Assessment of Different Parameters Used in the SNGR Method M. Mesbah, J. Meyers, M. Baelmans, W. Desmet K.U.Leuven, Department of Mechanical Engineering, Celestijnenlaan 300 A, B-3001, Leuven, Belgium e-mail:
[email protected]
Abstract This paper focuses on the Stochastic Noise Generation and Radiation (SNGR) method, that aims at reconstructing aero-acoustic sources from cost-efficient RANS flow calculations. The SNGR method is applied to a 2D-axisymmetric high-speed subsonic round jet with D = 25 mm and M = 0.86. An averaged flow solution is obtained from a RANS simulation, using a standard k– turbulence model. A stochastic turbulent velocity field is generated based upon these flow data and is represented as a finite sum of Fourier modes. The acoustic source terms resulting from these fluctuating velocity fields are subsequently used for the computation of the acoustic intensity in the far field by solving the axisymmetric linearized Euler equations. All simulation results are compared with the experiments of Lush (J. Fluid Mech., Vol. 46, No. 3, 1971, pp.477500). Further, sensitivity analysis is performed to assess the influence of crucial SNGR parameters such as the extent of the source domain, the integral length scale, and the random number series.
1 Introduction Aero-acoustics deals with the study of the interaction between acoustic waves and flows. Aero-acoustic sound is sometimes considered as a nuisance, such as noise generated by fluid machines, e.g., jet engines, vacuum cleaners, hair dryers, exhaust pipes and ventilation systems. In order to reduce the noise emission in these engineering applications, the ability to predict and understand flow-generated noise is of utmost importance. Computational Aero-Acoustics (CAA) covers the multi-physical discipline of combined simulation techniques for fluid mechanics and acoustics. As such, it can be used to simulate the acoustic noise generation and radiation from flow phenomena. There are several CAA approaches, which can be categorized into two groups: direct and hybrid methods. In the first group, the full compressible Navier-Stokes equations are solved using LES or DNS on a domain that includes the acoustic source region and extends to the far field. However, the use of these direct methods is infeasible for most industrial applications due to their excessive computational cost. The second group of approaches consists of hybrid methods. They divide the problem into generation of sound on the one hand and propagation of the generated sound on the other hand. In this context, the Stochastic Noise Generation and Radiation (SNGR) model has been developed. In the SNGR method, originally presented by Bechara et. al. [1], a random velocity field is generated by a finite sum of discrete Fourier modes based upon averaged data of the flow field. In a further development, Bailly et. al. [2] add a time dependent term in these Fourier modes. Billson et. al. [5] solve a convection equation for the filtered velocity field in each time step in order to include convection effects. In the SNGR method, the generated random velocity field is used to determine the source for acoustic perturbation. Thereby, the Euler equations, which are linearized around a mean flow, are considered as an exact wave operator. Furthermore, it is assumed that the turbulent field is not influenced by the acoustic waves. To summarize, calculations associated with SNGR methods may be discerned in three steps: 1. A Reynolds-Averaged Navier-Stokes solution (RANS), e.g., closed with a k– turbulence model, pro389
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vides time-averaged information of the flow field. 2. An isotropic turbulent velocity field based on random Fourier modes is generated using the data obtained from the previous step. 3. In the last step the Linearized Euler Equations (LEE), with unsteady source terms induced by the fluctuating velocity field, are solved. It is clear that SNGR holds the promise to drastically reduce the computational cost of CAA simulations in comparison to DNS or LES. However, the reconstruction of the turbulent velocity field is a crucial step in the SNGR method. Therefore, the aim of this work is to assess the sensitivity of the far field noise to different parameters used to generate the turbulence field. Furthermore, several improvements to the SNGR method are proposed. This paper is organized as follows. Section 2 displays the two-dimensional axisymmetric LEE with source terms in a conservative form. Section 3 introduces a method to generate a turbulent velocity field. Section 4 describes the numerical scheme and boundary conditions employed in the LEE solver. The set-up and results of the reference simulation obtained from a RANS are discussed in Section 5. This part also contains the first simulation of acoustic intensities in the far field. The effect of different parameters on far field results are further elaborated in section 6. Finally, conclusions are summarized in section 7.
