Some general properties of the exact acoustic fields in horns and baffles

The propagation of the fundamental, longitudinal acoustic mode in a duct of variable cross-section is considered, and the “Webster” wave equations for the sound pressure and velocity are used to establish some general properties of the exact acoustic fields. The equipartition of kinetic and compression energies is shown (section 2.1) to hold at all stations only for (i) a duct of constant cross-section and (ii) an exponential horn; these are the two cases for which the wave equations for the acoustic velocity and pressure coincide. It is proved (section 2.3) that there are only five duct shapes, forming two dual families, which have constant cut-off frequency(ies): namely, (I) the exponential duct, which is self-dual, and is the only shape with constant (and coincident) cut-offs both for the velocity and pressure; (II) the catenoidal horns, of cross-section S～cosh², sinh², which, with their duals (III) the inverse catenoidal ducts S～sech², csch², have one constant cut-off frequency, respectively, for the acoustic pressure and velocity. The existence of at least one constant cut-off frequency implies that the corresponding wave equation can be transformed into one with constant coefficients, and thus the acoustic fields calculated exactly in terms of elementary (exponential, circular and hyperbolic) functions; this property also applies to the imaginary transformations of the above shapes, viz., the sinusoidal S～sin² and inverse sinusoidal S～csc² ducts, that have no cut-off frequency, i.e., are acoustically “transparent”. It is shown that elementary exact solutions of the Webster equation exist only (section 3.1) for these seven shapes: namely, the exponential, catenoidal, sinusoidal and inverse ducts; it is implied that for all other duct shapes the exact acoustic fields involve special functions, in infinite or finite terms, e.g., Bessel and Hermite functions respectively for power-law and Gaussian horns. Examples of the method of analysis are given by calculating, in elementary form, the exact acoustic fields in inverse catenoidal ducts, for all cases of (a) propagating waves above, (b) non-oscillating modes below and (c) transition fields at the cut-off frequency. The inverse catenoidal ducts consist of (A) the horn of cross-section S～sech², ressembling the “soliton” of non-linear water wave fame, and (B) the baffle of cross-section S～csch², which also matches two exponentially converging ducts, but has infinite, instead of finite, flare at the origin. The geometrical and acoustic properties of these ducts are illustrated by sets of six plots, in Figure 1(a) for the sech-horn and in Figure 1(b) for the csch-baffle; the exact acoustic fields are described by amplitude and phase decompositions of the sound velocity and pressure, plotted as functions of position along the duct, for four frequencies ranging from the cut-off condition to the ray limit (or W.K.B.J. approximation).     

Experimental study of the influence of annular nozzle on acoustic characteristics of detonation sound wave generated by pulse detonation engine

Acoustic characteristics of the detonation sound wave generated by a pulse detonation engine with an annular nozzle, including peak sound pressure, directivity, and A duration, are experimentally investigated while utilizing gasoline as fuel and oxygen-enriched air as oxidizer. Three annular nozzle geometries are evaluated by varying the ratio of inner cone diameter to detonation tube exit diameter from 0.36 to 0.68. The experimental results show that the annular nozzles have a significant effect on the acoustic characteristics of the detonation sound wave. The annular nozzles can amplify the peak sound pressure of the detonation sound wave at 90° while reducing it at 0° and 30°. The directivity angle of the detonation sound wave is changed by annular nozzles from 30° to 90°. The A duration of the detonation sound wave at 90° is also increased by the annular nozzles. These changes indicate that the annular nozzles have an important influence on the acoustic energy distribution of the detonation sound wave, which amplify the acoustic energy in a direction perpendicular to the tube axis and weaken it along the direction of the tube axis.     

Phase prediction of the response of choked nozzles to entropy and acoustic disturbances

The development and transmission of sound through the exit of an aero-engine combustor is often investigated by modelling the complex geometry as a convergent-divergent nozzle. However, these analytical acoustic predictions are usually limited to the compact case, where the length of the nozzle is insignificant compared to the wavelength of the flow perturbations, or to cases where the variation of the mean velocity through the nozzle may be treated as linear or piece-wise linear. Considering terms up to first order in frequency for the conservation of mass, momentum and energy, this paper investigates an alternative approach by deriving effective lengths for the passage of the flow perturbations through a supercritical convergent-divergent nozzle. The effects due to the presence of a normal shock wave are also studied using a linearised form of the Rankine-Hugoniot relations. The analyses lead to predictions for the phase and magnitude of the transmitted acoustic waves from finite-length nozzles, and are valid for low non-dimensional frequencies. It has been found that these predictions agree well with the numerical results from inviscid simulations.     

