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Results for “asymptotic expansion”

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.

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At least 127 records · Page 7
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NASA NTRS1973

Acoustic propagation in ducts with varying cross sections and sheared mean flow.

An analysis is presented of the wave propagation and attenuation in two-dimensional ducts with slowly varying cross sections which carry sheared mean flow. A uniformly valid asymptotic expansion of the acoustic waves is obtained in terms of the maximum slope of the wall by using the method of multiple scales. The result is a first-order differential equation describing the amplitude variation with the axial distance. This equation is used to evaluate the effects of variations of the duct cross section and growing boundary layers on the different acoustic modes.

Nayfeh, A. H.↗
NASA NTRS1973

A study of selected radiation and propagation problems related to antennas and probes in magneto-ionic media

Research consisted of computations toward the solution of the problem of the current distribution on a cylindrical antenna in a magnetoplasma. The case of an antenna parallel to the applied magnetic field was investigated. A systematic method of asymptotic expansion was found which simplifies the solution in the general case by giving the field of a dipole even at relatively short range. Some useful properties of the dispersion surfaces in a lossy medium have also been found. A laboratory experiment was directed toward evaluating nonlinear effects, such as those due to power level, bias voltage and electron heating. The problem of reflection and transmission of waves in an electron heated plasma was treated theoretically. The profile inversion problem has been pursued. Some results are very encouraging, however, the general question of stability of the solution remains unsolved.

Source record↗
NASA NTRS1973

Asymptotic solution of the problem for a thin axisymmetric cavern

The boundary value problem which describes the axisymmetric separation of the flow around a body by a stationary infinite stream is considered. It is understood that the cavitation number varies over the length of the cavern. Using the asymptotic expansions for the potential of a thin body, the orders of magnitude of terms in the equations of the problem are estimated. Neglecting small quantities, a simplified boundary value problem is obtained.

Serebriakov, V. V.↗
NASA NTRS1973

Asymptotic solution to the tangential low thrust energy increase trajectory

A approximate analytic solution to the low thrust constant acceleration energy increase trajectory using a tangential steering program is presented. The solution is based on a two-variable asymptotic expansion of the equations of motion formulated in regularized variables. The high accuracy of the second order solution derived is demonstrated by comparing it with numerically integrated trajectories.

Schwenzfeger, K. J.↗
NASA NTRS1974

Viscous, radiating hypersonic flow about a blunt body

The viscous, radiating hypersonic flow past an axisymmetric blunt body is analyzed based on the Navier-Stokes equations, plus a radiative equation of transfer derived from the Milne-Eddington differential approximation. The fluid is assumed to be a perfect gas with constant specific heats, a constant Prandtl number of order unity, a viscosity coefficient varying as a power of the temperature, and an absorption coefficient varying as the first power of the density and as a power of the temperature. The gray gas assumption is invoked, thereby making the absorption coefficient independent of the spectral frequency. Limiting forms of the solutions are studied as the freestream Mach number freestream Reynolds number and the temperature ratio across the shock wave, go to infinity, and as the Bouguer number and the density ratio across the shock wave go to zero. The method of matched asymptotic expansions is used in the analysis, and it is shown that there is a far-field precursor, composed of two regions, in which the fluid mechanics can be neglected for all practical purposes but included for completeness.

Passamaneck, R. S.↗
NASA NTRS1974

Aerodynamic influence coefficient method using singularity splines

A numerical lifting surface formulation, including computed results for planar wing cases is presented. This formulation, referred to as the vortex spline scheme, combines the adaptability to complex shapes offered by paneling schemes with the smoothness and accuracy of loading function methods. The formulation employes a continuous distribution of singularity strength over a set of panels on a paneled wing. The basic distributions are independent, and each satisfied all the continuity conditions required of the final solution. These distributions are overlapped both spanwise and chordwise. Boundary conditions are satisfied in a least square error sense over the surface using a finite summing technique to approximate the integral. The current formulation uses the elementary horseshoe vortex as the basic singularity and is therefore restricted to linearized potential flow. As part of the study, a non planar development was considered, but the numerical evaluation of the lifting surface concept was restricted to planar configurations. Also, a second order sideslip analysis based on an asymptotic expansion was investigated using the singularity spline formulation.

