation of a boundary curve that separates the long and finite wavelength instabilities. Computing points along this boundary using the exact solution (when possible) is more efficient than the typical numerical approaches, such as finite difference or spectral methods. Above the curve, a long wavelength instability was predicted, suggesting that convection would occur principally through one large flow cell in the layer, whereas below the curve, finite wavelength instabilities occur which suggest multiple finite-sized circulation cells. For many common liquids with layer depths greater than 100 mm, finite wave instability is predicted under terrestrial conditions; however, with little exception, long wavelength instability is predicted in microgravity for the identical fluid systems.
Among the Murchison meteoritic amino acids, isovaline stands out as being both nonbiological (non-protein) and having a relatively high abundance. While approximately equal amounts of D- and L-isovaline have been reported in Murchison and other CM meteorites, the molecule's structure appears to prohibit its racemization in aqueous solutions. We recently have investigated the low-temperature solid-phase chemistry of both isovaline and valine with an eye toward each molecule's formation, stability, and possible interconversions of D and L enantiomers. Ion-irradiated isovaline- and valinecontaining ices were examined by IR spectroscopy and highly-sensitive liquid chromatography/time-of-flight mass spectral methods to assess both amino acid destruction and racemization. Samples were studied in the presence and in the absence of water-ice, and the destruction of both isovaline and valine was measured as a function of radiation dose. In addition, we have undertaken experiments to synthesize isovaline, valine, and their amino acid isomers by solid-phase radiation-chemical pathways other than the oft-invoked Strecker process. This presentation will review and summarize some of our recent findings. -- Our work has been supported by a grant to the Goddard Center for Astrobiology through the NASA Astrobiology Institute. Experiments were performed in the Cosmic Ice Laboratory (RLH, MHM, AL) and the Astrobiology Analytical Laboratory (JPD, DPG) at the NASA Goddard Space Flight Center.
Reliable noise prediction capabilities are essential to enable novel fuel efficient open rotor designs that can meet the community and cabin noise standards. Toward this end, immersed boundary methods have reached a level of maturity where more and more complex flow problems can be tackled with this approach. This paper demonstrates that our higher-order immersed boundary method provides the ability for aeroacoustic analysis of wake-dominated flow fields generated by a contra-rotating open rotor. This is the first of a kind aeroacoustic simulation of an open rotor propulsion system employing an immersed boundary method. In addition to discussing the methodologies of how to apply the immersed boundary method to this moving boundary problem, we will provide a detailed validation of the aeroacoustic analysis approach employing the Launch Ascent and Vehicle Aerodynamics (LAVA) solver. Two free-stream Mach numbers with M=0.2 and M=0.78 are considered in this analysis that are based on the nominally take-off and cruise flow conditions. The simulation data is compared to available experimental data and other computational results employing more conventional CFD methods. Spectral analysis is used to determine the dominant wave propagation pattern in the acoustic near-field.
We derived new vector gradients based models of crustal magnetic field at the lunar surface with data from the Lunar Prospector (LP) satellite using two model parameterization approaches: a global set of 35820 1° spaced (~30 km) equal area monopoles at 20 km below the surface (O’Brien and Parker, 1994; Olsen et al., 2017) and combined results of subsets of 100000 0.66° spaced monopoles at the same depth. We use the scheme of iteratively reweighted least-squares inversion to compute the initial model. Then the amplitudes of these monopoles are determined by minimizing the misfit to the components together with the global average of |Br| at the ellipsoid surface (i.e. applying a L1 model regularization of Br). In previous approaches using vector fields for modeling, we found that external field contamination leads to spurious anomalies in the downward continued field models even with stringent data selection criteria and ad-hoc noise removal techniques (e.g., satellite’s position in the Moon’s wake w.r.t. the solar wind and in the Earth’s magnetotail, internal/external dipoles fields removal, low-order polynomial removal, joint equivalent source cross-validation technique and visually removing remaining anomalous segments). On the other hand, with the use of gradients-only data (along-track first finite differences), we were able to completely bypass the ad-hoc techniques. Similar processing of Kaguya magnetic data, which have only higher altitude coverage over the most of the Moon except in the region of the South Pole–Aitken (SPA) basin, completely misses some of the anomalies seen in the Lunar Prospector data. The combined Lunar Prospector and Kaguya gradient-based models also severely degrade the derivation of the anomaly fields in many regions. Euler analysis of isolated anomaly features from the Reiner Gamma swirl suggests top depths of about 0.3 to 1.5 km and center depths of 10-14 km; in the region between Stein and Vallier craters north of the SPA basin our analysis suggests top depths of around 1.5 km and center depths of 13-15 km. With the spectral depth determination techniques, the SPA basin region yields depths to the base of magnetization ranging between 15 and 40 km. Three-dimensional modeling and the bulk magnetization determinations of the sources constrained by the Euler and spectral methods is underway.
