The Link Between Wall Pressure Spectra and Radiated Sound from

AIAA 2010-3904

16th AIAA/CEAS Aeroacoustics Conference

16th AIAA/CEAS Aeroacoustics Conference, 7-9 June, 2010, Stockholm, Sweeden

The link between wall pressure spectra and radiated sound from turbulent boundary layers Xavier Gloerfelt∗ Arts et M´etiers ParisTech, DynFluid Laboratory, 75013 Paris, France

Aerodynamic noise from turbulent boundary layers (TBL), also known as flow noise, is a fundamental topic in flow-induced noise. The large range of turbulent structures of the boundary layer provides the hydrodynamic energy which is converted into acoustic waves either directly (quadrupole noise) or indirectly (diffraction of evanescent waves by a singularity or structural response of the surface). A significant amount of computational resources is needed to resolve wall turbulence, and becomes prohibitive in most engineering wall bounded flows. That is why statistical models for the noise radiation are still desirable. In the present study, two numerical databases for a flat plate TBL (AIAA paper 200934011 ) and for a high-Reynolds number flow over a NACA0012 airfoil (AIAA paper 200932012 ) are used to characterize wall pressure fluctuations. The simulations are direct noise computations (DNC) where both the aerodynamic and the acoustic fields are resolved, so that they constitute idealized experiences that serve as reference for the evaluation of various statistical methods. Various formulations of the acoustic analogy are tested for the flat plate TBL, and the wall pressure is analysed in the wavenumber-frequency space in both configurations, since this the basis of wall pressure models. Estimates of the far-field radiation from the wall spectra are then discussed.

I.

Introduction

The wall pressure beneath a turbulent boundary layer (TBL) constitutes a fingerprint of the vortical structures evolving in the TBL. The knowledge of the wall pressure characteristics gives some insight into the features and dynamics of the flow. It also determines the structural response of a surface submitted to this turbulent buffeting, and is consequently the principal entry to evaluate the indirect sound radiation induced by the vibrations of the surface. The wall pressure is also directly linked with the direct acoustic emissions by virtue of the Lighthill-Curle theory of aerodynamic noise.3, 4 The direct radiation in the absence of scatterer is referred to as flow noise, i.e. the self-noise generated by wall turbulence. This purely aerodynamic sources have an increasing contribution to internal noise in terrestrial or aeronautic vehicles. With the recent progresses in reducing the other sources, such as road or motor noises for a car, or jet and fan noises for a plane, cabin noise due to the TBL has become a matter of concern for manufacturers, in order to improve the comfort of passengers during cruising trip. One difficulty in flow noise predictions is that the sources are very weak, but are efficiently transmitted, so that the direct contribution is not necessarily negligible with respect to the indirect vibroacoustic contribution. The measure of the direct acoustic radiation is thus hardly achievable,5, 6 and most of the knowledge relies on the acoustic analogy (see the reviews by Blake7 and Dowling8 ). A major contribution is given by Powell,9 who showed that the surface integrals of Curle’s formulation yield essentially quadripolar sources when a semi-infinite plane is present. A quantitative evaluation of flow noise levels is nevertheless lacking. The most important mechanism for enhancing the sound from convected wall turbulence is the introduction of scatterers. For instance, if the surface is terminated with a trailing edge, radiation away from the body becomes dipole for a rigid surface. Most of the models of flow noise relies on expressions on the wall pressure frequency-wavenumber spectra. Since the pressure is a non-localized variable, the wall pressure provides a fingerprint of the TBL structures ∗ Assistant

Professor, [email protected].

1 of 30 American Institute of Aeronautics and Astronautics Paper 2010-2904 Copyright © 2010 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

(a) (b) Figure 1. Instantaneous snapshots of the fluctuating pressure (top row) and of the norm of the vorticity

(bottom row) for the two studied configurations: (a) flat plate turbulent boundary layer at M∞ = 0.5, 300 < Reθ < 2000, (b) flow around a NACA0012 airfoil at M∞ = 0.2, Rec = 2.32 × 106 , α = 2.5o .

(reducing by one the dimension of the problem). Moreover spectra are used since the TBL pressure fluctuations can be considered statistically stationary in time and locally homogeneous in space. The pressure fluctuations beneath a TBL have been the subject of numerous experimental and theoretical studies,10–15 and can give insight into the turbulent motions responsible for the noise emissions. In the low Mach number limit, the acoustic far-field can be deduced from the knowledge of the wall spectra by making use of the stationary phase method to select the wavenumbers contributing to the noise spectrum.16 In the same spirit, the diffraction theory of Chandiramani-Chase17–21 is based on the knowledge of the wall pressure over the airfoil, in order to predict the radiated sound. That is why experimental22–24 and numerical25, 26 investigations of trailing edge noise often provide an analysis of the wall pressure. Its spectral content, notably the hydrodynamic evanescent waves, provides the energy with is converted into acoustics at the trailing edge. The aim of the present study is to link the wall pressure over rigid surfaces to the direct noise radiation by performing numerical experiments. Firstly, an archetype of flow noise is considered, namely a TBL over an infinite flat plate, where the sound is caused by a combination of quadripolar sources and surface shear stresses, the latter arising from the viscous interactions at the wall. The noise is directly related to the flow fluctuations having a supersonic wavenumber, which produce oscillating solutions that propagates to the far-field. Secondly, we discuss the sound radiated because the turbulent structures past a sharp edge, referred to as trailing edge noise (TEN). In this case, a second more efficient source is added to the direct emissions: the trailing edge behaves as a scatterer that can convert the evanescent waves of the energetic hydrodynamic part of the pressure into propagating modes. This mechanism provides a dipolar sources that dominates the quadrupole system at low speeds. The first part of the paper summarizes some results for the TBL case. Acoustic analogies in the temporal or frequency space are used to investigate the contributions of the different source terms, and to test some physical or numerical hypotheses. The wall pressure is then analysed, and will serve to apply various models in the wavenumber-frequency space for the noise emissions. In a second part, the wall pressure for the flow over an airfoil is studied. It constitutes the principal entry for diffraction models, such as ChandiramaniChase,17–21 or Amiet27, 28 formulations.

2 of 30 American Institute of Aeronautics and Astronautics Paper 2010-2904

II.

Boundary layer noise

The first configuration is the spatial development of an equilibrium three-dimensional turbulent boundary layer (TBL), calculated by means of compressible large eddy simulation (LES). The free-stream Mach number is taken to be M∞ = 0.5, and the Reynolds number based on the momentum thickness ranges from 300 to 2000. A.

Large eddy simulation of a turbulent boundary layer

The direct noise computation (DNC) relies on LES of the compressible Navier-Stokes equations. The effect of the non-resolved subgrid-scales are taken into account by the regularization effect of an explicit selective filter. The numerical algorithm is detailed in Ref.1, 29 High-order low dissipative and low dispersive methods are employed with non reflecting boundary conditions, to allow the exit of acoustic waves without spurious reflections. A large sponge zone, combining grid stretching and a Laplacian filter, has been added to dissipate the turbulent structures before they hint the outflow boundary. This reduces considerably the levels of lowfrequency oscillations generated by the exit of vortical packets. Another tricky point is the turbulence seeding at the inlet plane. The similation of the complete transition would require an expensive lengthening of the computational domain. A solution consists in superimposing synthetic fluctuations on a mean TBL profile. This method has been seen to produce a level of spurious noise that can mask the direct radiation from the TBL.30 We thus decided to bypass transition by introducing a small step upstream of the computational domain. The step lies in the spanwise direction and is located between x1 /δ0 = 5.65 and x2 /δ0 = 8.95, and 0 ≤ x2 /δ0 ≤ 0.26, where δ0 is the thickness of the incoming laminar boubdary layer. This step has a large width/height aspect ratio and is hence expected to have few impact on the global mean flow while providing a discontinuity igniting the transition to turbulence.29 The small step will induce some spurious noise, since a rough wall is known to generate higher noise levels than a smooth wall.31 Nevertheless, the trick used to ignite transition toward a fully turbulent step has the great advantage to be steady, and is thus more silent than the introduction of unsteady perturbations.29 The mesh size is taken to be uniform in the spanwise direction. In a similar manner, the mesh in the streamwise direction is uniform but grid stretching, with a ratio of 1.02, is used in the sponge zone over the last 100 points. In the wall normal coordinate, the grid size is stretched using a geometric progression of 2.5%. The friction velocity uτ is calculated when the flow is fully turbulent. This reference velocity allows in particular to express simulation parameters in wall units, e.g. u¯+ = u ¯/uτ or x+ i = uτ xi /ν. Using the value at the middle of the computational domain uτ = 7.38 m/s at x1 /δ0 =143.5, the mesh sizes correspond to + + 6 ∆x+ 1 =37, ∆x2 =0.98 and ∆x3 =14.7. The total number of grid points is 1372 × 300 × 131 ∼ 54 × 10 . The −9 time step ∆t ≃ 5.8 × 10 s corresponds to a Courant-Friedrichs-Levy number CFL = c∞ ∆t/∆x2 equals to 1. The computation is initiated with a Blasius’ boundary layer with δ0 = 3.18 × 10−4 m, soit Re0θ = 480. The domain of interest, excluding the sponge zone, has dimensions 287δ0 × 400δ0 × 12.2δ0 . The freestream air is characterized by a Mach number of 0.5, a static pressure of p∞ =101300 Pa, and a temperature T∞ =298.15 K. The spatial development of the equilibrium TBL has been validated in a previous paper.1 The transient stabilisation of the computation lasted 400 000 iterations and the statistics were evaluated for the 300 000 following iterations. Now 1 000 000 iterations have been realized, doubling the useful database. Moreover, during these 300 000 new iterations, the primitive flow variables have been recorded every 30 iterations on the volume [29.4 61.6]δref × [0 4.52]δref × [−1.72 1.72]δref, where δref =1.13 mm is the boundary layer thickness at the middle of the computational domain, used as a reference value throughout the study. This database will notably be used to apply the acoustic analogy formulations and represents an amount of roughly 4 To. As shown in Fig.2, the mean flow data and the turbulent intensities compare favorably to the DNS reference data.32, 33 In addition, the noise generated by the turbulent flow has been captured in the same run (see Fig.1(a)). The pressure fluctuations p′ are plotted in the central plane (x1 , x2 , x3 = 0) as a colorscale (with levels between -5 and 5 Pa). The weak acoustic wavefronts (± 4 Pa) have a relatively large wavelength. They are mainly oriented towards the upstream direction, which is reinforced by the Doppler effect from the outer flow at M∞ =0.5. An animation could clearly demonstrates that these waves are not emanating from the exit region (located three times farther than the limit plotted in Fig.1(a)). The wave pattern is highly correlated with the lobes of the hydrodynamic pressure presenting a large wavelength which can be associated with large scale arrangements, such as the hairpin packets.34 On this picture, a high-frequency source originates from the small step located at the inflow of the computational domain to bypass the laminar-turbulent transition.


