REFRACTION CORRECTIONS FOR SURFACE INTEGRAL METHODS IN JET AEROACOUSTICS

Transcription

1 REFRACTION CORRECTIONS FOR SURFACE INTEGRAL METHODS IN JET AEROACOUSTICS F. L. Pan Purdue University, West Lafayette, IN, A. Uzun Florida State University, Tallahassee, FL, and A. S. Lyrintzis Purdue University, West Lafayette, IN, One of the major challenges in computational jet aeroacoustics is the accurate modelling and prediction of acoustic fields in order to reduce and control the jet noise. Surface integral methods (e.g. Kirchhoff method and Ffowcs Williams - Hawkings (FW-H) method) can be used in Computational Aeroacoustics (CAA) in order to extend the near-field CFD results to far-field. The surface integral methods can efficiently and accurately predict aerodynamically generated noise provided the control surface surrounds the entire source region. However, for jet noise prediction, shear mean flow exists outside the control surface that causes refraction. The mean flow refraction corrections had been done in the past using simple geometric acoustics (GA). A more rigorous method based on Lilley s equation is investigated here due to its more realistic assumption of the acoustic propagation. Jet noise computational results based on Large Eddy Simulation (LES) for the near-field for an isothermal jet at Reynolds number 400,000 are used for the evaluation of the solution on the FW-H control surface. The proposed methodology for prediction of far-field sound pressure level shows that the acoustic field within the zone of silence is consistent with the experimental results. Nomenclature a = local sound speed normalized by its ambient value, a ā a o Ai = Airy function G ω = free-space Green s function in time domain G ω = fundamental solution of Fourier transformed wave equation defined by Eq. (21) AIAA , Presented at the 10th AIAA/CEAS Aeroacoustics Conference, May 2004, Manchester United, United Kingdom. Graduate Student, School of Aeronautics and Astronautics, Student Member AIAA. Post-Doctoral Research Associate, School of Computational Science and Information Technology, Member AIAA. Professor, School of Aeronautics and Astronautics, Associate Fellow AIAA. 1

2 G ω = fundamental solution of Fourier transformed wave equation defined by Eq. (22) k o = streamwise wavenumber, k o = ω a o L = wave operator M = Mach number or mean Mach number profile n = critical azimuthal wavenumber ˆn j = outward directed unit normal vector p = local pressure q = arbitrary function defined by Eq. (23) Q = arbitrary function defined by Eq. (23) r = magnitude of radiation vector, r j r δ = turning point defined by Eq. (23) R = distance between observer and source point Re D = Reynods number based on jet diameter R ω = arbitrary function defined by Eq. (23) or (24) S = closed surface defined by f(y, τ) St = Strouhal number t = time or observer reception time u i = Cartesian surface velocity components P ij = compressive stress tensor with the constant p oδ ij subtrated U = mean velocity or mean velocity profile ( x, t) = observer coordinates and time <> = root mean square evaluation Greek φ = acoustic variable ω = cyclic frequency δ ij = Kronecker delta ρ = fluid density θ = source emission angle ϑ = sound propagation angle ϕ = difference in azimuthal angles of the observer and source point coordinates ɛ = arbitrary function defined by Eq. (23) η = arbitrary function defined by Eq. (23) Γ = acoustic source distribution Π = acoustic pressure fluctuation normalized by ρā 2 τ = retarded or emission time, t R/a o ξ = arbitrary function defined by Eq. (24) ζ = arbitrary function defined by Eq. (18) Superscript = pertubation quantity (e.g.ρ = ρ ρ o) = source time derivative = mean flow quantity = Fourier transformed quantity ˆ = unit vector Subscript av = average parameter c = corrected parameter cr = critical angle parameter J = quantity on jet centerline m = measured parameter o = ambient or freestream quantity 2

3 r = inner product with ˆr j ref = reference parameter s = quantity evaluated at source position ret = quantity evaluated at retarded time τ n = inner product with ˆn j I. Introduction Computational aeroacoustics (CAA) is concerned with the prediction of the aerodynamic sound source and the propagation of the generated sound. The sound is generated by nonlinear process, which is governed by the full, time-dependent, and compressible Navier-Stokes equations. Indeed, the acoustic far-field is linear and irrotational, one can consider two distinct characteristics of sound, i.e. the nonlinear generation of sound, and the linear propagation of sound. This implies that instead of solving the full nonlinear flow equations out in the far-field for sound propagation, one can extend the nonlinear near-field acoustic sources to the linear far-field sound through the traditional Lighthill s acoustic analogy 1 or surface integral methods, i.e. Kirchhoff s method 2 and Ffowcs Williams - Hawkings (FW-H) method. 3, 4 The latter methods are more attractive as the volume integration is avoided. The surface integral methods assume that the sound propagation is governed by the simple wave equation, and can efficiently and accurately predict aerodynamically generated sound provided the control surface surrounds the entire nonlinear source region. 5, 6 However, in jet noise prediction, the control surface cannot enclose the entire aerodynamic generated source region. Thus, the linear wave equation is invalid when the sound propagates through the region near the jet axis, downstream of the control surface. In the past, researchers utilizing surface integral methods to predict jet noise had ignored sound generated at and outside of the downstream end of the control surface. 5 When acoustic waves propagate through the jet mean flow downstream of the control surface, it crosses velocity-shear interfaces, which deflects the radiated sound away from the mean flow direction. This effect is called refraction. 7 For isothermal or hot jets, the effect of refraction gives rise to a zone of silence, a region where a substantial reduction of sound occurs. Early investigations 7 9 found that the waves propagating in the zone of silence initially start as evanescent waves near the source. The presence of the zone of silence was also measured experimentally Surface integral methods are not able to predict the refraction outside the control surface and the resulting zone of silence. The mean flow refraction corrections to the surface integral method have been investigated using simple geometric acoustics (GA) approximation. 6, 13 This approximation is valid only at high-frequency region. However, azimuthal variations between the sources in the sheared layer and the observers were not taken into account and hence the accuracy of this correction may be limited. It is worthwhile to attempt a more rigorous method in order to achieve more reliable prediction. A linearized homogeneous third-order convective wave equation, namely Lilley s equation, 14 has been solved asymptotically using the stationary phase method 3

