Search NASA⌕ Search

SEARCH · Search NASA

Results for “FOURIER SERIES”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 217 records · Page 12
All sourcesNASA NTRSDOE OSTI
NASA NTRS1950

Vibration of a Wing of Finite Span in a Supersonic Flow

An investigate ion was made of the disturbed motion of a gas for the harmonic vibrations of a thin slightly cambered wing of finite span moving forward with supersonic velocity. This problem was considered by E. A. Krasilshchikova who applied the method of Fourier series and obtained a solution of the space problem for the condition that the Mach cones drawn through the leading edge of the wing intersect the wing or are tangent to it. In this paper, a different method of solution is given, which is free from the previously mentioned condition. In particular, the vibrations of a triangular wing lying within the Mach cone are considered.

Haskind, M. D.↗
NASA NTRS2004

Oceanic Tidal Mixing As a Contributor to Milankovitch-scale Climate Change

We propose that changes in the magnitude of oceanic tidal mixing on long time scales is an important, but previously unrecognized, contributor to global climate change. it is well known that Earth's orbital and rotational state changes significantly on 10(exp 4)-10(exp 5) year time scales, and that this influences the spatial and temporal pattern of incident radiation. It is widely supposed that climatic variations on these same time scales are, in large part, a response of the ocean-atmosphere-cryosphere system to this radiative forcing. Our proposal is that variations in the luni-solar tidal potential, induced by these same orbital and rotational variations, influences oceanic mixing and thus modulates meridional heat transport, by amounts which are competitive with the radiative forcing. There are some obvious differences between tidal potential and insolation. First is that the Sun and Moon both contribute to tides, whereas the radiation is entirely of solar origin. Second is that the Earth is transparent to gravity but opaque to radiation. Clipping associated with this opacity makes the radiation pattern temporal spectrum rather more complex than the tidal spectrum. A third point is that solar radiation directly delivers energy to Earth's surface whereas tidal mixing will only expedite lateral transport of heat in association with oceanic thermohaline circulation. The diurnal average insolation pattern is best parameterized via a Fourier series in time of year and Legendre polynomials in sine of latitude. Our present focus will be on the annual average terms. The Legendre degree n=0 term describes the global average insolation, and is nearly constant. The degree n=l term describes differences between northern and southern hemispheres, and the annual mean is zero. The degree n=2 term is the main contributor to the equator to pole variations, and varies with obliquity and orbital eccentricity, with the obliquity variation dominating. The lowest order decomposition of the tidal potential recognizes 3 constituents: semi-diurnal, diurnal, and long period, with solar and lunar contributions to each. Our present focus will be on long term variations in the mean square amplitude of the semi-diurnal constituent, with averaging over all the short period variations. For the solar tide that includes the day and year. For the lunar tide it includes the day, month, year, and the apsidal (8.85 year) and nodal (18.6 year) periods. We present calculations of the variations in radiative and tidal forcing for the past 3 million years. The two signals are quite similar. Both vary by approximately 1% of their respective mean values, are dominated by obliquity variations, and exhibit only secondary influence from orbital eccentricity.

Munk, Walter↗
NASA NTRS2005

Mode-expansion method for predicting radar signature above rough ocean surfaces at low-grazing angle

The Mode-Expansion Method (MEM) is introduced to calculate the electromagnetic (EM) waves scattered by 2-D rough water surfaces at low-grazing angles. The Electric Field Integral Equation (EFIE) is used in defining the problem and is simplified by using the Impedance Boundary Condition (IBC). The surface currents are expressed as the sum of modes expanded as the Fourier series with incident wave as the dominant mode. It is shown that, by the MEM and for the geometry with transmitting and receiving waves at low-grazing angles, very few modes are needed in solving the forward scattering field with reasonable accuracy.

electromagnetic (EM)↗
NASA NTRS2006

Low-Velocity Impact Response of Sandwich Beams with Functionally Graded Core

The problem of low-speed impact of a one-dimensional sandwich panel by a rigid cylindrical projectile is considered. The core of the sandwich panel is functionally graded such that the density, and hence its stiffness, vary through the thickness. The problem is a combination of static contact problem and dynamic response of the sandwich panel obtained via a simple nonlinear spring-mass model (quasi-static approximation). The variation of core Young's modulus is represented by a polynomial in the thickness coordinate, but the Poisson's ratio is kept constant. The two-dimensional elasticity equations for the plane sandwich structure are solved using a combination of Fourier series and Galerkin method. The contact problem is solved using the assumed contact stress distribution method. For the impact problem we used a simple dynamic model based on quasi-static behavior of the panel - the sandwich beam was modeled as a combination of two springs, a linear spring to account for the global deflection and a nonlinear spring to represent the local indentation effects. Results indicate that the contact stiffness of thc beam with graded core Increases causing the contact stresses and other stress components in the vicinity of contact to increase. However, the values of maximum strains corresponding to the maximum impact load arc reduced considerably due to grading of thc core properties. For a better comparison, the thickness of the functionally graded cores was chosen such that the flexural stiffness was equal to that of a beam with homogeneous core. The results indicate that functionally graded cores can be used effectively to mitigate or completely prevent impact damage in sandwich composites.

