Vibroacoustic responses of a heavy fluid loaded cylindrical shell excited by a turbulent boundary layer

A fully coupled structural–acoustic model of a cylindrical shell under external turbulent boundary layer excitation is herein developed. The numerical process requires computation of the wall pressure cross spectral density function as well as sensitivity functions for the fluid-loaded cylindrical shell. A semi-empirical model from literature is used to describe the wall pressure field induced by the turbulent boundary layer in the wavenumber–frequency domain. An analytical expression of the wall pressure field for a flat surface is adapted to describe the wall pressure field for a cylindrical surface. Circumferential sensitivity functions are derived using a wavenumber-point reciprocity principle. Results for the near-field and far-field acoustic pressure spectra are presented. Contributions of individual circumferential modes to the acoustic pressure spectra are examined, showing distinct trends below and above the ring frequency. The proposed method is computationally efficient and provides an effective approach to investigate vibroacoustic responses for maritime platforms.     

Dynamic interaction of a spherical radiator in a fluid-filled cylindrical borehole within a poroelastic formation

Harmonic acoustic radiation from a modally oscillating spherical source positioned at the center of a fluid-filled cylindrical cavity embedded within a fluid-saturated porous elastic formation is studied in an exact manner. The formulation utilizes the Biot theory of dynamic poroelasticity along with the cylindrical to spherical wave-field transformations, and the pertinent boundary conditions to obtain a closed-form series solution. The analytical results are illustrated with a numerical example in which the spherical source, with its polar axis oriented along the main axis of a water-filled borehole and embedded within a water-saturated Ridgefield sandstone formation, is excited in vibrational modes of various orders. The magnitude of the reflected component of acoustic pressure along the axis of the borehole for a pulsating (n = 0), an oscillating (n = 1), and also a multipole (n = 0–3) spherical source as a function of the excitation frequency is calculated and discussed for representative values of the parameters characterizing the system. Special attention is paid to the effects of source excitation frequency, size, surface velocity profile, and internal impedance as well as borehole interface permeability condition on the reflected pressure magnitudes. Limiting cases are considered and fair agreements with well-known solutions are obtained.     

Finite Difference Method for the Acoustic Radiation of an Elastic Plate Excited by a Turbulent Boundary Layer: A Spectral Domain Solution

A finite difference method is developed to study, on a two-dimensional model, the acoustic pressure radiated when a thin elastic plate, clamped at its boundaries, is excited by a turbulent boundary layer. Consider a homogeneous thin elastic plate clamped at its boundaries and extended to infinity by a plane, perfectly rigid, baffle. This plate closes a rectangular cavity. Both the cavity and the outside domain contain a perfect fluid. The fluid in the cavity is at rest. The fluid in the outside domain moves in the direction parallel to the system plate/baffle with a constant speed. A turbulent boundary layer develops at the interface baffle/plate. The wall pressure fluctuations in this boundary layer generates a vibration of the plate and an acoustic radiation in the two fluid domains. Modeling the wall pressure fluctuations spectrum in a turbulent boundary layer developed over a vibrating surface is a very complex and unresolved task. Ducan and Sirkis [1] proposed a model for the two-way interactions between a membrane and a turbulent flow of fluid. The excitation of the membrane is modeled by a potential flow randomly perturbed. This potential flow is modified by the displacement of the membrane. Howe [2] proposed a model for the turbulent wall pressure fluctuations power spectrum over an elastomeric material. The model presented in this article is based on a hypothesis of one-way interaction between the flow and the structure: the flow generates wall pressure fluctuations which are at the origin of the vibration of the plate, but the vibration of the plate does not modify the characteristics of the flow. A finite difference scheme that incorporates the vibration of the plate and the acoustic pressure inside the fluid cavity has been developed and coupled with a boundary element method that ensures the outside domain coupling. In this paper, we focus on the resolution of the coupled vibration/interior acoustic problem. We compare the results obtained with three numerical methods: (a) a finite difference representation for both the plate displacement and the acoustic pressure inside the cavity; (b) a coupled method involving a finite difference representation for the displacement of the plate and a boundary element method for the interior acoustic pressure; (c) a boundary element method for both the vibration of the plate and the interior acoustic pressure. A comparison of the numerical results obtained with two models of turbulent wall pressure fluctuations spectrums - the Corcos model [3] and the Chase model [4] - is proposed. A difference of 20 dB is found in the vibro-acoustic response of the structure. In [3], this difference is explained by calculating a wavenumber transfer function of the plate. In [6], coupled beam-cavity modes for similar geometry are calculated by the finite difference method. This revised version was published online in July 2006 with corrections to the Cover Date.     

