Stewart J. Leib

Ohio Aerospace Institute, Brook Park, Ohio 

Modeling Sound Propagation Through 

Non-Axisymmetric Jets 

NASA/CR—2014-218107

March 2014
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Stewart J. Leib

Ohio Aerospace Institute, Brook Park, Ohio 

Modeling Sound Propagation Through 

Non-Axisymmetric Jets 

NASA/CR—2014-218107

March 2014

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared under Contract NNC07BA13B
Acknowledgments

This work was supported by the NASA Fundamental Aeronautics Program High-Speed Project. The author would like to 

thank Mr. E. Brian Fite and Dr. James Bridges for their support and Mr. Daniel Ingraham for 

his comments on an early draft of this report. 

Available from

NASA Center for Aerospace Information

7115 Standard Drive

Hanover, MD 21076–1320

National Technical Information Service

5301 Shawnee Road

Alexandria, VA 22312

Available electronically at http://www.sti.nasa.gov

This work was sponsored by the Fundamental Aeronautics Program 

at the NASA Glenn Research Center.

Level of Review: This material has been technically reviewed by NASA technical management OR expert reviewer(s).

This report contains preliminary findings, 

subject to revision as analysis proceeds.
NASA/CR—2014-218107 1

Modeling Sound Propagation Through Non-Axisymmetric Jets 

Stewart J. Leib

Ohio Aerospace Institute

Brook Park, Ohio 44142

Abstract

A method for computing the far-field adjoint Green’s function of the generalized acoustic analogy 

equations under a locally parallel mean flow approximation is presented. The method is based on 

expanding the mean-flow-dependent coefficients in the governing equation and the scalar Green’s 

function in truncated Fourier series in the azimuthal direction and a finite difference approximation in the 

radial direction in circular cylindrical coordinates. The combined spectral/finite difference method yields 

a highly banded system of algebraic equations that can be efficiently solved using a standard sparse 

system solver. The method is applied to test cases, with mean flow specified by analytical functions, 

corresponding to two noise reduction concepts of current interest: the offset jet and the fluid shield. 

Sample results for the Green’s function are given for these two test cases and recommendations made as 

to the use of the method as part of a RANS-based jet noise prediction code.

1.0 Introduction

Exhaust system noise emission continues to be an issue for commercial and military aircraft and is,

therefore, the subject of ongoing research efforts as to its understanding, prediction and reduction. 

National and international regulations are imposing significant limitations on the noise footprint of 

aircraft operations, which are driving the development of noise-reduction concepts. Nearly all of these 

concepts involve the use of non-circular geometries to attempt to reduce overall noise emissions, or tailor 

the directivity characteristics in a desired manner. Robust and reliable prediction tools are needed to 

assess the noise-reduction potential of proposed concepts. In addition, since each concept typically 

involves numerous parametric options, noise prediction codes that can provide results relatively quickly, 

i.e., in a time period short enough to include them in the design process, are especially needed. Although 

progress continues to be made on direct numerical simulation of noise from turbulent flows, such 

calculations are still much too time consuming to be used in the design and optimization processes for 

new aircraft exhaust systems and reduced-order methods, therefore, are needed. 

Most practical noise prediction codes are based on an acoustic analogy, wherein the sound-generating 

turbulent flow is replaced by a set of equivalent sources with the sound propagating through a specified 

base flow. The acoustic analogy approach was pioneered by Lighthill (Refs. 1 and 2). Subsequent 

formulations, such as that Lilley (Ref. 3), explicitly account for mean flow interaction effects in the linear 

differential wave operator. The JeNo code (Refs. 4 and 5) is based on the Lilley (Ref. 3) analogy and is an 

example of the type of statistically based prediction code which can be used for parametric studies in

round jets. 

