Mean Square Pressure Research Articles

Two point correlations of jet noise

A large body of careful experimental measurements of two-point broad band correlations of far-field jet noise has been carried out and was briefly reported recently by Lucio Maestrello in NASA TM X-72835. The rather sharp directional lobes and marked absence of axisymmetry were striking and motivated the present effort to bring theory to bear. The model of jet-noise generation is an approximate version of an earlier work of Ribner, based on the foundations of Lighthill. The model incorporates isotropic turbulence superimposed on a specified mean shear flow, with assumed space-time velocity correlations, but with source convection neglected. The particular vehicle is the Proudman format, and the previous work (mean-square pressure) is extended to display the two-point space-time correlations of pressure. The shape of polar plots of correlation is found to derive from two main factors: (1) the non-compactness of the source region, which allows differences in travel times to the two microphones—the dominant effect—and (2) the directivities of the constituent quadrupoles —a weak effect. The non-compactness effect causes the directional lobes in a polar plot to have pointed tips (cusps) and to be especially narrow in the plane of the jet axis. In these respects, and in the quantitative shapes of the normalized correlation curves, results of the theory show generally good agreement with Maestrello's experimental measurements.

Method of predicting Leq created by urban traffic

The time-averaged overall mean-square sound pressure created by statistically stationary traffic travelling a finite, straight road segment is determined explicitly. This result is extended to a system of roads by using digital simulation. Theoretical predictions for a typical urban conurbation show encouraging agreement with measured values. Therefore the technique appears to offer a practical means of evaluating community plans before the introduction of road systems and changes in trunk routes and traffic controls, etc., are realized.

Space variance for rectangular modes

In a recent paper, the free-wave and eigenmode models of the sound field in a reverberation chamber were compared, and evidence was presented to support the thesis that the two models were consistent. Here a further piece of this evidence is given. We consider the spatial variance of the mean-square pressure in a reverberant sound field of one frequency which has a mean value of unity. This variance, when calculated from the free-wave model is known to have the value of unity. It is also know that, for one oblique rectangular mode, the corresponding value of the variance is 2.375. It is shown here that for N such modes, whose frequencies coincide but whose phases are uniformly distributed, the variance approaches 1 as N approaches ∞.

Origin and character of vascular murmurs: model studies.

The fluctuating mechanical stress imposed upon the arterial-wall downstream of a stenosis by turbulent blood flow was studied in a model experiment. The dependence of mean-square wall pressure and spectral density of wall pressure upon Reynolds number, diameter ratio, and distance from the stenosis was obtained. The mean-square pressure was found to have strong Reynolds-number dependence at low Reynolds number, and the distance downstream of the stenosis at which the mean-square fluctuating pressure reaches its maximum intensity was found to depend upon jet velocity and fluid viscosity only. A high Reynolds-number scaling of the spectral density of wall pressure was obtained.

Effect of high-amplitude flow pulsations on heat transfer

We present results of an experimental study of the effect of flow pulsations on turbulent heat transfer in the transverse direction in propagation of sound waves. A significant increase in heat-transfer intensity was recorded at mean-square pressure pulsation amplitude p=168–180 dB at frequency f=100–150 Hz. The dependence of heat-transfer intensity on acoustic field characteristics is obtained.

Aerodynamic pressures on vortex-shedding struts

New measurements will be presented of the aerodynamic pressures induced at trailing edges of rigid struts. These pressures are generated with a sinuous time dependence when the wake of the strut is characterized by a von Karman vortex street. The variation of the surface pressure with upstream distance from the trailing edge is given for a series of edge configurations. These include varying ratios of trailing edge thickness to the thickness of the boundary layer which separates at the edge, as well as variations in the edge geometry. The mean-square pressures are found to depend on x−1, where x is the upstream distance from the edge, within ten edge-thicknesses of the edge. Spatial surveys of the disturbances in the wake provide intensity and length scales which reflect the shed vortex strength and structure and they therefore relate to the magnitudes of the pressures. A dimensional analyses is presented which specifies this relationship. The resulting dimensionless chordwise distribution of fluctuating pressure is shown to be given by an empirical function which is similar for all edges examined.

