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NASA NTRS1974

Spherical means of solutions of partial differential equations in a conical region

The spherical means of the solutions of a linear partial differential equation Lu = f in a conical region are studied. The conical region is bounded by a surface generated by curvilinear ti surfaces. The spherical mean is the average of u over a constant ti surface. The conditions on the linear differential operator, L, and on the orthogonal coordinates (ti, eta, zeta) are established so that the spherical mean of the solution subjected to the appropriate boundary and initial conditions can be determined directly as a problem with only space variable. Conditions are then established so that the spherical mean of the solution in one concial region will be proportional to that of a known solution in another conical region. Applications to various problems of mathematical physics and their physical interpretations are presented.

Ting, L.↗

NASA NTRS1975

Spherical means of solutions of partial differential equations in a conical region

The spherical means of the solutions of a linear partial differential equation Lu = f in a conical region are studied. The conical region is bounded by a surface generated by curvilinear xi lines and by two truncating xi surfaces. The spherical mean is the average of u over a constant xi surface. Conditions on the linear differential operator, L, and on the orthogonal coordinates xi, eta, and zeta are established so that the problem for the determination of the spherical mean of the solution subjected to the appropriate boundary and initial conditions can be reduced to a problem with only one space variable. Conditions are then established so that the spherical mean of the solution in one conical region will be proportional to that of a known solution in another conical region. Applications to various problems of mathematical physics and their physical interpretations are presented.

Ting, L.↗

NASA NTRS1977

Transmission of sound through nonuniform circular ducts with compressible mean flows

An acoustic theory is developed to determine the sound transmission and attenuation through an infinite, hard-walled or lined, circular duct carrying compressible, sheared, mean flows and having a variable cross section. The theory is applicable to large as well as small axial variations, as long as the mean flow does not separate. Although the theory is described for circular ducts, it is applicable to other duct configurations - annular, two dimensional, and rectangular. The theory is described for the linear problem, but the technique is general and has the advantage of being applicable to the nonlinear case as well as the linear case. The technique is based on solving for the envelopes of the quasi-parallel acoustic modes that exist in the duct instead of solving for the actual wave. A computer program was developed. The mean flow model consists of a one dimensional flow in the core and a quarter-sine profile in the boundary layer. Results are presented for the reflection and transmission coefficients in ducts with varying slopes and carrying different mean flows.

Nayfeh, A. H.↗

NASA NTRS1977

A finite element algorithm for sound propagation in axisymmetric ducts containing compressible mean flow

An accurate mathematical model for sound propagation in axisymmetric aircraft engine ducts with compressible mean flow is reported. The model is based on the usual perturbation of the basic fluid mechanics equations for small motions. Mean flow parameters are derived in the absence of fluctuating quantities and are then substituted into the equations for the acoustic quantities which were linearized by eliminating higher order terms. Mean swirl is assumed to be zero from the restriction of axisymmetry. A linear rectangular serendipity element is formulated from these equations using a Galerkin procedure and assembled in a special purpose computer program in which the matrix map for a rectangular mesh was specifically coded. Representations of the fluctuating quantities, mean quantities and coordinate transformations are isoparametric. The global matrix is solved by foreward and back substitution following an L-U decomposition with pivoting restricted internally to the blocks. Results from the model were compared with results from several alternative analyses and yielded satisfactory agreement.

Abrahamson, A. L.↗

NASA NTRS1977

Monthly mean forecast experiments with the GISS model - Correction

One of the three monthly mean forecasts computed previously by Spar et al., (1976) with the GISS (Goddard Institute for Space Studies) general circulation model was started from incorrect initial conditions. For the January 1975 prediction, a climatological mean January data set was accidentally substituted for the required January 1, 1975 initial data. In the present paper, the forecast for January 1975 is computed with the proper initial conditions, and the corrected results are presented. It is shown that the use of correct initial conditions improves the gross forecast mean energetics. In general, the use of correct initial conditions has resulted in a modest improvement in the monthly mean forecast for January 1975, but has not altered any of the previous conclusions by Spar et al.

Spar, J.↗

NASA NTRS1977

The mean magnetic field of the sun - Method of observation and relation to the interplanetary magnetic field

The mean solar magnetic field as measured in integrated light has been observed since 1968. Since 1970 it has been observed both at Hale Observatories and at the Crimean Astrophysical Observatory. The observing procedures at both observatories and their implications for mean field measurements are discussed. A comparison of the two sets of daily observations shows that similar results are obtained at both observatories. A comparison of the mean field with the interplanetary magnetic polarity shows that the IMF sector structure has the same pattern as the mean field polarity.