2
The Linearized Euler Equations with source terms
The SNGR method relies on the fact that linearized Euler equations are considered as an exact wave operator for acoustic perturbations. Typically, different methods are presented in the literature to linearize the Euler equations [1, 2, 4], leading to different source terms. Bechara et. al. [1] decompose the flow variables into the mean, turbulent and acoustic parts. In this derivation, two main assumptions are made, i.e., the turbulent field is assumed to be incompressible, and the acoustic perturbations are taken isentropic. Later, Bailly et. al. [2] introduced an alternative set of source terms for these LEE. Billson et. al. [4] linearize the Euler Equations by using Favre averaging and decomposing the flow variables into the mean and fluctuation parts. In this case the source terms not only depend on the unsteady Reynolds stress but also on the unsteady total enthalpy. In this research, LEE with the source terms described in Reference [2], are employed as wave operator. These equations in 2-dimensional axisymmetric coordinates, may be written in conservative form as ∂U ∂E 1 ∂(rF) + + + H = S. (1) ∂t ∂x r ∂r where U represents the unknown acoustic variables. The terms E and F are the 2-dimensional flux vectors, and S is the source terms. The vector H contains all terms that are linear with respect to the acoustic variables.
Pa
U = Ua Va
¯ U Pa + γ P¯ Ua Pa ¯ E= U Ua + ρ¯ ¯ Va U
¯ V Pa + γ P¯ Va V¯ Ua F= V¯ Va + Pρ¯a
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¯
¯
¯
Ua ∂(r V¯ ) r ∂r
¯
¯
U P − Pa ∂∂x + Va ∂∂rP − Pa 1r ∂(r∂rV ) ) (1 − γ)(Ua ∂∂x
Va ∂∂rU − −
Va V¯ + Pρ¯a r
−
Pa ∂ P¯ ρ¯2 c2 ∂r
−
Pa ∂ P¯ ρ¯2 c2 ∂x ¯
¯
V U + Ua ∂∂x − Va ∂∂x
0
∂Ut ∂Ut ∂Ut t S = Ut ∂U ∂x + Vt ∂r − Ut ∂x + Vt ∂r ∂Vt ∂Vt ∂Vt t Ut ∂V ∂x + Vt ∂r − Ut ∂x + Vt ∂r
,
(2)
.
Here, U and V are the velocity components in axial (x) and radial (r) direction, respectively, and P is the pressure. Further, (·) is used to denote time-averaging, while the subscripts ‘t’ and ‘a’ stand for turbulent and acoustic variables. The sound speed is denoted by c and γ is the ratio of specific heats. Since these equations are derived based on the assumption that the acoustic perturbations are isentropic, it follows that Pa = c2 ρa . As a result, acoustic density fluctuations are retrieved as acoustic pressure fluctuations in Equation (2).
3
Stochastic Turbulence Modelling
A method to simulate a space-time stochastic turbulent velocity field has been developed by Bailly et. al. [2]. To this end, an isotropic turbulent field is defined as a finite sum of discrete Fourier modes, i.e., Ut (x, t) = 2
N X
n=1
u en cos[kn .(x − tUc ) + ψn + ωn t]σ n .
(3)
where u en , ψn , and σn are the amplitude, phase, and direction of the nth Fourier mode, respectively. Moreover, Uc is the local convection velocity and ωn is the angular frequency of the nth mode. The angular frequency ωn is randomly selected with a Gaussian probability density function, i.e., P (ωn ) =
1 √
ω0n 2π
exp[−
(ω − ω0n )2 ], 2 2ω0n
(4)
0
where ω0n = u kn is a function of the wave number kn =k kn k, and the estimated turbulent velocity 0 u ' (2k/3)1/2 , which is obtained from RANS. The wave vector kn is chosen randomly on a sphere with radius kn . Figure 1 shows a sketch of the geometry. The probability density functions of all other parameters are given in Table 1. Assuming incompressibility, one further requires for all modes in frequency space, that kn .σ n = 0.
(5)
Hence, the wave vector kn , and the spatial direction of the turbulent field σn of the nth mode are perpendicular.
