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Pressure-based integral formulations of Lighthill–Curle's analogy for internal aeroacoustics at low Mach numbers
Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Pressure-based integral formulations of Lighthill–Curle's analogy for internal aeroacoustics at low Mach numbers N. Papaxanthos a,n, E. Perrey-Debain a, S. Bennouna a,b, B. Ouedraogo a, S. Moreau a, J.M. Ville a a Laboratoire Roberval UMR CNRS 7337, Sorbonne Universités, Université de Technologie de Compiègne, CS 60319, 60203 Compiègne, France b Valeo Thermal Systems, 8 rue Louis Lormand, 78320 La Verrière, France
a r t i c l e i n f o
abstract
Article history: Received 18 July 2016 Received in revised form 7 January 2017 Accepted 17 January 2017 Handling Editor: A.V. Metrikine
The use of unsteady incompressible-flow simulations has become very popular for aeroacoustic noise predictions at low Mach numbers, as it provides a good compromise between computational time and reliable predictions. The acoustic radiation of the aerodynamic sources is calculated in a second step by solving an appropriate system of acoustic equations. In order to predict the noise produced by confined flows, two integral formulations of Lighthill–Curle's analogy are developed. Both formulations require only the knowledge of the incompressible-flow pressure. The first one, which is based on Ribner's reformulation of Lighthill's source terms, is exact and shall serve as a reference to the second approximate formulation which involves only the pressure on the boundary of the fluid domain. The two formulations are shown to be in excellent agreement for the case of a long straight duct obstructed by a diaphragm which makes the simplified integral formulation a reliable alternative to usual computational methods. The sound power levels as well as the modal contributions compare favorably with measurements. Moreover, it is shown that the computed radiated sound is independent of the outlet condition of the flow simulation. & 2017 Elsevier Ltd All rights reserved.
Keywords: Aeroacoustics Low Mach Internal flows Lighthill's analogy BEM Large eddy simulation HVAC noise
1. Introduction The prediction from the conception phase of the generated sound by a turbulent flow inside ducts, such as heating, ventilation and air-conditioning (HVAC) systems or pipelines, is a major competitive challenge for designers. In the car industry, the development of silent motorization as for hybrid or electric vehicles has made the noise resulting from air conditioning systems a dominant one in the vehicle. In this respect, our concern deals with the presence of confined obstacles with low Mach number flows that results in unsteady aerodynamic fluctuations which generate sound waves propagating along the ducting. Computational aeroacoustics (CAA) has achieved substantial progress over the past decades especially through advances in computational fluid dynamics (CFD) and the development of processor performance. CAA becomes in industries more used than semi-empirical predictions based on power laws established by Lighthill [1] or by others later [2,3]. Using purely
n
Corresponding author. E-mail address: [email protected] (N. Papaxanthos).
http://dx.doi.org/10.1016/j.jsv.2017.01.030 0022-460X/& 2017 Elsevier Ltd All rights reserved.
Please cite this article as: N. Papaxanthos, et al., Pressure-based integral formulations of Lighthill–Curle's analogy for internal aeroacoustics at low Mach numbers, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j. jsv.2017.01.030i
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computational means, a hierarchy of methods, ranging from stochastic noise generation (SNGR) with unsteady Reynoldsaveraged Navier–Stokes (RANS) based turbulence models at the lower end [4] to the resolution of all flow scales with the direct numerical simulation (DNS) at the higher end [5], provide estimation of the radiated sound from unsteady-type flows. At low Mach and high Reynolds numbers, the unsteady incompressible-flow simulation coupled with an acoustic analogy has proven to be a good compromise with reasonable computational time and accurate prediction [6]. This two-step resolution, called hybrid approach, exhibits reliable results for various industrial applications as the prediction of the noise emitted by simplified elements of HVAC systems [7,8], by trailing edges [9] or by suction valves [10]. A fundamental assumption for the hybrid approach-based prediction is the one-way coupling of flow and sound, thus, the feedback mechanisms between the acoustic and aerodynamic fields are ignored in the analysis. Modern aeroacoustics began in the early 1950s with Lighthill's analogy which brought to light equivalent acoustic quadrupole sources in the flow field as responsible for the flow noise [1]. Curle extended next the analogy to account for the presence of solid surfaces immersed in the unsteady flow field [11]. For low Mach number flows, Curle demonstrated that the scattered field by the rigid body dominates the direct field from quadrupoles if the surface is compact (i.e. of small dimensions compared to the acoustic wavelength). This finding has been largely used to calculate at little expense the radiated sound from flow/compact obstacle interaction [12,13]. Numerous studies have clarified this conversion of the incident field from quadrupoles into a dominant scattered field [14,15]. It is shown that the scattering effect is not restricted to compact obstacles and can also occur in the presence of non-compact surfaces with singularities, such as edges, protuberances or corners. Based on this observation, a recent simplification of the integral form of Lighthill's equation has been proposed [16,17]. Note that contrary to compact obstacles, non-compact surfaces also modify the sound radiation by reflexion and diffraction without any contribution to its generation. The calculation of the flow noise with non-compact surfaces from incompressible CFD data can be accomplished in various manners. Lighthill's equation can be implemented in a finite element framework [18,19]. Green's function tailored to the geometry may be used to account for all the reflections and scattering, this latter can be found analytically for simple geometries [9,20] or numerically for more complex geometries [10,21]. Schram proposes a boundary element method (BEM) approach that computes the acoustic component of the wall pressure [22]. Martínez-Lera et al. solve Lighthill's equation through a boundary value problem for the scattered pressure [16]. In this paper, two integral formulations based on the mere knowledge of the incompressible-flow pressure are derived. In the first one, the classical volume term of Lighthill– Curle's analogy takes a more suitable form from a numerical point of view [23,24]. The second is based on a simplification which uses the fact that scattering phenomena account for most of the radiated sound power [16,17]. The derivation of the two formulations is presented in Section 2. Their validation is supported by a realistic test case: the aerodynamic noise due to the insertion of a diaphragm in a rectangular duct. Before the description of the flow simulation in Section 4, a short presentation of the experimental setup and procedure is made in Section 3. Section 5.1 shows the numerical results and the comparisons with the measurement of the sound power levels and the modal contributions. In Section 5.2, a spatial Fourier transform is performed on one surface of the duct and brings to light spurious pressure fluctuations from the CFD simulation depending on the outlet boundary condition. The approach which is developed in this paper is shown to be not sensitive to this numerical phenomenon and allows to produce consistent results regardless of the outlet condition in the flow modeling.
2. Integral formulations of Lighthill's equation Lighthill's equation is an inhomogeneous Helmholtz equation satisfied by the pressure which is derived simply by rearranging the Navier–Stokes equations [1]. In the frequency domain (the Fourier convention eiωt is adopted), Lighthill's equation is:
( Δ + k2) p = q
where q = −
∂ 2Tij ∂xi ∂xj
(1)
is the source term and is composed of the double spatial derivative of Lighthill's tensor Tij, k = ω/c0 is the wavenumber with c0 being the sound velocity. For isentropic low Mach number flows, it is commonly accepted that Lighthill's tensor takes the simplified form Tij ≃ ρ0 ui uj − τij with ρ0 being the fluid density and ui the i-component of the velocity. The viscous contribution τij is usually neglected for high Reynolds number flows but it is left here as it is provided by an incompressible-flow modeling. In Fig. 1 is sketched an arbitrary confined domain Ω bounded by two ducts from where the fluid enters and exits the domain. The interaction of the flow and obstacles or constrictions generates sound which propagates through the inlet and outlet ducts. In each duct, which are considered identical for simplicity, the pressure consists of acoustic waves only so it can be expressed as a sum of radiating modes:
p (x ) =
∑ am Φm (x ) m
on ∂Ωα
(2)
where the duct mode Φm (x ) obeys the Helmholtz equation and am is a modal coefficient, α takes the value of 1 or 2 depending on the fictitious surface. Using the orthonormality of the modes and the fact that ∂nΦm = − ik m Φm (km is the axial Please cite this article as: N. Papaxanthos, et al., Pressure-based integral formulations of Lighthill–Curle's analogy for internal aeroacoustics at low Mach numbers, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j. jsv.2017.01.030i
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Fig. 1. An arbitrary confined domain Ω with an inlet duct of section ∂Ω1 and an outlet duct of section ∂Ω2 .
wavenumber), the normal derivative of the pressure can be expressed in terms of the pressure as follows:
∂p (x ) = − i ∑ k m Φm (x ) ∂n m
∫∂Ω
p (y ) Φm* (y ) dΓy.