Sound radiation from a simple source in circular and helical motion

The analysis of sound radiation from a time-varying simple source in uniform circular and helical motion is made. The expressions for the mean-square farfield sound pressure and the overall sound-power output are derived. The retarded time approach is used and illustrates the relative significance of the time rate of pulsations and the acceleration due to rotation of the source. Subject Classification; 20.15, 20.55; 30.50.

Wave directional spectra from synthetic aperture observations of radio scatter

We have measured the directional distribution of waves produced by a quasi-stationary, homogeneous wind field. This was done by observing radio backscatter from the LORAN A transmitter on Wake Island. The LORAN frequency of 2 MHz is in Bragg resonance with 7 S ocean waves. A directional resolution of ±3° has been attained from a synthesized antenna formed by moving the receiver at constant speed along the airport taxiway. The directional distribution of the 7 S ocean waves as inferred from the variable scatter cross-section compared favorably with directional moments measured with a pitch-and-roll buoy. The directional wave energy is found to fall off monotonically with angle θ relative to the wind, in accord with a model cos S( 1 2 θ ) proposed by Longuet-Higgins, Cartwright and Smith, Ocean wave spectra, pp 111–136, Prentice-Hall (1963); s increases with wind speed and decreases with frequency, and can be plotted against a single parameter μ = μ ∗/cκ , where √ ( u ∗/ρ ) is wind stress, c is phase velocity, and κ = 0·4 is von Karman's constant. The radio measurements give an upper limit of 0·02 for the upwind/downwind ration in wave energy flux. The ratio of up-down to cross-wind components in mean-square slope computed for the cos S( 1 2 θ ) model is compared to the ratio inferred from measurements of sun glitter. The (second-order) mean-square pressure fluctuations at great depth are computed from our estimates of oppositely traveling wave energy.

A stochastic model of the noise field emitted from traffic in steady flow

The noise emitted from steady statistically stationary traffic flow is studied analytically from the point of view of a stochastic acoustic model, in order to relate the noise field in terms of the details of the flow, and the acoustic characteristics of the vehicles therein. By making a number of simple assumptions regarding the vehicle as a source of sound, it has been found possible to represent the acoustic emission from a traffic stream comprised of indivudual vehicles in uniform motion as that from a stream of non-directional acoustic point sources of constant power. The model enables the statistical field properties to be obtained explicity. The statistical properties obtained in this paper are the time-averaged value and variance of the temporal mean-square sound pressures recorded at a fixed point near the flow and the time-averaged frequency spectrum of the noise field, the predicted values of which are found to agree reasonably well with experiment.

Guidelines for determination of the average sound power radiated by discrete-frequency sources in a reverberation room

When a source of discrete-frequency sound is placed in a reverberation room, the sound-power output depends on the source position and frequency. In addition, the measured sound-pressure level depends on microphone position. In this paper, some experimental data are presented that are relevant to both the microphone path length required for an accurate determination of the space/time average mean-square pressure in the room, and the variations of sound-power output with source position in the room. Based on theoretical and experimental results, some guidelines are proposed for the determination of the sound-power radiated by discrete-frequency sources in a reverberation room.

Distortion of sonic bangs by atmospheric turbulence

Recorded pressure signatures of supersonic aircraft often show intense, spiky perturbations superimposed on a basicN-shaped pattern. A first-order scattering theory, incorporating both inertial and thermal interactions, is developed to explain the spikes. Scattering from a weak shock is studied first. The solution of the scattering equation is derived as a sum of three terms: a phase shift corresponding to the singularity found by Lighthill; a small local compression or rarefaction; a surface integral over a paraboloid of dependence, whose focus is the observation point and whose directrix is the shock. The solution is found to degenerate at the shock into the result given by ray acoustics, and the surface integral is identified with the scattered waves that make up the spikes. The solution is generalized for arbitrary wave-forms by means of a superposition integral. Eddies in the Kolmogorov inertial subrange are found to be the main source of spikes, and Kolmogorov's similarity theory is used to show that, for almost all timestafter a sonic-bang shock passes an observation point, the mean-square pressure perturbation equals$(\Delta p)^2 (t_c/t)^{\frac{7}{6}}$, where Δpis the pressure jump across the shock andtcis a critical time predicted in terms of meteorological conditions. For an idealized model of the atmospheric boundary layer,tcis calculated to be about 1 ms, a figure consistent with the qualitative data currently available. The mean-square pressure perturbation just behind the shock itself is found to be finite but enormous, according to first-order scattering theory. It is conjectured that a second-order theory might explain the shock thickening that actually occurs.

