Corcos Model Research Articles

A measurement of flow noise spectrum of an axisymmetric body

The pressure fluctuation on the surface of a submerged body has been recognized as a dominant noise source. There have been many studies concerning the flow induced noise on a flat plate. However, the noise over an axisymmetric body has not been well reported. This paper addresses the way in which we have investigated the mechanism of noise generation due to an axisymmetric body. The associated experiments and signal processing methods are introduced. A 3-dimensional axisymmetric body whose length and diameter were 2 m and 10.4 cm, was prepared as a test specimen. The wall pressure on the surface of the body was measured in a large scale low noise wind tunnel at KIMM(Korea Institute of Machinery and Metals). To measure the wall pressure, we used two microphone arrays which were tangential and normal to the flow. Based on the measured signal, frequency-wavenumber spectrum which explains the structure of turbulence noise, was estimated. Tangential to the flow, there exists convective ridge at a relatively higher wavenumber region; this can cause spatial aliasing. To circumvent this problem, the cross spectrum was interpolated. The interpolation has been performed by unwrapping the phase and smoothing the cross spectrum. The phase unwrapping was done based on the Corcos model; the phase of cross spectrum decreases linearly with the distance between microphones. Aforementioned signal processings are possible by employing the experimental results that the estimated wavenumber spectrum quite resembles the Corcos model. We try to modify the Corcos model which is applicable to the flat plate, by altering the magnitude of cross spectrum to fit the experimental data more accurately. We proposed that this wavenumber spectrum model is suitable for the 3-dimensional axisymmetric body. Normal to the flow, there exists a little correlation between signals of different microphones. The circumferential wavenumber spectrum contains uniform power along the wavenumbers.

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.

Sound Radiated in a Cavity by a Vibrating Plate: Comparison between the Results of a Finite Element Method (FEMAK) and a Boundary Element Method (NoVAlea2D)

The coupling between a cavity and a vibrating plate is written as a strong coupling where both the continuity of stresses and particle velocities at the interface are taken into account. A FEM analysis reveals an important coupling between the first mode of the plate and the cavity. The pressure distribution in the cavity shows a strong quarter of a wave length pattern indicating the influence of the first Dirichlet mode in the cavity. The vibrating plate is then excited by a turbulent flow modelled by its spectral density using the Corcos model. A comparison is made between the results obtained from a FEM and a BEM formulation. Good agreements are observed concerning the plate displacements. The comparison of the cavity sound pressure levels is less good.

Vibro-acoustic response of a pipe excited by a turbulent internal flow

This article presents an experimental study of the vibro-acoustic response of a pipe excited by a fully-developed turbulent air flow. First, the wall pressure field acting on the internal pipe wall is investigated. The power spectral density of the wall pressure fluctuations is analyzed after cancellation of contaminating background noise. The convection velocity and correlation lengths are calculated from measured cross-spectra, and the cross-spectra are expressed in Corcos model form. Second, the vibro-acoustic response of the pipe is analyzed by referring to the structural modes of the pipe.

Effects of Convection and Decay of Turbulence on the Wall Pressure Wavenumber-Frequency Spectrum

Cross-spectral and cross-correlation data from experiments and numerical simulations have shown the turbulent wall pressure convection velocity to vary with the streamwise sensor spatial separation. This variation is due to the spatial decay rates of turbulent structures in the inner and outer regions of the boundary layer. Its effect is shown to have a significant impact on the distribution of energy in the wavenumber-frequency spectrum Φ (k1, ω). The standard Corcos model is known to over predict the wavenumber-frequency spectrum at low wavenumbers. This is shown to result from its constant convection velocity assumption. The spectral levels at sub convective and lower wavenumbers are shown to be directly influenced by the spatial variation in convection velocity. Convection velocity measurements from past investigations that cover the range 285 ≤ Rθ ≤ 29,000 are compared, and an outer variable scaling is shown to effectively collapse the data.

