July 2014 

NASA/TM–2014-218506 

The Prediction and Analysis of Jet Flows and 

Scattered Turbulent Mixing Noise about Flight 

Vehicle Airframes 

Steven A. E. Miller 

Langley Research Center, Hampton, Virginia 
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July 2014 

NASA/TM–2014-218506 

The Prediction and Analysis of Jet Flows and 

Scattered Turbulent Mixing Noise about Flight 

Vehicle Airframes 

Steven A. E. Miller 

Langley Research Center, Hampton, Virginia 
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 

Acknowledgments 

This research is conducted by the National Aeronautics and Space Administration 

Fundamental Aeronautics Program High Speed Project. Matthew J. Smith is 

gratefully acknowledged for supporting this research through the National Institute of 

Aerospace cooperative agreement number NNL09AA00A. James Bridges, Clifford F. 

Brown, and Mark P. Wernet of the NASA Glenn Research Center are gratefully 

acknowledged for use of their measurement data. 

The use of trademarks or names of manufacturers in this report is for accurate reporting and does not constitute an 

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and Space Administration.
Abstract

Jet flows interacting with nearby surfaces exhibit a complex behavior in

which acoustic and aerodynamic characteristics are altered. The physical understanding and prediction of these characteristics are essential to

designing future low noise aircraft. A new approach is created for predicting scattered jet mixing noise that utilizes an acoustic analogy and

steady Reynolds-averaged Navier-Stokes solutions. A tailored Green’s

function accounts for the propagation of mixing noise about the airframe and is calculated numerically using a newly developed ray tracing

method. The steady aerodynamic statistics, associated unsteady sound

source, and acoustic intensity are examined as jet conditions are varied about a large flat plate. A non-dimensional number is proposed to

estimate the effect of the aerodynamic noise source relative to jet operating condition and airframe position. The steady Reynolds-averaged

Navier-Stokes solutions, acoustic analogy, tailored Green’s function, nondimensional number, and predicted noise are validated with a wide variety of measurements. The combination of the developed theory, ray

tracing method, and careful implementation in a stand-alone computer

program result in an approach that is more first principles oriented than

alternatives, computationally efficient, and captures the relevant physics

of fluid-structure interaction.

1
Nomenclature

Symbols

A Empirical constant

C Diffraction coefficient for

corner-diffracted rays

c Speed of sound

cl Empirical constant associated

with turbulent length scale

cτ Empirical constant associated

with turbulent time scale

c∞ Free-stream speed of sound

D Nozzle exit diameter

D Diffraction coefficient for edge

diffracted ray

Dj Fully expanded jet diameter

Dm Diffraction coefficient for creeping ray

E

0 First end point of edge segment

E

F Second end point of edge segment

f1 First damping function of discontinuity

correction

f2 Second damping function of

discontinuity correction

k Wave number or turbulent kinetic energy

(TKE)

ks Turbulent kinetic energy associated

with fine-scale mixing noise

Ma Acoustic Mach number

Md Design Mach number

Mj Fully expanded Mach number

M∞ Free-stream Mach number

N Axis of the diffraction

cone and unit vector of edge

O Observer location

p Static pressure

pa Acoustic pressure

pj Fully expanded static pressure

po Stagnation pressure

pˆD Complex pressure amplitude

of diffracted ray

L Length of plate used for point

source validation

ls Turbulent length scale

pˆI Complex pressure amplitude of

incident ray

qˆs Statistical source term

r Distance from edge to observer

S Source location

S Spectral density of acoustic pressure

St Strouhal number

T Transfer function used for creeping rays

Tj Fully expanded static temperature

To Stagnation temperature

t Proportionate distance along edge segment

u Mean streamwise velocity component

uj Fully expanded jet velocity

V Diffraction location

W Width of plate used for point

source validation

x Vector observer position

xI Jet impingement location

xp Axial distance from the jet centerline

to the plate

y Source vector

yp Radial distance from the plate to

jet centerline

β Wedge angle

δ Damping coefficient of

discontinuity correction

δij Kronecker delta function

δη Spreading angle of the jet

Γ Non-dimensional number

γ Ratio of specific heats

φo Polar angle of diffracted ray

φs Polar angle of incident ray

Ψ Observer angle from the upstream axis

ν Wedge index

ρ Density or

Distance from source to observer

θ Oblique angle between the edge and

incident ray

τs Turbulent time scale

ω Radial frequency

Abbreviations

AMELIA Advanced model for extreme lift

and improved aeroacoustics

BBSAN Broadband shock-associated noise

BVP Boundary value problem

CFD Computational fluid dynamics

CJES Compact jet engine simulator

DNS Direct numerical simulation

EPNL Effective perceived noise level

ESM Equivalent source method

FSC Fast scattering code

FUN3D Fully unstructured Navier-Stokes

three-dimensional solver

HBPR High bypass ratio

HWB Hybrid-wing body

JSIT Jet-surface interaction test

NPR Nozzle pressure ratio

LEE Linearized Euler equations

LBPR Low bypass ratio

PIV Particle image velocimetry

OASPL Overall sound pressure level

PIV Particle image velocimetry

PSD Power spectral density

RANS Reynolds-averaged Navier-Stokes

SHJAR Small hot jet acoustic rig

SPL Sound pressure level

SMC Small metal chevron

SST Shear stress transport

TTR Total temperature ratio

2
1 Introduction

Jet powered flight vehicles generate high intensity acoustic radiation that has a negative impact on

the surrounding community, military personnel, and can cause sonic fatigue. The need to reduce jet

noise and its impact on the surrounding environment cause communities to create noise regulations.

These restrictions have contributed to motivating the research and development of noise reduction

technology. One such technology integrates the propulsion system with the flight vehicle airframe

as shown by Hill and Thomas [1]. Envia and Thomas [2] outline the difficulties of reducing noise

with continued airline traffic growth. Advanced aircraft configurations are required for continued

noise reduction.

Examples of advanced aircraft configurations using new technologies include the hybrid-wing

body (HWB) aircraft (see Thomas et al. [3] and Czech et al. [4]), the advanced model for extreme

lift and improved aeroacoustics (AMELIA) (see Horne et al. [5]), and future supersonic aircraft (see

Welge et al. [6, 7] and Morgenstern et al. [8]). These latter aircraft have low-bypass ratio engines

that are similar to single stream jets and are discussed by Sokhey and Kube-McDowell [9]. All of

these concepts demonstrate situations where the unsteady aerodynamic noise source and resultant

acoustic propagation are difficult to predict with current methodology.

Unfortunately, contemporary jet noise prediction models are incapable of predicting the noise

from full-scale jet engine exhaust, let alone are capable of predicting noise when airframe surfaces are

present. Two recent promising prediction approaches for installed jets include large eddy simulation

(LES) by Paliath and Premasuthan [10] or Bogey [11] and a wave-packet scattering approach by

Papamoschou and Mayoral [12]. The LES predictions of Paliath and Premasuthan [10] are very

accurate but come at an extremely high computational cost. A direct numerical simulation (DNS)

of a jet, by Freund [13] for example, is even more costly for high Reynolds numbers. Wave-packet

approaches, such as those by Papamoschou and Mayoral [12], are considerably faster than LES

but likely require calibration with measurement data for each jet operating condition. New jet

noise prediction methods, that are fast and more first principles oriented, need to be created to

address new aircraft configurations and advanced nozzle designs. In this paper, one such method

is proposed that is based on the acoustic analogy of Miller [14], steady Reynolds-averaged NavierStokes (RANS) solutions, and a tailored Green’s function.

1.1 Jet Noise Sources and Spectra

High intensity turbulence within jet plumes causes mixing noise at all frequencies and directions. It

is labeled ‘mixing’ in the far-field power spectral density shown in Fig. 1. The vertical axis represents

Sound Pressure Level (SPL) per unit Strouhal number (St). St is based on the fully expanded jet

velocity and fully expanded jet diameter. Ψ is the angle in degrees measured relative to the forward

nozzle centerline axis. Turbulent mixing is the predominant source of sound at subsonic jet Mach

numbers and is always the dominant source of sound in the downstream direction at supersonic

jet Mach numbers. Mollo-Christensen [15], Tam et al. [16], Viswanathan [17], and others have

shown that large-scale coherent turbulent structures are responsible for the highest intensity noise

in the downstream direction. These structures are highly coherent over multiple jet diameters.

The incoherent turbulence causes acoustic radiation in all jet directions and is responsible for

the highest noise intensity in the sideline and upstream jet directions. Semi-empirical fine-scale

mixing noise models pioneered by Tam [18] and Tam and Auriault [19] accurately predict the

sideline and upstream mixing noise. More recent descriptions by Morris and Farassat [20] yield an

equivalent explanation by using an acoustic analogy approach. The unsteady aerodynamic source

and associated jet mixing noise are the focus of this paper. Jet noise has been reviewed by many

3
investigators, including Ffowcs Williams [21, 22] and Goldstein [23], and specifically for supersonic

jet noise by Tam [18].

When the jet Mach number is supersonic and the nozzle is operating off-design, a shock-cell

structure forms and its presence creates additional noise due to high intensity turbulence convecting

through the shock waves. Acoustic waves propagate to the far-field from these interactions and

combine constructively and destructively to create broadband shock-associated noise (BBSAN),

shown in Fig. 1. BBSAN is observed as strong broad spectral peaks with additional lower intensity

peaks at higher frequencies that may dominate mixing noise. Notable models for the prediction

of BBSAN are by Harper-Bourne and Fisher [24], Tam [25], Tam [26], and Morris and Miller [27].

Acoustic waves within the jet flow propagate upstream and trigger aerodynamic instabilities that

close a feed-back loop that create discrete tones called ‘screech,’ first observed and described by

Powell [28], and are also shown in Fig. 1. For additional details on screech tones see Tam [19] and

for a comprehensive review see Raman [29]. Shock-associated noise is often observed in the spectra

under investigation in this study.

When the propulsion system is integrated with the airframe the noise sources are altered relative to the free jet. Sawyer [30], Al-Qutub and Budair [31], and others examined the changes in

aerodynamics when the jet exhaust is far from the airframe. The turbulent statistics of the jet

are affected when the jet plume is multiple nozzle diameters away from the airframe. Donaldson

and Snedeker [32] and Lamont and Hunt [33] showed that the turbulent statistics and shock-cell

structure of jets impinging on oblique and perpendicular surfaces are significantly affected. The

unsteady aerodynamic source of sound is dependent on the turbulent statistics and jet structure.

Capturing variations of the acoustic source due to changes within the jet flow-field are essential for

accurate predictions of jet noise. In this paper, we attempt to capture these changes with the use of

steady RANS solutions that include the nozzle and airframe geometry.

The far-field jet noise spectrum changes significantly when the propulsion system is integrated

with the airframe. This is illustrated in Fig. 2 at an inlet angle of 50 deg. for a fully expanded

Mach number Mj = 1.29 and total temperature ratio TTR = 1.00 off-design jet. The measured

free jet prediction is shown and consists of mixing and shock-associated noise. When a large flat

plate extends 20 nozzle diameters, D, downstream and is 2D from the nozzle centerline axis in

the radial direction, the measured spectrum changes dramatically as shown by the ‘jet & airframe’

measurement. The observer is located on the opposite side of the flat plate. Predictions for the free

jet mixing noise and BBSAN can be made with the method of Miller [14] and Morris and Miller [27],

illustrated by the free jet mixing noise and free jet BBSAN prediction respectively. The jet-surface

interaction noise, which occurs at low frequencies typically smaller than St < 0.1, can be predicted

by Miller [34]. Studies of Curle [35], Ffowcs Williams and Hawkings [21], Ffowcs Williams and

Hall [36], Crighton and Leppington [37], and Amiet [38], suggest that the most dominant source

of interaction noise is generated by the turbulence within the shear layer of the jet convecting

downstream past the trailing edge of the wing or airframe. The jet-surface interaction noise is a

combination of unsteady loading on the airframe surface within the boundary layer (see Curle [35])

and an interaction and subsequent diffraction of the jet turbulent flow about the trailing edge (see

Ffowcs Williams and Hall [36]). Note that at this particular angle and jet condition, the sound

intensity is a combination of jet-surface interaction noise, mixing noise, and BBSAN. The main

focus of this paper is the prediction of mixing noise scattered by the airframe.

4
St

SPL per unit St

10-2 10-1 10010 90 1

105

120

135

150

Mixing

BBSAN

Screech Ψ=50o

Peak BBSAN

St

SPL per unit St

10-2 10-1 10010 90 1

105

120

135

150

Mixing

BBSAN

Screech

Ψ=90o

Peak BBSAN

St

SPL per unit St

10-2 10-1 10010 90 1

105

120

135

150

Ψ=150o

 Mixing

Screech

Figure 1. Sound pressure level per unit St resulting from an off-design heated supersonic jet. Ψ is

the angle from the upstream jet axis to the observer centered about the nozzle exit. The observers

are located 100 nozzle exit diameters from the center of the nozzle exit.

