Time-domain inflow boundary condition for turbulence–airfoil interaction noise prediction using synthetic turbulence modeling

Journal of Sound and Vibration 340 (2015) 138–151

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Time-domain inflow boundary condition for turbulence–airfoil interaction noise prediction using synthetic turbulence modeling Daehwan Kim a, Seung Heo b, Cheolung Cheong b,n a b

Agency for Defense Development, 330 Sinseong-ro Haemi-myeon Seosan-si Chungnam, 356-823 Korea School of Mechanical Engineering, Pusan National University, 30, Janjeon-dong, Geumjeong-gu, Busan 609-735, Republic of Korea

a r t i c l e in f o

abstract

Article history: Received 30 December 2013 Received in revised form 8 October 2014 Accepted 29 November 2014 Handling Editor: P. Joseph Available online 31 December 2014

The present paper deals with development of the synthetic turbulence inflow boundary condition (STIBC) to predict inflow broadband noise generated by interaction between turbulence and an airfoil/a cascade of airfoils in the time-domain. The STIBC is derived by combining inflow boundary conditions that have been successfully applied in external and internal computational aeroacoustics (CAA) simulations with a synthetic turbulence model. The random particle mesh (RPM) method based on a digital filter is used as the synthetic turbulence model. Gaussian and Liepmann spectra are used to define the filters for turbulence energy spectra. The linearized Euler equations are used as governing equations to evaluate the suitability of the STIBC in time-domain CAA simulations. First, the velocity correlations and energy spectra of the synthesized turbulent velocities are compared with analytic ones. The comparison results reveal that the STIBC can reproduce a turbulent velocity field satisfying the required statistical characteristics of turbulence. Particularly, the Liepmann filter representing a non-Gaussian filter is shown to be effectively described by superposing the Gaussian filters. Each Gaussian filter has a different turbulent kinetic energy and integral length scale. Second, two inflow noise problems are numerically solved using the STIBC: the turbulence–airfoil interaction and the turbulence-a cascade of airfoils interaction problems. The power spectrum of noise due to an isolated flat plate airfoil interacting with incident turbulence is predicted, and its result is successfully validated against Amiet's analytic model (Amiet, 1975) [4]. The prediction results of the upstream and downstream acoustic power spectra from a cascade of flat plates are then compared with Cheong's analytic model (Cheong et al., 2006) [30]. These comparisons are also in excellent agreement. On the basis of these illustrative computation results, the STIBC is expected to be applied to investigate more complicated inflow noise problems including the effects of non-uniform mean flow, nonlinear interaction, and real airfoil shapes. & 2014 Elsevier Ltd. All rights reserved.

1. Introduction Inflow noise is generated by an interaction between turbulence and an airfoil. It is the main contributor to broadband noise emissions from aero-engines [1] due to stators or outline guide vanes (OGVs) interacting with rotor-wake. Inflow noise

n

Corresponding author. Tel.: þ82 51 510 2311; fax: þ 82 51 514 7640. E-mail address: [email protected] (C. Cheong).

http://dx.doi.org/10.1016/j.jsv.2014.11.036 0022-460X/& 2014 Elsevier Ltd. All rights reserved.

