Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “wave equation”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 595 records · Page 33
All sourcesNASA NTRSDOE OSTI
NASA NTRS1975

Radiation of cylindrical duct acoustic modes with flow mismatch

Motivated by the problem of Mach number and temperature discontinuity mismatch between the tailpipe and the surrounding medium, and the effect of this flow mismatch on aircraft engine noise, the radiation of spinning turbo-acoustic modes from a round duct is investigated. The model, based on a plug flow exhaust jet, solves the convected wave equations using Fourier transforms, inversion being accomplished via a coupled Wiener-Hopf and Carrier-Koiter technique. It is concluded that the Mach number discontinuity results in a conflict between convection and refraction effects, while the temperature mismatch results in a sharp beaming of sound off to the side.

Savkar, S. D.↗
NASA NTRS1975

Strong scintillations in astrophysics. II - A theory of temporal broadening of pulses

A theory of temporal broadening of pulses propagating in a turbulent medium is developed on the basis of the Markov approximation. The theory may be applied to quite general turbulence spectra and to thin or thick turbulent regions. Since the basis of the theory is the wave equation, no reliance is placed on geometrical optics and no assumptions are made about the scattered angular spectrum. The observed smearing is found to be the combination of three effects: the dispersion effect, the pure refractive effect, and the diffraction effect. The last of these dominates for typical pulsar parameters. Pulse shapes are calculated for both Gaussian and Kolmogorov turbulence spectra and it is shown how these scale with the various turbulence parameters.

Lee, L. C.↗
NASA NTRS1976

Acoustic pulse propagation in a duct with flow

The effect of a mean flow on acoustic pulse propagation in a wind tunnel of rectangular cross-section is considered both experimentally and theoretically. Pulse shapes in the wind tunnel are recorded by microphones located upstream and downstream of a source. A theoretical model of pulse propagation from a point source located in an infinitely-long rectangular duct carrying a uniform flow is considered. The convective wave equation with boundary conditions appropriate to hard walls is solved by using the Laplace transform. For the input pulse shape used in the experiments, the solution is obtained by numerically evaluating the inverse Laplace transform. The results of experiment and theory are seen to be in good agreement.

Cole, J. E., III↗
NASA NTRS1976

Effects of mean flow on duct mode optimum suppression rates

The nature of the solution to the convected acoustic wave equation and associated boundary conditions for rectangular ducts containing uniform mean flow is examined in terms of the complex mapping between the wall admittance and characteristic mode eigenvalues. It is shown that the Cremer optimum suppression criteria must be modified to account for the effects of flow below certain critical values of the nondimensional frequency parameter of duct height divided by sound wavelength. The implications of these results on the design of low frequency suppressors are considered.

Kraft, R. E.↗
NASA NTRS1977

Unsteady supersonic aerodynamic theory for interfering surfaces by the method of potential gradient

A generalized solution of the hyperbolic wave equation was further developed to relate the velocity components at a field point to the potential gradient distribution in the dependence domain. Singular integrals were evaluated in closed form, with numerical integration methods for more complex but analytic functions. Idealization of the lifting surfaces by trapezoidal elements with two sides parallel to the streamlines is computationally efficient. Streamwise integrals were performed analytically, and spanwise integrals were neccessary only on element leading and trailing sides. All integrands vanish on the Mach cone. Pressure distribution on a double delta wing and generalized aerodynamic coefficients for three AGARD planforms were calculated and compared with available results.

Jones, W. P.↗
NASA NTRS1976

Fluid dynamics applications of the Illiac IV computer

The Illiac IV is a parallel-structure computer with computing power an order of magnitude greater than that of conventional computers. It can be used for experimental tasks in fluid dynamics which can be simulated more economically, for simulating flows that cannot be studied by experiment, and for combining computer and experimental simulations. The architecture of Illiac IV is described, and the use of its parallel operation is demonstrated on the example of its solution of the one-dimensional wave equation. For fluid dynamics problems, a special FORTRAN-like vector programming language was devised, called CFD language. Two applications are described in detail: (1) the determination of the flowfield around the space shuttle, and (2) the computation of transonic turbulent separated flow past a thick biconvex airfoil.