3

20

2.5 u+,v+,w+

25

u

+

15 10

2 1.5 1 0.5

5 1 0 10

1

10

2

10

(a)

3

y+

0 0

4

10

10

0.2

(b)

0.4

0.6 y/δ

0.8

1

1.2

Figure 2. Mean profiles from the compressible LES. (a) Profiles of the mean longitudinal velocity in wall

), and x1 /δref =41.5 (Reθ =1551) coordinates for two longitudinal locations x1 /δref =73 (Reθ =2229) ( ( ). compared with Spalart’s DNS at Reθ = 141032 (◦), and Jimenez’s DNS at Reθ =155133 (◦). (b) Turbulent intensities at x1 /δref =73: urms /uτ ( present LES; ◦ Spalart; ◦ Jimenez), vrms /uτ ( present LES; + Spalart; + Jimenez), wrms /uτ ( present LES; Spalart; Jimenez).

This was expected since, as previously noted, a rough wall is known to generate higher noise levels than a smooth wall.31, 35 Nevertheless, this unwanted source has a relatively regular frequency and weak levels, so that it can be ignored in the following analysis. B. 1.

Direct acoustic radiation Theoretical background

Powell’s reflection principle Powell9 has proposed a new formulation of the Lighthill-Curle3, 4 acoustic analogy that shed some light on the nature of the sources adjacent to an infinite plane. The derivation is briefly recalled since this is one of the basis of boundary layer noise. f0 O

Σ1

y S

Σ0

Σ1

n’

V’0

Figure 3. Illustration of Powell’s reflection principle.9

The Ffowcs Williams and Hawkings (FWH) formulation36 for a steady permeable surface is: ZZ ZZZ ZZ [Tij ]τ ∗ ∂2 1 ∂ρun ∂ 1 ′ 4πp (x, t) = dΣ(y) + dy − [ρui un − Pij nj ]τ ∗ dΣ(y) ∂xi ∂xj r r ∂τ τ ∗ ∂xi r Σ0 V0 Σ0 ZZ ZZ ∂un ∂ 1 1 ρ∞ dΣ(y) − [pni ]τ ∗ dΣ(y) (1) − r ∂τ τ ∗ ∂xi r Σ1

Σ1


with Pij = pδij − τij . The subscript τ ∗ indicates that source terms are evaluated at the retarded time t − r/c∞ , where r = |x − y| is the distance between the source and the listener point. For the image sources closed by the surface Σ′0 + Σ′1 + Σ′2 , the left hand side is zero since the fictitious source is inside the surface (H(f )=0): ZZ ZZZ ′ ZZ Tij τ ∗ ′ 1 ∂ρ′ u′n 1 ′ ′ ′ ∂ ∂2 ′ ′ dΣ (y ) + dy − ρ ui un − Pij′ n′j τ ∗ dΣ′ (y′ ) 0= ′ ′ ′ ∂xi ∂xj r r ∂τ τ ∗ ∂xi r Σ′0

V0′

−

ZZ Σ′1

1 ′ ∂un ρ r′ ∞ ∂τ

dΣ′ (y′ ) −

τ∗

Σ′0

∂ ∂xi

ZZ Σ′1

1 ′ ′ [p ni ]τ ∗ dΣ′ (y′ ) r′

(2)

For a horizontal plane defined by x2 =0, the unit normal is ni = (n1 , n2 , n3 ) = (0, 1, 0), and ρui un = (ρu2n , ρus un ) for n = 2, and s = 1, 3. We add (1) and (2), noting that on the boundary, we get n = −n′ , ui = u′i , and p = p′ : ZZ ZZ ZZZ [Tij ]τ ∗ ∂un 2 2 ∂ρun ∂2 ρ∞ dΣ(y) − dΣ(y) dy − 4πp′ (x, t) = ∂xi ∂xj r r ∂τ τ ∗ r ∂τ τ ∗ V0 +V0′

∂ + ∂xs

ZZ

Σ1

Σ0

2 [ρus un + τsj nj ]τ ∗ dΣ(y) r

(3)

Σ0

where us denotes the fluctuations tangential to the plane (s = 1, 3). Assuming now a rigid surface Σ0 + Σ1 , we have: ZZZ ZZ [Tij ]τ ∗ ∂ 2 ∂2 ′ dy + [τsj nj ]τ ∗ dΣ(y) (4) 4πp (x, t) = ∂xi ∂xj r ∂xs r V0 +V0′

Σ0

If the viscous terms are neglected (see discussion below), the Powell reflection principle is obtained:9 ”The pressure dipole distribution on a plane, infinite and rigid surface accounts for the reflection in that surface of the volume distribution of acoustic quadrupole generators of a contiguous inviscid fluid flow, and for nothing more, when these distributions are determined in accordance with Lighthill’s concept of aerodynamic noise generation and its natural extension.” ZZZ [Tij ]τ ∗ ∂2 4πp′ (x, t) ≃ dy (5) ∂xi ∂xj r V0 +V0′

This formulation can also be retrieved by using an exact Green’s function for the half-plane in Green’s formula.8 The exact Green function simply uses the image source at −y2 to ensure ∂G/∂n=0 at the wall. The formulation (5) has interesting corollaries concerning the nature of the source terms in the acoustic analogy. First, the multipole hierarchy of source terms is only justified for compact sources, often encountered in low speed flows. Powell’s principle shows how cancellations near a boundary can convert the dipolar term of Curle’s formulation into a quadrupolar system, which has a weaker radiation efficiency. The acoustic conversion is indeed expected to vary as M 5 , as for free turbulence, whereas compact surfaces yield a M 3 scaling law. Since the surface is assumed rigid, surface terms are only equivalent sources, taking into account the reflection or the diffraction by the surface.37 Role of viscous shear stresses The role of surface shear stresses has long been an area of controversy. A term implying the tangential viscous stresses, τsj , remains in Eq.(5): ZZ ZZ 1 ∂ τ12 τ32 1 ∂ dΣ + dΣ 2π ∂x1 r 2π ∂x3 r Σ

Σ

Its dipolar nature has led some authors, as Vecchio38 or Landahl,39 postulate that this term is dominant for low speed flows. On the contrary, numerous authors40, 41 have neglected the viscous contribution in 5 of 30 American Institute of Aeronautics and Astronautics Paper 2010-2904

the discussion of Powell’s formula. For instance Howe42 suggests that the viscous terms have essentially a dissipative role (see the elegant model of Howe42, 43 ). They can distort the long wavelengthes, and should be included in the propagation operator. On the contrary, the numerical studies of Shariff and Wang,44 or Hu, Morfey and Sandham45, 46 indicate that the dipole wall shear-stress can be a true source of sound as the Mach number approaches zero. The latter contribution is shown to take over at low frequencies at a Mach number of order M=0.1.46 A similar conclusion is drawn in the numerical study of Wang, Lele & Moin47 for transition noise at M=0.02: ”the surface sound due to viscous stress fluctuations dominates the far field for a low-Mach-number flow, in agreement with Landahl39 ”. Haj-Hariri and Akylas48 provide theoretical estimates showing that the viscous dipoles are weak in a large Reynolds number flow and cannot dominate the inviscid quadrupole contribution. They showed that only viscous fluctuations having wavenumbers kδ lower than M1/2 /Re1/2 can contribute. For M=0.2 and Re∼ 104 -107 , M1/2 /Re1/2 ∼ 10−3 -3×10−5 , corresponding to exceedingly large wavelengthes of 103 -105 times the boundary layer thickness. In our case, M=0.5 and Re=U∞ δref /ν ≃ 13 000, so that M1/2 /Re1/2 ∼ 6×10−3; and the wavelength would have to be approximately five hundred δ! One of the aim of the application of the acoustic analogy to the TBL database in the following is thus to estimate quantitavely the contribution of viscous shear stresses for relatively high freestream velocity (M=0.5), and a low Reynolds number. Propagation effects By neglecting the slow thickening of the TBL in the flow direction, the effect of the mean flow on the acoustic propagation is well-described by the Pridmore-Brown operator49, 50 for a simple shear flow with mean shear dU1 /dx2 . Suzuki and Lele51 have described the different interactions for a source embedded in a boundary layer mean flow profile, by deriving low- and high-frequency approximations of the Green function for the Pridmore-Brown operator. A shadow zone is formed upstream of the source because of the refraction effects, and in the high-frequency limit, the refraction of acoustic rays lead channeled waves inside the boundary layer downstream of the source. A scattering effect can induce a diffusion of the noise emissions. An analysis in the wavenumber-frequency domain allow to relate directly the acoustic radiation to the source fluctuations having supersonic wavenumber components, in the absence of any scatterer. A wavenumber is said supersonic if it is lower than the acoustic wavenumber κ0 = ω/c∞ p, and has thus a phase speed greater than the sound speed. The range of supersonic wavenumber k = k12 + k32 < ω/c∞ defines the acoustic domain, bounded by the circle k = κ0 in the wavenumber space. When a mean flow, defined by a Mach√number M, is present, the supersonic wavenumbers correspond to (κ0 − k1 M)2 − (k12 + k32 ) > 0. Noting β = 1 − M 2 the Prandtl-Glauert parameter, the acoustic domain is represented in the wavenumber space by the interior of the ellipse: 2 k1 + κ0 M/β 2 k32 + (6) 2 2 =1 (κ0 /β 2 ) (κ0 /β) with center at (−κ0 M/β 2 , 0), major radius κ0 /β 2 , and minor radius κ0 /β, plotted in Fig.4. k3 κ0 β2

−

κ0 1+M

κ0 1−M O κ0 M − 2 β

k1 κ0 β

Figure 4. Acoustic domain when a mean flow is present.


2.

