Category Noise Sources in Turbulent Shear Flows

Statistical properties of the wall pressure spectrum: corre­lations and wavenumber-frequency spectra

According to the ‘weak coupling approximation’ introduced above, in the present approach we consider a boundary layer developing on an infinitely extended rigid flat plate in a low Mach number flow without mean pressure gradients. In this framework, taking into account that the boundary layer thickness increases slowly in the streamwise direction, it is possible to con­sider the pressure field statistically homogeneous on the plane of the plate and statistically stationary in time. The homogeneous plane is described by the Cartesian axes that, for the sake of clarity, are defined as x, x2, being x aligned with the free stream velocity. The frame of reference adopted is depicted in Figure 4.

 Figure 4. Frame of reference adopted to describe the statistics of pressure fluctuations.

Considering the fluctuating component of the pressure field p(x1,x2,t), the space time correlation can be written as:

Rpp(€i, b ,т) = -1 E[p(xi, x2,t)p(xi + £bx2 + 6 ,t + t )] (40)

ap

where ap is the pressure variance and the symbol E[•] denotes the expected value. When the ergodic hypothesis holds, time averages can be used. This is an important hypothesis when pointwise pressure measurements are per­formed. In this case the pressure is a function of time only and the cross­correlation is given by a much simpler expression:

Rpp(t) = -1 <p(t)p(t + t) >t (41)

ap

where the symbol < • >t now denotes the time average. Taking the Fourier transform of Eqs. 40 and 41 one obtain the wavenumber-frequency spectrum Фр(к1,к2,ш) and the frequency spectrum Фp(ш). In this notation ш is the radian frequency and k, k2 are the components of a two dimensional wavevector. By taking the frequency Fourier transform of Eq. 40 it is possible to obtain the cross-spectrum rp(£i,£2,w) that is defined in the space-frequency domain. The experimental determination not being very difficult, Гр represents a key ingredient for the theoretical models that are presented below.

In the framework of the statistical modeling, a relevant role is played by the phase velocity w/k, being k the magnitude of the wavevector, whose magnitude spans from the order of the flow speed to sonic or supersonic values.

Relevant properties of the turbulent boundary layer

A short review of the main parameters characterizing the turbulent boundary layer and used for the scaling of the wall pressure spectra is given in the following. Extensive discussions can be found in several textbooks [see e. g. Schlichting (1979)]; therefore we limit ourselves to reviewing some relevant parameters that influence the overall statistical properties of the wall pressure fluctuations field.

At the wall, the boundary layer exerts a shear stress tw, and there is a strong connection between this shearing and the behavior of the flow in the immediate vicinity of the wall. As the distance from the wall increases, the influence of the wall shear on the fluid motion diminishes and the flow properties may be described in terms of the local free stream velocity U and the thickness of the boundary layer S, this symbol denoting the so-called Blasius thickness. In this region, the flow behavior is usually called wake — like. Thus, depending upon the distance from the wall, two important flow regions can be identified. A layer close to the wall, where the velocity depends upon the fluid viscosity and the local wall shear, and an outer layer, where the velocity depends on the external properties of the flow (i. e. UTO, S and the upstream history of the layer). In the near wall region, the velocity increases linearly for increasing distance from the wall. In the outer layer the velocity defect evolves according to the well-known logarithmic law. Of course, due to the turbulent nature of the velocity field, the two regions boundaries can be defined only statistically.

In the linear region, the velocity gradient is independent of the distance from the wall. This assumption yields the following relationship:

Ui = — (23)

where the subscript 1 denote the velocity component on the streamwise (x) direction and m is the dynamic viscosity of the fluid.

In the logarithmic region the turbulence activity is the greatest and the velocity gradients are proportional to the distance from the wall. This gives rise to the logarithmic velocity profile described by the following equation:

where ln(-) is the natural logarithm of •. The quantity UT is called the friction velocity and it is defined as

being p the fluid density at ambient temperature. The coefficient k in Eq. 24 is the so-called Von KArman constant, equal to approximately 0.4 for any type of wall. B is a coefficient that depends only on the degree of surface roughness. The notation commonly used to represent the dimensionless quantities, is the following:

In Figure 3 a simplified scheme of the turbulent boundary layer is re­ported for completeness.

