The prediction of jet noise ground effects using an acoustic analogy and a tailored Green's function

Journal of Sound and Vibration 333 (2014) 1193–1207

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The prediction of jet noise ground effects using an acoustic analogy and a tailored Green's function Steven A.E. Miller n The National Aeronautics and Space Administration, Langley Research Center, Aeroacoustics Branch, 2 N. Dryden St. MS 461, Hampton, VA 23681, USA

a r t i c l e i n f o

abstract

Article history: Received 20 May 2013 Received in revised form 17 October 2013 Accepted 23 October 2013 Handling Editor: Y. Auregan Available online 18 November 2013

An assessment of an acoustic analogy for the mixing noise component of jet noise in the presence of an infinite surface is presented. The reflection of jet noise by the ground changes the distribution of acoustic energy and is characterized by constructive and destructive interference patterns. The equivalent sources are modeled based on the twopoint cross-correlation of the turbulent velocity fluctuations and a steady ReynoldsAveraged Navier–Stokes (RANS) solution. Propagation effects, due to reflection by the surface and refraction by the jet shear layer, are taken into account by calculating the vector Green's function of the linearized Euler equations (LEE). The vector Green's function of the LEE is written in relation to that of Lilley's equation; that is, it is approximated with matched asymptotic solutions and Green's function of the convective Helmholtz equation. The Green's function of the convective Helmholtz equation in the presence of an infinite flat plane with impedance is the Weyl–van der Pol equation. Predictions are compared with measurements from an unheated Mach 0.95 jet. Microphones are placed at various heights and distances from the nozzle exit in the peak jet noise direction above an acoustically hard and an asphalt surface. The predictions are shown to accurately capture jet noise ground effects that are characterized by constructive and destructive interference patterns in the mid- and far-field and capture overall trends in the near-field. Published by Elsevier Ltd.

1. Introduction An acoustic analogy is developed based on the Euler equations for the prediction of jet mixing noise in the vicinity of an arbitrary geometry surface. The analogy is based on the development by Morris and Farassat [1] but has unique differences. The Euler equations, written in terms of the logarithm of the pressure, are rearranged into a left-hand side operator consistent with the linearized Euler equations (LEE) and right-hand side equivalent sources. The far-field pressure is written in terms of an integral solution of the governing equations. It appears as a volumetric integral of the equivalent sources and the vector Green's function of the LEE. Any adjoint vector Green's function solver for the LEE can be used within the analogy. Equivalent source models are created for both the dilatation and unsteady force per unit volume source terms. The equivalent sources follow the models of Tam and Auriault [2] and Morris and Boluriaan [3]. The model is dependent on steady Reynolds-Averaged Navier–Stokes (RANS) solutions but could easily be based on an empirical flow-field or unsteady

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0022-460X/$ - see front matter Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.jsv.2013.10.028

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Nomenclature As Bs c cl cu cτ D E F f fi g gl K lx ly lz M Mj p Rp Rs r S TTR t u

constant associated with dilatation source constant associated with unsteady force per unit volume source speed of sound constant of integral length scale constant of integral velocity scale constant of integral time scale nozzle exit diameter variation of two-point cross-correlation of velocity fluctuations with space and time Rudwick boundary loss factor frequency unsteady force per unit volume associated with velocity fluctuations Green's function of the convective Helmholtz equation Green's function of Lilley's equation turbulent kinetic energy integral length scale in the streamwise direction integral length scale in the y direction integral length scale in the z direction Mach number fully expanded Mach number pressure reflection coefficient flow resistivity radial direction spectral density total temperature ratio time streamwise velocity component in the x direction

us v w x x y z zs γ δ ε η θ π n πg ρ τ τs ϕ Ψ Ω ω

integral scale of velocity radial velocity component in the y direction radial velocity component in the z direction or numerical distance observer position streamwise direction source position vector from source to observer surface impedance ratio of specific heats Dirac delta function dissipation rate of turbulent kinetic energy vector between two source locations ðξ; η; ζÞ dilatation rate or observer angle from nozzle downstream axis logarithm of the normalized pressure vector Green's function of the linearized Euler equations density retarded time integral time scale azimuthal angle observer angle from the nozzle inlet axis specific dissipation rate of turbulent kinetic energy radian frequency

Subscript j s 1

jet fully expanded quantity property of turbulence ambient value

simulation. This allows the nozzle pressure ratio, temperature ratio, and nozzle geometry to be connected directly to the aerodynamic solution and resultant jet noise. A model of the two-point cross-correlation of the unsteady velocity fluctuations within the jet connects the turbulence statistics to the noise prediction. It is based on measurement and the turbulence statistics generated by a two-equation closure of the steady RANS equations. The refraction of the sound through the mean flow, diffraction by surfaces, or reflection by the ground or airframe, are all handled by the solution of the vector Green's function of the LEE. For simplicity in this investigation, the vector Green's function of the LEE is written in terms of Lilley's [4] equation. An approximate solution of Lilley's equation is formed about Green's function of the convective Helmholtz equation. Analytic forms of the Green's function of the convective Helmholtz equation are used to account for sound reflection on an infinite flat surface which is representative of the ground. The impedance model of Delany and Bazley [5] is used to simulate the damping of the reflected acoustic waves, but can be neglected for an acoustically hard wall. The form of the Green's function of the convective Helmholtz equation is the Weyl– van der Pol [6] formula. The geometry chosen to exercise the model coincides with the experiment by Miles [7]. The methodology presented in this paper is developed to address many of the issues encountered by previous investigators. Experimental measurements conducted near reflecting surfaces pose unique difficulties. These are summarized well by the studies of Seiner et al. [8], Butzel [9], Schlinker et al. [10], Pao et al. [11], Huber and Sogeti [12], and many others, who performed experiments with microphones at or near various surface locations relative to a free jet as well as aircraft flyover experiments. With careful microphone placement and corrections, accurate measurements can be conducted. A large body of prediction methodologies have been developed to address reflection and impedance effects. In particular, Illston et al. [13] performed a combined theoretical and experimental study on jet noise, erosion, and ground effects. Their methodology used a Harper-Bourne [14] like prediction approach by integrating the source strength per unit length along the jet centerline axis. The Aircraft Noise Prediction Program (ANOPP) (see Zorumski [15] for details) accounts for ground