Many areas require a very high-order accurate numerical solution of conservation laws for complex shapes. This paper deals with the extension to three dimensions of the Spectral Finite Volume (SV) method for unstructured grids, which was developed to solve such problems. We first summarize the limitations of traditional methods such as finite-difference, and finite-volume for both structured and unstructured grids. We then describe the basic formulation of the spectral finite volume method. What distinguishes the SV method from conventional high-order finite-volume methods for unstructured triangular or tetrahedral grids is the data reconstruction. Instead of using a large stencil of neighboring cells to perform a high-order reconstruction, the stencil is constructed by partitioning each grid cell, called a spectral volume (SV), into 'structured' sub-cells, called control volumes (CVs). One can show that if all the SV cells are partitioned into polygonal or polyhedral CV sub-cells in a geometrically similar manner, the reconstructions for all the SVs become universal, irrespective of their shapes, sizes, orientations, or locations. It follows that the reconstruction is reduced to a weighted sum of unknowns involving just a few simple adds and multiplies, and those weights are universal and can be pre-determined once for all. The method is thus very efficient, accurate, and yet geometrically flexible. The most critical part of the SV method is the partitioning of the SV into CVs. In this paper we present the partitioning of a tetrahedral SV into polyhedral CVs with one free parameter for polynomial reconstructions up to degree of precision five. (Note that the order of accuracy of the method is one order higher than the reconstruction degree of precision.) The free parameter will be determined by minimizing the Lebesgue constant of the reconstruction matrix or similar criteria to obtain optimized partitions. The details of an efficient, parallelizable code to solve three-dimensional problems for any order of accuracy are then presented. Important aspects of the data structure are discussed. Comparisons with the Discontinuous Galerkin (DG) method are made. Numerical examples for wave propagation problems are presented.
Spectral collocation methods have proven to be efficient discretization schemes for many aerodynamic and fluid mechanic problems. The high order accuracy and resolution shown by these methods allows one to obtain engineering accuracy solutions on coarse meshes, or alternatively, to obtain solutions with very small error. One drawback to these techniques was the requirement that a complicated physical domain must map into a simple computational domain for discretization. This mapping must be smooth if the high order accuracy and expontential convergence rates associated with spectral methods are to be preserved. Additionally even smooth stretching transformations can decrease the accuracy of a spectral method, if the stretching is severe. A further difficulty with spectral methods was in their implementation on parallel processing computers, where efficient spectral algorithms were lacking. The above restrictions are overcome by splitting the domain into regions, each of which preserve the advantages of spectral collocation, and allow the ratio of the mesh spacing between regions to be several orders of magnitude higher than allowable in a single domain. Such stretchings would be required to resolve the thin viscous region in an external aerodynamic problem. Adjoining regions are interfaced by enforcing a global flux balance which preserves high-order continuity of the solution, regardless of the type of the equations being solved.