Maccormack, R. W.↗
NASA NTRS1977

Unsteady supersonic aerodynamic theory by the method of potential gradient

A generalized solution of the hyperbolic wave equation has been derived. The method used has been further developed to relate the velocity components at a field point to the potential gradient distribution in the dependence domain. Singular integrals have been evaluated in closed form, while numerical integration methods are suggested for treating more complex but analytic functions. Idealization of the lifting surfaces by trapezoidal elements with two sides parallel to the streamlines is computationally efficient because line integrations along the other two sides need only be considered. Furthermore, all the integrands vanish on the Mach cone and the need for determining the hyperbolic curves of intersection of the cone with the lifting surface is avoided. Generalized aerodynamic coefficients for three AGARD planforms have been calculated and compared with the available results.

Jones, W. P.↗
NASA NTRS1977

A contribution to computer analysis of coupled-cavity traveling wave tubes

A flexible accurate large-signal computer program has been developed for the design of coupled-cavity traveling wave tubes. The program is written for a TSS-360 time sharing system. The beam is described by a disk model and the slow wave structure by a sequence of cavities or cells. The computational approach is arranged so that each cavity may have different geometrical or electrical parameters than its neighbors. This allows the program user to simulate a tube of almost arbitrary complexity. Input and output couplers, severs, complicated velocity tapers, and other features peculiar to one or a few cavities may be modeled by a correct choice of input data. The beam-wave interaction is handled by a new approach in which the RF fields are expanded in solutions to the TM wave equation retaining all significant space harmonics. The program was used to perform a design study of the TWT developed for the CTS satellite. Good agreement was obtained between the predictions of the program and the measured performance of the flight tube. The internal check on power balance was satisfied within plus or minus 0.2 per cent of input beam power.

Connolly, D. J.↗
NASA NTRS1977

Stability of higher-order Bragg interactions in active periodic media

The stability of waves in unbounded, longitudinally periodic media is studied for index and gain coupling. Time-independent periodic media are found to support both stable and absolutely unstable waves. The wave characteristics depend upon average gain or loss, coupling type, and Bragg order. The extended coupled waves equations provide explicit values of threshold, frequency, and temporal growth rate for instabilities at all Bragg resonances through the dispersion relation. Applications to multiharmonic periodicities and complex couplings are briefly discussed with particular note taken of possible reductions of the stability thresholds and removal of threshold degeneracies. Comparisons are made to the longitudinally bounded case of distributed feedback lasers.

Jaggard, D. L.↗
NASA NTRS1978

Numerical spatial marching techniques for estimating duct attenuation and source pressure profiles

A numerical method was developed that could predict the pressure distribution of a ducted source from far field pressure inputs. Using an initial value formulation, the two-dimensional homogeneous Helmholtz wave equation (no steady flow) was solved using explicit marching techniques. The Von Neumann method was used to develop relationships which describe how sound frequency and grid spacing effect numerical stability. At the present time, stability considerations limit the approach to high frequency sound. Sample calculations for both hard and soft wall ducts compare favorably to known boundary value solutions. In addition, assuming that reflections in the duct are small, this initial value approach was successfully used to determine the attenuation of a straight soft wall duct. Compared to conventional finite difference or finite element boundary value approaches, the numerical marching technique is orders of magnitude shorter in computation time and required computer storage and can be easily employed in problems involving high frequency sound.

Baumeister, K. J.↗
NASA NTRS1978

Numerical spatial marching techniques for estimating duct attenuation and source pressure profiles

A numerical method is developed that could predict the pressure distribution of a ducted source from far-field pressure inputs. Using an initial value formulation, the two-dimensional homogeneous Helmholtz wave equation (no steady flow) is solved using explicit marching techniques. The Von Neumann method is used to develop relationships which describe how sound frequency and grid spacing effect numerical stability. At the present time, stability considerations limit the approach to high frequency sound. Sample calculations for both hard and soft wall ducts compare favorably to known boundary value solutions. In addition, assuming that reflections in the duct are small, this initial value approach is successfully used to determine the attenuation of a straight soft wall duct. Compared to conventional finite difference or finite element boundary value approaches, the numerical marching technique is orders of magnitude shorter in computation time and required computer storage and can be easily employed in problems involving high frequency sound.

Baumeister, K. J.↗
NASA NTRS1978

Unsteady vortical and entropic distortions of potential flows round arbitrary obstacles

The analysis concerns the alterations produced when small amplitude disturbances, including entropy and vorticity disturbances, are imposed on steady potential flows. For the most general nonacoustic incident distortion field that can be imposed on the uniform upstream flow, it is shown that the perturbation velocity at any point of the resulting unsteady compressible and vortical flow consists of a part that is a known function of the imposed upstream distortion field and the mean flow variables and a potential part that can be found by solving a linear inhomogeneous wave equation with a dipole-type source term whose strength is a known function of the imposed upstream distortion field. The theory is applied to the unsteady flow past a corner, and a closed-form analytical solution is found.