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K3
λ3 θn
σn
λ1
λ3
σn αn
λ1
Kn
K1
K2
ϕn
Figure 1: Schematic of the wave vector kn and the perpendicular vector of σn P (ϕn ) = 1/(2π) P (ψn ) = 1/(2π) P (αn ) = 1/(2π) P (θn ) = (1/2) sin(θ)
0 ≤ ϕn ≤ 2π 0 ≤ ψn ≤ 2π 0 ≤ αn ≤ 2π 0 ≤ θn ≤ π
Table 1: Probability density functions
The Von Karman-Pao spectrum, as shown in Figure 2, is employed to simulate the energy spectrum for isotropic turbulence 2k (k/ke )4 k E(k) = A (6) exp(−2 ), 2 17/6 3ke [1 + (k/ke ) ] kη where kη = ε1/4 ν −3/4 is the Kolmogorov wave number. The parameter A is a numerical constant and ke is the wave number at which the spectrum reaches its maximum energy. The parameters A and ke determine the shape of the spectrum and the distribution of energy over different wave numbers. The integral of the energy spectrum over all wave numbers should be equal to the total turbulent kinetic energy. Z ∞ k= E(k) dk (7) 0
The numerical constant A can be obtained from Equation (7) using an infinite Reynolds number assumption 55 Γ(5/6) A= √ ' 1.453. 9 π Γ(1/3) In order to calculate ke , we assume that the turbulent length scale L—which is estimated from the energy of the large eddies, and the dissipation rate—equals the integral length scale. The turbulent length scale L is defined as 0 u3 , (8) L = fL ε where fL is a factor of order unity, introduced here for the sake of later convenience. The integral length scale Λ, is defined as Z ∞ π E(k) π dk, (9) Λ = 0 2 E11 (0) = 0 2 k 2u 2u 0 where E11 is the one dimensional spectrum. Therefore, ke =
9π A . 55 L
(10)
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E(k)
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k1 ke
kn
Wave number
kN
Figure 2: Von Karman-Pao spectrum
In the present paper, the spectrum E(kn ) is discretized into N equidistant points, with a minimum wave number k1 and maximum wave number kN . The selection of k1 and kN will be further discussed in Section 5. The amplitude u en of each discrete mode is related to the energy spectrum E(kn ), i.e., p (11) u en = E(kn )∆k, where ∆k is the spacing of the wave number distributions (see Figure 2). Further, it is clear that k=
N X
n=1
u e2n .
(12)
4 LEE Solver In the last step of the SNGR method, the LEE have to be solved. The acoustic waves usually have small amplitudes and radiate from the source region over long distances and in all directions into the far field. Hence, the numerical scheme must have a minimal numerical dispersion and dissipation. Moreover, boundary conditions should not reflect outgoing acoustic waves back into the domain, as these reflections are not physical and decay very slowly. In this section the scheme and boundary conditions used for two-dimensional cartesian grids are modified for use in the axisymmetric case.
4.1 Numerical Scheme All spatial derivatives are discretized by a seven-point finite difference scheme, corresponding to the DispersionRelation-Preserving (DRP) scheme of Tam et. al. [8]. As a result, the discretized formulation of Equation (1) reads 3 3 X 1 X ∂Ui,j αl ri,j+l Fi,j+l − Hi,j + Si,j . αl Ei+l,j − =− ∂t ri,j l=−3
l=−3
(13)
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The coefficients αl are taken from Reference [8]. A filtering procedure is essential to eliminate spurious numerical oscillations that could contaminate the computed solution. To this end, artificial selective damping [7] is used. The damping terms are added to the right hand side of equation (13), and correspond to Di,j
3 3 1 X 1 X dl Ui+l,j + ( dl ri,j+l Ui,j+1 ), =− Res ri,j l=−3
(14)
l=−3
where Res is the artificial mesh Reynolds number, defined as c∆x/ν. This Reynolds number is inversely proportional to the amount of damping applied to the simulations. Following Reference [7], Res = 3 is selected for the present computations, which provides satisfactory results for all simulations. The coefficients dl which are chosen to filter the short waves, are taken from Reference [7]. For the time integration, the second-order five-stages Low-Dispersion-Dissipation Rung-Kutta scheme of Bogey et. al. [9] is used.
4.2 Boundary Conditions It is important to formulate boundary conditions such that outgoing acoustic waves are not reflected back into the calculation domain. To this end, the boundary conditions of Tam [6] are implemented. These boundary conditions are slightly altered here, in order to correspond to the axisymmetric LEE (Equation (1)). For the free-stream boundary conditions—also called the radiation boundary conditions—this leads to [
∂ ∂ 1 1 ∂ + cos(ϕ) + sin(ϕ) + ]U = 0, V (ϕ) ∂t ∂x ∂r ζ
(15)
Here, ζ and ϕ are the 2D polar coordinates in the axisymmetric plane. These coordinate system is chosen as follows. An origin is selected in the core of the source region. The coordinate ζ reflects the magnitude of the position vector with respect to this origin. Further, ϕ corresponds to the angle between the position vector and the direction of the symmetry axis. Further, the effective velocity V (ϕ) in polar coordinates is defined by V (ϕ) = U cos(ϕ) + V sin(ϕ) +
q
c2 − [V cos(ϕ) − U sin(ϕ)]2 .