D. Kim et al. / Journal of Sound and Vibration 340 (2015) 138–151

139

is also one of the noise components of a wind turbine. Passive control methods have been tried to reduce the inflow noise. In aero-engines, swept/leaned stators blades and acoustical treatments are representative of these methods [2]. Recently, the effects of leading edge serrations on inflow noise have been investigated [3]. However, it is essential for the further reduction of the noise to understand the mechanism of broadband noise generation, which relies on effective and accurate prediction methods. There arise many difficulties in predicting inflow broadband noise because a wide range of time and length scales is involved in broadband noise generation and propagation. Development of a broadband noise prediction method to resolve these issues is, therefore, vital for the future success in controlling inflow broadband noise. Analytical models have been developed to predict turbulence–airfoil interaction noise. The Amiet [4] model can be used to predict noise generated by a thin airfoil interacting with inflow turbulence. Blandeau et al. [5] presented acoustic power formulas by using Amiet's model to estimate the broadband noise radiated from an isolated flat plate in two-dimensional turbulent flow. These analytical models have limitations in their applications because they have been derived assuming a flat plate airfoil and thus cannot treat the noise radiated from a real airfoil with non-zero thickness, camber, and leading edge radius. In contrast to the analytical approaches, time-domain simulations can predict turbulence–airfoil interaction noise more accurately by considering more complex airfoil configurations and realistic flows. Until now, harmonic gust–airfoil interaction problems have been extensively investigated in many studies. Lockard and Morris [6] described the interaction of a vortical gust with airfoils of NACA series using a Navier–Stockes code. Hixon et al. [7] discussed computational aeroacoustics methods that treat noise from isolated airfoils interacting with finite amplitude gusts. Golubev et al. [8–10] focused on the effects of a gust with high amplitude and high frequency on the gust–airfoil interaction. Although the mentioned studies [6–10] reported the successful computation of tonal noise, extension of these methods to broadband noise is not straightforward. In fact, it is essential for extending time-domain approaches applied for tonal noise to inflow broadband noise to develop an appropriate inflow turbulence model in the time-domain. A turbulent field synthesized using the stochastic method must satisfy the features of turbulence such as turbulent kinetic energy, length scale, etc. Furthermore, the generation of synthesized turbulence must be integrated with time-domain acoustic propagation models. Stochastic methods to synthesize a turbulent velocity field can be divided into three categories: the method based on the superposition of Fourier modes, those based on digital filters and synthetic eddy models. First method discretizes a prescribed turbulent energy spectrum to synthesize turbulent velocities. Initially, Kraichnan [11] introduced the method to generate the stochastic velocity fields based on random Fourier modes. Bechara et al. [12] modified Kraichnan's model [11] to treat more general turbulent spectra and proposed synthetic turbulent velocities as a sum of cosine functions. Their stochastic model directly determines an amplitude of each mode from a given turbulent energy spectrum and uses logarithmic wavenumber distribution. Bailly et al. [13] added the convection effect to the synthesized turbulent velocities developed by Bechara et al. [12]. In contrast to the model by Bechara et al. [12], they adopted a different wavenumber distribution to improve its resolution in high wavenumbers. Billson [14,15] also included the convection effect in the synthesized turbulent velocities by solving a convection equation at each time step. Based on the work of Kraichnan [11], Smirnov et al. [16] proposed the random flow generation algorithm which can be used to prescribe inlet conditions as well as initial conditions for spatially developing inhomogeneous, anisotropic turbulent flow. Their work was extended by Huang et al. [17] to deal with arbitrary turbulent spectra. However, the methods based on Fourier modes require a large number of modes to generate a turbulent velocity field that satisfies the statistical features of the turbulence. Golubev et al. [18,19] proposed a method that generates synthetic turbulence in an arbitrary computational region upstream of the surface instead of the upstream boundary, which can increase frequency resolution by locating the turbulence source in the nearsurface region with refined mesh. Second methods that generate turbulence by filtering arbitrarily distributed random values using a digital filter have been devised for improving numerical efficiency. Klein et al. [20] proposed a method for filtering white noise that cannot produce divergence-free velocity field which is required to satisfy the dispersion relation of vorticity wave. The filtering method developed by Careta et al. [21] can generate divergence-free velocity fields, but it cannot consider the convection effect. More recently, Ewert et al. [22–25] developed a random particle mesh (RPM) method which can account for the convection effect and synthesize divergence-free velocities. In various applications, the RPM methods are combined with steady Reynolds averaged Navier–Stokes (RANS) simulations to obtain features of turbulence such as the kinetic energy and dissipation rate of turbulent flow, which are in turn used as inputs to determine the digital filter. This is then incorporated in computational aeroacoustics (CAA) codes. In many applications of the RPM method, as described in Refs. [22–25], only the Gaussian filter was used. General (non-Gaussian) turbulent spectra can be produced by superposing Gaussian spectra [26]. Dieste and Gabard [27] proposed non-Gaussian filters to deal with various turbulent spectra and predicted noise due to the interaction of turbulence with a flat plate. Synthetic turbulent velocities were used along the flat plate as the wall boundary condition. The third method uses the synthetic eddy model under the assumption that turbulence can be regarded as a superposition of random eddies. Jarrin [28] firstly introduced the synthetic eddy model without focusing on the divergence-free condition. Recently, Sescu [29] extended Jarrin's model to satisfy the divergencefree condition and applied the model for generating the realistic turbulent flow. The main objective of the present paper is to develop a stochastic method based on the RPM model to generate synthetic turbulence at inflow boundary, which can be directly incorporated in time-domain CAA simulations for the prediction of inflow broadband noise. Two types of synthetic turbulent inflow boundary conditions (STIBCs) were developed by merging the filter-based RPM method with two types of inflow boundary conditions: one for the external flow problem and another for the internal flow problem. The former was derived by combining the RPM method with the inflow radiation boundary condition based on the far-field asymptotic solution of the Euler equations, and the latter was devised by coupling the RPM

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method with the characteristic boundary condition based on the characteristic waves of the Euler equations. As described above, Dieste and Gabard [27] used the RPM method to predict broadband interaction noise between a turbulent stream and a flat plate. We extend this approach by developing the STIBC. The extension is made by coupling the RPM method with inflow boundary conditions instead of applying the synthetic turbulence velocities as wall boundary conditions. This extension enables the developed method to deal easily with inflow noise problems involving real airfoil configurations as well as a flat plate under the assumption of a frozen turbulence. In order to investigate inflow noise radiated from real airfoils, careful attention must be paid to non-uniform mean flow around an airfoil. The trajectory of the random particles has to be determined through a non-uniform mean flow. This can be done by using a body-fitted curvilinear grid system that can describe the streamline around an airfoil. In this respect, coupling the RPM method with inflow boundary conditions is more efficient for treating real airfoils. In addition, the present method effectively generates a non-Gaussian filter such as the Liepmann or von Kármán filter that can produce the more realistic energy spectra required for a specific problem by superposing Gaussian filters, each of which has its own turbulent kinetic energy and integral length scale. Two classical inflow noise problems were solved with time-domain CAA simulations to investigate the suitability of the STIBC for inflow broadband noise simulations in external and internal flows. The first is the broadband noise from a flat plate, and the second is the broadband noise from a cascade of flat plates. The predicted results of the acoustic power spectra were validated against analytical solutions. Analytical solutions for each inflow problem were obtained from a modified version [5] of Amiet's model [4] and the analytic formula by Cheong et al. [30]. In both problems, the STIBC showed good performance in generating synthetic turbulence, which is essential for accurate prediction of inflow noise. In the following section, the RPM model [27] is briefly reviewed, and the STIBCs are derived by combining the RPM model and inflow boundary conditions. In Section 3, the STIBC is applied with the linearized Euler equations to check the validity of the present method in generating the inflow turbulence of the required stochastic features. In Section 4, the benchmark problems on inflow broadband noise are solved using the present numerical methods and the prediction results are compared with analytic solutions.