(3)
α
Eq. (3) is known as a Dirichlet-to-Neumann (DtN) condition and is usually written T(p) [25]. In the finite domain its boundaries, the pressure is given by:
C (x ) p (x ) =
⎛
∫Ω qG dΩy + ∫∂Ω ⎜⎝ p ∂∂nGy
−G
∂p ⎞ ⎟ dΓy ∂ny ⎠
Ω and on
(4)
where G = G (x, y ) refers to the classical free-field Green's function. Eq. (4) is the integral representation of Lighthill's equation. The volume integral describes the incident field from quadrupoles while the surface integral accounts for the reflexion and scattering of waves by the surfaces. One recalls that the normal derivative of the pressure is null on rigid walls and that the coefficient C (x ) corresponds to the solid angle divided by 4π and is equal to 1 for a listener in Ω and 1/2 when x lies on boundaries (except corners and edges). The integral formulation (4) can be written in a more appropriate form in view of a BEM resolution:
SΩ [ p](x ) =
∫Ω qG dΩy
(5)
where the RHS contains the source terms given by the flow simulation and the LHS refers to the scattering operator associated with the open cavity Ω :
SΩ [ p](x ) = C (x ) p (x ) −
∫∂Ω p ∂∂nGy dΩy + ∫∂Ω
GT (p) dΩy.
1,2
(6)
However, despite its simplicity, the resolution of Eq. (5) is challenging. In addition to the significant demand in memory to backup the source terms q, their spatial derivatives are not estimated adequately by CFD softwares which usually brings overestimation of the generated sound [26]. In order to ease the computation of the volume integral (5), an alternative way for solving Lighthill's equation based on the knowledge of the incompressible-flow pressure is presented. Two integral formulations are developed, the first one is exact and shall serve as reference to a simplified version. The starting point is to split the pressure as the sum of two terms:
p = p0 + pc
(7)
where p0 is the incompressible-flow pressure provided by the incompressible-flow simulation, pc is called the pressure correction term. By combining the decomposition (7) and Lighthill's equation, and noting that p0 is the solution to Poisson's equation Δp0 = q , it results the following inhomogeneous Helmholtz equation for the pressure correction term:
(Δ + k2) pc = − k2p0 .
(8)
Therefore, pc can be seen as the response to the fluctuations of p0. Ribner calls Eq. (8) the dilatation equation with the source term the fluid pulsation [23]. According to the relation ∂npc = T (p) − ∂np0 and thanks to the linearity of the DtN condition, the inversion of Eq. (8) yields the exact integral form:
SΩ [ pc ](x ) = − k2
⎛ ∂p ⎞ ⎜ G 0 − GT (p0 ) ⎟ dΓy. ∂ny ⎠ 1,2 ⎝
∫Ω p0 G dΩy + ∫∂Ω
(9)
Please cite this article as: N. Papaxanthos, et al., Pressure-based integral formulations of Lighthill–Curle's analogy for internal aeroacoustics at low Mach numbers, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j. jsv.2017.01.030i
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The resolution of Eq. (9), call it Formulation A, requires the storage of the incompressible-flow pressure in the entire domain and p0 and ∂np0 on the fictitious surfaces. Note, by adopting the same decomposition (7), Schram develops in Ref. [22] an alternative formulation that also avoids the numerical errors related to the volume integral calculation. Another formulation can be derived from Eq. (9) when manipulating the volume integral. Indeed, noting that −k2G = Δ (G − G0 ), where G0 is the static Green's function (k ¼0), and by integrating by parts twice, one obtains:
SΩ [ pc ](x ) ≃
∫∂Ω p0
∂ ( G − G0 ) ∂ny
dΓy +
⎛ ∂p ⎞ ⎜ G 0 0 − GT (p0 ) ⎟ dΓy. ∂ny ⎠ 1,2 ⎝
∫∂Ω
(10)
Call it Formulation B. The volume integral:
Π=
∫Ω q ( G − G0 ) dΩy
(11)
can be seen as the compressible part of the incident field from quadrupoles and is intentionally omitted in the RHS of Eq. (10). In Ref. [16], Martínez-Lera et al. have found the quantity Π negligible for the case of a non-compact wing in an unbounded domain as a result of the proximity of the main sources to the trailing edge. The fact that the reflected part of the incident wave field accounts for most of the radiated power is discussed in detail in Ref. [11] for compact obstacles and in Ref. [17] for confined flows with obstacles. Note that in Ref. [16], the acoustic radiation is computed via a boundary value problem that merely requires pressure data on boundaries, as in the case of Formulation B.
3. Experimental setup and measurement procedure The test rig is made of two main parts described in detail with the measurement procedure in Ref. [27]. The flow generation part consists of a fan, mufflers, a plenum chamber and an air flow rate meter. The measurement part is of rectangular cross-section (a portion is visible in Fig. 2) and includes a total of 96 flush mounted microphones located upstream and downstream of the test section and a particle image velocimetry (PIV) system. The last is composed of a pulsed laser with an operating frequency of 15 Hz and two cameras that allow the measurement of the three components of the velocity. Two anechoic terminations enclose the measurement part. The acoustic measurements are accomplished with the so-called multiport method which permits a multimodal characterization of the passive and active acoustic properties of obstacles [28]. The passive part is controlled by the duct and obstacle geometries and determines how the sound propagates through the system. The active part identifies the aeroacoustic sources due to the interaction between the flow and the obstacle. The experimental procedure is conducted in two steps. First, the passive properties are measured by using a movable external source which generates an acoustic field uncorrelated with the one generated from the test section. The impedance of the upstream and downstream terminations are deduced from this first step. The active properties of the obstacle are obtained in a second step with the use of a correlation technique that suppresses the flow disturbance [29]. The passive and active generation processes can be related to the following linear equation:
Pout = SPin + Ps
(12)
where Pin and Pout are respectively the incoming and outgoing pressure wave vectors and Ps is the source vector (the active part). Each vector is 2N -dimension (N is the number of propagating modes) as it includes informations about the propagating waves from both sides of the obstacle. The free-reflection scattering matrix S (the passive part) is of dimension 2N × 2N and contains the reflection and transmission characteristics of the test section.
Fig. 2. View on the measurement part, the obstacle (here a diaphragm) and the test section are made with plexiglass.
