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Broadband noise prediction using large eddy simulation and a frequency domain method
Applied Acoustics xxx (2016) xxx–xxx
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Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Broadband noise prediction using large eddy simulation and a frequency domain method Fan Tong a,⇑, Wei-Yang Qiao a, Wei-Jie Chen a, Liang-Feng Wang a, Xun-Nian Wang b a b
School of Power and Energy, Northwestern Polytechnical University, Xi’an 710072, China State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang 621000, China
a r t i c l e
i n f o
Article history: Received 4 May 2016 Received in revised form 26 July 2016 Accepted 1 November 2016 Available online xxxx Keywords: Broadband noise Large eddy simulation Acoustic analogy Frequency domain method Span correction
a b s t r a c t A new LES-acoustic analogy method for accurate flow and broadband noise prediction is proposed. A frequency domain method for the generalized Lighthill acoustic analogy theory is derived in detail and the final equations for code is provided which can help to bridge the gap between the flow field prediction and the acoustic field prediction for those who are interested in acoustic results but lack of acoustic prediction ability. The hybrid method (LES and acoustic analogy method) for broadband noise prediction is validated using the rod-airfoil interaction problem. Both flow field results and acoustic field results are compared with experimental results and other numerical results in detail. The flow field results agree very well with the experimental results and other numerical results. The predicted acoustic results and experimental results also reach good agreement both in far field acoustic pressure Power Spectral Density (PSD) and noise directivity. The method developed in the paper proves to be an effective tool for broadband noise prediction. In addition, the different span correction methods for the acoustic pressure spectrum are also discussed. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction The strong market demand for quieter aircraft encourages airplane and turbo engine manufacturers to provide more environmentally friendly and quieter aircraft and turbo engines. Noise prediction capability is of great importance for the design of future aircraft and turbo engines to make sure that the increasingly more rigorous regulations can be met. Turbo engine noise involves tonal noise and broadband noise. For modern ultra-high by-pass ratio engines, fan noise is becoming more and more significant. The tonal part of the fan noise and the broadband part of the fan noise both carry about half of the sound power. The underlying generation mechanism of fan tonal noise is relatively well understood. However, the fan broadband noise generation mechanism is much more complicated and only partly understood. And for military aircraft, the broadband noise from the exhaust jet often dominates all other sources. Therefore, broadband noise is of great academic and technical importance. There are several methods to predict broadband noise: empirical prediction models [1–3], analytical prediction models [4,5],
⇑ Corresponding author. E-mail address: [email protected] (F. Tong).
fully numerical prediction methods [6] and hybrid prediction methods [7–10]. Empirical prediction models are generally simple and fast but they cannot assess the acoustic performance between different detailed designs of engine components since they can only associate a few parameters. The analytical prediction models can provide satisfactory results fast. Nevertheless, they cannot guide the detailed designs due to the many simplifications. Fully numerical prediction methods can provide flow-to-far field simulations with little simplification of the geometry. However, they require huge computational resources. The hybrid prediction methods, which use CFD (like LES, DES, URANS) to obtain noise sources information and use acoustic analogy [11–14] to obtain far field acoustic information, can take detailed design parameters into consideration and require much less computational resources compared with fully numerical prediction methods. For these reasons, the hybrid prediction methods are very promising methods to predict aircraft/engine broadband noise. In order to predict noise sources, especially broadband noise sources, highly accurate unsteady CFD solutions must be obtained. LES numerical method is adopted in this study. The rod-airfoil configuration is particularly suitable for the assessment of CFD codes in modeling broadband noise sources. Its relevance has been thoroughly discussed by Jacob et al. [15]. The rod-airfoil configuration combines the periodic vortex shedding with a random
http://dx.doi.org/10.1016/j.apacoust.2016.11.001 0003-682X/Ó 2016 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Tong F et al. Broadband noise prediction using large eddy simulation and a frequency domain method. Appl Acoust (2016), http://dx.doi.org/10.1016/j.apacoust.2016.11.001
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F. Tong et al. / Applied Acoustics xxx (2016) xxx–xxx
perturbation due to the wake’s transition into turbulence. The airfoil undergoes a broadband perturbation which is dominated by a preferred shedding frequency, somewhat like that observed in turbomachinery applications. The ability of a combined CFD/ acoustic approach to predict the spectral broadening around the shedding frequency and its harmonics is a relevant measure of its ability to model broadband sources [15]. Due to these reasons, the rod-airfoil configuration is widely used for assessing the different CFD/acoustic approaches. Casalina et al. [16] used 2D unsteady RANS combined with FW-H equation to predict rod-airfoil interaction noise. The vortex shedding frequency is significantly over predicted. Magagnato [17] and Boudet et al. [8,18] performed the first 3D LES on rod-airfoil interaction noise. Boudet’s LES was performed with the Turb’Flow code and the near field is in good agreement with experiment. However the far field acoustic results are poorly converged. Peth et al. [19] used LES and the Linearized Perturbed Compressible Equations (LPCE) to predict rod-airfoil interaction noise. The shedding frequency is slightly under-predicted and the far field pressure Power Spectral Density (PSD) peak is under predicted by 5–7 dB. More recently, Jacob et al. [20], Giret et al. [7] and Greschner [9] also carried out detailed analysis of rod-airfoil interaction noise prediction. Many researchers have validated their in-house codes using this benchmark case and most of their acoustic analogy methods are based on time domain method [7–9,10,20]. At the same time, although many of the CFD solvers have demonstrated impressive ability in flow field prediction, the accurate prediction of aeroacoustic noise, especially broadband noise, is a more challenging work compared to steady or unsteady flow field predictions. Because of this, only a few CFD solvers have integrated the basic noise prediction ability, for example the familiar Fluent solver. Whereas there are also many CFD solvers whose noise prediction ability is limited and not mature (e.g. CFX). Obviously, developing the broadband noise prediction tools will help to extend CFD solvers’ functionality and add our option and freedom. Epikhin [21] presented a Dynamic Library for computational aeroacoustics applications using the OpenFOAM source package, however, the details of the method are not available which may hinder its further application and development by others. Moreover, the Dynamic Library is specific for OpenFOAM source package. For other codes the Dynamic Library does not work. Therefore, there is a need to develop a broadband noise prediction tool for various CFD codes which are not ready for direct broadband noise prediction. The current study proposed a new LES-acoustic analogy method for accurate flow and broadband noise prediction. A frequency domain method for generalized Lighthill acoustic analogy theory is derived in detail. Compared with time domain method, the frequency domain method has several advantages. For time domain methods, the determination of the retarded time is computationally intensive. In addition, the variables and their time derivatives must be interpolated to the retarded time for every grid point, retarded time and observer position, which also leads to intensive calculation. In contrast, frequency domain methods are more computationally efficient. The frequency domain methods can be carried out further faster when harmonic noise is of interest where only several selected frequencies need to be calculated. In this paper, the final equations of the frequency domain method for code are also provided which can help to bridge the gap between the flow field prediction and the acoustic prediction for those who are interested in acoustic results but lack of acoustic prediction ability. The code of the frequency domain method is then successfully validated through the comparison with experimental results and other numerical results.
2. Methodology 2.1. Numerical method for flow field LES is used to compute the broadband noise sources with the commercial code CFX [22]. In LES, the large three-dimensional unsteady turbulent motions are directly represented, whereas the effects of the smaller-scale motions are modelled. The rationale behind LES technique is a separation between large and small scales. Large scales of the flow contain the main part of the total fluctuating kinetic energy and characterize the flow. The driving physical mechanisms are carried by the large scales. The large scales of the flow are sensitive to the boundary conditions and so are anisotropic [23]. In contrast, small scales of the flow contain only a few percent of the total kinetic energy and have weak influence on the mean movement. Their main function is viscous dissipation, however, they can also have an effect on higher scales. In LES, the unresolved small scales of the flow may be isotropic or anisotropic which depends on the user and how they decided to model the unresolved scales. A filtering operation is defined to decompose the flow variable U into the sum of a filtered (or and a residual (or subgrid-scale, SGS) comresolved) component U ponent U0 . The equations for the filtered field can be derived from the Navier-Stokes equations, however a closure problem arises because of the SGS stress tensor. The closure can be obtained by modeling the SGS stress tensor. There are several methods to model the SGS stress. Smagorinsky proposed the original Smagorinsky model [24], Nicoud and Ducros proposed the WALE model [25] (wall-adapted local eddyviscosity model), Germano and Lilly presented the Dynamic Smagorinsky-Lilly model [26,27]. The Smagorinsky model and WALE model are algebraic models and the model coefficient is constant. In contrast, the Dynamic Smagorinsky-Lilly model uses information contained in the resolved turbulent velocity field to evaluate the model coefficient. Thus, the model coefficient is no longer a constant value and adjusts automatically to the flow type. In this paper, the Dynamic Smagorinsky-Lilly model and the Second Order Backward Euler transient scheme are adopted. 2.2. Numerical method for acoustic field The acoustic prediction method is based on Goldstein’s generalized Lighthill equation [14]. Goldstein extended Lighthill’s acoustic analogy to include the effects of solid boundaries and moving medium. The fundamental equation governing the generation of sound in the presence of solid boundaries is presented below with slight changes to its original form for the current application. !
c20 q0 ðx ; tÞ ¼
Z
Z
T
T
Z
þ
AðsÞ T
T
q0 V 0N
Z
vðsÞ
T 0ij
! DG dAðy Þds þ Ds
@2G ! d y ds @yi @yj
Z
T
T
Z AðsÞ
fi
! @G dAðy Þds @yi
ð1Þ
where c0 is the ambient speed of sound and q0 is the acoustic density disturbance. At sufficient distance from the source, c20 q0 is equal !
to the acoustic pressure p0 . x ¼ ðx1 ; x2 ; x3 Þ is the observation coordi!
nate and y ¼ ðy1 ; y2 ; y3 Þ is the source coordinate. s is the source time (retarded time). V 0N is the velocity of the surface normal to itself relative to the fluid. The first term in Eq. (1) represents the sound generated due to volume displacement effects of the surface, the second term represents the sound generated due to the exertion of unsteady forces by the boundaries on the fluid, and the last term represents the generation of the sound by volume sources. In each
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F. Tong et al. / Applied Acoustics xxx (2016) xxx–xxx
of the three terms, the outer integral is over a range of source time T < s < T large enough to include all contributions to the noise signal at observer time t. For stationary surface, the first term becomes zero. When the incoming flow Mach number is small, the volume sources contribute to the noise relatively a little and can be neglected. When the first term and last term in Eq. (1) are ignored, the acoustic pressure at sufficient distance from the source can be expressed as follows !
p0 ðx ; tÞ ¼
Z
T
Z
T
AðsÞ
fi
! @G dAðy Þds @yi
ð2Þ
where G is the free space, moving medium, time dependent Green function [28]
dðt s r=c0 Þ G¼ 4pS
ð3Þ
Since only f i eixs is dependent on source time s; Eq. (12) can be rewritten as follows !
p0 ðx ; xÞ ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h iffi ðx1 y1 Þ2 þ b2 ðx2 y2 Þ2 þ ðx3 y3 Þ2
and
r is the phase radius with b2 ¼ 1 M2
r¼
Mðx1 y1 Þ þ S
M ¼ U 0 =c0
!