(16)
Apart from pressure waves, vorticity waves, which travel with the flow, reach the inflow and outflow boundaries. Therefore, inflow and outflow boundary conditions are different from the radiation boundary conditions, and correspond instead to
1 ∂Pa ∂Ua ∂Ua ∂Ua +U +V =− , ∂t ∂x ∂r ρ ∂x ∂Va ∂Va ∂Va 1 ∂Pa +U +V =− , ∂t ∂x ∂r ρ ∂r ∂Pa ∂Pa 1 ∂Pa + V (ϕ) cos(ϕ) + V (ϕ) sin(ϕ) + V (ϕ) Pa = 0. ∂t ∂x ∂r ζ
5
(17)
Set-up and Results of the Reference Simulation
A high-speed subsonic jet with Mach number M = 0.86, and nozzle diameter D = 25 mm , is simulated using the above described SNGR method. This setup corresponds to the experiment presented by Lush [11], which was used before as a point of reference for SNGR validation [10, 2]. In the following subsections some aerodynamic and acoustic results will be discussed.
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(a)
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400
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700 1300
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2800 2 (b)
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2500 6
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Figure 3: (a) U-velocity component ( ms ) , (b) Turbulent kinetic energy ( ms2 )
5.1 Flow Results The mean flow solution is obtained from a RANS simulation, using a k– model. This simulation is performed using the CFX4.3 solver (CFX International, AEA Technology, Harwell, Didcot, Oxon., UK,1997). The computations are performed for the axisymmetric case on a domain with size 5D × 20D. This domain is discretized using 4794 cells. Figures 3 (a) and (b) show the mean velocity U in the jet direction and the turbulent kinetic energy, respectively. The present results do well agree with results in Reference [10]. The results show that the calculated potential core (i.e. the length where the mean center-line velocity is constant) is about 4.75D, which is close to the experimentally observed values of 5D.
5.2 Acoustic Results Acoustic calculations are performed on an axisymmetric domain which extends 20D in the jet-normal direction, and 20D along the symmetry axis. The mesh contains 104,329 internal points with constant mesh size ∆x = ∆r = 1.53 × 10−3 m. The time step ∆t = 1.2 × 10−6 seconds, corresponding to a CFL number of 0.5. As already mentioned in Section 1, the aim of this research is the assessment of different parameters in SNGR methods based on the method presented in References [2, 3]. Hence, as a first reference simulation, the SNGR method is performed using the same set-up as in References [2, 3]. In those references, the setup consists of a source region, which is limited to points where the turbulent kinetic energy k > 0.3k max . Furthermore, an extra constraint on the source domain is posed by the size of the integral length scale L. Regions where L < 6∆x are excluded from the source region. This is connected to the discretization used here [8], where the artificial selective damping removes all fluctuations with wavelength λ < 6∆x. In this paper, first the criterion, k > 0.3k max , is used to determine the source region. In section 6, this criterion will be discussed. Further, the lower and upper wave number are k1 = kemin /5 and kN = 2π/7∆x, respectively, with, kemin the smallest value of ke over the source volume. The wave number range is discretized using N = 30 modes [3, 5]. The computation covers a time interval T = 3.84 × 10−3 seconds, which corresponds
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r 20 D
×
× × × × Rc=19.5 D
× ×
θ
×
D
x
20 D
Nozzle
Figure 4: Sketch of the acoustic computational domain
60
Pressure(Pa)
40 20 0 −20 −40 −60 0
0.2
0.4
0.6
0.8
1 1.2 Time (second)
1.4
1.6
1.8
2 −3
x 10
Figure 5: Recorded pressure signal
to 3,200 time steps. The pressure signals are recorded at locations Rc = 19.5D (centered at the jet exit) and θ = 15◦ , 20◦ , ..., 90◦ (see Figure 4). The analysis is performed over the last 1,500 time steps of the computation. The first 1,700 time steps are not considered, since they represent the transition to a statistical steady-state regime. Figure 5 shows an example of a recorded pressure signal. For reference, the acoustic intensity LI , is defined as LI = 10 log(
I ) Iref
(dB),
(18)
W where Iref = 10−12 ( m 2 ) and
I = Pa ·
p
Ua2 + Va2
W (m 2) .