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effects from various sources such as the fan, jet, and airframe. ANOPP uses ground reflection corrections described by Chien and Soroka [16] and the impedance model by Delany and Bazley [5]. Finally, McLaughlin et al. [17] performed a series of experiments and predictions using supersonic jets with and without a ground plane. They created a model to describe the source distribution within the jet plume and predictions showed that the interference patterns were correctly predicted at high frequencies. In the following sections, the model development is presented starting with the governing equations. An overview of the implementation of the method in a computer program, its connection to a steady RANS solver, a description of the model calibration, and method to find the steady RANS solution are shown. Results of the predictions are compared with the experiment of Miles [7]. 2. Mathematical model The acoustic analogy presented herein is based on that of Morris and Farassat [1]. The governing equations are the Euler equations, Dπ ∂ui þ ¼ 0; Dt ∂xi

(1)

Dui ∂π þ c2 ¼ 0; ∂xi Dt

(2)

where D=Dt is the material derivative, c is the speed of sound, u is the velocity, and π ¼ γ  1 lnðp=p1 Þ. By linearizing the governing equations about time-averaged values and retaining terms on the left-hand side that are linear in the fluctuations then the inhomogeneous LEE are formed: ∂π′ ∂π′ ∂u′i þ uj þ ¼ θ; ∂t ∂xj ∂xi

(3)

∂u′i ∂u′ ∂u ∂π′ þ u j i þu′j i þ c2 ¼ f i; ∂xi ∂t ∂xj ∂xj

(4)

∂u i ∂π ∂π ∂π′  uj  u′j  u′j ∂xj ∂xj ∂xj ∂xi

(5)

and

where θ¼  is a dilatation rate and f i ¼  c2

∂u′ ∂π ∂π ∂π′ ∂u  c2′  c2′  u j i  u′j i ∂xi ∂xi ∂xi ∂xj ∂xj

(6)

is an unsteady force per unit mass. In both the equivalent sources only the second-order fluctuations are considered to contribute to the sound field. Note that π′C γ  1 lnð1 þ p′=p1 Þ C γ  1 p′=p1 . The vector Green's function of the LEE is defined as Do π ng Dt

þ

∂ungi

¼ δðx yÞδðt  τÞδ0n ;

∂xi

(7)

and Do ungi Dt

þ ungj

∂π ng ∂u i þ c2 ¼ δðx  yÞδðt  τÞδin ; ∂xj ∂xi

where Do =Dt ¼ ∂=∂t þ u j ∂=∂xj . The periodic vector Green's function is Z 1 π ng ðxjy; ωÞexp½iωðt τÞ dω; π ng ðx; tjy; τÞ ¼

(8)

(9)

1

and, π ng ðxjy; ωÞ ¼

1 2π

Z

1 1

  π ng x; t y; τÞexp½ iωðt  τÞ dτ:

(10)

The fluctuating far-field pressure is a convolution integral of the vector Green's function and the equivalent sources Z 1 Z 1 3 ⋯ π 0g ðx; tjy; τÞθðy; τÞ þ ∑ π ng ðx; tjy; τÞf n ðy; τÞ dτ dy: (11) p′ðx; tÞ ¼ γp1 1

1

n¼1

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The spectral density is defined as the Fourier transform of the autocorrelation of the fluctuating pressure Z 1 Sðx; ωÞ ¼ p′ðx; tÞp′ðx; t þ τÞ exp½  iωτ dτ:

(12)

1

Using Eqs. (11) and (12) yields

Z

Sðx; ωÞ ¼ ρ21 c41 3

þ ∑

1 1

Z ⋯

1 1

π 0g ðx; y;  ωÞπ 0g ðx; y þ η; ωÞθðy; τÞθðy þ η; t þτÞ

3

∑ π ng ðx; y;  ωÞπ m g ðx; y þη; ωÞf n ðy; τÞf m ðy þη; t þ τÞ  exp½  iωτ dτ dη dy:

(13)

n¼1m¼1

By definition the periodic vector Green's function of the LEE π ng n ðx; y; ωÞ ¼ π ng ðx; y;  ωÞ;

(14)

and following Tam and Auriault [2], Green's function for two closely placed source points in the jet are related by   iωξ cos θ : π ng ðx; y þ η; ωÞ C π ng ðx; y; ωÞ exp c1

(15)

Using Eqs. (14) and (15) with Eq. (13) yields  Z 1  Z 1 3 3 Sðx; ωÞ ¼ ρ21 c41 ⋯ π ng0 ðx; y; ωÞπ 0g ðx; y; ωÞθðy; τÞθðy þ η; t þ τÞ þ ∑ ∑ π ngn ðx; y; ωÞπ m g ðx; y; ωÞf n ðy; τÞf m ðy þ η; t þ τÞ 1 1 n¼1m¼1    iωξ cos θ  exp½ iωτ dτ dη dy: (16) exp c1 We have now obtained a general formulation that involves the two-point cross-correlation of the equivalent sources and the vector Green's function of the LEE. A model is required for the two-point cross-correlation of the equivalent sources and the vector Green's function of the LEE. The dilatation rate source is modeled following Tam and Auriault [2] and written in the form of Morris and Boluriaan [3], θðy; τÞθðy þ η; t þτÞ ¼ A2s

ðus =c1 Þ4 Eðη; τÞ; τ2s

(17)

where As is a constant. The two-point cross-correlation of the unsteady force per unit volume is modeled following Morris and Boluriaan [3], f n ðy; τÞf m ðy þ η; t þ τÞ ¼ B2s

ðus =c1 Þ2 u4s 2

lx

Eðη; τÞ;