Statistical Properties of Reverberant Sound Fields

Reverberant sound fields are used in several standard acoustical measurements, in which the sound field is sampled at different points. Here, the distributions of such sampled values are considered for the reverberant sound fields derived from different types of signal. It is shown that the values approach equality only where the field contains many frequency components simultaneously. For finite numbers of components, exact expressions in closed form are found for the distributions. For the field derived from a single frequency component, the values of mean-squared pressure follow a gamma distribution. For two or more frequency components of different mean amplitude, the distribution is related to a gamma distribution, and some computed values of it are given for the case of two components. Some experimental data are given. The results bear on the accuracy to be expected in reverberant field measurements.

Electroacoustic Transducer Efficiency of R, L, and C Components

It has long been recognized that electrical components under ac excitation behave as electroacoustic transducers (e.g., thermophone, transformer). The development of high-frequency power supplies for microminiaturized electronic packaging revived interest in resistors, inductors, and capacitors as sources of acoustical noise. The sound-power levels of 3 types of resistors, 3 types of inductors, and 4 types of capacitors were measured in a 0.75-m3 a reverberation chamber. Each component was excited by 13-octave bands of noise in the frequency range 1–16 kcps. From the PWL data and the input electrical power, the transducer efficiency and the average mean-square sound pressure at a distance of 1 m in a free field were determined. At 4 kcps, the efficiencies are approximately 10−14, 10−9, and 10−8 for representative R, C, and L components, respectively. For some C and L components, the sound pressures were sufficiently high to require noise-control measures. The effectiveness af various noise-control treatments (e.g., capsulation, potting) was evaluated.

Internal Sound in a Cylindrical Shell, due to an External Random Pressure Field

The outer skin of a space vehicle is exposed to a random pressure field. This pressure field causes the skin of the vehicle to vibrate and radiate sound to the interior of the vehicle. In the paper, we idealize the space vehicle by a finite cylindrical shell. Relations are derived for the mean-square pressure inside the vehicle. These relations explicitly depend on the spatial and time correlation function of the random pressure acting on the outside of the shell. It is shown that the effect of the internal fluid on the response of the shell is to cause a change in the inertia of the shell. The change in the inertia assumes various magnitudes and could be positive or negative, depending on the particular mode shape. Moreover, it is shown that there are optimum frequencies at which most of the sound is propagated to the inside of the vehicle. A set of curves showing the effect of the various parameters on the root-mean-square level of the internal acoustic pressure is given. Comparisons with field measurements are given where pertinent and available.

Experiments on jet flows and jet noise far-field spectra and directivity patterns

The results of measurements of far-field sound emitted from jets are reported. The narrow-band power spectral density of the sound in the far field was measured for three jet diameters, three Mach numbers, and five angular positions. The intensity distribution of mean-square pressure fluctuation in the far field in several wide frequency ranges were also measured. The similarity relations found from the experiments are reported.

On the Effect of Atmospheric Turbulence on Sound Propagated over Ground

The sound field above a plane boundary has been calculated by assuming that both the direct and reflected waves fluctuate in amplitude and phase. The pressure field is described in terms of a time-dependent mean-square pressure (p̄2) whose statistical properties are studied. The mean and variance of (p̄2) have been calculated by assuming that the turbulent fluctuations are normally distributed. The results are compared with some experimental data obtained outdoors over a large rigid boundary. Most of the experimental data have been found to be in good agreement with the calculations.