(19)
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Sound Intesity (dB)
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115
110
105
0
20
40
60
Emission Angle (θ°)
80
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Figure 6: Sound intensity at different angles θ. This simulation is performed with N = 30, k1 = 2π 7∆x ,
kemin 5 , kN
=
k
fL = 1, Source region k > 0.3, and Random series ‘A’. (— ∗ ): measurement; (— 4): Simulation 1 max using the far-field assumption, i.e., Equation (20); (— +): Simulation 1, using Equation (19). (See Table 2).
In case the recorded signal is situated in the far-field, Equation (19) can be approximated by I≈
Pa2 ρc
W (m 2) .
(20)
The difference between Equations (19) and (20) is mainly significant when the observation point is not located far enough from the source region. Hence, the acoustic intensity is calculated by both Equations (19) and (20), plotted in Figure 6. The results show that the observation points are far enough from the source region, indicating that the assumption of far-field is valid. One should further remark that for low angles θ < 30◦ , the observation points are still situated in or close to the source region (see Figure 3). Hence, a slight difference in the results between Equations (19) and (20) is visible in Figure 6 for θ < 30◦ . In case of increasing integral length scale factor, i.e. fL = 6, this difference increases, roughly ranging from 0.5 to 3 dB for θ < 30◦ . In order to compare the results with Lush’s experimental data [11], the measured acoustic intensity LI is corrected here by i.e., in Equations (18) I is scaled with (120D/19.5D)2 . As can be appreciated from Figure 6, the acoustic intensity is overestimated. Moreover, the difference between experiment and simulation increases when the observation angle is larger than 75◦ , i.e., close to the inflow boundaries. This might be attributed to the fact that the source region cannot be considered as a source point in the present computational domain. As a result, the boundary conditions (Equation (17)) are not perfectly non-reflecting. Apart from this boundary condition effect, one should notice that the simulation overpredicts the measurement with approximately 7 dB. Similar studies using the SNGR methodology likewise show overpredictions of the generated far field sound, roughly corresponding to 10% of the experimental sound level [2, 5].
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Simulation No.
Source Region k > k
Length Scale Factor fL
Random Series
1 2 3 4 5 6 7 8 9
0.3 0.15 0.45 0.6 0.3 0.3 0.3 0.3 0.3
1 1 1 1 3 6 1 1 1
A A A A A A B C D
max
Table 2: Overview of simulations. These simulations are performed with N = 30, k1 =
kem in 5 ,
kN =
2π 7∆x .
6 Assessment of Different Parameters in the Source-Generating Algorithm In order to further evaluate the significance of the 7 dB gap between the SNGR simulation and the experiment, a sensitivity analysis is performed on some input parameters for the SNGR model. Table 2 presents an overview of the different simulations performed. The first simulation in Table 2 corresponds with the reference simulation of the previous subsection (cfr. Figure 6). Different effects are investigated, i.e., first, the extent of the source domain, second, the effect of the integral length scale, and third, the influence of the random numbers, related to the repeatability of the calculation. As discussed above, a practical way to choose the extent of the source region is by requiring a minimum value for k/kmax (see Figure 7). In Simulations 1 to 4 (cfr. Table 2) k/k max is varied. Its effect on the sound intensity can be appreciated in Figure 8. It can be clearly observed that the sound intensity increases with increasing source domain. This is readily attributed to the related increase in total energy over the full source domain. With respect to the directivity of the sound intensity, one can see that none of the simulations provides satisfactory results, i.e., a large source domain seems to predict a reasonable slope at high values for θ, but a too-negative slope for 0◦ < θ < 40◦ ; in contrast, a small source domain rather predicts the slope at 0◦ < θ < 40◦ well, while the slope at θ < 40◦ is not steep enough. It is important to notice that the source-domain limitation based on a k/k max -criterion is arbitrary and should only be used to limit—in a sensible way—the computational effect. However, the selection of k/k max should not influence the far-field acoustic intensity. If the selection of k/kmax does influence the result, the criterion is too restrictive and a lower value for k/k max should be used to limit the source region. Hence, as can be observed from Figure 7, k/k max should be selected lower than 0.15. From this point of view, the SNGR algorithm over-predicts the actual measurements by 9 dB. Finally, one might notice that a better criterion to select the source region might be based on the local Reynolds number ReL , (similar to Reference [5]). However, since such a criterion should not restrict the part of the energy which still has a contribution to the total acoustic intensity. Therefore, the same 9 dB overprediction can be expected. Next, simulations 1, 5 and 6 are used to investigate the effect of the energy spectrum on the far field noise. The results are shown in Figure 9. As can be observed, by increasing the length scale factor fL (see Equation (8)) from 1 to 6, the acoustic sound intensity decreases. The difference between fL = 1 and fL = 6 roughly amounts to 5 dB. Figure 9 shows that the sound directivity is not sensitive to the length scale factor. In order to understand how increasing the integral length scale causes these results, variations of amplitude of each mode are displayed for a typical source point in Figure 10. The integrated kinetic energy corresponding to fL = 1, 3, and 6 is 1208, 1385, and 1446 (m2 /s2 ), respectively. Although the total energy captured by the spectrum increases, the acoustic intensity in the far field decreases. Hence, it seems that modes with higher
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14
Figure 7: Source region for different value of
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k kmax
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Figure 8: Effect of source region extent on sound intensity and directivity. The simulations are performed 2π ∗ ): measurement; (— ◦ ): k k > with N = 30, k1 = kemin 5 , kN = 7∆x , fL = 1, and Random series A. (— max
0.15; (— 4):
k
kmax
> 0.3; (— 5):
k
k max
> 0.45; (— ):
k
kmax
> 0.6.