5
St

SPL per unit St (dB re 20 

μ Pa)

10-2 10-1 100 60

70

80

90

100

110

120

130

140

Measurement: Free-Jet

Measurement: Jet & Airframe

Prediction: Free-Jet Mixing Noise

Prediction: Free-Jet BBSAN

Prediction: Jet & Airframe Mixing Noise

Prediction: Jet-Surface Interaction Noise

Screech Tones

Mixing Noise Shielding

Mixing Noise

Jet-Surface Interaction

 Noise

BBSAN

Figure 2. The free jet and installed jet predictions at R/D = 100 and Ψ = 50 deg. compared

with measurement. The jet operates at Mj = 1.29 and TTR = 1.00 from the SMC016 nozzle with

D = 0.0508 m and the plate is located at xp/D = 20 and yp/D = −2.

1.2 The Prediction of Jet Mixing Noise

Many methods have been developed to predict jet mixing noise. Lighthill [39, 40] pioneered the

acoustic analogy approach, which is an exact rearrangement of the equations of motion into a wave

operator and inhomogeneous equivalent source term. Lilley [41] created an acoustic analogy that

included refraction effects in the propagation operator allowing for realistic shear layer refraction

effects to be accounted for. Khavaran [42] developed a prediction method that followed the ‘MGB’

code of Mani et al. [43] but differed significantly by informing the equivalent sources with a steady

RANS solution. The approach of Khavaran [42] has the benefit of including the nozzle geometry

in the calculation and shock waves within the jet plume. An alternative approach to the acoustic

analogy was created by Tam and Auriault [19] who developed a semi-empirical theory for fine-scale

mixing noise based on observations of Tam et al. [16]. Morris and Farassat [20] and later Morris

and Boluriaan [44] examined the model of Tam and Auriault [19] and formulated a similar source in

terms of an acoustic analogy. Recently, Miller [14] used the model basis of Morris and Farassat [20]

and Tam and Auriault [19] and predicted single and dual stream jet noise for a wide range of jet

Mach numbers and temperature ratios. This latter theory is modified for predictions of jet mixing

noise about the flight vehicle airframe. These predictions are shown in Fig. 2 as lines labeled ‘Mixing

Noise’ and require knowledge of how sound is scattered by the flight vehicle airframe.

1.3 The Scattering of Sound from Flight Vehicles

Separate techniques have been developed to predict the scattering of noise from jets about flight

vehicle airframes. They are typically based on the solution of the Helmholtz or convective wave

equation. One popular approach uses the boundary element method (BEM) that solves an integral equation for the Helmholtz equation. It was recently applied successfully by Huang and

6
Papamoschou [45]. It has also been applied by Agarwal and Dowling [46] to predict monopole

acoustic shielding of a blended wing body aircraft. Solving the linear system associated with the

BEM approach can become computationally expensive because the order of the matrix depends on

the size of the aircraft and the need to meet the Nyquist criteria.

The equivalent source method (ESM) follows the same approach as the BEM which replaces

the boundary value problem (BVP) with a distribution of multipoles located inside the scattering

body. An excellent overview is given by Dunn [47] and applied in NASA’s Fast Scattering Code

(FSC) by Tinetti et al. [48]. The fundamental advantage of the ESM over other boundary methods

is that the resulting linear system of equations can be considerably smaller than those arising from

the BEM. We use predictions of FSC [49] to help validate the approach adopted in this paper.

Another approach for acoustic scattering is based on the geometric theory of diffraction (acoustic

ray theory). This approach offers several advantages. For example, the method’s computational

cost is frequency independent and rays need to only be traced once for all frequencies of interest

(see Agarwal et al. [50] for details). Also, it is not computationally dependent on the geometry size

but only the complexity. These are key advantages relative to other methods, which have great

computational costs for high frequencies and large geometries. Unfortunately, there are a limited

number of source models for scattering. In this work, we will connect a custom ray tracing solver

(that provides a tailored Green’s function) with the acoustic analogy of Miller [14].

1.4 ab initio ad finem

The remainder of this paper is organized as follows. A mathematical model is reviewed that is based

on the acoustic analogy of Miller [14, 51]. The calculation of the vector Green’s function within

the acoustic analogy is described using the geometric theory of diffraction. To quantify the propulsion airframe aeroacoustic installation effects on the aerodynamic noise source, a non-dimensional

number, Γ, is proposed that is dependent on the jet operating conditions and airframe location.

Steady RANS solutions of free and installed jets are compared with particle image velocimetry

(PIV) measurements of Bridges and Wernet [52]. The non-dimensional number, Γ, is evaluated

and compared with noise predictions using the validated steady RANS solutions. The ray tracing

approach is validated with measurements by Ahtye and McCulley [53] of a point source near a

flat plate. Ray tracing is then used in conjunction with an acoustic analogy to predict far-field

mixing noise generated from a jet shielded by a flat plate. These predictions are validated with

measurements of Brown et al. [54]. Finally, a summary and conclusion are presented.

2 Mathematical Analysis

2.1 An Acoustic Analogy Approach

An acoustic analogy is an arrangement of the governing equations of motion into an operator

that supports wave propagation and is equated to equivalent sources. The acoustic analogy of

Miller [14, 51] is based on the analogy of Morris and Farassat [20] and is used as the basis of

this prediction approach. Following their approach, the Euler equations are arranged into the

inhomogeneous LEE in which the acoustic propagation and equivalent sources are separated. A

convolution integral of the vector Green’s function of the LEE and the equivalent sources define

the fluctuating pressure in the far-field. Using an approximation by Tam and Auriault [19], two

closely placed source points in the jet are related, and the spectral density is written in terms of the

two-point cross-correlation of the equivalent sources and the vector Green’s function of the LEE.

The spectral density, S, of the far-field fluctuating acoustic pressure is,

7
S(x, ω) = ρ

2

∞c

4

∞

Z ∞

−∞

Z ∞

−∞

Z ∞

−∞

2π

3/2

c

2

∞lxlylzτsx

2

c

2∞x2 + (ux1 + vx2 + wx3 + c∞x)2τ2

s ω2

×

(

π

∗0

g

(x, y, ω)π

0

g

(x, y, ω)A

2

s

(us/c∞)

4

τ

2

s

+

X

3

n=1

X

3

m=1

π

∗n

g

(x, y, ω)π

m

g

(x, y, ω)B

2

s

(us/c∞)

2u4

s

l

2

x

)

× exp "−



l

2

xx

2

1 + l

2

yx

2

2 + l

2

zx

2

3



ω

2

4c

2∞x2

#

dy,

(1)

where π

n

g

is the n

th component of the vector Green’s function of the LEE. The other variables are

defined in the nomenclature.

Details regarding the development of Eqn. 1 are discussed in Miller [14]. The scales of turbulence

in Eqn. 1, lx, ly, lz, us, and τs, are required to conduct a prediction. These can be found by simple

empirical models or unsteady CFD simulations. Here, they are related to a steady RANS solution

by, lx = clK3/2/, τs = cτK/, and us = cu(2K/3)1/2.

The coefficients are cτ = 0.30, cu = 1.00, and cl = 1.00. These coefficients are based on a

reference jet operating at the sonic condition, TTR = 3.20, and a 0.0508 m convergent nozzle at

the sideline location of R/D = 100. They have been calibrated by the same methodology of Tam

and Auriault [19]. Their variation from Tam and Auriault is attributed to the use of a different

acoustic analogy and steady RANS solver. The cross-stream turbulent length scales, ly and lz, are

set to 3/10 of lx, which corresponds to experimental observation. Coefficients associated with the

turbulent length scales are never altered irrespective of the jet Mach number, temperature, nozzle

geometry, or any other parameter.

A solution for π

n

g

is now required. One method suitable for calculating the vector Green’s

function of the LEE is by finding the Green’s function of Lilley’s [41] equation. For a parallel

axisymmetric mean flow the Green’s function of the LEE can be written in terms of the Green’s

function of Lilley’s equation. Morris and Boluriaan [44] show that the vector Green’s function of

the LEE is related to the Green’s function of Lilley’s [41] equation by,

π

0

g

(x, y, ω) = ω

2

gl(x, y, ω) − 2iuω

∂gl(x, y, ω)

∂yx

− u

2 ∂

2

gl(x, y, ω)

∂y2

x

, (2)

π

1

g

(x, y, ω) = −



iω + u

∂

∂yx



∂

∂yx

gl(x, y, ω), (3)

π

2

g

(x, y, ω) = −



3

∂u

∂yr

∂

∂yx

−



iω + u

∂

∂yx



∂

∂yr



gl(x, y, ω), (4)

π

3

g

(x, y, ω) = −



iω + u

∂

∂yx



1

yr

∂

yθ

gl(x, y, ω), (5)

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 the Green’s function of Lilley’s [41] equation instead

of the full vector Green’s function of the LEE. Morris and Miller [55] used a similar technique

and found an analytic solution of the vector Green’s function of the LEE related to the Green’s

function of the Helmholtz equation for quiescent flow (see Eqn. 23 of Morris and Miller). Tam and

Auriault [56] and Raizada [57] also used this technique with success. These methods use an adjoint

approach and are relatively computationally inexpensive. However, these methods do not include

8
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.

An asymptotic solution of Lilley’s equation for low frequencies was found by Goldstein [58–60]

and for high frequencies by Balsa et al. [61]. Miller [51] examined the asymptotic solutions of

Lilley’s equation, the Green’s function of the convective Helmholtz equation, g, and a matching

function, and proposed,

gl(x, y, ω) ≈

i

c

2∞c

cα

0 ω

c∞

c

exp 

−iω

c∞

{x cos θ + r sin θ cos (φ0 − φ)}



qeg, (6)

where c is the local speed of sound, cβ = 1 × 10−4, qe = exp [−cβω/c∞] for Re[g

∗

o

] ≤ 0, c0 is

Ribner’s convection coefficient, cα is the amplification power coefficient and is otherwise unity, and

the remaining terms are described in Miller [51].

The benefit of this approach is existing numerical solvers can provide the Green’s function of

the convective Helmholtz equation. For example, the Fast Scattering Code (FSC) by Tinetti and

Dunn [49], any BEM (see an overview by Katsikadelis [62]), or FastBEM by Liu [63]. This provides

the ability to predict jet noise scattering from any surface as long as the noise sources are minimally

affected by the airframe. Here, we find a tailored form of g using the geometric theory of diffraction

described in the next section.

2.2 A Tailored Green’s Function

The propagation of sound about the flight vehicle airframe is governed by g. Here, a ray tracing

approach is used to provide the tailored Green’s function. For free jet predictions the free-field

Green’s function of the Helmholtz equation is used,

g =

exp[ikr]

4πr

, (7)

where r = x − y is a vector from the source to observer. When a scattering geometry such as the

airframe is integrated with the jet, the tailored Green’s function of the Helmholtz equation is found

using a ray tracing approach. In this case, the tailored Green’s function is the sum of contributions

from the incident and diffracted field,

g = gI +

XgD, (8)

where gI is the incident ray, equivalent to the free-field Green’s function in Eq. 7, and gD are the

diffracted rays. The diffracted rays consist of edge and corner diffracted rays.

2.3 The Geometric Theory of Acoustic Rays

In this section, an overview of the ray tracing method is presented that is used to evaluate Eqn. 8.

A shadow region exists where there is no direct line of sight from source to observer. The theory,

developed by Keller [64], has been recently used in the area of propulsion airframe aeroacoustics to

aid in the study of shielding effects (see Suzuki [65], Agarwal [50], van Rens [66], and Lummer [67]

for example). Kinsler et al. [68] showed that under certain circumstances energy on defined paths

can be approximated as rays rather than waves.

We assume a solution for the convective Helmholtz equation of the form, ˆp(x) = A(x, ω) exp[iωτ (x)],

where ˆp(x) is the Fourier transform of the complex pressure, τ (x) = T(x)/c∞ is the quantity known

as the eikonal (see Kinsler et al. [68] for details), and ω is the radial frequency. Constant values

of T(x) define surfaces of constant phase. Substitution of the assumed solution into the governing

9
equation and equating the real and imaginary parts of the resulting equation yields two equations

that are coupled and nonlinear. The high frequency limit allows for a simple solution to be obtained. It is assumed that an asymptotic expansion of the pressure amplitude, A(x, ω), exists as a

power series (see Pierce [69] for details) in inverse powers of radial frequency,

A(x, ω) = X∞

n=0

An(x)

ωn

. (9)

By using Eqn. 9 and eliminating all terms except those with the two highest orders of magnitude

of ω, a simplified equation is obtained, iω 2∇τ∇A0 + ∇2τA0



− ω

2A0

h

(∇τ )

2 − c

−2

i

= 0 . It is

assumed that A0 is an approximation of A for large radial frequencies, limω→∞A(x, ω) = A(x).