Feedback control of sound radiation from a plate excited by a turbulent boundary layer

In this paper it is shown that the problem of applying active control to a simply supported plate excited by a turbulent boundary layer can be presented in a form which allows the application of optimal control theory. Linear regulator theory and stochastic linear regulator theory are briefly summarized. The approach taken follows that of Baumann et al. who show that the optimal control theory can be applied to structural acoustic problems if the sound power radiated by the structure is estimated from the vibration signal by means of a ‘‘radiation filter.’’ A modal model for the vibration of an elastic structure is presented and the state space realization of the model is given. It is shown that a simple Corcos model of the turbulent boundary layer excitation can be modeled by using a number of ‘‘excitation filters.’’ The state space realization of these filters can be combined with the state space model of a vibrating plate to give a model to which optimal control theory can be applied. In similar fashion the sound power radiated by the plate can be estimated by means of a radiation filter matrix. A detailed account of the calculation of such a radiation filter matrix is given. The state space forms of the excitation filters and radiation filter matrix can be combined with the state space model of the plate to give a system state space realization which can be used in the solution of a linear quadratic regulator problem. Optimal control analysis is then applied to the state equations and optimal reductions in the far-field radiated power are obtained for various arrangements of control forces. These results give insight into the possibilities for obtaining useful reductions in turbulent boundary layer induced noise in aircraft by the application of feedback control.

Calculation of turbulent boundary layer wall pressure spectra

The suitability of various wave-vector–frequency spectral models for predicting the point wall pressure spectrum due to developed turbulent boundary layer flow over planar, rigid surfaces is investigated. The early Corcos model [G. Corcos, J. Fluid Mech. 18, 353–378 (1964)] along with the incompressible and compressible models developed by Chase [D. Chase, J. Sound Vib. 112, 124–147 (1987)] are given particular attention. The effect of finite-sized measurement transducers is included in the integration of the theoretical wave-vector–frequency spectrum over the in-plane wave numbers in order to arrive at an attenuated point pressure frequency spectrum that can be compared directly to existing experimental data obtained in air, water, and glycerine. The selected experiments cover a wide range of fluid properties and Reynolds numbers. It is found that the Corcos model does not predict the measured data as well as the Chase model. An optimum set of empirical constants needed to exercise the Chase models are presented, but found to depend on the experiment considered. The need to consider viscous effects in future modeling is borne out when dealing with viscous fluids like glycerine.

Turbulent boundary layer wall pressure spatial filtering with sensors constructed of polyvinylidene fluoride (PVDF)

Piezoelectric pressure transducers constructed of electrically polarized polyvinylidene fluoride (PVDF) were assessed as spatial filters for a spatially homogeneous turbulent boundary layer induced wall pressure field. Rectangular-shaped transducers were constructed of lengths ranging from 0.25 to 7.3 times the displacement thickness of the boundary layer in the streamwise direction and widths of 0.25 the displacement thickness. Measurements of the power spectral levels of the wall pressure fluctuations are compared for each transducer with the predicted levels determined by the integration over the spatial extent of each transducer of the Corcos model of the wall pressure cross spectrum. [Work supported by ONR.]

Wave-vector filter analysis of turbulent flow

The Corcos model for convective turbulence is equivalent to the problem of an ensemble of plane waves passing through spatial windows with rhombic iso-amplitude contours. The sharp corners introduce spurious spatial effects analogous to transients in the time domain. The elliptic model was proposed to suppress these artifacts by smoothing the contours [R. H. Mellen, J. Acoust. Soc. Am. 88, 2891–2893 (1990)]. Comparison of the two models is extended to further illustrate some of the implications for wave-vector filter experiments.

Flow noise reduction techniques for a planar array of hydrophones.