(18)

where Bs is a constant. A model must be formed for the space–time decay function Eðη; τÞ. Following Ribner's [18] postulate, the spatial and temporal terms of Eðη; τÞ are separable, " # " # " #   jτj ðξ  uτÞ2 ðη  vτÞ2 ðζ wτÞ2 Eðη; τÞ ¼ exp exp  exp exp : (19) 2 2 2 τs l l l x

y

z

Eqs. (17) through (19) are used in Eq. (16) and integrals involving the variables of integration τ, η, ξ, and ζ are performed analytically. Assumptions required for analytical integration involving η and τ require τs 4 0, li 4 0, c1 4 0, x 40, and for these quantities to be real. These assumptions are reasonable as long as the nozzle exit is at x ¼0 and in the y–z plane. Also, the turbulent kinetic energy at the boundary of the nozzle is zero due to the no-slip condition, which is enforced by modern steady RANS solvers. These integrations result in ( Z 1 Z 1 Z 1 2π 3=2 c21 lx ly lz τs x2 ðus =c1 Þ4 2 4 Sðx; ωÞ ¼ ρ1 c1  π gn0 ðx; y; ωÞπ 0g ðx; y; ωÞA2s 2 2 2 2 2 τ2s  1  1  1 c1 x þ ðux1 þvx2 þ wx3 þc1 xÞ τ s ω 3 2 2 ) 2 2 2 2 2 2 2 4 lx x1 þ ly x2 þ lz x3 ω 3 3 2 ðus =c1 Þ us 5 dy: (20) þ ∑ ∑ π ngn ðx; y; ωÞπ m  exp4  g ðx; y; ωÞBs 2 4c21 x2 lx n¼1m¼1 Additional details regarding the development of Eq. (20) are discussed in Miller [19]. The integral scales of turbulence in Eq. (20), lx, ly, lz, us, and τs, are required to make a prediction. These could be found by simple empirical models or unsteady CFD simulations. Here, they are related to a steady RANS solution by lx ðyÞ ¼ cl KðyÞ3=2 =εðyÞ;

(21)

τs ðyÞ ¼ cτ KðyÞ=εðyÞ;

(22)

and, us ðyÞ ¼ cu

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KðyÞ=3:

(23)

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The coefficients are cτ ¼ 0:30, cu ¼ 1:00, and cl ¼1.00. These coefficients are based on a reference jet operating at the sonic condition, TTR¼3.20, and a 0.0508 m convergent nozzle at the sideline location of R=D ¼ 100. They have been calibrated by the same methodology of Tam and Auriault [2]. Their variation from Tam and Auriault is attributed to the use of a different acoustic analogy and steady RANS solver. The cross-stream integral length scales, ly and lz, are set to 3/10 of lx, which corresponds to experimental observation. These coefficients are never altered irrespective of the nozzle Mach number, diameter, temperature, nozzle geometry, or any other parameter. n A solution for πg is now required. One method suitable for calculating the vector Green's function of the LEE is by finding Green's function of Lilley's [4] equation. For a parallel axisymmetric mean flow Green's function of the LEE can be written in terms of Green's function of Lilley's equation. Morris and Boluriaan [3] show that the vector Green's function of the LEE is related to Green's function of Lilley's [4] equation by π 0g ðx; y; ωÞ ¼ ω2 g l ðx; y; ωÞ  2iuω

∂g l ðx; y; ωÞ ∂2 g l ðx; y; ωÞ  u2 ; ∂yx ∂y2x

(24)

 ∂ ∂ π 1g ðx; y; ωÞ ¼  iω þ u g ðx; y; ωÞ; ∂yx ∂yx l

(25)

   ∂u ∂ ∂ ∂ π 2g ðx; y; ωÞ ¼  3  iωþ u g ðx; y; ωÞ; ∂yr ∂yx ∂yx ∂yr l

(26)

 ∂ 1 ∂ π 3g ðx; y; ωÞ ¼  iωþ u g ðx; y; ωÞ; ∂yx yr yθ l

(27)

where the subscript of the source vector y denotes the direction in which the partial derivative is taken. The problem is now reduced to finding Green's function of Lilley's equation instead of the full vector Green's function of the LEE. Morris and Miller [20] used a similar technique and found an analytic solution of the vector Green's function of the LEE related to Green's function of the Helmholtz equation (see Eq. (23) of Morris and Miller). Tam and Auriault [21] and Raizada [22] also used this technique with success. These methods used an adjoint approach and are relatively computationally inexpensive. However, these methods do not include a scattering surface such as the airframe or ground, thus analytic or numerical approximations of Lilley's equation that have the potential to include a scattering surface are sought. Analytical solutions of Lilley's [4] equation can be found but only for very special mean flows. These are instructive but not helpful for more realistic jets. An asymptotic solution of Lilley's equation for low frequencies was found by Goldstein [23–25] and for high frequencies by Balsa et al. [26]. By examination of the asymptotic solutions, Green's function of the convective Helmholtz equation, g, and a matching function, an approximate Green's function of Lilley's equation is created,  

i c1  iω  g l ðx; y; ωÞ ¼ 2 cα exp x cos θ þ r sin θ cos ðϕ0 ϕÞ qe g; (28) c1 c1 c0 ω c where c is the local speed of sound, cβ ¼ 1  10  4 , qe ¼ exp½ cβ ω=c1  for Re½g no  r0 and is otherwise unity, and, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1  ðu=c1 Þ cos θÞ2  cos 2 θ: g no ¼ c 2 =c21

(29)

c0 is Ribner's convection coefficient, c0 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1  u=c1 cos θÞ2 þ c20α ðu=c1 Þ2 ;

(30)



where c0α ¼ ð4=25Þ uj =c1 1 for uj =c1 41 and is otherwise zero. The convective amplification power coefficient is cα ¼ 3 tanh½fDj =uj :

(31)

Eq. (28) is compared with the asymptotic solutions of Balsa et al. [26] and Goldstein [23–25] in the Appendix for various positions and emission angles. The benefit of this approach is existing numerical solvers that can provide the Green's function of the convective Helmholtz equation. For example, the Fast Scattering Code (FSC) by Tinetti and Dunn [27], any Boundary Element Method (BEM) (see an overview by Katsikadelis [28]), or FastBEM by Liu [29]. This permits the opportunity of finding the jet noise scattering from any surface as long as the aerodynamic noise sources are minimally affected by the scattering geometry. For free jet predictions Green's function of the Helmholtz equation is used, g ¼ exp½ikz=ð4πzÞ. If an infinite plane exists parallel to the jet centerline then the tailored Green's function of the Helmholtz equation is represented by the Weyl–van der Pol [6] formula, g¼



 exp½ikR2  exp½ikz  þ Rp þ 1 Rp F ðwÞ ; 4πz 4πR2

(32)

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where R2 is the distance from an equivalent source point on the opposite side of the infinite plane to the observer and Rp is the reflection coefficient, Rp ¼

Z s ðf Þ cos θ  ρ1 c1 ; Z s ðf Þ cos θ þ ρ1 c1

(33)

and F(w) is the Rudwick boundary loss factor,   pffiffiffi FðwÞ ¼ 1 þ i π w exp w2 erfc½  iw;