Throughout the major portion of the fully developed turbulent boundary layer, the mean velocity profile over both smooth and rough walls satisfies

a defect law of the following form:

where Ue is the external velocity, outside the boundary layer. The function W(y/S) has been introduced by Coles (1956) and it is given by:

It is well know that the definition of the Blasius thickness S is not suitable for turbulent boundary layers. It is better to introduce more objective definitions. Very briefly we remind the definition of displacement thickness S* based on a mass balance in the boundary layer and given by the following expression:

Of course also S* is an outer scale because its magnitude is of the order of the depth of the viscous sublayer. Typically, S* is approximately equal to a fraction of S, from 1/8 to 1/5, depending on the surface roughness and the pressure gradient. Similarly, another length scale can be defined on the basis of the momentum balance. It is called the momentum thickness в and it is given by the following expression:

The ratio of the two length scales is called the shape factor:

S*

H=в <31>

According to the laws of the wall described above, it is possible to determine explicit relationships among set of boundary layer thickness and the friction factor. We refer to more specific textbooks for the details [e. g. Schlichting (1979)].

By integrating along y, between 0 and S, the momentum balance equa­tion written on x, it is possible to determine an equation relating integral quantities characterizing the turbulent boundary layer. This relationship, often denoted as the Von Karman integral equation, reads:

Cf _ de в Ґ 2 + H dP 2 dx 2 у I pU^J dx

being Cp the static pressure coefficient.

Empirical relationships are used also to determine the inner properties of the turbulent boundary layer once the outer scales are known either exper­imentally or numerically. In this case, by the knowledge of в, it is possible to empirically determine Cf and then UT. This approach is of common use since the estimation of UT by the direct measurement or computation of tw might be very difficult in practice.

We finally remind that the velocity profile at high Reynolds numbers can be described by a power law of the following form [Schlichting (1979)]:

Ul = (У1 "

UT V 6 )

where typically n ~ 7 for smooth walls and 4 for rough walls. By considering the thickness definitions, the following relations are obtained:

6*

6

For n = 7 it is obtained 6*/6 = 1/8. Also Eqs. 37, 38 and 39 can be used for a qualitative estimation of the boundary layer integral properties.

The wall pressure statistics

The random forces resulting from pressure fluctuations in the turbulent boundary layer over structural surfaces cause vibration. This surface motion becomes a source of noise which must be considered in the design of a vehicle. Therefore, the development of methods aimed at predicting interior noise levels, pressure fluctuations, and structural loading has become important in the design for instance of commercial aircraft, payload-carrying aerospace launchers, high speed trains. As pointed out by Graham (1996), in order to take into account this aspect in the design phase, there is a need for simple models capable of enhancing our physical understanding of the noise generation process and to provide relatively simple predictive formula to be utilized in the design process.

The methods of modeling and predicting sound and vibrations from a structure subject to a random pressure load, presume that the forcing func­tion for the surface has been estimated. It can be shown [see e. g. Blake (1986) and Graham (1997)] that the excitation term is directly related to the boundary layer wavenumber-frequency spectrum that, therefore, has become the subject of many investigations. In the present discussion, we
do not enter into the details of the structural aspects, but we limit our­selves to reviewing the main features concerning the wavenumber-frequency spectrum analysis, modeling and prediction.

Prediction of the far field pressure spectrum: a novel ap­proach

In a recent paper Morino, Leotardi & Camussi (2010) proposed a novel approach for estimating the far field pressure Power Spectrum (PSD) by the knowledge of the PSD of the pressure on the boundary surface, provided that the region where the flow is rotational and/or nonlinear is adequately thin. In order to accomplish this, the PSD of the pressure at any given point (either in the field or on the boundary) is evaluated in terms of the Power Spectral Density (PSD) of the transpiration velocity over the boundary surface. This contribution is denoted as given by equivalent sources xb.