Applications of acoustic analogies

Since the estimates of the wall pressure spectra beneath a TBL are generally given in the frequencywavenumber space, we will try to confront the results of an acoustic analogy formulated in the frequencywavenumber space against the results of the DNC. To work in that space, the TBL flow has to be considered homogeneous in the two spatial directions parallel to the plate, and statistically stationary. Since these assumptions are only approximative for a simulation over a limited domain and for a rather short finite duration, the numerical applications in frequency-wavenumber space are particularly tedious. To clearly identify the effects of the different hypotheses, we decided to use two supplementary steps by applying the analogy first in the time domain, and then in the frequency domain. The different formulations written for a convected wave operator are presented before. Convected acoustic analogy in the time domain A uniform mean flow U∞ in the x1 -direction is considered in the acoustic region. The Ffowcs Williams - Hawkings (FWH) equation36 for a steady surface, defined by f = 0, can be written for a convected wave operator52, 53 as: 2 i i i 2 ∂ ∂2 h ˜ 1 D∞ ∂ h˜ ∂ h˜ [(p − p )H(f )] = − Qδ(f ) (7) − 2 T H(f ) + F δ(f ) − ∞ ij i 2 ∂xi c∞ Dt2 ∂xi ∂xj ∂xi ∂t where the material derivative is defined by D∞ /Dt = ∂/∂t + U∞ ∂/∂x1 , H and δ are the Heaviside and Dirac functions respectively. The convected sources terms, with a tilde notation, are given by:  2   T˜ij = ρ(ui − U∞ δi1 )(uj − U∞ δj1 ) + (p − c∞ (ρ − ρ∞ ))δij − τij (8) ∂f ˜ = ρui ∂f  and Q  F˜i = [ρ(ui − 2U∞ δi1 )uj + pδij − τij ] ∂xj ∂xi

where δij denotes the Kronecker symbol. The function f = 0 is scaled so that ∂f /∂xj =nj is the j-component of the unit normal vector pointing toward the observer domain (f > 0). The viscous shear stresses are given for a Newtonian fluid as: ∂ui 2 ∂uk ∂uj (9) + − δij τij = 2µ ∂xi ∂xj 3 ∂xk

where µ is the dynamic molecular viscosity, obtained with Sutherland’s law. The 3-D free-space Green function for the convected wave operator of Eq.(7) is: ˜ t|y, τ ) = − δ(t − τ − τ ∗) G(x, 4πrβ

with τ ∗ =

rβ − M∞ (x1 − y1 ) c∞ β 2

(10)

x = (x1 , x2 , x3 ) denotes the position of the observation point (listener) at the time t, and y = (y1 , y2 , y3 ) p is the location of a source point at the time τ . We note rβ = (x1 − y1 )2 + β 2 (x2 − y2 )2 + β 2 (x3 − y3 )2 with β 2 = 1 − M2∞ . A temporal formulation of the convected FWH analogy is derived by applying the space derivatives of the right hand side of Eq.(7) to the Green function in the convolution step. These derivatives are then written as derivatives for the observer time t, yielding: ! # ZZZ " T˜ij bi ri bj rj bj rj bi ri 1 ∂2 2 ′ − M∞ δi1 + M∞ δj1 δj1 − M∞ δj1 dy 4πH(f )p (x, t) = 2 4 2 c∞ β ∂t rβ2 rβ rβ rβ ∗ t−τ

V0 (y)

+

1 ∂ c∞ β 2 ∂t

ZZZ "

V0 (y)

+

Z Z Z "

3bi ri bj rj bj rj bi ri − M∞ δi1 − bi δij − M∞ δj1 rβ2 rβ rβ

3bi ri bj rj − bi δij r2

V0 (y)

Z Z

˜ # Tij rβ3

dy

+

t−τ ∗

∂ ∂t

ZZ Σ

ρun rβ

1 ∂ ρ (ui − U∞ δi1 ) un − Pij nj bi ri − M∞ δi1 c∞ β 2 ∂t rβ rβ Σ # " ZZ bi ri (ρ (ui − U∞ δi1 ) un − Pij nj ) dΣ(y) − rβ3 ∗ −

T˜ij rβ2

#

dy t−τ ∗

dΣ(y) t−τ ∗

dΣ(y)

t−τ ∗

t−τ

Σ

!


(11)

where b1 = 1, and bα = β 2 , for α=2 or 3, and ri = xi −yi . We note the surface stresses Pij = [(p−p∞ )δij −τij ] and un = uj nj . For a rigid surface, un =0, simplifying the expression. The subscript t − τ ∗ indicates that source terms are evaluated at the retarded time. The second and third volume integrals, and the third surface integral scale as O(1/rβ2 ) or O(1/rβ3 ); they have a negligible contribution in the far-field. Convected acoustic analogy in the frequency domain With application of the Fourier transform, defined as: Z ∞ Z ∞ 1 −iωt f (x, ω) = f (x, t)e dt, f (x, t) = f (x, ω)e+iωt dω (12) 2π −∞ −∞

where the arguments of f allow to distinguish between the function and its Fourier transform, Eq.(7) becomes: 2 2 ∂ ∂ 2 ∂ 2 + κ0 − 2iM∞ κ0 − M∞ 2 [H(f )p′ (x, ω)] ∂x2i ∂x1 ∂x1 i i ∂ h˜ ∂2 h ˜ ˜ ω)δ(f ) (13) Tij (x, ω)H(f ) + Fi (x, ω)δ(f ) − iω Q(x, =− ∂xi ∂xj ∂xi ˜ of this convected Helmholtz where κ0 = ω/c∞ is the acoustic wavenumber. The free-space Green function G equation in the frequency domain is: κ0 exp −i 2 [rβ − M∞ (x1 − y1 )] β ˜ (14) G(x|y, ω) = − 4πrβ

The convolution of the Green function with the inhomogeneous Helmholtz equation (13) yields the integral solution: ZZ Z ZZ ZZ ˜ ˜ ∂ 2 G(x|y, ω) ∂ G(x|y, ω) ′ ˜ ˜ ˜ ˜ Tij (y, ω) H(f )p (x, ω) = − dy − dΣ − iω Q(y, ω)G(x|y, ω) dΣ Fi (y, ω) ∂yi ∂yj ∂yi Σ

V0 (y)

Σ

(15) where the source terms are given by (8), and the derivatives of the Green function Eq.(14) are: ” “ ! κ −i β02 [rβ −M∞ (x1 −y1 )] ˜ ∂ G(x|y, ω) bi ri e M∞ δi1 iκ0 bi ri + = × − ∂yi 4π β2 rβ2 rβ rβ3 ˜ ω) ∂ G(x|y, e = ∂yi ∂yj 2

+

” “ κ −i β02 [rβ −M∞ (x1 −y1 )]

iκ0 β2

(16)

! bi ri bj rj bj rj bi ri 2 − M∞ δi1 + M∞ δj1 δj1 − M∞ δj1 4π rβ2 rβ rβ ! ! 3bi ri bj rj bi ri bj rj 3bi ri bj rj bi δij bi δij (17) + − 2 − M∞ δj1 2 − M∞ δi1 2 − rβ3 rβ rβ rβ r4 rβ κ2 × − 04 β

The frequency domain formulation avoids the evaluation of the retarded time, and the spatial derivatives are applied on the Green function avoiding the numerical differentiation of aerodynamic quantities, which could lead to numerical errors. Convected acoustic analogy in the frequency-wavenumber space By defining the 2-D spatial Fourier transform for the direction parallel to the wall as: ZZ ∞ ZZ ∞ 1 −ik.x f (k, ω) = f (x, ω)e dx, f (x, ω) = f (k, ω)e+ik.x dk with k = (k1 , 0, k3 ) , (18) (2π)2 −∞ −∞ Eq. (13) is transformed to: 2 ∂ ∂2 2 [H(f )p′ (k, ω)] = −k12 + 2 − k32 + κ20 − 2M∞ κ0 k1 + M2∞ k12 [H(f )p′ (k, ω)] = + γ ˜ (k) ∂x2 ∂x22 h h i i ∂ ∂ ∂ ˜ ω)δ(f ) ikj + δj2 T˜ij (k, ω)H(f ) + iki + δi2 F˜i (k, ω)δ(f ) − iω Q(k, − iki + δi2 ∂x2 ∂x2 ∂x2 (19)


avec γ˜ 2 = (κ0 − k1 M∞ )2 − k12 − k32 . The wavenumbers are supersonic when γ˜ 2 > 0, and subsonic when γ˜ 2 < 0. The root of γ˜ 2 is chosen in such a way that when γ˜ 2 is positive it has the same sign as ω, and when γ˜ 2 is purely imaginary it has a negative imaginary part, ensuring evanescent waves: ( p γ˜ = sgn(ω) (κ0 − k1 M∞ )2 − (k12 + k32 ) if γ˜ 2 > 0 p (20) if γ˜ 2 < 0 γ˜ = −i |(κ0 − k1 M∞ )2 − (k12 + k32 )| The free-space Green function is given by: ˜ 2 |y2 , k, ω) = − 1 G(x 2π This yields:

Z

∞

−∞

e

ik2 (x2 −y2 )

1 dk2 =− × 2iπ k22 − γ˜ 2 2π

(

e−i˜γ (x2 −y2 ) /2˜ γ ei˜γ (x2 −y2 ) /2˜ γ

˜ 2 |y2 , k, ω) = − i e−i˜γ |x2 −y2 | G(x 2˜ γ

when x2 − y2 > 0 when x2 − y2 < 0 (21)

Convolution with Eq.(19) and integration by parts lead to: Z ∞ i ′ H(f )p (k, ω) = − (ki + sgn(x2 − y2 )˜ γ δi2 ) (kj + sgn(x2 − y2 )˜ γ δj2 ) T˜ij (y2 , k, ω)e−i˜γ |x2 −y2 | dy2 2˜ γ −∞ ∂ i ˜ ˜ iki + δi2 Fi (y2 , k, ω) − iω Q(y2 , k, ω) e−i˜γ |x2 −y2 | (22) − 2˜ γ ∂x2 This formula can be applied for a surface defined by y2 =cste. Similar formulations in terms of the threedimensional Fourier transform of a source Tij (y2 , k, ω) are derived in Ref.7, 8, 40, 54, 55 An interesting point in the formula (22) concerns the possible coincidence at the acoustic wavenumber. The wavenumber γ˜, appearing in the denominator is then zero, and a wavenumber-frequency spectrum will be singular at k = κ0 (or ∓κ0 /(1 ± M) for the convected operator). A comprehensive discussion of the behaviour of the spectrum near this singularity is provided by Howe.31 Three main phenomena can remove the singularity: • the finite size of the domain (theoretical works suppose an infinite plane). This point is first studied by Bergeron;56 • the viscous effect. Howe42 suggests that the viscous terms provide the necessary dissipation near the acoustic wavenumber; • the curvature of the surface.31, 41 Meecham41 has shown that a local surface curvature can result in a dipole sound. The main contribution seems to be due to the finite size effect, but this is still an open question. Ffowcs Williams55 has derived an estimate for k = κ0 by assuming the far field contribution of long wavelengthes from a bounded surface panel. Results of wave extrapolation methods The acoustic analogy is now applied to the TBL database. The primitive flow variables ρ, ui , and p have been stored during the 300 000 last iterations of the LES every 30 iterations for a volume of [29.4 61.6]δref × [0 4.52]δref × [−1.72 1.72]δref, which corresponds to the surface under the white dashed line in the midplane view of Fig.5(d) multipled by the whole span of the LES. In the results presented in this paper, only every four samples in time are used to achieve reasonable RAM memory occupation, leading to a sample time ∆tanalogy = 120∆tLES. It is easy to get a result which resembles an acoustic field with the acoustic analogies, since a wave operator is simply applied to a source term, it is however difficult to have confidence in the results. Spatial or temporal truncations, periodicity assumptions, frequency aliasing, or numerical errors can indeed dramatically alter the computed acoustic field. That is why, we decide to proceed gradually. Since all the compressible information is available from the DNC simulation, we first use the formulations in a wave extrapolation method (WEM) framework. This consists in using a fictitious surface which encloses the physical source region, reducing by one dimension the numerical effort. Several numerical parameters or physical assumptions can then be tested: effects of the sampling rate, of the periodicity condition in the spanwise direction, of the use of an open extrapolation surface (spatial truncation), of the relatively short signal duration (temporal truncation) for frequency domain applications... 9 of 30 American Institute of Aeronautics and Astronautics Paper 2010-2904

We begin with the application of the convected temporal FWH-WEM, Eq.(11), from the horizontal surface at y2ex /δref = 4.53 (white dashed line in Fig.5). In this case, the acoustic field for a listener at x is obtained for a single time t. The time derivations in Eq.(11) are evaluated with a second-order central differenciation, and the source term at the different retarted times are deduced from the LES values thanks to a fourth-order Lagrange interpolation. Results for a listener grid of 101×75 points, covering a portion of the LES domain, are depicted in Fig.5. The influence of the spanwise periodicity used in the LES is investigated. The extrapolation surface is replicated several times in the y3 -direction in order to take into account the image sources due to the periodicity present in the LES. We can see in Fig.5(a) that the acoustic levels are underestimated when the surface is replicated 3 times. When the number of replications, hereafter noted nn3 , becomes large, low frequencies are more pronounced, since they propagate over longer distances. The agreement with the DNC results are better for nn3 =7 even if several discrepancies with the reference solution still arise. They are related to the limited extent of the extrapolation surface: acoustic waves generated by sources upstream or downstream of the surface in the LES are not taken into account. The truncation effect yield also unwanted lobes near the upstream and downstream extremities of the surface. Besides the mean pressure used to bring out the acoustic perturbation is not far from ambiguity. In the temporal WEM approach, the mean contribution is automatically removed, whereas a temporal mean is calculated during the LES run. This latter mean varies very slightly in time, because the LES convergence is not perfect, and because very low frequency variations in the outflow boundary can pollute the acoustic signal (which very 2 weak: 4 Pa for a dynamic pressure q∞ = 21 ρ∞ U∞ = 17728 Pa, i.e. around 0.02%).