Please cite this article as: N. Papaxanthos, et al., Pressure-based integral formulations of Lighthill–Curle's analogy for internal aeroacoustics at low Mach numbers, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j. jsv.2017.01.030i
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Fig. 3. The computational domain. The pressure on the highlighted surface is used in Section 5.2, as well as the point x 0 . (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
4. Turbulent flow simulation The geometry (see Fig. 3 and the view of the test section Fig. 2) consists of a constriction of rectangular shape with dimensions D × D/2 × D/12.5 centered in a duct of rectangular cross-section 2D × D with D¼10 cm. The duct inlet is located 30 cm upstream of the obstacle whereas the duct outlet is 1.6 m downstream to ensure that remaining hydrodynamic fluctuations are negligible on the outlet section. The structured mesh used for the CFD, illustrated in Fig. 4, comprises 9 million cells. The mesh is built using 4 resolution levels ranging from 0.5 mm to 4 mm. The smallest cells are located in the vicinity of the diaphragm edges in order to capture correctly the physics of the shear layers. To ensure adequate resolution near solid surfaces, 14 boundary layers consisting of local refinement are added with a growth rate of 1.5 and a total thickness of 1 mm. The flow simulation is carried out with the commercial software Star-CCM + . A RANS k –ϵ simulation is first performed to give an initial condition for the incompressible large eddy simulation (LES). Inlet boundary conditions, velocity components and turbulence kinetic energy are prescribed using realistic data provided by PIV measurement. The measured mass flow rate is 565 kg/h which corresponds to an average velocity over the duct section of 6.5 m/s. The corresponding Mach number is 0.019 and the Reynolds number 4.2⁎104 based on the transverse dimension D. At the outlet section, an outflow condition (constant flow rate) is applied. A large eddy simulation with a Smagorinsky subgrid scale model is carried out with a simulated physical time of 0.32 s and a time step δt = 10−5 s. Aerodynamic quantities are stored each 10 δt which is sufficient to ensure that the frequency range of interest (up to 3400 Hz) is covered. The physical time of 0.32 s is divided into eight segments of 0.05 s each with 1/4 overlapping. A window function is then applied on each segment. The flow might be regarded as statistically converged after 0.1 s so the first three segments are discarded. Fig. 5 shows an instantaneous snapshot of the velocity at x1 = D in the region −D ≤ x3 ≤ 5D . The steady flow upstream from the diaphragm is uniform with fine boundary layers near the duct walls. From the obstacle to D downstream, a jet is formed emanating from the aperture with the creation of a periodic vortex shedding from the upstream diaphragm lips. The additional contraction of the jet, called the vena contracta, is clearly visible. As the jet progresses away from the diaphragm, one distinguishes large coherent structures which arise from the arrangement of smaller eddies. A steady state is then slowly recovered after a certain distance. Complete descriptions of the flow through a diaphragm can be found in several studies, such as in Ref. [5,30]. Fig. 6 displays a comparison of the mean velocity vectors from the PIV measurement and the LES in the plane x1 = D . The velocity vectors are determined by the following expression:
〈u〉 = 〈u2 〉e 2 + 〈u3 〉e 3.
(13)
Results are similar though slight discrepancies can be identifiable in the recirculation regions where the rotational speed is more pronounced in the measurement. The mean turbulence kinetic energy gives an indication of the location and intensity of the unsteady phenomena and is defined as:
k=
〈u1′ 2〉 + 〈u2′ 2 〉 + 〈u3′ 2 〉 2
(14)
where u′i is the fluctuating part of the velocity and stems from the Reynolds decomposition ui = 〈ui 〉 + u′i . Fig. 7 shows the mean turbulence kinetic energy over the duct cross-section located 3 cm downstream of the diaphragm. A quite fair agreement is obtained between the LES prediction and the PIV result. High energy corresponds to small eddies convected in the shear layers. Differences are observed especially near the right angles of the constriction which might be explained by the fact that the vortex dynamics is not perfectly captured by the simulation. Fig. 8 shows the rms value of the component T33 of Lighthill's tensor in the duct at x1 = D for several frequencies. It serves to illustrate the location of the dominant sources of sound. The other components of Lighthill's tensor are for all frequencies lower than T33/2. At low frequency, the longitudinal fluctuations reach their maximum values at about 2D away from the obstacle and are due in part to large structures impinging on the duct walls. As the frequency increases, predominant
Fig. 4. View of the mesh near the diaphragm at x1 = D . The 4 resolution levels are visible with the finest one surrounding the diaphragm edges.
Please cite this article as: N. Papaxanthos, et al., Pressure-based integral formulations of Lighthill–Curle's analogy for internal aeroacoustics at low Mach numbers, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j. jsv.2017.01.030i
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0.1
0.1
0.075
0.075
x2 (m)
x2 (m)
Fig. 5. Snapshot of the instantaneous velocity magnitude at t¼ 0.25 s in the plane x1 = D , amplitudes ranging from 0 (blue) to 43 m/s (red). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
0.05
0.05
0.025
0.025
0
0.2
0
0.25
0.2
0.25 x3 (m)
x (m) 3
Fig. 6. Mean velocity vectors in the plane x1 = D , (a) PIV, (b) LES, amplitudes ranging from 0 (blue) to 41 m/s (red). Cameras do not have a direct access to the flow inside the constriction and this is visible in (a). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
0.1
x2 (m)
2
x (m)
0.1
0.05
0 0
0.05
0.1 x1 (m)
0.15
0.2
0.05
0 0
0.05
0.1
0.15
0.2
x1 (m)
Fig. 7. Turbulence kinetic energy on the duct cross-section located 3 cm downstream from the diaphragm, (a) PIV, (b) LES, amplitudes ranging from 0 (blue) to 143 J/kg (red). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Fig. 8. Component T33 (the rms value) of Lighthill's tensor in the duct at x1 = D and −0.3D ≤ x3 ≤ 3.2D . (a) f = 200 Hz (max. =205 Pa), (b) f = 1000 Hz (max. =99 Pa), (c) f = 2000 Hz (max. =55 Pa), (d) f = 3000 Hz (max. =25 Pa). The scale is linear and goes from blue (T33 = 0 ) to red (max. of T33). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Please cite this article as: N. Papaxanthos, et al., Pressure-based integral formulations of Lighthill–Curle's analogy for internal aeroacoustics at low Mach numbers, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j. jsv.2017.01.030i
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structures become smaller and are located in the shear layers around the vena contracta. In Fig. 8d, the presence of the refinement boxes in the meshing process is visible as it lowers the amplitude of the fluctuations. However, this has no impact on the acoustic results.
5. Results and discussions Due to the difference in size of acoustic and turbulent wavelengths, the requirement in terms of grid resolution is less severe for the acoustic computation than for the CFD. Formulations A and B are then solved by collocation on a coarse mesh which is composed of around 8500 triangular surface cells of maximum dimension 2 cm, this corresponds to λ/5 at 3400 Hz. For the sake of simplicity, the acoustic domain is chosen identical to the CFD domain. It is possible to build a smaller acoustic domain in order to reduce the computational time by moving the outlet surface closer to the turbulence zone, but this must be handled with care [31,32]. The temporal boundary-data p0 and ∂np0 are first summed on the coarse mesh centroids using a conservative mapping from Star-CCMþ then Fourier transformed. For the computation of the reference integral Formulation A, the incompressible-flow pressure over all cells of the CFD mesh is Fourier transformed before the computation of the volume integral is performed. 5.1. Radiated sound power The sound power radiated from the two boundaries ∂Ω1 and ∂Ω2 is computed with: 2
W=
∑∫ α=1
∂Ωα
Re {pvn* } 2
dΓy
(15)
where p = p0 + pc and vn is the normal component of the acoustic velocity. Eq. (15) is used to compare the results of the Formulations A and B and this is illustrated in Fig. 9a. With only surface integrals, Formulation B gives identical results up to 2500 Hz to those of the exact Formulation A. At high frequencies, slight differences become discernible. Thus, the compressible part of the incident field from quadrupoles, i.e. the quantity Π , can be neglected over the major part of the frequency range, as discussed in [16,17]. Note that the boundary terms which involve the gradients of the incompressibleflow pressure in Formulations A and B are negligible, however, the terms T (p0 ) have to be taken into account. From now, all presented results are computed with Formulation B. The computed sound power level (SWL) in Fig. 9b is the result of an averaging process over five segments and is compared with the measurement. It yields a reliable prediction with globally less than 5 dB differences over the whole frequency range. At frequencies 860 Hz and 1720 Hz, discernible jumps correspond to the apparition of the first spanwise and transverse duct acoustic modes respectively. They are less pronounced from measurements which can be explained by a certain amount of vibration and damping from the duct walls. Above 2500 Hz, the calculation curve decreases more than the experimental one and this might be due to the fact that the volume integral Π is neglected. For completeness, the cut-off frequencies of the rectangular duct are reminded here:
fm =
c0 ⎛⎜ p ⎞⎟2 ⎛⎜ q ⎞⎟2 + ⎝ D⎠ 2 ⎝ 2D ⎠
(16)
60
60
50
50 SWL (dB/Hz)
SWL (dB/Hz)
and reported in Table 1. Mode indices p and q are the following spanwise (x1) and transverse (x2) directions respectively and we took c0 = 343 m/s. The correspondence m ↔ (p, q) is given by ordering the cut-off frequencies in the ascending order. The projection on duct modes of the pressure at the duct endings allows the direct calculation of the contribution of each
40
30
20
40
30
500
1000
1500
2000 f (Hz)
2500
3000
20
500
1000
1500
2000
2500
3000
f (Hz)
Fig. 9. (a) Computed SWL from 1 realization, ( ) Formulation A, ( ) Formulation B. (b) Comparison with the measurement, ( ) computed SWL from 5 realizations and Formulation B, ( ) measured SWL, ( ) computed SWL with the use of the tailored Green's function of the duct.