The approach briefly described therein is based upon a formulation that falls within the general class of potential-vorticity decompositions for the velocity field of the type

v = V<f + w, (6)

where w is any particular solution of the equation

Vxw = Z. (7)

with Z := Vx v denoting the vorticity field.

The decomposition given in Eq. 6 is valid for any vector field and Eq. 7 is a necessary and sufficient condition for the validity of Eq. 6. Here, we assume w to be defined so as to have

w = 0 (8)

outside of the vortical region, , which is defined as the region where the vorticity Z is not negligible.

For incompressible flows, the continuity equation reads

V – v = 0 (9)

Combining with v = V<p + w, one obtains

V2^ = a, where a = —V – w (10)

In order to complete the problem, the boundary conditions have to be con­sidered. For viscous flows, the boundary condition over SB is the no-slip condition:

The numerical formulation of the above equations can be determined both in the physical and in the Fourier domain but it is not reported here for

the sake of brevity. We just point out that, after discretization using piece­wise constant approximation and Fourier transform, the following linear relationship can be achieved:

Pf = HpB. (21)

the symbol t denotes the Fourier transform of the discretized counterpart of the pressure and the equation represents the desired relationship between the field pressure (subscript F) and the boundary pressure (subscript B).

By using classical Wiener-Khintchine relationships, the above equation can be expressed in terms of the PSD matrix Sv. Thus, using Eq. 21, we have

SPF = H* SPB HT (22)

which is the desired relationship between the PSD matrix SPF of the pressure at NV arbitrary points in the region K3W and the PSD matrix SPB of the pressure at NB points on SB.

The expression in Eq. 22 allows one to evaluate the field-pressure PSD from the boundary-pressure PSD, thereby providing a link between two sets of experimental data (PSD of field pressure and PSD of surface pressure), often considered independent.

The community noise problem

With the term ‘Community noise’ we mean the far field noise generated at the flow side of a plate moving in a still fluid. Even though very difficult, several theoretical studies have been carried out with the aim of predicting the features of the pressure field radiated by a plane turbulent boundary layer. This topic was first investigated by Curle (1955) and Powell (1960a) using Lighthill’s analogy [Lighthill (1952)]. In the following, the integral formulations underlying those original approaches are briefly reviewed along with order of magnitude considerations to establish the importance of the radiative effects.

1.1 Integral formulations

The prediction of the propagation of acoustic waves in the far-field can be attained through an acoustic analogy approach and the search for a solution of the propagation equation derived therein. The reference theory is that of Lighthill (1952) that is based on the rearrangement of the Navier- Stokes equations to form an exact, inhomogeneous, wave equation, whose

Acoustic pressure waves radiating in the

Community noise

fluctuations at the wa

Acoustic pressure waves radiating in interior of the body

boundary of the body

Figure 1. A scheme of the overall mechanisms generating sound waves from a turbulent boundary layer overflowing an elastic flat plate.

Acoustic pressure waves induced by the turbulent region (Curie s.

FWH. Howe s analogies)

Turbulence pressure fluctuations at the wall:

Statistical modeling of the

wavenumber-frequency spectra

Figure 2. A scheme of the theoretical problems faced in the present chapter.

source terms are significant in the neighborhood of vortical regions of the flow. As pointed out above, the sound is supposed to be a sufficiently small component of the whole motion that its effect on the main flow can be neglected. This hypothesis can be accepted in low Mach number (M) flows as well as in the absence of resonating systems and multiphase flows.

Detailed discussion about such an approach, elucidating also the appli­cability limits, are given in other chapters of the present book. Here we limit ourselves to recall the final form of Lighthill’s equation that can be written as:

d^_c2y2p = d2Tjj dt2 Oxidxj

where Tij represents the Lighthill stress tensor that, neglecting the viscous terms, is denoted as follows:

Tij = puiUj + (p – pc2)Sij

Here, c is the speed of sound, p and p are density and pressure perturbations, u the fluid velocity, x the spatial coordinate and t the time. This equation is valid within and without a source region. Where linear acoustics is valid, the acoustic pressure can be found from the relation p = c2p.