This paper presents techniques for reducing turbulent boundary layer pressure fluctuations received by an array of hydrophones embedded within a plane layer of elastomer. Reduction of flow noise can be achieved by spatial filtering through the use of large finite hydrophones or an array of hydrophones, or by filtering high wave-number components through the use of an elastomer layer. The theoretical model used is a plane elastomer layer backed by an infinite elastic plate with finite thickness; the other side of the layer is exposed to turbulent flow. The baseline turbulent wall pressure spectrum that represents the flow excitation function is based on the Corcos model modified to fit a realistic low wave-number estimate. The occurrence of the convective ridge wave number is estimated by using a convective flow speed that is a function of frequency. Results presented are numerically calculated noise reductions, relative to the calculated noise level for a flush-mounted point hydrophone. Noise reductions that can be achieved by various techniques and their sensitivity to the flow noise model chosen, as well as the combined wave-number filtering mechanism, are presented. Some design approaches for achieving specific noise reductions using various combinations of hydrophone size, elastomer layer thickness, and array characteristics for flow over a plane surface will be discussed.

Flow noise reduction techniques for a planar array of hydrophones

This paper presents techniques for reducing turbulent boundary layer pressure fluctuations using wave-number filters. Reduction of flow noise can be achieved by filtering wave numbers using a finite hydrophone or an array of hydrophones, or by filtering wave numbers using a layer of elastomer. The baseline turbulent wall pressure spectrum that represents the flow excitation function is based on the Corcos model modified to fit a realistic estimate at the low wave-number region. The occurrence of the convective ridge wave number is estimated by using a convective flow speed that is a function of frequency. Major results presented are numerically calculated noise reductions compared to the calculated noise level for a flush-mounted point hydrophone.

On wave‐vector filter analysis of turbulent flow

The Corcos model for convective turbulence is equivalent to the problem of an ensemble of plane waves passing through spatial windows with rhombic iso‐amplitude contours. The sharp corners introduce spurious spatial effects analogous to transients in the time domain. The elliptic model was proposed to suppress these artifacts by smoothing the contours [R. H. Mellen, J. Acoust. Soc. Am. 88, 2891–2893 (1990)]. Comparison of the two models is extended to further illustrate some of the implications for wave‐vector filter experiments.

Surface roughness effects on flow noise

The Corcos wall pressure model was modified to incorporate surface roughness effects. The longitudinal cross-spectral density part of the Corcos model was modeled from rough wall data taken from the literature. The wall pressure frequency spectrum was nondimensionalized by the wall shear stress and boundary layer displacement thickness and shown to fit both smooth and rough wall data. The boundary layer parameters were obtained from a finite-difference boundary layer code developed by the author to account for surface roughness effects by the discrete roughness approach [G. H. Christoph and R. H. Pletcher, AIAA J. 21, 509–515 (1983)]. The blockage of the roughness height, shape and spacing is explicitly accounted for in this approach. Flow noise was calculated for a variety of roughness element heights and spacings in air and in water. The calculations clearly show that element spacing as well as height is important and that the viscous sublayer height, not the boundary layer thickness, determines the influence of roughness on flow noise. Thus roughness is typically more important in water than in air because of the thinner viscous sublayer. Also, the effect of transducer size and viscoelastic layer thickness are shown for rough surfaces.

Boundary-layer pressures and the Corcos model: a development to incorporate low-wavenumber constraints

This paper re-examines the theoretical arguments that indicate the structure of the pressure field induced on a flat surface by boundary-layer turbulence at low Mach number. The long-wave elements are shown to be dictated by the acoustics of the flow, and the limit of the acoustic range is the coincidence condition of grazing waves where the spectrum is singular and proportional to the logarithm of the flow scale. The surface spectrum is shown to be proportional to the square of frequency at low-enough frequency and to the square of wavenumber at those low wavenumbers with subsonic phase speed.The similarity model successfully used by Corcos for the main convective elements of the field is used in this paper to model the turbulent sources of pressure, not the pressure itself, so that a Corcos-like description of the pressure spectrum is derived that is consistent with constraints imposed by the governing equations. This results in a fairly compact specification of the pressure spectrum with yet-undetermined constants, which must be derived from experiment. Despite an extensive search of published data on the pressure field, it is concluded that existing information is an inadequate basis for setting those constants and that new free field experiments are needed. Boundary layers formed on gliders or buoyant underwater bodies offer the most promising source of such data.The paper concludes with a study of how large flush-mounted transducers discriminate against the local flow noise field and i t is shown that they do so at a rate of 9 decibels per doubling of transducer diameter. This different conclusion from Corcos’ correct 6 decibel rate for small transducers is entirely due to the low- wavenumber constraints on the spectrum, which are misrepresented in the simple similarity model. This result, which conforms with the constraints imposed by the weak compressibility of the fluid, is the same as that later suggested by Corcos for transducers that are large on the boundary-layer scale.