2. Synthetic turbulence inflow boundary condition (STIBC) In this section, the RPM method in Ref. [27] is briefly presented. Then the STIBCs for external and internal flows are derived by merging the algorithm for synthesizing turbulence with inflow radiation [31,32] and characteristic boundary conditions [33], respectively. The turbulence is assumed to be isotropic, homogeneous, and incompressible. The turbulence is also assumed to be non-evolving, i.e., fixed in a frame of reference moving with the mean flow. 2.1. Random particle mesh (RPM) method The two-dimensional turbulent velocities ut ¼(ut, vt)T located at x in Cartesian coordinates can be synthesized in Lagrangian coordinates in the form: Z ut ðx; t Þ ¼ (1) G
ðx  x0 ðx0 ; t ÞÞU ðx0 ; t Þdx0 ; Ω0

 T where G
¼ ∂G=∂y;  ∂G=∂x , G denotes the digital filter, and U represents a random field. x0 (x0,t) denotes the trajectory of the fluid element located at x0 at an initial time t0. By discretizing the initial volume of the fluid, Ω0, at t0 using nonoverlapping elements Ω0n(n ¼1,2,…,N), Eq. (1) can be represented by a finite sum of Ω0n. Furthermore, if Ω0n is small relative to the turbulent integral length scale Λ, G can be considered to be constant over each element Ω0n. This yields the final discretized form of the RPM method to synthesize turbulent velocities as follows: N   ut ðx; t Þ ¼ ∑ G
x  xn ðt Þ U n ðt Þ;

(2)

n¼1

where xn is the position of Ω0n as it convects with the mean flow. The trajectory of xn is defined as the center of mass of Ω0n. The linearized Euler equations are used to deal with inflow turbulence problems in the present paper, so that the position of a random vortex at time t can be defined as xn ðt Þ ¼ x0 þut;

(3)

where u denotes uniform meanflow, ðu; vÞT . In Eq. (2), the strength of the random value Un is defined as the averaged value of U over Ω0n: Z U n ðt Þ ¼ U ðx0 ; t Þdx0 : (4) Ω0n

For the frozen turbulence we are considering, U(x0, t) is constant irrespective of time for the observer moving with Ω0n, which leads to Un(t) ¼Un(0). Un is a Gaussian random variable with zero mean, and its variance is equal to the area of Ω0n. Consequently, using Eq. (2), turbulent velocities at x are synthesized by a finite sum of N vortices with each located at xn.

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2.2. Coupling with the inflow boundary condition Now the RPM method in Eq. (2) is incorporated with the time-domain CAA inflow boundary conditions to develop the STIBC. Two kinds of the STIBCs are developed to deal with external and internal inflow broadband noise problems. The former is based on Tam's radiation boundary condition [31,32], and the latter is based on Giles' inflow boundary condition [33]. These boundary conditions have been widely used for CAA simulations in free-field or in-duct conditions [3,7,34,35]. Incorporating the RPM method into so-called buffer-zone boundary conditions as done by Wohlbrandt et al. [36] is not taken into account in this study, because turbulent velocities need to be synthesized over more grid points in the bufferzone, which may increase computational time. Tam's radiation boundary condition [31,32] was developed based on asymptotic far-field solutions of the linearized Euler equations as the form 2 03 2 03 2 03 2 03

ρ

ρ

ρ

ρ

6 07 6 07 6 07 6 07 u 7 6u 7 6u 7 1 ∂6u 7 ∂6 1 7 6 0 7 þ sin θ ∂ 6 0 7 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 0 7 ¼ 0;   6 0 7 þ cos θ 6 6 7 6 7 7 ∂x4 v 5 ∂y4 v 5 2 x2 þ y2 6 V θ ∂t 4 v 5 4v 5 0 0 0 0 p p p p

(5)

where     2 1=2 ; V θ ¼ u cos θ þ v sin θ þ c2  u cos θ v sin θ Here, θ ¼ tan  1 ðy=xÞ, c is speed of sound, ρ and p represent density and pressure, respectively, and u and v also denote velocity components in the x- and y-directions, respectively. And u and v are mean velocities. The superscript ‘0 ’ in Eq. (5) denotes the fluctuating acoustic quantities. The main function of this radiation boundary condition of Eq. (5) causes the acoustic disturbance propagating in the upstream direction to leave the computation domain without any reflection back into the computational domain. To simulate the inflow broadband noise problem, this inflow boundary condition needs to generate the inflow turbulence of the required statistical features without reflecting outgoing disturbance. Hixon et al. [32] and Nallasamy et al. [33] developed the procedures to achieve this goal. Following these procedures, the radiation boundary condition with the synthetic turbulent velocity can be derived in the form: 2 3 2 3 2 3 2 3 0 0 0 0

ρ

ρ

ρ

ρ

6 0 7 6 0 7 6 0 7 6 0 7 6 u  ut 7 6 u  ut 7 6 u  ut 7 1 ∂ 6 u  ut 7 1 7 þ cos θ ∂ 6 0 7 þ sin θ ∂ 6 0 7 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 0 7 ¼ 0;   6 0 6 7 6 7 6 7 6 ∂x4 v  vt 5 ∂y4 v  vt 5 2 x2 þ y2 4 v  vt 7 V θ ∂t 4 v  vt 5 5 p0 p0 p0 p0