In the presence of solid boundaries, an integral solution of Eq. 1 is based on the introduction of a closed control surface S that may coincide with the surface of a moving body or mark a convenient interface between fluid regions of widely differing mean properties. When S coincides with the solid boundary, the solution of the equation is carried out by imposing suitable boundary conditions on it. The oldest strategy proposed to solve the propagation equation relies on the use of a proper Green’s function obtained as a solution of Eq. 1 when the source term is replaced by the impulse point source. The most general representation of this kind is due to Ffowcs Williams & Hawkings (1969), and is applicable to a control surface in arbitrary motion. This equation is obtained by deriving a wave type equation similar to that by Lighthill for a region made up of two subregions bounded by the control surface S. The region inside S contains fluid and/or solid boundaries, the region outside contains only fluid.

Without entering into the details, the integral form of the Ffowcs Williams and Hawkings equation can be obtained again making use of the free space Green’s function, leading to the outgoing wave solution. To the purpose of the present discussion, we can consider the case of a stationary control surface, leading the FWH equation to reduce to a simpler formula [see also

Howe (1998)] that was given previously by Curle (1955) and that we report in the following[23]:

As indicated by Howe (1998), Curle’s equation written for a rigid surface can be used to determine the order of magnitude of the sound generated by an acoustically compact body within a turbulent flow (e. g. a cylinder or an airfoil moving in an incompressible flow). This analysis applies also for non-compact bodies when turbulence interacts with compact structural elements, such as surface discontinuities, edges, corners.

The contribution from the quadrupole volume integral in Eq. 3 to the acoustic power П radiated in the far field, can be estimated to be

П <x v[24] M5

The quadrupole effect predicted by Eq. 4 is the same as in the absence of the body (it is the famous Lighthill’s ‘eight power’ law). On the other hand, at low M, the total power radiated by the dipole term (the first surface integral of equation 3) can be estimated to be:

thus exceeding the quadrupole power by a factor ~ 1/M[25] >> 1. The conclusion is that at low M the dipole term is largely dominant. This is the reason why surfaces with disconuities (such as sharp edges, steps, cavities) are much more noisy than smooth walls.

A different conclusion can be driven in the case of non-compact struc­tures, that is, for objects whose size is not small compared to the acoustic wavelength, as is the case of an infinite rigid plate. Curle’s approach can again be used, and the presence of the infinite surface can be taken into account by introducing image vortices [Powell (1960b)]. Powell suggests
to use a Green’s function that is basically obtained by superimposing the free-space G with its image. In this way Powell shows that the pressure exerted on a plane boundary is the result of reflections of the quadrupole generators of the flow itself. In other words it is demonstrated that the surface integral is not a true dipole source but it represents the effect of image quadrupoles. Therefore, as concluded by Howe [Howe (1998)], the apparently strong contribution from the surface pressure dipoles actually reduces to a term of quadrupole strength, thus much less efficient, at low M, in terms of radiated pressure power. In the airframe noise context, if the effect of panel vibrations is not accounted for, it is reasonable to ignore the pure quadrupole radiation from the boundary layers, in comparison with that from edges and other inhomogeinities, such as wing trailing edge, flap side-edges, undercarriage gears and cavities. This is proven even for aircraft of large dimensions. As an example, the noise from the fuselage is expected to be more than 10dB below the level of the trailing edge noise.

However [Hubbard (1991)] the far field acoustic radiation due to panel vibrations might be a significant source of airframe noise in real (full-scale) aircraft. Furthermore [Howe (1998)] the presence of roughness breaks the Powell cancellation mechanism thus leading the dipole contribution to be­come relevant.

It should be pointed out that some recent numerical experiments [Hu, Morfey & Sandham (2002), Hu, Morfey & Sandham (2003) and Shariff & Wang (2005)] have focused on the role of the wall shear stress, rather than pressure, as sound source. They have shown that unsteady shear stresses can be an efficient sound source of dipole type that can be dominant at low Mach numbers and at very low frequencies.