Numerical modeling of the disturbances of the separated flow in a rounded compression corner

Numerical modeling of the time-dependent supersonic flow over a compression corner with different roundness radii is performed on the basis of the solution of the two-dimensional Navier-Stokes equations in the regimes corresponding to local boundary layer separation. The development of unstable disturbances generated by local periodic injection/suction in the preseparated boundary layer is calculated. The results are compared with those of similar calculations for a flat plate. It is shown that the natural oscillations of the boundary-layer second mode stabilize in the separation zone and grow intensely downstream of the reattachment point. The acoustic modes excited within a separation bubble are studied using numerical calculations and an asymptotic analysis.     

Short-wave modes for wave ducts in quasi-stratified media

Propagation of harmonic waves in three-dimensional ducts, with slowly varying properties in the horizontal directions, is treated. The wave length is assumed to be small in comparison with the width of the duct. A set of asymptotic solutions to the wave equation (generalisations of normal modes of horizontally homogeneous wave-guides) is constructed. The method extends the applicability region of the method of “vertical modes and horizontal rays” and gives a simple estimation of the latter. Oscillations bounded by the wave-guide's boundaries or by the caustics are treated, and a unified formal procedure for construction of appropriate asymptotic solutions is proposed. The vicinity of ‘horizontal caustics’ is investigated. The analysis is carried out for the examples of acoustic and internal gravitational waves in fluids.     

Development of a numerical wave tank for analysis of nonlinear and irregular wave field

A numerical wave-absorption filter has been developed for an open boundary condition in the analysis of nonlinear and irregular wave evolution. The filter is composed of a simulated sponge layer and Sommerfeld's radiation condition at the outer edge of the layer. The wave-absorption characteristics of the filter have been investigated by applying the linear potential theory and a two-dimensional nonlinear boundary element model. In both cases, the filter is found to he applicable for a wide range of wave parameters. In order to realize an idealized “numerical wave tank”, the present model also incorporates a nonreflective wave generator in the computational domain composed of a series of vertically aligned point sources. Numerous numerical experiments demonstrate that the present approach is effective in generating an arbitrary wave profile without reflection not only at the open boundaries but also at the wave generator.     

Acoustic scattering from baffled membranes that are backed by elastic cavities

An elastic membrane backed by a fluid-filled cavity in an elastic body is set into an infinite plane baffle. A time harmonic wave propagating in the acoustic fluid in the upper half-space is incident on the plane. It is assumed that the densities of this fluid and the fluid inside the cavity are small compared with the densities of the membrane and of the elastic walls of the cavity, thus defining a small parameter . Asymptotic expansions of the solution of this scattering problem as →0, that are uniform in the wave number k of the incident wave, are obtained using the method of matched asymptotic expansions. When the frequency of the incident wave is bounded away from the resonant frequencies of the membrane, the cavity fluid, and the elastic body, the resultant wave is a small perturbation (the “outer expansion”) of the specularly reflected wave from a completely rigid plane. However, when the incident wave frequency is near a resonant frequency (the “inner expansion”) then the scattered wave results from the interaction of the acoustic fluid with the membrane, the membrane with the cavity fluid, and finally the cavity fluid with the elastic body, and the resulting scattered field may be “large”. The cavity backed membrane (CBM) was previously analyzed for a rigid cavity wall. In this paper, we study the effects of the elastic cavity walls on modifying the response of the CBM. For incident frequencies near the membrane resonant frequencies, the elasticity of the cavity gives only a higher order (in ) correction to the scattered field. However, near a cavity fluid resonant frequency, and, of course, near an elastic body resonant frequency the elasticity contributes to the scattered field. The method is applied to the two dimensional problem of an infinite strip membrane backed by an infinitely long rectangular cavity. The cavity is formed by two infinitely long rectangular elastic solids. We speculate on the possible significance of the results with respect to viscoelastic membranes and viscoelastic instead of elastic cavity walls for surface sound absorbers.     