ð6Þ
Z Z A
T
T
fi
! @G dsdAðy Þ @yi
ð7Þ
Apply Fourier transform to Eq. (7), we can obtain the expression of acoustic pressure in frequency domain !
p0 ðx ; xÞ ¼
Z
þ1
Z Z
1
A
T
T
fi
! @G dsdAðy Þeixt dt @yi
ð8Þ
Since only G is related to reception time t and A is independent of reception time t or source time s , Eq. (8) can be changed to !
p0 ðx ; xÞ ¼
Z Z A
T
Z
T
þ1 1
fi
! @G ixt e dtdsdAðy Þ @yi
ð9Þ
Insert the time dependent Green’s function from Eq. (3) into Eq. (9) to obtain 0
!
p ðx ; xÞ ¼
Z Z A
T
Z
T
þ1 1
! @ 1 fi dðt s r=c0 Þ eixt dtdsdAðy Þ @yi 4pS !
Note that f i is dependent on source time s and source location y and independent on reception time t, so Eq. (10) can be written as follows
p0 ðx ; xÞ ¼
Z þ1 ! @ 1 fi dðt s r=c0 Þeixt dt dsdAðy Þ @y 4 p S T 1 i
Z Z A
T
ð11Þ Take consideration of the integration property of d function, we can obtain !
p0 ðx ; xÞ ¼
Z Z A
T
T
fi
! @ 1 ixðsþr=c0 Þ dsdAðy Þ e @yi 4pS
ð13Þ
!
p0 ðx ; xÞ ¼
Z
!
A
f i ðy ; xÞ
! @ eixr=c0 dAðy Þ @yi 4pS
ð14Þ
The Eq. (14) can be presented in a more simple form !
p0 ðx ; xÞ ¼
Z
!
A
f i ðy ; xÞ
! @Gx dAðy Þ @yi
ð15Þ
where Gx ¼ e 4pS 0 is the form of Green function in the ! frequency domain. The force f i ðy ; xÞ can be obtained from LES @Gx and @y can also be acquired through the following equation derivation. Note that ixr=c0
@Gx e ¼ @yi
@r @S i cx0 S @y @y i
i
4pS2
ðFor i ¼ 1; 2; 3Þ
ð16Þ
Then take partial derivative of r and S with yi and we can obtain the following equations
. @ r y1 x1 ¼ M 1 M2 @y1 S
ð17Þ
@ r ðy2 x2 Þ ¼ S @y2
ð18Þ
@ r ðy3 x3 Þ ¼ S @y3
ð19Þ
@S ðy x1 Þ ¼ 1 @y1 S
ð20Þ
@S ðy x2 Þð1 M 2 Þ ¼ 2 @y2 S
ð21Þ
@S ðy x3 Þð1 M 2 Þ ¼ 3 @y3 S
ð22Þ
From Eqs. (17)–(22) and Eq. (16), we can obtain
ð10Þ
!
! @ eixr=c0 dAðy Þ @yi 4pS
i
In the current work, the stationary airfoil noise is of interest. So the integral symbol AðsÞ in the area integration is independent of source time s and AðsÞ can be changed to A. Eq. (2) can be reformed as follows
p0 ðx ; tÞ ¼
T
A
f i eixs ds
ixr=c
ð4Þ
ð5Þ
b2
T
The inner integration is the Fourier transform of the force f i ; thus we can obtain the final expression of acoustic pressure in frequency domain.
where S is the amplitude radius
S¼
Z Z
ð12Þ
ixr=c0
@Gx e ¼ @y1
ixr=c0
@Gx e ¼ @y2
ixr=c0
@Gx e ¼ @y3
n o h i ðy1 x1 Þ ix=c0 ð1 M 2 Þ=S þ ixSM=c0 4pS2 ð1 M 2 Þ n h io ðy2 x2 Þ ix=c0 ð1 M 2 Þ=S 4pS2 n h io ðy3 x3 Þ ix=c0 ð1 M 2 Þ=S 4pS2
ð23Þ
ð24Þ
ð25Þ
Finally, the acoustic pressure can be calculated by the following equation
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! @G ! ! @Gx @Gx x þ f 2 y; x þ f 3 ðy ; xÞ dAðy Þ @y1 @y2 @y3 A h i n o 2 Z ðy1 x1 Þ ix=c0 ð1 M2 Þ=S þ ixSM=c0 ! ixr=c0 4f 1 ðy ; xÞ ¼e 4pS2 ð1 M2 Þ A n h io ðy2 x2 Þ ix=c0 ð1 M2 Þ=S ! þ f 2 ðy ; xÞ 4pS2 n h io3 ðy3 x3 Þ ix=c0 ð1 M2 Þ=S ! ! 5dAðy Þ þ f 3 ðy ; xÞ ð26Þ 4pS2 !
p0 ðx ; xÞ ¼
Z
!
f 1 ðy ; xÞ
!
If the viscous stress tensor is ignored, f i ðy ; xÞ can be calculated as follows !
!
!
f i ðy ; xÞ ¼ p0 ðy ; xÞn i
Fig. 2 shows the sketch of the computational domain. The computational domain extends 26c in the stream-wise direction, 20c in the cross-stream direction and 2d in the spanwise direction. The sketch of the computational mesh near the airfoil is shown in Fig. 3. Fig. 4 shows the dimensionless wall-cell sizes Dyþ along the airfoil. It can be seen that the Dyþ is below 1. In addition, there are 240 grid points around the rod and 400 grid points around the airfoil resulting in the dimensionless wall-cell sizes Dxþ 90. In the spanwise direction, there are 33 grid points and the dimensionless wall-cell sizes Dzþ 60. The total grid number is about 5.15 million. At the same time, a coarser mesh which has also 33 grid points in the spanwise direction but with less grid points in
ð27Þ
!
where p0 ðy ; xÞ is the airfoil surface pressure fluctuation spectrum !
and n i is the unit normal vector in ith direction. At last, it should be pointed out that the current approach uses a free-space Green’s function and does not account for the effects of scattering of sound by solid bodies or propagation/refraction effects by the flow.
Fig. 3. Sketch of the computational Mesh.
3. Benchmark experiment and numerical setup The rod-airfoil configuration in the benchmark experiment by Jacob et al. [15] is shown in Fig. 1. The configuration involves a symmetric NACA0012 (chord: c = 100 mm) airfoil located one chord downstream of a rod (rod diameter: d = 10 mm). The airfoil and rod both have a span of 300 mm. In the reference configuration, the incoming flow velocity is 72 m/s and the free stream turbulence intensity is about 0.8%. The Reynolds number based on rod diameter is about 4:8 104 and the Reynolds number based on airfoil chord is about 4:8 105 :
Fig. 1. Sketch of the experimental set-up [15].
Fig. 4. Dimensionless wall-cell sizes Dyþ distribution along the airfoil at mid-span.
Fig. 2. Sketch of the computational domain.
Please cite this article in press as: Tong F et al. Broadband noise prediction using large eddy simulation and a frequency domain method. Appl Acoust (2016), http://dx.doi.org/10.1016/j.apacoust.2016.11.001
F. Tong et al. / Applied Acoustics xxx (2016) xxx–xxx
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the airfoil plane was also studied. The total grid number of the coarse mesh is 3.46 million. Fig. 5 shows the velocity profiles of the rod wake predicted by the two different meshes. It can be seen that both predicted results agree reasonably well with the experimental results. However, the coarse mesh slightly over predicts the wake deficit while the fine mesh yields much better results. Therefore, the fine mesh was used hereafter as it is expected that the fine mesh can give more accurate prediction of the broadband noise sources. The highest frequency that can be resolved in the simulation is determined by the grid. Michel [29] has proposed a grid Strouhal number to estimate the highest frequency that can be resolved by the grid. An axial grid Strouhal number is defined as
St s ¼ f Dxs =U c < 0:25
ð28Þ
where the Dxs is the grid size in the mean flow direction, U c is the convection speed of the disturbance in the flow and U c 0:8U 0 , f is the frequency of the emitted sound. The max grid size around the airfoil is Dxs 0:8 mm for the fine grid. It can be estimated that the maximum resolvable frequency is f 16 kHz: The inflow conditions and the flow parameters are based on the experiment conditions. The incoming free stream velocity is 72 m/s. The outlet static pressure is set to 98,900 Pa. For spanwise direction, both use of periodic boundary conditions and slip (or symmetry) boundary conditions have been documented [7–9,17,19,20]. For the periodic boundary conditions, the periodicity is forced to be met for the boundaries and all the flow field quantities of the limiting planes are fully correlated. For slip (or symmetry) boundary conditions, one component of velocity (z-component) vanishes whereas the two planes in the spanwise direction are not forced to be correlated. The slip condition only affects the vicinity of the boundary. It is expected that the slip (or symmetry) boundary condition will lead to a lower level of coherence than periodic boundary condition. Lockard [30] has pointed out that only calculations based on the full length of the model span were able to capture the complete decay in the spanwise correlation. However, in practical calculations, this is often not feasible because of the huge computational resources it demanded. Since most of the studies adopted the periodic boundary conditions in the spanwise direction [7,17,19,20], the same boundary conditions are used in this paper for more direct compar-
Fig. 5. Velocity profiles of the rod wake of two different meshes.