(34)

where erfc is the complementary error function. We numerically evaluate erfc using the identity erfc½  iw ¼ 1  erf½  iw. w is the numerical distance, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 (35) w ¼ ð1 þ iÞ 2πfR2 =c1 ½ cos θ þðρ1 c1 Þ  1 : 2 There are many models for the surface impedance Zs. Following ANOPP, [15] one appropriate choice is the model of Delany and Bazley [5], Z s ¼ ρ1 c1 ½1 þ 0:0571Xðf Þ  0:754 þ i0:87Xðf Þ  0:732 ;

(36)

where Xðf Þ ¼ ρ1 f =Rs . Rs is the flow resistivity. The asymptotic solution of Zs for Rs approaching infinity yields the hard-wall solution. For additional information regarding the development of Eqs. (32) through (36), examine Weyl–van der Pol [6], Attenborough and Ver [30], and Delany and Bazley [5]. 3. Results The developed mathematical models are now assessed by comparing the resulting predictions with the experimental measurements of Miles [7]. Here, a convergent nozzle with a 0.3302 m (13 in.) exit diameter was placed in an open environment with the jet centerline axis 3.89 m (12.75 ft) above the ground. The nozzle centerline axis was parallel to the ground. The ground was described as asphalt and has a flow resistivity of Rs ¼27,000 Pa s/m2, which is the average flow resistively described by Attenborough et al. [30]. The ambient environment had a temperature of 288.9 K and was described as dry. Corrections have been made for atmospheric absorption. Microphones were placed relative to the nozzle exit at an angle ψ ¼ 145 degrees from the upstream axis. Three sets of microphones were located at this observer angle at ground shadow distances (distance from a point on the ground under the nozzle measured to a point under the microphone) of 7.62, 15.24, and 22.86 m. At each of these observer distances, microphones were placed at 0.0061, 1.52, and 3.89 m above the ground. Lossless sound pressure level spectra were calculated in 1/3 octave bands from 50 to 20,000 Hz. The coordinate system shown in Fig. 1 illustrates the relative positions of the microphones with respect to the nozzle. The nozzle is shown on the left side of the figure and the jet plume develops from the left to right direction along the positive x-axis. The y-axis is normal to the ground and the z-axis is parallel to the ground in the sideline observer direction. The height of the nozzle is specified by the distance along a normal from the ground plane to the center of the nozzle exit. The observer angle, ψ, is the angle from the upstream nozzle centerline axis to the observer. The azimuthal angle, ϕ, is measured about the x-axis originating from the x–y plane. Values of x are specified to define the microphone position relative to the nozzle exit along with the microphone height.

Fig. 1. The coordinate system showing the nozzle, the ground plane, and the observer. The positive y-axis points away from the ground below the nozzle. The origin of the coordinate system resides at the center of the nozzle exit.

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3.1. The steady RANS solution The developed prediction model requires the knowledge of a mean flow. The steady RANS solution is found with the Fully Unstructured Navier–Stokes (FUN3D) solver from The NASA Langley Research Center. For more information on FUN3D see Anderson and Bonhaus [31] or Nielsen [32]. The steady RANS equations are closed with the Menter [33] Shear Stress Transport (SST) turbulence model. Because the nozzle is axisymmetric and the ground is not specified as a boundary condition in the CFD calculation, the resultant steady flow-field is axisymmetric. The flow-field within the first quadrant of the y–z plane is obtained with the CFD calculation. After the calculation is complete, only the x–y plane is retained. The computer program that evaluates the prediction model rotates the steady RANS solution about the centerline axis when performing azimuthal integration. The integration range extends from 0:001 r x=D r 20 and 0:001 r r=D r 4 with 400 grid points in the streamwise direction and 250 grid points in the radial direction. For the jet conditions examined in this study, doubling the range or number of grid points in either direction has no significant effect on the predicted noise intensity. By using this approach, the evaluation of Eq. (20) is restricted to the jet plume where the aerodynamic source of noise is located. No additional processing of the steady RANS solution is performed. The methodology is compatible with other steady RANS solvers and has been demonstrated with the NPARC Alliance Wind-US solver (see Nelson and Power [34] for details). The computational domain is unstructured and contains a region inside the nozzle, a region outside the nozzle upstream of the nozzle exit, and a region for the plume of the jet. The plume region extends 100 nozzle diameters downstream from the nozzle exit and 50 nozzle diameters in the radial direction from the jet centerline. A complete description of the methodology to construct the computational domains used for these types of simulations is found in Miller and Veltin [35]. The nozzle geometry included in the steady RANS simulation is based on the SMC000 nozzle of NASA Glenn Research Center. See Bridges et al. [36] for information on the development of the SMC series of nozzles. The nozzle contour is shown in Fig. 2 along with a portion of the computational domain in the jet near-field. The number of grid points in the domain is 205,251 and the number of elements is 204,000. Boundary conditions are chosen to replicate the experiment of Miles [7]. The 0.3302 m diameter convergent nozzle operates with a fully expanded Mach number of 0.95 and total temperature ratio of 1.00. This results in a fully expanded exit velocity of 300.01 m/s, fully expanded static temperature of 248.20 K, fully expanded speed of sound of 315.80 m/s, and fully expanded density of 1.42 kg/m3. The characteristic frequency of the jet is 913.6 Hz and the Reynolds number based on the fully expanded jet diameter is 8:944  106 . A portion of the jet flow-field predicted by FUN3D is shown in Fig. 3. Contours of 2.5

y/D

2 1.5 1 0.5 0 -5

-4.5

-4

-3.5

-3

-2.5

-2 x/D

-1.5

-1

-0.5

0

0.5

Fig. 2. A slice of the computational domain showing the convergent contoured nozzle. The x- and y-axes have been normalized by the nozzle exit diameter. The nozzle geometry corresponds to the NASA Glenn Research Center SMC000 nozzle. 6 4

y/D

2 0 -2 -4 -6

0

2

4

6

8

10

12

14

16

18

20

22

x/D Fig. 3. Contour maps of the steady RANS solution closed by the Menter [33] SST turbulence model. The coordinates are normalized by the nozzle diameter. In the top half plane contours of the streamwise velocity component, u m/s, are shown. In the bottom half plane contours of turbulent kinetic energy, TKE (m2/s2), are shown.