Measurements of the fluctuating pressure at the wall beneath a thick turbulent boundary layer

Measurements of the turbulent pressure field at the wall beneath a thick (5-inch) turbulent boundary layer produced by natural transition on a smooth surface are reported. The data include the mean-square pressure, parallel to the stream, and spatial correlation of the pressure transverse to the stream.The root-mean-square wall pressure was 2.19 times the wall shear stress. The power spectra of the pressure were found to scale with the free-stream speed and the boundary-layer displacement thickness. A few tests with a rough surface showed that the increase in root-mean-square wall pressure was greater than the increase in wall shear stress.The space-time correlation measurements parallel to the stream direction exhibit maxima at certain time delays corresponding to the convection of pressure-producing eddies at speeds varying from 0.56 to 0.83 times the stream speed. The lower convection speeds are measured when the spatial separation of the pressure transducers is small, or when only the pressure fluctuations at high frequencies are correlated. Higher convection speeds are observed when the spatial separation of the pressure transducers is large, or when only low frequencies are correlated. The result that low-frequency pressure fluctuations have the highest convection speed is in agreement with the measurements of Corcos (1959, 1962) in a fully turbulent tube flow. Analysis of these measurements also shows that both large- and small-scale pressure-producing eddies decay after travelling a distance proportional to their scale. More precisely, a pressure-producing eddy of large or small wavelength λ decays and vanishes after travelling a distance of approximately 6λ.The transverse spatial correlation of the wall-pressure fluctuations was measured and compared with the longitudinal scale. Both the transverse and the longitudinal scale of the pressure fluctuations were of the order of the boundary-layer displacement thickness. The transverse and longitudinal scales of both large- and small-scale wall-pressure fluctuations were also measured and were also found to be approximately the same.

Pressure Field within Homogeneous Anisotropic Turbulence

Integral expressions are derived for the mean-square fluctuation pressure, pressure correlation, and certain moments of the pressure correlation within general anisotropic, homogeneous turbulence. Application is made to idealized flow models exhibiting certain anisotropy characteristics similar to those observed in boundary layer turbulence. Pressure fluctuations arising from the turbulence itself and from interaction of the turbulence with a mean flow exhibiting constant shear are compared with the isotropic case. It is found that, in contrast to the completely isotropic case, the anisotropic pressure correlation is negative for some difference coordinate values and exhibits a “tail” which falls off with distance x as x−3 at separations for which the velocity correlations have fallen essentially to zero. It is concluded that in general the principal normalized moments of the pressure correlation will tend to be less than the corresponding velocity correlation moments. For isotropic turbulence this implies that the pressure correlation must fall off with distance faster than the velocity correlation. For the models of anisotropic turbulence treated it is found that departure from isotropy results in lower mean-square pressure fluctuations for a given mean kinetic energy of turbulence.

Near Sound Field of Turbulence

The general expressions for the velocity potential, particle velocity, sound pressure, mean-square sound pressure, intensity, and sound power describing the sound field of a turbulent region are derived starting from Lighthill's fundamental relation for the density fluctuations in the medium outside the turbulent region. The mean-square sound pressure and intensity of the lateral and longitudinal quadrupole components of the near- and far-sound field are discussed in detail and for shear flows a correction to the sound pressure measured in the near field is given. These expressions should be useful in the interpretation of noise data taken close to jets and turbulent boundary layers.

Sound Field Within a Region of Anisotropic Turbulence

The pressure fluctuations within a region of anisotropic turbulence are considered, taking into account the effect of mean shear. Expressions for the pressure correlation function, mean-square pressure, mean-square-pressure-gradient, and certain integral moments of the correlation function are given for the general anisotropic case in the form of single space integrals over velocity correlation functions. As an example, a model of turbulence with oriented vorticity interacting with a uniform mean shear is treated. The relation of the associated pressure correlation-function-moments to the acoustic driving force to be anticipated on a plate bounding a turbulent flow is discussed. This work was supported by the Wright Air Development Center under Contract AF 33(616)-2331.