Mercer, J. E.↗
NASA NTRS1973

Aircraft Range Optimization Using Singular Perturbations

An approximate analytic solution is developed for the problem of maximizing the range of an aircraft for a fixed end state. The problem is formulated as a singular perturbation and solved by matched inner and outer asymptotic expansions and the minimum principle of Pontryagin. Cruise in the stratosphere, and on transition to and from cruise at constant Mach number are discussed. The state vector includes altitude, flight path angle, and mass. Specific fuel consumption becomes a linear function of power approximating that of the cruise values. Cruise represents the outer solution; altitude and flight path angle are constants, and only mass changes. Transitions between cruise and the specified initial and final conditions correspond to the inner solutions. The mass is constant and altitude and velocity vary. A solution is developed which is valid for cruise but which is not for the initial and final conditions. Transforming of the independent variable near the initial and final conditions result in solutions which are valid for the two inner solutions but not for cruise. The inner solutions can not be obtained without simplifying the state equations. The singular perturbation approach overcomes this difficulty. A quadratic approximation of the state equations is made. The resulting problem is solved analytically, and the two inner solutions are matched to the outer solution.

Oconnor, Joseph Taffe↗
NASA NTRS1973

Some conclusions regarding the aeroelastic stability of hingeless helicopter blades in hover and in forward flight

In this paper results and conclusions obtained from the study of the aeroelastic instability of hingeless helicopter blades are presented. First, the large amplitude coupled flap-lag equations of motion of a hingeless elastic helicopter blade are solved using an asymptotic expansion procedure in multiple time scales. Both hover and forward flight cases are considered. Stability boundaries and amplitudes of nonlinear response are obtained. From these, the importance of the nonlinear coupling and the effect of the periodic coefficients is determined. Next, using a system of linearized coupled flap-lag-pitch equations in hover, various divergence mechanisms for hingeless blades are shown. Finally, the flutter boundaries for coupled flap-lag-pitch are obtained. The effect of the torsional degree of freedom on the flap-lag type of instability is investigated. Similarly the effect of lag on the flap-pitch type of instability is considered. In addition, the effect of various blade parameters on the stability boundaries is shown.

Friedmann, P.↗
NASA NTRS1974

Conical flow near singular rays

The steady flow of an ideal gas past a conical body is investigated by the method of matched asymptotic expansions, with particular emphasis on the flow near the singular ray occurring in linearized theory. The first-order problem governing the flow in this region is formulated, leading to the equation of Kuo, and an approximate solution is obtained in the case of compressive flow behind the main front. This solution is compared with the results of previous investigations with a view to assessing the applicability of the Lighthill-Whitham theories.

Zahalak, G. I.↗
NASA NTRS1974

Inviscid drag at transonic speeds

A systematic asymptotic expansion procedure is introduced into the Euler equations to obtain the transonic, small-disturbance potential equation and corresponding relation for the drag. The relation consists of an integral around any contour enclosing the body and an integral along all shocks contained within the contour. Interpretations are given for the origin of the pressure drag due to shock waves, lift, and wind-tunnel boundaries. Numerical procedures are discussed for evaluating these quantities from finite-difference relaxation calculations. Computed results are included for various contours, and results are compared with some experimental data.

Murman, E. M.↗
NASA NTRS1974

Analysis of the interaction of a weak normal shock wave with a turbulent boundary layer

The method of matched asymptotic expansions is used to analyze the interaction of a normal shock wave with an unseparated turbulent boundary layer on a flat surface at transonic speeds. The theory leads to a three-layer description of the interaction in the double limit of Reynolds number approaching infinity and Mach number approaching unity. The interaction involves an outer, inviscid rotational layer, a constant shear-stress wall layer, and a blending region between them. The pressure distribution is obtained from a numerical solution of the outer-layer equations by a mixed-flow relaxation procedure. An analytic solution for the skin friction is determined from the inner-layer equations. The significance of the mathematical model is discussed with reference to existing experimental data.

Melnik, R. E.↗
NASA NTRS1974

Asymptotic solution for solar electric low thrust orbit raising with eclipse penalty

This paper describes an analytic solution for the solar electric low thrust orbit raising problem derived by application of the two-variable asymptotic expansion method. Included in the formulation are the effects of (1) high initial eccentricity, (2) large change of inclination, (3) variable mass, and (4) earth shadow penalty including the rotation of the shadow axis relative to the osculating orbit. An advantage of this mathematical strategy is that the effects of various steering policies can be evaluated readily from a physical point of view.