4 in the far-field. 7 The high-frequency limit solution 15 of Lilley s equation is revisited and applied to refraction corrections. The frequency range in which the approximation solution of Lilley s equation holds, is also studied. The remainder of the paper describes the surface integral methodologies, i.e. the Kirchhoff and FW-H methods, both in time and frequency domain. Then, the two refraction corrections approaches, i.e. the simple GA method and Lilley s equation method are discussed. An example of a monopole source is used to demonstrate the utility and efficienty of the refraction corrections schemes. An application involving an overall sound pressure level (OASPL) calculation based on a LES computation of an isothemal jet at Re D = 400,000 is shown followed by the concluding remarks. II. Surface Integral Methodologies Kirchhoff s Method. Kirchhoff s formula was first published in It is an integral representation (i.e. surface integral around a control surface) of the solution to the wave equation. Kirchhoff s formula, although primarily used in the theory of diffraction of light and in other electromagnetic problems, has also many applications in studies of acoustic wave propagation. Farassat and Myers 16 derived a Kirchhoff formulation for a moving, deformable, piecewise smooth surface. Assume the linear, homogeneous wave equation 1 a 2 2 φ t 2 2 φ = 0 (1) x i x i is valid for some acoustic variable, φ, and sound speed, a o in the entire region outside of a closed and bounded smooth surface. For stationary surfaces, the Kirchhoff formula is [ ] 1 1 φ [φ] 4πφ( x, t) = φ cos θ ds + ret S R a n ret S R 2 ds (2) where cos θ = ˆR ˆn, θ is the source emission angle, ˆn is the outward directed unit normal vector of the closed control surface S, R is the distance between the observer and the surface source. The surface integrals are over the control Kirchhoff surface, S, and evaluated at retarded or emission time, τ described by the following equation τ = t R a o (3) where t is observer or reception time. The dependent variable φ is normally taken to be the disturbance pressure, but can be any quantity which satisfies the linear wave equation. With the use of a Fourier transform, F{f(t)} = f(ω) = f(t)e iωt dt 4

5 F 1 { f(ω)} = f(t) = 1 f(ω)e iωt dt (4) 2π equation (2) can be expressed in the frequency domain (i.e, starting from Helmholtz equation) as [ ( 4π φ( x, 1 iω ω) = e iωr/a cos θ R a φ φ ) + φ ] cos θ n R 2 ds (5) S where φ is the Fourier transform of φ, and ω is the cyclic frequency. Both time and frequency expressions can be extended for a uniform rectilinear motion. 17 Equations (2) and (5) work well for aeroacoustic predictions when the control surface is placed in a region of the flow field where the linear wave equation is valid. However, this might not be possible for some cases. Additional nonlinearities (i.e. quadrupoles) can be added outside the control Kirchhoff surface. 18 Kirchhoff s method has also been applied for open surfaces 19 and can be extended to include mean flow refraction effects based on the GA method. 13 Porous FW-H Method. A general version of the FW-H equation is needed because the usual practice is to assume that the FW-H integration surface corresponds to a solid body and is impenetrable. For the simple case of a stationary surface we can define (following Brentner and Farassat 20 and Francescantonio 21 ) new variables U i and L i as U i = ρu i ρ o (6) and L i = P ij ˆn j + ρu i u n (7) where subscript o implies ambient conditions, superscript implies disturbances (e.g. ρ = ρ + ρ o ), ρ is the density, u is the velocity, and P ij is the compressive stress tensor with the constant p o δ ij subtracted. Now by taking the time derivative of the continuity equation and subtracting the divergence of momentum equation, followed with some rearranging, the integral form of FW-H equation can be written as (Formulation I) where p ( x, t) = p T ( x, t) + p L( x, t) + p Q( x, t) (8) 4πp T ( x, t) = t S 4πp L( x, t) = 1 [ Lr a t S R ] [ ] ρo U n ds (9) R ret ] ds + ret S [ Lr R 2 ds (10) ret and the quadrupole noise pressure p Q ( x, t) can be determined by any method currently available. The subscript r or n indicates a dot product of the vector with the unit vector in the radiation direction ˆr or the unit vector in the surface normal direction ˆn i, respectively. It should be noted that the three pressure terms have a physical meaning for rigid surfaces: p T ( x, t) is known as thickness 5