Scherrer, P. H.↗

NASA NTRS1978

Experiments in monthly mean simulation of the atmosphere with a coarse-mesh general circulation model

The Hansen atmospheric model was used to compute five monthly forecasts (October 1976 through February 1977). The comparison is based on an energetics analysis, meridional and vertical profiles, error statistics, and prognostic and observed mean maps. The monthly mean model simulations suffer from several defects. There is, in general, no skill in the simulation of the monthly mean sea-level pressure field, and only marginal skill is indicated for the 850 mb temperatures and 500 mb heights. The coarse-mesh model appears to generate a less satisfactory monthly mean simulation than the finer mesh GISS model.

Lutz, R. J.↗

NASA NTRS1978

Simulations of the monthly mean atmosphere for February 1976 with the GISS model

Monthly mean simulations of the global atmosphere were computed for February 1976 with the GISS model from observed initial conditions. In a replication experiment, two of these computations generated slightly different monthly mean states, apparently due to the schedule of interruptions on the computer. The root-mean-square errors of replication over the Northern Hemisphere were found to be about 2 mb, 20 m, and 1 K for sea-level pressure, 500 mb height, and 850 mb temperature, respectively. The monthly mean 500 mb forecast results for February 1976 over the Northern Hemisphere were consistent with those from earlier GISS model experiments.

Spar, J.↗

NASA NTRS1978

Monthly mean simulation experiments with a course-mesh global atmospheric model

Substitution of observed monthly mean sea-surface temperatures (SSTs) as lower boundary conditions, in place of climatological SSTs, failed to improve the model simulations. While the impact of SST anomalies on the model output is greater at sea level than at upper levels the impact on the monthly mean simulations is not beneficial at any level. Shifts of one and two days in initialization time produced small, but non-trivial, changes in the model-generated monthly mean synoptic fields. No improvements in the mean simulations resulted from the use of either time-averaged initial data or re-initialization with time-averaged early model output. The noise level of the model, as determined from a multiple initial state perturbation experiment, was found to be generally low, but with a noisier response to initial state errors in high latitudes than the tropics.

Spar, J.↗

NASA NTRS1978

Fuel delivery system including heat exchanger means

A fuel delivery system is presented wherein first and second heat exchanger means are each adapted to provide the transfer of heat between the fuel and a second fluid such as lubricating oil associated with the gas turbine engine. Valve means are included which are operative in a first mode to provide for flow of the second fluid through both first and second heat exchange means and further operative in a second mode for bypassing the second fluid around the second heat exchanger means.

Coffinberry, G. A.↗

NASA NTRS1979

A wave-envelope of sound propagation in nonuniform circular ducts with compressible mean flows

An acoustic theory is developed to determine the sound transmission and attenuation through an infinite, hard-walled or lined circular duct carrying compressible, sheared, mean flows and having a variable cross section. The theory is applicable to large as well as small axial variations, as long as the mean flow does not separate. The technique is based on solving for the envelopes of the quasi-parallel acoustic modes that exist in the duct instead of solving for the actual wave, thereby reducing the computation time and the round-off error encountered in purely numerical techniques. The solution recovers the solution based on the method of multiple scales for slowly varying duct geometry. A computer program was developed based on the wave-envelope analysis for general mean flows. Results are presented for the reflection and transmission coefficients as well as the acoustic pressure distributions for a number of conditions: both straight and variable area ducts with and without liners and mean flows from very low to high subsonic speeds are considered.

Nayfeh, A. H.↗

NASA NTRS1980

High-frequency sound propagation in a spatially varying mean flow

An equation for acoustic ray paths in a spatially varying mean flow was examined to determine some of the characteristics of the flow gradient effects on sound propagation. In a potential flow, the acoustic rays are deflected in the direction of increasing mean flow, and the gradient of the mean flow speed is the dominant factor causing the ray deflection. In contrast, in a sheared mean flow, the vorticity is the dominant factor in deflection of the acoustic rays.