Free-surface fluctuations in hydraulic jumps: Experimental observations

A hydraulic jump is the rapid and sudden transition from a high-velocity supercritical open channel flow to a subcritical flow. It is characterised by the dynamic interactions of the large-scale eddies with the free-surface. New series of experimental measurements were conducted in hydraulic jumps with Froude numbers between 3.1 and 8.5 to investigate these interactions. The dynamic free surface measurements were performed with a non-intrusive technique while the two-phase flow properties were recorded with a phase-detection probe. The shape of the mean free surface profile was well defined and the turbulent fluctuation profiles highlighted a distinct peak of turbulent intensity in the first part of the jump roller, with free-surface fluctuation levels increasing with increasing Froude number. The dominant free-surface fluctuation frequencies were typically between 1 and 4 Hz. A comparison between the acoustic sensor signals and conductivity probe data suggested that the air–water “free-surface” detected by the acoustic sensor corresponded to about the boundary between the turbulent shear layer and the upper free-surface layer. Simultaneous measurements of free surface and bubbly flow fluctuations for Fr = 5.1 indicated that the frequency ranges of both sensors were similar (F < 5 Hz) whatever the position downstream of the toe. The present results highlighted that the dynamic free-surface measurements can be conducted successfully using acoustic displacement meters, and the time-averaged depth measurements was a physical measure of the free-surface location in hydraulic jumps.     

Modal acoustic impedance force on a spherical source near a rigid interface

The modal acoustic radiation load on a spherical surface undergoing angularly periodic axisymmetric harmonic vibrations while immersed in an acoustic halfspace with a rigid (infinite impedance) planar boundary is analyzed in an exact fashion using the classical technique of separation of variables. The formulation utilizes the appropriate wave field expansions, the classical method of images and the appropriate translational addition theorem to simulate the relevant boundary conditions for the given configuration. The associated acoustic field quantities such as the modal impedance matrix and the modal acoustic radiation force acting on the spherical surface are determined. The analytical results are illustrated with a numerical example in which the spherical surface, excited in vibrational modes of various orders, is immersed near an impervious rigid wall. The presented solution could eventually be used to validate those obtained by numerical approximation techniques.     

超声速平面剪切层声辐射涡模态数值分析

对Mc = 1.2二维超声速空间发展平面自由剪切层, 进行了扰动模态及流动结构的数值分析. 采用时空三阶改进MacCormack格式, 差分求解可压缩扰动Navier-Stokes方程, 直接数值模拟入口不同基频谐波扰动的非线性演化特征. 采用空间线性稳定性理论证明, 计算所促发的扰动波是声辐射涡模态. 扰动参数及特征函数分析显示, 声辐射涡模态是弱色散的快／慢两种外部模态, 在扰动对流Mach数为超声速一侧呈膨胀／压缩状辐射. 单频受迫扰动可无相差地促发多模态混合扰动波, 而在自然扰动条件下, 剪切层的稳定性受慢模态主导.     

Selective enhancement of oblique waves caused by finite amplitude second mode in supersonic boundary layer

Nonlinear interactions of the two-dimensional(2D) second mode with oblique modes are studied numerically in a Mach 6.0 flat-plate boundary layer, focusing on its selective enhancement effect on amplification of different oblique waves. Evolution of oblique modes with various frequencies and spanwise wavenumbers in the presence of 2D second mode is simulated successively, using a modified parabolized stability equation(PSE) method, which is able to simulate interaction of two modes with different frequencies efficiently. Numerical results show that oblique modes in a broad band of frequencies and spanwise wavenumbers can be enhanced by the finite amplitude 2D second mode instability wave. The enhancement effect is accomplished by interaction of the 2D second mode, the oblique mode, and a forced mode with difference frequency. Two types of oblique modes are found to be more amplified, i.e., oblique modes with frequency close to that of the 2D second mode and low-frequency first mode oblique waves. Each of them may correspond to one type of transition routes found in transition experiments. The spanwise wavenumber of the oblique wave preferred by the nonlinear interaction is also determined by numerical simulations.     

Nonlinear wave processes in a hypersonic shock layer

Nonlinear disturbance development in a hypersonic flat-plate shock layer (M_∞ = 21, Re _L = 1.44×10⁵) exposed to external-flow slow-mode acoustic perturbations at one or several frequencies is studied on the basis of the numerical solution of the Navier-Stokes equations. The mean flow distortion by disturbances and the nonlinear self-interaction between spectral modes is investigated by varying the initial amplitudes of the acoustic waves introduced. The appearance of combination frequencies, both summarized and subtracted, and their interaction with each other is shown to exist.     

