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NASA NTRS1984

Applications of CONMIN to wing design optimization with vortex flow effect

Slender wings on supersonic cruise configurations are expected to be thin and highly swept. As a result, edge-separated vortex flow is inevitable and must be accounted for in aerodynamic analysis and design. The present method is based on the method of suction analogy to calculate the total aerodynamic characteristics. The method requires the solution of the attached flow problem, the latter being solved by a low-order panel method in subsonic and supersonic flow. In essence, the lifting pressure is calculated by using a pressure-doublet distribution satisfying the Prandtl-Glauert equation. From the pressure distribution, the leading-edge suction is calculated. The latter is assumed to be the vortex lift through the method of suction analogy. For a cambered wing, the location of vortex-lift action point is important in predicting the aerodynamic characteristics. It is also seen that the effect of camber shape appears nonlinearly in all aerodynamic expressions. To design the camber shape, the camber slope is represented by a cosine Fourier series at each of several spanwise stations. The Fourier coefficients are the design variables. To design a leading-edge flap in the vortex flow (i.e., a vortex flap), the coordinates of corner points and the deflection angle are the design variables. The process of wing design is to determine the camber shape and twist distribution such that an objective function, typically the drag, is minimized, subject to various constraints.

Lan, C. E.↗

NASA NTRS2010

Broken Ergodicity in Ideal, Homogeneous, Incompressible Turbulence

We discuss the statistical mechanics of numerical models of ideal homogeneous, incompressible turbulence and their relevance for dissipative fluids and magnetofluids. These numerical models are based on Fourier series and the relevant statistical theory predicts that Fourier coefficients of fluid velocity and magnetic fields (if present) are zero-mean random variables. However, numerical simulations clearly show that certain coefficients have a non-zero mean value that can be very large compared to the associated standard deviation. We explain this phenomena in terms of broken ergodicity', which is defined to occur when dynamical behavior does not match ensemble predictions on very long time-scales. We review the theoretical basis of broken ergodicity, apply it to 2-D and 3-D fluid and magnetohydrodynamic simulations of homogeneous turbulence, and show new results from simulations using GPU (graphical processing unit) computers.

Morin, Lee↗

NASA NTRS1974

Derivation of transformation formulas between geocentric and geodetic coordinates for nonzero altitudes

Four formulas, for the nonzero altitude transformation from geodetic coordinates (geodetic latitude and altitude) to geocentric coordinates (geocentric latitude and geocentric distance) and vice versa, are derived. The set of four formulas is expressed in each of the three useful forms: series expansion in powers of the earth's flattening; series expansion in powers of the earth's eccentricity; and Fourier series expansion in terms of the geodetic latitude or the geocentric latitude. The error incurred in these series expansions is of the order of one part in 3 x 10 to the 7th power.

Long, S. A. T.↗

NASA NTRS1980

FFT methods in signal processing of the coal interface detector radar

The FM radar for the coal interface detector, operating in the frequency band 2 to 4 GHz, is intended for the display of thicknesses between 2 cm and 20 cm. Because of such a short range, the thickness information is contained in the very few lowest spectral components of the output signal. To overcome this inconvenience, the Fourier series of the output signal was augmented to approximate a Fourier integral. This modification in the signal processing resulted in a higher spectral density, which in turn enabled an easier identification of the interface position in the laboratory. The orientation and spacing of the receiving and transmitting antennas is found to have an important influence on the system performance.

Kajfez, D.↗

NASA NTRS1983

Finite element or Galerkin type semidiscrete schemes

A finite element of Galerkin type semidiscrete method is proposed for numerical solution of a linear hyperbolic partial differential equation. The question of stability is reduced to the stability of a system of ordinary differential equations for which Dahlquist theory applied. Results of separating the part of numerical solution which causes the spurious oscillation near shock-like response of semidiscrete scheme to a step function initial condition are presented. In general all methods produce such oscillatory overshoots on either side of shocks. This overshoot pathology, which displays a behavior similar to Gibb's phenomena of Fourier series, is explained on the basis of dispersion of separated Fourier components which relies on linearized theory to be satisfactory. Expository results represented.