(6)

where subscript ‘t’ indicates that the disturbances are associated with the turbulent quantities to be synthesized. Eq. (6) can be used to generate the required turbulence as well as to function as a nonreflective boundary condition for upstream-going acoustic waves in the external flow. Giles' inflow boundary conditions were developed using the properties of characteristic waves supported by the Euler equations. The two-dimensional version of the inflow boundary condition is given by 2 3 1 1 2 03 2 3  12 0 2 2 ρ 2c 2c 7 c1 6 c 6 07 6 0 6 7 0 21ρc 21ρc 7 6u 7 6 76 c2 7 6 0 7¼6 76 7; (7) 6v 7 6 0 7 7 1 0 0 76 4 5 6 4 c3 5 ρc 4 5 p0 c4 1 1 0 0 2 2 where c1, c2, c3 and c4 represent the characteristics resulting from the Euler equations. And ρ is mean density. Here, since the first three terms are incoming characteristics, and c4 is outgoing characteristic for Giles' inflow boundary condition; c1, c2 and c3 are updated with new information from incoming disturbances, whereas c4 is either extrapolated from the interior or it remains unchanged. The time derivatives of the inflow characteristics are computed from a 4th-order formulation for the incoming characteristics (c1, c2 and c3). ∂c1 ∂c1 ¼ v ; ∂t ∂y

(8a)

∂c2 ∂c2 1 ∂c3 1 ∂c4 ¼ v  ðc þ uÞ  ðc  uÞ ; ∂t ∂y 2 ∂y 2 ∂y

(8b)

∂c3 1 ∂c2 ∂c3 ¼  ðc uÞ v : 2 ∂t ∂y ∂y

(8c)

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The derivative of the characteristic in the y-direction can be expressed from Eq. (8) with the inflow turbulent velocities in the form 2 ∂c 3 2 ∂ρ0 3 1 2 3 2 ∂y c 0 0 1 6 ∂c 7 6 ∂y 7 76 ∂u0 ∂ut 7 6 27 6 6 ∂y 7 6 0 ρ c 0 76 ∂y  ∂y 7 0 76 0 6 ∂c 7 ¼ 6 7 (9) 6 37 6 0 6 ∂v ∂vt 7: ρc 0 1 7 56 ∂y  ∂y 7 6 ∂y 7 4 4 5 4 5 0 ∂c4 ∂p 0  ρc 0 1 ∂y

∂y

The time derivatives of the inflow variables in Eq. (8) are then computed as the following equation where inflow turbulent velocity is introduced as the acceleration terms in Eq. (6). 32 3 2 ∂ρ0 3 2 1 1 1 ∂c1  2 0 2 3 2 2 0 2c 2c 7 c ∂t ∂t 6 6 ∂u0 7 6 6 1 1 76 ∂c2 7 6 ∂ut 7 6 7 6 0 7 0 2ρc 2ρc 76 ∂t 7 6 ∂t 7 6 ∂t 7 6 76 ∂c 7 þ 6 7: (10) 6 ∂v0 7 ¼ 6 3 1 t 5 6 ∂t 7 6 0 7 4 ∂v 0 0 7 76 ∂t 5 ∂t ρc 4 05 4 4 5 ∂c4 ∂p 1 1 0 0 0 ∂t ∂t 2 2 It can be seen that turbulent velocity terms are incorporated in the form of their spatial and time derivatives in Eqs. (6), (9) and (10). The spatial derivatives can be defined by the second derivative of the digital filter G in Eq. (2). The additional spatial derivatives of the synthesized velocities in Eq. (2) lead to     N ∂2 G x x ðt Þ N ∂2 G x  x ðt Þ ∂ut ∂ut n n U n ðt Þ; ðx; t Þ ¼ ∑ ðx; t Þ ¼ ∑ U n ðt Þ; (11a) ∂x∂y ∂x ∂y ∂y2 n¼1 n¼1   N ∂2 G x x ðt Þ ∂vt n ðx; t Þ ¼  ∑ U n ðt Þ; ∂x ∂x2 n¼1

  N ∂2 G x  x ðt Þ ∂vt n U n ðt Þ: ðx; t Þ ¼  ∑ ∂x∂y ∂y n¼1

(11b)

For frozen turbulence, synthesized turbulent velocities satisfy the convection equation. The time derivatives can therefore be expressed in the form ∂ut ðx; t Þ ¼  ðu U ∇Þut ðx; t Þ: ∂t

(12)

