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A result obtained by Williams (1977) for two-dimensional airfoils oscillating in an arbitrary subsonic parallel flowfield is reformulated to show that the pressure distribution induced by any deformation can be construed from the particular solutions for heaving and pitching motions. Specific formulas are presented for an oscillating control surface with a sealed gap.
The paper presents a review of the historical development in unsteady transonic aerodynamics, along with the foundations and accomplishments of several approaches to solve the equations of unsteady transonic flow. The discussion covers the linearized unsteady flow theory, numerical solution of the exact equations for an inviscid compressible gas, nonlinear small disturbance theory of transonic flow and linearization of the unsteady component about the nonlinear solution for the steady state, local linearization solution for unsteady transonic flow, unsteady transonic flow theory for slender wings and bodies, and three-dimensional unsteady transonic flows. The relation between the calculated results and experiment is examined. It is shown that the newly emerging numerical methods are capable of solving the nonlinear equations for two-dimensional flow and can be extended to three-dimensional flows.
The study of differential equations, or dynamical systems in general, has two fundamentally different approaches. We are most familiar with the construction of solutions to differential equations. Another approach is to study the statistical behavior of the solutions. Ergodic Theory is one of the most developed methods to study the statistical behavior of the solutions of differential equations. In the theory of satellite orbits, the statistical behavior of the orbits is used to produce 'Coverage Analysis' or how often a spacecraft is in view of a site on the ground. In this paper, we consider the use of Ergodic Theory for Coverage Analysis. This allows us to greatly simplify the computation of quantities such as the total time for which a ground station can see a satellite without ever integrating the trajectory, see Lo 1,2. More over, for any quantity which is an integrable function of the ground track, its average may be computed similarly without the integration of the trajectory. For example, the data rate for a simple telecom system is a function of the distance between the satellite and the ground station. We show that such a function may be averaged using the Ergodic Theorem.
The results of an intensive analysis of a differential orbit improvement method utilizing observational data for a 550-kilometer altitude, near-circular, near-equatorial satellite orbit are presented. Observations of the Small Astronomy Satellite (SAS-1) are in the form of direction cosines as measured at two ground interferometer tracking stations near the Equator during the first 22 orbital revolutions (approximately 37 hours) after launch of the spacecraft. Numerical results, in both tabular and graphical form, are displayed for numerous iterated fittings of various observational arcs by differential correction of the orbital elements. Parameters varied in these comparative cases include the time duration of the observational data block, the number of pairs of direction cosine data and the number of tracking station passes included in the solution, the distribution of such passes between the two available tracking stations, and the acceptance criterion for the observational residuals in the least squares fitting procedure. It was found that three observational pairs of direction cosine data, the minimum number possible for a uniquely determined solution in theory, are sufficient to promote covergence to an accurate solution, if properly selected.
A simple recursive solution for a class of fading memory tracking filters is presented. A fading memory filter provides estimates of filter states based on past measurements, similar to a traditional Kalman filter. Unlike a Kalman filter, an exponentially decaying weight is applied to older measurements, discounting their effect on present state estimates. It is shown that Kalman filters and fading memory filters are closely related solutions to a general least squares estimator problem. Closed form filter transfer functions are derived for a time invariant, steady state, fading memory filter. These can be applied in loop filter implementation of the Deep Space Network (DSN) Advanced Receiver carrier phase locked loop (PLL).
The computation of inviscid supersonic flow over a two-dimensional airfoil is considered. There are two main nonlinear approaches which lead to approximate solutions. Small-amplitude theory gives valid solutions provided the airfoil thickness is not too great and the Mach number is not too high. The second type of approximation, shock expansion theory, employs the fact that even for flows with strong shocks the effect of the down-running characteristics remains small. This leads to an analytic solution at the airfoil. It is pointed out that the approximate solution is accurate enough for many cases of interest. The numerical method furnishes a rapid correction to the solution in those cases where it is not. The characteristic-streamline coordinate system is useful both for the computation of the approximate solution and the corrections, and is also convenient for displaying and interpreting the results.
Liapunov stability theory applied to class of partial differential equations, and generation of contraction groups related to equivalent norms
Kepler's Equation is solved using an integrative algorithm developed using homotropy theory. The solution approach is applicable to both elliptic and hyperbolic forms of Kepler's Equation. The results from the proposed algorithm compare quite favorably with those from existing iterative schemes.
