Recurrence formulae for the Hansen's developments.
Recurrence formulas for calculating Fourier expansions in elliptic motion in terms of eccentric and mean anomaly
Vinh, N. X.↗
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Finite-amplitude instability of the compressible laminar wake - Weakly nonlinear theory.
Nonlinear finite-amplitude instability of two dimensional compressible laminar wake flows, using Fourier expansion and Landau equation
Liu, J. T. C.↗
Nonlinear cellular motions in Poiseuille channel flow
In a number of nonlinear solutions to the equations of channel flow the velocity is decomposed into a mean part plus a nonlinear disturbance. The idea that nonlinear effects place a limitation on the amplitudes of the disturbance flow is considered. In the reported investigation the disturbance flow is represented by drastically limited Fourier expansions in the downstream coordinate. The resulting equations are solved numerically with high accuracy to obtain a good representation of the cross-stream structure of the solution. The results of the investigation show that indeed the nonlinear terms always limit the amplitude of the disturbance flow in this approximation.
Zahn, J.-P.↗
On the Laplace relation among the satellites of Uranus
Three of Uranus' satellites are involved in an interaction similar to the Laplace relation among the Galilean satellites of Jupiter. However, there are no pair-wise commensurabilities in the Uranus system as there are among the Galilean satellites. Thus analysis by Fourier expansion of disturbing functions becomes too unwieldy for practical calculation. Instead, the unexpanded perturbations are numerically averaged over a time scale short compared to the important variations in the Laplace relation. Perturbations of Miranda's orbit are estimated to be of observable magnitude for reasonable masses of Ariel and Umbriel.
Greenberg, R.↗
The seasonal and interannual variability of total ozone as revealed by the BUV Nimbus-4 experiment
The BUV/Nimbus-4 total ozone data is analyzed with emphasis on the seasonal and interannual variability for the period April 1970 to April 1972. An objective analysis using a Fourier expansion shows the annual wave dominates at mid and high latitudes where the semiannual wave becomes significant in the tropics. A small interannual difference is detected and is most likely due to changes in the general circulation.
Hilsenrath, E.↗
Analysis of solar radiation pressure induced coupled librations of a gravity stabilized axisymmetric satellite
This paper presents an analysis of solar radiation pressure induced coupled librations of gravity stabilized cylindrical spacecraft with a special reference to geostationary communication satellites. The Lagrangian approach is used to obtain the corresponding equations of motion. The solar induced torques are assumed to be free of librational angles and are represented by their Fourier expansion. The response and periodic solutions are obtained through linear and nonlinear analyses, using the method of harmonic balance in the latter case. The stability conditions are obtained using Routh-Hurwitz criteria. To establish the ranges of validity the analytic response is compared with the numerical solution. Finally, values of the system parameters are suggested to make the satellite behave as desired. Among these is a possible approach to subdue the solar induced roll resonance. It is felt that the approximate analysis presented here should significantly reduce the computational efforts involved in the design and stability analysis of the systems.
Shrivastava, S. K.↗
Large-eddy simulation of the shear-free turbulent boundary layer
The shear-free turbulent boundary layer is calculated by the large-eddy simulation technique. The filtered Navier-Stokes equations are used; the method of integration employs Fourier expansions in the homogeneous directions and finite differences in the cross-stream direction. Results indicate that the simulation is capable of predicting the primary Reynolds-number effects.
Biringen, S.↗
A new numerical method for the simulation of three dimensional flow in a pipe
A new numerical technique for simulating three dimensional, unsteady, incompressible pipe flows is presented and its utility and accuracy is shown. Each vector function in the expansion of the velocity field is divergence free and satisfies the boundary conditions for viscous flow. Some of the benefits of the expansion technique are that pressure is eliminated from the dynamics, only two unknowns per mesh point are required, implicit treatment of the viscous terms is provided at no extra computational cost, and no fractional time steps are required. The method uses spectral expansions: Fourier series in the azimuthal and streamwise directions, and Jacobi polynominals in the radial direction.
