Jet noise modelling for shielding calculations using RANS solution source localisation and WEM propagation C. J. O’Reilly, H. J. Rice Department of Mechanical and Manufacturing Engineering, Parsons Building, University of Dublin, Trinity College, Dublin 2, Ireland. e-mail:
[email protected]
Abstract In this paper, a method to determine the jet noise source input for shielding calculations is described. The method involves three steps – localisation, propagation and simplification. Firstly, for the frequency of interest, a small localised source region is determined, using Goldstein-Rosenbaum’s acoustic model, with a numerical RANS solution providing local flow properties. A number of distinct incoherent noise sources may be located in this region, using the turbulence length scales. In the second step, each source is propagated out through the jet flow using the wave expansion method (WEM) to solve the Helmholtz equation, with local Mach values acquired from the RANS solution. The third step is to define a Kirchhoff surface outside the jet flow, from which the sound may be propagated further by simply using the Helmholtz-Kirchhoff integral, which can account for any mean flow present. Although the method is quite crude, it is relatively robust as it is a shielding factor or ratio, rather than an absolute value, which is of interest in design evaluation. Preliminary results, presented in this paper, provide encouragement that this new three-step jet noise source modelling method can be used to provide a source input for use in airframe shielding calculations, that is equivalent – in terms of source frequency, distribution and directivity – to the noise produced by a jet flow.
1 Introduction The reduction of near-ground aircraft operating noise levels has become an important design consideration for aircraft manufacturers, in recent times. Threshold values for noise certification of new aircraft have reduced and local authorities now impose direct noise penalties, and limitations on operating hours, on airline companies. With it anticipated that the number of commercial aircraft in the skies will more than double in the next twenty years, the search in now on for industrially viable quieter aircraft designs. Aircraft noise may be divided into two categories – airframe noise and engine noise. A novel approach, currently under consideration as part of the European FP6 Project ’Novel Aircraft Concepts REsearch’ (NACRE), to reduce the engine noise that propagates towards the ground, is to position the engines high on the rear of the fuselage, where a new U-shaped tail section or empennage design, would potentially act as a noise shield or barrier. At present, no method exists to suitably evaluate such a configuration from an acoustic point-of-view. There is, therefore, an industrial requirement to develop design tools to quantify any noise reduction created, by such a configuration. It must be remembered though, that acoustic evaluation is only one component of an overall design evaluation and so, this acoustic evaluation method must be appropriate in terms of computational time and resources. The shielding factor is the criteria of interest for design purposes. The shielding factor, FS , is the ratio of the shielded sound pressure level, SP L, at an acoustic receiver, to the unshielded or reference sound pressure level, or simply SP LShielded (1) FS = SP LU nshielded 667
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In order to evaluate the shielding factor, one must describe the noise source and then examine how it propagates to the receiver, directly (unshielded) and by diffracting around an obstacle (shielded). Turbulent Noise Sources Jet Nozzle Exit Fully Developed Flow Potential Core Mixing Region
Shielding Barrier
Acoustic Receiver
Figure 1: Noise generation in turbulent mixing region for a simple jet, with a noise shield This current paper is concerned with expressing the noise field produced by a jet flow as a noise source input for shielding calculations. It should be noted that jet noise is not the only noise source from the engine, but it is the only source to be considered in this present paper. The objectives, as such are to: 1. Develop a jet noise modelling methodology that is applicable to realistic jet nozzle geometries, and 2. Provide a simplified jet noise source, which is appropriate as a source input for shielding calculations. The shielding factor is highly dependent on the source frequency, the source position (relative to the barrier and receiver) and the directivity of acoustic field it produces; and so it is important when describing any noise source input to account for these three factors – a jet noise source is no different in this respect. Note, that for a single source, the source amplitude has no bearing on the shielding factor as it will simply cancel out in the ratio, however, for multiple source inputs, the relative strengths of the sources will affect the shielding factor. The jet noise source model, for design purposes, must be applicable to realistic jet nozzle geometries. Windtunnel test data from the NACRE project will be used for calibration/validation of the method, and the test nozzle and flow conditions will be employed in the development of the source model.