Goldstein, M. E.↗
NASA NTRS1979

Sound radiation from hyperboloidal inlet ducts

This paper presents rigorous solutions for the problem of sound radiation from various inlet ducts including hyperboloidal (or hyperbolic) inlet ducts and circular ducts with wide flange. The numerical results include the complex conversion (or reflection) coefficients and the radiation directivity for the various incident wave modes - spinning modes as well as axisymmetric modes. The analysis utilizes hyperboloidal wave functions which are defined here as a class of eigensolutions of the wave equation for oblate spheroidal coordinates, and is valid for the whole frequency range including frequencies above and below the cutoff frequencies of duct modes involved.

Cho, Y. C.↗
NASA NTRS1979

Acoustic power balance in lined ducts

It is shown that the two common definitions of acoustic energy density and intensity in uniform unlined ducts carrying uniform flow are compatible to the extent that both energy densities can be used in an appropriate variational principle to derive the convected wave equation. When the duct walls are lined both energy densities must be modified to account for the wall energy density. This results in a new energy conservation equation which utilizes a modified definition of axial power and accounts for wall dissipation. Computations in specific cases demonstrate the validity of the modified acoustic energy relation.

Eversman, W.↗
NASA NTRS1979

Noise transmission - Turboprop problem

An analytical model is presented to predict low frequency (1000 Hz and below) noise transmission into aircraft. Noise transmission is determined by solving the acoustic wave equation for the interior noise field, the equivalent 'smeared' monocoque shell equation for overall fuselage vibrations and skin-stringer panel equation for local vibrations of panels and stringers. The solution to this system of equations is obtained by using a Galerkin-like procedure. The modes and frequencies for skin-stringer panels are determined by a transfer matrix method. Results include noise transmission through discretely stiffened panels and cabin noise due to turbulent boundary layer and propeller passage harmonics for a turboprop type aircraft.

Vaicaitis, R.↗
NASA NTRS1979

Scattering and distortion of the unsteady motion on transversely sheared mean flows

It is shown that the pressure and velocity fluctuations of the unsteady motion on a transversely sheared mean flow can be expressed entirely in terms of the derivatives of two potential functions. One of these is a convected quantity that can be specified as a boundary condition and is related to a transverse component of the upstream velocity field. The other can be determined by solving an inhomogeneous wave equation whose source term is also a convected quantity that can be specified as a boundary condition in any given problem. The general theory is used to study the interaction of an unsteady flow with a semi-infinite plate embedded in a shear layer. The acoustic field produced by this interaction is calculated in the limits of low and high frequency. The results are compared with experimental one-third octave sound pressure level radiation patterns. The agreement is found to be excellent, especially in the low frequency range, where the mean-flow and convective effects are shown to have a strong influence on the directivity of the sound.

Goldstein, M. E.↗
NASA NTRS1979

Turbulence generated by the interaction of entropy fluctuations with non-uniform mean flows

The turbulence generated by random entropy fluctuations in an accelerating stream is analyzed. The results are obtained by using rapid distortion theory together with a high frequency solution of a previously developed wave equation that governs the small-amplitude unsteady vortical and entropic motion on steady potential flows (Goldstein, 1978). Simple results are obtained for the case of symmetric contraction, expansion or combination of the two. It is shown that the energy of the entropy-generated turbulence increases more rapidly with the contraction ratio of a subsonic flow than that of any imposed upstream turbulence. This result indicates that the entropy-generated turbulence may be more significant than the hydrodynamically generated turbulence in the turbine stages of aircraft engines.

Goldstein, M. E.↗
NASA NTRS1980

Higher order mode propagation in nonuniform circular ducts

Higher order mode propagation in a nonuniform circular duct without mean flow was investigated. An approximate wave equation is derived on the assumptions that the duct cross section varies slowly and that mode conversion is negligible. Exact closed form solutions are obtained for a particular class of converging-diverging circular duct which referred to as 'circular cosh duct.' Numerical results are presented in terms of the transmission loss for the various duct shapes and frequencies. The results are applicable to multimodal propagation, single mode propagation, and sound radiation from certain types of contoured inlet ducts, or of sound propagation in a converging-diverging duct of somewhat different shape from a cosh duct.

Cho, Y. C.↗