We refer the reader to classical textbooks [such as Howe (1998)] and to the notes of the other authors included in this book, for further details on the integral approaches.

Roberto Camussi, and Alessandro Di Marco

Universita Roma Tre,

Dipartimento di Ingegneria Meccanica e Industriale,

Via della Vasca Navale 79, 00146, Roma, Italy

Abstract Boundary layer noise concerns the generation of acous­tic waves as an effect of the interaction of a fluid with a moving surface. Several issues are related to the noise generation mecha­nisms in such a configuration. In the present description we focalize mainly onto the case of an infinite flat plate and two main distinct situations are considered. The first one deals with the prediction of the far field noise as accomplished from the classical integral the­ories, and the main formulations, including Curle’s approach, are briefly reviewed. A novel approach based on the computation of the surface transpiration velocity is also presented. The second aspect concerns the interior noise problem and it is treated from the view point of the fluid dynamic effects rather than from that of the struc­tural dynamics. Attention is focused on the statistical properties of the wall pressure fluctuations and a review of the most effective theoretical models predicting statistical quantities is given. The discussion is completed by a short review of the pressure behavior in realistic situations, including the separated boundary layers in incompressible and compressible conditions and the effect of shock waves at transonic Mach numbers.

1 Introduction

Aerodynamic noise from a turbulent boundary layer is a fundamental topic in flow-induced noise and is of interest for both fundamental studies and applied research. The action of the pressure fluctuations indeed provides the driving force to excite surface vibrations and produce acoustic radia­tion. Many engineering problems are connected with this topic. Fatigue loading on panels of an aircraft fuselage and the vibrational generation of acoustic radiation into an aircraft cabin enclosed by the boundary surface,

R. Camussi (Ed.), Noise Sources in Turbulent Shear Flows: Fundamentals and Applications, CISM International Centre for Mechanical Sciences, DOI 10.1007/978-3-7091-1458-2_6,

are two examples among many. Generally speaking, in high speed trans­port technology, the understanding of the physical mechanisms underlying the generation of pressure fluctuations at the wall, has received increasing attention in view of the use of lightweight and flexible structures. In the field of aerospace launch vehicles design, this problem is of great relevance since vibrations induced in the interior can cause costly damages to the payload while panel vibrations of the external surface must be avoided to prevent fatigue problems and structural damages. In the context of ma­rine transportation, this topic has become quite important in the case e. g. of high-speed ships for passenger transportation where requirements of on board comfort have to be satisfied. This concern has become of great im­portance for ground transportation as well, notably for high speed trains design. In this case, the effect of pressure fluctuations induced by flow sep­arations (e. g. due to the pantograph cavity) becomes the dominant noise producing mechanism, this situation being of relevance in the automotive industry in general, since large flow separations are unavoidable on cars.

The vibration of a panel induced by a random pressure load leads to acoustic radiation into the flow as well. Also this problem is of relevance for many engineering applications including, for example, the generation of noise from piping systems or the transmission of pressure waves by under­water vehicles, the so-called acoustic-signature.

Due to its importance, since the early 1960s, researchers have been study­ing this subject using different approaches including experimental investi­gations, numerical simulations and theoretical speculations.

When a solid surface is overflown by a turbulent boundary layer, several relevant mechanisms contributing to the generation of sound waves, can be identified. To simplify the description, consider the case of a panel subject to a flow on one side. The pressure field on the surface flow side consists of the sum of the turbulence pressures which would be observed on a rigid wall and the acoustic pressures which would be generated by the plane motion in the absence of turbulence. At a first approximation, these two effects can be studied separately. This idea represents the so-called weak coupling approximation and can be derived from an acoustic analogy analysis of the problem [see e. g. Dowling (1983) and Howe (1992)]. The hypothesis that the basic turbulence structure is unaffected by the acoustic motions is indeed the basis of the acoustic analogies and can be accepted if the acoustic velocities are small in comparison with the turbulence velocities. This position, even though not always satisfied, has become accepted as a standard method even at supersonic flow speed [see also Graham (1997)]. The main reason for this is that fully coupled computations are, at present, prohibitive for any length scale of practical relevance even with the most powerful computer resources. On the other hand, the engineering design still nowadays requires simple models which allows fast understanding and rapid computations.