Leonard, A.↗
A new numerical method for the simulation of three-dimensional flow in a pipe
A new numerical technique for simulating three dimensional, unsteady, incompressible pipe flows is presented and its utility and accuracy is shown. Each vector function in the expansion of the velocity field is divergence free and satisfies the boundary conditions for viscous flow. Some of the benefits of the expansion technique are that pressure is eliminated from the dynamics, only two unknowns per mesh point are required, implicit treatment of the viscous terms is provided at no extra computational cost, and no fractional time steps are required. The method uses spectral expansions: Fourier series in the azimuthal and streamwise directions, and Jacobi polynominals in the radial direction. Previously announced in STAR as N82-31644
Leonard, A.↗
Cloud streets during KonTur: A comparison of parallel/thermal instability modes with observations
Estimates of cloud street geometry produced by a model of the parallel/thermal instability modes of shallow convection are compared with observations obtained during the 1981 KonTur experiment. Good agreement between the modeled and observed orientation angles, wavelengths and Reynolds numbers are found when the streets are assumed to derive their energy from the average shear and the lowest order sine terms of a Fourier expansion of the mean wind profile (or equivalently from the lowest order cosine terms of the mean shear profile). The modes associated with the cosine terms of the wind profile (or the sine terms of the wind shear profile) do not agree well with the observations. These results suggest that the boundary layer rolls observed during KonTur might have developed owing to a combined parallel/thermal instability originating primarily from the cosine terms of the ambient roll parallel wind shear.
Shirer, H. N.↗
Construction of approximate analytical solutions to a new class of non-linear oscillator equation
The principle of harmonic balance is invoked in the development of an approximate analytic model for a class of nonlinear oscillators typified by a mass attached to a stretched wire. By assuming that harmonic balance will hold, solutions are devised for a steady state limit cycle and/or limit point motion. A method of slowly varying amplitudes then allows derivation of approximate solutions by determining the form of the exact solutions and substituting into them the lowest order terms of their respective Fourier expansions. The latter technique is actually a generalization of the method proposed by Kryloff and Bogoliuboff (1943).
Mickens, R. E.↗
Cloud streets during KonTur - A comparison of parallel/thermal instability modes with observations
Estimates of cloud street geometry produced by a model of the parallel/thermal instability modes of shallow convection are compared with observations obtained during the 1981 KonTur experiment. Good agreement between the modeled and observed orientation angles, wavelengths and Reynolds numbers are found when the streets are assumed to derive their energy from the average shear and the lowest order sine terms of a Fourier expansion of the mean wind profile (or equivalently from the lowest order cosine terms of the mean shear profile). The modes associated with the cosine terms of the wind profile (or the sine terms of the wind shear profile) do not agree well with the observations. These results suggest that the boundary layer rolls observed during KonTur might have developed owing to a combined parallel/thermal instability originating primarily from the cosine terms of the ambient roll parallel wind shear.
Shirer, H. N.↗
Steady-state phase error for a phase-locked loop subjected to periodic Doppler inputs
The performance of a carrier phase locked loop (PLL) driven by a periodic Doppler input is studied. By expanding the Doppler input into a Fourier series and applying the linearized PLL approximations, it is easy to show that, for periodic frequency disturbances, the resulting steady state phase error is also periodic. Compared to the method of expanding frequency excursion into a power series, the Fourier expansion method can be used to predict the maximum phase error excursion for a periodic Doppler input. For systems with a large Doppler rate fluctuation, such as an optical transponder aboard an Earth orbiting spacecraft, the method can be applied to test whether a lower order tracking loop can provide satisfactory tracking and thereby save the effect of a higher order loop design.
Chen, C.-C.↗
On the nonlinear development of the most unstable Goertler vortex mode
The nonlinear development of the most unstable Gortler vortex mode in boundary layer flows over curved walls is investigated. The most unstable Gortler mode is confined to a viscous wall layer of thickness O(G -1/5) and has spanwise wavelength O(G 11/5); it is, of course, most relevant to flow situations where the Gortler number G is much greater than 1. The nonlinear equations covering the evolution of this mode over an O(G -3/5) streamwise lengthscale are derived and are found to be of a fully nonparallel nature. The solution of these equations is achieved by making use of the numerical scheme used by Hall (1988) for the numerical solution of the nonlinear Gortler equations valid for O(1) Gortler numbers. Thus, the spanwise dependence of the flow is described by a Fourier expansion, whereas the streamwise and normal variations of the flow are dealt with by employing a suitable finite difference discretization of the governing equations. Our calculations demonstrate that, given a suitable initial disturbance, after a brief interval of decay, the energy in all the higher harmonics grows until a singularity is encountered at some downstream position. The structure of the flowfield as this singularity is approached suggests that the singularity is responsible for the vortices, which are initially confined to the thin viscous wall layer, moving away from the wall and into the core of the boundary layer.