Continuous Mode Transition in High-speed Boundary-layers

Mode interaction is studied via direct numerical simulations of a Mach 4.5 boundary layer with discrete and continuous modes imposed at the inflow. An approximate decoupling procedure is developed to create separate vortical, acoustic and entropic continuous mode components. Oblique horizontal vorticity modes induce boundary layer disturbances that grow with downstream distance, similarly to their incompressible counterpart. One salient difference is that a low frequency vorticity mode, alone, is found to induce transition by spawning two-dimensional, unstable discrete modes. The discrete modes are non-linearly excited at high harmonics of the inlet perturbation. Adding a Mack second mode, in addition to the vorticity mode, causes even earlier transition, suggesting that, in supersonic flow, unstable discrete modes play a crucial role in breakdown of boundary-layer streaks.     

Similarity solution of thermal boundary layers for laminar narrow axisymmetric jets

In this work, the similarity equation describing the thermal boundary layers of laminar narrow axisymmetric jets is derived based on boundary layer assumptions. The equation is solved exactly. Some properties of the thermal jet are discussed. By introducing new-defined non-dimensional coordinates, the similarity solution results in a “universal” format. The results can also be applied in the boundary layer problem of species diffusion.     

高超声速边界层中模态转化的数值研究

在高超声速边界层中,第一模态和第二模态是与转捩有关的两个主要不稳定模态.除了不稳定模态,还存在一类稳定模态,其相速度在前缘接近快声波的相速度称为快模态.在感受性过程中,这类模态对激发边界层中不稳定模态起着很重要的作用.前缘感受性理论解释了边界层外扰动激发边界层中第一模态波的机理.针对高超声速平板边界层,利用相似性解剖面作为基本流,采用线性稳定性理论和直接数值模拟的方法研究了快模态和慢模态的稳定性行为.研究发现模态转化的位置与马赫数有关.根据线性稳定性理论的结果定义了临界频率.当扰动频率高于临界频率,第一模态与第二模态同支;而当扰动频率低于临界频率,第一模态与第二模态的共轭模态同支.借助稳定性方程的伴随方程分析了直接数值模拟的结果.直接数值模拟结果表明不论上游是快模态还是慢模态,当它们经过第二模态的不稳定区,它们都会演化成第二模态. 这可用模态在非平行流中传播的特征来解释.      

Three-dimensional receptivity of boundary layers

The paper presents a review of results of some recent (mainly experimental) studies devoted to a quantitative investigation of the problem of receptivity of the 2D and 3D boundary layers with respect to various 3D (in general) external perturbations. The paper concentrates on the mechanisms of excitation and development of stationary and travelling instability modes in a 3D boundary layer on a swept wing, as well as in 2D boundary layers including the Blasius flow and a self-similar boundary layer with an adverse pressure gradient. In particular, the following problems of the boundary-layer receptivity are discussed: (i) receptivity to localized 3D surface roughness, (ii) receptivity to localized 3D surface vibrations, (iii) acoustic receptivity in presence of 3D surface roughness, and (iv) acoustic receptivity in the presence of 3D surface vibrations. All experiments described in the paper were conducted using controlled disturbance conditions with the help of simulation of the stationary and non-stationary perturbations by means of several disturbance generators. This approach gives us the possibility to obtain quantitative results which are independent of any uncontrolled background perturbations of the flow and the experimental model. In contrast to the data obtained at “natural” environmental conditions these results can be directly compared with calculations without any significant assumptions about the physical nature of the disturbances under investigation. The complex (amplitude and phase) coefficients of the boundary-layer receptivity to external perturbations, obtained as functions of the disturbance frequency and the spanwise wavenumber (or the wave propagation angle), represent the main results of the experiments described. These results can be used for the evaluation of the initial amplitudes and phases of the instability modes generated by various external perturbations, as well as for quantitative verification of linear receptivity theories. Several examples of the comparison of experimental results with calculations are also presented in this paper. A brief analysis of the state-of-art in the field is performed and some general properties of different receptivity mechanisms are discussed.     