2 Jet Noise Source Model The fundamental equations relating the noise produced by a jet to the turbulence in the jet mixing region were derived by Lighthill [1], who realised that the exact equations of motion may be written in the form of the homogeneous acoustic wave equation with a quadrupole source term, thus showing that there is an exact analogy between the density fluctuations that occur in any real flow and the small amplitude density fluctuations that would result from a moving quadrupole source distribution, convected at a velocity , in a fictitious stationary acoustic medium [2]. Turbulent flow can be divided into well-correlated quadrupole regions so that the acoustic pressure amplitude within a region adds linearly, while only the R.M.S amplitudes combine from uncorrelated regions. The size of these correlated quadrupole regions is taken to be roughly the size of a typical energy-bearing turbulent eddy, or correlation length scale.
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A number of specialised formulations for jet noise have been developed based on Lighthill’s general theory. Simple models using dimensional arguments have been developed, by Proudman [3] and by Lilley [4], however, this approach has difficulties in dealing with more complicated, realistic, jet exit geometries such as a coaxial jet. Alternatively, to use Lighthill’s equation to predict the sound intensity, semi-empirical methods may be employed, by evaluating the two-point space-time correlations of the second derivatives of the Reynolds stresses, and then integrating over the turbulent region. Under certain simplifying assumptions, this may be achieved experimentally or numerically, with a RANS solution. Turbulent flow may be modelled by a number of different methods, however, highly accurate Direct Numerical Solution (DNS) or Large Eddy Simulation (LES) methods, which attempt to solve all time and spatial scales, are computationally expensive and are not practical in a lot of situations - this present jet noise modelling study included. A more practical method is to solve RANS (Reynolds-averaged Navier-Stokes) equations, to obtain time or space averaged results. RANS calculations introduce additional terms in the governing equations that need to be modelled in order to achieve a ’closure’ for the unknowns. RANS solutions of the flow are, therefore, used in this present research. Goldstein and Rosenbaum’s model [5] is one such semi-empirical method that, makes a number of assumptions including axisymmetric turbulence, and so enables the longitudinal length scale to be greater than the transverse length scale, something which has been observed experimentally [6] but which earlier acoustic models, such as Ribner’s [7], crucially neglect. All the contributing correlations may then be evaluated, for an assumed axisymmetric noise pattern. The quadrupole correlations may be separated into those comprising of turbulence-only ’self-noise’ terms and those that arise from the interaction of the turbulence with the mean shear-flow, or ’shear-noise’ terms. The respective directional patterns are combined to yield a basic directivity of the eddy noise generators. The overall directivity pattern is a combination of this basic quadrupole correlation pattern, and the dominating and competing effects of convection and refraction – convection acts to beam the sound downstream, while refraction acts to bend the sound away from the jet axis – resulting in a heart-shaped pattern. An alternative approach, again using a RANS solution, is to stochastically synthesise turbulence to obtain source terms and then propagate these using linearised Euler equations. This approach, however, is time consuming, and so for this present jet source modelling study, it is not considered. In this present work, the modelling of jet noise to provide a source input for shielding calculations has been broken into a three-step process - (1) localisation with the Goldstein-Rosenbaum model, (2) propagation beyond the jet flow, using the wave expansion method and (3) simplification by defining a Kirchhoff surface. This is a fast and computationally light approach.