Separation of the real and imaginary terms and further simplification result in,

(∇τ )

2 =

1

c

2

or (∇T)

2 =

c

2

∞

c

2

= n

2

(10a)

2∇τ∇A + ∇2τA = 0 or ∇ · A

2∇τ



= 0. (10b)

Equation 10a is the eikonal equation where n = c

−1

c∞ is the index of refraction. The second

part of Eqn. 10a is formed from multiplying by A and using ∇ · (φB~ ) = φ∇ · B~ + ∇B~ · φ. It can

be shown from Eqn. 10a that (see Kinsler [68] for details), ∇T = nsˆ, where ˆs is the unit vector

that gives the local direction of propagation. To determine how the amplitude A varies along a ray,

Eqn. 10b is solved using ray tube areas (see Pierce [69] for details). Rays travel in the direction

∇T = ˆs and only pass through the end caps of volume elements, thus the integral over the surface

of the side of the ray tube vanishes. Using these assumptions we obtain,

A

2

(x)S(x)(∇τ · ~n)x − A

2

(x0)S(x0)(∇τ · ~n)x0 = 0, (11)

where ~n is the surface normal vector. If the ray is passing through a homogeneous medium (c =

constant), then (ˆs · ~n)x = (ˆs · ~n)x0

and the pressure amplitude at x along the ray tube is, A(x) =

A(x0) [S(x0)/S(x)]1/2. Energy within a ray tube (SA2) remains constant within the limitation

set forth by the eikonal equation. Incident rays that impinge on surface edges, corners, or slightly

graze the surface create diffracted rays that propagate to the shadow region. There are multiple

rays that account for the diffracted field; edge diffracted rays, creeping rays, and corner diffracted

rays.

2.3.1 Edge Diffracted Rays

Edge diffracted rays are based on a modified form of Fermat’s principle. Fermat’s principle states

that an edge diffracted ray from S to O is a curve that has stationary optical length among all

curves from S to O with one point on the edge (see Keller [64]). The curves correspond to paths

that can be traversed in the least time. Sommerfeld [70] showed that waves diffracting by a semiinfinite edge diffract in a direction normal to the edge in the form of a cylindrical wave. Fermat’s

principle for edge diffraction implies that the diffracted and incident ray have corresponding angles

relative to the point of diffraction, as long as they share the same medium. These principles of

diffracted rays are illustrated in Fig. 3.

After the ray is diffracted, it has similar properties to the incident ray, but is dependent on the

point of diffraction. The acoustic pressure field at the diffraction point on the edge is determined

using the solution, ˆp(V ) = A(S, ω)|V − S|

−1

exp[iωτ ], where S is the source location and V is

10
the diffraction location on the edge. The acoustic pressure field at the diffraction point determines

the initial amplitude and phase of the diffracted ray. The phase of the diffracted ray is the eikonal

added to the phase at the diffraction point. The amplitude of the diffracted ray is governed by

the ray tube area relationship. Therefore, the energy per unit area of the diffracted ray tube is

proportional to r

−1 and the amplitude is proportional to r−1/2

. Assuming that the amplitude of

the diffracted ray is proportional to the amplitude at the point of diffraction, the acoustic pressure

of the diffracted ray is,

pˆ(O) = Dpˆ(V )



ρ

r(r + ρ)

1/2

exp[iωτ ], (12)

where D is the diffraction coefficient, r is the distance from the edge to observer, |O − E|, and ρ

is the distance from source to observer |O − S|. Keller [64] showed that Eqn. 12 satisfies Sommerfeld’s [70] exact solution for diffraction of a wave by a half-plane when it is asymptotically expanded

for large frequencies. The resultant diffraction coefficient is,

D =

sin (π/n) exp[iπ/4]

n(2πk)

1/2 sin(θ)

"

1

cos π

n − cos

φs−φo

n

+

1

cos π

n − cos

φs+φo+π

n

#

, (13)

where k = 2πf /c∞ is the wave number, φs is the polar angle to the incident ray, φo is the polar

angle to the diffracted ray, and θ is the oblique angle between the edge and incident ray. Here, n is a

parameter of the wedge angle, β = nπ, where β is the angle of the wedge. The angles are illustrated

in Figs. 3 and 4. Analysis of a half-plane corresponds to n = 2. Adopting a simplification from

Agarwal et al. [50], a wedge index ν = π/β is used to redefine the diffraction coefficient as,

D =

ν sin (νπ) exp[iπ/4]

(2πk)

1/2 sin(θ)



1

cos (νπ) − cos (ν (φs − φo)) +

1

cos (νπ) − cos (ν (φs + φo + π))

. (14)

The contribution to the acoustic field from a single diffracted ray can be calculated using

Eqns. 12 and 14 and the variables illustrated in Figs. 3 and 4.

xe

ye

ze

θ

θ

S

V

x

y

z

Figure 3. Edge diffraction and associated variables.

11
xe

ye

S(rs,φs,zs)

O(ro,φo,zo)

φo

φs

V β

Figure 4. Variables associated with diffraction on a sharp edge.

2.3.2 Creeping Rays

Creeping rays, also known as surface diffracted rays, are produced when a ray is incident tangentially

to a smooth boundary or interface (see Levy and Keller [71] for details). After the incident ray

grazes the surface tangentially, it travels along the surface and continuously sheds a diffracted ray,

as shown in Fig. 5. The diffracted rays are continuously emitted at angles tangent to the surface.

Creeping rays are based upon an alternate form of Fermat’s principle, that a surface diffracted ray

from a point S to a point O is a curve that makes stationary the optical length among all curves

from S to O having an arc on the boundary. This implies that S to ST and OT to O are straight

lines tangent to the surface at ST and OT respectively, and ST to OT is a geodesic curve on the

surface (see Fig. 4).

O

S

ST

t OT

Figure 5. Incident ray diffracting from a smooth surface.

A solution for a surface diffracted ray, similar to the edge diffracted ray, is proposed by Levy

and Keller [71],

pˆ(O) = T(ST , OT )ˆp(ST )



ρ

r(r + ρ)

1/2

exp[iωτ ], (15)

12
where T(ST , OT ) is a transfer function that relates the diffracted field at the two tangent points

ST and OT (see Agarwal et al. [72]). The function T(ST , OT ) is given by,

T(ST , OT ) = X

m

Dm(ST )Dm(OT ) exp ikt −

Z t

0

αm(τ )dτ  dσ(ST )

dσ(OT )

1/2

, (16)

where Dm(ST ) is the diffraction coefficient at ST , αm is the decay coefficient, t is the distance

along the curve, and [dσ(ST )/dσ(OT )]1/2is the attenuation of the ray field due to the divergence

of two nearby creeping rays. These values are problem specific and are based on the diffraction

geometry, material properties, and acoustic wave number. These values have been evaluated for

simple problems such as a cylinder and a sphere by Levy [71]. Equation 16 shows that the creeping

ray exponentially decreases with the distance along the path. If t << r, where r is the distance

to the observer, the creeping ray converges to the solution of an edge diffracted ray. If t >> r,

the amplitude of the ray decreases exponentially and has a negligible contribution to the diffracted

field relative to an edge diffracted ray.

2.3.3 Corner Diffracted Rays

Incident rays that impinge on a sharp corner of the airframe are diffracted as spherical waves as

described by Keller [73]. The cross-sectional area of the corner diffracted ray tube is proportional

to r

−2 and the amplitude is proportional to r−1

. Following the same derivations for edge diffracted

rays, the complex acoustic pressure of the corner diffracted ray is,

pˆ(O) = Cpˆ(V )

|O − V |

exp[iωτ ], (17)

where C is the corner diffraction coefficient. The coefficient C is analogous to D and depends on

the geometry, direction of the incident and diffracted ray, and wavenumber. C is proportional to

k

−1 and the corner diffracted field decreases faster than the edge diffracted field as k increases.

Many forms of C exist and due to their multitude and complex nature they are not surveyed here

(see Kraus [74], Felsen [75], and Siegel [76] for details).

2.4 A Non-Dimensional Theory

We now turn our attention to the development of a non-dimensional number to quantify the effect

that the airframe has on the jet mixing noise source. Parameters are identified that alter the noise

source with varying jet operating condition and airframe position relative to the nozzle exit. The

first parameter is the fully expanded diameter of the jet which represents the equivalent nozzle exit

diameter for a shock free flow,

Dj = D

 

1 + γ−1

2 M2

j

1 + γ−1

2 M2

d

!(γ+1)/4(γ−1) 

Md

Mj

1/2

, (18)

where D is the nozzle exit diameter and γ is the ratio of specific heats.

Additionally, xp is the distance from the nozzle exit to the trailing edge of the airframe1 parallel

to the jet centerline. yp is the characteristic length from the nozzle centerline to the nearest airframe

surface. xI is the axial distance from the nozzle exit to the jet impingement location, is a function

of the jet spreading rate, and is approximated as,

1We elect to use the terms ‘airframe’ and ‘flat plate’ interchangeably in terms of this analysis.

13
xI =

yp − D/2

tan[δη]

, (19)

where δη is the spreading angle of the jet. Values of δη are not readily available without numerical

calculations or measurement. An empirical model developed by Lau [77] is adopted for δη,

δη = 0.177 1 − 0.294M2

j





1 +

1

2

(M2

j − 1)(Tj/To − 1.4)2



, (20)

which is valid for a wide range of single-stream jet Mach numbers and temperature ratios. The

flows of this investigation fall within the range of validity of Eqn. 20. Using these variables a

non-dimensional number is proposed,

Γ = 

Dj

yp

 xp

xI



. (21)

Using the approximation of xI from Eqn. 19, Eqn. 21 is written as,

Γ =

Djxp tan[δη]

yp(yp − D/2), (22)

where yp > D/2. Physically, Eqn. 22 is the ratio of the product of jet and airframe length scales

divided by the cross-stream length scale and interaction distance. Small values of Γ imply that

airframe effects on the jet aerodynamic noise sources are negligible. Likewise, large values of Γ

imply that the effects of the airframe on the aerodynamic noise sources of the jet are very large.

Equation 22 is evaluated in the next section using different jet conditions and airframe surface

positions, and compared with numerical predictions for the variation of noise from the aerodynamic

source.

3 Application

3.1 Application of the Acoustic Analogy

Equation 1 is implemented in a computer program. The program uses structured or unstructured

steady RANS solutions and converts the specific dissipation, Ω, to dissipation  = 0.09KΩ. The

steady RANS solution is interpolated using the inverse weighted distance method onto a new

structured grid that encompasses the jet plume. Numerical integration is performed using the new

structured domain to approximate the integrals of the model equations. Interpolated values reside

on the x-y plane and are subsequently rotated about the jet centerline axis to form a cylindrical

region encompassing the plume. Integration ranges and grid point density are varied until the

solution is grid independent. The streamwise integration range extends from 10−4D ≤ x ≤ 20D

and in the radial direction from 10−4D ≤ r ≤ 2D. The structured domain contains 300 grid points

in the streamwise direction and 100 grid points in the radial direction. The angle between each

plane in the azimuthal direction is 15 degrees. Doubling the number of grid points in the streamwise

or radial direction or doubling the number of azimuthal planes, relative to the converged solution,

has negligible effect on the solution.

Equation 1 is evaluated by approximating the integrals as summations. At each grid point the

local volume is multiplied by the integrand and the pre-factor. The integrand contains arguments

that are obtained from the steady RANS solutions. Evaluation of Eqn. 1 also requires a tailored

Green’s function that is unique from each source point, y, to each observer, x on a radial frequency

14
basis. The integrand of Eqn. 1 contains the vector Green’s function of the LEE shown in Eqns. 2

through 5. Equations 2 through 5 are dependent on a steady RANS solution and the Green’s

function of Lilley’s equation, gl. Here, glis written in terms of the Green’s function of the convective

wave equation. In the absence of a scattering surface such an aircraft airframe, g is calculated using

Eqn. 7. When a scattering surface is present, Eqn. 8 is used, where the arguments depend on the

incident or diffracted ray path. Here, the developed ray theory is used to find g for each source

point and frequency. This is performed within the computer program by calling subroutines that

are described in the next section.

3.2 Application of the Ray Theory

The developed ray theory is implemented in the same computer program as the acoustic analogy.

Some specific computational strategies are used to evaluate the developed ray theory and are

described here. The implementation is summarized as follows: read the scattering geometry, find the

three-dimensional projection of the geometry, find the edge of the geometry, test if the observer is in

the shadow zone, find all diffracted rays associated with an observer, and sum the existing diffracted

and incident rays in the complex field. The geometry of the airframe is read into the memory of the

computer and includes the source and observer locations. Next, the three-dimensional projection

of the scattering object based on the calculated ray paths is found. This projection is created via

a ‘rotation matrix’ and is used to temporarily rotate the airframe to a new reference frame. The

rotation is performed by creating a z-axis (z

0

) based on the vector between the observer and the

source location. If O = [O1, O2, O3] is the observer location and S = [S1, S2, S3] is the source

location then, z

0 = S − O/ |S − O| . Corresponding x0 and y0 axes are arbitrary, orthonormal, and

follow a right-hand rule convention. Here, we adopt the following rule of orthogonality, z

0

· x

0 = 0.

We form a simple solution, x

0 = [z0

2

, −z

0

1

, 0] and y

0 = z0 × x0

.

Using the new axes, a rotation matrix is formed and multiplied by the node locations of the

airframe geometry to determine the projected coordinates in the x

0 − y0 plane,



xproj

yproj

=



x

0

1 x

0

2 x

0

3

y

0

1

y

0

2

y

0

3







x

y

z



 . (23)

The edge, or diffraction outline, of the airframe geometry is now determined based on the source

and observer location. Edges that are unique to elements are on the boundary in the projected

domain and are only specified once to describe an element. For example, if two elements share an

edge then that edge is not unique and not on the boundary of the geometry for a particular source

and observer. An example is shown in Fig. 6 that illustrates the edge in the projected plane for a

discretized sphere.