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Table 1 Cut-off frequencies of the duct below 3400 Hz. fm (Hz) (p,q) m
857.5 (1,0) 1
1715 (2,0) 2
1715 (0,1) 3
1917 (1,1) 4
2425 (2,1) 5
2572.5 (3,0) 6
3092 (3,1) 7
mode to the total SWL. It is also possible to calculate the modal contributions using the tailored Green's function of the infinite duct which leads to some interesting results. For the simple geometry of a rectangular infinite rigid duct, the tailored Green's function is expressed as follows [33]:
Gtai .(x, y ) = −
∑ m
1 Φm (x ) Φm* (y ). 2ik m
(17)
By neglecting the volume integral of the ensuing integral formulation, we arrive at:
p (x ) ≃
∫∂Ω
p obs.
∂Gtai . dΓy ∂ny
(18)
where ∂Ωobs. refers to the boundary of the obstacle only. The corresponding SWL is shown in Fig. 9b and reveals that the knowledge of the pressure fluctuations on the obstacle is sufficient to compute the radiated sound. This is evidenced by numerous studies based on an equivalent dipole source distribution in the duct [2,3]. From this result and by neglecting the phase difference between the upstream and downstream faces of the diaphragm, it can be demonstrated that the plane mode contribution to the total SWL is proportional to the square of the drag force acting on the obstacle:
W0 ≃
|F3 |2 4Aρ0 c0
(19)
where A is the area of the duct section. Note that Eq. (19) yields identically radiated powers upstream and downstream of the obstacle, and this is observed both experimentally and numerically. In Fig. 10 are displayed the contribution of the plane mode (Fig. 10a), the first spanwise mode (Fig. 10b) and the first transverse mode (Fig. 10c) to the total SWL. Computed and measured results show a fair agreement. The plane mode contribution computed with Eq. (19) gives identical results than its exact contribution from Eq. (18) and is removed from Fig. 10a. The fine solid line in Fig. 10a is computed by simply taking the incompressible-flow pressure on the surface of the obstacle in Eq. (18) and shows growing discrepancies as the frequency increases. 5.2. Description of the pressure correction term The aim of this section is to highlight the informations contained in the pressure correction term. A wavenumber frequency analysis is operated on the single rectangular surface of the duct (called it Γ) which is shown in Fig. 3. The spatial Fourier transform is computed as follows: 40 SWL (dB/Hz)
60
30 20 10
40
500
1000
1500
2000 f (Hz)
2500
3000
500
1000
1500
2000 f (Hz)
2500
3000
30 20 10
500
1000
1500
2000 f (Hz)
2500
3000
SWL (dB/Hz)
SWL (dB/Hz)
50
30 20 10
Fig. 10. Contributions to the total SWL, (a) plane mode, (b) first spanwise mode (m¼ 1) and (c) first transverse mode (m¼3). ( ) computation, ( measurement, in (a) is also shown the plane mode contribution using only the incompressible-flow pressure in the surface integral: ( ).
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)
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Fig. 11. Wavenumber frequency spectra, (a) computed with P = p0 , (b) computed with P = pc , (c) computed with P ¼p, k1 = 44 rad/m. Amplitudes ranging from blue to red. The pink arrow in (a) indicates the wide background fluctuations. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Φ (x 0, k ) =
1 (2π )2
∫Γ SPP (x 0, ξ ) exp ( − ik. ξ ) dξ
(20)
where SPP = 〈P (x 0 ) P (x 0 + ξ )*〉 is the space correlation between point x 0 and an arbitrary point on the surface, ξ is the separation vector and k = (k1, k3 ) is the two-dimensional wavevector with the component k1 fixed to 44 rad/m. The latter is chosen empirically as it permits a favorable detection of modes. The point x 0 is placed 15 cm behind the diaphragm in the turbulent zone and is indicated by a cross in Fig. 3. The pressure term P in the space correlation can be either the incompressible-flow pressure (P = p0 ), the pressure correction term (P = pc ) or the true pressure (P ¼p). Fig. 11 shows the corresponding wavenumber frequency spectra in logarithmic scale. The analysis of the incompressible-flow pressure (Fig. 11a) shows an intense zone that corresponds mainly to convective structures impinging on the upper wall of the duct. With only the pressure correction term in the space correlation (Fig. 11b), several curves can be identified which correspond to the axial wavenumbers of the propagating modes. The superimposed black lines are built with:
km =
⎛ 2πfm ⎞2 k2 − ⎜ ⎟ . ⎝ c0 ⎠
(21)
Fig. 11c merely illustrates the combination of the two pressures. Note the multiple oblique and vertical lines are due to the Fourier transform over the bounded domain Γ. It is clear from the above analysis that the correction pc represents mainly acoustic waves generated by the turbulent flow. Another information emerges in Fig. 11. One observes in Fig. 11a a non-zero value along the frequency axis as k3 → 0 (pointed out by an arrow). It corresponds to wide background fluctuations which affect a large part of the computational domain. Similar fluctuations are mentioned from another commercial CFD Software in Ref. [34]. This spurious phenomenon is canceled by the pressure correction term as it does not appear anymore in Fig. 11c. Another CFD simulation has been carried out by changing only the outflow condition to a pressure outlet condition. The corresponding wavenumber frequency spectrum constructed with P = p0 is displayed in Fig. 12. Although it is less discernible, the incompressible-flow pressure is also disturbed by the background fluctuations especially at high frequencies. Despite these spurious pressure fluctuations, which differ depending on the outlet condition of the flow simulation, outgoing acoustic waves are well predicted once the pressure p = p0 + pc is recovered. This is confirmed in Fig. 13 where the radiated sound power computed
Fig. 12. Wavenumber frequency spectrum computed with P = p0 . The outlet condition of the flow simulation is changed to a static pressure condition, k1 = 44 rad/m. The scale is the same as Fig. 11. The pink arrow indicates the wide background fluctuations. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Please cite this article as: N. Papaxanthos, et al., Pressure-based integral formulations of Lighthill–Curle's analogy for internal aeroacoustics at low Mach numbers, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j. jsv.2017.01.030i
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10
60
SWL (dB/Hz)
50
40
30
20
500
1000
1500
2000
2500
3000
f (Hz)
Fig. 13. Computed SWL, (
) from flow simulation with the outflow condition, (
) from flow simulation with the pressure condition.
from the two different CFD simulations are shown. When solving the incompressible Navier–Stokes equations, the pressure is defined up to an arbitrary constant function, spatially and time dependent, as soon as Neumann boundary conditions for the pressure are applied on the boundary of the computational domain [35,36]. The resulting wide background fluctuations affect the inlet and outlet surfaces which explains why p0 is non-negligible on those surfaces.
6. Conclusion An approach combining an incompressible-flow modeling and a BEM resolution is applied to the calculation of the radiated noise from the interaction of a low Mach number flow and a ducted diaphragm. Within reasonable assumptions, it is shown that the volume integral (11), which can be interpreted as the compressible part of the incident pressure field generated by a distribution of quadrupole sources, can be neglected. This assumption leads to an approximated integral formulation presented at Eq. (10). The latter offers the advantages that it demands little data transfer and storage as only pressure on the boundaries of the domain is required. The method seems particularly suited for the prediction of the noise produced by confined low speed flows in the presence of obstacles as sound sources generally remain in the close vicinity of the trailing edges of the obstacles. Numerical predictions are found independent of the outlet condition of the flow simulation and in very good agreement with the measurement. The numerical implementation of a Dirichlet-to-Neumann boundary condition at the duct inlet and outlet permits to calculate the modal contributions to the total sound power level and this is also validated by comparisons with the measurement.
Acknowledgment This work is part of the CEVAS (Conception d'Equipements de Ventilation d'Air Silencieux) project, funded by the Picardie region and FEDER (Fonds Européen de DEveloppement Régional).