Apetre, N. A.↗
NASA NTRS2006

Application of the Convolution Formalism to the Ocean Tide Potential: Results from the Gravity and Recovery and Climate Experiment (GRACE)

A computationally efficient approach to reducing omission errors in ocean tide potential models is derived and evaluated using data from the Gravity Recovery and Climate Experiment (GRACE) mission. Ocean tide height models are usually explicitly available at a few frequencies, and a smooth unit response is assumed to infer the response across the tidal spectrum. The convolution formalism of Munk and Cartwright (1966) models this response function with a Fourier series. This allows the total ocean tide height, and therefore the total ocean tide potential, to be modeled as a weighted sum of past, present, and future values of the tide-generating potential. Previous applications of the convolution formalism have usually been limited to tide height models, but we extend it to ocean tide potential models. We use luni-solar ephemerides to derive the required tide-generating potential so that the complete spectrum of the ocean tide potential is efficiently represented. In contrast, the traditionally adopted harmonic model of the ocean tide potential requires the explicit sum of the contributions from individual tidal frequencies. It is therefore subject to omission errors from neglected frequencies and is computationally more intensive. Intersatellite range rate data from the GRACE mission are used to compare convolution and harmonic models of the ocean tide potential. The monthly range rate residual variance is smaller by 4-5%, and the daily residual variance is smaller by as much as 15% when using the convolution model than when using a harmonic model that is defined by twice the number of parameters.

models↗
NASA NTRS2010

Numerical Mean Element Orbital Analysis with Morbiter

The Morbiter software numerically averages an osculating orbit s equations of motion (EOM) to arrive at the mean orbit s EOMs, which are then numerically propagated to obtain the long-term orbital ephemerides. The long-term evolution characteristics, and stability, of an orbit are best characterized using a mean element propagation of the perturbed, two-body variational equations of motion. The average process eliminates short period terms, leaving only secular and long period effects. Doing this avoids the Fourier series expansions and truncations required by the traditional analytic methods.

Ely, Todd A.↗
NASA NTRS2012

Theoretical Studies of N2-broadened Half-widths of H2O Lines Involving High j States

Based on the properties of the energy levels and wave functions of H2O states, one can categorize H2O lines into individually defined groups such that within the same group, the energy levels and the wave functions associated with two paired lines have an identity property while those associated with different pairs have a similarity property. Meanwhile, by thoroughly analyzing processes used to calculate N2-broadened half-widths, it was found that the 'Fourier series' of W(sup a)(sub L(sub 1))(sub K(sub 1))(sub K(sub 1)) (t; j(sub f) T(sub f) and W(sup a)(sub L(sub 1))(sub K(sub 1))(sub K(sub 1)) (t; j(sub i) T(sub i), and a factor P(sub 222) (j(sub f) T(sub f) j(sub i) T(sub i)) are the key items in the Robert-Bonamy formalism to distinguish contributions to ReS2(r(sub c)) among different transitions of j(sub f) T(sub f) - j(sub i). However, these items are completely determined by the energy levels and the wave functions associated with their initial and final states and they must bear the latter's features as well. Thus, it becomes obvious that for two paired lines in the same group, their calculated half-widths must be almost identical and the values associated with different pairs must vary smoothly as their ji values vary. Thus, the pair identity and the smooth variation rules are established within individual groups of lines. One can use these rules to screen half-width data listed in HITRAN and to improve the data accuracies.

water↗
NASA NTRS2014

Modeling Sound Propagation Through Non-Axisymmetric Jets

A method for computing the far-field adjoint Green's function of the generalized acoustic analogy equations under a locally parallel mean flow approximation is presented. The method is based on expanding the mean-flow-dependent coefficients in the governing equation and the scalar Green's function in truncated Fourier series in the azimuthal direction and a finite difference approximation in the radial direction in circular cylindrical coordinates. The combined spectral/finite difference method yields a highly banded system of algebraic equations that can be efficiently solved using a standard sparse system solver. The method is applied to test cases, with mean flow specified by analytical functions, corresponding to two noise reduction concepts of current interest: the offset jet and the fluid shield. Sample results for the Green's function are given for these two test cases and recommendations made as to the use of the method as part of a RANS-based jet noise prediction code.