6 noise, p L ( x, t) is called loading noise and p Q ( x, t) is called quadrupole noise. For a porous surface the terms lose their physical meaning, but the last term p Q ( x, t) still denotes the quadrupoles outside surface S. An alternative way 21 is to move the time derivative inside the integral: (Formulation II) [ ] 4πp ρ o U n T ( x, t) = ds (11) R 4πp L( x, t) = 1 a S S [ Lr R ] ret ret ds + S [ Lr R 2 ] ds (12) ret The dot over a variable implies source-time differentiation of that variable. It appears that Formulation I (Eqs. 8-10) has less memory requirements, because it does not require storage of the time derivatives, and requires less operations per integral evaluation. For our case of a stationary Kirchhoff surface the integral can be reused at the next time step. Formulation I was used by Strawn et al. 22 for rotorcraft noise predictions using a nonrotating control surface with very good results. On the other hand taking the time derivative inside could prevent some instabilities for moving control surfaces. Thus Formulation II (Eqs ) might be more robust for a moving control surface, Formulation II was used for rotorcraft noise prediction by Brentner and Farassat 20 with a rotating control surface with very good results. However, a more detailed comparison of the two formulations would be very helpful. With the use of Fourier transform pair in Eq. (4), both formulations can be written in the frequency domain for a stationary surface as 4π p T ( x, ω) = iω e ρ iωr/a oûn ds (13) S R 4π p L( x, ω) = iω Lr e iωr/a a R ds + L r ds (14) R2 S where p, Ûn, and L r are the Fourier transforms of p, U n, and L r, respectively. and ω is the cyclic frequency. Both the above formulations provide a Kirchhofflike formulation if the quadrupoles outside the control surface (p Q ( x, t) term) are ignored. The equivalence of the porous FW-H equation and Kirchhoff formulation was proven by Pilon & Lyrintzis 18 and Brentner & Farassat. 20 The reader is referred to the above references for the details. The major difference between Kirchhoff s and FW-H formulation is that Kirchhoff s method needs p, p n, p t as input, whereas the porous FW-H needs p, ρ, ρu i. Also, the porous FW-H method puts less stringent requirements on the placement of surface in the linear region than Kirchhoff s method, as shown in Ref. 20 for a rotorcraft noise problem. The method has been extended to include refraction corrections based on GA. 6 S 6

7 ϑο Fig. 1. θ source Sound emission and propagation angle of a source due to refraction. U III. Refraction Corrections Geometric Acoustics. The refraction problem has drawn attention, because of its importance in the open jet wind tunnels experiments. Early development of solution of the refraction by a thick cylindrical shear layer with a source on the jet axis was given by Tester and Morfey, 8 and for a plane shear layer by Amiet. 23 Recent development on the source off the jet axis was given by Morfey and Joseph. 24 These refraction corrections approaches are geometrical algebra in close form solutions. The application of simple geometric acoustics (GA) closed-form approximation used in this work is based on the thick cylindrical shear layer refraction correction. 23 The simple GA approximation is valid only when the acoustic wavelength, λ is small compared with the shear layer thickness, δ. It is assumed here that the downstream end of the cylindrical control surface is located far enough downstream of the jet potential core that the shear layer thickness is large compared with the acoustic wavelength. Regardless, Morfey and Szewczyk 25 have shown that jet mixing noise can be effectively modeled with the simple GA method even when δ/λ < 1. One of the advantages of utilizing their method is that it can be applied to a general source. This flexibility was arrived due to the separate corrections of the angle and amplitude. We consider only mean flow effects on the sound radiation and model the outer shear layer as a stratified flow. 23 The stratified flow is described in Eq. (15), a o cos ϑ o = U + a o cos θ (15) where U is the steady velocity at the downstream end of the control surface, θ is the sound emission angle with respect to the jet axis, and ϑ o is the propagation angle in the stagnant, ambient air, as shown in Fig. 1. On reaching the shear layer, the sound is refracted and hence both amplitude and angle change. The refraction consists of a collection of point acoustic sources acting at radial location, and the parallel sheared mean flow is determined at each location. Equation (15) can be rearranged to show that the critical angle is defined by 7

8 cos θ cr = M (16) Below the critical angle, θ cr, no sound can reach the observer and this gives the zone of silence. This result is purely kinematic and a thick cylindrical shear layer is considered here to capture more detailed information of the refraction. The amplitude correction can be found from the following equation, P c = (1 M cos ϑ)2 (17) P m where P c is the corrected pressure, and P m is the measured pressure. Note that the Mach number of the mean sheared flow, M varies along the control surface in radial direction. The correction should be valid provided the radial distance is large compared to the shear layer s wavelength and the source dimensions. (18) An additional correction is needed, which converts the present source position to the retarded source position; it can be done by a factor multiplication, r ret (1 M cos ϑ) = r c (1 + M 2 ζ 2 ) (18) where ζ = (1 M cos ϑ) 2 cos 2 ϑ A final correction is necessary to accurately account for the mean flow refraction. Imposing the local zone of silence condition described earlier allows a surface source at a relatively large radial location to radiate sound into and through the shear flow. This is because the local zone of silence decreases in size with the radial location of the source. The simple correction is to set the source strength to zero if the observation point is located closer to thet jet axis than the source point on the FW-H surface. It should be noted that it is questionable that the energy argument used to derive the result for the large thickness case is still feasible for zero thickness case. More details can be found in Ref. 23. The GA corrections can be applied to surface integral methods 6, 13 for the downstream portion of the control surface using the local velocity distribution. The velocity is assumed to be parallel to the axis. Another issue is the azimuthal variations between observer point and source, which is not accounted using this simplified approach. This azimuthal variation parameters will be introduced by using the alternative correction method in the following section. Lilley s Equation. Lilley s equation 14 is derived from the inhomogeneous wave equation. The equation describes explicitly the combined effects of sound convection and refraction by a unidirectional transversely sheared mean flow. Lilley s equation can be written as LΠ = Γ (19) 8