Fig. 6. Residual curves during calculation.
ison. The adiabatic no-slip conditions are imposed on the rod and the airfoil. The computational time step is 1 105 s and the total acquired physical time for acoustic processing is about 0.22 s which corresponds to a time interval during which the incoming flow passes about 159 times of the airfoil chord. The simulation was run on High Performance Cluster of Northwestern Polytechnical University with 32 cores. The simulation time was about six weeks. The relatively long simulation time helps to increase the stability of the signal and is beneficial for the calculation of the acoustic results. The code employs a dual time-stepping algorithm and we chose the number of sub-iterations to ensure that the variables plateaued and the residual dropped by 2–3 orders of magnitude within each time step. Fig. 6 shows the residual curves of different variables during the calculation. The root mean square (RMS) residual for three momentum equations is about 1 105 to 2 105 and the RMS residual for continuity equation is about 1 106 to 2 106 . Fig. 7 shows the time history of the airfoil lift
Fig. 7. Time history of the airfoil lift and drag coefficient.
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and drag coefficient. It can be seen from Figs. 6 and 7 that the flow field converges reasonably well. In the present study, the solid surface was selected as the integral surface. Therefore, only the dipole sources are considered since the dipole sources dominate the sound sources in the current low Mach number condition. In the following part of the paper, it will be shown that the noise prediction results agree well with experimental results although only the dipole sources are considered. 4. Results and discussion Present numerical flow field results and far field acoustic results are compared with experimental results by Jacob et al. [15] and other numerical results. Both the steady flow field and unsteady flow field are considered. The far field noise spectrum and noise directivity are also compared with experimental data. 4.1. Comparison positions The profiles of the mean velocity and the rms (root mean square) value of velocity fluctuations obtained from LES are compared with experimental results as well as other numerical results. Fig. 8 shows the sketch of measurement positions. The profiles at three different locations are compared, i.e. plane A at x/c = 0.25 which corresponds to the far wake of the rod, plane B at x/c = 0.25 which is slightly upstream of the airfoil thickest point, plane C at x/c = 1.1 which corresponds to the near wake of the airfoil. It should be noted that x/c = 0 corresponds to airfoil leading edge. Streamwise velocity power spectrum density is also compared with available experimental results at two different locations, i.e. P1 and P2 (Fig. 8). P1 is located at (0.25c, 0.08c) and P2 is located at (0.25c, 0.16c). P1 is 0.2d off the airfoil surface and P2 is 1d off the airfoil surface. In addition, the far field acoustic pressure spectrum at a distance of R = 1.85 m of 90° angular angle (just above the airfoil) from the airfoil center is also compared with experimental results. Moreover, the Strouhal number of the vortex shedding frequency is compared with experimental results. 4.2. Steady flow field Velocity profiles and turbulent intensity profiles are compared with experiment results by Jacob et al. [15] and previous LES (or DES) results by Jacob [15], Giret [7] and Greschner [9] at three different locations in Fig. 8. Fig. 9(a) and (b) shows the mean velocity profile and turbulent intensity profile at plane A. It can
be found that the present numerical results agree well with the experimental results and other numerical results. The rod wake mean velocity profile is well predicted by the current simulation. The minimum value of u/U0 is about 0.709 in experimental results while the predicted minimum value of u/U0 is 0.699 and the relative error is about 1.41%. It can be seen from Fig. 9(b) that, overall the turbulent intensity profile predicted by present LES reaches an agreement with experiment results and other numerical results. The maximum value of turbulent intensity of experiment and the present simulation is about 0.176 and 0.193, respectively. The predicted value is 9.66% higher than that of the experiment. Fig. 9(c) shows the mean velocity profile at plane B. The present LES performs similarly to Giret’s LES results and the mean profile is over predicted by about 5.33%. This over prediction is also found in many LES results [8,20]. Fig. 9(d) shows the turbulent intensity profile at plane B. The peak value of turbulent intensity of experiment and the present simulation is about 0.164 and 0.223, respectively. The present simulation over predicts the turbulent intensity peak by about 36.0%. Part reason for this large discrepancy is that the peak value of the turbulent intensity is hard to measure in the experiment due to the spatial resolution of the hot wire. Fig. 9(e) and (f) shows the mean velocity profile and turbulent intensity profile at plane C, which represents the near wake of the airfoil. Compared with experimental results, the maximum wake deficit is under predicted by about 7.90% in Fig. 9(e). The maximum value of the turbulent intensity is 0.135 for experiment and 0.152 for present simulation. The maximum value of the turbulent intensity is over predicted by about 12.6% in Fig. 9(f). There is a discrepancy between the experimental turbulent intensity results and numerical turbulent intensity results near the location of y/c = 0.2, which can also be seen in Giret’s numerical results. Overall, the present LES gives satisfactory prediction of mean velocity profiles and turbulent intensity profiles. 4.3. Unsteady flow field The unsteady flow field is also investigated. The Strouhal number is defined as St ¼ fd=U 0 . The experimental value of vortex shedding Strouhal number is 0:19 0:002. The predicted value of Strouhal number is 0:202. The predicted result agrees reasonably well with the experimental result which indicates that the physics of the flow separation on the rod are correctly reproduced in the current simulation. The instantaneous span-wise vorticity and iso-surfaces of the Q-criterion are shown in Figs. 10 and 11, respectively. It can be seen that the main vortices shedding from the rod form regular Karman vortex street which impinges onto the airfoil and partly broken into small structures. Moreover, many smaller vortices can be observed in the rod wake, due to the transition to turbulence in the shear layer of the rod. The broken of large vortices is successfully captured by the current simulation. It is expected that the rod wake interacting with the leading edge of the airfoil is the main mechanism of the turbulence airfoil interaction noise. 4.4. Velocity spectrum
Fig. 8. Sketch of measurement positions.
The power spectrum density of the streamwise velocity component is compared with the experimental results in Fig. 12. The agreement with experimental data is satisfactory. It can be seen from Fig. 12(a) that the current simulation can capture the broadened peak which means the interaction between the random part of the flow and the quasi deterministic vortex shedding is well
Please cite this article in press as: Tong F et al. Broadband noise prediction using large eddy simulation and a frequency domain method. Appl Acoust (2016), http://dx.doi.org/10.1016/j.apacoust.2016.11.001
F. Tong et al. / Applied Acoustics xxx (2016) xxx–xxx
(a) Mean velocity proile at plane A
(b) Turbulent intensity proile at plane A
7
(d) Turbulent intensity proile at plane B
(e) Mean velocity proile at plane C
Fig. 9. Comparison of mean velocity profile and turbulent intensity profile with experiment and other numerical simulations.
predicted. However, the peak of the power spectrum density is over predicted by about 1.6 dB and the vortex shedding frequency is over predicted by 6.3%. Fig. 12(b) shows the power spectrum density of the streamwise velocity at P2, which is one rod diameter off the airfoil surface. It can be seen that vortex shedding frequency is much less obvious than that at P1. The spectrum at P2
is also well predicted by the current simulation. The peak of the power spectrum density is over predicted by about 3.0 dB and the predicted power spectrum density is about 2 dB lower than that of the experimental result in high frequency range (around St = 0.8) where the predicted spectrum seems to decay slightly more quickly.
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Fig. 10. Span-wise vorticity contour.
Lc Lexp Spp ðf Þ exp ¼ Spp ðf Þ sim þ 20 lg þ 10 lg ; Lsim Lc
Lsim < Lc 6 Lexp ð32Þ
Lexp ; Spp ðf Þ exp ¼ Spp ðf Þ sim þ 20 lg Lsim
The spanwise length of the LES computational domain is often limited by computational resources. As a result, the spanwise length in the numerical simulation Lsim is usually shorter than the experimental spanwise length Lexp . In order to compare the predicted sound pressure level with experimental data, the sound pressure level obtained from numerical simulations must be corrected. Different correction methods have been developed by Kato et al. [31], Seo et al. [32], Boudet et al. [8], and Perot et al. [33]. The correction of the power spectral density of the pressure in the far field can be expressed as follows [8],
! R Lexp R Lexp Cðjz2 z1 j; f Þdz1 dz2 0 0 ¼ Spp ðf Þ sim þ 10 lg R L R L sim sim Cðjz2 z1 j; f Þdz1 dz2 0 0 ð29Þ
where Cðjz2 z1 j; f Þ is the coherence function at the frequency f between two points along the solid surface. The coherence length Lc is defined as
Z
þ1
z1
Cðjz2 z1 j; f Þdz2
ð30Þ
Simplifications of Eq.(29) can be made by assuming the coherence function of specific mathematical form. Kato [31] assumes the coherence function of the rectangular function form and the correction function can be simplified as follows
Spp ðf Þ
exp
Lexp ¼ Spp ðf Þ sim þ 10 lg ; Lsim
pffiffiffiffi Lc 6 Lsim = p
ð34Þ
pffiffiffiffi Lc pLexp Spp ðf Þ exp ¼ Spp ðf Þ sim þ 10 lg þ 10 lg ; Lsim Lsim pffiffiffiffi pffiffiffiffi Lsim = p < Lc 6 Lexp = p
ð35Þ
Lexp Spp ðf Þ exp ¼ Spp ðf Þ sim þ 10 lg Lsim
4.5. Acoustic results
Lc ¼
Lc 6 Lsim
ð33Þ
Seo [32] assumes the coherence function of the Gaussian function form and the correction function can be simplified as follows
Fig. 11. Iso-surfaces of the Q-Criterion (Q = 2:5 106 s2).
ðSpp ðf ÞÞexp
Lexp < Lc
ð31Þ
Lexp ; Spp ðf Þ exp ¼ Spp ðf Þ sim þ 20 lg Lsim
Lexp =
pffiffiffiffi
p < Lc
ð36Þ
Perot [33] assumes the coherence function of the exponential function form and gives a correction method independent on spanwise coherence length but dependent of the distance between observer point and the source point
arctan Spp ðf Þ exp ¼ Spp ðf Þ sim þ 10 lg
Lexp R
arctan arctan Lsim R
Lsim R
ð37Þ
Actually, the coherence length Lc is a function of frequency and the corrections should also be a function of frequency. Unfortunately, the exact Lc as a function of frequency is often not available so a reasonable estimate of coherence length is usually used in practice. According to Jacob’s experiment results, the spanwise pressure correlation length Lc on the rod is about 6.5d [15]. Fig. 13 shows the comparison of different correction methods. It should be noted that in Fig. 13 the following parameters are used, Lsim ¼ 2d, R ¼ 1:85 m and Lexp ¼ 30d. For the current problem, Lc ¼ 6:5d and Lc =Lsim ¼ 3:25. So Kato’s method will lead to a correction of 16.87 dB and Seo’s method will lead to a correction of 19.37 dB while Perot’s method gives a correction of 11.42 dB.