Near-field acoustics of clustered rocket engines

This paper presents an approach for the prediction and characterization of the near-field acoustic levels from closely spaced clustered rocket engines. The calculations are based on the method proposed by Eldred wherein the flow field from the clustered rockets is divided into two zones. Zone 1 contains the isolated nozzles that produce noise independently and extends up to a distance where the individual flows completely mix to form an equivalent single nozzle flow. Zone 2 is occupied by the single mixed stream starting from the station where the jets merge. The acoustic fields from the two zones are computed separately on the basis of the NASA-SP method developed for a single equivalent nozzle. A summation of the spectra for the two zones yields the total effective sound pressure level for the clustered engines. Under certain conditions of nozzle spacing and flow parameters, the combined sound pressure level spectrum for the clustered nozzles displays a double peak. Test cases are presented here to demonstrate the importance of hydrodynamic interactions responsible for the double peak in the sound spectrum in the case of clustered rocket nozzles, and the role of ground reflections in the case of noninterfering jets.     

On the acoustic modes in a cylindrical duct with an arbitrary wall impedance distribution

The present paper considers the propagation of sound in a cylindrical duct, with a wall section of finite length covered by an acoustic liner whose impedance is an arbitrary function of position. The cases of (i) uniform wall impedance, and wall impedance varying along the (ii) circumference or (iii) axis of the duct, or (iv) both simultaneously, are explicitly considered. It is shown that a nonuniform wall impedance couples modes with distinct azimuthal l or axial m wave numbers, so that their radial wave numbers k can no longer be calculated separately for each pair (m,l). The radial wave numbers are the roots of an infinite determinant, in the case when the wall impedance varies either (i) circumferentially or (ii) radially. If the wall impedance varies (iv) both radially and circumferentially, then the radial wave numbers are the roots of a doubly infinite determinant, i.e., an infinite determinant in which each term is an infinite determinant. The infinite determinants specifying the radial wave numbers are written explicitly for sound in a cylindrical nozzle with a uniform axial flow, in which case the radial eigenfunctions are Bessel functions; the method of calculation of the radial wave numbers applies equally well to a cylindrical nozzle with shear flow and/or swirling flows, with the Bessel functions replaced by other eigenfunctions. The radial wave numbers are calculated by truncation of the infinite determinants, for several values of the aspect ratio, defined as the ratio of length to diameter. It is shown that a nonuniform wall impedance will give rise to additional modes compared with a uniform wall impedance. The radial wave numbers specify the eigenfrequencies for the acoustic modes in the duct; the imaginary parts of the eigenfrequencies specify the decay of the sound field with time, and thus the effectiveness of the acoustic liner.     

Acoustic disturbance from gas non-uniformities convected through a nozzle

The non-steady flow generated by convection of gas containing non-uniform temperature regions or “entropy spots” through a nozzle is examined analytically as a source of acoustic disturbance. The first portion of the investigation treats the “compact nozzle”, the case where all wave lengths are much longer than the nozzle. Strengths of transmitted and reflected one-dimensional waves are given for supersonic and subsonic nozzles and for one configuration of supersonic nozzle with normal shock at the outlet. In addition to a wave reflected from the nozzle inlet, the supersonic nozzle discharges two waves, one facing upstream and the other facing downstream. For reasonable values of the nozzle inlet Mach number, the pressure amplitude of each wave increases directly as the discharge Mach number.The acoustic perturbations from a supercritical nozzle of finite length, in which the undisturbed gas velocity increases linearly through the nozzle, are analyzed for several inlet and discharge Mach number values and over a wide frequency range. The results which agree with the compact analysis for low frequency, deviate considerably as the frequency rises, achieving pressure fluctuation levels of several times the compact values. It is shown that this result originates in a phase shift between the two waves emitted downstream and that the pressure fluctuations for moderate frequencies may be approximated from the compact analysis with an appropriate phase shift.In all cases, the pressure fluctuations caused by a 2% fluctuation in absolute inlet temperature are large enough to require consideration in acoustic analysis of nozzles or turbine blade channels.     