9 where Π denotes the acoustic pressure fluctuation normalized by ρā 2, ρ is the mean flow density, ā is the mean speed of sound, Γ represents the acoustic source distribution. L is the wave operater originally derived by Pridmore-Brown (1958). L = D ( ) D 2 Dt Dt 2 ā2 + 2ā 2 ū x (20) where D Dt = t + ū x is the convective derivative relative to mean flow. ū is the mean flow velocity. In the past, many researchers 7, 9, 26 have tried to obtain the Greens function expression of Lilley s equation using high-frequency, far-field asymptotic approximation. This approximation is valid only when the acoustics wavelength of the aerodynamic noise is much shorter than the characteristic length scale of the mean flow, and also the mean velocity profile is parallelly well-defined. At the end of the jet potential core, for example, the mean velocity profiles are significantly distorded, and the parallel mean flow assumption would fail. Based on this assumption, three different closed-forms of Green s function solution with locally parallel and axisymmetric mean flow were obtained. The first expression was developed by Goldstein 7 in which the source was placed on several wavelengths off the jet axis. At the same time, Balsa 9 developed an expression with the source located near the jet centerline axis. The third form was derived by Goldstein 27 with the implementation of ray-theory for parallel round jets. If a monople source distribution is considered, the Green s function associated to the Fourier transformed solution of Lilley s wave equation, G ω (x x s ) satisfies the following equation L[G ω (x x s )e iωt ] = D Dt [ā2 δ(x x s )e iωt ] (21) where ω is source frequency, x s is source position, δ is the Kronecker delta. The fundamental solution of Fourier transformed Lilley s equation, which is mathematically called the reduced Green s function, describes the effect of the surrounding mean flow on radiated sound. Mean velocity gradients and temperature gradients are known to refract the acoustic energy within the mean shear layer. For simplicity, an isothermal jet condition is considered with no temperature gradients. Hence, the refraction is caused only by non-uniform mean flow whithin the shear layer, which varies radially at the outflow of the control surface. One can solve the reduced Green s function using a high-frequency asymptotic approximation. As mentioned in Ref. 15, if the distance between the source and the jet centerline axis is sufficiently large (several factors of 1/k o, where k o is streamwise wavenumber, k o = ω/a o ), then the critical azimuthal wavenumber, n can be scaled to the order of k o. This scaling was considered by Goldstein 7 and the resulting acoustic field was referred to as the asymmetric, high-frequency 9

10 approximation. n = O(k o ) as k o On the other hand, as the source moves closer to the jet centerline axis, the scaling becomes n = O(1) as k o Here, the critical wavenumber, n scales like 1/r J, where r J is the jet nozzle radius. This near-axis source problem was studied by Balsa 9 and the resulting acoustic field was referred to as the quasi-symmetric, high-frequency approximation. The following Eq. (22) gives the ratio of the reduced Green s function to the free-space Green s function, which was implemented in the refraction corrections codes developed in this research. G ω (x x s ) G ω (x x s ) R ω (x x s ) a(r s )(1 M(r s ) cos θ) (22) It should be noted that this equation is valid only when both k o and R. The former assumption needs to be well defined and the lowest wavenumber, k o value should be correctly determined. An analysis was done for this evaluation in Ref. 15. The same results are obtained here, which show the asymptotic highfrequency approximation for solutions of Lilley s equation remains accurate with Strouhal number as small as 1/2. Based on the asymmetric and quasi-symmetric behaviors of acoustic field, two distinct R ω (x x s ) functions from Eq. (22) are derived in separate approaches and are used to describe the acoustic field accordingly. The asymmetric, highfrequency solutions are used for off-axis source locations, whereas the quasisymmetric solutions are applied to near-axis locations. For the asymmetric, far-field approximation, R ω (x x s ) is defined by the following equation. where R ω (x x s ) n= n= [ 2 k o ] 1 2 ηn (r s ) Ai[η n (r s )]ɛ (23) r s Q n (r s ) ɛ = e in ϕ+iko(ζn(r) R sin2 θ) ( ) 2 nko Q n (r) = q 2 (r) r [1 ] 2 M(r) cos θ q(r) = cos a(r) 2 θ [ ] 2 3 η n (r) = 2 k 3 oζ n (r) 10

11 r ζ n (r) = Q n (r)dr, Q 2 n(r δ ) = 0 r δ where a ā/a o is the local sound speed normalized by its ambient value, M = U/a o is the Mach number based on the ambiet speed of sound, and ϕ is the azimuthal difference between observer and source point, r s is the source location, Ai is the Airy function. Note that cube root in the definition of η n in Eq. (23) is taken such that η n < 0 for r > r δ and η n > 0 for r < r δ, For the quasi-symmetric, far-field approximation, R ω (x x s ) is defined by the following equation. where R ω (x x s ) e iko(ξ(r) R sin2 θ ξ(r s) cos ϕ) ξ(r) = r 0 q(r)dr (24) It should be noted that r δ defined in Eq. (23) does not appear in Eq. (24). r δ is the solution for which the nonlinear equation, Q n is zero, and is called turning point. r δ is determined by solving Q n numerically. Due to its dependency on Mach number, which varies along the radial direction, the use of asymmetric approximation turns out more computationally expensive compared to the quasi-symmetric approximation. Figures 2 (a)-(b) show the comparisons of G ω /G ω at various azimuthalangle parameters, ϕ for both asymmtric and quasi-symmetric approximations. The asymmetric and quasi-symmetric approximations are indicated by the solid and dashed lines, respectively. We used the case suggested in Ref. 15, i.e. a point source at r s = 0.75 using a mean flow profile given by equation ū(r) ā(r) = M(r) a(r) = M Jsech 2 (2r) (25) The Strouhal number is 0.5 and can be calculated as St = k o π r J M J (26) where r J = 0.5 is the jet radius, M J = 0.9 is the Mach number at jet centerline. The results are identical to Ref. 15. One can see how G ω /G ω varies with different azimuthal angle parameters, ϕ. This implies that simply ignoring the azimuthal variations between the source and the observer points will result to inaccurate predictions. IV. Results and Discussion Point Source. A simple monopole is used to demonstrate the application of refraction corrections in jet noise prediction. The frequency domain version of the Ffowcs Williams-Hawkings (FW-H) integral was used to calculate the acoustic far-field of the monopole. A cylindrical control surface (L k = 40, r k = 5) as shown in Fig. 4., was used in this test case. The monopole is placed at the middle of the surface and the surface pressure is defined as 11