Low wavenumber wall pressure measurements using a rectangular membrane as a spatial filter

The response of a rectangular membrane to a convecting random pressure field is interpreted to reveal the inherent wavenumber filtering characteristics of the device. After experimental determination of its resonant response characteristics, one such membrane is used to measure the low wavenumber components of the wall pressure fluctuations beneath a plane turbulent boundary layer. The measurements were made at wavenumbers far below the convective region ( k 1 = ω/ U c ) but above the acoustic region ( k 1 ⩽ ω/ c o ). Possible contamination by acoustic and convective ridge effects is considered. The low wavenumber measurements are also compared with values of the wavenumber-frequency pressure spectral density Φ p( k,ω) obtained by Fourier transforming cross-spectral density data obtained by Blake in the same wind tunnel. From this comparison it is seen that this transformation of Blake's data (based on a Corcos model) grossly overestimates the magnitude of Φ p( k,ω) in the low-wavenumber region. The measured values of Φ p( k, c o) are about 36 dB down from convective ridge levels at the same frequency. The data are also compared with earlier results obtained by Jameson. In a similar frequency range the current data levels are approximately 10 dB higher than those of Jameson.

Acoustic radiation from plates excited by flow noise

Theoretical formulation of noise radiation from thin, rectangular, flexible plates excited by turbulent wall pressure fluctuations. An extensive parametric study including surface roughness and both plate and flow parameters is presented. From the results, presented graphically, it is possible to predict the spectrum of the total radiated power provided that the power spectral density of the wall-pressure fluctuations, the flow velocity, the plate dimensions and the plate modal damping parameters are known. Along with the parametric study, an estimate of the dependence of the noise radiation on plate boundary conditions based on a consideration of the wavenumber-frequency selectivity of the plate is presented. The parametric study is based on the assumption that the wave number/frequency spectrum of the wall-pressure fluctuations can be described by a Fourier transform of a Corcos model of the cross spectral density. The results indicate that at frequencies above the hydrodynamic-coincidence frequency the radiated power is dependent on the low-wave-number spectrum of the excitation. The dependence on the high-wave-number spectrum, however, is limited to a narrow band of frequencies below the hydrodynamic-coincidence frequency.

On the Utilization of a Flexible Bar as a Continuous Spatial Filter

In order to determine empirically the wavenumber-frequency spectra of the wall pressure fluctuations beneath a turbulent boundary layer at nonconvective wavenumbers, a spatial filter with high wavenumber resolution is necessary. Such a filter can be realized by using a continuous system such as a flexible bar or by using a discrete array of finite-sized transducers. The bar offers the advantage that high wavenumber resolution can be achieved without the use of the elaborate equipment necessary for the transducer arrays. In order to use the bar as a spatial filter all that is required is a measurement of the modal displacement response at some spatial position. In this paper, the theoretical formulation of the bar as a continuous spatial filter is developed. Calibration procedures and measurement problems such as those associated with vibrational noise are discussed. Finally, measurements of the low-wavenumber (nonconvective) spectra of the wall pressure fluctuations obtained by using the bar as a spatial filter are reported. One important result of this study is that it demonstrates that the primary wavenumbers associated with the pressure fluctuations contributing to the bar response are not the convective wavenumbers. This implies that Corcos's model, which has been empirically verified only for wavenumbers near the convective wavenumbers [D. M. Chase, J. Acoust. Soc. Amer. 46, 1350–1365 (1969)], may not be adequate for predicting panel response. The measured low-wavenumber (nonconvective) spectra indicate the limitations of Corcos' model in this region. [Work supported by grant from the National Aeronautics and Space Administration.]