Flandro, G. A.↗
NASA NTRS1974

Prediction of unsteady airloads for oblique blade-gust interaction in compressible flow

The techniques of Galilean-Lorentz transformation and matched asymptotic expansions are used to simplify the procedure of calculating the lift and pressure distribution induced on an infinite-span thin wing interacting with an oblique sinusoidal gust in subsonic flow. This technique requires that the product of the flow Mach number and the reduced frequency be small. Under this condition, the inner region of the transformed space behaves as an incompressible flow, so that existing incompressible flow theories can be used as a basis to construct closed-form solutions for the airload induced on the wing. This approach is an extension of the GASP approximation developed by Amiet and Sears (1970). Results are obtained for both the magnitude and the phase of the unsteady lift due to interaction with gust. These results are compared with available numerical results. Some discrepancies are noted and discussed.

Chu, S.↗
NASA NTRS1974

Stability analysis of nonlinear systems by multiple time scaling

The asymptotic solution for the transient analysis of a general nonlinear system in the neighborhood of the stability boundary was obtained by using the multiple-time-scaling asymptotic-expansion method. The nonlinearities are assumed to be of algebraic nature. Terms of order epsilon to the 3rd power (where epsilon is the order of amplitude of the unknown) are included in the solution. The solution indicates that there is always a limit cycle which is stable (unstable) and exists above (below) the stability boundary if the nonlinear terms are stabilizing (destabilizing). Extension of the solution to include fifth order nonlinear terms is also presented. Comparisons with harmonic balance and with multiple-time-scaling solution of panel flutter equations are also included.

Morino, L.↗
NASA NTRS1975

Analysis of transonic flow about lifting wing-body configurations

An analytical solution was obtained for the perturbation velocity potential for transonic flow about lifting wing-body configurations with order-one span-length ratios and small reduced-span-length ratios and equivalent-thickness-length ratios. The analysis is performed with the method of matched asymptotic expansions. The angles of attack which are considered are small but are large enough to insure that the effects of lift in the region far from the configuration are either dominant or comparable with the effects of thickness. The modification to the equivalence rule which accounts for these lift effects is determined. An analysis of transonic flow about lifting wings with large aspect ratios is also presented.

Barnwell, R. W.↗
NASA NTRS1975

On upstream blocking in a viscous diffusive stratified flow

The effect of diffusion of specie upon the flow about a transverse flat plate moving horizontally in a viscous stratified medium is considered. Asymptotic expansions are used to define a parameter regime where a viscous-diffusive-buoyancy balance is dominant. The solution, expressed in terms of an inverse Fourier transform, is numerically integrated. The results show that, as in the non-diffusive problem, a region of closed streamlines exists ahead of the body. However, unlike the case where diffusion is neglected, the density field within this recirculating region is uniquely determined and found to be statically stable. It is also found that varying the relative amount of diffusion affects not only the density distribution, but the velocity profile as well, indicating a strong coupling between the vorticity and specie equation.

Koop, C. G.↗
NASA NTRS1975

Analytic solution for low thrust out of ecliptic mission design

Description of an analytic trajectory solution for low thrust out of ecliptic mission design based on the two-variable asymptotic expansion technique. A practical steering program utilizing normal thrusting symmetrical about the nodes is assumed. A two-term expansion yields an essentially exact solution of the problem which is in a form suitable for evaluation on small desk-top computers. It is shown that for a circular base orbit at 1 AU, a 15-kw solar electric system with a specific impulse of 3000 sec is capable of delivering an 1100-kg net payload to a heliographic latitude of about 63 deg.

Flandro, G. A.↗
NASA NTRS1975

Stability analysis of nonlinear autonomous systems - General theory and application to flutter

The analysis makes use of a singular perturbation method, the multiple time scaling. Concepts of stable and unstable limit cycles are introduced. The solution is obtained in the form of an asymptotic expansion. Numerical results are presented for the nonlinear flutter of panels and airfoils in supersonic flow. The approach used is an extension of a method for analyzing nonlinear panel flutter reported by Morino (1969).

Smith, L. L.↗