12 asymmetric quasi-symmetric asymmetric quasi-symmetric a) θ (degree) b) θ (degree) asymmetric quasi-symmetric asymmetric quasi-symmetric c) θ (degree) d) θ (degree) Fig. 2. G ω/g ω at (a) ϕ = 0; (b) ϕ = 30; (c) ϕ = 90; (d) ϕ =

13 normalized amplitude with no refraction simple GA method Lilley s equation, k o =1.0 Lilley s equation, k o = θ (deg) Fig. 3. Refraction effects. p(x s, ω) = 1 rs iω e ao (27) r s The acoustics wavenumber, k o of 1.0 and 1.5 are considered. We are using 20 points per wavelength. The distance between the midpoint of the cylinder s axis and the observer is taken to be 60r J, where r J is 1. The result with no refraction correction is shown in Fig. 3. A normalized amplitude of 1 is expected with no mean flow outside the FW-H surface. The refraction corrections are then applied to the right base of the cylinder using the simple GA method and Lilley s equation. The shear flow is modelled with velocity profile shown in Fig. 5. The mean flow velocity profile is only applied to the right base of the cylindrical control surface. At surface radius of 5, the corresponding Mach number is while the centerline Mach number is 0.5. This is a reasonable distribution for jet velocites. 6 Since velocity becomes negligible near the radial distance of 5, the refraction correction is applied only for r < 5. When solving asymptotic solutions for the Lilley s equation, which is based on the high-frequency assumption, a minimum Strouhal number must be considered. The Strouhal number can be determined from Eq. (26). As demonstrated in the last section, at Strouhal number of 1/2, the solutions are still very close to the exact solutions of Lilley s equation. Below this Strouhal number limit, asymptotic approximation does not apply and the refraction correction must be done by an alternative way, which we choose to be the simple GA method. Since the asymmetric solution is more accurate, it is used for all source elements except for a near-axis source, where the asymmetric solution breaks down and a quasi-symmetric solution is used. As shown in Fig. 3, the zone of silence is predicted in a region where there is a substantial reduction in the normalized pressure amplitude. For the monopole 13

14 r S ^ n R Observer (X,t) θ r k Point Source x Lk Fig. 4. Point source location and control surface geometry. case, the Lilley s equation results a smooth amplitude decrease in the zone of silence, wheareas the simple GA method shows an abrupt decrease at this region. A similar trend was obtained from the Kirchhoff method, and hence only the FW-H result is shown here. The azimuthal variation between the source and observer points has been ignored in the simple GA method, whereas the variation parameters can be evaluated with Lilley s equation, which has been discussed in the previous section. The azimuthal variations between the sources and observer points are important because the FW-H control surface utilized is three-dimensional. It is worthwhile to note that the limitation of the high-frequency applications of Lilley s equation to jet noise can be overcome by adding the axial length of the control surface to capture more of the low frequency noise. Also, the refraction effects inside the control surface are captured better with a longer surface. However, higher computational cost will result from the increase of the surface length. Jet Noise Prediction Using LES. The surface integral methods can extend the near-field flow data to far-field acoustic properties, e.g. acoustic pressure. The near-field input for the surface integral methods will be obtained from a 28, 29 Large Eddy Simulation (LES) calculation. In this study, the LES calculation is peformed for a turbulent isothermal round jet at a Mach number of 0.9 and Reynolds number of 400,000. The LES calculation employs an implicit sixth-order tri-diagonal spatial filter 30 as the implicit subgrid-scale (SGS) model. The jet centerline temperature was chosen to be the same as the ambient temperature, T o and set to 286K. A fully curvilinear grid consisting of approximately 16 million grid points was used in the simulation. The grid had 390 points along the streamwise x direction and 200 points along both the y and z directions. The physical portion of the domain in this simulation is extended to 35 jet radii in the streamwise direction and 30 jet radii in the transverse directions. The coarsest resolution in this simulation is estimated to be about 400 times the local Kolmogorov length scale. The boundaries of the computational domain are applied using the Tam and 14