(a)

(b)

(c) (d) Figure 5. Acoustic fields extrapolated with the convected temporal FWH-WEM from the horizontal

surface at y2ex /δref = 4.53 (white dashed line): (a) surface replicated 3 times in the spanwise direction; (b) surface replicated 7 times in the spanwise direction; (c) surface replicated 11 times in the spanwise direction; (d) DNC reference at the same time tuτ /δref =16.4. The colorscale levels are between -3 Pa (blue) and 4 Pa (red) for each plot.

Next the WEM integral method (15) in the frequency domain is implemented. The source terms sampled over 2250∆tanalogy are Fourier-transformed. Then the integral is evaluated with a trapezoidal rule for the whole range of positive frequencies. An inverse transform is applied at the end, and the resulting acoustic pressure is doubled to take into account the negative frequencies. Thus when applying a frequency-space formulation, the signal is calculated for all the times that compose the initial sampling. It is then more


(b): at point A p’ (Pa)

5 0 −5

13

14

15

16

17

t u /δ τ

18

19

20

21

18

19

20

21

ref

(c): at point B p’ (Pa)

5 0 −5

(a)

13

14

15

16

17

t uτ/δref

Figure 6. Acoustic field extrapolated with the convected spectral FWH-WEM from the horizontal surface at y2ex /δref = 4.53 at tuτ /δref =16.4: (a) surface replicated nn3 =3 times in the spanwise direction [pressure levels between -3 and 4 Pa]; (b) time history at the point A (x1 , x2 , x3 )/δref = (32.7, 10.6, 0) ( on Fig(a)); (c) time history at the point B (x1 , x2 , x3 )/δref = (44.2, 18.6, 0) (• on Fig(a)). FWH-WEM with nn3 =5 ( ) compared with DNC signal at the same location ( ).

expensive to obtain a pressure map than with the temporal method. For the purpose of comparison, we compute such a map on a grid of 101×75 points. The result is plotted in Fig.6, and can be compared with Fig.5(a), obtained at the same time and also with nn3 =3. The great similarities give us confidence in the frequency-space procedure (a discussion on the windowing effect is given below). The time histories of the acoustic pressure at two points A and B are compared in Fig.6(b) and (c) with the DNC fluctuating pressure at the same point. The quantitative agreement for nn3 =5 is satisfactory considering the effect of the truncation of the open extrapolation surface described previously. The signal at point B is closer to the DNC reference than the signal at point A, because the latter is near the upstream surface extremity, and also because it feels more the influence of the high-frequency spurious waves scattered by the small step located at the beginning of the computational domain to ignite the laminar-turbulent transition. This second effect is clearly visible on the power spectra depicted in Fig.8. (b): at point B

(a): at point A

5 p’ (Pa)

p’ (Pa)

5 0 −5 15

0 −5

16

17

t uτ/δref

18

19

20

15

16

17

t uτ/δref

18

19

20

Figure 7. Convected FWH-WEM in frequency domain from y2ex /δref = 4.53. Influence of the spanwise

periodicity on the acoustic pressure at points A (x1 , x2 , x3 )/δref = (32.7, 10.6, 0) (a), and B (x1 , x2 , x3 )/δref = (44.2, 18.6, 0) (b). The extrapolation surface is replicated 1 ( ), 3 ( ), 5 ( ), 7 ( ), 11 ) times in the spanwise direction. (

The influence of the number of times nn3 that the extrapolation plane is replicated in the spanwise direction is shown in Fig.7 for the time histories at the points A and B. Similar conclusions can be drawn as with the temporal application. When nn3 =1 or 3, the amplitude is too weak, whereas low-frequency modulations are promoted by a large number nn3 =11. The power spectral densities of the acoustic timetrace in Fig.8 allow to quantify the low-frequency level added by the replication. Note that the high frequencies are almost not affected. We have chosen a value nn3 =5 for the results presented hereafter. Furthermore, we have tested the influence of the extrapolation surface location by plotting in Fig.8 and 9 the extrapolation from y2ex /δref = 3.43. The results for the temporal evolution and for the spectral content are close to the case y2ex /δref = 4.53. Theoretically, it is expected that the results are independent of the shape or location of the extrapolation surface if it encloses all the physical sources. This is not exactly the case here since the extrapolation surface is open and is relatively close to the boundary layer edge, cutting the upper part of the hydrodynamic pressure lobes which results from large-scale arrangements, as seen for


(a): at point A

(b): at point B 100 PSD(p’) (Pa /Hz ref 2e−5)

80 60

80 60

2

PSD(p’) (Pa2/Hz ref 2e−5)

100

40 20 0

−1

0

10

40 20 0

1

10

−1

10

ωδref/U∞

0

10

1

10

10

ωδref/U∞

Convected FWH-WEM in frequency domain. Power spectral densities of the acoustic pressure at points A (x1 , x2 , x3 )/δref = (32.7, 10.6, 0) (a), and B (x1 , x2 , x3 )/δref = (44.2, 18.6, 0) (b) for different parameters: (a) y2ex /δref = 4.53 and nn3 =1 ( ), y2ex /δref = 4.53 and nn3 =3 ( ), y2ex /δref = ), y2ex /δref = 4.53 and nn3 =7 ( ), DNC ( ); (b) y2ex /δref = 4.53 and nn3 =3 4.53 and nn3 =5 ( ( ), y2ex /δref = 4.53 and nn3 =5 ( ), y2ex /δref = 4.53 and nn3 =11 ( ), y2ex /δref = 3.43 and nn3 =5 ( ), y2ex /δref = 0 and nn3 =5 ( ), DNC ( ). Figure 8.

instance in Fig.5(d). Note that when the extrapolation surface is taken at the rigid plane y2ex /δref = 0, the transpiration velocities identically vanish. The resulting acoustic signal has a similar overall amplitude in Fig.9, but the spectral analysis in Fig.8 reveals that the slope of the high-frequency range is slower. The grid stretching in the vertical direction can explain a part of the difference, since it implies a smaller frequency cut-off in the DNC, and thus in the WEM, which use DNC data away from the wall.

p’ (Pa)

5 0 −5 15

16

17

t uτ/δref

18

19

20

Convected FWH-WEM in frequency domain at point B (x1 , x2 , x3 )/δref = (44.2, 18.6, 0) for nn3 =5 from y2ex /δref = 4.53 ( ), from y2ex /δref = 3.43 ( ), and from y2ex /δref = 0 ( ), compared to the DNC reference ( ). Figure 9.

Finally the effect of windowing is investigated. Windowing can attenuate the influence of spatial and temporal truncations. This is especially sensitive in the frequency domain applications since operations are implemented on the time Fourier transform of complex sources. To that purpose, we use a Tukey window,57 also known as the tapered cosine window. For a signal x, it is defined as:  2π α 1  when 0 6 x 6 α2  2 1 + cos α x − 2 w0 (x) = 1 when α2 6 x 6 1 − α2   1 2π α when 1 − α2 6 x 6 1 2 1 + cos α x − 1 + 2

where α is the ratio of taper to constant sections and is between 0 and 1. For α=1, it becomes a Hann window. The windowing function is plotted for α=0.1 and 0.05 in Fig.10(a). A test has been conducted for a short signal (750 samples). The results of the frequency-domain acoustic analogy applied to the T11 sources in a plane are shown in Fig.10(b). Without windowing, the acoustic signal obtained after an inverse Fourier transform is severely distorted at the beginning of the temporal range. The Tukey window is able to remove these oscillations, but the beginning of the signal is then damped, as seen for α=0.5. Lower values of α allow to reduce the part of the signal affected by the windowing. Figure 10(c) indicates better results for α=0.05 than for α=0.1. A Tukey window with α=0.05 is consequently used in the following, both for the 12 of 30 American Institute of Aeronautics and Astronautics Paper 2010-2904

temporal and spatial windowing. Obrist et al.58 have shown for instance that the use of a Tukey window can reduce dramatically the effects of spatial truncation. 1

0.2 0.1

p’ (Pa)

Amplitude

0.8 0.6 0.4

0 −0.1

0.2 0 0

−0.2 16

0.2

0.4

0.6 x

0.8

16.5

17

17.5

1

18 t uτ/δref

18.5

19

19.5

20

(b)

(a) 0.2

p’ (Pa)

0.1 0 −0.1 −0.2 16

16.5

17 t uτ/δref

17.5

18

(c) Effect of temporal truncation on the application of the FWH analogy in the frequency domain: influence of the parameter α of the Tukey window on the temporal evolution of the acoustic pressure at (x1 , x2 , x3 )/δref = (44.2, 18.6, 0) from volume sources T11 in the plane y2 /δref = 0.042 using Eq.(15). (a) Window function amplitude versus normalized samples x for α=0.05 ( ), 0.1 ( ), 0.5 ( ), 1, i.e. Hann window ( ). (b) Acoustic pressure with no windowing (α=0, i.e. ), for α=0.1 ( ), and for α=0.5 ( ) for N =750 samples. A rectangular window) ( longer signal (N =2250), free of windowing effect in the plotted temporal range, yields a reference (· · · · · · *). (c) Refined view on the perturbed signal to show the influence of the window parameter: α=0.05 ( ), α=0.1 ( ), reference (· · · · · · *). Figure 10.