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F. Tong et al. / Applied Acoustics xxx (2016) xxx–xxx
Fig. 13. Comparison of different spanwise correction methods.
(a) PSD at P1
Fig. 12. Comparison of PSD of the streamwise velocity component at P1 and P2.
The sound pressure level (SPL) of the far field acoustic pressure corrected by different methods at 1.85 m for the angular position 90° is shown in Fig. 14 where the abscissa is scaled to Strouhal number. The sound pressure level is defined as
PSDðf Þ SPLðf Þ ¼ 10 log 10 p2ref
9
! ð38Þ
where pref ¼ 2 105 Pa and the PSD is normalized to be per unit f and Welch’s method was used to calculate the PSD in Eq. (27). It can be seen from Fig. 14 that the peak value of SPL corrected by Kato’s method is 90.44 dB, 92.94 dB by Seo’s method and 84.99 dB by Perot’s method. The peak value of SPL is about 91.44 dB in the experiment. Perot’s correction method obviously under predicts the noise peak level by 6.45 dB and the predicted
Fig. 14. Comparison of SPL of the acoustic pressure at 90° and at 1.85 m corrected by different methods.
broadband part of the noise in high frequency range (near St = 1) is also lower than that of experiment results by 3–4 dB. The Seo’s correction method over predicts the peak value of SPL by 1.5 dB and the broadband part of the noise by about 4 dB (near St = 1). The Kato’s correction method performs well both for the noise peak level prediction and the broadband part of the noise prediction in a large frequency range. The peak value of SPL is under predicted by 1 dB while the broadband part of noise is over predicted by about 1.5 dB in high frequency range (near St = 1) and about 4 dB between St = 0.2–0.3. This big discrepancy is mainly due to the over-predicted peak frequency (vortex shedding frequency) which leads to the shift of the whole predicted spectrum. The vortex shedding Strouhal number is 0:19 0:002 in experiment whereas the predicted value of Strouhal number is 0:202 with a 6.3% over predict. To address this effect, Fig. 15 shows the
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Fig. 15. Comparison of SPL of the acoustic pressure at 90° and at 1.85 m corrected by Kato’s method (with peak frequency corrected).
comparison of SPL after the predicted peak frequency are corrected to the experimental peak frequency and the agreement improves significantly. The difference between predicted results and experimental results is within 2–3 dB over a wide frequency range. The Kato’s correction method can give very good results if the peak frequency could be predicted more accurately. In addition, the Kato’s correction method is widely used and demonstrates satisfactory prediction results [7,9,15]. As a result, the Kato’s correction method [31], precisely Eq. (32), is used hereinafter to correct the SPL of acoustic pressure obtained from the numerical simulation. Fig. 16 compares the SPL of the far field acoustic pressure at 1.85 m for angular position of 90° with the experimental data [15] and other numerical results [7,9,20] available. It can be seen from Fig. 16 that the present prediction results agree well with experimental data. The discrepancy between the prediction and experimental results is within 2 dB at the peak noise frequency and the broadband part of the noise also agrees well with the experimental results. It can be seen that the present prediction method shows similar or improved results than other numerical results both in the main peak noise and the broadband part of the noise. The acoustic directivity results are compared with experimental data in Fig. 17. The prediction results of the present method agree fairly well with experimental results. The discrepancy between predicted results and experimental results is within 3 dB for most of the angular positions. The maximum discrepancy is about 4.5 dB near angular position of 110°. The acoustic directivity results is over predicted by 2–4.5 dB between angular position of 90° to 120° and 240° to 270°. Excellent agreement is obtained between angular position of 60° to 90° and 270° to 300°. Through the comparison of far field acoustic pressure spectrum and the acoustic directivity between the predicted results and experimental results, it can be found that the acoustic results obtained from LES and the current acoustic analogy method agree fairly well with the experimental results. This indicates that the current LES and the frequency domain method of generalized Lighthill acoustic analogy theory are capable to predict broadband noise sources of complex flows and the resulting broadband noise.
Fig. 16. Acoustic pressure results compared with experimental data, at 90° and 1.85 m.
5. Conclusions A new LES-acoustic analogy method for accurate flow and broadband noise prediction is proposed and assessed. A frequency domain method for the generalized Lighthill acoustic analogy theory is developed. The frequency domain method is derived in detail and the final equations for code is provided which can help to bridge the gap between the flow field prediction and the acoustic prediction for those who are interested in acoustic results but lack of acoustic prediction ability. The rod-NACA0012 airfoil configuration is selected because of its detailed experimental results and its relevance to model broadband noise. The LES is performed by CFX and the acoustic prediction method is based on the generalized Lighthill equation. Both the flow field results and the acoustic field results are compared with the experimental results and other numerical results.
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F. Tong et al. / Applied Acoustics xxx (2016) xxx–xxx
11
Appendix A Nomenclature c c0 d f G Lsim Lexp Lc R S St s St Spp t
!
chord length ambient speed of sound rod diameter frequency filter function or Green function spanwise length in numerical simulation spanwise length in experiment correlation length distance between observer point and the source point amplitude radius grid Strouhal number Strouhal number power spectral density of the pressure in the far field observation time coherence function retarded time fluid variable filtered variable unresolved part of the variable acoustic density disturbance acoustic pressure observation coordinate
!
source coordinate
C
s
Fig. 17. Acoustic directivity compared with experimental data.
U U U0
q0 p0 x
The flow field results show that the current LES can give accurate prediction of the broadband noise sources although the vortex shedding frequency is slightly over predicted. Both the mean velocity profiles and turbulent intensity profiles are well predicted. Moreover, the predicted velocity spectrum agrees very well with the experimental data. The acoustic field results show that the predicted acoustic pressure spectrum results match very well with the experimental data. The discrepancy between the predicted results and experimental results is within 2 dB at the peak noise frequency and the broadband part of the noise also agrees well with the experimental results. Due to the discrepancy of span length between the numerical and experimental setup, the acoustic pressure spectrum obtained from numerical results needs to be corrected before compared with experimental data. It is found that the span correction methods have an important effect on the level of the predicted acoustic results. It is shown that the assumption of coherence function of the rectangular function form by Kato is reasonable and gives better results than that of Seo’s and Perot’s method. Overall, the LES and acoustic analogy method for broadband noise prediction developed in this paper performs well and can give a good prediction of the broadband noise.
Acknowledgements This study is supported by National Natural Science Foundation of China and the project number is 51276149, 51476134. This research is also supported by State Key Laboratory of Aerodynamics of China and the project number is SKLA20160201. Additional acknowledgements to Key Laboratory of Aerodynamic Noise Control of China Aerodynamics Research and Development Center for the project of ANCL20160102. The simulation work was supported by Center for High Performance Computing of Northwestern Polytechnical University, China.
y V 0N
velocity of the surface normal to itself
References [1] Heidmann MF. Interim prediction method for fan and compressor source noise. Report, NASA TM X-71763, USA; 1975. [2] Raney JP. Development of a new computer system for aircraft noise prediction. In: Proceedings of 2nd aero-acoustics conference of American Institute of Aeronautics and Astronautics, Hampton, VA, U.S.A., 24–26, March 1975, paper no. 75–536. [3] Hough JW, Weir DS. Aircraft noise prediction program (ANOPP) fan noise prediction for small engines. Report, NASA CR 198300, USA; 1996. [4] Ventres CS. Turbofan noise generation. Volume 1: analysis. Report, NASA CR167952, USA; 1983. [5] Meyer HD, Envia E. Aeroacoustic analysis of turbofan noise generation. Report, NASA CR 4715, USA; 1996. [6] Clair V, Polacsek C, Garrec TL, et al. CAA methodology to simulate turbulenceairfoil noise. In: 18th AIAA/CEAS aeroacoustics conference, Colorado Springs, CO, USA, 04–06, June 2012, paper no. 2012–2189. [7] Giret JC, Sengissen A, Moreau S, et al. Prediction of the sound generated by a rod-airfoil configuration using a compressible unstructured LES solver and a FW-H analogy. In: 18th AIAA/CEAS aeroacoustics conference, Colorado Springs, CO, USA, 04–06, June 2012, paper no. 2012–2058. [8] Boudet J, Grosjean N, Jacob MC. Wake-airfoil interaction as broadband noise source: a large-eddy simulation study”. Int J Aeroacoust 2005;4: 93–116. [9] Greschner B, Thiele F, Jacob MC, et al. Prediction of sound generated by a rod– airfoil configuration using EASM DES and the generalised Lighthill/FW-H analogy. Comput Fluids 2008;37(37):402–13. [10] Galdéano S, Barré S, et al. Noise radiated by a rod-airfoil configuration using DES and the Ffowcs-Williams & Hawkings’ analogy. In: 16th AIAA/CEAS aeroacoustics conference, Stockholm, Sweden, 7–9, June, 2010, paper no. 2010-3702. [11] Lighthill MJ. On sound generated aerodynamically. I. General theory. Proc R Soc 1952;211:564–87. [12] Lighthill MJ. On sound generated aerodynamically. II. Turbulence as a source of sound. Proc R Soc 1954;222:1–32. [13] Ffowcs Williams JE, Hawkings DL. Sound generated by turbulence and surfaces in arbitrary motion. Philos Trans R Soc 1969;264:321–42. [14] Goldstein ME. Unified approach to aerodynamic sound generation in the presence of solid boundaries. J Acoust Soc Am 1974;56:497–509. [15] Jacob MC, Boudet J, Casalino D, et al. A rod-airfoil experiment as benchmark for broadband noise. Model Theor Comput Fluid Dyn 2005;19:171–96.