Boundary conditions for Tricomi equation in symmetric positive form, showing analytical and finite difference solutions
An analytic solution for the magnetic field in the space defined by a spherical moon and its downstream cylindrical cavity formed by the solar wind is derived for interplanetary magnetic fields both parallel and perpendicular to the cavity axis. By superposition, the solution is obtained for arbitrary orientations of the interplanetary field. The theory is quasi-static and is formulated in terms of a scalar magnetic potential. Thus the moon model consists of a core of arbitrary size and infinite electrical conductivity surrounded by a nonconducting shell; the cavity volume is assumed to be nonconducting. The variation of the magnetic field on the lunar surface, both on the sunlit and on the dark side hemispheres, and on the cavity boundary is presented for various values of core radius. The solution yields the distribution of currents on the lunar sunlit surface and the surface of the cavity. Theoretical transfer functions are presented and their variations with position on the lunar surface and with core size are discussed.
An analytic solution for the magnetic field in the space defined by a spherical moon and its downstream cylindrical cavity formed by the solar wind is derived for interplanetary magnetic fields both parallel and perpendicular to the cavity axis. By superposition, the solution is obtained for arbitrary orientations of the interplanetary field. The theory is quasi-static and is formulated in terms of a scalar magnetic potential. Thus, the moon model consists of a core of arbitrary size and infinite electrical conductivity surrounded by a nonconducting shell; the cavity volume is also assumed to be nonconducting. The variation of the magnetic field on the lunar surface (both sunlit and dark hemispheres) and on the cavity boundary is presented for various values of core radius.
Finite-density zero-pressure models of universe based on general relativity theory - approximate solution for luminosity distance
Static spherically symmetric charged mass distribution exact solutions derived for electrons in scalar tensor theory of gravity by Hamilton-Jacobi method
The Helmholtz equation is solved within a three-dimensional rectangular duct with a nonlocal radiation boundary condition at the duct exit plane. This condition accurately models the acoustic admittance at an arbitrarily-located computational boundary plane. A linear system of equations is constructed with second-order central differences for the Helmholtz operator and second-order backward differences for both local admittance conditions and the gradient term in the nonlocal radiation boundary condition. The resulting matrix equation is large, sparse, and non-Hermitian. The size and structure of the matrix makes direct solution techniques impractical; as a result, a nonstationary iterative technique is used for its solution. The theory behind the nonstationary technique is reviewed, and numerical results are presented for radiation from both a point source and a planar acoustic source. The solutions with the nonlocal boundary conditions are invariant to the location of the computational boundary, and the same nonlocal conditions are valid for all solutions. The nonlocal conditions thus provide a means of minimizing the size of three-dimensional computational domains.
Rational approximations and solution of control theory problems involving Ricatti equations
The overall contractual effort and the theory and numerical solution for the Reacting and Multi-Phase (RAMP2) computer code are described. The code can be used to model the dominant phenomena which affect the prediction of liquid and solid rocket nozzle and orbital plume flow fields. Fundamental equations for steady flow of reacting gas-particle mixtures, method of characteristics, mesh point construction, and numerical integration of the conservation equations are considered herein.
For a lifting body of revolution, the exact solutions to the gas dynamical equation are given for the flow field in a stratified atmosphere. The boundary condition is satisfied by using the (linearized) slender-body theory. The solution allows one to make a prediction of the magnitude and the location of the sonic boom in a stratified atmosphere as well as to handle cutoff effects.
Application of singular perturbation techniques to trajectory optimization problems of flight mechanics is discussed. The method of matched asymptotic expansions is used to obtain an approximate solution to the aircraft minimum time-to-climb problem. Outer, boundary-layer, and composite solutions are obtained to zeroth and first orders. A stability criterion is derived for the zeroth-order boundary-layer solutions (the theory requires a form of boundary-layer stability). A numerical example is considered for which it is shown that the stability criterion is satisfied and a useful numerical solution is obtained. The zeroth-order solution proves to be a poor approximation, but the first-order solution gives a good approximation for both the trajectory and the minimum time-to-climb. The computational cost of the singular perturbation solution is considerably less than that of a steepest descent solution. Thus singular perturbation methods appear to be promising for the solution of optimal control problems.