Results of FWH analogy Since we know better the effects of the numerous parameters, we can proceed with the analysis of the source terms in the FWH analogy. Note that FWH source terms are not the true sources of sound, as enlighted for instance by Powell’s principle. They are just equivalent sources, but they constitute the basis of the TBL noise estimates, so that we can try to evaluate the legitimacy of the assumptions made in the literature. The frequency-domain formulation (15) with the volume sources T˜ij on [29.4 61.6]δref × [0 4.52]δref × [−1.72 1.72]δref, and the surface sources at y2 =0. Note that only the shear stresses are sources on the rigid surface (un =0). The source duration is 2250∆tanalogy, the volume is replicated nn3 =5 times in the spanwise direction, and Tukey’s windows are used with α = 0.05 for temporal windowing, and α = 0.1 for spatial windowing. Fig.11 presents the contributions of the different components of the convected Lighthill’s tensor T˜ij , defined by Eq. (8). They can be ordered as following: T11 dominates clearly, then 2T12 and T22 provide a slight part of the intermediate frequency range. The other terms 2T13 , T33 , and 2T23 are negligible, except for the very high frequencies, which is certainly an aliasing effect due to the Fourier transform. The heavy domination of the T11 term is explained by the convected formulation used, where the shear term (following the terminology of Ribner59 ) has a leading role over the self-noise term ρu′1 u′1 . It would be interesting to separate the contribution of these two terms, as in the theoretical fundations of Chase’s model.60 The viscous stresses taking place in the integral equation (15) are τ2j , defined by (9), since the unit normal to the plate is n2 = (0, 1, 0). Among these three terms, τ21 is highly dominating as seen in Fig.12. The two others are negligible. Note also that the overall level of the viscous stresses is ten times smaller than that of the volume sources. The viscous stresses are thus an effective source even at Mach 0.5, but their contribution is modest. The principal source τ21 is also related to the mean shear dU1 /dx2 .



100

p’ (Pa)

5

0

−5 15

80 60 40 20 0

16

17

t u /δ

18

19

−1

10

20

τ ref

0

1

10

ωδ /U ref

10

∞

Figure 11. Application of the FWH analogy in the frequency domain: contributions of the volume

source terms to the acoustic pressure at (x1 , x2 , x3 )/δref = (44.2, 18.6, 0): T11 ( ); T22 ( ); T33 ( ); 2T12 ( ); 2T13 ( ); 2T23 ( ); sum of Tij ( ). Time traces on the left, and corresponding PSD on the right.


80

p’ (Pa)

0.5

0

−0.5 15

60 40 20 0 −20

16

17

t uτ/δref

18

19

−1

10

20

0

1

10

ωδ /U ref

10

∞

Figure 12. Application of the FWH analogy in the frequency domain: contributions of the surface

viscous stress terms to the acoustic pressure at (x1 , x2 , x3 )/δref = (44.2, 18.6, 0): τ21 ( ); τ22 ( τ23 ( ); sum of τ2j ( ). Time traces on the left, and corresponding PSD on the right.

);


100

p’ (Pa)

5 0 −5

80 60 40 20 0

15

16

17

t uτ/δref

18

19

20

−1

10

0

1

10

ωδ /U ref

10

∞

Figure 13. Application of the FWH analogy in the frequency domain: contributions of the surface

pressure source terms to the acoustic pressure at (x1 , x2 , x3 )/δref = (44.2, 18.6, 0): p at y2 = 0 ( ); sum of surface stresses p − τij ( ); p at y2ex /δref = 4.53 ( ). Time traces on the left, and corresponding PSD on the right.


The other surface term in the analogy implies the wall pressure. Its contribution in Fig.13 is almost undistinguishable from the curve representing the sum of all surface stresses, reinforcing the impression that the viscous contribution is not important. The level of the wall pressure term is close to the overall level, as illustrated by the FWH WEM result, and already noted in Fig.9. PSD(p’) (Pa2/Hz ref 2e−5)

100

p’ (Pa)

5 0 −5

80 60 40 20 0

13

14

15

16

17

t uτ/δref

18

19

20

21

−1

10

0

1

10

ωδ /U ref

10

∞

Figure 14. Application of the FWH analogy in the frequency domain at (x1 , x2 , x3 )/δref = (44.2, 18.6, 0). On the left, pressure histories from: FWH analogy (volume+surface) ( ); FWH WEM from y2ex /δref = 4.53 ( ); DNC at the same location ( ). On the right, power spectral densities ); FWH analogy (surface only) of the acoustic pressure from: FWH analogy (volume+surface) ( ( ); FWH WEM from y2ex /δref = 4.53 ( ); DNC ( ).

By adding the surface and volume integrals, the total sound field is reconstructed. It is compared to the DNC reference and to the FWH WEM result in Fig.14. The fluctuations amplitude is well reproduced in the low- and middle-frequency ranges, as seen on the PSD in the right side of Fig.14. The main discrepancies are visible for the very high frequencies. They can partly be explained by the fact that the volume sources are more prone to aliasing. Besides, as previously observed, higher frequencies are noticeable for the surface pressure alone, so that it was inferred that the widening of the grid cut-off can also be invoked. The contribution of the entropy term ((p − p∞ ) − c2∞ (ρ − ρ∞ ))δij in the Lighthill tensor Tij has not yet been calculated. In the absence of external entropy sources, this term is essentially related to local variations √ of the sound speed c = γrT , and is thus proportional to temperature variations, which are limited in the subsonic case investigated. The contribution of the entropy term is thus expected to be negligible. C. 1.

Characteristics of wall-pressure fluctuations in the acoustic domain Spectral resolution

In the present study, the analysis of wall pressure fluctuations aims at characterizing the turbulent motions responsible for the noise emissions. It is therefore crucial to take care of the spectral resolution, especially in the low wavenumber region. This is indeed strenuous to measure the low wavenumbers in the acoustic region, either because of the finite extent of the model or of the microphone array, or because of the contamination by the background noise of the wind tunnel facility. The question whether the spot sometimes observed in the acoustic region is coming from the boundary layer self-noise or the environment noise is still open. With the development of supercomputers, the numerical simulation constitutes a new efficient approach to investigate the features of the wall pressure field. But only incompressible solvers have been used in the past,61–65 so that a discussion about the supersonic wavenumber range is questionable. When the wall is plane and the boundary layer is growing slowly, the pressure fluctuations can be considered statistically stationary in time and homogeneous in x1 - and x3 -directions. It is then possible to analyse the wall pressure in the wavenumber-frequency space. The wavenumber-frequency spectra are obtained by a three-dimensional Fourier transform (x1 , x3 , t) → (k1 , k3 , ω) using the powerful FFT algorithm. Correlations or cross-spectra are deduced using the number of inverse Fourier transforms required. Once the statistical steady state is reached, the wall pressure is stored during 585 000 iterations every 30 time steps of the Navier-Stokes solver (∆tuτ /δref ≃ 1.1 × 10−3 ). The total time of storage corresponds to Lt uτ /δref = 22.08. The 19 500 samples are divided in Nseg =11 overlapping segments, each containing ∗ 3250 samples. The frequency resolution is ∆ω = 1.71uτ /δref = 0.013U∞/δref , and the maximum frequency ∗ is ωmax = 5549uτ /δref = 41U∞ /δref . The frequency cut-off due to the spatial resolution is however far lower. By estimating roughly that the spatial scheme is able to resolve spatial structures discretized by 4 ∗ . Using the Taylor hypothesis with points, the wavenumber cut-off is kcut-off = 2π/4∆x = 23.6/δref = 4.1/δref 15 of 30 American Institute of Aeronautics and Astronautics Paper 2010-2904

∗ Uc =0.6, we estimate a cut-off frequency ωcut-off = 335uτ /δref = 2.5U∞ /δref . The wavenumber resolution is ∗ ∆k1 = 2π/Lx1 = 0.085/δref = 0.015/δref in the streamwise direction and covers a range 0 ≤ k1 δref ≤ 94.6 ∗ ∗ , and 0 ≤ k3 δref ≤ 236.6 (0 ≤ k1 δref ≤ 16.5). In the spanwise direction, ∆k3 = 2π/Lx3 = 1.82/δref = 0.32/δref ∗ (0 ≤ k3 δref ≤ 41.4).

2.

Wall pressure coherence

Since the work of Corcos,66 the coherence has frequently been assumed to be a decreasing function of a Strouhal number, based on the separation ξi , the pulsation ω, and a characteristic velocity. Corcos has taken an exponential in the x1 - and x3 -directions, and modeled the loss of coherence by the product: ω|ξ1 |

ω|ξ3 |

Spp (ξ, ω) = φpp (ω) e− α1 Uc e− α3 Uc eiωξ1 /Uc

(23)

1

1

0.8

0.8 Γ(0,ξ ,ω)

0.6

0.6

3

Γ(ξ1,0,ω)

However, several authors11, 67, 68 have noted the fact that the coherence does not tend to unity as the frequency tends to zero due to the limitation in eddy size imposed by the boundary layer thickness, and the alteration of large strutures by the TBL shear. The coherence is evaluated for our LES database, and compared to classical estimates. To this end, we first perform a time Fourier transform of the wall pressure, and then evaluate the cross-spectra by a two-dimensional spatial inverse transform of the wavenumber spectra. A cross-spectrum is thus obtained for each frequency, and is a complex function composed of a co-spectrum and a quad-spectrum: Spp (ξ1 , ξ3 , ω) = C(ξ1 , ξ3 , ω) − iQ(ξ1 , ξ3 , ω)

0.4

0.4 0.2

0.2 0 0

5

10

15

α(ξ ,ω)=ωξ /U 1

(a)

1

cp

20

0 0

25

(b)

2

4 6 α(ξ3,ω)=ωξ3/Ucp

8

Figure 15. Coherence Γ as a function of the phase angle: (a) for successive streamwise separations ∗ ξ1 /δref =0.38 (o), 1.90 (), 3.42 (△), 4.94 (×), 6.45 (⋄), 7.97 (*), 9.49 (▽), 11.01 (+). (a) for successive ∗ spanwise separations ξ3 /δref =0.15 (o), 0.76 (), 1.37 (△), 1.97 (×), 2.58 (⋄), 3.19 (*), 3.80 (▽), 4.40 (+). ω|ξ | − α Ui

A exponential fit e

i

c

(

) is superimposed in the ξ1 - (α1 = 1/0.12) or ξ3 -direction (α3 = 1/0.72).

p Coherence Γ is defined as |Spp (ξ1 , ξ3 , ω)|. A phase angle can also be introduced as α(ξ1 , ξ3 , ω) = tan−1 (Q(ξ1 , ξ3 , ω)/C(ξ1 , ξ3 , ω)). The longitudinal and transverse coherences are represented in Fig.15 versus the phase angle. A good collapse is observed at high frequencies for the different separations ξi . Exponential −

ω|ξi |

curves e αi Uc fit well the high-frequency coherence loss. The small scales responsible for the high-frequency content are indeed rapidly destroyed as they are convected. The values 1/α1 =0.12 and 1/α3 =0.72 deduced from Fig.15 are in very good agreement with the values 0.11 and 0.714 obtained by Corcos from miscellaneous experimental databases a . 3.