An approach by Glassner [78] is modified to test if the observer is within the shadow region

relative to the source. This approach solves two equations. One equation describes the ray path and

the second the airframe. If a real solution exists then there is an intersection of the ray and airframe

geometry, thus the observer is in the shadow zone. Here, we modify the approach of Glassner and

use the edge determined in the projected plane, as previously described, which is based upon

two-dimensional polygons. The point of intersection is determined by using the equation for the

three-dimensional projected plane. It can then be determined if the intersection point is within

the boundaries of the edge on the two-dimensional plane. Figure 7 illustrates this process for a ray

intersecting a spherical object. A new reference frame, ˆu − vˆ, is established with the origin at the

point of intersection on the two-dimensional plane. Each edge segment is examined for intersection

15
xproj

yproj

-10 -5 0 5 10

-10

-5

0

5

10

Figure 6. The unique edges from the projected plane.

relative to the positive ˆu-axis. If only one edge segment intersects the positive ˆu-axis then the

point is within the two-dimensional boundary. If there are zero or two edge segments crossing the

positive ˆu-axis, the point is outside the two-dimensional boundary. An example is shown in Fig. 7,

where the first intersection point crosses the positive ˆu-axis and the second intersection point has

zero edge segments crossing the positive ˆu-axis.

The equation for a cone about an arbitrary axis is,

(O − V )

T N = (O − V )T

(O − V ) cos θ, (24)

where O is the observer location on the surface of the cone, V is the point of diffraction and vertex

of the cone, N is the axis of the cone and unit vector of the edge, and θ is the angle of the cone.

Relevant angles of diffraction are displayed in Fig. 3. All observer points O that satisfy Eqn. 24

are on the surface of the cone and have a diffracted ray edge point V . The angle θ is,

θ = cos−1



[V − S] · N

|V − S|



. (25)

Combining Eqns. 24 and 25 yield an equation for the vertex vector location V ,

p

(V1 − S1)

2 + (V2 − S2)2 {N1(O1 − V1) + N2(O2 − V2) + N3(O3 − V3)}

=



(V1 − S1)

2 + (V2 − S2)2 + (V3 − S3)2

	

{N1(V1 − S1) + N2(V2 − S2)} .

(26)

Equation 26 is parameterized by t on a straight edge segment. The parameterized equations

are,

Vi = E0

i + (EFi − E0i

)t, (27)

where E0and EFare the first and second end point of the edge segment respectively and t is

the proportionate distance along the edge segment. Figure 3 shows these quantities and the cone

parameters. Using the parameters in Eqns. 26 and 27, t is found in the region 0 ≤ t ≤ 1 using the

secant method. If a real solution for t exists for 0 ≤ t ≤ 1, the diffraction point, source location, and

16
xproj

yproj

-10 -5 0 5 10

-5

0

5

10

Intersection

Point 2

Intersection

Point 1

u

v

v

u

(u2,v2)

(u1,v1)

Figure 7. Example ray intersection in a two-dimensional projection.

observer location are used to calculate all angle parameters required for the diffraction coefficient of

Eqn. 14. The complex value of the acoustic pressure field from the diffracted ray is then calculated

using Eqn. 12. This process is repeated for each segment identified in the projected plane. A similar

procedure is followed for the other types of diffracted rays.

For each source, observer, and airframe position the incident and diffracted rays are summed

and their phase relationships are retained. For an observer location O, the total complex amplitude

of acoustic pressure from a monopole at source S is,

pˆ(O) = ˆpI (O) +XpˆD(O), (28)

where ˆpI (O) is the complex pressure amplitude of the incident ray and ˆpD(O) are the complex

pressure amplitudes of the diffracted rays (see Eqns. 12, 15, and 17). If the observer is in a shadow

region, the incident ray is zero and the total diffracted field is the sum of all diffracted rays.

Equation 28 is the tailored Green’s function, g, with appropriate monopole source strength.

For particular source, observer, and airframe locations a discontinuity exists within the diffraction coefficient, D. The last terms of Eqn. 14 are,

. . .

1

cos (νπ) − cos (ν (φs − φo)) +

1

cos (νπ) − cos (ν (φs + φo + π)). (29)

Which are singular when φs − φo = ±π or φs + φo + π = ±π, where φs and φo are the polar

angles of the incident ray and the diffracted ray, as shown in Fig. 4, respectively. For example,

φs = 3π/2 results in a discontinuity. Here, the denominator of the first term in Eqn. 29, decreases as

φo approaches π/2 which is directly below the diffraction point. In this case, values of φo less than

π/2 are in the shadow region. To correct for the discontinuity, terms in Eqn. 29 are multiplied by a

damping function to change the behavior from ±∞ to 0+ and 0− respectively as the denominator

approaches zero. The damping function does not alter the terms far from the neighborhood of the

singularity. The proposed functions for multiplication of the first and second term of Eqn. 29 are,

17
f1 = 1 − exp h

−δ {cos[νπ] − cos[ν(φo − φs)]}

2

i

, (30)

and,

f2 = 1 − exp h−δ {cos[νπ] − cos[ν(φo + φs + π)]}

2

i

, (31)

where δ is a damping coefficient. Figure 8 shows the corrected term when multiplied by f1 for the

first term in Eqn. 29. This example corresponds to φs = 3π/2, ν = 5/9, and 0 ≤ φo ≤ π. The

uncorrected function is discontinuous at φo = π/2 and the corrected function is continuous and

finite. Note, there is little deviation between the two functions far from φo = π/2. The narrow

region affected by the correction function is dependent on the damping coefficient. Decreasing the

value of δ increases the range of values affected by the damping function. A value of δ = 30 yields

results that agree with measurement and are shown in the next section. The coefficient δ is held

constant for all comparisons.

φs

First Term of Diffraction Coefficient

0 0.5 1 1.5 2 2.5 3

-60

-40

-20

0

20

40

60

Uncorrected Function

Corrected Function

Figure 8. Example discontinuity correction.

18
4 Results

4.1 Steady RANS Solutions

Steady RANS solutions are examined to study the aerodynamics and turbulent statistics of free

and installed jets. The solutions are compared with PIV of free and installed jets The steady RANS

equations are solved with a computational fluid dynamics (CFD) approach using the NASA Fully

Unstructured Navier-Stokes three-dimensional [79] (FUN3D) solver. The Menter [80] shear stress

transport (SST) turbulence model is used to close the RANS equations. The model utilizes the

strengths of the Jones and Launder k- model [81] and the Wilcox k-ω model [82,83]. In this work,

the noise source is dependent upon quantities obtained from the steady RANS solution.

The coordinate system and geometry of the nozzle and plate are illustrated in Fig. 9. Coordinates x, y, and z are normalized by the nozzle exit diameter, D. The origin of the coordinate

system is the center of the nozzle exit. The positive x-axis is the jet centerline, the y-axis is normal

to the plate, and the z-axis is parallel to the plate. The plate is located multiple nozzle diameters

from the jet centerline axis, yp, and multiple nozzle diameters from the nozzle exit to the trailing

edge, xp.

Airframe Surface

x

p

yp

y

x

z

Jet Centerline

Lipline

xI

δη

Axisymmetric

 Nozzle D

yp-D/2

Figure 9. The coordinate system.

The plate positions examined relative to the nozzle exit are shown in Table 1. The first column

contains normalized streamwise distances from the nozzle exit to the edge of the flat plate and

the second column shows normalized radial distances to the flat plate. Table 2 shows the jet

operating conditions examined for each plate position. For each operating condition, free and

installed jet steady RANS solutions are obtained. The operating conditions include five subsonic

jets, a transonic jet, and three supersonic jets. Four of the conditions are heated. The nozzle

geometries, flow conditions, and plate positions coincide with the PIV measurements of Bridges

and Wernet [52].

The computational domain consists of a mixed element (structured-unstructured) grid. An

19
Table 1. Plate Locations.

Axial Distance (xp/D) Radial Distance (yp/D)

4 1, 2, 4, 6, 8, 10

10 1, 2, 4, 6, 8, 10

20 1, 2, 4, 6, 8, 10

Table 2. Jet Operating Conditions.

Nozzle Setpoint NPR TTR Mj

SMC000 3 1.197 1.000 0.513

SMC000 7 1.861 1.000 0.985

SMC000 23 1.102 1.814 0.376

SMC000 27 1.357 1.926 0.678

SMC000 29 1.888 2.118 1.000

SMC000 46 1.219 2.862 0.548

SMC016 11606 2.733 1.000 1.290

SMC016 11610 3.670 1.000 1.500

SMC016 11617 4.320 1.000 1.610

unstructured grid is used to resolve the flow around the airframe and a structured grid is used to

resolve the jet plume. All the CFD calculations are three-dimensional. Symmetry of the steady

RANS solution and computational domain allow for large computational savings. Figure 10 shows

an outline of the three-dimensional computational domain with the plate located two nozzle diameters, D, off-set from the jet axis and parallel to the jet centerline (yp/D = −2). po and To,

calculated from the nozzle pressure ratio (NPR) and TTR, are specified at the nozzle inlet plane.

The ambient static pressure is held constant at the far-field boundary. A symmetric boundary

condition is enforced on the x-y plane at z = 0. All other boundaries are defined with a free-stream

condition of M∞ = 0.01 and ambient pressure. The SMC000 nozzle profile and a portion of the

computational domain are shown in Fig. 11. This portion of the domain represents a slice in the

x-y plane at z = 0 normalized by D. The full computational domain extends 75D downstream,

100D cross-stream along the y-axis, and 50D cross-stream along the z-axis. This forms a rectangular prism. Note that the far-field z plane boundary condition implies a symmetric flow solution.

Typically, the number of grid points in the computational domain with the plate present is approximately 2.2 million and the number of elements is approximately 2.8 million. The number of grid

points in the computational domain with only the nozzle present is approximately 1.2 million and

the number of elements is approximately 1.5 million.

Validation of the free jet steady RANS solution is performed by comparing the streamwise

velocity component, u, and turbulent kinetic energy, k, with the PIV measurements of Bridges

and Wernet [52]. Here, mean velocities, k, and Reynolds stresses are important statistics for noise

prediction. We present the first two for validation of the steady RANS solutions and in subsequent

analysis.

A comparison of the steady RANS solution of a subsonic cold jet operating at Mj = 0.513 from

the convergent SMC000 nozzle is shown in Fig. 12. Here, contours show qualitative differences of

u and k. u and k are normalized by the fully expanded velocity and its square, respectively. The

20
X

Y

Z

Inlet Plate Outlet

(y-z plane)

Free Stream

Symmetry

(y-x plane)

Figure 10. An outline of the computational domain used for the CFD calculations. The flow of

the jet is in the positive x direction.

x/D

y/D

-5 -4 -3 -2 -1 0 1 2 3

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Figure 11. A portion of the computational domain on the x-y plane at z = 0 showing the

structured/unstructured grid of the convergent nozzle and flat plate located 2D from the centerline.

21
steady RANS solutions are shown on the top half planes and the corresponding PIV data are shown

on the bottom half planes. Note that the measured data near the nozzle exit is not available due

to the placement of the PIV cameras and flow seeding. u as shown in Fig. 12(a) has noticeable

variation from the PIV. The predicted thickness of the potential core past 6D is smaller than

measurement and the fall-off past 8D is higher. The predicted peak k occurs near 5.5D and the

PIV peak k occurs further downstream at 6.25D. Furthermore, the predicted magnitude of peak k

is smaller than measurement. However, the global qualitative agreement compares favorably with

other solutions produced by similar CFD codes (for example see Georgiadis et al. [84]).

x/D

y/D

0 2 4 6 8 10 12 14

-2

0

2

u/uj 0.95

0.85

0.75

0.65

0.55

0.45

0.35

0.25

0.15

0.05

(a) Normalized streamwise component of velocity.

x/D

y/D

0 2 4 6 8 10 12 14

-2

0

2

k/uj

2

0.026

0.022

0.018

0.014

0.01

0.006

0.002

(b) Normalized turbulent kinetic energy.

Figure 12. Contours of the steady RANS solution and PIV data. The steady RANS solutions are

shown on the top half planes and PIV data are shown on the bottom half planes. The jet operates

at Mj = 0.513 and TTR = 1.00.

Comparisons are conducted for the free jet by examining aerodynamic data along the centerline

and axial locations at x/D = 1, 4, and 16. Figure 13 shows the centerline variation of u/uj and k/u2

j

of both the steady RANS solution and PIV data. Figure 14 shows radial profile comparisons of the

same quantities. In Fig. 13(a), the predicted jet potential core length is larger than measurement

by 2D. The predicted u follows the same approximate 1/x decay as the measured data along the

centerline, shown in Fig. 13(a), and also matches the radial decay at x/D = 1 and 4, shown in

Fig. 14(a). Predicted k along the centerline matches the measurement in terms of peak magnitude

but the peak occurs one diameter further upstream than measurement. The peak magnitudes of

22
k at x/D = 1 and x/D = 4 are larger than measurement by 0.0015 and 0.001 k/u2

j

respectively

and are located 0.25D closer to the centerline than experiment. The solution under-predicts both

u and k relative to measurement far downstream from the nozzle exit.

x/D

u/uj

0 5 10 15 20 0

0.2

0.4

0.6

0.8

1

(a) Streamwise velocity component, u.

x/D

k/u

2j

0 5 10 15 20 0

0.005

0.01

0.015

0.02

0.025

Fun3D

PIV

(b) Turbulent kinetic energy, k.