The predicted upper and lower bounds power spectra for all of the cases and response items given in Volume 1 are plotted. The flight test power spectra are shown on each prediction plot for the nominal value of angle of attack that most closely agrees with the flexible angle for the prediction. The flight test and prediction conditions are given in tabular form for all cases considered.
The framework for constructing a high-order, conservative Spectral (Finite) Volume (SV) method is presented for two-dimensional scalar hyperbolic conservation laws on unstructured triangular grids. Each triangular grid cell forms a spectral volume (SV), and the SV is further subdivided into polygonal control volumes (CVs) to supported high-order data reconstructions. Cell-averaged solutions from these CVs are used to reconstruct a high order polynomial approximation in the SV. Each CV is then updated independently with a Godunov-type finite volume method and a high-order Runge-Kutta time integration scheme. A universal reconstruction is obtained by partitioning all SVs in a geometrically similar manner. The convergence of the SV method is shown to depend on how a SV is partitioned. A criterion based on the Lebesgue constant has been developed and used successfully to determine the quality of various partitions. Symmetric, stable, and convergent linear, quadratic, and cubic SVs have been obtained, and many different types of partitions have been evaluated. The SV method is tested for both linear and non-linear model problems with and without discontinuities.
Hyperbolic systems of conversation laws are considered which are discretized in space by spectral collocation methods and advanced in time by finite difference schemes. At any time-level a domain deposition method based on an iteration by subdomain procedure was introduced yielding at each step a sequence of independent subproblems (one for each subdomain) that can be solved simultaneously. The method is set for a general nonlinear problem in several space variables. The convergence analysis, however, is carried out only for a linear one-dimensional system with continuous solutions. A precise form of the error reduction factor at each iteration is derived. Although the method is applied here to the case of spectral collocation approximation only, the idea is fairly general and can be used in a different context as well. For instance, its application to space discretization by finite differences is straight forward.
Solving initial value problems with higher-order methods can improve the accuracy of the simulation results or the efficiency of the calculation. In this paper, we apply the spectral deferred correction (SDC) method to solve the initial value problem of the point kinetics equations (PKE). SDC is a stable, robust, and efficient high-order time-integration method capable of an arbitrary order of accuracy. For our implementation we show that it is A-stable for orders up to 8 and the order of accuracy is verified for PKE problems with a range of different reactivities. A 5.-order SDC method was then implemented to solve the exact PKE (EPKE) in the Transient Multilevel (TML) method of MPACT. The error from solutions of the EPKE is shown to be negligible. (authors)
Conjugate residual methods for the solution of spectral equations are described. An inexact finite-difference operator is introduced as a preconditioner in the iterative procedures. Application of these techniques is limited to problems for which the symmetric part of the coefficient matrix is positive definite. Although the spectral equation is a very ill-conditioned and full matrix problem, the computational effort of the present iterative methods for solving such a system is comparable to that for the sparse matrix equations obtained from the application of either finite-difference or finite-element methods to the same problems. Numerical experiments are shown for a self-adjoint elliptic partial differential equation with Dirichlet boundary conditions, and comparison with other solution procedures for spectral equations is presented.
For decades, uncertainty quantification techniques based on the spectral approach have been demonstrated to be computationally more efficient than the Monte Carlo method for a wide variety of problems, particularly when the dimensionality of the probability space is relatively low. The time-dependent generalized polynomial chaos (TD-gPC) is one such technique that uses an evolving orthogonal basis to better represent the stochastic part of the solution space in time. Here in this paper, we present a new numerical method that uses the concept of enriched stochastic flow maps to track the evolution of the stochastic part of the solution space in time. The computational cost of this proposed flow-driven stochastic chaos (FSC) method is an order of magnitude lower than TD-gPC for comparable solution accuracy. This gain in computational cost is realized because, unlike most existing methods, the number of basis vectors required to track the stochastic part of the solution space does not depend upon the dimensionality of the probability space. Four representative numerical examples are presented to demonstrate the performance of the FSC method for long-time integration of second-order stochastic dynamical systems in the context of stochastic dynamics of structures.