A finite element approach for predicting nozzle admittances

A finite element method is used to predict the admittances of axisymmetric nozzles. It is assumed that the flow in the nozzle is isentropic and irrotational, and the disturbances are small so that linear analyses apply. An approximate, two dimensional compressible model is used to describe the steady flow in the nozzle. The propagation of acoustic disturbances is governed by the complete linear wave equation. The differential form of the acoustic equation is transformed to an integral equation by using Galerkin's method, and Green's theorem is applied so that the acoustic boundary conditions can be introduced through the boundary residuals. The boundary conditions are described for both straight and curved sonic lines. A two dimensional FEM with linear elements is used to solve the acoustic equation. A one dimensional FEM is also used to solve the reduced equation of Crocco, and the solution verifies the sufficiency of the boundary residual formulation. Comparison between computed admittances and experimental data is shown to be quite good.     

Interaction between a composite compression wave and a vortex in a thermodynamically nonideal medium

A numerical analysis is presented of two-dimensional interaction between a transverse vortex and a composite compression wave that can exist in a thermodynamically nonideal medium. It is shown that the interaction of a composite wave involving a “neutrally stable” shock with a vortex generates weakly damped outgoing acoustic waves; i.e., the shock is a source of sound. This phenomenon increases the post-shock acoustic noise level in an initially turbulent flow.     

Acoustic and aerodynamic design and characterization of a small-scale aeroacoustic wind tunnel

The aeroacoustic wind tunnel at Brandenburg University of Technology at Cottbus is a newly commissioned research facility for the experimental study of sound generation from bodies immersed in a fluid flow. The paper discusses the design criteria for the open jet wind tunnel that provides a maximum wind speed of 72 m/s at continuous operation and may be operated with nozzles of different dimension between 35 cm diameter (circular nozzle) and 12 cm by 14.7 cm (rectangular nozzle). Experiments may be performed either in a reverberant or in an anechoic environment. Both the aerodynamic and the acoustic design of the wind tunnel components are discussed in detail. Background noise measurements in the completed facility revealed very low levels comparable to other wind tunnels. The results of aerodynamic wind tunnel calibration confirmed a uniform flow quality in the jet and a very low axial turbulence intensity which is less than 0.2% for the 35 cm nozzle and less than 0.1% for the other nozzles. A final benchmark is provided by results of successful trailing edge noise measurements on an SD7003 airfoil that are presented and compared to results from the literature.     

On the validity criteria of the phenomenological theory of electron absorption and amplification of sound in semiconductors

By using the kinetic equation for the electron distribution function in the acoustic wave field, the criteria are established for the validity of the phenomenological (“hydrodynamic”) theory of electron absorption and amplification of sound (acoustic waves) in the semiconductors. It is shown that the upper frequency limit of the validity of this theory lies, generally speaking, in the region of considerably lower (maybe by orders of magnitude) frequencies than it follows from the usually accepted criterion ql ? 1.     

Control of spray spot in cold spray technology. Part 1. Gas dynamic aspects

This paper presents a study of supersonic jets formed by approaches that are new for cold spray technique: the main flow is swirled, the nozzles with permeable profiles and with exit slots on the supersonic section are engineered. The flow swirling achieved in the nozzle prechamber retains downstream to substrate surface. The system of vortices created within the permeable nozzles changes the shock wave features of the overexpanded jet and the geometry of the bow shock wave ahead of the substrate surface. These new features of flow may affect particle motion and particlesubstrate interaction under the conditions of cold spray process; this offers tools for obtaining the necessary shape of a spray spot.     

Effect of chevron count and penetration on the acoustic characteristics of chevron nozzles

Experimental investigations have been carried out on chevron nozzles to assess the importance of chevron parameters such as the number of chevrons (chevron count) and chevron penetration. Acoustic measurements such as overall sound pressure level, spectra, directivity, acoustic power, and broadband shock noise have been made over a range of nozzle pressure ratio from sub-critical to underexpansion levels. Shadowgraph images of the shock-cell structure of jets from various chevron nozzles have also been captured for different nozzle pressure ratios. The results indicate that a higher chevron count with a lower level of penetration yields the maximum noise suppression for low and medium nozzle pressure ratios. Of all the geometries studied, chevron nozzle with eight lobes and 0° penetration angle gives the maximum noise reduction. Chevron nozzles are found to be free from screech unlike regular nozzles. Acoustic power index has been calculated to quantitatively evaluate the performance of the various chevron nozzles. Chevron count is the pertinent parameter for noise reduction at low nozzle pressure ratios, whereas at high nozzle pressure ratios, chevron penetration is crucial. The results illustrate that by careful selection of chevron parameters substantial noise reduction can be achieved.     