Figure 13. The steady RANS a) u and b) k along the jet centerline compared with the measurement

of Bridges and Wernet [52]. The jet operates at Mj = 0.513 and TTR = 1.00 from the convergent

SMC000 nozzle.

Comparisons are presented for the supersonic over-expanded jet operating at Mj = 1.29 from

the Md = 1.5 convergent-divergent SMC016 nozzle. Again, the aerodynamic data is shown along

the centerline and radial locations at x/D = 1, 4, and 16. Figure 15 shows the centerline variation

of normalized u and k of both the steady RANS solution and PIV. Figure 16 shows radial profile

comparisons of the same quantities. As observed in Fig. 15, the predicted potential core length is

approximately 2D larger relative to measurement. Within the potential core, the predicted shockcell structure location compares favorably with measurement. The steady RANS solution overpredicts the rate of decay along the centerline as shown in Fig. 15(a) and matches the radial decay

in the potential core region at x/D = 1 and 4 as shown in Fig. 16(a). The steady RANS solution

under-predicts the radial decay through the shear layer greater than 0.5D from the centerline.

In the fully developed region of the flow at x/D = 16, the steady RANS solution only slightly

over-predicts the radial velocity profile. This is a favorable prediction relative to the k profile at

x/D = 16 shown in Fig. 16(b) and the subsonic jet predictions shown at x/D = 16 in Fig. 14.

On the centerline, peak k/u2

j

is approximately 0.003 below measurement and occurs 2D further

downstream than measurement as shown in Fig. 15(b). The k prediction agrees with experiment

in peak magnitude at x/D = 4 but over-predicts k/uj

2 at x/D = 1 and x/D = 16 by 0.015

and 0.005, respectively. Like the subsonic cold jet radial comparisons of k, radial comparisons

at far downstream locations over-predict measurement. These trends are representative of all jet

conditions shown in Table 2. The predictions are favorable relative to measurement based upon

this author’s experiences with steady RANS solutions for the jet conditions examined, especially so

in the potential core region of the flow-field where inviscid terms dominate the equations of motion.

The steady RANS solutions of a Mj = 0.513 and TTR = 1.00 jet from the convergent SMC000

23
y/D

u/uj

-1 -0.5 0 0.5 1

0

0.2

0.4

0.6

0.8

1

(a) Streamwise velocity component, u.

y/D

k/u

2j

-1 -0.5 0 0.5 1

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Fun3D x/D=1

Fun3D x/D=4

Fun3D x/D=16

PIV x/D=1

PIV x/D=4

PIV x/D=16

(b) Turbulent kinetic energy, k.

Figure 14. Radial profiles of steady RANS a) u and b) k compared with the measurement of

Bridges and Wernet [52]. The jet operates at Mj = 0.513 and TTR = 1.00 from the convergent

SMC000 nozzle.

nozzle in the presence of a flat plate are now compared with the free jet operating at the same

condition. Figure 17 shows radial profiles of k at x/D = 10 for multiple plate positions. The plate

is located at yp/D = −1, −2, −4, −6, −8, and −10 perpendicular to the jet centerline and extends

to xp/D = 10 and 20 downstream from the nozzle exit. Figure 17 shows that as the plate is moved

closer to the jet the k distribution is increasingly deformed. The peak magnitude closest to the

plate decreases and the peak magnitude furthest from the plate increases. This trend is amplified

when increasing the xp/D location from 10 to 20 for most of the cases examined. Furthermore,

the jet plume is being deformed and is drawn toward the plate due to a coanda like effect. For

example, in Fig. 17(a) the xp/D = 10 and free jet case both have a k local minimum at y/D = 0.

However, as the plate is moved closer as in Fig. 17(f) the local minimum of k for xp/D = 10 is now

located at y/D = −0.16. The effect of extending the plate from xp/D = 10 to xp/D = 20 amplifies

the effect of the deformation of the jet plume as shown in Fig. 17(a). The peak k close to the plate

is lower by 0.0005 k/uj

2 and the peak k furthest from the plate is higher by 0.001 k/uj2

. Similar

changes of the k distribution are observed for all jet conditions examined. These numerical results

show the changes in the aerodynamic characteristics of the jet plume induced by nearby surfaces.

These changes affect the noise source intensity and position as the plate location is altered.

24
x/D

u/uj

0 5 10 15 20 0

0.2

0.4

0.6

0.8

1

1.2

(a) Streamwise velocity component, u.

x/D

k/u

2j

0 5 10 15 20 0

0.005

0.01

0.015

0.02

0.025

Fun3D

PIV

(b) Turbulent kinetic energy, k.

Figure 15. The steady RANS a) u and b) k along the jet centerline compared with the measurement

of Bridges and Wernet [52]. The jet operates at Mj = 1.29 and TTR = 1.00 from the SMC016

nozzle.

y/D

u/uj

-1 -0.5 0 0.5 1

0

0.2

0.4

0.6

0.8

1

1.2

(a) Streamwise velocity component, u.

y/D

k/u

2j

-1 -0.5 0 0.5 1

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Fun3D x/D=1

Fun3D x/D=4

Fun3D x/D=16

PIV x/D=1

PIV x/D=4

PIV x/D=16

(b) Turbulent kinetic energy, k.

Figure 16. Radial profiles of steady RANS a) u and b) k compared with the measurement of

Bridges and Wernet [52]. The jet operates at Mj = 1.29 and TTR = 1.00 from the SMC016 nozzle.

25
y/D

k/u

2j

-1 -0.5 0 0.5 1

0.015

0.02

0.025

0.03

Isolated Jet

xp/D=10

xp/D=20

(a) yp/D = −10

y/D

k/u

2j

-1 -0.5 0 0.5 1

0.015

0.02

0.025

0.03

Isolated Jet

xp/D=10

xp/D=20

(b) yp/D = −8

y/D

k/u

2j

-1 -0.5 0 0.5 1

0.015

0.02

0.025

0.03

Isolated Jet

xp/D=10

xp/D=20

(c) yp/D = −6

y/D

k/u

2j

-1 -0.5 0 0.5 1

0.015

0.02

0.025

0.03

Isolated Jet

xp/D=10

xp/D=20

(d) yp/D = −4

y/D

k/u

2j

-1 -0.5 0 0.5 1

0.015

0.02

0.025

0.03

Isolated Jet

xp/D=10

xp/D=20

(e) yp/D = −2

y/D

k/u

2j

-1 -0.5 0 0.5 1

0.015

0.02

0.025

0.03

Isolated Jet

xp/D=10

xp/D=20

(f) yp/D = −1

Figure 17. Radial variation of k at x/D = 10 due to a jet operating at Mj = 0.513 and TTR =

1.00. Parts a) through f) show solutions at yp/D = −10, -8, -6, -4, -2, and -1 and xp/D = 10 and

20 (D = 0.0508 m).

26
4.2 The Variation of Γ with Jet Operating Condition and Airframe Position

The results of the aerodynamic assessment show that nearby solid surfaces change the aerodynamic

characteristics of the jet flow-field, even if the jet centerline is many diameters from a surface.

Thus, the method by which a jet engine is integrated with the airframe of a flight vehicle can have

significant effect on the aerodynamic source of sound. This is a highly complicated problem that

has received significant attention (see Czech [4] and Thomas [3,85] for example). It is characterized

by a large number of parameters that are highly interdependent. In this section, the model of

Tam and Auriault [19] is used to quantify the relative effect of propulsion airframe aeroacoustics

installation effects on the aerodynamic source. Γ is used as a basic guide to ascertain whether

the aerodynamic source is affected by the airframe relative to the equivalent free jet aerodynamic

source.

The model for spectral density, S, of Tam and Auriault [19] is,

S(x, ω) = 4π

 π

ln 2

3/2

Z ∞

−∞

Z ∞

−∞

Z ∞

−∞

qˆ

2

s

l

3

s

c

2

τs

|pa(x; y, ω)|

2

exp h−

ω

2

l

2

s

u

24 ln 2 i

1 + ω2τ

2

s



1 −

u

c∞

cos θ

2

dy, (32)

where ˆqs = (4/9)A2

c

2ρ2k2

s

is a statistical source term and ks is associated with turbulence that

produces fine-scale mixing noise. A is a constant coefficient, c is the speed of sound, u is the mean

streamwise velocity component, x is the observer position, y is the source position, and ω is the

radial frequency. ks is set equal to k computed from the steady RANS solution.

If predictions are restricted to the sideline direction and it is assumed that sound refraction

by the jet shear layer has negligible effect on S, then a simplified form of pa(x; y, ω) can be

constructed. As shown by Morris and Farassat [20], the adjoint acoustic pressure at Ψ = 90 deg.

(the jet sideline at Φ = 90 deg.) is, |pa(x2; y, ω)|

2 = ω2/(64π4

c

4

∞x

2

). The scales of turbulence in

Eqn. 32 are related to the steady RANS solution of the jet by simple dimensional models. The

coefficients are calibrated with the SMC000 nozzle operating at Mj = 1.00 and TTR = 1.00. The

values are A = 3548, cl = 0.018, and cτ = 0.015, and vary from those calculated by Tam and

Auriault [19] because the steady RANS solver and turbulence model differ.

The effects that the flat plate has on acoustic intensity originating from the jet aerodynamic

source are investigated by using the steady RANS solutions and Tam and Aurialt’s model. The free

jet predictions are subtracted from the installed jet predictions on a power spectral density (PSD)

basis for comparison. The first comparison consists of the cold subsonic jet at Mj = 0.513 and is

presented in Fig. 18. In Fig. 18(a), the plate is located at yp/D = −1, −2, −4, −6, −8, and −10 from

the jet centerline and extends xp/D = 10 downstream of the nozzle exit. In Fig. 18(b), the plate is

located at yp/D = −1 from the jet centerline and extends xp/D = 4, 10, and 20 downstream of the

nozzle exit. When the plate is xp/D = 10 downstream of the nozzle exit, the trends in Fig. 18(a)

show only a small effect on the noise spectrum as the plate approaches the centerline until the

plate is at yp/D = −2. The noise deviation from the free jet reaches a maximum of -1.2 dB at the

lowest frequency and 0.75 dB at the highest frequency when the plate is located at yp/D = −2. For

all plate locations further than 2D from the centerline, the noise deviations are within ±0.75 dB

relative to the free jet case. When the plate is located at yp/D = −1, the noise intensity is -6.9 dB

relative to the free jet case at the lowest frequency. The effect of varying the plate length relative

to the nozzle exit is displayed in Fig. 18(b). At the lowest frequency, the noise intensity deviation

from the free jet reaches a maximum of -6.5 dB, -6.9 dB, and -7.9 dB as the plate is extended to

xp/D = 4, 10, and 20 downstream from the nozzle exit, respectively. At the highest frequency, the

maximum deviation for each plate extension does not exceed 0.75 dB.

27
St

∆SPL per unit St (dB re 20 

μ Pa)

10-2 10-1 100 101 -8

-6

-4

-2

0

2

Isolated Jet

yp/D = -10

yp/D = -8

yp/D = -6

yp/D = -4

yp/D = -2

yp/D = -1

(a) xp/D = 10

St

∆SPL per unit St (dB re 20 

μ Pa)

10-2 10-1 100 101 -8

-6

-4

-2

0

2

Isolated Jet

x

p

/D = 4

x

p

/D = 10

x

p

/D = 20

(b) yp/D = −1

Figure 18. Predicted attenuation using the model of Tam and Auriault [19] for the free and

installed jet at R/D = 100 and Ψ = 90 deg. The jet operates at Mj = 0.513 and TTR = 1.00 from

the convergent SMC000 nozzle with exit diameter D = 0.0508 m. Part a) shows plate locations at

yp/D = -1, -2, -4, -6, -8, and -10 laterally from the jet centerline and xp/D = 10. Part b) shows

the plate located at yp/D = -1 and xp/D = 4, 10, and 20.

An over-expanded supersonic Mj = 1.29 jet from the convergent-divergent SMC016 nozzle at

various plate positions is now examined. A comparison is shown in Fig. 19(a) for the variation of

∆SPL as the surface approaches the jet centerline. Figure 19(b) shows the effect that the surface

has on ∆SPL as the plate length is extended relative to the nozzle exit. In Fig. 19(a), the noise

intensity deviation from the free jet reaches a maximum of -1 dB at the lowest frequency and

0.1 dB at the highest frequency when the plate is located at yp/D = −2. For all plate locations

further than 2D from the centerline, the noise deviations are within -0.6 dB relative to the free

jet case at the lowest frequency and -0.02 dB at the highest frequency. When the plate is located

at yp/D = −1, the noise reaches a maximum -6.4 dB difference relative to the free jet case. In

Fig. 19(b), the noise deviation relative to the free jet reaches a maximum of -5.7 dB, -6.4 dB, and

-7.7 dB at the lowest frequency as the plate extends xp/D = 4, 10, and 20 downstream from the

nozzle exit, respectively. The maximum dB difference relative to the free jet for each plate position

is slightly smaller in magnitude at lower frequencies relative to the subsonic jet. There is no increase

in noise intensity at high frequencies in contrast to the subsonic case.