15 5 4 3 r Mach number Fig. 5. Mean-flow profile. Dong s 3-D radiation and outflow boundary conditions, 31 as illustrated in Fig. 6. A sponge zone 32 was used in which grid stretching and artificial damping are applied to suppress the unwanted reflections from the outflow boundary. Since the actual nozzle geometry is not included in the present calculations, randomized perturbations in the form of a vortex ring are added to the jet velocity profile at a short distance downstream of the inflow boundary in order to excite the 3-D instabilities in the jet and cause the potential core of the jet to break up at a reasonable distance downstream of the inflow boundary. The infow forcing used here was proposed by Bogey and Bailly. 33 There are 16 azimuthal modes in total and the first 4 modes are not included in the vortex ring forcing. A more detailed study of the inflow vortex ring forcing can be found in Ref. 34. The mean shear layer imposed on the inflow boundary has a streamwise velocity profile ū(r) = U J 2 [ [ ( 1 + tanh 10 1 r )]] r J (28) where U J is the jet centerline velocity at nozzle exit, r J is the jet nozzle radius. There are about 14 grid points in the initial jet shear layer. The standard fourthorder explicit Runge-Kutta scheme is used for time advancement. A Ffowcs William - Hawkings (FW-H) control surface was placed around the jet as illustrated in Fig. 7. The control surface starts 1 jet radius downstream of the inflow boundary and extends to 31 jet radii along the streamwise direction. Hence the total streamwise length of the control surface is 30r J. The control surface is initially at a distance of 3.9 jet radii from the jet centerline, and opens up with downstream distance. The LES flow field data are gathered on the control surface in time domain at every 5 times steps over a period of 25,000 time steps during the LES run. The initial transients exited the domain over 15

16 Tam & Dong s radiation boundary conditions Tam & Dong s radiation bcs Tam & Dong s outflow boundary conditions Vortex ring forcing Sponge zone Tam & Dong s radiation boundary conditions Fig. 6. Fig. 7. Schemetic of the boundary conditions. Schematic of the control surface surrounding the jet flow. the first 10,000 time steps. The total acoustic sampling period corresponds to a time scale in which an ambient sound wave travels about 14 times the domain length in the streamwise direction. Assuming at least 6 points per wavelength are needed to accurately resolve an acoustic wave, it is found that the maximum frequency resolved with the grid spacing around the control surface corresponds to a Strouhal number of approximately 3.0. This is the grid cutoff frequency used in computation domain. Based on the data sampling rate, the number of temporal points per period in this maximum frequency is 8. To investigate results of the refraction corrections, a frequency domain of the FW-H method is developed and has LES data as an input at outflow surface. A 4608-point Fast Fourier Transform (FFT) analysis is performed to obtain bn, and L b r at each the variables required in FW-H calculation, which are pb0, U frequency. These terms are calculated in the retarded time algorithm using in Eq. (3), which can be solved by performing simple linear interpolation. 16

17 Fig. 8. Schematic showing the center of the arc and how the angle θ is measured from the jet axis. It should be noted that the refraction corrections evaluated on the downstream control surface are using the frequency domain FW-H equation. These results are added to the time domain FW-H results for an open surface. 29 Thus, an inverse FFT is needed to recover the signal for the downstream refraction surface in the time domain. Numerical results of far-field overall sound pressure level (OASPL) are calculated using Eq. (29) along an arc of radius 60r J from the jet nozzle. OASPL = 20log 10 ( E P ref ) (29) where P ref is the non-dimensional reference pressure calculated from the ambient temperature, T o = 286K, ambient pressure, p o = 1.01KPa, M = 0.9 at jet centerline nozzle exit, and dimensional reference pressure, 2x10 5 Pa. E is defined as < P av (f)df >, where P av is calculated from the average acoustic pressure signals over the equally spaced 36 azimuthal points on a full circle at a given θ location on far-field arc of radius 60r J from the jet nozzle. The center of the arc is chosen as the jet nozzle exit and the angle θ is measured from the jet axis as illustrated in Fig. 8. Table 1. Some jet noise experiments with Mach numbers similar to that our LES M Re D Reference ,000 Mollo-Christensen et al ,000 Lush ,600 Stromberg et al ,000 SAE ARP 876C ,000 Our 3-D LES 28 17

18 Simple GA refraction Lilley s refraction Open control surface Closed control surface SAE ARP 876C prediction Mollo-Christensen et al. data (cold jet) Lush data (cold jet) Stromberg et al. data (cold jet) OASPL (db) θ (deg) Fig. 9. Overall sound pressure levels at 60r J from the nozzle exit. One should remove the low frequencies resulting from the outflow boundary which may not be damped out effectively by the sponge zone. In our jet case, this frequency with the corresponding Strouhal number of 0.05 was found. Hence, the OASPL spectra integration only includes the resolved frequencies range of 0.05 St 3.0. The computations of surface integral method were performed on IBM Power 3 machine at Purdue University. Advanced features offered by FORTRAN 90 such as dynamics memory allocations are implemented in the code to enable the computations. The codes are run in parallel using MPI implementation. The outflow surface comprises of a total of 135 x 135 patches. The walltime required for computing acoustic signals at each arc location (with 36 azimuthal locations) from the outflow FW-H surface is around 5.8 hours using eight 375 MHz processors. The solutions of the rest of the FW-H surfaces are added to the solutions of the outflow surface that includes the refraction corrections. In Fig. 9, the solid line represents the OASPL result with no refraction correction at observer location along a far-field arc of x = 60r J. Table (1) summarizes the Mach numbers and Reynolds numbers of the experiments and the ARP database 37 that we are comparing to. It should be noted that the experimental jets were cold jets, whereas our jet is an isothermal jet. However, as indicated in the ARP database, the effect of the temperature is very small, i.e. about 0.3 db. From the OASPL plot for an open surface, it is shown that the prediction from ARP database for a cold jet at Re D of 400,000 is slightly lower (about 1dB) than the LES calculations for high angles, i.e. θ 40 o. Also the OASPL are comparable to the experimental values at these angles. However, at shallow angles, i.e. θ 40 o, the OASPL from the LES drops significantly due to the 18