Acoustic Radiation from a Finite Length Cylindrical Shell Excited by an Internal Acoustic Source: Solution Based on a Boundary Element Method and a Matched Asymptotic Expansion

This paper deals with acoustic radiation by a thin elastic shell, closed by two perfectly rigid discs, immersed in water and filled with air. The system is driven by an internal acoustic source. The shell has a length L, is clamped along one of its boundaries and is freely supported along the other boundary. Using the infinite domain Green's function, the radiated acoustic pressure is modeled by a hybrid layer potential (linear combination with nonreal coefficient of a simple layer and a double layer). Using Green's tensor of the in vacuo shell operator, the shell displacement is expressed as the sum of the field generated by the acoustic pressures and that due to boundary sources. Finally, the Green's function of the interior Neumann problem is used to express the acoustic pressure inside the shell in terms of the acoustic source and shell normal displacement: this representation fails for any frequency equal to one of the resonance frequencies of the shell interior. To overcome this, a light fluid approximation, which is allowed because the inner fluid is a gas, is adopted. Around each resonance frequency, an inner approximation is defined which matches the classical outer approximation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Sensitivity of a supersonic boundary layer to acoustic disturbances

The results obtained by the authors in [1] are extended to the case of arbitrary angles of incidence of the external wave. This is not a trivial generalization, since the acoustic scattering undergoes a qualitative change. It is possible to distinguish two excitation channels: the first is connected with the diffraction of the acoustic wave by the spatial inhomogeneity resulting from the displacing action of the boundary layer, and the second with the presence of concentrated acoustic field sources associated with the scattering of the wave at the leading edge. The latter makes the principal contribution to the initial amplitude of the unstable modes when the angles of incidence of the sound are substantially different from zero. At low angles of incidence there is a singularity which can be revealed by introducing narrow intervals in the neighborhood of the limiting values of the wave numbers, where the two excitation channels are approximately equivalent. It is possible to obtain composite expressions for the initial amplitudes of the unstable modes uniformly valid for all angles of incidence of the acoustic wave.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 40–47, January–February, 1992.     

The Numerical Decomposition of Turbulent Fluctuations in a Compressible Boundary Layer

In many flows the turbulence is weakly compressible even at large Mach number. For example, in a compressible boundary layer Ma<5, the differences relative to an incompressible boundary layer understood as being caused by density variations that accompany variations temperature across the layer. Turbulent fluctuations in a boundary layer are therefore expected to be dominated by the effects nonconstant temperature, and low Mach number theories in which fluctuations are not dominant should be applicable to the fluctuating field. However, the analysis of compressible boundary layer DNS data reveals presence of significant acoustic fluctuations. To distinguish acoustic and thermal effects, a numerical decomposition procedure compressible boundary layer fluctuations is applied to determine the and nonacoustic fluctuations. Except for very near the wall, where decomposition procedure is not valid, it is found that the fluctuations are only weakly coupled to the acoustic fluctuations at numbers as high as 6. Received 13 March 2000 and accepted 21 May 2001     

Assessment of global linear stability analysis using a time‐stepping approach for compressible flows

Global linear stability analysis combined with computational fluid dynamics (CFD) is considered useful for understanding the physics of fluid flows. However, the numerical techniques of global linear stability analysis for compressible flows have not been well established in comparison with those for incompressible flows. In this study, we develop and assess a set of appropriate numerical techniques required to conduct a global linear stability analysis for compressible flows. For the eigensystem analysis, the Arnoldi method combined with time integration is in effect to preserve the memory (RAM) size of the computer. The compact difference scheme is used for the CFD analysis from the viewpoints of computing accurate global modes and saving memory by reducing the number of grid points to obtain the necessary spatial resolution. To assess the proposed method, two‐dimensional compressible flow problems, including regularized cavity flow, flow around a square cylinder, and the compressible mixing layer, are analyzed, and it is confirmed that the proposed method can obtain accurate mode shapes, growth rate, and frequency of the corresponding global modes. In addition, influences and an appropriate formulation of the outflow boundary conditions are investigated. Results reveal that the outflow boundary causes spurious unstable modes in the global linear stability analysis, and the radiation and outflow boundary condition and the extension of the computational domain with grid stretching keep the spurious unstable modes to a minimum. Copyright © 2015 John Wiley & Sons, Ltd.