2.1 Step One – Localisation
Nozzle Exit y Flow Axis-of-Symmetry
Potential Core Mixing Region
θ r
Figure 2: Coordinate system for acoustic model
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With the Goldstein-Rosenbaum model, for an isolated turbulent eddy, located at y, the acoustic intensity spectrum, Iω , from that turbulent eddy, at a radius r and angle θ from the jet axis, with the coordinate origin located at the nozzle exit, is given by µ
2
ρ0 L1 L22 u2t1 ω 4 √ 40 2π 1.5 c50 r2 ωf
Iω (r, θ/y) =
+
³
ρ0 L1 L42 u2t1 π 1.5 c50 r2
exp ´2
∂U1 ∂y2
¶ −ω 2 C 2 8ωf2
Dse µ
ω4
ωf
exp
(2)
¶ −ω 2 C 2 4ωf2
Dsh
where c0 and ρ0 are the far-field speed of sound and density respectively, and ω is the particular frequency of interest. The first term on the right-hand-side of equation 2 is a self-noise term and the second term is a shearnoise term. Dse and Dsh are the self and shear basic directivity patterns from the quadrupole correlations and given by ³ ´ 2 2 Dse = 1 + 2 M 9 − N cos θ sin θ + 13
h
M2 7
³
+ M − 1.5N 3 − 3N +
³
1 2
Dsh = cos2 θ cos2 θ + where ∆=
L2 L1
³
;
³
³
M = 1.5 ∆ −
1 ∆2 1 ∆
´
1.5 ∆2
− 2N sin2 θ
´´2
−
∆2 2
´i
sin4 θ
´
³
´ ³
N = 1 − u2t2 / u2t1
;
´
C is a convection term that represents moving quadrupoles at a velocity Uc , thus if Mc is the Mach convection number C = 1 − Mc cosθ (3) The convection velocity is chosen, here, to be 0.67 of the jet exit or maximum velocity [6]. A numerical RANS solution may be used to directly provide the remaining local and statistical flow properties [8]. The velocity gradient, ∂U1 /∂y2 , can be taken straight from the solution and the length scales L1 and L2 , and the characteristic frequency of turbulence, ωf , can be defined as L1 =
3/ 2
u2t1 ε
;
L2 =
3/ 2
u2t2 ε
;
ωf = 2π κε
(4)
If a κ-ε turbulence closure is used for the RANS solution, the Reynolds stresses may be approximated1 as u2t1 = 89 κ ;
u2t2 = 49 κ
(5)
Integrating equation 2 for all receiver angles provides the acoustic power, W , from a given location, y, and so provides a means to identify a localised source region Zπ
Iω (r, θ/y) r2 sin θ dθ
Wω (/y) = 2π
(6)
0
For a frequency of interest, a number of incoherent simple sources (multipoles) may be located within this localised source region by making use of the correlation length scales. These individual sources may now be propagated with the wave expansion method. 1
Artificially imposes a ∆ value, or L2 /L1 = 1/3, as the κ-² turbulence model cannot predict the anisotropic length scales.
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2.2 Step Two – Propagation Caruthers et al. proposed the Wave Expansion Method (WEM) [9] as an efficient discrete frequency domain interpolation method, to solve the Helmholtz equation. They showed for a radiation problem, an accurate solution could be obtained with less than three nodes-per-wavelength, compared with ten nodes-per-wavelength with a finite difference method. The WEM was applied to a turbofan inlet [10], demonstrating how the WEM could be formulated in an axisymmetric form by azimuthally decomposing the three-dimensional field by Fourier expansion. This example also showed that axisymmetric mean flow solution could be introduced into the field. For axisymmetric acoustic propagation the acoustic pressure field may be decomposed azimuthally by Fourier expansion, with azimuthal variation of order l, so that p (x, r, φ) =
∞ X
p (x, r) e−jlφ
(7)
l=0
As ∂p/∂θ = −l2 p, the cylindrical homogeneous Helmholtz equation becomes µ
¶
∂ 2 p 1 ∂p ∂ 2 p l + 2 + k2 − 2 p = 0 + 2 ∂r r ∂r ∂x r
(8)
By separating p (x, r) into the product of a function of x and a function of r, a solution for equation 8 may be expressed as (2) p (x, r) = exp (−jkx x) Hl (kr r) (9) (2)
where kx2 + kr2 = k 2 cos2 θ + k 2 sin2 θ = k 2 and Hl is a Hankel function, of the second kind, of order l. Equation 9 represents a plane wave solution to the axisymmetric Helmholtz equation, with unit amplitude and propagating in direction θ. The WEM allows one to compute the acoustic pressure at a discrete point, p0 , from amplitude and phase data of M neighbouring points. The pressure at each point in a domain may be approximated by the superposition of the field generated by N hypothetical plane waves of strength γn and with unit propagation in direction θn .