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Pressure-based integral formulations of Lighthill–Curle's analogy for internal aeroacoustics at low Mach numbers N. Papaxanthos a,n, E. Perrey-Debain a, S. Bennouna a,b, B. Ouedraogo a, S. Moreau a, J.M. Ville a a Laboratoire Roberval UMR CNRS 7337, Sorbonne Universités, Université de Technologie de Compiègne, CS 60319, 60203 Compiègne, France b Valeo Thermal Systems, 8 rue Louis Lormand, 78320 La Verrière, France
a r t i c l e i n f o
abstract
Article history: Received 18 July 2016 Received in revised form 7 January 2017 Accepted 17 January 2017 Handling Editor: A.V. Metrikine
The use of unsteady incompressible-flow simulations has become very popular for aeroacoustic noise predictions at low Mach numbers, as it provides a good compromise between computational time and reliable predictions. The acoustic radiation of the aerodynamic sources is calculated in a second step by solving an appropriate system of acoustic equations. In order to predict the noise produced by confined flows, two integral formulations of Lighthill–Curle's analogy are developed. Both formulations require only the knowledge of the incompressible-flow pressure. The first one, which is based on Ribner's reformulation of Lighthill's source terms, is exact and shall serve as a reference to the second approximate formulation which involves only the pressure on the boundary of the fluid domain. The two formulations are shown to be in excellent agreement for the case of a long straight duct obstructed by a diaphragm which makes the simplified integral formulation a reliable alternative to usual computational methods. The sound power levels as well as the modal contributions compare favorably with measurements. Moreover, it is shown that the computed radiated sound is independent of the outlet condition of the flow simulation. & 2017 Elsevier Ltd All rights reserved.
Keywords: Aeroacoustics Low Mach Internal flows Lighthill's analogy BEM Large eddy simulation HVAC noise
1. Introduction The prediction from the conception phase of the generated sound by a turbulent flow inside ducts, such as heating, ventilation and air-conditioning (HVAC) systems or pipelines, is a major competitive challenge for designers. In the car industry, the development of silent motorization as for hybrid or electric vehicles has made the noise resulting from air conditioning systems a dominant one in the vehicle. In this respect, our concern deals with the presence of confined obstacles with low Mach number flows that results in unsteady aerodynamic fluctuations which generate sound waves propagating along the ducting. Computational aeroacoustics (CAA) has achieved substantial progress over the past decades especially through advances in computational fluid dynamics (CFD) and the development of processor performance. CAA becomes in industries more used than semi-empirical predictions based on power laws established by Lighthill [1] or by others later [2,3]. Using purely
n
Corresponding author. E-mail address: [email protected] (N. Papaxanthos).
http://dx.doi.org/10.1016/j.jsv.2017.01.030 0022-460X/& 2017 Elsevier Ltd All rights reserved.
Please cite this article as: N. Papaxanthos, et al., Pressure-based integral formulations of Lighthill–Curle's analogy for internal aeroacoustics at low Mach numbers, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j. jsv.2017.01.030i
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computational means, a hierarchy of methods, ranging from stochastic noise generation (SNGR) with unsteady Reynoldsaveraged Navier–Stokes (RANS) based turbulence models at the lower end [4] to the resolution of all flow scales with the direct numerical simulation (DNS) at the higher end [5], provide estimation of the radiated sound from unsteady-type flows. At low Mach and high Reynolds numbers, the unsteady incompressible-flow simulation coupled with an acoustic analogy has proven to be a good compromise with reasonable computational time and accurate prediction [6]. This two-step resolution, called hybrid approach, exhibits reliable results for various industrial applications as the prediction of the noise emitted by simplified elements of HVAC systems [7,8], by trailing edges [9] or by suction valves [10]. A fundamental assumption for the hybrid approach-based prediction is the one-way coupling of flow and sound, thus, the feedback mechanisms between the acoustic and aerodynamic fields are ignored in the analysis. Modern aeroacoustics began in the early 1950s with Lighthill's analogy which brought to light equivalent acoustic quadrupole sources in the flow field as responsible for the flow noise [1]. Curle extended next the analogy to account for the presence of solid surfaces immersed in the unsteady flow field [11]. For low Mach number flows, Curle demonstrated that the scattered field by the rigid body dominates the direct field from quadrupoles if the surface is compact (i.e. of small dimensions compared to the acoustic wavelength). This finding has been largely used to calculate at little expense the radiated sound from flow/compact obstacle interaction [12,13]. Numerous studies have clarified this conversion of the incident field from quadrupoles into a dominant scattered field [14,15]. It is shown that the scattering effect is not restricted to compact obstacles and can also occur in the presence of non-compact surfaces with singularities, such as edges, protuberances or corners. Based on this observation, a recent simplification of the integral form of Lighthill's equation has been proposed [16,17]. Note that contrary to compact obstacles, non-compact surfaces also modify the sound radiation by reflexion and diffraction without any contribution to its generation. The calculation of the flow noise with non-compact surfaces from incompressible CFD data can be accomplished in various manners. Lighthill's equation can be implemented in a finite element framework [18,19]. Green's function tailored to the geometry may be used to account for all the reflections and scattering, this latter can be found analytically for simple geometries [9,20] or numerically for more complex geometries [10,21]. Schram proposes a boundary element method (BEM) approach that computes the acoustic component of the wall pressure [22]. Martínez-Lera et al. solve Lighthill's equation through a boundary value problem for the scattered pressure [16]. In this paper, two integral formulations based on the mere knowledge of the incompressible-flow pressure are derived. In the first one, the classical volume term of Lighthill– Curle's analogy takes a more suitable form from a numerical point of view [23,24]. The second is based on a simplification which uses the fact that scattering phenomena account for most of the radiated sound power [16,17]. The derivation of the two formulations is presented in Section 2. Their validation is supported by a realistic test case: the aerodynamic noise due to the insertion of a diaphragm in a rectangular duct. Before the description of the flow simulation in Section 4, a short presentation of the experimental setup and procedure is made in Section 3. Section 5.1 shows the numerical results and the comparisons with the measurement of the sound power levels and the modal contributions. In Section 5.2, a spatial Fourier transform is performed on one surface of the duct and brings to light spurious pressure fluctuations from the CFD simulation depending on the outlet boundary condition. The approach which is developed in this paper is shown to be not sensitive to this numerical phenomenon and allows to produce consistent results regardless of the outlet condition in the flow modeling.
2. Integral formulations of Lighthill's equation Lighthill's equation is an inhomogeneous Helmholtz equation satisfied by the pressure which is derived simply by rearranging the Navier–Stokes equations [1]. In the frequency domain (the Fourier convention eiωt is adopted), Lighthill's equation is:
( Δ + k2) p = q
where q = −
∂ 2Tij ∂xi ∂xj
(1)
is the source term and is composed of the double spatial derivative of Lighthill's tensor Tij, k = ω/c0 is the wavenumber with c0 being the sound velocity. For isentropic low Mach number flows, it is commonly accepted that Lighthill's tensor takes the simplified form Tij ≃ ρ0 ui uj − τij with ρ0 being the fluid density and ui the i-component of the velocity. The viscous contribution τij is usually neglected for high Reynolds number flows but it is left here as it is provided by an incompressible-flow modeling. In Fig. 1 is sketched an arbitrary confined domain Ω bounded by two ducts from where the fluid enters and exits the domain. The interaction of the flow and obstacles or constrictions generates sound which propagates through the inlet and outlet ducts. In each duct, which are considered identical for simplicity, the pressure consists of acoustic waves only so it can be expressed as a sum of radiating modes:
p (x ) =
∑ am Φm (x ) m
on ∂Ωα
(2)
where the duct mode Φm (x ) obeys the Helmholtz equation and am is a modal coefficient, α takes the value of 1 or 2 depending on the fictitious surface. Using the orthonormality of the modes and the fact that ∂nΦm = − ik m Φm (km is the axial Please cite this article as: N. Papaxanthos, et al., Pressure-based integral formulations of Lighthill–Curle's analogy for internal aeroacoustics at low Mach numbers, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j. jsv.2017.01.030i
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Fig. 1. An arbitrary confined domain Ω with an inlet duct of section ∂Ω1 and an outlet duct of section ∂Ω2 .
wavenumber), the normal derivative of the pressure can be expressed in terms of the pressure as follows:
∂p (x ) = − i ∑ k m Φm (x ) ∂n m
∫∂Ω
p (y ) Φm* (y ) dΓy.
