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NASA NTRS1985

Prediction of sound fields in acoustical cavities using the boundary element method

A method was developed to predict sound fields in acoustical cavities. The method is based on the indirect boundary element method. An isoparametric quadratic boundary element is incorporated. Pressure, velocity and/or impedance boundary conditions may be applied to a cavity by using this method. The capability to include acoustic point sources within the cavity is implemented. The method is applied to the prediction of sound fields in spherical and rectangular cavities. All three boundary condition types are verified. Cases with a point source within the cavity domain are also studied. Numerically determined cavity pressure distributions and responses are presented. The numerical results correlate well with available analytical results.

Kipp, C. R.↗

NASA NTRS1987

Prediction of acoustical behavior in cavities using an indirect boundary element method

An indirect boundary element method is developed to predict sound fields in acoustical cavities. An isoparametric quadratic boundary element is utilized. The formulations of pressure, velocity and/or impedance boundary conditions are developed and incorporated into the method. The capability to include acoustic point sources within the cavity is also implemented. The method is applied to the prediction of sound fields in spherical and rectangular cavities. All three boundary condition types are verified. Cases having a point source within the cavity domain are also studied. Numerically determined cavity pressure distributions and responses are presented. The numerical results correlate well with available analytical results.

Kipp, C. R.↗

NASA NTRS1988

Analysis of the EM scattering from arbitrary open-ended waveguide cavities using axial Gaussian Beam tracking

The electromagnetic (EM) scattering from a planar termination located inside relatively arbitrarily shaped open-ended waveguide cavities with smoothly curved interior walls is analyzed using a Gaussian Beam (GB) expansion of the incident plane wave fields in the open end. The cavities under consideration may contain perfectly-conducting interior walls with or without a thin layer of material coating, or the walls may be characterized by an impedance boundary condition. In the present approach, the GB's are tracked only to the termination of the waveguide cavity via beam reflections from interior waveguide cavity walls. The Gaussian beams are tracked approximately only along their beam axes; this approximation which remains valid for relatively well focussed beams assumes that an incident GB gives rise to a reflected GB with parameters related to the incident beam and the radius of curvature of the wall. It is found that this approximation breaks down for GB's which come close to grazing a convex surface and when the width of the incident beam is comparable to the radius of curvature of the surface. The expansion of the fields at the open end depend on the incidence angle only through the expansion coefficients, so the GB's need to be tracked through the waveguide cavity only once for a wide range of incidence angles. At the termination, the sum of all the GB's are integrated using a result developed from a generalized reciprocity principle, to give the fields scattered from the interior of the cavity. The rim edge at the open end of the cavity is assumed to be sharp and the external scattering from the rim is added separately using Geometrical Theory of Diffraction. The results based on the present approach are compared with solutions based on the hybrid asymptotic modal method. The agreement is found to be very good for cavities made up of planar surfaces, and also for cavities with curved surfaces which are not too long with respect to their width.

Burkholder, R. J.↗

NASA NTRS1990

Electromagnetic scattering from two-dimensional thick material junctions

The problem of the plane wave diffraction is examined by an arbitrary symmetric two dimensional junction, where Generalized Impedance Boundary Conditions (GIBCs) and Generalized Sheet Transition Conditions (GSTCs) are employed to simulate the slabs. GIBCs and GSTCs are constructed for multilayer planar slabs of arbitrary thickness and the resulting GIBC/GSTC reflection coefficients are compared with exact counterparts to evaluate the GIBCs/GSTCs. The plane wave diffraction by a multilayer material slab recessed in a perfectly conducting ground plane is formulated and solved via the Generalized Scattering Matrix Formulation (GDMF) in conjunction with the dual integral equation approach. Various scattering patterns are computed and validated with exact results where possible. The diffraction by a material discontinuity in a thick dielectric/ferrite slab is considered by modelling the constituent slabs with GSTCs. A non-unique solution in terms of unknown constants is obtained, and these constants are evaluated for the recessed slab geometry by comparison with the solution obtained therein. Several other simplified cases are also presented and discussed. An eigenfunction expansion method is introduced to determine the unknown solution constants in the general case. This procedure is applied to the non-unique solution in terms of unknown constants; and scattering patterns are presented for various slab junctions and compared with alternative results where possible.