The comparisons described are representative of the other jet conditions analyzed in this study.

As the jet approaches the plate there is a consistently larger difference in the noise spectrum relative

to the free jet at lower frequencies. This is due to the surface having a larger effect on the flow

further downstream from the nozzle exit. It has been shown by Brooks et al. [86] and Podboy [87],

for example, that peak noise sources are located near the nozzle exit at higher frequencies and lower

frequency peak noise sources are located multiple nozzle diameters downstream. The presence of

the plate has a larger effect on the jet flow multiple diameters from the nozzle exit and therefore

has a larger effect on the aerodynamic source at lower frequencies.

28
St

∆SPL per unit St (dB re 20 

μ Pa)

10-2 10-1 100 101 -8

-6

-4

-2

0

2

Isolated Jet

yp/D = -10

yp/D = -8

yp/D = -6

yp/D = -4

yp/D = -2

yp/D = -1

(a) xp/D = 10

St

∆SPL per unit St (dB re 20 

μ Pa)

10-2 10-1 100 101 -8

-6

-4

-2

0

2

Isolated Jet

x

p

/D = 4

x

p

/D = 10

x

p

/D = 20

(b) yp/D = −1

Figure 19. Predicted attenuation using the model of Tam and Auriault [19] for the free and

installed jet at R/D = 100 and Ψ = 90 deg. The jet operates at Mj = 1.29 and TTR = 1.00 from

the convergent SMC016 nozzle with D = 0.0508 m. Part a) shows plate locations at yp/D = -1, -2,

-4, -6, -8, and -10 laterally from the jet centerline and xp/D = 10. Part b) shows the plate located

at yp/D = -1 and xp/D = 4, 10, and 20.

Next, the effect of the plate on the spatial location of the acoustic source is examined by comparing contours of source intensity with varying plate position. This is accomplished by evaluating

the integrand of Eqn. 32. Contours of SPL per unit St for the cold subsonic Mj = 0.513 jet at a

frequency of 1 kHz (St ≈ 0.3) are shown in Fig. 20. In this comparison the plate length extends

to xp/D = 10 and the plate is located at yp/D = −10, −6, −4, −2, and −1 from the jet centerline.

When the plate is located at yp/D = −10 in Fig. 20(a), the noise source distribution is unaffected

by the presence of the plate. The two peak noise sources are located 9.5D downstream from the

nozzle exit and are symmetric in magnitude about the jet centerline. There is no significant deformation of the source distribution due to the presence of the plate further than 2D from the

jet centerline. As the plate approaches the jet at yp/D = −2, as shown in Fig. 20(d), the peak

magnitude of the source closest to the plate is decreased by approximately 1 dB and the location

is unchanged. When the plate is located at yp/D = −1, as shown in Fig. 20(e), the peak noise

source furthest from the plate is decreased in magnitude by approximately 1 dB and is shifted 0.5D

upstream relative to the yp/D = −10 case shown in Fig. 20(a). The peak noise source distribution

closest to the plate is significantly deformed. The peak magnitude is decreased by approximately

6 dB and is shifted 1D upstream relative to the yp/D = −10 case. An additional strong noise

source is formed close to the plate further downstream at x/D = 14. This is a distinguishable

difference from the other comparisons and is due to an increase in k past the trailing edge of the

plate. Figure 20 shows that the magnitude of the peak acoustic source closest to the plate decreases

as the plate surface approaches the jet centerline, and the magnitude of the peak acoustic source

furthest from the plate is not significantly altered until the plate is located at yp/D = −1. The

location of the peak noise sources is unaltered until the plate is located at yp/D = −1. Varying

the azimuthal angle, Φ, will have little effect on the results using this analysis. These comparisons

29
represent trends that are consistent for the other jet conditions analyzed.

Overall, the results show that the plate has a larger effect on the acoustic source as it approaches the jet centerline and as the plate length extends further downstream. As the effective

jet impingement area of the plate is increased, the acoustic intensity radiating from the jet mixing

noise decreases. Note, this does not account for additional sources produced by the jet interacting

with the surface.

x/D

y/D

0 2 4 6 8 10 12 14 16 18 20 -1

-0.5

0

0.5

1 SPL per St

98

97

96

95

94

93

92

91

90

89

(a) yp/D = −10

x/D

y/D

0 2 4 6 8 10 12 14 16 18 20 -1

-0.5

0

0.5

1

SPL per St

98

97

96

95

94

93

92

91

90

89

(b) yp/D = −6

x/D

y/D

0 2 4 6 8 10 12 14 16 18 20 -1

-0.5

0

0.5

1 SPL per St

98

97

96

95

94

93

92

91

90

89

(c) yp/D = −4

x/D

y/D

0 2 4 6 8 10 12 14 16 18 20 -1

-0.5

0

0.5

1 SPL per St

98

97

96

95

94

93

92

91

90

89

(d) yp/D = −2

x/D

y/D

0 2 4 6 8 10 12 14 16 18 20 -1

-0.5

0

0.5

1 SPL per St

98

97

96

95

94

93

92

91

90

89

(e) yp/D = −1

Figure 20. Contours of acoustic source strength at f = 1000 Hz (St ≈ 0.3). The jet operates at

Mj = 0.513 and TTR = 1.00. Parts a) through e) show the acoustic source with the plate located

at yp/D = −10, -6, -4, -2, and -1 respectively and xp/D = 10.

The non-dimensional number Γ is used to quantify the effect of a nearby surface on the jet

noise source. The overall sound pressure level (OASPL) is predicted over a frequency range of 20

Hz to 100 kHz using the steady RANS solutions and Eqn. 32 at Ψ = 90 deg. The installed jet

predictions of OASPL are subtracted from the free jet predictions on a PSD basis. Figure 21 shows

the absolute value of ∆OASPL as a function of Γ. The data collapse shows a critical value of Γ

30
of approximately unity. As the cross-stream length scales yp(yp − D/2) decrease to the equivalent

of the product of the jet and airframe length scales (Djxp tan[δη]), Γ increases from 0 to 1. In

Fig. 21, small values of Γ approaching approximately 1 have ∆OASPL less than 0.5 dB and are

considered negligible. As the jet spreading angle, fully expanded jet diameter, or surface length

increase, the product of the jet and airframe length scales surpasses the cross-stream length scales

and Γ increases beyond unity. As a result of the dominance of the jet and airframe length scales in

this region, 1 < Γ < ∞, the physical quantities of the aerodynamic flow are sufficiently altered and

have a large effect on the jet noise source. As shown in Fig. 21, values of Γ > 1 result in a range

of ∆OASPL from 2 dB to 2.75 dB. It is observed that ∆OASPL increases as Γ increases.

Γ

∆OASPL (dB re 20 

μ Pa)

10-2 10-1 100 101

-0.5

0

0.5

1

1.5

2

2.5

3

SP3

SP7

SP27

SP11606

SP11610

Figure 21. ∆OASPL, relative to the free jet, for each condition shown in Tables 1 and 2.

When Γ << 1, the airframe is expected to have a negligible effect on the jet noise source. For

instance, a supersonic on-design Mj = 1.5 cold jet with an airframe surface located at xp/D = 20

and yp/D = 4 corresponds to Γ ≈ 0.099 and ∆OASPL = 0.083 dB in Fig. 21. Since Γ << 1 and

there is a negligible difference relative to the free jet noise source, the free jet aerodynamic source

model along with a tailored Green’s function can be used for a prediction that includes propagation

about the airframe. For example, a Mj = 0.985 jet with an airframe surface located at xp/D = 4

and yp/D = −1 corresponds to Γ = 5.063 and ∆OASPL = 2.47 dB in Fig. 21. For large values

of Γ, some noise reduction is attributed to the change in the jet noise source and not to shielding

effects.

The parameter Γ can include additional terms involving multiple jet streams or account for

the boundary layer thickness on the airframe. An equivalent parameter for the fully expanded jet

diameter and jet spreading angle can be developed for a dual-stream jet. However, a new model

will be required for the estimation of the jet spreading angle. Inclusion of boundary layer effects

will increase the effect of the cross-stream length scales. The airframe boundary layer thickness

can be subtracted from the cross-stream length scales. A model to estimate the boundary layer

thickness would be required. The general effect of the airframe on the jet noise source as a function

of Γ is expected to be consistent for these cases but would likely have a different critical value.

31
4.3 Acoustic Validation

4.3.1 Ray Tracing Validation

This section describes comparisons of predicted and measured shielding of a point source by rectangular plates. Measurements of Ahtye and McCulley [53] are used to validate the ray tracing method

and its implementation. Unfortunately, the measured data is not available in a digital form and is

digitized electronically from technical reports. There is some error due to the digitization process.

The predictions are also compared with NASA’s Fast Scattering Code (FSC) developed by Tinetti

and Dunn [49].

4.3.2 Measurement Overview

Rectangular plates and cylinders are used to validate the ray tracing method. Here, we focus

on rectangular plates because of their sharp edges that are similar to airfoil trailing edges. The

rectangular plates have square and sharp tapered edges. The angles of the square and sharp edge

are β = 270 and 340 degs. The dimensions of the square and sharp edge are shown in Fig. 22. The

length and width of the plates is L = 2.0 m and W = 0.5 m.

Figure 23 shows the coordinate system used for the point source validation. The point source is

placed above the plate and is simulated with an inverse tapered horn connected to a loudspeaker. A

microphone is used to conduct measurements at various locations along the x-axis below the plate.

The microphone location on the traverse axis is designated by the angle θ relative to the source.

θ = 0 deg. is directly below the source and positive θ signifies a microphone position in the positive

x direction. See Ahtye and McCulley [53] for a full description of the measurements, sound source,

microphone, directivity, amplitudes, and calibration details. Ahtye and McCulley encountered

a number of difficulties during their experiment. Notable difficulties include: simulating a pure

omni-directional sound source, reflections from acoustic wedges lining the anechoic chamber walls,

possible error in the simulated point source location, and impedance loading of the source by

reflection from the plate. The sensitivity of the point source location relative to prediction will be

discussed.

(a) Square Edge (b) Sharp Edge

Figure 22. Rectangular plate edge dimensions.

Surface grids are created that correspond to the rectangular plates used in the measurement.

The number of elements in the domain is 54 and the number of nodes is 80. Note, ray tracing

solutions of rectangular geometry are independent of the number of nodes used. Figure 24 shows

the surface grid, the calculated extracted edge, and the source location. The acoustic source used

in the simulation consists of a stationary monopole of unit strength located at x = 0, y = 0,

and z = W/2. After identifying the edge of the geometry, the incident and diffracted rays are

determined for each microphone location, as shown in Fig. 25. The diffracted rays are summed for

32
L

w

x

y

z

0

θ

Shadow Region

Microphone Traverse

S(xs,ys,zs)

θ = 0 °

Figure 23. Coordinate system of the point source validation experiment.

the total complex acoustic pressure and converted to sound pressure level (SPL). The corresponding

incident free-field is subtracted from the diffracted field to determine the shielding in ∆SPL.

The predicted and measured ∆SPL is shown as a function of the traverse angle, θ, at various

frequencies. The traverse is located at z = −10W, y = 0, and ranges from −4W ≤ x ≤ 4.5W. The

range of available measured data varies by case. Angles, θ = ±45 deg., correspond to the edge of

the geometric shadow region. The Helmholtz number, kW, is shown for each comparison, where k

is the wave number and W is the width of the plate.

−2

−1

0

1

2

−1

−0.5

0

0.5

1

−2

−1.5

−1

−0.5

0

0.5

x (m) y (m)

z (m)

Figure 24. The diffraction edge is identified based on the source and observer location.

Comparisons of the square edge case, β = 270 deg., are shown in Fig. 26. The source excitation

frequencies are 1, 2, 4, and 12 kHz. The predictions illustrate that the overall trends of diffraction

are captured by ray tracing but with some noticeable error. The maxima and minima are generally

coincident for all frequency ranges presented with exception of the highest frequency. The captured

pattern of the diffracted field resembles the constructive and destructive interference between the

multiple diffracted waves produced by the edges. Also, the predicted diffraction pattern is sym33
−5

0

5

10

−1

−0.5

0

0.5

1

−5

−4

−3

−2

−1

0

1

x (m) y (m)

z (m)

Direct Rays

Diffracted Rays

Observers

Source

Figure 25. Incident and edge diffracted acoustic rays radiating from a monopole source above a

flat plate.

metric, that is due to the geometry and source directivity symmetry. The slight asymmetry of the

measured data is likely due to the difficulty of placing the source and its inherent directivity. The

ray tracing method consistently under-predicts the measured shielding for all frequencies, where

FSC over-predicts the shielding for 1, 2, and 4 kHz. The largest deviation from the ray traced

and measured peaks ranges from 4-7 dB, except near θ = 45 deg., while FSC over-predicts the

measurements by 4-5 dB and is within ±8 dB at 4 kHz. The discontinuous jump present in the ray

tracing prediction at θ = 45 deg. is due to the addition of direct rays to the diffracted field outside

of the shadow region. These discontinuities in the diffracted field were also observed by Agarwal et

al. [72].