Durgun, K.↗

NASA NTRS1981

An analytic formulation for heating source memory in the thermospheric composition

Numerical calculations from a spectral circulation model are utilized to construct an analytic Green's function formulation describing the meridional, time-dependent thermospheric composition and temperature response during magnetic storms. The purpose is to develop a formulation that embodies source memory while being sufficiently simple to serve as a heuristic guide for empirical modeling. By passing from the discrete Fourier series representation, utilized for the numerical circulation model, to a continuous Fourier integral representation, explicit waves are obtained for the thermospheric response times. The response times are altitude and species dependent and can exceed two days below 200 km. Thus, for certain storm scenarios, pronounced source memory signatures for the composition and temperature are predicted. Response times obtained from the formulation are shown to give a response consistent with previously published neutral composition data from AE-C for the February 1974 storm when an ap dependent heat source is employed.

Porter, H. S.↗

NASA NTRS1984

Doppler line profile analysis for a multichannel Fabry-Perot interferometer

A new method of instrument calibration and data analysis is presented for single-etalon interferometric measurements of winds, temperatures, and emission line intensities. The technique has been developed for the multichannel Fabry-Perot interferometer on the Dynamics Explorer spacecraft. A numerical representation of the instrumental transfer function is used based on a truncated Fourier series with empirically determined coefficients. The numerical form is compared with the conventional analytic form. The Fourier coefficients describing the instrument function are generated at the wavelength of a stable He-Ne laser and are translated to other wavelengths using an interpolation technique for both phase and power. A quasi-linear least-squares fitting process involving matrices provides for a rapid and accurate data reduction.

Killeen, T. L.↗

NASA NTRS2010

Broken Ergodicity in MHD Turbulence

Ideal magnetohydrodynamic (MHD) turbulence may be represented by finite Fourier series, where the inherent periodic box serves as a surrogate for a bounded astrophysical plasma. Independent Fourier coefficients form a canonical ensemble described by a Gaussian probability density function containing a Hermitian covariance matrix with positive eigenvalues. The eigenvalues at lowest wave number can be very small, resulting in a large-scale coherent structure: a turbulent dynamo. This is seen in computations and a theoretical explanation in terms of 'broken ergodicity' contains Taylor s theory of force-free states. An important problem for future work is the case of real, i.e., dissipative flows. In real flows, broken ergodicity and coherent structure are still expected to occur in MHD turbulence at the largest scale, as suggested by low resolution simulations. One challenge is to incorporate coherent structure at the largest scale into the theory of turbulent fluctuations at smaller scales.

Shebalin, John V.↗

NASA NTRS1990

Frequency analysis via the method of moment functionals

Several variants are presented of a linear-in-parameters least squares formulation for determining the transfer function of a stable linear system at specified frequencies given a finite set of Fourier series coefficients calculated from transient nonstationary input-output data. The basis of the technique is Shinbrot's classical method of moment functionals using complex Fourier based modulating functions to convert a differential equation model on a finite time interval into an algebraic equation which depends linearly on frequency-related parameters.

Pearson, A. E.↗

NASA NTRS1991

Frequency analysis via the method of moment functionals

Pearson, A. E.↗

NASA NTRS1971

Computerized series solution of relativistic equations of motion.

A method of solution of the equations of planetary motion is described. It consists of the use of numerical general perturbations in orbital elements and in rectangular coordinates. The solution is expanded in Fourier series in the mean anomaly with the aid of harmonic analysis and computerized series manipulation techniques. A detailed application to the relativistic motion of the planet Mercury is described both for Schwarzschild and isotropic coordinates.

Broucke, R.↗

NASA NTRSDate not supplied

TPSAS-NF1676L-17943-DND

SAGE II used the solar occultation technique to measure atmospheric ozone for 21 years between 1984 and 2005 and has continued to play a key role in numerous international assessments. Traditionally, the determination of trends in ozone using SAGE II data has been done via time series analysis of monthly mean data within specific latitude bands. However, given the sparse sampling of SAGE II data, this will introduce temporal and/or spatial biases, particularly in the period after late 2000 when the instrument operated at half-duty cycle. Herein we present a new method for performing time series analysis of unevenly sampled data (in particular SAGE II); namely, we utilize the daily mean data and perform a simultaneous temporal and spatial analysis, making use of a Fourier series to constrain latitudinal variation. This methodology is applied to both the previous (v6.2) and current (v7.0) versions of the SAGE II data. Results are discussed that show the improvements to the data between versions 6.2 and 7.0 that make time series analysis produce reduced residuals as well as the preliminary results of the time series itself, which reveal some of the limitations in currently used proxies for the regression.