Ricoy, M. A.↗

NASA NTRS1990

Scattering by an impedance sphere coated with a chiral layer

The scattering of a plane, linearly polarized electromagnetic wave by a sphere on whose surface an impedance boundary condition holds, and that is covered with a concentric layer of chiral material, is considered. Exact, explicit expressions are derived for the scattered field coefficients. The co-polarized and cross-polarized components of the far backscattered field are determined and discussed. The value of this canonical problem as a benchmark for computer codes is pointed out.

Uslenghi, Piergiorgio L. E.↗

NASA NTRS1991

High-frequency techniques for RCS prediction of plate geometries

Several different high-frequency methods for modeling the radar cross sections (RCSs) of plate geometries are examined. The Method of Equivalent Currents and a numerically derived corner diffraction coefficient are used to model the RCS of a rectangular, perfectly conducting plate in nonprincipal planes. The Uniform Theory of Diffraction is used to model the RCS of a rectangular, perfectly conducting plate in principal planes. For the soft polarization case, first-order and slope-diffraction terms are included. For the hard polarization case, up to four orders of diffraction are included. Finally, the Uniform Theory of Diffraction for impedance wedges and the Impedance Boundary Condition are used to model the RCS of a coated, rectangular plate in principal planes. In most of the cases considered, comparisons are made between theoretical and experimental results.

Balanis, Constantine A.↗

NASA NTRS1992

Broadband vertical interconnects using slot-coupled shielded microstrip lines

A full-wave space-domain integral equation analysis of aperture coupled shielded microstrip lines is presented based on the Equivalence Principle. The formulation includes the capability to model multilayered substrates through the derivation of the associated dyadic Green's functions which represent the layers through impedance boundary conditions. The method of moments is used to solve for the line currents and slot voltage with even and odd mode excitations which are then interpreted through transmission line analysis to determine the two-port scattering parameters. A parametric study together with experimental data is presented which demonstrates the behavior of the coupler and the accuracy of the technique.

Vandenberg, Norman L.↗

NASA NTRS1996

Finite Difference Time Marching in the Frequency Domain: A Parabolic Formulation for Aircraft Acoustic Nacelle Design

An explicit finite difference iteration scheme is developed to study harmonic sound propagation in aircraft engine nacelles. To reduce storage requirements for large 3D problems, the time dependent potential form of the acoustic wave equation is used. To insure that the finite difference scheme is both explicit and stable, time is introduced into the Fourier transformed (steady-state) acoustic potential field as a parameter. Under a suitable transformation, the time dependent governing equation in frequency space is simplified to yield a parabolic partial differential equation, which is then marched through time to attain the steady-state solution. The input to the system is the amplitude of an incident harmonic sound source entering a quiescent duct at the input boundary, with standard impedance boundary conditions on the duct walls and duct exit. The introduction of the time parameter eliminates the large matrix storage requirements normally associated with frequency domain solutions, and time marching attains the steady-state quickly enough to make the method favorable when compared to frequency domain methods. For validation, this transient-frequency domain method is applied to sound propagation in a 2D hard wall duct with plug flow.

Baumeister, Kenneth J.↗

NASA NTRS1996

Time-Dependent Parabolic Finite Difference Formulation for Harmonic Sound Propagation in a Two-Dimensional Duct with Flow

An explicit finite difference real time iteration scheme is developed to study harmonic sound propagation in aircraft engine nacelles. To reduce storage requirements for future large 3D problems, the time dependent potential form of the acoustic wave equation is used. To insure that the finite difference scheme is both explicit and stable for a harmonic monochromatic sound field, a parabolic (in time) approximation is introduced to reduce the order of the governing equation. The analysis begins with a harmonic sound source radiating into a quiescent duct. This fully explicit iteration method then calculates stepwise in time to obtain the 'steady state' harmonic solutions of the acoustic field. For stability, applications of conventional impedance boundary conditions requires coupling to explicit hyperbolic difference equations at the boundary. The introduction of the time parameter eliminates the large matrix storage requirements normally associated with frequency domain solutions, and time marching attains the steady-state quickly enough to make the method favorable when compared to frequency domain methods. For validation, this transient-frequency domain method is applied to sound propagation in a 2D hard wall duct with plug flow.