In view of such considerations, in the following discussions the wall can be considered as a rigid plate and the panel vibrations considered apart.

The problem of the boundary layer noise can then be treated consider­ing two different aspects, namely the so-called community noise and the interior noise. The first term applies to the effect of the acoustic waves generated by the wall turbulence and evolving in the far field from the flow side of the surface. The second one pertains with the transmission of noise at the side of the surface in still air. In both cases, the attempt to predict the noise emission is based on the correct representation, in a statistical sense, of the random load acting on the surface. For this reason, most of the discussions that follow are concerned with the clarification of the prop­erties of the wall pressure field and the predictability of its main statistical properties. In Figure 1 an overall view of the mechanisms generating sound waves including the definitions adopted therein is reported. Figure 2 evi­dences the topics faced in the present chapter. The problem related to the interior noise is treated in more details in the second part of this chapter where the theoretical background regarding the noise transmission trough solid structures is presented.

Concluding Remarks

The analytical approach declined in the chapter for various broadband – noise mechanisms can be understood as a post-processing technique to be associated with some flow description. It is found reliable for thin and mod­erately cambered airfoils provided that aerodynamic input data are avail­able. The latter appear as the crucial point of the method. When they are provided by unsteady flow computations, the models can be applied as an alternative till full numerical methods including the prediction of the sound field can be applied at reasonable cost. The analytical method also suffers

 Figure 25. Measured sound spectra of the flat plate at the Reynolds number of 4,000 at 0° and 5° angles of attack, versus vortex-shedding sound predictions. From Roger & Moreau (2010).

from limitations inherent to the necessary assumptions. The limitations are the best motivation for further developments and improvements. Among other topics, connections between the wall-pressure statistics and the mean – flow parameters in the boundary layers with adverse pressure gradients still need being elucidated for trailing-edge noise predictions. For turbulence – impingement noise, the effect of departures from homogeneity are not fully understood either. Finally, for all investigated mechanisms, including three­dimensional effects related to blade geometry in turbomachines is required to accurately cover engineering applications.

Vortex-Shedding Noise Results

Vortex-shedding sound is a less documented topic. Indicative results and modeling attempts are mentioned in this section, for a flat plate with rectangular trailing-edge. Complementary elements are found in the book by Blake (1986). Typical far-field sound spectra are first plotted in Fig. 24 for an inclined flat plate tested in an open-jet anechoic wind tunnel. Because the flow is deflected by the mean lift acting on the plate, precise values of the geometrical angle of attack are indicative. Only the qualitative features of the flow and the related acoustic signature make sense, described here for increasing values of the angle from ai = 0 to a maximum value a4. Even higher values would correspond to stall. The relative thickness of the plate is 3%. At zero angle of attack, a large peak is heard at the expected Strouhal number of 0.2 built on the plate thickness h. As the angle increases that peak reduces whereas a wide hump grows at lower frequencies. The higher the angle of attack, the lower and wider the associated frequency range. The bump is attributed to trailing-edge noise; indeed the flow separates at the leading edge and reattaches on the suction side of the plate, triggering a turbulent boundary layer, similar to the case of the CD airfoil, Fig. 22-c. The drop of the peak is caused by a loss of coherence and a progressive deactivation of the vortex shedding by increasing upstream disturbances. The frequency shift is attributed to different local flow accelerations at dif­ferent angles of attack. It is concluded that both vortex-shedding sound and trailing-edge noise can coexist if the frequency range of the latter does not cover the one of the former. Yet competition obviously takes place.

The same plate of 3% thickness tested on a different nozzle produces the results of Fig. 25, taken from Chang et al (2006) and Roger & Moreau (2010) for illustrating the accompanying prediction methodology. In the first configuration at zero angle of attack, vortex shedding is well structured and not significantly influenced by disturbances developing upstream in the boundary layers. The associated sound can be predicted with eq. (19). The needed upwash spectrum is provided by the model calibrated in Fig. 17. The agreement is perfect in this case, dealing with the narrow-band signature only.