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the streamwise velocity component are shown in the top half plane and contours of turbulent kinetic energy are shown in the lower half plane. The spatially independent variables are normalized by the nozzle exit diameter. Doubling the number of grid points in the streamwise and cross-stream directions yields the same solution. Global residuals of the final CFD solution show reductions of over eight orders of magnitude. The ground alters the mean flow and turbulent statistics of the jet plume. The influence that the ground has on the jet plume's development is dependent on the distance from the jet centerline to the ground. If the jet is closer to the ground then the mean flow and turbulent statistics will be more-so affected. Smith and Miller [37] conducted a numerical investigation to

u (m/s) 1.5

0

50

100

150

200

300

350

TKE

u

1

250

y/D

0.5

Free Jet yp/D = 10 yp/D = 08 yp/D = 06 yp/D = 04 yp/D = 02

0

-0.5

-1

-1.5

0

500

1000

1500

2000

2500

TKE (m2/s2) u (m/s) 1.5

0

50

100

150

250

300

350

TKE

u

1

200

y/D

0.5

Free Jet yp/D = 10 yp/D = 08 yp/D = 06 yp/D = 04 yp/D = 02

0

-0.5

-1

-1.5

0

500

1000

1500

2000

2500

TKE (m2/s2) Fig. 4. Comparison of various streamwise velocity component and turbulent kinetic energy radial profiles at (a) x=D ¼ 10 and (b) x=D ¼ 16. The jet operates at Mj ¼0.985 and TTR¼ 1.00. The nozzle is convergent with a 0.0508 m exit diameter. The nozzle centerline is yp =D diameters from a large semi-infinite flat plate.

S.A.E. Miller / Journal of Sound and Vibration 333 (2014) 1193–1207

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100

ANOPP Prediction

SPL (dB re 20 μ Pa)

90

80

70

60

102

103 f (Hz)

104

Fig. 5. Free-field predictions of the SAE ARP876 [38] method contained within the Aircraft Noise Prediction Program (ANOPP) and the developed acoustic analogy. The jet conditions are M j ¼ 0:95, TTR ¼1.00, D ¼0.3302 m, R=D ¼ 100, ψ ¼ 1451 (inlet angle), and azimuthal angle ϕ ¼ 901.

assess the effects of the flight vehicle airframe on the jet plume, noise source, and far-field noise. Here, we show one set of numerical results from Smith and Miller to demonstrate that the jet flow within the experiment of Miles [7] is relatively unaffected by the ground below. Fig. 4 shows comparisons of the streamwise velocity component and turbulent kinetic energy radial profiles at (a) x=D ¼ 10 and (b) x=D ¼ 16. The jet operates at Mj ¼0.985 and TTR¼1.00 (which is very similar to the experiment of Miles [7]). The nozzle is convergent with a 0.0508 m exit diameter. The nozzle centerline is placed at yp =D 2, 4, 6, 8, and 10 diameters from a large semi-infinite flat plate and the edge is 20 diameters downstream of the nozzle exit. At both x=D locations there is less than a percent change in turbulent kinetic energy between the free jet solution and the solutions where the plate is greater than 6 diameters away. The variation of u at plate distances greater than 6 jet diameters from the plate shows a similar agreement. Noticeable differences of the mean flow and turbulent statistics are only present when the jet centerline axis is very close to the plate. In the measurements of Miles [7] the distance from the jet centerline to the ground is 11.18 nozzle diameters. It is unlikely that the mean flow or turbulent statistics in the experiment are altered by the ground relative to the free jet within the dominant sound producing region. 3.2. Aeroacoustic predictions A free-field calculation is required to validate the prediction method. Since experimental data are not available for this jet operating condition and nozzle diameter in the free-field, the result of the developed acoustic analogy is compared with the SAE ARP876 [38] methodology within NASA's ANOPP [15]. The results are presented as lossless Sound Pressure Level (SPL) versus one-third octave band center frequency. At the observer angle, ψ ¼ 1451, measured from the upstream axis of the jet, the agreement between the prediction and SAE ARP876 is satisfactory at all frequencies as shown in Fig. 5. Predictions of jet noise about the ground plane are now presented. For these comparisons, the nozzle height is 3.89 m qffiffiffiffiffiffiffiffiffiffiffiffiffiffi and the observer angle is ψ ¼ 1451. The ground shadow distance is defined as x21 þx23 and is representative of the distance between the nozzle exit and microphone along the ground. The ground shadow distance between the nozzle exit and microphone is R=D ¼ 69:23. Fig. 6(a) shows the prediction of the jet noise compared with the experiment of Miles [7]. In this comparison the prediction of the jet noise above an acoustically hard surface (ideal) is denoted by a line with circles, above an asphalt surface (impedance) as a line with deltas, without the ground surface (free jet) as a line with diamonds, and the measurement is denoted by a line with squares. The y-axis is sound pressure level (SPL) in one-third octave bands and the x-axis is frequency. All following figures use this convention. The ground reflections create destructive and constructive interference at various frequencies. For example, at 80 Hz the magnitude of the predicted and measured constructive interference agree well. An example of destructive interference occurs at 125 Hz. At frequencies higher than 2500 Hz, the ideal prediction and the measurement agree very well. The predicted impedance noise intensity under-predicts the measurement and is less than a dB higher than the free jet prediction in this range. At low frequencies, constructive interference is observed both in the predictions and measurement. None of the predictions match the magnitude of the measurement but the hard surface prediction matches the peak frequency. It is instructive to compare the hard surface

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Fig. 6. Ground shadow distance R=D ¼ 69:2308 and microphone heights (a) 3.89 m, (b) 1.52 m, and (c) 0.0061 m (ground).