Convection velocities

Frequency analysis Cross-spectra yield a phase speed Ucp = ωξ1 /α, depending on the frequency. It is depicted in Fig.16(b) and 17(a), and can be compared to the narrow-band correlations measured by Blake.12 A comparison with the more recent experiments of Farabee and Casarella,67 and Leclercq and Bohineust68 a Note

that a too high value of 1/α1 =0.19 was found in AIAA paper 2009-34011 because the cross-spectral density rather than its square root was erronesouly used.


1

0.8

0.8

0.6

0.6

∞

1

cp

U /U

2

Γ (ξ1,0,ω)

is done in Fig.16. The phase velocities are seen to increase rapidly from zero as the frequency is increased and then reach a plateau. For large separations, the increase is abrupt and an asymptote around 0.75U∞ is reached. The value of this limit is also in fair agreement with both the measurements of Bull11 and Blake.12 For the smallest separation plotted, a phase velocities of 0.55 is noted indicating that the less correlated structures, probably small scales, are convected more slowly. This is coherent with the picture of smaller scales closer to the wall.

0.4

0.4 0.2

0.2 0 0

100

ωδref/uτ

(a)

200

0 0

300

100

(b)

ωδ/uτ

200

300

Figure 16. (a) Coherence versus the phase angle for successive streamwise separations compared to

measurements by Leclercq and Bohineust68 at Reθ =7467 (o). (b) Phase velocities versus frequency compared to the experimental results of Farabee and Casarella67 at Reθ =2945 (△), and Leclercq and Bohineust68 (o). The successive streamwise separations are given in Fig.15 for the LES simulation. They are ξ1 /δ=0.10, 0.24, 0.42, 0.9, and 1.63 in Leclercq and Bohineust’s experiment, and 0.21≤ ξ1 /δ ≤5 in Farabee and Casarella’s experiment.

Broadband analysis The study of spatio-temporal correlations1 allows to evaluate a mean broadband convection speed, by following the maxima of the longitudinal space-time correlation for different fixed space ¯c (ξ1 ) = ξ1 /τ . Alternately, an instantaneous broadband increments ξ1 as a function of the time delay, i.e. U convection velocity may be defined as Uc (ξ1 ) = dξ1 /dτ , by evaluating the slope of the curve corresponding to minimum decay rate of the pressure correlation. These convection speeds are referred to as broadband since it is the same for all frequencies. They are plotted in Fig.17(b). The average velocity varies from 0.6U∞ for zero separation to 0.84U∞ for large separations. Willmarth and Woolridge10 indicate values of 0.53 at small ξ1 and 0.83 at large ξ1 , and Bull11 reported values of 0.53 and 0.825 respectively. The instantaneous velocity Uc (red broken line) is increasing in fair agreement with the measurements.10–12 The instantaneous convection velocity approaches 0.85U∞ . This reflects the decay of slower small-scale pressure sources, and the gradual predominance of higher-speed sources for large distances. 1

0.9 0.8

0.6

Uc/U∞

Ucp/U∞

0.8

0.4

0.6

0.2 0 0

(a)

0.7

0.5

1

*

ωδ /U

1.5

∞

2

0.5 0

2.5

(b)

10

20

30 ξ1/δ*

40

50

60

Figure 17. (a) Phase velocities Ucp versus frequency for successive streamwise separations (given in

¯c (ooo) and instantaneous Uc ( Fig.15). (b) Mean U ) broadband convection velocities as a function ¯c . the longitudinal separation. The blue line is a curve fitting of the average velocity U


10 log10[φ(ω)U∞/(τw δ)]

0

τ 10

0 −10 −20

−2

10

(a)

−5

2

10

w

10 log [φ(ω)u2/(τ2 ν)]

20

−15 −20 −25 −30 −1 10

0

ων/u2τ

−10

10

(b)

0

10

1

ωδ/U∞

10

2

10

Frequency power spectra of wall pressure for successive streamwise absissae: ∗ ∗ ∗ x1 /δref =75.9 (Reθ =848); x1 /δref =151.8 (Reθ =1180); x1 /δref =227.8 (Reθ =1491); ∗ ∗ ∗ x1 /δref =303.7 (Reθ =1781); x1 /δref =379.6 (Reθ =2058); x1 /δref =455.6 (Reθ =2328). (a) Scaling by inner variables: τw as pressure scale and ν/u2τ as timescale. Experimental spectra are superimposed: ++ Gravante et al.75 (Reθ =2953); ** Farabee and Casarella67 (Reθ =2945); oo Schewe13 (d=2 mm, Reθ =1400); oo Schewe13 (d=4 mm, Reθ =1400). The semi-empirical model of Goody74 for RT =20 is plotted with a black dashed line, and the red vertical dashed line indicates the frequency cut-off of the LES. (b) Scaling by outer variables: τw as pressure scale and δref /U∞ as timescale. : Goody’s model (RT =20). Figure 18.

4.

Frequency-wavenumber analysis

The determination of Ucp , α1 , and α3 , together with the boundary layer thicknesses, constitutes the input for the semi-empirical model of wall-pressure fluctuations in frequency-wavenumber space. The most popular are Corcos’ model66 and Chase’s model.60, 69 Other models have been proposed and tested: Maestrello (used in,70 §2.1 of 71 ), Efimtsov (§2.2 of 71 §2.3.2 of 72 ), Smol’yakov and Tkachenko (§2.5 of 71 or §2.3.4 of 72 ), Witting (§2.3 of 71 ), Leclercq and Bohineust.68 Blake,7 Bull,73 Graham,72 or Hwang et al.71 propose comparisons between the different semi-empirical models. Point spectra The single-point spectra for successive longitudinal locations are depicted in Fig.18 with different scalings. When the inner variables are used, as in Fig.18(a), the high-frequency part of the spectra are seen to collapse to some degree. The different spectra superimpose well with experiments in the intermediate range 0.1 ≤ ων/u2τ ≤ 0.6. For very high frequencies a decay law with ω −5 is expected. This law is satisfied by the model of Goody74 which has been plotted with the black dashed line. The semi-empirical model of Goody74 is an extension of the Chase-Howe model.16, 69 To take account of the Reynolds number dependence, Goody proposed the following semi-empirical formula : C2 (ωδ/U∞ )2 φ(ω)U∞ = 7 2 3.7 τw δ [(ωδ/U∞ )0.75 + C1 ] + C3 RT−0.57 (ωδ/U∞ )

(24)

where C1 = 0.5, C2 = 3 and C3 = 1.1 are empirical constants, RT = (δ/U∞ /(ν/u2τ )) is the ratio of the outer to inner boundary layer time scale, which represents the effect of the Reynolds number, since RT ∝ uτ δ/ν. The model has been plotted for RT = 20, which would correspond to Reθ =1500. The use of outer variables in Fig.18(b) allows a good collapse at middle frequencies. Note that in both cases, the spectra from the LES fall off too fast at very high frequencies. The vertical thick dashed line indicates the frequency cut-off of the simulation as defined in the previous section, and thus explain the discrepancies for ων/u2τ ≥ 0.67. We have computed the mean square pressure,1 and we find prms /τw ≃ 2.4, in fair agreement with the value of 2.35 reported in the DNS of Spalart,32 or 2.5 in the experiments of Schewe13 for similar Reynods number (Reθ ≃ 1400, Reτ ≃ 500). Consequently, the too rapid decay at high frequencies in the present simulation is not penalizing since the energetic contribution of the high-frequency region seems negligible.


Frequency-wavenumber spectra The wavenumber-frequency spectrum is obtained by three-dimensional fast Fourier transform as explained in Ref.1 The contour maps of the two-dimensional spectra are depicted in Fig.19, with cuts for particular frequencies. The principal peak characterizes the convected nature of the wall-pressure field (the line passing through gives the mean convection velocity). The asymmetry towards positive streamwise wavenumbers also indicates more energy in the forward moving waves. Even if the low wavenumber region is hardly resolved, no peak is noticed near the acoustic wavenumber κ0 = ω/c∞ . This may be due to the very low levels of direct acoustic radiations. −4

2

−5

1

−6

*

ωδ /U

∞

Φpp(k1,0,f)

3

0 −1

−8

−2

−9

−3 −3

(a)

−7

−2

−1

0*

k1δ

1

2

−10 −4

3

−2

0

(b)

2 k δ*

4

6

8

1

Figure 19. Wavenumber frequency spectra: (a) Contour plots (19 contours between -10 and -1 dB every 0.5 dB). The black broken line represents the relation kc = ω/Uc with Uc = 0.6U∞ , and the blue broken line for Uc = 0.85U∞ . The red lines denotes the convected acoustic wavenumber κ0 = ω/(U∞ ± c∞ ); (b) Wavenumber frequency spectra for different fixed nondimensional frequency ωδ ∗ /U∞ : 0.25 ( △), 0.83 ( o), 1.65 ( ), 2.48 ( ▽). The red solid line indicates the convective wavenumber kc = ω/Uc and the red broken line indicates the convected acoustic wavenumber for the frequency ωδ ∗ /U∞ =0.83.

The convective ridge in the k1 -k3 plane is represented in Fig.20 for a low nondimensional frequency ωδ ∗ /U∞ =0.31, and a higher frequency 1.69. Some noise is visible on the contours because of the reduced spanwise extent in the simulation and the limited time duration but its shape is satisfactory and exhibits the well-known convective crest. When the convection by the mean flow is taken into account, the acoustic domain is the interior of the ellipse Eq.(6), represented in Fig.20. When the frequency is too low, the acoustic domain is overwhelmed by the convective ridge. The separation is enhanced for lower speed flows, so that it would be interesting to have the same plots for a M=0.1 TBL. At the higher frequency plotted in Fig.20(b), the acoustic domain is hardly detectable, but a noisy activity is well correlated with the interior of the ellipse, as in the experiments of Arguillat et al.,14, 76 or Ehrenfried and Koop.15 The cut for ωδ ∗ /U∞ =1.69 and k3 = 0 is shown in Fig.21, and the insert indeed indicates small peaks near κ0 /(1 + M∞ ). Nevertheless, almost no trace is visible for the nearby frequency ωδ ∗ /U∞ =1.65, so that the view of Fig.20(b) exaggerate the emergence of the acoustic spot. D. 1.

Link between wall pressure and radiated pressure Comparison with models for the wall pressure

The frequency-wavenumber spectra are firstly compared to two classical estimates, in order to evaluate their behaviour at low wavenumbers. The first one is due to Corcos,66 who used the exponential coherence loss (23) to express the spectrum as: Φpp (k, ω) =

α1 kc α3 kc φpp (ω) 2 2 2 2 2 π kc + α1 (k1 − kc ) kc + α23 k32

(25)

where kc = ω/Uc is the convective wavenumber. The Corcos model only includes the convective properties of 19 of 30 American Institute of Aeronautics and Astronautics Paper 2010-2904

(a)

(b)

Figure 20. Contour plots of the frequency wavenumber spectrum Φpp (k1 , k3 , ω) of the wall pressure fluctuations with a logarithmic colorscale: (a) ωδ ∗ /U∞ =0.31; (b) ωδ ∗ /U∞ =1.69.