The inflow boundary conditions of Eqs. (6) and (10) combined with Eqs. (11) and (12) can generate the inflow turbulence of the statistical properties required in a given problem as well as nonreflecting outgoing disturbances. In the next section, the synthesized turbulence generated at the inflow boundary using the STIBC is shown to have the required statistical features by solving the linearized Euler equations. 3. Validation of the synthesized turbulent velocity field The suitability of the STIBC is investigated by comparing autocorrelations and energy spectra between numerical solutions obtained from time-domain simulations and analytical ones. Two-dimensional linearized Euler equations are adopted as the governing equations and are solved by using high-order finite difference CAA techniques in the time-domain. The linearized Euler equations are given by 2 0 3 2 3 2 3 uρ0 þ ρu0 vρ0 þ ρv0 ρ 6 07 6 7 6 7 ρu 7 6 ρuu0 þp0 7 ∂ 6 ρvu0 7 ∂6 6 0 7þ ∂ 6 7þ 6 7 (13) 0 0 0 7 ¼ 0; 6 7 6 7 6 ∂t 4 ρv 5 ∂x4 ρuv 5 ∂y4 ρvv þp 5 up0 þ γ pu0 vp0 þ γ pv0 p0 where γ is thepspecific ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi heat ratio of the air. The reference physical scales for the non-dimensionalization of Eq. (13) are as follows: W ¼ u2 þv2 for velocities, ρ for density, ρW 2 for pressure, c for length scale, and c/W for time. Here c is the chord length of an airfoil. High-order finite difference CAA techniques in the time-domain are used to solve Eq. (13). The seven-point stencil dispersion-relation-preserving (DRP) scheme [31] is used to compute the spatial derivative, and the optimized four-level Adams–Bashforth method [31] is used for time integration. The CAA program based on this scheme has been applied and validated over various applications [37–41] including broadband noise generation and propagation. The objectives of CAA simulation in this section are to capture the features of synthesized turbulence and to validate the STIBC by comparing numerical results with analytical ones. The computation domain is outlined in Fig. 1 where there is no airfoil considered at this stage. The synthesized turbulence is generated at the inflow boundary by using the STIBC, and it convects through the computational domain at the speed of the uniform meanflow. The turbulence finally exits the computational domain through the outflow boundary. The two-dimensional Gaussian spectrum [11] is first tested in the

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143

Fig. 1. Schematic diagram of the computational domain to investigate the suitability of the STIBC for capturing the features of the prescribed turbulent energy spectrum.

following simulation. The Gaussian energy spectrum is !

Λ2 κ 2 Eðκ Þ ¼ 2 K Λ κ exp  ; π π 2

where K denotes the kinetic energy, and

4 3

(14)

κ is the wavenumber. The corresponding digital filter [27] is given in the form GðrÞ ¼

rffiffiffiffiffiffiffi 2K

 π r2 exp  2 ; π 2Λ

(15)

where r(¼|x  x0 |) represents the distance between the position x where the velocities are synthesized and the location of a random vortex x0 . For the simplicity, v is set to be zero. The test computation is conducted under the conditions that the Mach number Mð ¼ W=cÞ equals 0.5 and the turbulence length scale is set to be Λ/c¼0.1. The computation domain is discretized with uniform grid spacing Δx/c ¼ Δy/c¼0.01. Since the seven-point DRP scheme can resolve the wavenumber range up to κΔx ¼1.1, the wavenumber resolved in the current computation is κc¼ 110 which corresponds to the maximum reduced frequency ωc/W¼110. Here, ω is the angular frequency. In order to obtain stable computational solutions, the time step is set to be Δt/(c/W)¼0.001 which satisfies the 2 criteria of Δt/(c/W) o0.0024 for acoustic wave and Δt/(c/W) o0.0094 for entropy and vorticity waves as given in Eqs. (5.14) and (5.15) of Ref. [31]. Periodic boundary condition is applied at y/c ¼1 and y/c ¼0. To generate turbulent velocity, the STIBC based on Giles' inflow boundary condition in Eq. (10) is applied at x/c ¼ 1.5. Here, the periodic boundary condition is also applied to the vortex region to ensure the periodicity of the synthesized turbulent velocities. Giles' outflow boundary condition [33] is used at x/c¼1.5. The random vortices are generated only at the left-most section of the vortex region and convect with mean velocity u as shown in Fig. 1. While passing through the vortex region, the strengths of random vortices Un are constant. In the present paper, the Gaussian random generator in Ref. [42] is used to generate a random value. Random vortices are arranged in a Cartesian distribution with distance Δr between each vortex in the vortex region as shown in Fig. 1. The term rmax represents the maximum distance from the grid points on the inflow boundary to the location of vortex. Dieste [43] suggested that rmax be at least 2.43Λ for the Gaussian filter since random vortices located at more than rmax rarely affect the synthetic velocities. Dieste [43] also recommended that Δr should be less than Λ/6 for retaining the accuracy of the synthetic velocities. In the present paper, Δr and rmax are set to be Λ/8 and 2.5Λ, respectively. As a result, the number of random values used to synthesize velocity at a single point x is about N ¼400. At the beginning of the simulation, the turbulent velocity fields are initialized over one-third of the computational domain. In Fig. 2, instantaneous distributions of turbulent kinetic energy are shown over the entire computational domain at times t/(c/W)¼0, 1, 2, and 3. It is confirmed that the synthetic turbulence is well generated at the inflow boundary, convects at the speed of the meanflow u and passes out through the outflow boundary from the computational domain without any reflection. Next the quality of the synthesized turbulence field is evaluated by comparing the statistical feature of turbulence in terms of two-point autocorrelations and the corresponding one-dimensional energy spectra. The two-point autocorrelations of the velocity field are computed using the following equations given in Ref. [44]:



Rxx ðr x Þ ¼ uð0; tÞuð0 þr x ; tÞ ; y=c ¼ 0:5; r x =c A ½  0:5; 0:5 ; (16a)

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Fig. 2. Instantaneous distribution of turbulent kinetic energy ð1=2Þu2t over the entire computational domain at (a) t/(c/W) ¼0 (initial condition), (b) t/(c/W) ¼1, (c) t/(c/W) ¼ 2 and (d) t/(c/W) ¼3.