prediction and free jet prediction with the measurement. At 300 Hz the peak noise intensity occurs and both the ideal prediction and measurement are 6 dB higher than the free jet prediction. This doubling of noise intensity is predicted by more traditional approaches where a point source is placed above and below the ground plane. At higher frequencies there is between a 4 and 6 dB increase of noise intensity beyond the free jet prediction. A second comparison is shown in Fig. 6(b) at R=D ¼ 69:23 but the microphone height is lowered to 1.59 m. As the microphone is lowered towards the ground the constructive and destructive interference pattern increases in frequency. This trend matches the analytic solution of the Helmholtz equation with a point source above an infinite flat surface. Both predictions have matching constructive and destructive interference patterns; however, these are not aligned with experiment. For example, the first measured destructive interference frequency is near 300 Hz but both predictions are located near 175–200 Hz. In addition, the amount of constructive interference in the prediction under-predicts that of experiment. The overall error between the hard-wall prediction and measurement is approximately 2–4 dB at frequencies higher than 1000 Hz and has considerable error approaching 10 dB at lower frequencies. This is consistent with the impedance ground surface prediction that is generally 2 dB lower over the same frequency ranges. When the observer is located near the ground, predictions become more challenging. The observer height is set at a height of 0.0061 m (as reported by Miles). The shadow distance between the microphone and nozzle remains at R=D ¼ 69:23. A comparison between the predictions and experiment is shown in Fig. 6(c). The predictions show much more interference relative to experiment, which is probably due to the fact that microphones do not measure pressure at a point but over a small volume of space, and that no air gap exists between the microphone and the ground. The interference pattern is strongly affected by the microphone location, especially when located so close to the ground surface. At low frequencies the prediction shows an extra constructive and destructive pattern relative to the experiment. This is unlike Fig. 6(a), where the measurement shows this constructive low frequency peak. The hard-wall prediction and measurement have approximately the same peak frequency near 250 Hz and the same peak intensity. The impedance prediction has a higher peak frequency and is lower in intensity by 2 dB relative to measurement. Measurement and the hard wall prediction have a lower peak frequency than the free jet prediction. Above 1000 Hz, the hard wall prediction and measurement are 4 and 6 dB higher than the free jet prediction respectively. At some frequencies below 300 Hz there is more than a 6 dB

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Fig. 7. Ground shadow distance R=D ¼ 46:1538 and microphone heights (a) 3.89 m, (b) 1.52 m, and (c) 0.0061 m (ground).

increase between the measurement and free jet prediction. At the ground plane the hard wall prediction agrees well with the measurement. Overall, in the far field at R=D ¼ 69:23 and at heights high above the ground, the ground effects have been captured by the prediction method. The ground shadow distance from the nozzle exit to the microphone is set at R=D ¼ 46:15 and the microphone heights are set at 3.69, 1.52, and 0.0061 m. Comparisons between predictions and experiment are shown in Fig. 7. At the height of 3.69 m in Fig. 7(a), it is shown that the prediction with the rigid wall matches the measurement within 2 dB above 100 Hz. The impedance prediction is lower than the rigid wall prediction and measurement at almost all frequencies by up to 4 dB. At frequencies larger than the peak frequency, the hard wall prediction and measurement are consistently between 2 and 4 dB higher than the free jet prediction. Only at the first constructive peak do we observe a 6 dB increase of the hard wall prediction over the free jet prediction. Interestingly, the impedance prediction is only slightly more intense at high frequencies and up to 2 dB lower at low frequencies than the free jet prediction. Fig. 7(b) shows comparisons for the 1.52 m high microphone. Compared to Fig. 6(b) the predictions are in better agreement with measurement. These comparisons show the same trends as the 3.69 m case but are approximately 2 dB lower overall relative to measurement at mid to high frequencies. At higher frequencies the prediction is typically 0–6 dB lower than experiment. The impedance prediction is almost identical to the free jet prediction at high frequencies but follows the hard wall prediction at low frequencies. At the ground microphone position the predictions do not compare favorably relative to previous cases. The peak frequencies of both predictions are higher than measurement by approximately a few hundred Hz and 2–6 dB lower. Once again a constructive interference pattern in the prediction is seen near 30 Hz. The hard wall prediction and measurement have a lower peak frequency than the impedance and free jet predictions. The predicted peak intensity of the hard wall prediction is 2 dB lower than measurement. Measurement is consistently between 4 and 7 dB higher than the free jet prediction. For the final predictions the microphone positions are moved well into the near-field of the jet. Near-field acoustic predictions from jet flows are often very difficult. The shadow distance from the nozzle exit to the microphones is set at R=D ¼ 23:08. Fig. 8(a) shows the predictions when the microphone height is at 3.89 m. The peak magnitude of the

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Fig. 8. Ground shadow distance R=D ¼ 23:0769 and microphone heights (a) 3.89 m, (b) 1.52 m, and (c) 0.0061 m (ground).

impedance based model agrees very well with experiment and the hard-wall acoustic model is over-predicted by 1 dB. The peak frequency is approximately 450 Hz. The lowest frequency of the experiment is 40 Hz and is too high to observe any destructive interference. Unfortunately, the high frequency fall-off of the predictions are larger than the measurement. At this near-field position the hard wall prediction noise intensity is at most 2 dB higher than the free jet prediction and the impedance prediction almost entirely coincides with the free jet prediction. A second near-field case is shown in Fig. 8(b) where predictions are compared with experiment at R=D ¼ 23:08 m and a height of 1.52 m. The predictions are nearly identical except for the slight attenuation due to the ground. Like the previous case the hard wall prediction is slightly more intense than the free jet and impedance predictions at mid to high frequencies. Both predictions that contain ground effects over-predict the lower frequencies are within 1 dB of the peak of the experiment, and greatly under-predict the energy above 900 Hz. In the experiment there is very little apparent constructive or destructive interference. A higher fidelity convective wave equation Green's function solver could potentially correct the predictions in terms of the increased fall-off at higher frequencies and also correct for the extra interference at low frequencies. A true near-field jet noise prediction method would also be required. A final prediction is shown in Fig. 8(c). The microphone height is 0.0061 m and the shadow distance is R/D¼ 23.08 m. This is a difficult position to make an accurate prediction as the observer is on the ground and is within the near-field of the jet. The peak frequency of the hard-wall prediction is 100 Hz higher than the experiment and the impedance peak frequency is 200 Hz lower than the experiment. The attenuation between the hard-wall prediction and the impedance prediction is much larger than previously seen. Unfortunately, the prediction using a ground surface with realistic impedance greatly under-predicts the measurement. The impedance prediction is approximately 2 dB higher than the free jet prediction at most frequencies. Near the ground the ideal prediction is 4–6 dB higher than the free jet prediction at mid to high frequencies. It should be noted how extremely sensitive to the microphone position and ground impedance the predictions are. Even a slight change of microphone height of 0.001 m near the ground can greatly influence the constructivedestructive interference patterns. Small changes in the impedance can also greatly change the predicted values. In the nearfield of the jet, roughness or small irregularities on the ground surface can greatly change the experimental result. This causes obvious difficulty when predictions are assumed about a perfectly flat surface and are amplified in the near-field of