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[16] Casalino D, Jacob MC, Roger M. Prediction of rod-airfoil interaction noise using the Ffowcs-Williams and Hawkings analogy. Am Inst Aeronaut Astronaut 2003;41(2):182–91. [17] Magagnato F, Sorgueven E, Gabi M. Far field noise prediction by large eddy simulation and Ffowcs-Williams Hawkings analogy. In: Proceedings of the 9th AIAA/CEAS aeroacoustics conference, Hilton Head, South Carolina, 12–14, May, 2003, paper no. 2003–3206. [18] Boudet J, Casalino D, Jacob MC, et al. Prediction of sound radiated by a rod using large-eddy simulation. In: Proceedings of the 9th AIAA/CEAS aeroacoustics conference, Hilton Head, South Carolina, 12–14, May 2003, paper no. 2003–3217. [19] Peth S, Seo JH, Moon YJ, et al. Computation of aerodynamic noise from rod wake-airfoil interactions. In: Proceedings of the European conference on computational fluid dynamics, Egmond aan Zee, The Netherlands, 5–8, September 2006. [20] Jacob MC, Ciardi M, Gamet L, et al. Assessment of CFD broadband noise predictions on a rod-airfoil benchmark computation. In: 14th AIAA/CEAS aeroacoustics conference, Vancouver, British Columbia, Canada, 5–7, May 2008, paper no. 2008–2899. [21] Epikhin A, Evdokimov I, Kraposhin M, et al. Development of a dynamic library for computational aeroacoustics applications using the OpenFOAM open source package. Proc Comput Sci 2015;66:150–7. [22] ANSYS CFX. Reference guide, release 14.0. ANSYS Inc.; 2011. [23] Wagner CD, Hüttl T, Sagaut P. Large-eddy simulation for acoustics. 1st ed. Cambridge University Press; 2012. p. 90. [24] Smagorinsky J. General circulation experiments with the primitive equations. Month Weath Rev 1963;93:99–165.
[25] Nicoud F, Ducros F. Subgrid-scale stress modelling based on the square of the velocity gradient tensor. Flow Turbulence Combust 1999;62: 183–200. [26] Germano M, Piomelli U, Moin P, et al. A dynamic subgrid-scale eddy viscosity model. Phys Fluids A 1991;3(7):1760–5. [27] Lilly DK. A proposed modification of the Germano subgrid-scale closure method. Phys Fluids A 1992;4(3):633–5. [28] Hanson DB, Parzych DJ. Theory for noise of propellers in angular inflow with parametric studies and experimental verification. Report, NASA CR 4499, USA; 1993. [29] Michel U, Eschricht D, Greschner B, et al. Advanced DES methods and their application to aeroacoustics. In: Peng SH, Doerffer P, Haase W, editors. Progress in hybrid RANS-LES modelling. Berlin, Heidelberg: Springer; 2010. p. 59–76. [30] Lockard DP, Khorrami MR, Choudhari MM, et al. Tandem cylinder noise predictions. In: Proceedings of the 13th AIAA/CEAS aeroacoustics conference, Rome, Italy, 23–25, May, paper no. 2007–3450. [31] Kato C, Ikegawa M. Large eddy simulation of unsteady turbulent wake of a circular cylinder using the finite element method. Adv Numer Simul Turbulent Flows 1991;1:49–56. [32] Seo JH, Moon YJ. Aerodynamic noise prediction for long-span bodies”. J Sound Vib 2007;306(3–5):564–79. [33] Perot F, Auger JM, Giardi H, et al. Numerical prediction of the noise radiated by a cylinder. In: Proceedings of the 9th AIAA/CEAS aeroacoustics conference, Hilton Head, South Carolina, 12–14, May 2003, paper no. 2003–3240.
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Contents lists available at ScienceDirect
Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Broadband noise prediction using large eddy simulation and a frequency domain method Fan Tong a,⇑, Wei-Yang Qiao a, Wei-Jie Chen a, Liang-Feng Wang a, Xun-Nian Wang b a b
School of Power and Energy, Northwestern Polytechnical University, Xi’an 710072, China State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang 621000, China
a r t i c l e
i n f o
Article history: Received 4 May 2016 Received in revised form 26 July 2016 Accepted 1 November 2016 Available online xxxx Keywords: Broadband noise Large eddy simulation Acoustic analogy Frequency domain method Span correction
a b s t r a c t A new LES-acoustic analogy method for accurate flow and broadband noise prediction is proposed. A frequency domain method for the generalized Lighthill acoustic analogy theory is derived in detail and the final equations for code is provided which can help to bridge the gap between the flow field prediction and the acoustic field prediction for those who are interested in acoustic results but lack of acoustic prediction ability. The hybrid method (LES and acoustic analogy method) for broadband noise prediction is validated using the rod-airfoil interaction problem. Both flow field results and acoustic field results are compared with experimental results and other numerical results in detail. The flow field results agree very well with the experimental results and other numerical results. The predicted acoustic results and experimental results also reach good agreement both in far field acoustic pressure Power Spectral Density (PSD) and noise directivity. The method developed in the paper proves to be an effective tool for broadband noise prediction. In addition, the different span correction methods for the acoustic pressure spectrum are also discussed. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction The strong market demand for quieter aircraft encourages airplane and turbo engine manufacturers to provide more environmentally friendly and quieter aircraft and turbo engines. Noise prediction capability is of great importance for the design of future aircraft and turbo engines to make sure that the increasingly more rigorous regulations can be met. Turbo engine noise involves tonal noise and broadband noise. For modern ultra-high by-pass ratio engines, fan noise is becoming more and more significant. The tonal part of the fan noise and the broadband part of the fan noise both carry about half of the sound power. The underlying generation mechanism of fan tonal noise is relatively well understood. However, the fan broadband noise generation mechanism is much more complicated and only partly understood. And for military aircraft, the broadband noise from the exhaust jet often dominates all other sources. Therefore, broadband noise is of great academic and technical importance. There are several methods to predict broadband noise: empirical prediction models [1–3], analytical prediction models [4,5],
⇑ Corresponding author. E-mail address: [email protected] (F. Tong).
fully numerical prediction methods [6] and hybrid prediction methods [7–10]. Empirical prediction models are generally simple and fast but they cannot assess the acoustic performance between different detailed designs of engine components since they can only associate a few parameters. The analytical prediction models can provide satisfactory results fast. Nevertheless, they cannot guide the detailed designs due to the many simplifications. Fully numerical prediction methods can provide flow-to-far field simulations with little simplification of the geometry. However, they require huge computational resources. The hybrid prediction methods, which use CFD (like LES, DES, URANS) to obtain noise sources information and use acoustic analogy [11–14] to obtain far field acoustic information, can take detailed design parameters into consideration and require much less computational resources compared with fully numerical prediction methods. For these reasons, the hybrid prediction methods are very promising methods to predict aircraft/engine broadband noise. In order to predict noise sources, especially broadband noise sources, highly accurate unsteady CFD solutions must be obtained. LES numerical method is adopted in this study. The rod-airfoil configuration is particularly suitable for the assessment of CFD codes in modeling broadband noise sources. Its relevance has been thoroughly discussed by Jacob et al. [15]. The rod-airfoil configuration combines the periodic vortex shedding with a random
http://dx.doi.org/10.1016/j.apacoust.2016.11.001 0003-682X/Ó 2016 Elsevier Ltd. All rights reserved.
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perturbation due to the wake’s transition into turbulence. The airfoil undergoes a broadband perturbation which is dominated by a preferred shedding frequency, somewhat like that observed in turbomachinery applications. The ability of a combined CFD/ acoustic approach to predict the spectral broadening around the shedding frequency and its harmonics is a relevant measure of its ability to model broadband sources [15]. Due to these reasons, the rod-airfoil configuration is widely used for assessing the different CFD/acoustic approaches. Casalina et al. [16] used 2D unsteady RANS combined with FW-H equation to predict rod-airfoil interaction noise. The vortex shedding frequency is significantly over predicted. Magagnato [17] and Boudet et al. [8,18] performed the first 3D LES on rod-airfoil interaction noise. Boudet’s LES was performed with the Turb’Flow code and the near field is in good agreement with experiment. However the far field acoustic results are poorly converged. Peth et al. [19] used LES and the Linearized Perturbed Compressible Equations (LPCE) to predict rod-airfoil interaction noise. The shedding frequency is slightly under-predicted and the far field pressure Power Spectral Density (PSD) peak is under predicted by 5–7 dB. More recently, Jacob et al. [20], Giret et al. [7] and Greschner [9] also carried out detailed analysis of rod-airfoil interaction noise prediction. Many researchers have validated their in-house codes using this benchmark case and most of their acoustic analogy methods are based on time domain method [7–9,10,20]. At the same time, although many of the CFD solvers have demonstrated impressive ability in flow field prediction, the accurate prediction of aeroacoustic noise, especially broadband noise, is a more challenging work compared to steady or unsteady flow field predictions. Because of this, only a few CFD solvers have integrated the basic noise prediction ability, for example the familiar Fluent solver. Whereas there are also many CFD solvers whose noise prediction ability is limited and not mature (e.g. CFX). Obviously, developing the broadband noise prediction tools will help to extend CFD solvers’ functionality and add our option and freedom. Epikhin [21] presented a Dynamic Library for computational aeroacoustics applications using the OpenFOAM source package, however, the details of the method are not available which may hinder its further application and development by others. Moreover, the Dynamic Library is specific for OpenFOAM source package. For other codes the Dynamic Library does not work. Therefore, there is a need to develop a broadband noise prediction tool for various CFD codes which are not ready for direct broadband noise prediction. The current study proposed a new LES-acoustic analogy method for accurate flow and broadband noise prediction. A frequency domain method for generalized Lighthill acoustic analogy theory is derived in detail. Compared with time domain method, the frequency domain method has several advantages. For time domain methods, the determination of the retarded time is computationally intensive. In addition, the variables and their time derivatives must be interpolated to the retarded time for every grid point, retarded time and observer position, which also leads to intensive calculation. In contrast, frequency domain methods are more computationally efficient. The frequency domain methods can be carried out further faster when harmonic noise is of interest where only several selected frequencies need to be calculated. In this paper, the final equations of the frequency domain method for code are also provided which can help to bridge the gap between the flow field prediction and the acoustic prediction for those who are interested in acoustic results but lack of acoustic prediction ability. The code of the frequency domain method is then successfully validated through the comparison with experimental results and other numerical results.