r N hypothetical plane waves
p0
θn O
pm
x
Figure 3: Computational stencil for axisymmetric WEM
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The acoustic pressure, p0 , can be expressed as p0 =
N X
(2)
γn exp (−jkx,n x0 ) Hl
(kr,n r0 )
n=1
or in vector notation p0 = aγ
(10)
where a is a 1 × N vector and γ is a N × 1 vector. Similarly, the acoustic pressure at a neighbouring point, pm , where m = 1, 2, ..., M , is given by pm =
N X
(2)
γn exp (−jkx,n xm ) Hl
(kr,n rm )
n=1
or p = Aγ
(11)
where p is a M × 1 vector and A is a M × N matrix. Choosing N > M results in an infinite set of solutions to γ, all of which satisfy the Helmholtz equation in P 2 the computational stencil. The optimum solution is when N n=1 γn is minimal. Taking the pseudo-inverse of A to get A+ , imposes that the optimum solution, thus γ = A+ p and substituting back into equation 10 leads to p0 − aA+ p = 0
(12)
If κ0 is the local stiffness vector for a computational stencil, κ0 = −aA+ , then equation 12 may be written as ( ) o p n 0 1 κ0 =0 p Then for the overall computational lattice, with a source vector, f , added to the right-hand-side and where κ is the overall stiffness matrix, then κ℘ = f (13) which, with suitable boundary conditions, may be solved for ℘, a vector containing the acoustic pressure at each point in the overall domain lattice. The inclusion of flow in the axial direction is achieved by simply adjusting the local wave number, such that kf lowaxial = k/(1 + Mx cos θ)
(14)
where Mx is the local axial Mach number component. The sources may be located within the computational domain from Step 1 and the numerical RANS solution provides the local Mach number values. The sources must be simple monopoles, i.e. with uniform directivity, however by entering a number of coherent monopoles, source directivity may be introduced.
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2.3 Step Three – Simplification The noise generated by the turbulent jet flow has been localised into a number of multipole sources in Step 1, and propagated out through the jet flow in Step 2. In this final step, this jet noise must now be expressed in a manner, which can be imputed into shielding calculations. This may achieved by simply defining the acoustic pressure, p and normal pressure gradient, ∂p/∂n on a fictitious Kirchhoff surface, S, outside the jet flow. The Helmholtz-Kirchhoff integral, equation 15, may then be used [11] to propagate the sound further, in order to determine the shielded and unshielded SP L. 1 p (r,ω) = 4π
Z µ S
¶
∂G ∂p G −p dS ∂n ∂n
(15)
where p is the acoustic pressure at the receiver, n is an inward pointing unit vector, normal to the Kirchhoff surface, and G is a free-space Greens’ function that may take into account the effect of a mean flow.
3 Results 3.1 Localisation
Figure 4: Local acoustic power field [top 30 dB] spectrum from NACRE jet. 20kHz (top-left), 10kHz (top-right), 5kHz (bottom-left), 1kHz (bottom-right). Figure 4 shows, using a κ-ε turbulence closure RANS solution of the NACRE coaxial jet nozzle, and equation 6, the local source regions for particular frequencies. Observe, that the high frequency noise is produced near the jet exit, were the turbulent noise producing eddies are small in dimension and as lower frequencies are examined, the source region becomes larger and moves downstream, corresponding to the growth in size of the turbulent noise producing eddies, as they move downstream.
3.2 Propagation Figure 5 shows the local axial direction Mach values (axial flow field, with Mφ = Mr = 0) from the RANS solution, of the NACRE coaxial jet, and the corresponding sound field produced when a point source is embedded within this flow, using the axisymmetric WEM. It can be observed how the jet flow imposes
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directivity, due to refraction, on a point or monopole sound source. Results obtained using this method agree well with those obtained using ray tracing and by solving linearised Euler equation [12].
Figure 5: Axisymmetric WEM field values – local Mach number in X-direction from NACRE jet (left) and the acoustic pressure [dB] (right) produced from a point source embedded in this flow, on the jet axis at 10 equivalent primary diameters, with l = 0 azimuthal mode The Kirchhoff surface may easily be defined from these results. For example, the acoustic pressure and normal gradient values along the top, left and right boundaries of the WEM domain provides a suitable Kirchhoff surface input for shielding calculations.