−4 −5

−7

pp

1

Φ (k ,0,ω)

−6

−8 −9 −10 −4

−2

0

2

k δ*

4

6

8

1

Figure 21. Wavenumber frequency spectra for ωδ ∗ /U∞ =1.65 (

), and 1.69 ( ). The green solid line indicates the convective wavenumber kc = ω/Uc and the green broken lines indicate the positive and negative convected acoustic wavenumbers.


the wall-pressure turbulence, and is therefore limited to wavenumbers in the neighborhood of the convective ridge. The model derived by Chase60 tries to describe the wavenumber-frequency spectrum in the subconvective region. It is based on theoretical considerations by formulating a Poisson problem, where the wall-pressure fluctuations result from mean-shear/turbulence interactions (subscript M ) and from turbulence/turbulence interactions (subscript T ). It can be expressed as follows: 2 2 1 Φpp (k, ω) 2 2 (k+ δ) + 1/b = (26) CM (k1 δ) + CT (kδ) 5 ρ20 u3τ δ 3 (kδ)2 + 1/b2 [(k+ δ)2 + 1/b2 ] 2 2 with k+ = (ω − Uc k1 )2 /(huτ )2 + k 2 , CM =0.1553, CT =0.0047, b=0.75, and h=3. Chase69 has improved his model by refining the turbulence/turbulence interaction term, and by taking into account the acoustic contribution, leading to a secondary peak around the acoustic wavenumber κ0 = ω/c0 : CM (k1 δ)2 k 2 Φpp (k, ω) 1 = 5 2 3 3 2 2 ρ0 u τ δ [(k+ δ) + 1/b ] 2 |k 2 − κ20 | + ǫ2 κ20 2 2 c2 |k 2 − κ20 | c3 k 2 2 (k+ δ) + 1/b (27) × c1 + + 2 + CT (kδ) (kδ)2 + 1/b2 k2 |k − κ20 | + ǫ2 κ20

−4

−4

−5

−5

−6

−6

−7

−7

−8 −9

−8 −9

−10

−10

−11

−11

−12 −4

(a)

Φpp(k1,0,ω)

Φpp(k1,0,ω)

where c1 = 23 , c2 = c3 = 16 , and the amplitude of the acoustic peak is defined by the value of ǫ ≃ 0.2.

−2

0

2* k1δ

4

6

−12 −4

8

(b)

−2

0

2* k1δ

4

6

8

Figure 22. Wavenumber frequency spectra for different fixed nondimensional frequency ωδ ∗ /U∞ =0.127,

0.636, 1.145, 1.654, 2.164, 2.673, 3.182, and 3.691 ( ). (a) Comparison with Corcos’ model Eq.(25) for the same frequencies with φpp (ω) given by the Goody model Eq.(24) ( ). (b) Comparison with ), and with Chase’s model extended in the acoustic domain standard Chase’s model Eq.(26) ( Eq.(27) ( ).

The comparisons in Fig.22 indicate that all the models are underestimating the width of the convective peak, and do not reproduce its asymmetry. Some discrepancies also occur for the location of the peak since the variations of the phase speed are not taken into account (notably the lower value for high-frequencies as shown in Fig.17). Note also that the two last frequencies of the LES depicted are beyond the resolvability cut-off, thus are artificially damped. In the low wavenumber range, the Corcos model tends to overestimate the pressure levels, as already noted by several authors.7, 72, 73 The levels given by the Chase model are closer to those from LES. The value chosen for the acoustic peak (ǫ = 0.2) in the extented Chase model overpredict the acoustic component, which is overwhelmed by the tail of the convective for the spectra from LES. 2.

Comparison with models for the far-field noise

The relationship between the surface pressure on a rigid wall beneath a TBL and the radiated sound pressure above the layer is indirect. The reflection principle indeed implies that the surface pressure integral solely represents the image sources (minus the tangential viscous stresses). However the volume contribution from the turbulent source or its image from the plane are of the same order of magnitude on a statistical sense. Since the wall pressure distribution is the consequence of the same turbulent sources, and can be deduced from it, it seems natural to try to link the wall pressure spectrum to the far field spectrum. 21 of 30 American Institute of Aeronautics and Astronautics Paper 2010-2904

We start with the expression (22), derived in § II-B-2, linking the acoustic pressure to the Lighthill’s tensor in the wavenumber-frequency space, taken without convection. The free-space Green function is replaced by the half-plane version: ˜ 2 |y2 , k, ω) = − i e−i˜γ |x2 −y2 | + e−i˜γ |x2 +y2 | (28) G(x 2˜ γ which yields for x2 − y2 > 0:

˜ 2 |y2 , k, ω) = − i e−i˜γ y2 cos(˜ G(x γ x2 ) γ˜

(29)

Eq.(22) becomes: p′ (k, ω) = −

i γ

Z

∞

γ x2 ) dy2 (ki + γδi2 ) (kj + γδj2 ) Tij (y2 , k, ω)e−i˜γ y2 cos(˜

(30)

−∞

where γ 2 = κ20 − k 2 with k 2 = k12 + k32 . By noting κ20 γij (k/κ0 ) = (ki + γδi2 ) (kj + γδj2 ), and [δ.Tij (k, ω)] the integral over y2 , Eq.(30) becomes:77 γij (k/κ0 ) |p′ (k, ω)| = κ20 [δ.Tij (k, ω)] p 2 κ0 − k 2

The spatial dependence is recovered thank to a double inverse Fourier transform, leading to: ZZ ∞ 2 iκ0 [δ.Tij (k, ω)]γij (k/κ0 ) ix2 √κ20 −k2 2 p e p′ (x, ω) = d k κ20 − k 2 −∞

The stationary phase approximation is used to evaluate the contribution of the wavenumbers to the far field pressure: 2πκ20 ¯ ω)]γij (k/κ ¯ 0) [δ.Tij (k, |p′ (x, ω)| = r where k¯ is the acoustic trave wavenumber. By noting the spectrum of the turbulent motions as follows: Φijkl (k, ω) =

2π Tij (k, ω)Tij∗ (k, ω) A

where A denotes the x1 x3 -area of the turbulent zone, the spectrum of the radiated noise can be written:    X ¯ ¯ ¯ A Φijkl (k, ω)γij (k/κ0 )γkl (k/κ0 )  (31) Φr (x, ω) = 2 ρ2∞ u4τ (κ0 δ)4   r ρ2∞ u4τ δ 2 ijkl

Blake7 proposes a similar expression for the wall pressure spectrum, namely:   2 2 X  U Φ (k, ω)γ (k/κ )γ (k/κ ) ωδ ∞ ijkl ij 0 kl 0 Φw (k, ω) = ρ2∞ u4τ δ2   c∞ U∞ ρ2∞ u4τ δ 2 [1 − (k/κ0 )2 ]

(32)

ijkl

By equating (31) and (32), with certain hypotheses as the fact that the term [1 − (k/κ0 )2 ] in (32) only represents an added directivity, and that only the supersonic wavenumber contribution is taken for Φw , we get: 2 A ωδ ∗ Φw (k ≤ κ0 , ω)U∞ Φr (r, ω)U∞ /δ ∗ 4 (33) ≃ M ∞ ρ∞ u4τ r 2 U∞ ρ∞ u4τ M2∞ δ ∗ 3 This relationship between wall spectrum and the acoustic spectrum was given by Blake,77 and is applied to the LES database. The approximation is superimposed on the acoustic spectrum in Fig.23(a).

The results are also compared to more global statistical models. At low Mach numbers, several authors give a simple relation between the acoustic domain of the wavenumber-frequency spectrum Φpp (k1 , k3 , ω) and the frequency spectrum of the radiated sound.8, 16, 77 Of course, since the Mach number is not small in 22 of 30 American Institute of Aeronautics and Astronautics Paper 2010-2904

140

100

10

∞

τ

10 log [φ(ω)U /(ρ2u4M2δ*3)]

120

80 60 40 20 0 −20 −2 10

(b)

0

10

*

ωδ /U

∞

Figure 23. (b) Power spectral density of the pressure perturbations in the acoustic field (x2 /δref = 42.3) ∗ as a function of the Strouhal number ωδref /U∞ and for various streamwise locations: ( ▽)x1 /δref = ◦) x1 /δref = 19.8, ( ∗) x1 /δref = 33.1, ( +) x1 /δref = 46.4, ( △) x1 /δref = 59.7, ( 6.6, ( ) x1 /δref = 72.9. Different simplified models are superimposed: ( ) Eq.(35), ( ) Eq.(36), ( ) Eq.(37), ( ) Eq.(38).

the present study (M=0.5), this relation is complicated by convection and refraction effects. Considering the wavenumber-frequency spectrum of the fluctuating pressure pδ just outside the boundary layer, the radiated pressure is given by: ZZ ∞ pδ (x, t) = pδ (k, ω) exp [i(k.x + γ(k)(x2 − δ) − ωt)] d2 kdω −∞

where γ(k) = sgn(κ0 )|κ20 − k 2 |1/2 for k < κ0 , and γ(k) = i|κ20 − k 2 |1/2 for k > κ0 . We have noted k = (k1 , k3 ) and κ0 = ω/c∞ is the acoustic wavenumber. At large distances, the integral can be approximated by using the method of stationary phase, which yields: ZZ ∞ ′ 4π 2 cos2 θ hp2 (x, t)i ≈ κ0 κ′0 hpδ (κ0 x/|x|, ω)p∗δ (κ′0 x/|x|, ω ′ )i ei(ω−ω )t dωdω ′ 2 |x| −∞ where θ = cos−1 (x2 /|x|). The stationary assumption for the fluctuating pressure hpδ (k, ω)p∗δ (k, ω ′ )i = A/(2π)2 δ(ω −ω ′ )Φδ (k, ω) is then invoked. Moreover, in the low Mach number limit, for the low wavenumbers characteristic of the acoustic domain, the spectrum at the distance δ from the wall is supposed similar with the wall spectrum, i.e., Φδ (k, ω) = Φw (k, ω) for k < |κ0 |. The spectrum of the radiated pressure is thus:16 Φr (x, ω) =

2Aκ20 cos2 θ Φw (κ0 x/|x|, ω) |x|2

(34)

Dowling8 gives two empirical expressions for the wall-pressure spectrum in the acoustic domain. The first one is deduced for the modified Chase69 model, which is derived from considerations about the Lighthill theory: ∗ −1 2 4 2 ∗3 ωδ uτ δ −2 ρ∞ uτ M δ (35) Φw (0, ω) = 1.2 × 10 U∞ U∞ δ ∗ U∞ −1 3 ρ2 u 4 M 2 δ ∗ ωδ ∗ = 9.1 × 10−4 ∞ τ (36) U∞ U∞ 23 of 30 American Institute of Aeronautics and Astronautics Paper 2010-2904

Another expression is provided by Sevik (1986), who fitted experimental points on a buoyant body for a frequency range 3 < ωδ ∗ /U∞ < 30: Φw (0, ω) = 5.6

ρ2∞ u4τ M2 δ ∗ U∞

3

ωδ ∗ U∞

−4.5

(37)

This expression is used by Howe16 in the formula (34), which yields: 254AM4 (uτ /U∞ ) Φr (0, ω) ≈ 2 3 ρ∞ u τ δ |x|2 (ωδ/U∞ )5/2

(38)

taking θ = π/2. The results obtained with equations (35), (36), (37), and (38) and superimposed on the acoustic spectra from the LES in Fig.23. The results for the Sevik formula given by Howe, or by putting (37) in (34) are very close, and is generally considered to be a higher bound for the quadrupole noise from a turbulent boundary layer. The acoustic results that he used are indeed undoubtly contamined by a source of exterior noise since the results for smooth and rough surface are identical (which is contradictory). Moreover, the range of validity is rather for high frequencies ωδ ∗ /U∞ > 1, and low Mach numbers. We can see that that the model effectively largely overestimate the sound although the exponent for the frequency dependence seems roughly correct. The levels predicted by the modelling of the acoustic domain in Chase’s model are in better agreement, but the frequency dependence is less favorable. The reliability of the various models remains controversial even after several decades of investigation. The results from a DNC solver can thus help to revisit these models.