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the jet. Overall, the predictions are satisfactory considering the sensitivity of the problem, measurement, and conducted in the near-field at a jet emission angle that is relatively less understood than the sideline. 4. Conclusion An acoustic analogy is formulated and the arguments are the vector Green's function of the LEE and equivalent sources. The latter involves the two-point cross-correlation of the turbulent velocity fluctuations and scaling terms based on a steady RANS solution. The vector Green's function of the LEE is written in relation to that of Lilley's [4] equation, which is approximated with matched asymptotic solutions about Green's function of the convective Helmholtz equation. Green's function of the Helmholtz equation is found analytically for the infinite flat plane. Large savings have been made in terms of computational cost with a minimal loss of accuracy relative to finding the full solution of the vector Green's function of the LEE. The new model has been exercised and compared with the experiment of Miles [7]. The predictions are shown to capture jet noise ground effects that are characterized by constructive and destructive interference patterns in the mid- and far-field and capture overall trends in the near-field. However, there are discrepancies with respect to measurement of the magnitudes and frequencies of select constructive and destructive interference patterns. The choice to study ground effects is due to the availability of experimental data and the ease of finding the appropriate Green's function. This model formulation allows a multitude of well-known Green's function solvers to account for scattering of jet mixing noise as long as the plume is not highly influenced by the scattering surface. The problem has been greatly simplified by relating the vector Green's function of the LEE to Green's function of the convective Helmholtz equation while still retaining the benefits of Lilley's equation that accounts for jet shear layer refraction and amplification.

Acknowledgments This work is inspired by the creative ideas of Prof. Feri Farassat and Dr. Marvin E. Goldstein of the National Aeronautics and Space Administration (NASA), Boeing/A.D. Welliver Prof. Philip J. Morris, and Robert O. Lawton Distinguished Prof. C.K.W. Tam. Dr. Jeffrey H. Miles of the NASA Glenn Research Center at Lewis Field is gratefully acknowledged for the use of his experimental data. The author is grateful for continuous support from the NASA Fundamental Aeronautics Program High Speed Project. Appendix A. Comparisons of the approximate Green's function Comparisons of Eq. (28) are shown relative to the asymptotic solutions of Lilley's equation developed by Balsa et al. [26] and Goldstein [23–25]. In this example the jet operates at the sonic condition and has a total temperature ratio of one. The nozzle is convergent with a 0.0127 m exit diameter. The mean flow quantities of the jet are created with the method developed by Lau et al. [39]. Comparisons of Green's function will be shown at three locations relative to the nozzle exit. Fig. A1 shows cross-stream variation of the streamwise velocity component, u, and the static temperature, T, at streamwise locations, x/D¼ 1, 4, and 8. 600 u [m/s] @ x/D = 1 u [m/s] @ x/D = 4 u [m/s] @ x/D = 8 T [K] @ x/D = 1 T [K] @ x/D = 4 T [K] @ x/D = 8

500

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Fig. A1. The cross-stream variation of the streamwise velocity component, u, and the static temperature, T, at various streamwise locations, x=D. The jet operates at Mj ¼1.00 and TTR ¼ 2.20. The nozzle is convergent with exit diameter, D¼ 0.0127 m.

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gl gl*/ Max[ g lg l* ]

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St Fig. A2. The approximate Green's function of Lilley's equation compared with the low- and high-frequency asymptotic solutions. Comparisons are shown at three locations over a broad frequency range. Observers are located 100 jet diameters from the nozzle exit at 301, 451, and 901 from the jet downstream axis. The jet operates at Mj ¼ 1.00 and TTR ¼2.20. The nozzle is convergent with exit diameter, D ¼0.0127 m.

Fig. A2 shows g l g nl normalized by its magnitude versus non-dimensional frequency. Three comparisons are shown. The solid line is Eq. (28) and the dashed lines at low and high frequencies are asymptotic solutions. The locations within the jet are x=D ¼ 1 and r=D ¼ 1=2, x=D ¼ 4 and r=D ¼ 1=4, and x=D ¼ 8 and r=D ¼ 1=4. Observers are located 100 nozzle diameters from the nozzle exit at angles of 901, 451, and 301 relative to the downstream axis. At low frequencies the approximation relative to the asymptotic solution agrees favorably for different locations within the jet plume and for different emission angles. The approximate solution relative to the high-frequency asymptotic solution has the same slope but can be up to a single order of magnitude in error. Larger error in magnitude typically occurs at higher frequencies. References [1] P.J. Morris, F. Farassat, Acoustic analogy and alternative theories for jet noise prediction, AIAA Journal 40 (4) (2002) 671–680, http://dx.doi.org/10.2514/ 2.1699. [2] C.K.W. Tam, L. Auriault, Jet mixing noise from fine-scale turbulence, AIAA Journal 37 (2) (1999) 145–153, http://dx.doi.org/10.2514/2.691. [3] P.J. Morris, S. Boluriaan, The prediction of jet noise from cfd data, 10th AIAA/CEAS Aeroacoustics Conference, Manchester, Great Britain, May 10–12, AIAA Paper 2004-2977, 2004, http://dx.doi.org/10.2514/6.2004-2977. [4] G. M. Lilley, On the noise from jets, AGARD Conference Proceedings in Noise Mechanisms, Vol. 13, 1974, pp. 1–11. [5] M.E. Delany, E.N. Bazley, Acoustical properties of fibrous absorbent materials, Applied Acoustics 3 (1970) 105–116. [6] A. Banos, Dipole Radiation in the Presence of a Conducting Half-space, Pergamon Press, Oxford, New York, 1996. [7] J.H. Miles, Analysis of ground reflection of jet noise obtained with various microphone arrays over an asphalt surface, NASA Technical Memorandum X-71696, 1975. [8] J. Seiner, B. Jansen, L. Ukeiley, Acoustic fly-over studies of f/a-18 e/f aircraft during fclp mission, 9th AIAA/CEAS Aeroacoustics Conference, Hilton Head, S.C., May 12–14, AIAA Paper 2003-3330, 2003, http://dx.doi.org/10.2514/6.2003-3330. [9] L.M. Butzel, Prediction of jet exhaust noise on airframe surfaces during low-speed flight, Journal of Aircraft 19 (12) (1982) 1038–1044, http://dx.doi.org/ 10.2514/3.44809. [10] R. Schlinker, S. Liljenberg, D. Polak, K. Post, C. Chipman, A. Stern, Supersonic jet noise characteristics and propagation: engine and model scale, 13th AIAA/CEAS Aeroacoustics Conference, Rome, Italy, AIAA Paper 2007-3623, 2007, http://dx.doi.org/10.2514/6.2007-3623. [11] S. Pao, A. Wenzel, P. Oncley, Prediction of ground effects on aircraft noise, NASA Technical Paper 1104, 1978. [12] J. Huber, S. Sogeti, Jet noise assessment and sensitivity at aircraft level, 13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference), Rome, Italy, May 21–23, AIAA Paper 2007-3728, 2007, http://dx.doi.org/10.2514/6.2007-3728. [13] L.W. Illston, R.G.A. Angel, D.E. Patience, R.E. Burns, Assessment methods for near field noise and ground erosion, AIAA International Powered Lift Conference, AIAA Paper 1993-4870, 1993, pp. 208–219, http://dx.doi.org/10.2514/6.1993-4870. [14] M. Harper-Bourne, Predicting the jet near-field noise of combat aircraft, RTO AVT Symposium on Aging Mechanisms and Control, October 8–11, 2001. [15] W.E. Zorumski, Aircraft noise prediction program. Theoretical manual. Parts 1 and 2, NASA TM 83199, 1982. [16] C. Chien, W. Soroka, Sound propagation along an impedance plane, Journal of Sound and Vibration 43 (1) (1975) 9–20, http://dx.doi.org/10.1016/0022460X(75)90200-X. [17] D.K. McLaughlin, C. Kuo, D. Papamoschou, Experiments on the effect of ground reflections on supersonic jet noise, 46th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January 7–10, AIAA Paper 2008-22, 2008, http://dx.doi.org/10.2514/6.2008-22. [18] H.S. Ribner, The generation of sound by turbulent jets, Advances in Applied Mechanics 8 (1964) 104–182. [19] S.A.E. Miller, Towards a comprehensive model of jet noise using an acoustic analogy and steady rans solutions, 19th AIAA/CEAS Aeroacoustics Conference (34th AIAA Aeroacoustics Conference), Berlin, Germany, May 27–29, AIAA Paper 2013-2278, 2013, http://dx.doi.org/10.2514/6.2013-2278. [20] P.J. Morris, S.A.E. Miller, Prediction of broadband shock-associated noise using Reynolds-averaged Navier–Stokes computational fluid dynamics, AIAA Journal 48 (12) (2010) 2931–2961, http://dx.doi.org/10.2514/1.J050560.