Robert P Damadeo↗

NASA NTRS1987

Nonlinear diffusion in the solid phase

A Fourier series expansion method has been used to develop simple and relatively accurate solutions for nonlinear Fickian diffusion in the solid phase. For the fundamental harmonic, the first term in the Fourier expansion is qualitatively similar but quantitatively quite different from the conventional exponential function. In contrast, the Fourier coefficients of the higher order harmonics are combinations of fast-decay and slow-decay components. The simple analytical results agree well with numerical solutions for one sample calculation.

Tsang, Tung↗

NASA NTRS1971

Techniques of orbital decay and long-term ephemeris prediction for satellites in earth orbit

In the special perturbation method, Cowell and variation-of-parameters formulations of the motion equations are implemented and numerically integrated. Variations in the orbital elements due to drag are computed using the 1970 Jacchia atmospheric density model, which includes the effects of semiannual variations, diurnal bulge, solar activity, and geomagnetic activity. In the general perturbation method, two-variable asymptotic series and automated manipulation capabilities are used to obtain analytical solutions to the variation-of-parameters equations. Solutions are obtained considering the effect of oblateness only and the combined effects of oblateness and drag. These solutions are then numerically evaluated by means of a FORTRAN program in which an updating scheme is used to maintain accurate epoch values of the elements. The atmospheric density function is approximated by a Fourier series in true anomaly, and the 1970 Jacchia model is used to periodically update the Fourier coefficients. The accuracy of both methods is demonstrated by comparing computed orbital elements to actual elements over time spans of up to 8 days for the special perturbation method and up to 356 days for the general perturbation method.

Barry, B. F.↗

NASA NTRS1988

Stress and vibraton analyses of anisotropic shells of revolution

An efficient computational strategy is presented for reducing the cost of the stress and free vibration analyses of laminated anisotropic shells of revolution. The analytical formulation is based on a form of the Sanders-Budiansky shell theory including the effects of both the transverse shear deformation and the laminated anisotropic material response. The fundamental unknowns consist of the eight strain components, the eight stress resultants and the five generalized displacements of the shell. Each of the shell variables is expressed in terms of trigonometric functions (Fourier series) in the circumferential co-ordinate, and a three-field mixed finite element model is used for the discretization in the meridional direction. The shell response associated with a range of Fourier harmonics is approximated by a linear combination of a few global approximation vectors, which are generated at a particular value of the Fourier harmonic, within that range. The full equations of the finite element model are solved for only a single Fourier harmonic, and the response corresponding to the other Fourier harmonics is generated using a reduced system of equations with considerably fewer degrees of freedom.

Noor, Ahmed K.↗

NASA NTRSDate not supplied

Nonlinear Dynamic Control Derivative Analysis for Aircraft with Application to Transonic Truss-Braced Wing

This paper presents the development of a nonlinear dynamic control derivative estimation method. A nonlinear aerodynamic model is developed to account for the effects of large control surface deflections on aircraft aerodynamic forces and moments. A series of unsteady RANS CFD simulations is performed to simulate the control surface oscillations at various reduced frequencies. The time-domain data are transformed into the frequency-domain data by a Fourier series analysis. Transfer functions of the dynamic control derivatives are then estimated by a frequency-domain regression. The method is applied to the Transonic Truss-Braced Wing (TTBW) to estimate the dynamic control derivatives for the elevator, aileron, and rudder.

Aircraft Control Derivatives↗

NASA NTRSDate not supplied

CFD-Based Frequency Domain Method for Dynamic Stability Derivative Estimation with Application to Transonic Truss-Braced Wing

This paper presents a dynamic stability estimation technique obtained from high-fidelity CFD simulations of the Mach 0.8 Transonic Truss-Braced Wing (TTBW). A series of unsteady RANS CFD simulations in FUN3D is performed on the TTBW in pitch and plunge oscillations at various reduced frequencies. 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 regression. 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 respect to the pitch rate are deter-mined 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. The steady-state stability derivatives are then compared to the results obtained from a stability analysis code VSPAERO as well as steady-state FUN3D simulations. The comparison of the steady-state stability derivatives shows excellent agreement.

Dynamic Stability Derivatives↗

NASA NTRS1988

A Fourier analysis for a fast simulation algorithm

This paper presents a derivation of compact expressions for the Fourier series analysis of the steady-state solution of a typical switching converter. The modeling procedure for the simulation and the steady-state solution is described, and some desirable traits for its matrix exponential subroutine are discussed. The Fourier analysis algorithm was tested on a phase-controlled parallel-loaded resonant converter, providing an experimental confirmation.

King, Roger J.↗