Application of RANS-based method to predict acoustic noise of chevron nozzles

Passive noise control devices for jet flows, such as chevron nozzles, have been studied for a long time due to their large application in turbofan engines. The main purpose of their usage is the reduction of low-frequency noise generation and thus decreasing the noise perceived at the far field. This work is a numerical study of acoustic noise generated by jet flow operating at Mach number 0.9 and Reynolds number 1.38 × 10⁶, considering two chevron nozzle geometries that differ from each other by the penetration angle into the flow. The main aim was to demonstrate that Reynolds averaged Navier Stokes (RANS)-based methods are reliable means to obtain acoustical noise predictions for the industry with a considerably low computational cost. In order to achieve this objective, computational fluid dynamics (CFD) RANS simulations were performed with a cubic k-ɛ model and the acoustic noise spectrum for different angles of radiation was obtained via the Lighthill ray-tracing (LRT) method. The numerical results for the acoustic and flow fields were seen to be in reasonable agreement with the experimental data, suggesting that this methodology can be used as a fast and useful option to predict acoustic noise of jet flows from chevron nozzles.     

Attenuation of sound due to vortex shedding from a splitter plate in a mean flow duct

A theoretical analysis is given of an experiment being performed at the University of Southampton [1] as part of a programme to quantify the effectiveness of perforated screens in dissipating sound in the presence of tangential mean flow. In the experiment vorticity is generated at the trailing edge of a splitter plate in a mean flow duct by a plane sound wave incident from upstream, acoustic energy being ceded to the kinetic energy of the vortex field. An expression is derived for the dissipated sound power at arbitrary subsonic mean flow Mach number and frequency. The calculation is performed both by a consideration of the net flux of acoustic energy into the trailing edge region of the splitter plate, and by evaluating the rate of working of the vortex lift forces in the field of the acoustic particle velocity. In particular, it is shown that the absorption is independent of frequency, provided the frequency does not exceed the minimum cut-on frequency of transverse acoustic modes within the duct.     

轴对称环状静电模的漂移波湍流参量激发理论研究

本文所论述的轴对称环状静电模是指环形磁约束等离子体(如托卡马克)中环向模数为零的近理想静电流体模,它包含有测地声模和基频率与之较低的声模;也含有所谓的‘近零频带状流’.本文根据冷离子流体模型在圆形磁面构成的准环坐标系中的表示,对涉及以上三种模式的漂移波湍流参量激发理论,在一级环形效应近似下,进行了系统讨论,并证明了带状流的四个新命题.利用对漂移波能谱的参数化描写,注意到由漂移波能谱径向有限宽度所引发的特性,如波能传播量的双Landau奇点,揭示了有限宽度对径向δ谱所得结果的重要修正:如,对近零频带状流和测地声模的参量激发条件带来的严格限制.此外,还讨论了密度带状流在高q条件下被激发的可能性.本文选用合理的物理参数.采用图示方法详细地讨论了有关的数值结果.分析表明,测地声模和近零频带状流的参量激发不可能发生在同一小半径处;如果测地声模被参量激发,也应能观察到密度带状流.     

Near-field acoustic characteristics of screech jet exhausted from a nozzle with a hard reflecting plate

The standing wave in the near field of the screech jet exhausted from a nozzle with a hard plate works on the jet flow as the forcing wave by the location of a reflecting plate, and then jet flow is considered to be changed. Moreover, the reflector location from the nozzle changes the sound pressure contours of the near field. Intensity maps of the screech tone which indicate the propagation to the jet axial direction or the radial direction of the jet by the presence of the reflector plate have not been explored. In the present paper, acoustic characteristics in the near field of the screech tone with the reflecting plate are studied using an optical wave microphone, which can measure the sound propagating for both vertical and horizontal directions to the jet axis. As a result, the standing wave in the near field of the screech jet with the reflector has two types: One is the standing wave between the hydrodynamic pressure fluctuation propagating jet downstream and the sound pressure propagating upstream, and the other is the standing wave by the difference between the wavelength of the sound wave and the wavelength at the place close to the jet.     