The procedure has been repeated for the geometrical angle of attack of 5°. In this case vortex shedding is disturbed by a higher turbulence level in the boundary layers and again the emergence of the peak is reduced as already emphasized in Fig. 24. Because of computational cost issues the nu­merical simulations ignored the flow-deflection effect in the experiment and some uncertainty results in the definition of the input data. An empirical spectral shape has been again defined anyway and the effect of the uncer­tainty assessed by the upper and lower estimates of the acoustic signature

in Fig. 25. For completeness, the trailing-edge noise prediction achieved with the model of section 5.2 fed with measured wall-pressure statistics is reported, again showing a satisfactory agreement.

The flat-plate test-case illustrates a hybrid application in which input data are partly measured and partly provided by Computational Fluid Dy­namics. The good overall agreement achieved here is in favor of the physical consistency of the reversed Sears’ problem. However the practical applica­tion is made tricky by the non-homogeneity of the velocity in the near wake.

Trailing-Edge Noise Results

Trailing-edge noise results for an industrial Controlled-Diffusion (CD) airfoil tested at low-Mach number are presented and discussed in this section as illustration of the methodology. The experimental protocol is the same as for turbulence-impingement noise. The airfoil is instrumented by clustered remote-microphone probes so that the wall-pressure statistics (Фрр and £y) close to the trailing edge is measured directly (Roger & Moreau (2004)). This provides information on the sound sources. The far-field sound is also measured in the mid-span plane. Both wall-pressure spectra and acoustic spectra are compared in Fig. 22. Different flow regimes are obtained by setting different geometrical angles of attack, all with attached and stable laminar boundary layers on the pressure side. The acoustic spectra are artificially shifted up in order to make the comparison of spectral envelopes easier.

In case (a) a laminar unstable boundary layer develops on the suction side, with the expected Tollmien-Schlichting (TS) instability waves. The main bump and its distortion harmonics feature the range of unstable fre­quencies. Because the unsteady motion is coherent, both source and sound spectra exhibit additional tones resulting from acoustic back-reaction. Case (b) corresponds to a turbulent boundary layer triggered by a small leading – edge separation bubble. This regime is characterized by quite high fluctu­ating levels at low frequencies. In case (c) the flow remains attached but the boundary layer rapidly grows upstream of the trailing edge due to the formation of large vortical patterns in the aft part of the airfoil. This regime is referred to as ’distributed vortex shedding’ by Roger & Moreau (2004). In each case the overall similarity of source and sound spectra confirms the cause-to-effect relationship. The level differences illustrate the ratio Spp/^pp and emphasize the variations of radiation efficiency with flow regime and/or frequency, essentially attributed to the span-wise correlations length £y. TS waves are the most efficient in the frequency range of the bump, because they are associated with quite large values of £y. Case (b) has the minimum efficiency, nearly frequency-independent. In contrast case (c) is more effi­cient below 1 kHz, because the span-wise coherence takes quite high values as reported in the reference paper.

Sound predictions in dimensional variables achieved by the analytical model of section 5.2 taking the measured wall-pressure statistics as input

Figure 22. Compared wall-pressure (thick lines) and far – field sound (thin dotted) spectra for a CD airfoil at different flow regimes. Sound measured in the mid-span plane, nor­mal to chord length. (a) airfoil cross-section. (b) Tollmien – Schlichting waves (laminar boundary layer). (c) leading – edge separation. (d) distributed vortex shedding. Acoustic spectra scaled by R2/(Lc) and shifted up by 30 dB. Fre­quencies below the vertical dashed lines are not reliable because of background-noise issues.

data are compared with measurements in Fig. 23, for the same three afore­mentioned regimes (Moreau & Roger (2009)). The upper plot refers to the onset of TS waves. The tones of Fig. 22-b resulting from amplifica­tion by acoustic feedback have been removed from the spectrum because the statistical model only addresses direct sound production from random disturbances. The remaining hump-like part corresponding to the primary

 Figure 23. Predicted versus measured trailing-edge noise spectra of a thin cambered airfoil, from Moreau & Roger (2009). The three sets of data are vertically shifted from each other by 10 dB for clarity.