Leib, Stewart J.↗
NASA NTRS2009

Mean Element Propagations Using Numerical Averaging

The long-term evolution characteristics (and stability) of an orbit are best characterized using a mean element propagation of the perturbed two body variational equations of motion. The averaging process eliminates short period terms leaving only secular and long period effects. In this study, a non-traditional approach is taken that averages the variational equations using adaptive numerical techniques and then numerically integrating the resulting EOMs. Doing this avoids the Fourier series expansions and truncations required by the traditional analytic methods. The resultant numerical techniques can be easily adapted to propagations at most solar system bodies.

orbit propagations↗
NASA NTRS2016

Interferometer with Continuously Varying Path Length Measured in Wavelengths to the Reference Mirror

An interferometer in which the path length of the reference beam, measured in wavelengths, is continuously changing in sinusoidal fashion and the interference signal created by combining the measurement beam and the reference beam is processed in real time to obtain the physical distance along the measurement beam between the measured surface and a spatial reference frame such as the beam splitter. The processing involves analyzing the Fourier series of the intensity signal at one or more optical detectors in real time and using the time-domain multi-frequency harmonic signals to extract the phase information independently at each pixel position of one or more optical detectors and converting the phase information to distance information.

Ohara, Tetsuo↗
NASA NTRS2016

Three-Dimensional Generalized Logarithmic Spirals

The family of generalized logarithmic spirals including a control parameter is extended to the three-dimensional case. The in-plane motion is decoupled from the out-of-plane motion in such a way that the integrals of motion found in the planar problem are still preserved in the three-dimensional case. Designing a low-thrust orbit transfer decomposes in two stages: first, orbits are projected on a reference plane and the planar transfer is solved with a generalized logarithmic spiral. Second, the out-of-plane component of the motion is included in order to target the final orbit. The projection of the three-dimensional transfer orbit on the reference plane is a generalized logarithmic spiral. Arbitrary shape-based laws for the 3D motion can be considered. This paper explores a polynomial and a Fourier series shaping method, together with a polynomial steering law. A fictitious low-thrust sample return mission to Ceres is designed to show the versatility of the method.

Roa, Javier↗
NASA NTRS2020

Investigation of a Helicopter Individual Blade Control (IBC) System in Two Full-Scale Wind Tunnel Tests: Volume II—Tabulated Data

This volume presents the tabulated data acquired during the testing of an individual blade control (IBC) system for a full-scale BO-105 helicopter rotor in the NASA Ames 40- by 80-Foot Wind Tunnel. The data are divided into appendices according to the data type. Table 6 in Volume I presents a summary of the data acquired during the 1993 and 1994 IBC wind tunnel tests. The data have been tabulated according to test year, run number, and point number (e.g., 1993 IBC Test, Run 20, pt. 3). The 1993 and 1994 IBC tests were each started with a run number of 1. A new run number was assigned each time the wind tunnel and rotor were restarted, even if no data were acquired during the previous run. For this reason, some run numbers appear to be missing, yet all of the IBC data is included in the tabulated data, with one notable exception. The Fourier series data presented in Appendices H, L, and N are provided only for the data points identified with the ”•“ symbol in the last column of Table 6 in Volume I. This was done to keep the physical size of the data tables reasonable. The table below shows which data were included in these appendices; the data encompass the majority of the IBC inputs tested in the wind tunnel. Data for the points not included in Appendices H, L, and N have been retained in the BO-105 IBC database and are electronically available from NASA Ames Research Center.

Investigation↗
NASA NTRS2021

Relationship between Horizontal Wind Velocity and Normalized Surface Cross Section Using Data from the HIWRAP Dual-Frequency Airborne Radar

The High-Altitude Imaging Wind and Rain Airborne Profiler (HIWRAP) dual-frequency conically scanning airborne radar provides estimates of the range-profiled mean Doppler and backscattered power from the precipitation and surface. A velocity–azimuth display analysis yields near-surface estimates of the mean horizontal wind vector υh in cases in which precipitation is present throughout the scan. From the surface return, the normalized radar cross section (NRCS) is obtained, which, by a method previously described, can be corrected for path attenuation. Comparisons between υh and the attenuation-corrected NRCS are used to derive transfer functions that provide estimates of the wind vector from the NRCS data under both rain and rain-free conditions. A reasonably robust transfer function is found by using the mean NRCS (⟨NRCS⟩) over the scan along with a filtering of the data based on a Fourier series analysis of υh and the NRCS. The approach gives good correlation coefficients between υh and ⟨NRCS⟩ at Ku band at incidence angles of 30° and 40°. The correlation degrades if the Ka-band data are used rather than the Ku band.