(3)
α
Eq. (3) is known as a Dirichlet-to-Neumann (DtN) condition and is usually written T(p) [25]. In the finite domain its boundaries, the pressure is given by:
C (x ) p (x ) =
⎛
∫Ω qG dΩy + ∫∂Ω ⎜⎝ p ∂∂nGy
−G
∂p ⎞ ⎟ dΓy ∂ny ⎠
Ω and on
(4)
where G = G (x, y ) refers to the classical free-field Green's function. Eq. (4) is the integral representation of Lighthill's equation. The volume integral describes the incident field from quadrupoles while the surface integral accounts for the reflexion and scattering of waves by the surfaces. One recalls that the normal derivative of the pressure is null on rigid walls and that the coefficient C (x ) corresponds to the solid angle divided by 4π and is equal to 1 for a listener in Ω and 1/2 when x lies on boundaries (except corners and edges). The integral formulation (4) can be written in a more appropriate form in view of a BEM resolution:
SΩ [ p](x ) =
∫Ω qG dΩy
(5)
where the RHS contains the source terms given by the flow simulation and the LHS refers to the scattering operator associated with the open cavity Ω :
SΩ [ p](x ) = C (x ) p (x ) −
∫∂Ω p ∂∂nGy dΩy + ∫∂Ω
GT (p) dΩy.
1,2
(6)
However, despite its simplicity, the resolution of Eq. (5) is challenging. In addition to the significant demand in memory to backup the source terms q, their spatial derivatives are not estimated adequately by CFD softwares which usually brings overestimation of the generated sound [26]. In order to ease the computation of the volume integral (5), an alternative way for solving Lighthill's equation based on the knowledge of the incompressible-flow pressure is presented. Two integral formulations are developed, the first one is exact and shall serve as reference to a simplified version. The starting point is to split the pressure as the sum of two terms:
p = p0 + pc
(7)
where p0 is the incompressible-flow pressure provided by the incompressible-flow simulation, pc is called the pressure correction term. By combining the decomposition (7) and Lighthill's equation, and noting that p0 is the solution to Poisson's equation Δp0 = q , it results the following inhomogeneous Helmholtz equation for the pressure correction term:
(Δ + k2) pc = − k2p0 .
(8)
Therefore, pc can be seen as the response to the fluctuations of p0. Ribner calls Eq. (8) the dilatation equation with the source term the fluid pulsation [23]. According to the relation ∂npc = T (p) − ∂np0 and thanks to the linearity of the DtN condition, the inversion of Eq. (8) yields the exact integral form:
SΩ [ pc ](x ) = − k2
⎛ ∂p ⎞ ⎜ G 0 − GT (p0 ) ⎟ dΓy. ∂ny ⎠ 1,2 ⎝
∫Ω p0 G dΩy + ∫∂Ω
(9)
Please cite this article as: N. Papaxanthos, et al., Pressure-based integral formulations of Lighthill–Curle's analogy for internal aeroacoustics at low Mach numbers, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j. jsv.2017.01.030i
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The resolution of Eq. (9), call it Formulation A, requires the storage of the incompressible-flow pressure in the entire domain and p0 and ∂np0 on the fictitious surfaces. Note, by adopting the same decomposition (7), Schram develops in Ref. [22] an alternative formulation that also avoids the numerical errors related to the volume integral calculation. Another formulation can be derived from Eq. (9) when manipulating the volume integral. Indeed, noting that −k2G = Δ (G − G0 ), where G0 is the static Green's function (k ¼0), and by integrating by parts twice, one obtains:
SΩ [ pc ](x ) ≃
∫∂Ω p0
∂ ( G − G0 ) ∂ny
dΓy +
⎛ ∂p ⎞ ⎜ G 0 0 − GT (p0 ) ⎟ dΓy. ∂ny ⎠ 1,2 ⎝
∫∂Ω
(10)
Call it Formulation B. The volume integral:
Π=
∫Ω q ( G − G0 ) dΩy
(11)
can be seen as the compressible part of the incident field from quadrupoles and is intentionally omitted in the RHS of Eq. (10). In Ref. [16], Martínez-Lera et al. have found the quantity Π negligible for the case of a non-compact wing in an unbounded domain as a result of the proximity of the main sources to the trailing edge. The fact that the reflected part of the incident wave field accounts for most of the radiated power is discussed in detail in Ref. [11] for compact obstacles and in Ref. [17] for confined flows with obstacles. Note that in Ref. [16], the acoustic radiation is computed via a boundary value problem that merely requires pressure data on boundaries, as in the case of Formulation B.
3. Experimental setup and measurement procedure The test rig is made of two main parts described in detail with the measurement procedure in Ref. [27]. The flow generation part consists of a fan, mufflers, a plenum chamber and an air flow rate meter. The measurement part is of rectangular cross-section (a portion is visible in Fig. 2) and includes a total of 96 flush mounted microphones located upstream and downstream of the test section and a particle image velocimetry (PIV) system. The last is composed of a pulsed laser with an operating frequency of 15 Hz and two cameras that allow the measurement of the three components of the velocity. Two anechoic terminations enclose the measurement part. The acoustic measurements are accomplished with the so-called multiport method which permits a multimodal characterization of the passive and active acoustic properties of obstacles [28]. The passive part is controlled by the duct and obstacle geometries and determines how the sound propagates through the system. The active part identifies the aeroacoustic sources due to the interaction between the flow and the obstacle. The experimental procedure is conducted in two steps. First, the passive properties are measured by using a movable external source which generates an acoustic field uncorrelated with the one generated from the test section. The impedance of the upstream and downstream terminations are deduced from this first step. The active properties of the obstacle are obtained in a second step with the use of a correlation technique that suppresses the flow disturbance [29]. The passive and active generation processes can be related to the following linear equation:
Pout = SPin + Ps
(12)
where Pin and Pout are respectively the incoming and outgoing pressure wave vectors and Ps is the source vector (the active part). Each vector is 2N -dimension (N is the number of propagating modes) as it includes informations about the propagating waves from both sides of the obstacle. The free-reflection scattering matrix S (the passive part) is of dimension 2N × 2N and contains the reflection and transmission characteristics of the test section.
Fig. 2. View on the measurement part, the obstacle (here a diaphragm) and the test section are made with plexiglass.