2. Methodology 2.1. Numerical method for flow field LES is used to compute the broadband noise sources with the commercial code CFX [22]. In LES, the large three-dimensional unsteady turbulent motions are directly represented, whereas the effects of the smaller-scale motions are modelled. The rationale behind LES technique is a separation between large and small scales. Large scales of the flow contain the main part of the total fluctuating kinetic energy and characterize the flow. The driving physical mechanisms are carried by the large scales. The large scales of the flow are sensitive to the boundary conditions and so are anisotropic [23]. In contrast, small scales of the flow contain only a few percent of the total kinetic energy and have weak influence on the mean movement. Their main function is viscous dissipation, however, they can also have an effect on higher scales. In LES, the unresolved small scales of the flow may be isotropic or anisotropic which depends on the user and how they decided to model the unresolved scales. A filtering operation is defined to decompose the flow variable U into the sum of a filtered (or and a residual (or subgrid-scale, SGS) comresolved) component U ponent U0 . The equations for the filtered field can be derived from the Navier-Stokes equations, however a closure problem arises because of the SGS stress tensor. The closure can be obtained by modeling the SGS stress tensor. There are several methods to model the SGS stress. Smagorinsky proposed the original Smagorinsky model [24], Nicoud and Ducros proposed the WALE model [25] (wall-adapted local eddyviscosity model), Germano and Lilly presented the Dynamic Smagorinsky-Lilly model [26,27]. The Smagorinsky model and WALE model are algebraic models and the model coefficient is constant. In contrast, the Dynamic Smagorinsky-Lilly model uses information contained in the resolved turbulent velocity field to evaluate the model coefficient. Thus, the model coefficient is no longer a constant value and adjusts automatically to the flow type. In this paper, the Dynamic Smagorinsky-Lilly model and the Second Order Backward Euler transient scheme are adopted. 2.2. Numerical method for acoustic field The acoustic prediction method is based on Goldstein’s generalized Lighthill equation [14]. Goldstein extended Lighthill’s acoustic analogy to include the effects of solid boundaries and moving medium. The fundamental equation governing the generation of sound in the presence of solid boundaries is presented below with slight changes to its original form for the current application. !
c20 q0 ðx ; tÞ ¼
Z
Z
T
T
Z
þ
AðsÞ T
T
q0 V 0N
Z
vðsÞ
T 0ij
! DG dAðy Þds þ Ds
@2G ! d y ds @yi @yj
Z
T
T
Z AðsÞ
fi
! @G dAðy Þds @yi
ð1Þ
where c0 is the ambient speed of sound and q0 is the acoustic density disturbance. At sufficient distance from the source, c20 q0 is equal !
to the acoustic pressure p0 . x ¼ ðx1 ; x2 ; x3 Þ is the observation coordi!
nate and y ¼ ðy1 ; y2 ; y3 Þ is the source coordinate. s is the source time (retarded time). V 0N is the velocity of the surface normal to itself relative to the fluid. The first term in Eq. (1) represents the sound generated due to volume displacement effects of the surface, the second term represents the sound generated due to the exertion of unsteady forces by the boundaries on the fluid, and the last term represents the generation of the sound by volume sources. In each
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F. Tong et al. / Applied Acoustics xxx (2016) xxx–xxx
of the three terms, the outer integral is over a range of source time T < s < T large enough to include all contributions to the noise signal at observer time t. For stationary surface, the first term becomes zero. When the incoming flow Mach number is small, the volume sources contribute to the noise relatively a little and can be neglected. When the first term and last term in Eq. (1) are ignored, the acoustic pressure at sufficient distance from the source can be expressed as follows !
p0 ðx ; tÞ ¼
Z
T
Z
T
AðsÞ
fi
! @G dAðy Þds @yi
ð2Þ
where G is the free space, moving medium, time dependent Green function [28]
dðt s r=c0 Þ G¼ 4pS
ð3Þ
Since only f i eixs is dependent on source time s; Eq. (12) can be rewritten as follows !
p0 ðx ; xÞ ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h iffi ðx1 y1 Þ2 þ b2 ðx2 y2 Þ2 þ ðx3 y3 Þ2
and
r is the phase radius with b2 ¼ 1 M2
r¼
Mðx1 y1 Þ þ S
M ¼ U 0 =c0
!
ð6Þ
Z Z A
T
T
fi
! @G dsdAðy Þ @yi
ð7Þ
Apply Fourier transform to Eq. (7), we can obtain the expression of acoustic pressure in frequency domain !
p0 ðx ; xÞ ¼
Z
þ1
Z Z
1
A
T
T
fi
! @G dsdAðy Þeixt dt @yi
ð8Þ
Since only G is related to reception time t and A is independent of reception time t or source time s , Eq. (8) can be changed to !
p0 ðx ; xÞ ¼
Z Z A
T
Z
T
þ1 1
fi
! @G ixt e dtdsdAðy Þ @yi
ð9Þ
Insert the time dependent Green’s function from Eq. (3) into Eq. (9) to obtain 0
!
p ðx ; xÞ ¼
Z Z A
T
Z
T
þ1 1
! @ 1 fi dðt s r=c0 Þ eixt dtdsdAðy Þ @yi 4pS !
Note that f i is dependent on source time s and source location y and independent on reception time t, so Eq. (10) can be written as follows
p0 ðx ; xÞ ¼
Z þ1 ! @ 1 fi dðt s r=c0 Þeixt dt dsdAðy Þ @y 4 p S T 1 i
Z Z A
T
ð11Þ Take consideration of the integration property of d function, we can obtain !
p0 ðx ; xÞ ¼
Z Z A
T
T
fi
! @ 1 ixðsþr=c0 Þ dsdAðy Þ e @yi 4pS
ð13Þ
!
p0 ðx ; xÞ ¼
Z
!
A
f i ðy ; xÞ
! @ eixr=c0 dAðy Þ @yi 4pS
ð14Þ
The Eq. (14) can be presented in a more simple form !
p0 ðx ; xÞ ¼
Z
!
A
f i ðy ; xÞ
! @Gx dAðy Þ @yi
ð15Þ
where Gx ¼ e 4pS 0 is the form of Green function in the ! frequency domain. The force f i ðy ; xÞ can be obtained from LES @Gx and @y can also be acquired through the following equation derivation. Note that ixr=c0
@Gx e ¼ @yi
@r @S i cx0 S @y @y i
i
4pS2
ðFor i ¼ 1; 2; 3Þ
ð16Þ
Then take partial derivative of r and S with yi and we can obtain the following equations
. @ r y1 x1 ¼ M 1 M2 @y1 S
ð17Þ
@ r ðy2 x2 Þ ¼ S @y2
ð18Þ
@ r ðy3 x3 Þ ¼ S @y3
ð19Þ
@S ðy x1 Þ ¼ 1 @y1 S
ð20Þ
@S ðy x2 Þð1 M 2 Þ ¼ 2 @y2 S
ð21Þ
@S ðy x3 Þð1 M 2 Þ ¼ 3 @y3 S
ð22Þ
From Eqs. (17)–(22) and Eq. (16), we can obtain
ð10Þ
!
! @ eixr=c0 dAðy Þ @yi 4pS
i
In the current work, the stationary airfoil noise is of interest. So the integral symbol AðsÞ in the area integration is independent of source time s and AðsÞ can be changed to A. Eq. (2) can be reformed as follows
p0 ðx ; tÞ ¼
T
A
f i eixs ds
ixr=c
ð4Þ
ð5Þ
b2
T
The inner integration is the Fourier transform of the force f i ; thus we can obtain the final expression of acoustic pressure in frequency domain.
where S is the amplitude radius
S¼
Z Z
ð12Þ
ixr=c0
@Gx e ¼ @y1
ixr=c0
@Gx e ¼ @y2
ixr=c0
@Gx e ¼ @y3
n o h i ðy1 x1 Þ ix=c0 ð1 M 2 Þ=S þ ixSM=c0 4pS2 ð1 M 2 Þ n h io ðy2 x2 Þ ix=c0 ð1 M 2 Þ=S 4pS2 n h io ðy3 x3 Þ ix=c0 ð1 M 2 Þ=S 4pS2
ð23Þ
ð24Þ
ð25Þ
Finally, the acoustic pressure can be calculated by the following equation
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! @G ! ! @Gx @Gx x þ f 2 y; x þ f 3 ðy ; xÞ dAðy Þ @y1 @y2 @y3 A h i n o 2 Z ðy1 x1 Þ ix=c0 ð1 M2 Þ=S þ ixSM=c0 ! ixr=c0 4f 1 ðy ; xÞ ¼e 4pS2 ð1 M2 Þ A n h io ðy2 x2 Þ ix=c0 ð1 M2 Þ=S ! þ f 2 ðy ; xÞ 4pS2 n h io3 ðy3 x3 Þ ix=c0 ð1 M2 Þ=S ! ! 5dAðy Þ þ f 3 ðy ; xÞ ð26Þ 4pS2 !
p0 ðx ; xÞ ¼
Z
!
f 1 ðy ; xÞ
!
If the viscous stress tensor is ignored, f i ðy ; xÞ can be calculated as follows !
!
!
f i ðy ; xÞ ¼ p0 ðy ; xÞn i
Fig. 2 shows the sketch of the computational domain. The computational domain extends 26c in the stream-wise direction, 20c in the cross-stream direction and 2d in the spanwise direction. The sketch of the computational mesh near the airfoil is shown in Fig. 3. Fig. 4 shows the dimensionless wall-cell sizes Dyþ along the airfoil. It can be seen that the Dyþ is below 1. In addition, there are 240 grid points around the rod and 400 grid points around the airfoil resulting in the dimensionless wall-cell sizes Dxþ 90. In the spanwise direction, there are 33 grid points and the dimensionless wall-cell sizes Dzþ 60. The total grid number is about 5.15 million. At the same time, a coarser mesh which has also 33 grid points in the spanwise direction but with less grid points in
ð27Þ
!
where p0 ðy ; xÞ is the airfoil surface pressure fluctuation spectrum !
and n i is the unit normal vector in ith direction. At last, it should be pointed out that the current approach uses a free-space Green’s function and does not account for the effects of scattering of sound by solid bodies or propagation/refraction effects by the flow.