19 omission of the signal from the downstream surface. For the closed control surface, the OASPL is generally higher than that of the open control surface. The results increase significantly for lower angles. However, they become very high at low angles as refraction effects outside the control surface are not included. For higher angles, i.e. θ 70 o, there is also an overprediction due to an spurious line of dipoles on the downstream surface area, as the quadrupoles move downstream and exit the control surface. 30 The implementation of the refraction correction begins with generating a parallel shear mean flow. The mean flow velocity profile generated from LES is shown in Fig. 10. To reduce the required computational time for Lilley s equation, the mean flow velocity profile is modeled with the following equation M(r) = M J e Ar2 + B (30) where M J = 0.46, A, B are constants, i.e and , respectively. Equation (30) is represented with dash-line in Fig. 10. This profile is used for both the simple GA method and Lilley s equation calculations. This sheared mean flow has an influence on the signals emitted from the surface elements on the downstream outflow FW-H surface. Refraction corrections are performed on the surface elements inside the sheared flow, i.e. on the downstream surface. The variation of the velocities across the radial direction of the shear layer is taken into account at each surface element location. The refraction corrections schemes result in the damping of far-field signal contributed by the surface element. Within the high-frequency region of 0.50 St 3.0, this refraction corrections are performed at each frequency and on each surface element. As shown in Fig. 9, the OASPL with refraction corrections starts to decrease as θ goes below 20 o. This gives a zone of silence as expected from the experimental results. The slight decrease of the results with no refraction corrections is due to the existence of refraction effect inside the FW-H control surface. With the refraction corrections outside the control surface using GA method, the results show a decrease on the OASPL making closer to the experiments. The results from the Lilley s equation method exhibit a smoother decrease on the OASPL inside the zone of silence which is consistent with the results for a monopole source shown in Fig. 3. With the application of Lilley s equation, the computational walltime required increases to about 180 hours for each arc location inside the zone of silence using eight 375 Mhz processors run in parallel. The maximum time ratio for Lilley s equation method to GA method is 60. This significant increase in computational time is expected when solving the Lilley s equation because the mean flow velocity dependent turning point, r δ in Eq. (23) needs to be evaluated numerically. The scheme currently used in this research is Brent s method. The use of the simple GA method and Lilley s equation have shown a successful prediction of the zone of silence, which cannot be done by the traditional surface integral methods. 19

20 6 5 LES data Analytical data 4 r M Fig. 10. Jet mean-flow profile. V. Concluding Remarks In the past the surface integral methods have proven to be an efficient and accurate tool for the prediction of jet noise and other aeroacoustic phenomena. However, they fail to predict the zone of silence accurately. We have developed refraction corrections and utilized them with the surface integral methods, i.e. Kirchhoff and porous Ffowcs Williams - Hawkings (FW-H) method. The refraction corrections are based on simple geometric acoustics (GA) and Lilley s equation high-frequency, far-field asymptotic solutions. The methods were applied to the noise calculation of an isothermal jet of Reynolds number 400,000. The solution on the FW-H control surface was provided using LES. The above refraction corrections were applied on the downstream control surface to account for the mean shear flow outside the control surface. Both refraction corrections improve the OASPL results dramatically at shallow angles near the axis. The comparisons between the two methods were also made and it was shown that the simple GA method is much simpler and requires less computational time compared to Lilley s equation. On the other hand, the high-frequency asymptotic solutions for Lilley s equation has the potential of being more realistic which allows azimuthal variations between source and observer. Acknowledgments The simulations were carried out on Purdue University s 320 processor IBM- SP3 supercomputer. The third author acknowledges support by the Indiana 21st Century Research and Technology Fund, and the Aeroacoustics Research Consortium (AARC), a government and industry consortium managed by the Ohio Aerospace Institude (OAI). The third author would also like to thank Dr. Khavaran from NASA Glenn for suggesting the Lilley s equation method for refraction corrections. The first author is grateful to Dr. Gregory Blaisdell and 20