Comparisons for a sharp edge case are shown in Fig. 27 where β = 340 deg. Measured data for

this case are available at frequencies of 4 and 12 kHz. The discontinuity in the prediction is again

apparent at θ = 45 deg., where the shadow region ends. In contrast to the square edge case, the

ray tracing predictions show improved agreement in terms of the shielding magnitude, especially at

small angles. Within the range |θ| < 10 deg., the ray tracing predictions are within ≈ ±3 dB of the

measurement at the intensity peaks and ≈ ±4 dB at the minima. Ray traced predictions also agree

well with the FSC prediction in this range. The larger deviation of the maxima and minima (> 5

dB) occur closer to the edge of the shadow region. Some of the deviation in this region is attributed

to the discontinuity correction. Ray traced predictions are compared to FSC at f = 1 kHz and 2

kHz. At 1 kHz, the ray tracing prediction does not capture the same diffraction pattern as the FSC

prediction but is in close agreement of the peak magnitude at θ = 0 deg. This illustrates the low

frequency limit of ray tracing. At 2 kHz, the prediction of the diffraction pattern generally agrees

with the FSC prediction, except near the edge of the shadow zone. There is a 4 dB difference at

the maxima and a much larger difference (> 10 dB) at the minima. The larger difference at the

minima is due to a larger phase sensitivity for destructive wave interference.

34
θ

∆ SPL (dB re 20 

μ Pa)

-20 0 20 40 60 -35

-30

-25

-20

-15

-10

-5

0

5

Ray Theory

FSC

Experiment

(a) f = 1000 Hz and kW = 9

θ

∆ SPL (dB re 20 

μ Pa)

-20 0 20 40 60 -35

-30

-25

-20

-15

-10

-5

0

5

(b) f = 2000 Hz and kW = 18

θ

∆ SPL (dB re 20 

μ Pa)

-20 0 20 40 60 -35

-30

-25

-20

-15

-10

-5

0

5

(c) f = 4000 Hz and kW = 37

θ

∆ SPL (dB re 20 

μ Pa)

-20 0 20 40 60 -35

-30

-25

-20

-15

-10

-5

0

5

(d) f = 12000 Hz and kW = 110

Figure 26. Square edge shielding predictions compared to measurement with source at x = 0, y = 0,

and z = W/2.

35
θ

∆ SPL (dB re 20 

μ Pa)

-20 0 20 40 60 -35

-30

-25

-20

-15

-10

-5

0

5

Ray Theory

FSC

Experiment

(a) f = 1000 Hz and kW = 9

θ

∆ SPL (dB re 20 

μ Pa)

-20 0 20 40 60 -35

-30

-25

-20

-15

-10

-5

0

5

(b) f = 2000 Hz and kW = 18

θ

∆ SPL (dB re 20 

μ Pa)

-20 0 20 40 60 -35

-30

-25

-20

-15

-10

-5

0

5

(c) f = 4000 Hz and kW = 37

θ

∆ SPL (dB re 20 

μ Pa)

-20 0 20 40 60 -35

-30

-25

-20

-15

-10

-5

0

5

(d) f = 12000 Hz and kW = 110

Figure 27. Sharp edge shielding predictions compared to measurement with source at x = 0, y = 0,

and z = W/2.

36
4.3.3 Effect of Edge Angle

To quantify the effect of edge angle variation on the diffracted field, ray tracing predictions in

the shadow region are examined while varying β. FSC and measurements are also shown at angles

β = 270 and 340 degs. Figure 28 shows these comparisons at f = 4 kHz. Predictions demonstrate an

increased shielding effect as edge angle increases. The magnitude of the constructive interference

pattern decreases. The relative decrease and increase of the maxima and minima illustrate a

flattening effect as the edge angle increases. There is a maximum 7 dB decrease in magnitude as

the edge angle increases from β = 270 deg. to β = 340 deg. at θ = 0 deg., resulting in an average

0.1 dB per deg. decrease. The relative magnitude difference from crest to trough decreases as

|θ| increases. Measurement shows only a small difference in attenuation between the two angles.

Interestingly, the FSC predictions do not follow the same trend as the ray tracing method. FSC

predictions show an overall decrease in shielding for the sharp edge (β = 340 deg.) case. The

relative difference in magnitude from peak to trough does decrease, which slightly mimics the

flattening trend of the diffraction pattern demonstrated by the ray tracing method. Ray traced

predictions show a stronger sensitivity to the sharp edge angle relative to FSC and measurement.

θ

∆

μ

-30 -20 -10 0 10 20 30 -35

-30

-25

-20

-15

-10

-5

0

β = 270°

β = 290°

β = 310°

β = 330°

β = 340°

β

(a) Ray Tracing Method

θ

∆

μ

-30 -20 -10 0 10 20 30 -35

-30

-25

-20

-15

-10

-5

0

β = 270° (Square Edge)

β = 340° (Sharp Edge)

β

(b) Experiment

θ

∆

μ

-30 -20 -10 0 10 20 30 -35

-30

-25

-20

-15

-10

-5

0

β = 270° (Square Edge)

β = 340° (Sharp Edge)

β

(c) Fast Scattering Code

Figure 28. Noise attenuation dependence on edge angle at f = 4000 Hz.

37
4.3.4 Point Source Sensitivity

In the experiment of Ahtye and McCulley [53], the opening at the small end of the inverse tapered

horn has a diameter of DT = 6.3 mm. This finite distance causes additional uncertainty when a

point source is used with ray tracing predictions. At high frequencies the uncertainty produced by

this distance may explain the deviation in the diffraction patterns shown in Figs. 26(d) and 27(d).

At the highest frequency presented, the acoustic wavelength is within an order of magnitude of the

horn diameter. Thus, the source cannot be approximated as a point source, and also likely contains

a non-uniform directivity pattern. Table 3 describes the range of frequencies and their acoustic

wavelength ratios relative to the diameter of the tapered horn.

Table 3. Predictions and measured point source conditions.

f (Hz) kW k (m−1) λ (m) DT /λ

1000 9 18 0.35 0.02

2000 18 36 0.17 0.04

4000 37 74 0.09 0.07

8000 73 146 0.04 0.16

12000 110 220 0.03 0.22

Predictions using various point source locations within the diameter of the horn source are

examined to assess the sensitivity of the source placement. The simulated source placement is

varied along the x-axis from −0.5DT to 0.5DT . Figure 29 shows the diffraction pattern in the

shadow zone for source locations at 8 and 12 kHz. Varying the source location on the x-axis by

one half horn diameter significantly shifts the diffraction pattern. At 8 kHz the location of the

maxima and minima are phase shifted for each source location. A consistent 2-3 dB difference is

observed in magnitude due to the phase shift. The diffraction pattern from −DT /2 to DT /2 is

symmetric about θ = 0 deg. because of inherent symmetry. At 12 kHz the diameter of the horn tip

is approximately a quarter of a wavelength and the diffraction pattern shifts if the source is moved

DT /2 in either direction. A full DT displacement in source location results in a pattern that is

phase shifted with large deviations (> 6 dB) in magnitude relative to measurement. Frequencies

with wavelengths comparable to the radius of the horn result in a phase shifted diffraction pattern.

The sensitivity of the diffraction pattern to the simulated source location can explain some of the

disagreement observed relative to measurement at high frequencies.

The ray tracing method consistently under-predicts the shielding of the square edge plate and

demonstrates improved agreement for the sharp edge plate relative to measurement. The FSC

predictions are superior for the square edge case at higher frequencies, although the ray tracing

predictions are within the same approximate dB error range at lower frequencies. Recall, both the

present theory and FSC are not formulated for square edge geometries. The ray traced predictions

show the largest discrepancies at the lowest frequency and in regions near the edge of the shadow

zone. A Helmholtz number of kW ≈ 9 is identified as a low frequency limit. Predictions of

the diffracted field from a source varied within the finite distance of the horn diameter show the

diffraction pattern is sensitive within the diameter at high frequencies. Despite the discrepancies

relative to measurement, the ray tracing method is capable of predicting the correct trends of the

diffraction pattern in the shielded region.

38
θ

∆ SPL (dB re 20 

μ Pa)

-20 -10 0 10 20 -40

-35

-30

-25

-20

-15

S = [-0.5DT, 0.0, 0.5W]

S = [0.0, 0.0 , 0.5W]

S = [0.5DT, 0.0, 0.5W]

x

z

W

DT

(a) f = 8000 Hz and kW = 73

θ

∆ SPL (dB re 20 

μ Pa)

-20 -10 0 10 20 -40

-35

-30

-25

-20

-15

S = [-0.5DT, 0.0, 0.5W]

S = [0.0, 0.0 , 0.5W]

S = [0.5DT, 0.0, 0.5W]

x

z

W

DT

(b) f = 12000 Hz and kW = 110

Figure 29. Source location sensitivity at high frequencies.

5 Predictions Compared with the NASA Jet-Surface Interaction

Test (JSIT)

In this section, an assessment of the developed acoustic analogy combined with the ray tracing

approach for the mixing component of jet noise in the presence of a scattering surface is presented.

Measured data from the Jet-Surface Interaction Test (JSIT) of Brown et al. [54] are used to validate

the predictions of jets disturbed by flat plates. The multi-phase experiment performed at NASA

Glenn Research Center was intended to support researchers developing aircraft noise prediction

tools by supplying data for a wide range of surface geometries, positions, and jet conditions. The

first phase of the measurement involved capturing far-field noise from a jet near a simple planar

surface with varying positions relative to the nozzle exit. A flat plate mounted on a two-axis

traverse was used to simulate a shield or reflecting surface between the jet plume and the observer.

Comparisons with the shielded measurements are used to validate the developed model. The

surface was moved through axial positions 2 < xp/D < 20 and radial positions −1 < yp/D < −10.

Recall, xp is the axial distance from the nozzle exit to the trailing edge and yp is the distance from

the jet centerline to the plate surface. The exit diameter of the SMC series nozzles is D = 0.0508

m. The plate positions are illustrated in Fig. 30. In the measurement, the span of the plate is

100D and the nozzle centerline is in the middle. Also, in the measurement the edge of the plate

in the sideline and upstream direction are covered with welding blankets. The plate was made

of aluminum and was 0.0127 m thick with a 0.00635 m trailing edge angled at β = 320.8 deg.

The far-field noise data were acquired from an azimuthal array of 24 microphones centered on

the nozzle exit with a constant radius of 3.81 m (75D). The microphones were distributed along

the arc at 5 deg. intervals ranging from Ψ = 50 deg. to 165 deg. relative to the jet upstream

axis. The angles are illustrated in Fig. 30. A photograph of the experiment is shown in Fig. 31.

Predictions are compared at various angles on the microphone arc. The range of experimental

data used to validate the prediction model includes the plate configurations shown in Table 1 and

the jet operating conditions shown in Table 2. The measured data are processed to account for

atmospheric absorption (see Bass et al. [88,89] for details) and corrected to a propagation distance

39
of 100D.

Measurements have also been conducted by Bridges and Brown [90] for the same jet conditions

but without the flat plate geometry. These are used to illustrate the prediction capability for free

jets and the relative differences between the shielded predictions and measurements. Results are

presented as Sound Pressure Level (SPL) per unit St, where St = fDj/uj is the Strouhal number,

and Dj and uj are the fully expanded jet diameter and fully expanded velocity respectively. In the

notation presented, Small Hot Jet Acoustic Rig (SHJAR) refers to measurements of the free jet

and JSIT refers to the installed jet.

Flat Plate

x

p

yp

y

x

Jet Centerline

Axisymmetric

 Nozzle

D

Ψ

R/D = 75

Microphone Arc Array

Figure 30. Coordinate system of the JSIT.

The first comparisons are shown in Fig. 32 for a jet operating at Mj = 0.678 and TTR = 1.926

from the SMC000 nozzle. The plate is located at xp/D = 10 and yp/D = −2. The predictions

at observer angles Ψ = 50, 90, 110, and 150 deg. from the upstream jet axis are shown. The last

information displayed in Fig. 32 is the parameter Γ calculated from the jet condition and plate

location. All the following figures use this convention. At Ψ = 50 deg. in Fig. 32, the free jet

prediction matches the experiment. The installed jet prediction and measurement both have a

high frequency decay beginning at St ≈ 0.3. At Ψ = 90 deg., the free jet prediction deviates

from experiment at higher frequencies; however, the relative decrease in SPL from the shielding

prediction agrees with the experiment. The installed jet prediction and measurement show a

maximum decrease of 24 dB relative to the free jet prediction and measurement respectively at

St ≈ 10. At the downstream angle, Ψ = 110 deg., the maximum predicted shielding also agrees

well with the experiment. The free jet prediction shows larger deviations at lower frequencies (6 to

8 dB). There is no predicted attenuation at Ψ = 150 deg. at high St, but the measurement shows a

8 to 10 dB decrease of SPL. For all observer angles, the maximum shielding for both the prediction

and measurement occurs at the highest frequency. Jet structure interaction noise is present at lower

frequencies, which is alluded to by the large value of Γ = 0.469. Recall, the developed prediction

method accounts only for the jet mixing noise. Apart from the deviation in the lower frequency

range where the additional noise sources dominate, the shielded predictions agree well with the

experiment at all observer angles except downstream at Ψ = 110 and 150 degs. At Ψ = 50 and

40
90 degs., the predicted shielding is within ±4 dB of the measurement above the peak jet mixing

noise frequency. The prediction agrees within ±6 dB of the experiment at Ψ = 110 deg. There is

insignificant predicted shielding at the highest downstream angle of Ψ = 150 deg.