Kreider, Kevin L.↗

NASA NTRS1996

Finite Difference Time Marching in the Frequency Domain: A Parabolic Formulation for the Convective Wave Equation

An explicit finite difference iteration scheme is developed to study harmonic sound propagation in ducts. To reduce storage requirements for large 3D problems, the time dependent potential form of the acoustic wave equation is used. To insure that the finite difference scheme is both explicit and stable, time is introduced into the Fourier transformed (steady-state) acoustic potential field as a parameter. Under a suitable transformation, the time dependent governing equation in frequency space is simplified to yield a parabolic partial differential equation, which is then marched through time to attain the steady-state solution. The input to the system is the amplitude of an incident harmonic sound source entering a quiescent duct at the input boundary, with standard impedance boundary conditions on the duct walls and duct exit. The introduction of the time parameter eliminates the large matrix storage requirements normally associated with frequency domain solutions, and time marching attains the steady-state quickly enough to make the method favorable when compared to frequency domain methods. For validation, this transient-frequency domain method is applied to sound propagation in a 2D hard wall duct with plug flow.

Baumeister, K. J.↗

NASA NTRS2005

Mode-expansion method for predicting radar signature above rough ocean surfaces at low-grazing angle

The Mode-Expansion Method (MEM) is introduced to calculate the electromagnetic (EM) waves scattered by 2-D rough water surfaces at low-grazing angles. The Electric Field Integral Equation (EFIE) is used in defining the problem and is simplified by using the Impedance Boundary Condition (IBC). The surface currents are expressed as the sum of modes expanded as the Fourier series with incident wave as the dominant mode. It is shown that, by the MEM and for the geometry with transmitting and receiving waves at low-grazing angles, very few modes are needed in solving the forward scattering field with reasonable accuracy.

electromagnetic (EM)↗

NASA NTRS1990

Experimental validation study of an analytical model of discrete frequency sound propagation in closed-test-section wind tunnels

The principal objective is to assess the adequacy of linear acoustic theory with an impedence wall boundary condition to model the detailed sound field of an acoustic source in a duct. Measurements and calculations are compared of a simple acoustic source in a rectangular concrete duct lined with foam on the walls and anechoic end terminations. Measurement of acoustic pressure for twelve wave numbers provides variation in frequency and absorption characteristics of the duct walls. Close to the source, where the interference of wall reflections is minimal, correlation is very good. Away from the source, correlation degrades, especially for the lower frequencies. Sensitivity studies show little effect on the predicted results for changes in impedance boundary condition values, source location, measurement location, temperature, and source model for variations spanning the expected measurement error.

Mosher, Marianne↗

NASA NTRS1990

Evaluation of discrete frequency sound in closed-test-section wind tunnels

The principal objective of this study is to assess the adequacy of linear acoustic theory with an impedance wall boundary condition for modeling the detailed sound field of an acoustic source in a duct. This study compares measurements and calculations of a simple acoustic source in a rectangular concrete duct lined with foam on the walls and anechoic end terminations. Measuring acoustic pressure for 12 wave numbers provides variation in frequency and absorption characteristics of the duct walls. The cases in this study contain low frequencies and low wall absorptions corresponding to measurements of low-frequency helicopter noise in a lined wind tunnel. This regime is particularly difficult to measure in wind tunnels due to high levels of the reverberant field relatively close to the source. Close to the source, where the interference of wall reflections is minimal, correlation is very good. Away from the source, correlation degrades, especially for the lower frequencies. Sensitivity studies show little effect on the predicted results for changes in impedance boundary condition values, source location, measurement location, temperature, and source model for variations spanning the expected measurement error.

Mosher, Marianne↗

NASA NTRSDate not supplied

Time Domain Approach for Acoustic Liner Analysis

- Procedures for implementing the Ingard-Myers impedance condition in time domain - Instability issue of the Ingard-Myers Condition - Proposed stabilization of the Ingard-Myers condition - Numerical examples