Ryy ðr y Þ ¼ vð0; tÞvð0 þ r y ; tÞ ; y=c ¼ 0:5; r y =c A ½ 0:5; 0:5 ;

(16b)

where 〈 〉 denotes the ensemble average. The corresponding one-dimensional energy spectra Exx(κ1) and Eyy(κ1) can be obtained via twice of Fourier transforms of Eqs. (16a) and (16b), respectively [44]. To compute the ensemble average of the autocorrelations in Eq. (16), 5000 samples are used, which are obtained every 200 time steps. The numerical and theoretical results are compared in Fig. 3. In Fig. 3(a) and (b), it can be seen that the numerically computed Rxx and Ryy show good agreement with analytical results. Prediction results for Exx and Eyy are also in excellent agreements with analytical results as shown in Fig. 3(c) and (d). On the basis of these comparison results, the suitability of STIBC is ensured to produce synthetic turbulence with the prescribed turbulence spectrum. The next simulation is conducted using a non-Gaussian spectrum. Dieste and Gabard [27] proposed digital filters for the Liepmann and VonKármán turbulent spectra. To reduce computational time, they used the interpolated form of nonGaussian filters instead of their exact formula. Although they argued that the interpolated non-Gaussian filters provide successful computational results, their filters suffer from two main drawbacks. One is that denser grid spacing is required to resolve the high wavenumber components that may generate spurious waves if they are not resolved. The other is that nonGaussian filters have a singularity in the case of x¼x0 that is not present in the Gaussian filter. Following the procedure proposed by Malte and Ewert [26], in the present paper Gaussian filters are superposed to produce non-Gaussian spectra. The Liepmann spectrum is selected for benchmark simulation and can be written in the form E L ðκ Þ ¼

16 5 K L ΛL  3π

κ4

 2 3

1 þ κ 2 ΛL

;

(17)

where subscript ‘L’ denotes the Liepmann spectrum. In the simulation, ΛL/c is set to be 0.1. The target band of the wavenumber is in the rage from κΛL/2π ¼0.1 to κΛL/2π ¼1.0, which corresponds the reduced frequency range of ωc/W¼2π to 20π. The 4 Gaussian spectra of the type given in Eq. (14) are used to describe the Liepmann spectrum over the target wavenumber band. The least-squares method is adopted for the best fitting. The fitted kinetic energy and integral length scale of each Gaussian spectrum are given in Table 1, together with total kinetic energy which is about 83.4 percent of the original Liepmann spectrum. This is due to the fact that the kinetic energy is excluded in the wavenumber ranges out of the target wavenumber band. The fitted Liepmann spectrum is compared with exact one in Fig. 4. It can be seen that the superposed Gaussian spectra well reproduce the exact Liepmann spectrum in the target wavenumber range. Subsequently, these four Gaussian filters are used in the test simulation to generate inflow turbulence. The onedimensional energy spectra, Exx and Eyy, are computed in the identical way used in the previous simulation using the Gaussian spectrum. The numerically computed one-dimensional energy spectra are compared with the analytical ones in Fig. 5. It can be seen that the spectrum obtained by superposing Gaussian spectra closely follows the Liepmann spectrum. The present method can be easily extended to generate other non-Gaussian energy spectra with an appropriate computation cost by trading off between the number of Gaussian filters and the required accuracy. The current method

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Fig. 3. Comparisons of autocorrelations and one-dimensional energy spectra for the Gaussian filter: (a) R11; (b) R22; (c) E11; and (d) E22 (numerical results ───; analytical solutions —————). Table 1 Kinetic energy and integral length scale of four Gaussian spectra to produce the Liepmann spectrum. Gaussian spectrum No.

No. 1

No. 2

No. 3

No. 4

Kinetic energy Integral length scale

0.1453365KL 0.2ΛL

0.2086737KL 0.125ΛL

0.2406832KL 0.08ΛL

0.2394948KL 0.04ΛL

Fig. 4. Exact Liepmann spectrum (—————) versus the fitted Liepmann spectrum (───) with 4 Gaussian spectra.

is used to generate the inflow turbulence of the non-Gaussian energy spectra in the benchmark problem in the following sections. 4. Applications to turbulent inflow noise In this section, the STIBC is applied for the prediction of inflow noise, and two inflow noise problems are numerically solved to test the STIBC for external and internal flows. The first is broadband noise due to the interaction of turbulence with an airfoil, and the second is broadband noise generated by turbulence interacting with a cascade of airfoils. The airfoils are assumed to be flat plates since analytic solutions exist for broadband noise from incident turbulence interacting with a flat plate and a cascade of flat plates. CAA simulation results of these problems can be thus validated against the analytical solutions. Mach number of incident mean flow is set to be 0.5 with zero incident angle. Gaussian spectrum is used as incident turbulence energy spectrum for both problems. Liepmann spectrum is additionally used as the turbulence energy spectrum for the problem of turbulence–cascade interaction. All the detailed conditions of these turbulent energy spectra

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Fig. 5. Comparison of one-dimensional energy spectra for the Liepmann spectrum: (a) E11 and (b) E22 (numerical results (───) and analytical ones (—————)).

Fig. 6. Schematic diagram of the computation domains for solving inflow broadband noise by interaction between: (a) turbulence and a flat airfoil and (b) turbulence and a cascade of flat plates.

are identical to those used in Section 3. The schematic diagram of the computation domain for each inflow noise problem is illustrated in Fig. 6.