Fig. 3. Sketch of the computational Mesh.
3. Benchmark experiment and numerical setup The rod-airfoil configuration in the benchmark experiment by Jacob et al. [15] is shown in Fig. 1. The configuration involves a symmetric NACA0012 (chord: c = 100 mm) airfoil located one chord downstream of a rod (rod diameter: d = 10 mm). The airfoil and rod both have a span of 300 mm. In the reference configuration, the incoming flow velocity is 72 m/s and the free stream turbulence intensity is about 0.8%. The Reynolds number based on rod diameter is about 4:8 104 and the Reynolds number based on airfoil chord is about 4:8 105 :
Fig. 1. Sketch of the experimental set-up [15].
Fig. 4. Dimensionless wall-cell sizes Dyþ distribution along the airfoil at mid-span.
Fig. 2. Sketch of the computational domain.
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the airfoil plane was also studied. The total grid number of the coarse mesh is 3.46 million. Fig. 5 shows the velocity profiles of the rod wake predicted by the two different meshes. It can be seen that both predicted results agree reasonably well with the experimental results. However, the coarse mesh slightly over predicts the wake deficit while the fine mesh yields much better results. Therefore, the fine mesh was used hereafter as it is expected that the fine mesh can give more accurate prediction of the broadband noise sources. The highest frequency that can be resolved in the simulation is determined by the grid. Michel [29] has proposed a grid Strouhal number to estimate the highest frequency that can be resolved by the grid. An axial grid Strouhal number is defined as
St s ¼ f Dxs =U c < 0:25
ð28Þ
where the Dxs is the grid size in the mean flow direction, U c is the convection speed of the disturbance in the flow and U c 0:8U 0 , f is the frequency of the emitted sound. The max grid size around the airfoil is Dxs 0:8 mm for the fine grid. It can be estimated that the maximum resolvable frequency is f 16 kHz: The inflow conditions and the flow parameters are based on the experiment conditions. The incoming free stream velocity is 72 m/s. The outlet static pressure is set to 98,900 Pa. For spanwise direction, both use of periodic boundary conditions and slip (or symmetry) boundary conditions have been documented [7–9,17,19,20]. For the periodic boundary conditions, the periodicity is forced to be met for the boundaries and all the flow field quantities of the limiting planes are fully correlated. For slip (or symmetry) boundary conditions, one component of velocity (z-component) vanishes whereas the two planes in the spanwise direction are not forced to be correlated. The slip condition only affects the vicinity of the boundary. It is expected that the slip (or symmetry) boundary condition will lead to a lower level of coherence than periodic boundary condition. Lockard [30] has pointed out that only calculations based on the full length of the model span were able to capture the complete decay in the spanwise correlation. However, in practical calculations, this is often not feasible because of the huge computational resources it demanded. Since most of the studies adopted the periodic boundary conditions in the spanwise direction [7,17,19,20], the same boundary conditions are used in this paper for more direct compar-
Fig. 5. Velocity profiles of the rod wake of two different meshes.
Fig. 6. Residual curves during calculation.
ison. The adiabatic no-slip conditions are imposed on the rod and the airfoil. The computational time step is 1 105 s and the total acquired physical time for acoustic processing is about 0.22 s which corresponds to a time interval during which the incoming flow passes about 159 times of the airfoil chord. The simulation was run on High Performance Cluster of Northwestern Polytechnical University with 32 cores. The simulation time was about six weeks. The relatively long simulation time helps to increase the stability of the signal and is beneficial for the calculation of the acoustic results. The code employs a dual time-stepping algorithm and we chose the number of sub-iterations to ensure that the variables plateaued and the residual dropped by 2–3 orders of magnitude within each time step. Fig. 6 shows the residual curves of different variables during the calculation. The root mean square (RMS) residual for three momentum equations is about 1 105 to 2 105 and the RMS residual for continuity equation is about 1 106 to 2 106 . Fig. 7 shows the time history of the airfoil lift
Fig. 7. Time history of the airfoil lift and drag coefficient.
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and drag coefficient. It can be seen from Figs. 6 and 7 that the flow field converges reasonably well. In the present study, the solid surface was selected as the integral surface. Therefore, only the dipole sources are considered since the dipole sources dominate the sound sources in the current low Mach number condition. In the following part of the paper, it will be shown that the noise prediction results agree well with experimental results although only the dipole sources are considered. 4. Results and discussion Present numerical flow field results and far field acoustic results are compared with experimental results by Jacob et al. [15] and other numerical results. Both the steady flow field and unsteady flow field are considered. The far field noise spectrum and noise directivity are also compared with experimental data. 4.1. Comparison positions The profiles of the mean velocity and the rms (root mean square) value of velocity fluctuations obtained from LES are compared with experimental results as well as other numerical results. Fig. 8 shows the sketch of measurement positions. The profiles at three different locations are compared, i.e. plane A at x/c = 0.25 which corresponds to the far wake of the rod, plane B at x/c = 0.25 which is slightly upstream of the airfoil thickest point, plane C at x/c = 1.1 which corresponds to the near wake of the airfoil. It should be noted that x/c = 0 corresponds to airfoil leading edge. Streamwise velocity power spectrum density is also compared with available experimental results at two different locations, i.e. P1 and P2 (Fig. 8). P1 is located at (0.25c, 0.08c) and P2 is located at (0.25c, 0.16c). P1 is 0.2d off the airfoil surface and P2 is 1d off the airfoil surface. In addition, the far field acoustic pressure spectrum at a distance of R = 1.85 m of 90° angular angle (just above the airfoil) from the airfoil center is also compared with experimental results. Moreover, the Strouhal number of the vortex shedding frequency is compared with experimental results. 4.2. Steady flow field Velocity profiles and turbulent intensity profiles are compared with experiment results by Jacob et al. [15] and previous LES (or DES) results by Jacob [15], Giret [7] and Greschner [9] at three different locations in Fig. 8. Fig. 9(a) and (b) shows the mean velocity profile and turbulent intensity profile at plane A. It can
be found that the present numerical results agree well with the experimental results and other numerical results. The rod wake mean velocity profile is well predicted by the current simulation. The minimum value of u/U0 is about 0.709 in experimental results while the predicted minimum value of u/U0 is 0.699 and the relative error is about 1.41%. It can be seen from Fig. 9(b) that, overall the turbulent intensity profile predicted by present LES reaches an agreement with experiment results and other numerical results. The maximum value of turbulent intensity of experiment and the present simulation is about 0.176 and 0.193, respectively. The predicted value is 9.66% higher than that of the experiment. Fig. 9(c) shows the mean velocity profile at plane B. The present LES performs similarly to Giret’s LES results and the mean profile is over predicted by about 5.33%. This over prediction is also found in many LES results [8,20]. Fig. 9(d) shows the turbulent intensity profile at plane B. The peak value of turbulent intensity of experiment and the present simulation is about 0.164 and 0.223, respectively. The present simulation over predicts the turbulent intensity peak by about 36.0%. Part reason for this large discrepancy is that the peak value of the turbulent intensity is hard to measure in the experiment due to the spatial resolution of the hot wire. Fig. 9(e) and (f) shows the mean velocity profile and turbulent intensity profile at plane C, which represents the near wake of the airfoil. Compared with experimental results, the maximum wake deficit is under predicted by about 7.90% in Fig. 9(e). The maximum value of the turbulent intensity is 0.135 for experiment and 0.152 for present simulation. The maximum value of the turbulent intensity is over predicted by about 12.6% in Fig. 9(f). There is a discrepancy between the experimental turbulent intensity results and numerical turbulent intensity results near the location of y/c = 0.2, which can also be seen in Giret’s numerical results. Overall, the present LES gives satisfactory prediction of mean velocity profiles and turbulent intensity profiles. 4.3. Unsteady flow field The unsteady flow field is also investigated. The Strouhal number is defined as St ¼ fd=U 0 . The experimental value of vortex shedding Strouhal number is 0:19 0:002. The predicted value of Strouhal number is 0:202. The predicted result agrees reasonably well with the experimental result which indicates that the physics of the flow separation on the rod are correctly reproduced in the current simulation. The instantaneous span-wise vorticity and iso-surfaces of the Q-criterion are shown in Figs. 10 and 11, respectively. It can be seen that the main vortices shedding from the rod form regular Karman vortex street which impinges onto the airfoil and partly broken into small structures. Moreover, many smaller vortices can be observed in the rod wake, due to the transition to turbulence in the shear layer of the rod. The broken of large vortices is successfully captured by the current simulation. It is expected that the rod wake interacting with the leading edge of the airfoil is the main mechanism of the turbulence airfoil interaction noise. 4.4. Velocity spectrum
Fig. 8. Sketch of measurement positions.
The power spectrum density of the streamwise velocity component is compared with the experimental results in Fig. 12. The agreement with experimental data is satisfactory. It can be seen from Fig. 12(a) that the current simulation can capture the broadened peak which means the interaction between the random part of the flow and the quasi deterministic vortex shedding is well
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(a) Mean velocity proile at plane A
(b) Turbulent intensity proile at plane A
7
(d) Turbulent intensity proile at plane B
(e) Mean velocity proile at plane C
Fig. 9. Comparison of mean velocity profile and turbulent intensity profile with experiment and other numerical simulations.
predicted. However, the peak of the power spectrum density is over predicted by about 1.6 dB and the vortex shedding frequency is over predicted by 6.3%. Fig. 12(b) shows the power spectrum density of the streamwise velocity at P2, which is one rod diameter off the airfoil surface. It can be seen that vortex shedding frequency is much less obvious than that at P1. The spectrum at P2
is also well predicted by the current simulation. The peak of the power spectrum density is over predicted by about 3.0 dB and the predicted power spectrum density is about 2 dB lower than that of the experimental result in high frequency range (around St = 0.8) where the predicted spectrum seems to decay slightly more quickly.
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Fig. 10. Span-wise vorticity contour.