4 Discussion The Goldstein-Rosenbaum acoustic model provides a means to identify a local source region as was shown in Figure 4. However, this method includes a number of approximations and the accuracy of this localisation must be questioned. RANS solutions tend to over-predict the length of the potential core. Near-field windtunnel test (WTT) data will be used in future work to calibrate/validate these frequency source location predictions. The acoustic model also considers the far-field intensity from an isolated turbulent region, without examining the interaction between turbulent regions. It has been shown [8] though, that this model can determine a reasonably accurate prediction of the far-field noise when compared with experimental data, however, there is a tendency to over predict the intensity at angles near the jet axis. This is due to the fact that the GoldsteinRosenbaum model does not account for refraction effects in its formulation. Again the WTT data will be made use of here. The Goldstein-Rosenbaum model also relies on an adjustment factor to correctly predict the sound magnitude, however, for shielding calculations this is not necessary. A uniform convection velocity of 0.67 times the maximum flow velocity was used in equation 3, however, this approach may not be appropriate, for coaxial nozzles in particular. It is intended to employ a spatially dependent convection velocity in future work. An important element of this methodology, which remains to be clearly defined, is how to position simple sources from the local source power regions shown in figure 4. It is believed that it may be sufficient to place a number of incoherent sources on the jet-axis, however, this must investigated further. The result in figure 5 is illustrative of how the axisymmetric WEM can be used to investigate the directivity imposed on a source due to jet flow, and so only one monopole source was used in this instance. Flow was included in the axisymmetric WEM by simply adjusting the wave number, as given in equation 14. This however, only accounts for an axial flow and neglects flow in the radial and azimuthal directions.
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5 Conclusions A three-step jet noise modelling methodology is presented in this paper. The three steps are (1) localisation of the source region using an axisymmetric RANS flow solution and the Goldstein-Rosenbaum acoustic model, (2) propagation of the sources with the WEM and (3) simplification of the noise source term by defining a Kirchhoff surface beyond the jet flow. This methodology will provide an approximate jet noise source, for use in calculating the shielding factor, provided by the insertion of an obstacle between the jet and a receiver. Importantly, the three steps may be executed in a short space of time. The method is quite robust as it is a shielding factor or ratio, rather than an absolute value, which is of interest in design evaluation, and so much of the error, generated by assumptions used, will cancel out. It has been demonstrated that the method is applicable to realistic jet nozzle geometries, and the results presented are for the NACRE coaxial nozzle. Preliminary results, presented in this paper, provide encouragement that this new three-step jet noise source modelling methodology can be used to provide a source input for use in airframe shielding calculations, that is equivalent - in terms of source frequency, distribution and directivity - to the noise produced by a jet flow, as required.
Acknowledgements The authors would like to thank the European Commission and the NACRE project (contract no. AIP4-CT205-516068) partners for their funding and collaboration.
References [1] M. J. Lighthill. On sound generated aerodynamically – part I. general theory. Proceedings of the Royal Society of London, Series A, Vol. 211, (1952), pp.564-587. [2] M. Goldstein. Aeroacoustics. McGraw-Hill International Book Company (1976). [3] I. Proudman. The generation of noise by isotropic turbulence. Proceedings of the Royal Society of London, Series A, Vol. 214, (1952), pp.119-132. [4] G. M. Lilley. The radiated noise from isotropic turbulence revisited. NASA Contract Report, NASA Langley Research Center, Hampton, VA, USA, (1993), pp.93-75. [5] M. Goldstein and B. Rosenbaum. Effect of anisotropic turbulence on aerodynamic noise. Journal of the Acoustical Society of America, Vol. 54, No. 3, (1973), pp.630-645. [6] P. A. O. L. Davies, M. J. Fisher and M. J. Barratt. The characteristics of the turbulence in the mixing region of a round jet. Journal of Fluid Mechanics, Vol. 15, No. 3, (1963), pp.337-367. [7] H. S. Ribner. Quadrupole correlations governing the pattern of jet noise. Journal of Fluid Mechanics, Vol. 38, No. 1, (1969), pp.1-24. [8] W. B´echara, P. Lafon and C. Bailly. Application of κ-ε turbulence model to the prediction of noise for simple and coaxial free jets. Journal of the Acoustical Society of America, Vol. 97, No. 6, (1995), pp.3518-3531. [9] J. E. Caruthers, J. C. French and G. K. Ravinprakash. Greens function discretization for a numerical solution of the Helmholtz equation. Journal of Sound and Vibration, Vol. 187, No. 4, (1995), pp.553-568.
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[10] J. E. Caruthers, R. C. Engels and G. K. Ravinprakash. A wave expansion computational method for discrete frequency acoustics within inhomogeneous flows. AIAA/CEAS 2nd Aeroacoustics Conference, State College, PA, USA, (1996). [11] D. Blackstock. Fundamentals of physical acoustics. Wiley-Interscience, (2000), chapter 14. [12] C. Bailly and D. Juv´e. Numerical solution of acoustic propagation problems using linearized Euler equations. AIAA Journal, Vol. 38, No. 1, (2000), pp.22-29.