III.

Trailing edge noise

The second configuration concerns the flow around a NACA 0012 airfoil of chord 495 mm with a truncated trailing edge at Mach 0.2 with an angle of attack α of 2.5◦ . The Reynolds number based on the airfoil chord is equal to 2.32 × 106 . A.

Large eddy simulation of the flow over a NACA0012 airfoil

The DNC simulation is described in AIAA paper 2009-3201.2 Only the main features are summarized there. Same numerical alogorithms as the TBL case are used to preserve the propagative nature of the weak acoustic waves. The use of the multi-size mesh multi-time step algorithm allows to reduce the calculation cost by a factor 10, while preserving a significant number of points to discretize the fine scales of the turbulent boundary layers and to propagate the acoustic field in the far field (figure 1(b)). A NACA 0012 airfoil of chord 495 mm with a truncated trailing edge (the height of the truncation is 2.5 mm, ≈ 0.505%) at Mach 0.2 with an angle of attack α of 2.5◦ has been considered, corresponding to the nominal configuration of the EXAVAC experimental program, realized by the French office ONERA.24 This case is also close to the experiments of Brooks and Hodgson.22 Two 3-D LES simulations (run 1 & 2) have been performed to study the influence of grid parameters on the quality of the solution, compared to the experimental databases. CH-type grid systems are used. Two successive doublings are imposed at fixed locations away from the airfoil, so that the number of points is reduced by a factor 6.4 (206×106 to 32×106) for run 1, and by a factor 7.62 (212×106 to 27.8×106) for run 2. Grid spacings in wall units close to the trailing edge are kept identical in both runs: ∆ξ + ≈ 30, ∆η + ≈ 5, and ∆z + ≈ 20 in the streamwise, radial, and spanwise directions, respectively. Periodic conditions are applied in the spanwise direction, with an extent corresponding to approximatively 2% of the chord length. First validations against experiments indicate encouraging results. The physical mechanisms are well captured, even if the boundary layer displacement thickness near the trailing edge is slightly underestimated, due to the too coarse grid resolution, or insufficient convergence. This implies a small surestimation of the vortex shedding frequency. In this configuration, turbulent boundary layers develop over most of the airfoil and noise is produced as turbulence passes over the trailing edge. Furthermore, a narrowband noise component associated with the vortex shedding from the truncated trailing edge is expected.2, 78 All the analysis presented here is based on the LES database named ”run 1” in Ref.2


B.

Characteristics of wall-pressure fluctuations in the acoustic domain

The diffraction theory of Chandiramani-Chase17–21 is based on the knowledge of the pressure over the airfoil, in order to predict the radiated sound. That is why experimental22–24 and numerical25, 26 investigations of trailing edge noise often provide an analysis of the wall pressure. Its spectral content, notably the hydrodynamic evanescent waves, provides the energy with is converted into acoustics at the trailing edge. Frequency-wavenumber spectra 1500

1000

1000

500

500

k , m−1

1500

−500

−500

−1000

−1000

−1500 −20

−10

0 f , kHz

10

−1500 −20

20

−10

(a) 1500

1000

1000

500

500

k , m−1

1500

0 f , kHz

10

0 f , kHz

10

20

(b)

0

1

0

1

k , m−1

0

1

0

1

k , m−1

1.

−500

−500

−1000

−1000

−1500 −20

−10

0 f , kHz

10

−1500 −20

20

(c)

−10

20

(d)

Figure 24. Contour plots of the frequency wavenumber spectrum Φpp (k1 , 0, f ) of the wall pressure fluctuations in dB (ref 2×10−5 Pa2 m/Hz) for 4 locations along the pressure side (run 1): (a) X/C=0.4; (b) X/C=0.6; (c) X/C=0.8; (d) X/C=0.96. 20 contours are equally spaced between -20 and 20 dB. The dashed line represents the convective relation kc = ω/Uc with Uc = 0.7U∞ , and the dotted line represents the acoustic convected wavenumber κ0 = ω/(U∞ ± c∞ ).

Figure 24 shows the two-dimensional wavenumber-frequency spectra for 4 successive locations at 0.4, 0.6, 0.8 and 0.96. The procedure to compute the spectra is similar as that used for the TBL database. Here, a first step consists in transforming the ξ-coordinate in curvilinear abscissa, and then to interpolate with cubic spline on a regular mesh, since the original longitudinal distribution is more or less irregular. The domain of computation is centered around each locations and covers 100 points along the chord and 96 in the transverse direction. The wavenumber resolution is ∆ks = 41.9 m−1 and ∆kz = 300.6 m−1 . 7600 samples are divided in 4 overlapping segments, which have been recorded every 100 timesteps of the LES. The contour maps of Fig.24 presents a principal peak that characterizes the convected nature of the wall-pressure field. The mean convection velocity is retrieved (dashed line). The convection ridge is more asymmetric as the boundary layer develops, with a tendency to broaden when the frequency rises.79 Another 25 of 30 American Institute of Aeronautics and Astronautics Paper 2010-2904

striking point is the small crests that appear along the dotted lines in Fig.24(d). This represents sound waves propagating either in the positive or negative direction. The spots visible on the line corresponding to the upstream propagation correspond to the vortex shedding noise from the blunt trailing edge. This spot is still detectable for X/C=0.8, but rapidly vanishes. Terracol et al.25 observed that the vortex shedding peak is perceived on the whole chord by plotting point-spectra. On the other side, Garcia Sagrado et al.23 reported a peak only very close to the airfoil for a blunted NACA0012 airfoil ar Reynolds 4 × 105 . This point should obviously depend on the level on the peak above the background noise, and on the flow Mach number, since airfoil self-noise scales as M5 . A zoom is realized to represent exactly the wavenumber-frequency region measured by Bult´e et al.24 in the EXAVAC database. They used two arrays of Kulite sensors with 4.5 mm spacing in both the lateral and longitudinal direction. The experimental spectrum compares favorably with the LES result in Fig.25. The two components are visible, namely the convective motion aligned with a positive slope, and the acoustic component with a negative slope, which corresponds to the diffracted field.

(a)

(b)

Figure 25. Wavenumber-frequency spectrum Φpp (k1 , 0, f ) of the wall pressure fluctuations in dB (ref

2×10−5 Pa2 m/Hz). (a) Simulation # 1 at location X/C=0.96 (pressure side); (b) EXAVAC experiment (Fig.5 of Bult´ e et al.24 ).

The convective ridge in the k1 -k3 plane is represented in Fig.27 for f =4073 Hz. Some noise is visible on the contours because of the reduced spanwise extent in the simulation and the limited time interval but its shape conforms with expectations. The acoustic domain is also clearly distinguishable in Fig.26 inside the ellipse.

IV.

Conclusions

Acoustic analogy The temporal- and frequency-domain formulation of FWH analogy have been implemented for the TBL database. The principal findings of this analysis are summarized. A portion of the LES volume is used implying 3 limitations: the relatively short time duration of the database, its severe truncation in the streamwise direction, the use of periodic condition in the spanwise direction in the LES. Windowing has been tested to reduce aliasing effects due to truncations. The surface or volume are furthermore replicated in the spanwise direction to reproduce the periodicity. A large number of replications considerably enhances the overall noise level, and promotes the long wavelenghtes. For the volume integral of FWH analogy, the term T11 is highly dominating, whereas for the surface integral, the pressure term overwhelms the viscous stresses. Among these viscous stresses, τ21 is the sole non-negligible term. This a true source of sound even at Mach 0.5, but the contribution is limited to few percents of the acoustic radiation. Wavenumber-frequency analysis The wavenumber-frequency spectrum have been obtained by threedimensional fast Fourier transform. The principal peak in the two-dimensional spectra characterizes the


50

50

Φpp(k1,0,f) dB ref 0.2µ Pa

60

40 30 20

pp

1

Φ (k ,0,f) dB ref 0.2µ Pa

60

10

−2

0

2

*

k1δ

30 20 10 0 −4

4

−2

0

(a) 60

50

50


60

40 30

*

k1δ

2

4

6

(b)

40 30 20

pp

1

20

pp

1


0

40

10 0 −5

0

*

k1δ

5

10 0 −10

10

(c)

−5

0

k1δ*

5

10

15

(d)

△), 1.7×104 Hz ( o), 2.6 × 10 Hz ( ), 3.5 × 10 Hz ( ▽). The red solid line indicates the convective wavenumber kc = ω/Uc and the red broken line indicates the acoustic convected wavenumber for the frequency ωδ ∗ /U∞ =0.7. (a) X/C=0.4; (b) X/C=0.6; (c) X/C=0.8; (d) X/C=0.96. Figure 26. Wavenumber frequency spectra for different frequencies f : 8.7×103 Hz ( 4

4


(a)

(b)

(c)

(d)

Contour plots of the frequency wavenumber spectrum Φpp (k1 , k3 , f ) of the wall pressure fluctuations in dB (ref 2 × 10−5 Pa2 m/Hz) f =4073 Hz (run 1): (a) X/C=0.4; (b) X/C=0.6; (a) X/C=0.8; (a) X/C=0.96. Color levels between -7 and 4 dB.

Figure 27.

convected nature of the wall-pressure field (the line passing through gives the mean convection velocity). In the case of the flat plate TBL, even if the low wavenumber region is hardly resolved, no peak is noticed near the acoustic wavenumber κ0 = ω/c∞ . This may be due to the very low levels of direct acoustic radiations. On the contrary, two small crests appear near the trailing edge of the airfoil. This represents sound waves propagating either in the positive or negative direction. The spots visible on the line corresponding to the upstream propagation correspond to the vortex shedding from the blunt trailing edge. The detection of the acosutic spot should obviously depend on the level of the peak above the background physical or numerical noise, and on the flow Mach number, since airfoil self-noise intensity scales as M5 (M=0.2 in the present case), whereas it scales as M8 for the flat plate (M=0.5). Moreover there is no clear separation of the wavenumber-frequency plane into acoustic and convective domains when the Mach number is large, and the properties of P (k, ω) are not well understood.

Acknowledgments This work was granted access to the HPC resources of IDRIS and CCRT under the allocation 2009-41736 made by GENCI (Grand Equipement National de Calcul Intensif).

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