4.1. Turbulence–airfoil interaction noise The inflow noise radiated from an isolated flat plate interacting with incident turbulence is computed by solving Eq. (13) numerically with the time-domain CAA techniques. Numerical schemes and conditions such as time interval and grid spacing are identical to those used in Section 3. The results are compared with the analytic solutions in terms of acoustic power. The computational domain is shown in Fig. 6(a). The far-field boundaries are located away from the airfoil by five times the chord length. The STIBC of Eq. (6) is implemented at the inflow boundary. Tam's outflow boundary condition [31,32] is applied on the other boundaries through which three types of waves—acoustic, vorticity and entropy waves—pass. Due to the presence of the flat plates in this and following subsections, Tam and Dong's wall boundary condition [45] is applied so that the normal velocity is made to be zero on the flat plates. Numerical spurious short waves are known to be generated at discontinuous boundary such as walls [46]. The artificial selective damping terms [45] are thus carefully added to the governing equations of Eq. (13) to suppress the high-frequency spurious waves which cannot be resolved by numerical schemes using finite grid points. Snapshots of the velocity vectors in the vicinity of the flat plate and fluctuating pressure computed by the CAA simulation are shown in Fig. 7(a) and (b), respectively. Fig. 7(b) shows the pressure distribution of the dipole pattern but with broadband nature. The scattered pressure by the trailing edge can be also identified, but its magnitude is negligible compared to the leading edge noise. For the comparison of the numerical solution with the analytic one, far-field acoustic wave radiation from the flat plate is predicted using the Ffowcs Williams–Hawkings (FWH) integral equation in the frequency domain with the two-dimensional form of Green's function [47]. Since the noise generation mechanism due to the turbulence–airfoil interaction can be modeled using dipole sources distributed over the airfoil surface, the loading term in the classical FWH formulation [47] is only used for the noise prediction as described in Curle's theory [48]:    ! ^ !   Z   ∂G x  y ;ω ! ! p^ x ; ω ¼ p^ y ; ω ni ; (18) ∂yi S

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147

Fig. 7. Snapshots of (a) the velocity vector and (b) pressure fluctuation.

Fig. 8. Comparison of acoustic power spectra for the isolated flat plate: numerical result (───) and analytical solution (—————).

! ! where the hat ‘\widehat’ denotes the value in the frequency-domain, x and y present the position vectors of an observer and source, respectively, and n is an outward vector normal to the integral surface. i    ! ^ ! In Eq. (18) G x  y ; ω is the two dimensional Green's function in the frequency-domain. It contains the convection effect written in the form: !      2 i κ ! ^ ! r G x  y ; ω ¼ ei ðMκðx1  y1 ÞÞ=β H 0ð2Þ (19) β ; 4β β2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 x1  y1 þ β x2  y2 , and H ð2Þ where β ¼ 1  M 2 ; κ ¼ ω=c; r β ¼ 0 is a Hankel function of the second kind, zeroth order. First, a Fourier transform of the time histories of pressure data on the flat plate is computed using the CAA simulation results. Subsequently, the power spectral density (PSD) of far-field pressure, Spp, is computed using Eq. (18). The acoustic power spectrum can be computed by integration on a circular arc of radius r as follows [5]: PðωÞ ¼

r

Z 2π

2ρc

0

β Aðθ; MÞ Spp ðr; θÞ dθ ; ðAðθ; MÞ  M cos θÞ2 4

(20)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Aðθ; MÞ ¼ 1 M 2 sin 2 θ. The acoustic power spectrum is estimated by averaging the data over 400 samples from the CAA simulation result. Each sample is recorded at the location of r/c¼10 for 40,000 time steps. The estimated acoustic power spectrum is compared to that by the two-dimensional version of Amiet's model [5] which gives inflow noise radiation using one-dimensional upwash velocity spectrum obtained by integrating the two-dimensional velocity spectrum over the wavenumber in the y-axis direction. The two-dimensional velocity spectrum can be expressed using the energy spectrum E(κ) [45]. For the Gaussian energy spectrum in Eq. (14), it is written as

Eyy

0



1

Λ2 κ 2x þ κ 2y A; κ x ; κ y ¼ 3 K Λ κ exp@  π π





2

4 2 x

(21)

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where Eyy represents the two-dimensional upwash velocity spectrum, and κx and κy are the wavenumber component in the x- and y-directions, respectively. One-dimensional velocity spectrum is then obtained by integrating Eq. (21) over κy. Eyy ðκ x Þ ¼

2

π

!

K Λ κ 2x exp  2 3

Λ2 κ 2x : π

(22)

The comparison of acoustic power between two solutions is made in Fig. 8. As shown in Fig. 8, the acoustic power spectrum computed using the result of the CAA simulation with the STIBC shows good agreement with the analytical solution over all frequency ranges.