Denier, James P.↗
On the convergence of the coupled-wave approach for lamellar diffraction gratings
Among the many existing rigorous methods for analyzing diffraction of electromagnetic waves by diffraction gratings, the coupled-wave approach stands out because of its versatility and simplicity. It can be applied to volume gratings and surface relief gratings, and its numerical implementation is much simpler than others. In addition, its predictions were experimentally validated in several cases. These facts explain the popularity of the coupled-wave approach among many optical engineers in the field of diffractive optics. However, a comprehensive analysis of the convergence of the model predictions has never been presented, although several authors have recently reported convergence difficulties with the model when it is used for metallic gratings in TM polarization. Herein, three points are made: (1) in the TM case, the coupled-wave approach converges much slower than the modal approach of Botten et al; (2) the slow convergence is caused by the use of Fourier expansions for the permittivity and the fields in the grating region; and (3) is manifested by the slow convergence of the eigenvalues and the associated modal fields. The reader is assumed to be familiar with the mathematical formulations of the coupled-wave approach and the modal approach.
Li, Lifeng↗
On the nonlinear development of the most unstable Goertler vortex mode TLSP - Final report
The nonlinear development of the most unstable Goertler vortex mode in boundary layer flows over curved walls is investigated. The most unstable Goertler mode is confined to a viscous wall layer of thickness O(G -1/5) and has spanwise wavelength O(G 11/5); it is, of course, most relevant to flow situations where the Goertler number G is much greater than 1. The nonlinear equations covering the evolution of this mode over an O(G -3/5) streamwise lengthscale are derived and are found to be of a fully nonparallel nature. The solution of these equations is achieved by making use of the numerical scheme used by Hall (1988) for the numerical solution of the nonlinear Goertler equations valid for O(1) Goertler numbers. Thus, the spanwise dependence of the flow is described by a Fourier expansion, whereas the streamwise and normal variations of the flow are dealt with by employing a suitable finite difference discretization of the governing equations. Our calculations demonstrate that, given a suitable initial disturbance, after a brief interval of decay, the energy in all the higher harmonics grows until a singularity is encountered at some downstream position. The structure of the flowfield as this singularity is approached suggests that the singularity is responsible for the vortices, which are initially confined to the thin viscous wall layer, moving away from the wall and into the core of the boundary layer.
Denier, James P.↗
Compressible flow about symmetrical Joukowski profiles
The method of Poggi is employed for the determination of the effects of compressibility upon the flow past an obstacle. A general expression for the velocity increment due to compressibility is obtained. The general result holds whatever the shape of the obstacle; but, in order to obtain the complete solution, it is necessary to know a certain Fourier expansion of the square of the velocity of flow past the obstacle. An application is made to the case flow of a symmetrical Joukowski profile with a sharp trailing edge, fixed in a stream of an arbitrary angle of attack and with the circulation determined by the Kutta condition. The results are obtained in a closed form and are exact insofar as the second approximation to the compressible flow is concerned, the first approximation being the result for the corresponding incompressible flow. Formulas for lift and moment analogous to the Blasius formulas in incompressible flow are developed and are applied to thin symmetrical Joukowski profiles for small angles of attack.
Kaplan, Carl↗
Comparison of level-1 and level-2 buckling and postbuckling solutions
The accuracy of the solutions obtained with the Level-1 computational modules of DISDECO (Delft Interactive Shell Design Code) are investigated by comparing them with solutions obtained with ANILISA, a Level-2 computational module. Whereas the Level-1 solutions are based on the use of trigonometric functions to represent the out-of-plane displacement W and involve Galerkin type approximations to reduce the governing partial differential equations to simple algebraic expressions, the Level-2 solutions employ a truncated Fourier expansion in the circumferential direction and solve the resulting ordinary differential equations forming a nonlinear eigenvalue problem by a Stodola like technique. This involves the numerical integration of the set of ordinary differential equations, which in turn makes it possible to satisfy the specified boundary conditions exactly. It is shown that in many applications the Level-1 predictions concerning the critical buckling load of the perfect structure and its imperfection sensitivity are quite accurate and reliable. For those cases where this is not true, it is shown how the pre and postprocessing facilities of DISDECO can be used efficiently to arrive at a reliable prediction. This involves always the use of the Level-2 computational module ANILISA with its rigorous prebuckling solution and its feature which makes it possible to satisfy the specified boundary conditions exactly.
Arbocz, J.↗