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°
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Emission Angle (θ )
Figure 9: Effect of integral length scale on sound intensity and directivity. The simulations are performed 2π k with N = 30, k1 = kemin ∗ ): measurement; 5 , kN = 7∆x , Source region kmax > 0.3, and Random series A. (— (— +): fL = 1; (— ): fL = 3; (— ◦ ): fL = 6. wave numbers have predominant effects. Finally, the effects on sound intensity due to a variation of the random number series is investigated using simulations 1, 7, 8, and 9. It can be seen in Figure 11, that this causes a scatter in results which reaches 4 dB at some observation points. The average over the different random SNGR realizations, which is also presented in Figure 11, is clearly much smoother than the individual SNGR results.
7 Conclusions and future steps In this paper an application of the Stochastic Noise Generation and Radiation method to a near sonic two dimensional axisymmetric jet is presented. The mean-flow solution is obtained from a RANS simulation, using a k– model. A stochastic turbulent velocity field based on a finite sum of Fourier modes and on an ensemble-averaged steady-state flow solution is generated. The source terms for the LEE are provided using this turbulent velocity field. Further, acoustic intensities in the far field are computed by solving the LEE numerically. For the reference case, which roughly corresponds to the SNGR setting selected by Bailly et. al. [2, 3], the acoustic intensity is overestimated by 7 dB. In order to investigate this inconsistency, a sensitivity analysis is performed on the effect of some SNGR parameters such as the extent of the source domain, the integral length scale, and the random number series. The results show that an extension of the source region causes an increase in acoustic intensity. Further, the acoustic intensity decreases with increasing integral length scale factor fL , whereas acoustic directionality is not sensitive to these changes. Analysis of a variation in spectral-energy shape by increasing the integral length scale factor fL , reveals that energy associated with higher wave numbers has an increasing effect on the acoustic intensity. Finally when the random series is varied, a scatter in results is observed, which reaches 4 dB at some observation points. This inaccuracy can easily be removed by taking an average over some SNGR realizations. In the present article an average over 4 different random generated SNGR results was sufficient.
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Amplitude , un, (m/ sec)
6
5
4
3
2
1
0
0
100
200
300 Wave number (1/m)
400
500
600
Figure 10: The shape of spectrum for different integral length scale factor. k1 = (− − −):fL = 1; (−−):fL = 3. (−·):fL = 6;
kemin 5 ,
kN =
2π 7∆x ;
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0
20
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Figure 11: Effect of random number series on sound intensity and directivity. The simulations are performed k 2π with N = 30, k1 = kemin ◦ ):averaged sound intensity of 5 , kN = 7∆x , fL = 1, Source region k max > 0.3. (— simulations 1, 7, 8, and 9. (— ∗ ): measurement.
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P ROCEEDINGS OF ISMA2004
The obtained results clearly show the importance of some central parameters to the SNGR-predicted far field noise. However, results also indicate that the present sensitivity analysis should be extended. Parameters which are not considered, but might certainly be relevant are the number of modes N in the SNGR, i.e., one should quantify how many modes should be included before N -independent results are obtained. Further, a variation of the size of the computational domain, where effects on the far field noise and the reflectivity of the boundary conditions are studied, seems interesting. Finally, the interpretation of results with use of far-field pressure spectrums can provide further insights in the SNGR methodology. These topics certainly provide interesting material for further research.
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