Influence of initial conditions at the nozzle exit on the structure of round jet

Experimental data concerning the influence of initial conditions at the nozzle exit on the structure and development characteristics of round jets are reported. Features in the development of laminar and turbulent round jets emanating from variously elongated nozzles at identical Reynolds numbers are revealed. Smoke visualization pictures obtained for jets formed under different initial conditions (with different distributions of mean and pulsating flow velocities at the nozzle exit) are discussed. It is shown possible to make the zone of laminar flow in the jet stream more extended, and to delay the jet turbulization process in space, by making the flow-velocity profile more parabolic at the exit of elongated nozzle. Features in the development of vortical structures in a jet under an acoustic action are identified. It is shown that, for a turbulent round jet to be produced right at the nozzle exit, the nozzle length must be increased in excess of a certain value so that to provide for spatial growth of turbulent boundary layer thickness, finally ending in the formation of a fully turbulent flow velocity profile across the channel. This work was supported by the Russian Foundation for Basic Research (Grant No. 08-01-00027), the Ministry of Education and Science of the Russian Federation (project RNP 2.1.2.3370), and by the grants of President of the Russian Federation (NSH-454.2008.1 and. MK-420.2008.1).     

Results of radius scaling experiments and analysis of neon K-shellradiation data from an inductively driven Z-pinch

The K-shell radiated energy (yield) from neon Z-pinch implosions with annular, gas-puff nozzle radii of 1, 1.75, and 2.5 cm was measured for implosion times from 50 to 300 ns while systematically keeping the implosion kinetic energy nearly constant. The implosions were driven by the Hawk inductive-storage generator at the 0.65-MA level. Initial neutral-neon density distributions from the nozzles were determined with laser interferometry. Measured yields are compared with predictions from zero-dimensional (0-D) scaling models of ideal. One-dimensional (1-D) pinch behavior to both benchmark the scaling models, and to determine their utility for predicting K-shell yields for argon implosions of 200 to >300 ns driven by corresponding currents of 4 to 9 MA, such as envisioned for the DECADE QUAD. For all three nozzles, the 0-D models correctly predict the Z-pinch mass for maximum yield. For the 1and 1.75-cm radius nozzles, the scaling models accurately match the measured yields if the ratio of initial to final radius (compression ratio) is assumed to be 8:1. For the 2.5-cm radius nozzle, the measured yields are only one-third of the predictions. Analysis of K-shell spectral measurements suggest that as much as 70% (50%) of the imploded mass is radiating in the K-shell for the 1-cm (1.75-cm) radius nozzle. That fraction is only 10% for the 2.5-cm radius nozzle. The 0-D scaling models are useful for predicting 1-D-like K-shell radiation yields (better than a factor-of-two accuracy) when a nominal (≈10:1) compression ratio is assumed. However, the compression ratio assumed in the models is only an “effective” quantity, so that further interpretations based on the 0-D analysis require additional justification. The lower-than-predicted yield for the 2.5-cm radius nozzle is associated with larger radius and not with longer implosion time, and is probably a result of two-dimensional effects     

Interaction of a spherical acoustic wave with an elastic spherical shell

The interaction of a spherical acoustic wave with an elastic spherical shell is treated analytically. The solution includes the coupling between the acoustic sound field and vibration of the shell with any degree of fluid loading. The formulation for the far-field acoustic pressure is derived in terms of natural spherical wave functions, the properties of the acoustic medium, and the material constants of the shell. The far acoustic field is computed for a thin aluminum shell and several sound source locations over a large range of ka, where k is the wavenumber, and a is the shell radius. It is shown that the acoustic pressure depends significantly on whether the shell is in air or is submerged in water, particularly when the sound source is very near the surface. In air, the sound field of the shell is nearly identical to that of a rigid sphere but, in water, the shell is more compliant, which results in a damped radiation field that is characterized by vibrational resonances throughout the range of frequencies considered. As the sound sources is moved further away from the surface, however, this resonance response decreases very rapidly, and the sound field corresponds more closely to that of the shell in air.