A second comparison is shown in Fig. 33 for the same jet condition but with the plate located

at xp/D = 20 and yp/D = −2. There are additional sources present in the measurement due to jet

structure interaction noise. This is predicted by the larger value of Γ = 0.994. However, there is less

noise intensity increase at low frequencies compared to the xp/D = 10 case. At frequencies higher

than those of jet structure interaction noise, the prediction agrees within ±2 dB of the experiment

at the upstream and sideline observer angles. The prediction is also in better agreement with the

measurement at the downstream angles relative to other plate configurations. At Ψ = 130 deg., the

prediction is within ±4 dB of the measurement. In contrast with the previous two plate locations,

the shielding is captured by the prediction method at Ψ = 150 deg. The high frequency decay

occurs at a higher cut-off frequency, St ≈ 0.2, relative to experiment, St ≈ 0.1, but the attenuation

is within ±4 dB at frequencies St > 1.

In Fig. 34, comparisons are presented for a trailing edge position of xp/D = 20 and yp/D = −6.

Similar agreement is observed relative to the case with the surface located at yp/D = −2, except

at Ψ = 150 deg. At Ψ = 50 and 90 degs. there is strong agreement at high frequencies within

±2 dB. The predicted attenuation rate at frequencies greater than the peak frequency matches the

measurement at all angles except Ψ = 150 deg. There is no predicted shielding at the downstream

angle Ψ = 150 deg. In contrast to the yp/D = −2 case, the value of Γ = 0.09 is low and there

is only a small region at low frequencies where there is an increase in noise. These comparisons

display the trends for all plate locations listed in Table 1. The predictions of the cold subsonic

Mj = 0.985 jet also follow the same trends and show the same relative agreement as with the

Mj = 0.678 comparisons.

We now turn our attention to supersonic jets. Figure 35 shows spectra from a cold on-design

Mj = 1.5 jet from the SMC016 nozzle with the plate located at xp/D = 20 and yp/D = −2. There

is significantly less shielding for this plate configuration compared to the equivalent subsonic case.

When the potential core length extends further downstream from the nozzle exit, there is a larger

distribution of equivalent sources past the trailing edge of the plate that are not shielded. The jet

operating condition and plate location result in Γ = 0.461 and additional jet-surface interaction

noise is observed. The predicted SPL of the shielded configuration approaches measurement as the

frequency increases and the contribution of surface interaction noise diminishes. The deviation from

the experiment decreases with increasing frequency to within ±1 dB at observer angles Ψ = 50,

90, and 110 degs. At Ψ = 150 deg., the shielding is over-predicted by up to 6 dB. The maximum

attenuation at Ψ = 150 deg. agrees with the measurement, although the high frequency decay

occurs at a higher cut-off frequency, St = 0.4, than experiment, St = 0.1.

Predictions and measurements of spectra are shown in Fig. 36 for an over-expanded Mj = 1.29

jet from the SMC016 nozzle. The plate is located at xp/D = 20 and yp/D = −2. Since the

jet is off-design, shocks are present in the jet plume and resulting screech tones and BBSAN are

observed in the measurements. Although the predictions do not take shock-associated noise into

account, agreement is observed in the relative attenuation of the turbulent mixing noise. Outside

the frequency range affected by the BBSAN, the predicted shielding follows the same high frequency

rate of attenuation at all observer angles except Ψ = 150 deg. At Ψ = 150 deg., the predicted

attenuation begins at a higher cut-off frequency and has a larger attenuation rate. This jet condition

and plate location result in Γ = 0.602, and additional noise from the surface interaction is observed

in the measurement.

Though disagreement between prediction and measurement is observed for select conditions and

observer angles, most of the jet conditions and plate positions examined show that the overall trends

41
are captured by prediction. The comparisons shown are representative of all plate configurations

and jet conditions examined. We summarize the prediction capability for installed jet mixing noise

as follows:

1. The comparisons show that the acoustic analogy and tailored Green’s function provided by

ray tracing are capable of correctly capturing trends in measurement.

2. The predictions show strong agreement at higher frequencies. This is due to the use of ray

tracing.

3. The predictions show strong agreement for large xp/D, especially at upstream and sideline

observer angles.

4. The prediction method captures the decrease in shielding effectiveness as the jet exit velocity

increases.

Figure 31. A photograph of the jet structure interaction test courtesy of Podboy [87].

42
St

SPL per unit St (dB re 20 

μ Pa)

10-2 10-1 10010

60 1

70

80

90

100

110

120

Ψ = 150o

St

SPL per unit St (dB re 20 Pa)

10-2 10-1 10010

60 1

70

80

90

100

110

Ψ = 110o

St

SPL per unit St (dB re 20 Pa)

10-2 10-1 10010

50 1

60

70

80

90

100

Ψ = 90o

St

SPL per unit St (dB re 20 Pa)

10-2 10-1 10010

50 1

60

70

80

90

100

SHJAR

JSIT

Free Prediction

Shield Prediction

Ψ = 50o

=0.469

Figure 32. The free and installed jet predictions compared with measurements. The jet operates

at Mj = 0.678 and TTR = 1.926 from the SMC000 nozzle with D = 0.0508 m and observers

at R/D = 100. The plate is located at xp/D = 10 and yp/D = −2. These conditions result in

Γ = 0.469.

43
St

SPL per unit St (dB re 20 

μ Pa)

10-2 10-1 10010

40 1

50

60

70

80

90

100

110

Ψ = 130o

St

SPL per unit St (dB re 20 Pa)

10-2 10-1 10010

50 1

60

70

80

90

100

110

120

Ψ = 150o

St

SPL per unit St (dB re 20 Pa)

10-2 10-1 10010

40 1

50

60

70

80

90

100

110

Ψ = 90o

St

SPL per unit St (dB re 20 Pa)

10-2 10-1 10010

40 1

50

60

70

80

90

100

110

SHJAR

JSIT

Free Prediction

Shield Prediction

Ψ = 50o

=0.994

Figure 33. The free and installed jet predictions compared with measurements. The jet operates

at Mj = 0.678 and TTR = 1.926 from the SMC000 nozzle with D = 0.0508 m and observers

at R/D = 100. The plate is located at xp/D = 20 and yp/D = −2. These conditions result in

Γ = 0.994.

44
St

SPL per unit St (dB re 20 

μ Pa)

10-2 10-1 10010

50 1

60

70

80

90

100

110

Ψ = 130o

St

SPL per unit St (dB re 20 

μ Pa)

10-2 10-1 10010

60 1

70

80

90

100

110

120

Ψ = 150o

St

SPL per unit St (dB re 20 

μ Pa)

10-2 10-1 10010

40 1

50

60

70

80

90

100

Ψ = 90o

St

SPL per unit St (dB re 20 

μ Pa)

10-2 10-1 10010

40 1

50

60

70

80

90

100

SHJAR

JSIT

Free Prediction

Shield Prediction

Ψ = 50o

Γ=0.090

Figure 34. The free and installed jet predictions compared with measurements. The jet operates

at Mj = 0.678 and TTR = 1.185 from the SMC000 nozzle with D = 0.0508 m and observers

at R/D = 100. The plate is located at xp/D = 20 and yp/D = −6. These conditions result in

Γ = 0.090.

45
St

SPL per unit St (dB re 20 

μ Pa)

10-2 10-1 10010

80 1

90

100

110

120

130

Ψ = 150o

St

SPL per unit St (dB re 20 Pa)

10-2 10-1 10010

70 1

80

90

100

110

120

Ψ = 110o

St

SPL per unit St (dB re 20 Pa)

10-2 10-1 10010

70 1

80

90

100

110

120

Ψ = 90o

St

SPL per unit St (dB re 20 Pa)

10-2 10-1 10010

70 1

80

90

100

110

120

SHJAR

JSIT

Free Prediction

Shield Prediction

Ψ = 50o

=0.461

Figure 35. The free and installed jet predictions compared with measurements. The jet operates

at Mj = 1.50 and TTR = 1.00 from the SMC016 nozzle with D = 0.0508 m and observers at

R/D = 100. The plate is located at xp/D = 20 and yp/D = −2. These conditions result in

Γ = 0.461.

46
St

SPL per unit St (dB re 20 

μ Pa)

10-2 10-1 10010

60 1

70

80

90

100

110

120

Ψ = 130o

St

SPL per unit St (dB re 20 

μ Pa)

10-2 10-1 10010

70 1

80

90

100

110

120

130

Ψ = 150o

St

SPL per unit St (dB re 20 Pa)

10-2 10-1 10010

60 1

70

80

90

100

110

120

Ψ = 90o

St

SPL per unit St (dB re 20 Pa)

10-2 10-1 10010

60 1

70

80

90

100

110

120

SHJAR

JSIT

Free Prediction

Shield Prediction

Ψ = 50o

=0.602

Figure 36. The free and installed jet predictions compared with measurements. The jet operates

at Mj = 1.29 and TTR = 1.00 from the SMC016 nozzle with D = 0.0508 m and observers at

R/D = 100. The plate is located at xp/D = 20 and yp/D = −2. These conditions result in

Γ = 0.602.

47
6 Summary and Conclusion

We have presented a new approach for predicting the scattering of jet mixing noise about airframe

surfaces. The method is based on an acoustic analogy that involves the vector Green’s function of

the LEE and the source based on steady RANS solutions. The vector Green’s function is written

in terms of the Green’s function of the convective wave equation. This latter Green’s function is

found numerically with a ray tracing approach. Aerodynamic statistics of free and installed jets

are validated with PIV measurement and the predicted far-field noise is compared with microphone

measurement. The combination of the developed theory, ray tracing, and careful implementation in

a stand-alone computer program result in an approach that is more first principles than alternatives,

numerically efficient, and captures the relevant physics of fluid-structure interaction and airframe

geometry.

A non-dimensional number, Γ, is proposed that is used as a basic guide to determine if the

aerodynamic source is affected by the airframe relative to the equivalent isolated jet aerodynamic

source. The critical value of Γ predicts the on-set of jet structure interaction noise and when the

jet deforms due to the presence of the airframe.

We summarize the comparisons of prediction with measurement and the theory with the following conclusions.

Aerodynamic Analysis

1. The aerodynamic assessment shows that nearby solid surfaces change the aerodynamic characteristics of the jet flow-field, even if the jet centerline is multiple diameters from a solid

surface.

2. The effect of the airframe surface on the jet aerodynamics increases dramatically as the

airframe approaches the nozzle axis and as the trailing edge extends further downstream.

Unsteady Aerodynamic Noise Source Analysis

1. Changes within the aerodynamic flow-field due to the presence of the airframe have a direct

effect on the unsteady aerodynamic noise source.

2. The overall jet mixing noise intensity is decreased as the effective impingement surface area

of the airframe is increased.

3. Γ is used as a guide to determine the on-set of jet structure interaction noise.

Jet-Surface Shielding Analysis

1. The ray traced predictions are validated with measurement and predict the correct trends of

diffraction in the shadow region over a wide range of frequencies.

2. Predictions of scattered jet mixing noise are validated with measured data and capture the

correct trends relative to jet operating conditions and airframe position.

48
References

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55
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The Prediction and Analysis of Jet Flows and Scattered Turbulent Mixing 

Noise About Flight Vehicle Airframes 

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Miller, Steven A. E.

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14. ABSTRACT

Jet flows interacting with nearby surfaces exhibit a complex behavior in which acoustic and aerodynamic characteristics are altered. The physical 

understanding and prediction of these characteristics are essential to designing future low noise aircraft. A new approach is created for predicting scattered jet 

mixing noise that utilizes an acoustic analogy and steady Reynolds-averaged Navier-Stokes solutions. A tailored Green's function accounts for the propagation 

of mixing noise about the airframe and is calculated numerically using a newly developed ray tracing method. The steady aerodynamic statistics, associated 

unsteady sound source, and acoustic intensity are examined as jet conditions are varied about a large flat plate. A non-dimensional number is proposed to 

estimate the effect of the aerodynamic noise source relative to jet operating condition and airframe position.The steady Reynolds-averaged Navier-Stokes 

solutions, acoustic analogy, tailored Green's function, non-dimensional number, and predicted noise are validated with a wide variety of measurements. The 

combination of the developed theory, ray tracing method, and careful implementation in a stand-alone computer program result in an approach that is more first 

principles oriented than alternatives, computationally efficient, and captures the relevant physics of fluid-structure interaction. 

15. SUBJECT TERMS

Acoustic analogy; Diffraction; Fluid-structure interaction; Green's function; Jet; Jet flow; Jet noise; Radiation; Ray theory; 

Shielding

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