Cho, Y. C.↗

NASA NTRS1981

Mean-flow and turbulence measurements in the vicinity of the trailing edge of an NACA (63 sub 1)-012 airfoil

The Langley aircraft noise reduction laboratory was used to measure mean flow and turbulence in the vicinity of a cusped-trailing edge of two dimensional NACA 631-012 airfoil at zero angle of attack. Naturally transitioned flow and artificially tripped flow were investigated. Flow regions studied include the boundary layer and the near wake. Measurements were made at a free stream Reynolds number based on airfoil chord of 1.25 x 10 to the 6th power and a free stream Mach number of 0.1. Distributions of streamwise mean velocity, integral properties of the mean flow, turbulent intensities, and Reynolds shear stress are reported. For the naturally transitioned flow, the general trends observed are similar to those reported for a fully developed turbulent boundary layer over a flat plate under zero pressure gradient, with the exception of the notable streamwise variations in the turbulence properties for the airfoil flow. The main effect of flow tripping is to eliminate these streamwise variations. Observed changes in the mean flow and turbulence fields caused by tripping are expected on the basis of the Reynolds number based on the boundary layer thickness.

Yu, J. C.↗

NASA NTRS1980

The role of forced planetary waves in the annual cycle of the zonal mean circulation of the middle atmosphere

A severely truncated semispectral numerical model is used to simulate the annual cycle of the zonally averaged circulation in the middle atmosphere (16-96 km). The model includes only a single zonal harmonic wave component which interacts with the mean flow; the circulation is driven by diabatic heating and by a specified perturbation in the topography of the lower boundary, which is taken to be the 100 mb surface. A comparison of the annual cycle simulated by this model with the results of an analogous two-dimensional model indicates that planetary waves have relatively little influence on the zonal mean temperature profiles and on the solstice mean zonal winds at high latitudes. The primary effects of the forced waves are in decelerating the mean winds at low latitudes in the winter hemisphere to produce a region of weak westerlies, and in generating final warmings at the spring equinoxes.

Holton, J. R.↗

NASA NTRS1981

Wave transience in a compressible atmosphere. I - Transient internal wave, mean-flow interaction. II - Transient equatorial waves in the quasi-biennial oscillation

Analytical and numerical solutions are obtained in an approximate quasi-linear model, to describe the way in which vertically propagating waves give rise to mean flow accelerations in an atmosphere due to the effects of wave transience. These effects in turn result from compressibility and vertical group velocity feedback, and culminate in the spontaneous formation and descent of regions of strong mean wind shear. The numerical solutions display mean flow accelerations due to Kelvin waves in the equatorial stratosphere, with wave absorption altering the transience mechanism in such significant respects as causing the upper atmospheric mean flow acceleration to be very sensitive to the precise magnitude and distribution of the damping mechanisms. The numerical simulations of transient equatorial waves in the quasi-biennial oscillation are also considered.

Dunkerton, T. J.↗

NASA NTRS1981

A further study of the annual cycle of the zonal mean circulation in the middle atmosphere

The influence on the stratospheric mean circulation of planetary wavenumber 2 disturbances excited by steady forcing at the 100 mb level is investigated using a global semi-spectral primitive equation model. There exists a critical forcing amplitude below which the waves have little effect on the mean flow, while above which the waves produce subseasonal time scale vacillations in the winter hemisphere, including both major and minor warmings. Wave transience induced by the evolving mean flow distribution is responsible for generating the vacillations, while thermal damping serves to reduce the wave-driven changes in the mean flow.

Holton, J. R.↗

NASA NTRS1981

High-frequency sound propagation in a spatially varying mean flow

An equation for acoustic ray paths in a spatially varying mean flow has been examined to determine some of the characteristics of the flow gradient effects on sound propagation. In a potential flow the acoustic rays are deflected in the direction of increasing mean flow, and the gradient of the mean flow speed is the dominant factor causing the ray deflection. In contrast, in a sheared mean flow, the vorticity is the dominant factor in deflection of the acoustic rays.

Cho, Y. C.↗

NASA NTRS1981

A simple model of the Lagrangian-mean flow produced by dissipating planetary waves

A simple equation for the Lagrangian-mean flow induced by damped planetary waves is derived. The flow computed for stationary planetary waves of a beta-plane is found to be generally poleward and downward during winter and appears to be about twice as strong as the diabatic circulation in the lower stratosphere. An important factor in determining the high-latitude Lagrangian-mean flow field is the subtropical jet stream which blocks planetary wave propagation toward the equatorial regions. Computations using two types of Lagrangian-mean boundary conditions at the surface show that incorrect orographic forcing distorts the Lagrangian-mean flow up to three scale heights or more above ground.

Schoeberl, M. R.↗