R Meneghini↗
NASA NTRS2021

The Double-Drained Drop: Symmetric Draining of Liquids from Containers with Interior Corners

An analytic solution is found for the late stage draining of a wetting capillary liquid from an interior corner. The solution exploits the symmetry of volumetric sink conditions at opposing ends of such a `double-drained' interior corner flow with applications ranging from liquid recovery in microfluidic devices on earth to liquid fuels scavenging in large fuel tanks aboard spacecraft. At long times t, the liquid depth is h~t^n, the liquid volume is V~ t^(2n), and the maximum volumetric liquid removal rate is Q~t^(2n-1), where n =-1. Representative experiments are conducted at larger length scales aboard the International Space Station and at microfluidic length scales in a terrestrial laboratory. Both sets of experiments confirm the solutions. We show that the approach offers a method to predict maximum drain rates from related geometries where a single drain location provides the required symmetry. We also show that the `fixed length' solution is one of many practical similarity solutions to the governing nonlinear evolution equation for the transient liquid profile along the interior corner. Exponential asymptotic Fourier series solutions are also found for partial corner fillling as well as draining.

microgravity↗
NASA NTRSDate not supplied

Instabilities in the Wake of a Pseudorandom Roughness on a Supersonic Flat Plate

A roughness patch defined by a two-dimensional Fourier series was installed in a highly-polished flat plate model immersed in a supersonic quiet flow. The NASA Langley Research Center Supersonic Low Disturbance Tunnel is capable of providing a Mach 3.5 flow with low freestream acoustic noise and low turbulence levels. Measurements made with a hot-wire probe downstream of the roughness element showed the presence of instability modes that did not cause transition in the measurement region at a freestream unit Reynolds number of Re =12.6×10^6/m. The largest amplitude mode appeared to be similar to a symmetric mode. The same roughness patch immersed in a higher freestream acoustic noise environment showed little to no evidence of the instability modes. Instead, disturbances found in the boundary layer matched closely to the freestream spectra measured in the empty tunnel under the same conditions, indicating some evidence of freestream forcing in the boundary layer.

Roughness induced transition↗
NASA NTRS2022

Instabilities in the Wake of a Pseudorandom Roughness on a Supersonic Flat Plate

A roughness patch defined by a two-dimensional Fourier series was installed in a highly-polished flat plate model immersed in a supersonic quiet flow. The NASA Langley Research Center Supersonic Low Disturbance Tunnel is capable of providing a Mach 3.5 flow with low freestream acoustic noise and low turbulence levels. Measurements made with a hot-wire probe downstream of the roughness element showed the presence of instability modes that did not cause transition in the measurement region at a freestream unit Reynolds number of Re =12.6×10^6/m. The largest amplitude mode appeared to be similar to a symmetric mode. The same roughness patch immersed in a higher freestream acoustic noise environment showed little to no evidence of the instability modes. Instead, disturbances found in the boundary layer matched closely to the freestream spectra measured in the empty tunnel under the same conditions, indicating some evidence of freestream forcing in the boundary layer.

Roughness induced transition↗
NASA NTRSDate not supplied

High-Fidelity Flight Dynamic Analysis of Transonic Truss-Braced Wing

This paper presents a high-fidelity flight dynamic analysis of the Mach 0.8 Transonic Truss-Braced Wing (TTBW). Unsteady RANS CFD simulations of the Mach 0.8 TTBW in pitch, plunge, roll, and yaw oscillations are conducted in FUN3D. The time-domain data are transformed into the frequency-domain data by Fourier series. Transfer functions of the dynamic stability derivatives are then estimated by a frequency-domain re- gression. The dynamic stability derivatives with respect to the angle of attack are determined by the regression of the unsteady aerodynamic coefficients for the plunge motion. The dynamic stability derivatives with re- spect to the pitch rate are determined by the regression of the differential unsteady aerodynamic coefficients for the pitch motion upon the removal of the angle of attack contribution by the plunge motion. Similarly, dynamic stability derivatives with respect to the angle of sideslip, roll rate, and yaw rates are determined from the frequency domain regression. The longitudinal and lateral-direction flight dynamic models of the Mach 0.8 TTBW are constructed from these dynamic stability derivatives. The eigenvalues of the aircraft modes are analyzed to determine the vehicle stability.

Aircraft Stability and Control↗
NASA NTRS2022

Instabilities in the Wake of a Pseudorandom Roughness on a Supersonic Flat Plate

A roughness patch defined by a two-dimensional Fourier series was installed in a highly-polished flat plate model immersed in a supersonic quiet flow. The NASA Langley Research Center Supersonic Low Disturbance Tunnel is capable of providing a Mach 3.5 flow with low freestream acoustic noise and low turbulence levels. Measurements made with a hot-wire probe downstream of the roughness element showed the presence of instability modes that did not cause transition in the measurement region at a freestream unit Reynolds number of Re =12.6×10^6/m. The largest amplitude mode appeared to be similar to a symmetric mode. The same roughness patch immersed in a higher freestream acoustic noise environment showed little to no evidence of the instability modes. Instead, disturbances found in the boundary layer matched closely to the freestream spectra measured in the empty tunnel under the same conditions, indicating some evidence of freestream forcing in the boundary layer.

Roughness induced transition↗