Please cite this article as: N. Papaxanthos, et al., Pressure-based integral formulations of Lighthill–Curle's analogy for internal aeroacoustics at low Mach numbers, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j. jsv.2017.01.030i
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5
Fig. 3. The computational domain. The pressure on the highlighted surface is used in Section 5.2, as well as the point x 0 . (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
4. Turbulent flow simulation The geometry (see Fig. 3 and the view of the test section Fig. 2) consists of a constriction of rectangular shape with dimensions D × D/2 × D/12.5 centered in a duct of rectangular cross-section 2D × D with D¼10 cm. The duct inlet is located 30 cm upstream of the obstacle whereas the duct outlet is 1.6 m downstream to ensure that remaining hydrodynamic fluctuations are negligible on the outlet section. The structured mesh used for the CFD, illustrated in Fig. 4, comprises 9 million cells. The mesh is built using 4 resolution levels ranging from 0.5 mm to 4 mm. The smallest cells are located in the vicinity of the diaphragm edges in order to capture correctly the physics of the shear layers. To ensure adequate resolution near solid surfaces, 14 boundary layers consisting of local refinement are added with a growth rate of 1.5 and a total thickness of 1 mm. The flow simulation is carried out with the commercial software Star-CCM + . A RANS k –ϵ simulation is first performed to give an initial condition for the incompressible large eddy simulation (LES). Inlet boundary conditions, velocity components and turbulence kinetic energy are prescribed using realistic data provided by PIV measurement. The measured mass flow rate is 565 kg/h which corresponds to an average velocity over the duct section of 6.5 m/s. The corresponding Mach number is 0.019 and the Reynolds number 4.2⁎104 based on the transverse dimension D. At the outlet section, an outflow condition (constant flow rate) is applied. A large eddy simulation with a Smagorinsky subgrid scale model is carried out with a simulated physical time of 0.32 s and a time step δt = 10−5 s. Aerodynamic quantities are stored each 10 δt which is sufficient to ensure that the frequency range of interest (up to 3400 Hz) is covered. The physical time of 0.32 s is divided into eight segments of 0.05 s each with 1/4 overlapping. A window function is then applied on each segment. The flow might be regarded as statistically converged after 0.1 s so the first three segments are discarded. Fig. 5 shows an instantaneous snapshot of the velocity at x1 = D in the region −D ≤ x3 ≤ 5D . The steady flow upstream from the diaphragm is uniform with fine boundary layers near the duct walls. From the obstacle to D downstream, a jet is formed emanating from the aperture with the creation of a periodic vortex shedding from the upstream diaphragm lips. The additional contraction of the jet, called the vena contracta, is clearly visible. As the jet progresses away from the diaphragm, one distinguishes large coherent structures which arise from the arrangement of smaller eddies. A steady state is then slowly recovered after a certain distance. Complete descriptions of the flow through a diaphragm can be found in several studies, such as in Ref. [5,30]. Fig. 6 displays a comparison of the mean velocity vectors from the PIV measurement and the LES in the plane x1 = D . The velocity vectors are determined by the following expression:
〈u〉 = 〈u2 〉e 2 + 〈u3 〉e 3.
(13)
Results are similar though slight discrepancies can be identifiable in the recirculation regions where the rotational speed is more pronounced in the measurement. The mean turbulence kinetic energy gives an indication of the location and intensity of the unsteady phenomena and is defined as:
k=
〈u1′ 2〉 + 〈u2′ 2 〉 + 〈u3′ 2 〉 2
(14)
where u′i is the fluctuating part of the velocity and stems from the Reynolds decomposition ui = 〈ui 〉 + u′i . Fig. 7 shows the mean turbulence kinetic energy over the duct cross-section located 3 cm downstream of the diaphragm. A quite fair agreement is obtained between the LES prediction and the PIV result. High energy corresponds to small eddies convected in the shear layers. Differences are observed especially near the right angles of the constriction which might be explained by the fact that the vortex dynamics is not perfectly captured by the simulation. Fig. 8 shows the rms value of the component T33 of Lighthill's tensor in the duct at x1 = D for several frequencies. It serves to illustrate the location of the dominant sources of sound. The other components of Lighthill's tensor are for all frequencies lower than T33/2. At low frequency, the longitudinal fluctuations reach their maximum values at about 2D away from the obstacle and are due in part to large structures impinging on the duct walls. As the frequency increases, predominant
Fig. 4. View of the mesh near the diaphragm at x1 = D . The 4 resolution levels are visible with the finest one surrounding the diaphragm edges.
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0.1
0.1
0.075
0.075
x2 (m)
x2 (m)
Fig. 5. Snapshot of the instantaneous velocity magnitude at t¼ 0.25 s in the plane x1 = D , amplitudes ranging from 0 (blue) to 43 m/s (red). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
0.05
0.05
0.025
0.025
0
0.2
0
0.25
0.2
0.25 x3 (m)
x (m) 3
Fig. 6. Mean velocity vectors in the plane x1 = D , (a) PIV, (b) LES, amplitudes ranging from 0 (blue) to 41 m/s (red). Cameras do not have a direct access to the flow inside the constriction and this is visible in (a). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
0.1
x2 (m)
2
x (m)
0.1
0.05
0 0
0.05
0.1 x1 (m)
0.15
0.2
0.05
0 0
0.05
0.1
0.15
0.2
x1 (m)
Fig. 7. Turbulence kinetic energy on the duct cross-section located 3 cm downstream from the diaphragm, (a) PIV, (b) LES, amplitudes ranging from 0 (blue) to 143 J/kg (red). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Fig. 8. Component T33 (the rms value) of Lighthill's tensor in the duct at x1 = D and −0.3D ≤ x3 ≤ 3.2D . (a) f = 200 Hz (max. =205 Pa), (b) f = 1000 Hz (max. =99 Pa), (c) f = 2000 Hz (max. =55 Pa), (d) f = 3000 Hz (max. =25 Pa). The scale is linear and goes from blue (T33 = 0 ) to red (max. of T33). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
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structures become smaller and are located in the shear layers around the vena contracta. In Fig. 8d, the presence of the refinement boxes in the meshing process is visible as it lowers the amplitude of the fluctuations. However, this has no impact on the acoustic results.
5. Results and discussions Due to the difference in size of acoustic and turbulent wavelengths, the requirement in terms of grid resolution is less severe for the acoustic computation than for the CFD. Formulations A and B are then solved by collocation on a coarse mesh which is composed of around 8500 triangular surface cells of maximum dimension 2 cm, this corresponds to λ/5 at 3400 Hz. For the sake of simplicity, the acoustic domain is chosen identical to the CFD domain. It is possible to build a smaller acoustic domain in order to reduce the computational time by moving the outlet surface closer to the turbulence zone, but this must be handled with care [31,32]. The temporal boundary-data p0 and ∂np0 are first summed on the coarse mesh centroids using a conservative mapping from Star-CCMþ then Fourier transformed. For the computation of the reference integral Formulation A, the incompressible-flow pressure over all cells of the CFD mesh is Fourier transformed before the computation of the volume integral is performed. 5.1. Radiated sound power The sound power radiated from the two boundaries ∂Ω1 and ∂Ω2 is computed with: 2
W=
∑∫ α=1
∂Ωα
Re {pvn* } 2
dΓy
(15)
where p = p0 + pc and vn is the normal component of the acoustic velocity. Eq. (15) is used to compare the results of the Formulations A and B and this is illustrated in Fig. 9a. With only surface integrals, Formulation B gives identical results up to 2500 Hz to those of the exact Formulation A. At high frequencies, slight differences become discernible. Thus, the compressible part of the incident field from quadrupoles, i.e. the quantity Π , can be neglected over the major part of the frequency range, as discussed in [16,17]. Note that the boundary terms which involve the gradients of the incompressibleflow pressure in Formulations A and B are negligible, however, the terms T (p0 ) have to be taken into account. From now, all presented results are computed with Formulation B. The computed sound power level (SWL) in Fig. 9b is the result of an averaging process over five segments and is compared with the measurement. It yields a reliable prediction with globally less than 5 dB differences over the whole frequency range. At frequencies 860 Hz and 1720 Hz, discernible jumps correspond to the apparition of the first spanwise and transverse duct acoustic modes respectively. They are less pronounced from measurements which can be explained by a certain amount of vibration and damping from the duct walls. Above 2500 Hz, the calculation curve decreases more than the experimental one and this might be due to the fact that the volume integral Π is neglected. For completeness, the cut-off frequencies of the rectangular duct are reminded here:
fm =
c0 ⎛⎜ p ⎞⎟2 ⎛⎜ q ⎞⎟2 + ⎝ D⎠ 2 ⎝ 2D ⎠
(16)
60
60
50
50 SWL (dB/Hz)
SWL (dB/Hz)
and reported in Table 1. Mode indices p and q are the following spanwise (x1) and transverse (x2) directions respectively and we took c0 = 343 m/s. The correspondence m ↔ (p, q) is given by ordering the cut-off frequencies in the ascending order. The projection on duct modes of the pressure at the duct endings allows the direct calculation of the contribution of each
40
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20
40
30
500
1000
1500
2000 f (Hz)
2500
3000
20
500
1000
1500
2000
2500
3000
f (Hz)
Fig. 9. (a) Computed SWL from 1 realization, ( ) Formulation A, ( ) Formulation B. (b) Comparison with the measurement, ( ) computed SWL from 5 realizations and Formulation B, ( ) measured SWL, ( ) computed SWL with the use of the tailored Green's function of the duct.