time domain impedance boundary condition↗

NASA NTRS2014

Acoustics of Jet Surface Interaction-Scrubbing Noise

Concepts envisioned for the future of civil air transport consist of unconventional propulsion systems in the close proximity of the structure or embedded in the airframe. While such integrated systems are intended to shield noise from community, they also introduce new sources of sound. Sound generation due to interaction of a jet flow past a nearby solid surface is investigated here using the generalized acoustic analogy theory. The analysis applies to the boundary layer noise generated at and near a wall, and excludes the scattered noise component that is produced at the leading or the trailing edge. While compressibility effects are relatively unimportant at very low Mach numbers, frictional heat generation and thermal gradient normal to the surface could play important roles in generation and propagation of sound in high speed jets of practical interest. A general expression is given for the spectral density of the far field sound as governed by the variable density Pridmore-Brown equation. The propagation Greens function is solved numerically for a high aspect-ratio rectangular jet starting with the boundary conditions on the surface and subject to specified mean velocity and temperature profiles between the surface and the observer. It is shown the magnitude of the Greens function decreases with increasing source frequency andor jet temperature. The phase remains constant for a rigid surface, but varies with source location when subject to an impedance type boundary condition. The Greens function in the absence of the surface, and flight effect are also investigated.

Propulsion Noise↗

NASA NTRS2014

Acoustics of Jet Surface Interaction - Scrubbing Noise

Concepts envisioned for the future of civil air transport consist of unconventional propulsion systems in the close proximity to the structure or embedded in the airframe. While such integrated systems are intended to shield noise from the community, they also introduce new sources of sound. Sound generation due to interaction of a jet flow past a nearby solid surface is investigated here using the generalized acoustic analogy theory. The analysis applies to the boundary layer noise generated at and near a wall, and excludes the scattered noise component that is produced at the leading or the trailing edge. While compressibility effects are relatively unimportant at very low Mach numbers, frictional heat generation and thermal gradient normal to the surface could play important roles in generation and propagation of sound in high speed jets of practical interest. A general expression is given for the spectral density of the far field sound as governed by the variable density Pridmore-Brown equation. The propagation Green's function is solved numerically for a high aspect-ratio rectangular jet starting with the boundary conditions on the surface and subject to specified mean velocity and temperature profiles between the surface and the observer. It is shown the magnitude of the Green's function decreases with increasing source frequency and/or jet temperature. The phase remains constant for a rigid surface, but varies with source location when subject to an impedance type boundary condition. The Green's function in the absence of the surface, and flight effects are also investigated

Jet Noise↗

NASA NTRS2008

Application of Fast Multipole Methods to the NASA Fast Scattering Code

The NASA Fast Scattering Code (FSC) is a versatile noise prediction program designed to conduct aeroacoustic noise reduction studies. The equivalent source method is used to solve an exterior Helmholtz boundary value problem with an impedance type boundary condition. The solution process in FSC v2.0 requires direct manipulation of a large, dense system of linear equations, limiting the applicability of the code to small scales and/or moderate excitation frequencies. Recent advances in the use of Fast Multipole Methods (FMM) for solving scattering problems, coupled with sparse linear algebra techniques, suggest that a substantial reduction in computer resource utilization over conventional solution approaches can be obtained. Implementation of the single level FMM (SLFMM) and a variant of the Conjugate Gradient Method (CGM) into the FSC is discussed in this paper. The culmination of this effort, FSC v3.0, was used to generate solutions for three configurations of interest. Benchmarking against previously obtained simulations indicate that a twenty-fold reduction in computational memory and up to a four-fold reduction in computer time have been achieved on a single processor.

Dunn, Mark H.↗

NASA NTRS1996

Effective Boundary Treatment for the Biharmonic Dirichlet Problem

The biharmonic equation can be rewritten as a system of two Poisson equations. Multigrid solution of this system is expected to converge with the same amount of work as solving two Poisson equations, requiring less than 70 floating point operations (scalar multiply or addition) per fine grid point to reach a solution using an FMG algorithm. For periodic boundary conditions, this goal is attained by simple, straightforward application of multigrid. For Dirichlet boundary conditions, however, convergence is impeded by poor interaction with the boundaries. Attempts to overcome the slowness without specifically addressing the boundaries have resulted in multigrid algorithms not attaining the Poisson convergence rate. We present three methods of boundary treatment with which full multigrid efficiency can be obtained. All implement an approach described by Brandt, concentrating some additional effort near the boundary. The first approach simply adds a number of relaxation sweeps over points close to the boundary. The second uses joint relaxation on near-boundary points. The third method takes something from each of the first two methods, resulting in a solver more suitable for highly parallel applications.

Brandt, A.↗