21 Phoi-Tak Lew for providing useful suggestions and discussions. References 1 Lighthill, M. J., On Sound Generated Aerodynamically: I. General Theory, Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences, 1952, 211(1107), Kirchhoff, G. R., Zur Theorie der Lichtstrahlen, Annalen der Physik und Chemie, Vol. 18, 1883, pp Ffows Williams, J. E., and Hawkings, D. L., Sound Generation by Turbulence and Surfaces in Arbitrary Motion, Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences, 1969, 264(1151), Crighton, D. G., Dowling, A. P., Ffowcs Williams, J. E., Heckl, M., and Leppington, F. G., Modern Methods in Analytical Acoustics: Lecture Notes, Springer Verlag, London, Lyrintzis, A. S., Surface Integral Methods in Computational Aeroacoustics- From the (CFD) Near-Field to the (Acoustic) Far-Field, Aeroacoustics Vol. 2, number 2, 2003, pp Lyrintzis, A. S., and Uzun, A., Integral Techniques for Jet Aeroacoustics Calculations, AIAA Paper , presented at the 7th AIAA/CEAS Aeroacoustics Conference, Maastricht, The Netherlands, May Goldstein M. E Aeroacoustics. New York: McGraw-Hill. Chapter 6 8 Tester, B. J. and Morfey, C. L., Developments in Jet Noise Modelling - Theoretical Predictions and Comparisons with Measured Data, Journal of Sound and Vibration, Vol. 46, 1976, pp Balsa, T. F., The Far-Field of High Frequency Convected Singularities in Sheared Flows, with an Application to Jet Noise Prediction, Journal of Fluid Mechanics, Vol. 74, March 1976, pp Atvars, J., Schubert, L. K., Ribner, H. S., Refraction of Sound from a Point Source Placed in an Air Jet, Journal of the Acoustical Society of America, Vol. 37, 1965, pp Lush, P. A., Measurements of Subsonic Jet Noise and Comparison with Theory, Journal of Fluid Mechanics, Vol. 46, No.3, 1971, pp Tanna, H. K., An Experimental Study of Jet Noise. Part I: Turbulent Mixing Noise, Journal of Sound and Vibration, Vol. 50, 1977, pp Pilon, A. R., and Lyrintzis, A. S, Mean Flow Refraction Corrections for the Kirchhoff Method, AIAA Journal of Aircraft, Vol. 35, No. 4, Jul.-Aug pp Lilley, G. M., On the Noise from Jet, AGARD CP 131, 1974, pp Wundrow, D. W. and Khavaran, A. On the Applicability of High-Frequency Approximations to Lilley s Equation, NASA/CR , Farassat, F., and Myers, M. K., Extension of Kirchhoff s Formula to Radiation from Moving Surfaces, Journal of Sound and Vibration, Vol. 123, No. 3, 1988, pp Pilon, A., Development of Improved Surface Integral Methods for Jet Aeroacoustic Predictions, Ph.D Dissertation, Aerospace Engineering and Mechanics Dept., University of Minnesota, Minneapolis, MN, Pilon, A. R., and Lyrintzis, A. S., Development of an Improved Kirchhoff Method for Jet Aeroacoustics, AIAA Journal, Vol. 36, No. 5, May 1998, pp Freund, J. B., Lele S. K., and Moin P. Calculation of the the Radiated Sound Field Using an Open Kirchhoff Surface, AIAA Journal, Vol. 34, No. 5, May 1996, pp Brentner, K. S., and Farassat, F., An Analytical Comparison of the Acoustic Analogy and Kirchhoff Formulations for Moving Surfaces, AIAA Journal, Vol. 36, No. 8, Aug. 1998, pp Di Francescantonio, P., A New Boundary Integral Formulation for the Prediction of Sound Radiation, Journal of Sound and Vibration Vol. 202, No. 4, 1997, pp

22 22 Strawn, R. C., Ahmad, J., and Duque, E. P. N., Rotorcraft Aeroacoustics Calculations with Overset-Grid CFD Methods, Proceedings of the AHS 54th Annual Forum, Vol. I, Washington DC, May 1998,pp Amiet, R. K., Refraction of Sound by a Shear Layer, Journal of Sound and Vibration, Vol. 58, No. 4, 1978, pp Morfey, C. L. and Joseph, P. F., Shear Layer Refraction Corrections for Off-Axis Sources in a Jet Flow, Journal of Sound and Vibration, Vol, 239, No. 4, 2001, pp Morfey, C. L. and Szewczyk, V. M., Jet Noise Modelling by Geometric Acoustics, Part 1. Theory and Prediction Outside the Cone of Silence, ISVR Technical Report, No. 91, Sept Mani, R., Gliebe, P. R., and Balsa, T. F., High Velocity Jet Noise Source Location and Reduction, FAA-RD II, Goldstein, M. E., High Frequency Sound Emission from Moving Point Multipole Sources Embedded in Arbitrary Transversely Shear Mean Flows, Journal of Sound and Vibration, Vol. 80, 1982, pp Uzun, A., 3-D Large Eddy Simulation for Jet Aeroacoustics, Ph.D Dissertation, School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN, December Uzun, A., Lyrintzis, A. S., and Blaisdell, G. A., Coupling of Integral Acoustics Methods with LES for Jet Noise Prediction, AIAA Paper presented at the 42nd Aerospace Sciences Meeting and Exhibit, Reno, NV, Jan. 5-8, Visbal, M. R. and Gaitonde, D. V., Very High-order Spatially Implicit Schemes for Computational Acoustics on Curvilinear Meshes, Journal of Computational Acoustics, 2001, 9(4), Bogey, C. and Bailley, C., Three-dimensional Non-Reflective Boundary Conditions for Acoustic Simulations: Far Field Formulation and Validation Test Cases, Acta Acustica, 2002, 88(4), Colonius, T., Lele, S. K. ad Moin, P., Boundary Conditions for Direct Computation of Aerodynamic Sound Generation, AIAA Journal, 1993, 31(9), Bogey, C. and Bailly, C., LES of a High Reynolds, High Subsonic Jet: Effects of the Inflow Conditions on Flow and Noise, AIAA Paper , Lew, P., Uzun, A., Blaisdell, G.A. and Lyrintzis, A.S., Effects of Inflow Forcing on Jet Noise Using Large Eddy Simulation, AIAA Paper , Mollo-Christensen, E., Kolpin, M. A., and Martucelli, J. R., Experiments on Jet Flows and Jet Noise Far-field Spectra and Directivity Patterns, Journal of Fluid Mechanics, Vol. 18, 1964, pp Stromberg, J. L., McLaughlin, D. K., and Troutt, T. R., Flow Field and Acoustic Properties of a Mach Number 0.9 Jet at a Low Reynolds Number, Journal of Sound and Vibration, Vol. 72, No. 2, 1980, pp SAE ARP 876C: Gas Turbine Jet Exhaust Noise Prediction. Society of Automotive Engineers, November