4.2. Turbulence–cascade interaction noise It is revealed that the STIBC shows excellent performance to predict inflow noise radiated from a flat plate placed in a turbulent stream. Now, the STIBC is applied to predict inflow noise generated by a cascade of flat plates interacting with incident turbulence. The number of flat plates B equals 4, and the distance between adjacent plates, s/c is set to be 1.0 as shown in Fig. 6. The Liepmann filter is used for the incident turbulence energy spectrum as well as the Gaussian filter. Note that the Liepmann filter is constructed by superposing the 4 Gaussian filters, the procedure for which was already described in detail in Section 3. The pressure contours over the entire computational domain obtained from the CAA simulations using the STIBCs with the Gaussian and Liepmann spectra are given in Fig. 9. In the comparison between Fig. 9(a) and (b), the high-frequency components of the acoustic wave can be seen in Fig. 9(b) because the Liepmann spectrum contains higher turbulent energy in the high wavenumber range than does the Gaussian spectrum. The acoustic power spectra are computed from the time-domain CAA simulation by integrating the intensity spectrum over a distance Bs in the y-direction. Here, Bs/c equals 4.0. The formula for the sound intensity spectrum in a fluid moving at uniform mean flow is given in Ref. [49]. In calculating the intensity spectrum, the acoustic particle velocity is computed using the momentum equation in the linearized Euler equations. The final formula for the acoustic power spectrum is given by P

7

 )  2  ω u 7n 7n   ðωÞ ¼ Re ∑ Al ðωÞ  ;   αl þ 2 ω þ uαl ρ ω þ uα 7 2 c l ¼ 1 Bs

(

1

(23)

l

where Al(ω) represents the complex modal amplitude of the acoustic pressure that can be computed using the frequency– wavenumber transform of pressure over time and in the y-direction. Here, the subscript ‘l’ is the mode number in the y-direction, and the superscript ‘ 7’ denotes variables related to the upstream- and downstream-going quantities. In Eq. (23), the axial acoustic wavenumber αl7 can be expressed in terms of ω and βl from non-trivial single frequency solutions of the linearized Euler equations as

αl7 ¼ Mðω=cÞ 7

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðω=cÞ2  ð1 M 2 Þβl 1  M2

;

(24)

where βl (¼2πl/Bs) is the acoustic wavenumber in the y-direction. Acoustic power spectra are estimated by averaging 1000 samples obtained from the time-domain CAA simulation result. Each sample is obtained over 40,000 time steps. The estimated acoustic power spectra computed from the CAA simulations are compared to the analytical solutions obtained from the formula by Cheong et al. [30]. This formula requires as input the two-dimensional upwash velocity spectrum that is computed using the turbulent energy spectrum. For the Gaussian spectrum, Eq. (21) is used to compute the acoustic power. The upwash velocity spectrum for the Liepmann spectrum in Eq. (17) is given by 

Eyy κ x ; κ y



16 5 ¼ 2KΛ  3π

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi κ 2x þ κ 2y κ 2x  3 : 2 1 þ Λ κ 2x þ κ 2y

(25)

The normalized acoustic power spectra obtained from the CAA simulation and the analytic formula by Cheong et al. [30] are compared in Fig. 10, where the upper spectrum denotes the downstream acoustic power spectrum, and the lower one denotes the upstream acoustic power spectrum. The acoustic power spectra for incident turbulence of Gaussian spectrum are shown in Fig. 10(a), while those for Liepmann spectrum are shown in Fig. 10(b). The comparison results reveal that the acoustic power spectra from the CAA simulation are in good agreements with the analytical solutions in both of the cases for the targeted frequency ranges from ωc/W¼2π to 20π. At the same time, it can be seen that the method based on the superposition of Gaussian filters to generate the Liepmann filter also shows excellent performance in predicting the inflow noise in the time-domain.

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Fig. 9. Distribution of fluctuating pressure for (a) Gaussian spectra and (b) Liepmann spectra at the time t/(c/W) ¼50.0.

Fig. 10. Comparison of acoustic power spectra of a cascade of flat plates for (a) the Gaussian spectrum and (b) the Liepmann spectrum: numerical results, ─── and analytical solutions, —————.

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5. Conclusions The synthetic turbulent inflow boundary condition (STIBC) was developed for the effective simulation of inflow broadband noise, and it could be incorporated easily with time-domain CAA techniques. The STIBCs for internal and external flows were derived by combining the stochastic synthetic turbulence model with the radiation and characteristic boundary conditions, respectively. The filter-based RPM method was adopted to synthesize turbulence at the inflow boundary. The RPM was incorporated with the STIBCs in terms of the second derivative of the digital filters. The use of second-derivative terms of the digital filter could give the spatial and time derivatives of the synthetic velocities, which is essential for generating turbulent velocity at inflow boundary. The validity of the STIBC was confirmed by the time-domain CAA computation where the autocorrelations and the corresponding energy spectra of synthesized turbulence were shown to be in excellent agreement with the theoretical results. The Gaussian and Liepmann filters were used to synthesize turbulent velocities. Particularly, it was shown that the Liepmann filter could be made by superposing the Gaussian filters. Two benchmark inflow noise problems were solved using the CAA techniques combined with the STIBC. First, inflow noise radiated from a flat plate interacting with incident turbulence was predicted using the STIBC for external flow. The Gaussian spectrum was used as a turbulence energy spectrum. The prediction result was compared to that obtained from the two-dimensional version of Amiet's solution [5] in terms of the acoustic power spectrum. Excellent agreement was found between the two solutions. Second, the upstream and downstream acoustic powers radiated from a cascade of flat plates were predicted using the STIBC for internal flow. Both the Gaussian and Liepmann spectra were used as the turbulence energy spectrum. The prediction results were compared with the analytical solutions obtained using the approach by Cheong et al. [30]. Again, all the computation results were in excellent agreement. On the basis of these results, it is believed that the current CAA techniques combined with the STIBCs would be effective prediction and analysis tools for inflow broadband noise. Particularly, the effects on broadband noise generation and propagation of airfoil geometries, nonuniform mean flow, and nonlinear interaction can be investigated using the current methods, which are the aims of future research.

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