TS wave radiation with no feedback is reproduced accurately because the span-wise coherence can be measured over an extended frequency range. The other plots are shifted by -10 dB (case 1, Fig. 22-c) and -20 dB (case 2, Fig. 22-d) for clarity. In these cases the much weaker span-wise coher­ence at high frequencies is not accessible. Therefore it is as far as possible deduced from the measurements and continued by fitted theoretical trends; Corcos’ model is used in case 1 and an ad hoc model proposed by Roger & Moreau (2004) is used in case 2. Bumps in the predicted spectra around 3 kHz and 5.5 kHz as well as dips around 2 kHz and 4.5 kHz are attributed to chord-wise non compactness (airfoil chord 13 cm). They do not clearly appear or they are shifted in the measurements, possibly because of camber effects and additional sound scattering by the nozzle lips in the experiment. Anyway the good overall agreement reported in Fig. 23 shows that the pre­dictions are reliable when fed with directly measured input data. Similar results for a flat plate are found in next section.

Strouhal-Number Versus Helmholtz-Number Scaling

For self-similar flows the properties of which do not essentially vary with the Reynolds number, the amplitude and the frequencies of velocity fluctuations are both proportional to the mean flow speed. The spectral shape of the far-field sound extends wider for higher speeds, with increased levels. The flow statistics involved in the forcing source terms is expected to scale according to the Strouhal number St = fc/U0. As a result, plotting the reduced PSD Spp U0 (n 1) as a function of the Strouhal number and ig­noring non-compactness interferences must produce a perfect collapse of the curves, if the overall (frequency-integrated) acoustic intensity scales like Un. This property should be used systematically to scale broadband noise data in aeroacoustics. In the same time higher frequencies triggered by higher flow speeds make the airfoil chord less compact, and departure from the self­similarity is expected in the sound signature from the onset of interference fringes. Apart from this, any geometrical environment of a flow is character­ized by resonant frequencies which do not essentially depend on flow speed. It can be guessed that resonance and interference features, in particular related to chord-wise non-compactness, rather depend on the Helmholtz number He = kc. Indeed plotting the PSD as a function of frequency or

Helmholtz number preserves the non-compactness dips and humps at the same places but cannot produce a collapse over the entire frequency range. Plotting as a function of the Strouhal number provides a better overall col­lapse except that the dips are now at different Strouhal numbers for different speeds. Yet the Strouhal-number scaling is more physically consistent when tracking scaling laws of sound-generating mechanisms.

 Figure 21. Helmholtz-number scaling of turbulence-airfoil interaction noise. Symbols stand for the NACA-0012 of Pa­terson & Amiet (1976) and thick lines for the thin cambered airfoil tested at ECL. Grid-generated turbulence, observer at 90° in the mid-span plane.

The sample results collected in Fig. 21 are taken again from Paterson & Amiet (1976) and the ECL data for a thin cambered airfoil (Figs. 18 and 19). A good collapse is achieved by plotting Spp/U0 as a function of the Helmholtz number kc, if the plot is aimed at emphasizing the humps and the dips attributed to non-compactness. Yet unacceptable scatter is found at lower frequencies. The first dip occurs quite close to the value kc = 2 n for which the acoustic wavelength is equal to the chord length, which is somewhat expected for an observer at 90° . The slight frequency shift of the dip between both sets of data is attributed to the different geometrical design. The sound levels also differ because of different experimental con­ditions and because of the thickness effect of the NACA-0012. This simple example illustrates that aerodynamic noise in the presence of solid surfaces combines intricate flow and geometry effects. The duality He-versus-St has been identified in many applications, for instance by Neise & Barsikow (1982) who discussed the acoustic similarity laws of low-speed fans.