Lc Lexp Spp ðf Þ exp ¼ Spp ðf Þ sim þ 20 lg þ 10 lg ; Lsim Lc
Lsim < Lc 6 Lexp ð32Þ
Lexp ; Spp ðf Þ exp ¼ Spp ðf Þ sim þ 20 lg Lsim
The spanwise length of the LES computational domain is often limited by computational resources. As a result, the spanwise length in the numerical simulation Lsim is usually shorter than the experimental spanwise length Lexp . In order to compare the predicted sound pressure level with experimental data, the sound pressure level obtained from numerical simulations must be corrected. Different correction methods have been developed by Kato et al. [31], Seo et al. [32], Boudet et al. [8], and Perot et al. [33]. The correction of the power spectral density of the pressure in the far field can be expressed as follows [8],
! R Lexp R Lexp Cðjz2 z1 j; f Þdz1 dz2 0 0 ¼ Spp ðf Þ sim þ 10 lg R L R L sim sim Cðjz2 z1 j; f Þdz1 dz2 0 0 ð29Þ
where Cðjz2 z1 j; f Þ is the coherence function at the frequency f between two points along the solid surface. The coherence length Lc is defined as
Z
þ1
z1
Cðjz2 z1 j; f Þdz2
ð30Þ
Simplifications of Eq.(29) can be made by assuming the coherence function of specific mathematical form. Kato [31] assumes the coherence function of the rectangular function form and the correction function can be simplified as follows
Spp ðf Þ
exp
Lexp ¼ Spp ðf Þ sim þ 10 lg ; Lsim
pffiffiffiffi Lc 6 Lsim = p
ð34Þ
pffiffiffiffi Lc pLexp Spp ðf Þ exp ¼ Spp ðf Þ sim þ 10 lg þ 10 lg ; Lsim Lsim pffiffiffiffi pffiffiffiffi Lsim = p < Lc 6 Lexp = p
ð35Þ
Lexp Spp ðf Þ exp ¼ Spp ðf Þ sim þ 10 lg Lsim
4.5. Acoustic results
Lc ¼
Lc 6 Lsim
ð33Þ
Seo [32] assumes the coherence function of the Gaussian function form and the correction function can be simplified as follows
Fig. 11. Iso-surfaces of the Q-Criterion (Q = 2:5 106 s2).
ðSpp ðf ÞÞexp
Lexp < Lc
ð31Þ
Lexp ; Spp ðf Þ exp ¼ Spp ðf Þ sim þ 20 lg Lsim
Lexp =
pffiffiffiffi
p < Lc
ð36Þ
Perot [33] assumes the coherence function of the exponential function form and gives a correction method independent on spanwise coherence length but dependent of the distance between observer point and the source point
arctan Spp ðf Þ exp ¼ Spp ðf Þ sim þ 10 lg
Lexp R
arctan arctan Lsim R
Lsim R
ð37Þ
Actually, the coherence length Lc is a function of frequency and the corrections should also be a function of frequency. Unfortunately, the exact Lc as a function of frequency is often not available so a reasonable estimate of coherence length is usually used in practice. According to Jacob’s experiment results, the spanwise pressure correlation length Lc on the rod is about 6.5d [15]. Fig. 13 shows the comparison of different correction methods. It should be noted that in Fig. 13 the following parameters are used, Lsim ¼ 2d, R ¼ 1:85 m and Lexp ¼ 30d. For the current problem, Lc ¼ 6:5d and Lc =Lsim ¼ 3:25. So Kato’s method will lead to a correction of 16.87 dB and Seo’s method will lead to a correction of 19.37 dB while Perot’s method gives a correction of 11.42 dB.
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Fig. 13. Comparison of different spanwise correction methods.
(a) PSD at P1
Fig. 12. Comparison of PSD of the streamwise velocity component at P1 and P2.
The sound pressure level (SPL) of the far field acoustic pressure corrected by different methods at 1.85 m for the angular position 90° is shown in Fig. 14 where the abscissa is scaled to Strouhal number. The sound pressure level is defined as
PSDðf Þ SPLðf Þ ¼ 10 log 10 p2ref
9
! ð38Þ
where pref ¼ 2 105 Pa and the PSD is normalized to be per unit f and Welch’s method was used to calculate the PSD in Eq. (27). It can be seen from Fig. 14 that the peak value of SPL corrected by Kato’s method is 90.44 dB, 92.94 dB by Seo’s method and 84.99 dB by Perot’s method. The peak value of SPL is about 91.44 dB in the experiment. Perot’s correction method obviously under predicts the noise peak level by 6.45 dB and the predicted
Fig. 14. Comparison of SPL of the acoustic pressure at 90° and at 1.85 m corrected by different methods.
broadband part of the noise in high frequency range (near St = 1) is also lower than that of experiment results by 3–4 dB. The Seo’s correction method over predicts the peak value of SPL by 1.5 dB and the broadband part of the noise by about 4 dB (near St = 1). The Kato’s correction method performs well both for the noise peak level prediction and the broadband part of the noise prediction in a large frequency range. The peak value of SPL is under predicted by 1 dB while the broadband part of noise is over predicted by about 1.5 dB in high frequency range (near St = 1) and about 4 dB between St = 0.2–0.3. This big discrepancy is mainly due to the over-predicted peak frequency (vortex shedding frequency) which leads to the shift of the whole predicted spectrum. The vortex shedding Strouhal number is 0:19 0:002 in experiment whereas the predicted value of Strouhal number is 0:202 with a 6.3% over predict. To address this effect, Fig. 15 shows the
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Fig. 15. Comparison of SPL of the acoustic pressure at 90° and at 1.85 m corrected by Kato’s method (with peak frequency corrected).
comparison of SPL after the predicted peak frequency are corrected to the experimental peak frequency and the agreement improves significantly. The difference between predicted results and experimental results is within 2–3 dB over a wide frequency range. The Kato’s correction method can give very good results if the peak frequency could be predicted more accurately. In addition, the Kato’s correction method is widely used and demonstrates satisfactory prediction results [7,9,15]. As a result, the Kato’s correction method [31], precisely Eq. (32), is used hereinafter to correct the SPL of acoustic pressure obtained from the numerical simulation. Fig. 16 compares the SPL of the far field acoustic pressure at 1.85 m for angular position of 90° with the experimental data [15] and other numerical results [7,9,20] available. It can be seen from Fig. 16 that the present prediction results agree well with experimental data. The discrepancy between the prediction and experimental results is within 2 dB at the peak noise frequency and the broadband part of the noise also agrees well with the experimental results. It can be seen that the present prediction method shows similar or improved results than other numerical results both in the main peak noise and the broadband part of the noise. The acoustic directivity results are compared with experimental data in Fig. 17. The prediction results of the present method agree fairly well with experimental results. The discrepancy between predicted results and experimental results is within 3 dB for most of the angular positions. The maximum discrepancy is about 4.5 dB near angular position of 110°. The acoustic directivity results is over predicted by 2–4.5 dB between angular position of 90° to 120° and 240° to 270°. Excellent agreement is obtained between angular position of 60° to 90° and 270° to 300°. Through the comparison of far field acoustic pressure spectrum and the acoustic directivity between the predicted results and experimental results, it can be found that the acoustic results obtained from LES and the current acoustic analogy method agree fairly well with the experimental results. This indicates that the current LES and the frequency domain method of generalized Lighthill acoustic analogy theory are capable to predict broadband noise sources of complex flows and the resulting broadband noise.
Fig. 16. Acoustic pressure results compared with experimental data, at 90° and 1.85 m.
5. Conclusions A new LES-acoustic analogy method for accurate flow and broadband noise prediction is proposed and assessed. A frequency domain method for the generalized Lighthill acoustic analogy theory is developed. The frequency domain method is derived in detail and the final equations for code is provided which can help to bridge the gap between the flow field prediction and the acoustic prediction for those who are interested in acoustic results but lack of acoustic prediction ability. The rod-NACA0012 airfoil configuration is selected because of its detailed experimental results and its relevance to model broadband noise. The LES is performed by CFX and the acoustic prediction method is based on the generalized Lighthill equation. Both the flow field results and the acoustic field results are compared with the experimental results and other numerical results.
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Appendix A Nomenclature c c0 d f G Lsim Lexp Lc R S St s St Spp t
!
chord length ambient speed of sound rod diameter frequency filter function or Green function spanwise length in numerical simulation spanwise length in experiment correlation length distance between observer point and the source point amplitude radius grid Strouhal number Strouhal number power spectral density of the pressure in the far field observation time coherence function retarded time fluid variable filtered variable unresolved part of the variable acoustic density disturbance acoustic pressure observation coordinate
!
source coordinate
C
s
Fig. 17. Acoustic directivity compared with experimental data.
U U U0
q0 p0 x
The flow field results show that the current LES can give accurate prediction of the broadband noise sources although the vortex shedding frequency is slightly over predicted. Both the mean velocity profiles and turbulent intensity profiles are well predicted. Moreover, the predicted velocity spectrum agrees very well with the experimental data. The acoustic field results show that the predicted acoustic pressure spectrum results match very well with the experimental data. The discrepancy between the predicted results and experimental results is within 2 dB at the peak noise frequency and the broadband part of the noise also agrees well with the experimental results. Due to the discrepancy of span length between the numerical and experimental setup, the acoustic pressure spectrum obtained from numerical results needs to be corrected before compared with experimental data. It is found that the span correction methods have an important effect on the level of the predicted acoustic results. It is shown that the assumption of coherence function of the rectangular function form by Kato is reasonable and gives better results than that of Seo’s and Perot’s method. Overall, the LES and acoustic analogy method for broadband noise prediction developed in this paper performs well and can give a good prediction of the broadband noise.
Acknowledgements This study is supported by National Natural Science Foundation of China and the project number is 51276149, 51476134. This research is also supported by State Key Laboratory of Aerodynamics of China and the project number is SKLA20160201. Additional acknowledgements to Key Laboratory of Aerodynamic Noise Control of China Aerodynamics Research and Development Center for the project of ANCL20160102. The simulation work was supported by Center for High Performance Computing of Northwestern Polytechnical University, China.
y V 0N
velocity of the surface normal to itself
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Please cite this article in press as: Tong F et al. Broadband noise prediction using large eddy simulation and a frequency domain method. Appl Acoust (2016), http://dx.doi.org/10.1016/j.apacoust.2016.11.001