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[21] C.K.W. Tam, L. Auriault, Mean flow refraction effects on sound radiated from localized sources in a jet, Journal of Fluid Mechanics 370 (1998) 149–174, http://dx.doi.org/10.1017/S0022112098001852. [22] N. Raizada, Numerical prediction of noise from high speed subsonic jets using an acoustic analogy (M.S. thesis), The Pennsylvania State University, 2005. [23] M.E. Goldstein, Aeroacoustics, McGraw-Hill International Book Company, New York, 1976. [24] M.E. Goldstein, The low frequency sound from multipole sources in axisymmetric shear flows, with applications to jet noise, Journal of Fluid Mechanics 70 (3) (1975) 595–604, http://dx.doi.org/10.1017/S0022112075002212. [25] M.E. Goldstein, The low frequency sound from multipole sources in axisymmetric shear flows – part ii, Journal of Fluid Mechanics 75 (1) (1976) 17–28, http://dx.doi.org/10.1017/S0022112076000104. [26] T.F. Balsa, P.R. Gliebe, R.A. Kantola, R. Mani, E.J. Stringas, High velocity jet noise source location and reduction. Task 2. Theoretical developments and basic experiments, Defense Technical Information Center, 1978. [27] A.F. Tinetti, M.H. Dunn, The fast scattering code (fsc): validation studies and program guidelines, NASA/CR-2011-217158, 2011. [28] J.T. Katsikadelis, Boundary Elements Theory and Applications, Elsevier, Kidlington, Oxford, United Kingdom, 2002. [29] Y.J. Liu, Fast Multipole Boundary Element Method – Theory and Applications in Engineering, Cambridge University Press, Cambridge, 2009. [30] K. Attenborough, I.L. Ver, Sound-Absorbing Materials and Sound Absorbers in Noise and Vibration Control Engineering: Principles and Applications, 2nd ed. John Wiley & Sons Inc., Hoboken, NJ, USA, 2007 (Chapter 8), http://dx.doi.org/10.1002/9780470172568. [31] W.K. Anderson, D.L. Bonhaus, An implicit upwind algorithm for computing turbulent flows on unstructured grids, Computers and Fluids 23 (1) (1994) 1–22. doi:10.1.1.55.4585. [32] E.J. Nielsen, Aerodynamic design sensitivities on an unstructured mesh using the Navier–Stokes equations and a discrete adjoint formulation (Ph.D. thesis), Virginia Polytechnic Institute and State University, 1998. [33] F.R. Menter, Two-equation eddy-viscosity turbulence models for engineering applications, AIAA Journal 32 (8) (1994) 1598–1605, http://dx.doi.org/ 10.2514/3.12149. [34] C. Nelson, G. Power, An overview of the nparc alliance's wind-us flow solver, 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, AIAA Paper 2010-27, 2010, http://dx.doi.org/10.2514/6.2010-27. [35] S.A.E. Miller, J. Veltin, Assessment of computational fluid dynamics for supersonic shock containing jets, AIAA Journal 47 (11) (2009) 2738–2746, http://dx.doi.org/10.2514/1.44336. [36] J. Bridges, C.A. Brown, Validation of the small hot jet acoustic rig for aeroacoustic research, 11th AIAA/CEAS Aeroacoustics Conference, Monterey, CA, 23–25 May, AIAA Paper 2005-2846, 2005, http://dx.doi.org/10.2514/6.2005-2846. [37] M.J. Smith, S.A.E. Miller, The effects of surfaces on the aerodynamics and acoustics of jet flows, 19th AIAA/CEAS Aeroacoustics Conference (34th AIAA Aeroacoustics Conference), Berlin, Germany, May 27–29, AIAA Paper 2013-2041, 2013, http://dx.doi.org/10.2514/6.2013-2041. [38] SAE International, SAE ARP876, Revision D. Gas Turbine Jet Exhaust Noise Prediction, SAE International, Warrendale, PA, 1994. [39] J.C. Lau, P.J. Morris, M.J. Fisher, Effects of exit mach number and temperature on mean-flow and turbulence characteristics in round jets, Journal of Fluid Mechanics 105 (1) (1981) 193–218, http://dx.doi.org/10.1017/S0022112081003170.