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Table 1 Cut-off frequencies of the duct below 3400 Hz. fm (Hz) (p,q) m
857.5 (1,0) 1
1715 (2,0) 2
1715 (0,1) 3
1917 (1,1) 4
2425 (2,1) 5
2572.5 (3,0) 6
3092 (3,1) 7
mode to the total SWL. It is also possible to calculate the modal contributions using the tailored Green's function of the infinite duct which leads to some interesting results. For the simple geometry of a rectangular infinite rigid duct, the tailored Green's function is expressed as follows [33]:
Gtai .(x, y ) = −
∑ m
1 Φm (x ) Φm* (y ). 2ik m
(17)
By neglecting the volume integral of the ensuing integral formulation, we arrive at:
p (x ) ≃
∫∂Ω
p obs.
∂Gtai . dΓy ∂ny
(18)
where ∂Ωobs. refers to the boundary of the obstacle only. The corresponding SWL is shown in Fig. 9b and reveals that the knowledge of the pressure fluctuations on the obstacle is sufficient to compute the radiated sound. This is evidenced by numerous studies based on an equivalent dipole source distribution in the duct [2,3]. From this result and by neglecting the phase difference between the upstream and downstream faces of the diaphragm, it can be demonstrated that the plane mode contribution to the total SWL is proportional to the square of the drag force acting on the obstacle:
W0 ≃
|F3 |2 4Aρ0 c0
(19)
where A is the area of the duct section. Note that Eq. (19) yields identically radiated powers upstream and downstream of the obstacle, and this is observed both experimentally and numerically. In Fig. 10 are displayed the contribution of the plane mode (Fig. 10a), the first spanwise mode (Fig. 10b) and the first transverse mode (Fig. 10c) to the total SWL. Computed and measured results show a fair agreement. The plane mode contribution computed with Eq. (19) gives identical results than its exact contribution from Eq. (18) and is removed from Fig. 10a. The fine solid line in Fig. 10a is computed by simply taking the incompressible-flow pressure on the surface of the obstacle in Eq. (18) and shows growing discrepancies as the frequency increases. 5.2. Description of the pressure correction term The aim of this section is to highlight the informations contained in the pressure correction term. A wavenumber frequency analysis is operated on the single rectangular surface of the duct (called it Γ) which is shown in Fig. 3. The spatial Fourier transform is computed as follows: 40 SWL (dB/Hz)
60
30 20 10
40
500
1000
1500
2000 f (Hz)
2500
3000
500
1000
1500
2000 f (Hz)
2500
3000
30 20 10
500
1000
1500
2000 f (Hz)
2500
3000
SWL (dB/Hz)
SWL (dB/Hz)
50
30 20 10
Fig. 10. Contributions to the total SWL, (a) plane mode, (b) first spanwise mode (m¼ 1) and (c) first transverse mode (m¼3). ( ) computation, ( measurement, in (a) is also shown the plane mode contribution using only the incompressible-flow pressure in the surface integral: ( ).
Please cite this article as: N. Papaxanthos, et al., Pressure-based integral formulations of Lighthill–Curle's analogy for internal aeroacoustics at low Mach numbers, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j. jsv.2017.01.030i
)
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Fig. 11. Wavenumber frequency spectra, (a) computed with P = p0 , (b) computed with P = pc , (c) computed with P ¼p, k1 = 44 rad/m. Amplitudes ranging from blue to red. The pink arrow in (a) indicates the wide background fluctuations. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Φ (x 0, k ) =
1 (2π )2
∫Γ SPP (x 0, ξ ) exp ( − ik. ξ ) dξ
(20)
where SPP = 〈P (x 0 ) P (x 0 + ξ )*〉 is the space correlation between point x 0 and an arbitrary point on the surface, ξ is the separation vector and k = (k1, k3 ) is the two-dimensional wavevector with the component k1 fixed to 44 rad/m. The latter is chosen empirically as it permits a favorable detection of modes. The point x 0 is placed 15 cm behind the diaphragm in the turbulent zone and is indicated by a cross in Fig. 3. The pressure term P in the space correlation can be either the incompressible-flow pressure (P = p0 ), the pressure correction term (P = pc ) or the true pressure (P ¼p). Fig. 11 shows the corresponding wavenumber frequency spectra in logarithmic scale. The analysis of the incompressible-flow pressure (Fig. 11a) shows an intense zone that corresponds mainly to convective structures impinging on the upper wall of the duct. With only the pressure correction term in the space correlation (Fig. 11b), several curves can be identified which correspond to the axial wavenumbers of the propagating modes. The superimposed black lines are built with:
km =
⎛ 2πfm ⎞2 k2 − ⎜ ⎟ . ⎝ c0 ⎠
(21)
Fig. 11c merely illustrates the combination of the two pressures. Note the multiple oblique and vertical lines are due to the Fourier transform over the bounded domain Γ. It is clear from the above analysis that the correction pc represents mainly acoustic waves generated by the turbulent flow. Another information emerges in Fig. 11. One observes in Fig. 11a a non-zero value along the frequency axis as k3 → 0 (pointed out by an arrow). It corresponds to wide background fluctuations which affect a large part of the computational domain. Similar fluctuations are mentioned from another commercial CFD Software in Ref. [34]. This spurious phenomenon is canceled by the pressure correction term as it does not appear anymore in Fig. 11c. Another CFD simulation has been carried out by changing only the outflow condition to a pressure outlet condition. The corresponding wavenumber frequency spectrum constructed with P = p0 is displayed in Fig. 12. Although it is less discernible, the incompressible-flow pressure is also disturbed by the background fluctuations especially at high frequencies. Despite these spurious pressure fluctuations, which differ depending on the outlet condition of the flow simulation, outgoing acoustic waves are well predicted once the pressure p = p0 + pc is recovered. This is confirmed in Fig. 13 where the radiated sound power computed
Fig. 12. Wavenumber frequency spectrum computed with P = p0 . The outlet condition of the flow simulation is changed to a static pressure condition, k1 = 44 rad/m. The scale is the same as Fig. 11. The pink arrow indicates the wide background fluctuations. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Please cite this article as: N. Papaxanthos, et al., Pressure-based integral formulations of Lighthill–Curle's analogy for internal aeroacoustics at low Mach numbers, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j. jsv.2017.01.030i
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SWL (dB/Hz)
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2000
2500
3000
f (Hz)
Fig. 13. Computed SWL, (
) from flow simulation with the outflow condition, (
) from flow simulation with the pressure condition.
from the two different CFD simulations are shown. When solving the incompressible Navier–Stokes equations, the pressure is defined up to an arbitrary constant function, spatially and time dependent, as soon as Neumann boundary conditions for the pressure are applied on the boundary of the computational domain [35,36]. The resulting wide background fluctuations affect the inlet and outlet surfaces which explains why p0 is non-negligible on those surfaces.
6. Conclusion An approach combining an incompressible-flow modeling and a BEM resolution is applied to the calculation of the radiated noise from the interaction of a low Mach number flow and a ducted diaphragm. Within reasonable assumptions, it is shown that the volume integral (11), which can be interpreted as the compressible part of the incident pressure field generated by a distribution of quadrupole sources, can be neglected. This assumption leads to an approximated integral formulation presented at Eq. (10). The latter offers the advantages that it demands little data transfer and storage as only pressure on the boundaries of the domain is required. The method seems particularly suited for the prediction of the noise produced by confined low speed flows in the presence of obstacles as sound sources generally remain in the close vicinity of the trailing edges of the obstacles. Numerical predictions are found independent of the outlet condition of the flow simulation and in very good agreement with the measurement. The numerical implementation of a Dirichlet-to-Neumann boundary condition at the duct inlet and outlet permits to calculate the modal contributions to the total sound power level and this is also validated by comparisons with the measurement.
Acknowledgment This work is part of the CEVAS (Conception d'Equipements de Ventilation d'Air Silencieux) project, funded by the Picardie region and FEDER (Fonds Européen de DEveloppement Régional).
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Please cite this article as: N. Papaxanthos, et al., Pressure-based integral formulations of Lighthill–Curle's analogy for internal aeroacoustics at low Mach numbers, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j. jsv.2017.01.030i