Goldstein (Ref. 6) showed that the Navier-Stokes equations can be rewritten as a set of `base’ flow 

equations plus a formally linear (about the base flow) set of equations for the ‘residual’ component (the 

difference between the total and base flow), and that the latter can be put into the form of the linearized 

Navier-Stokes equations with source terms whose strengths are represented by a generalized Reynolds 

stress. The general formulation of (Ref. 6) was used by Goldstein and Leib (Ref. 7) to develop an acoustic 

analogy-based approach for jet noise predictions from information about the turbulence. The method was 

applied to round, unheated jets in References 7 and 8 and the results showed good agreement with

experimental data for subsonic and moderately supersonic jets at all polar angles throughout the range of 

interest.
NASA/CR—2014-218107 2

Implementations of acoustic analogy formulations typically rely on the computation of a Green’s 

function to describe sound propagation through the base flow coupled with models for the nominal source 

terms arising from the particular analogy formulation being used. For an axisymmetric, locally parallel, 

mean flow the problem for the Green’s function can be reduced to solving an uncoupled set of secondorder ordinary differential equations for its azimuthal Fourier modes. For non-axisymmetric jets, such 

reduction is not generally possible and, since most of the computational expense of acoustic analogy 

based noise prediction methods is incurred in computing the Green’s function, predictions for nonaxisymmetric jets are generally significantly more time consuming than those for axisymmetric ones. 

A reduced-order model for obtaining the Green’s function for certain classes of non-axisymmetric jets 

was presented in (Ref. 9). The approach uses a conformal mapping of the level surfaces of mean 

streamwise velocity and sound speed (assumed to be coincident) and their orthogonal curves to a semiinfinite periodic strip. The Green’s function can then be solved in terms of its Fourier modes (in the 

transformed plane), but the latter are now coupled, owing to the ‘azimuthal’ dependence of the Jacobian 

of the transformation. The method was applied to a set of rectangular jets (the Extensible Rectangular 

Nozzle (ERN) System (Ref. 10)) using a mapping to cylindrical elliptical coordinates along with the 

hybrid source model developed in Reference 8. Results from low-speed flow surveys of the jet plumes, 

carried out by Zaman (Ref. 11), were used to scale the various source components of the model. 

Extensive PIV flow measurements of these jets were later made by Bridges and Wernet (Ref. 12). 

Comparisons of the results of noise calculations using this method with data taken in the Small Hot Jet 

Acoustic Rig (SHJAR) at NASA Glenn (Ref. 13) showed that the model was capable of predicting the 

overall azimuthal directivity characteristics of these non-axisymmetric jets. 

The conformal mapping based method provides a reduced-order model for obtaining the Green’s 

function with computational resources comparable to those required for a round jet, but it is has some 

limitations as to the types of mean flows that can be treated. A number of the noise reduction concepts 

currently being considered are not amenable to the conformal mapping method and it is desirable to have 

a more general method that can treat a wider range of geometries. 

In this report, we present a more general method for computing the Green’s function of the acoustic 

analogy equations for a locally parallel mean flow that could potentially be used for most of the noise 

reduction concepts currently of interest to NASA. The method relies on a Fourier series representation of 

the azimuthal variation of the mean flow dependent coefficients appearing in the governing equation for 

the Green’s function, and solves for the latter in terms of its own azimuthal Fourier components. As in the 

conformal mapping based approach (and to a potentially much greater extent), the equations for the 

azimuthal Fourier modes of the Green’s function are coupled in the present method, requiring solution of 

a (possibly quite large) system of simultaneous ordinary differential equations. One purpose of this report 

is to demonstrate the use of the method on some relatively generic non-axisymmetric jets and determine 

the feasibility of using it in the context of a noise prediction scheme for non-axisymmetric jets. 

The overall plan of the paper is as follows. In Section 2.0 the equations and boundary conditions 

governing the far-field adjoint Green’s function for a locally parallel mean flow are given. The numerical 

method used to solve for the azimuthal Fourier modes of the Green’s function is described in Section 3.0. In 

Section 4.0 results are presented using this method for a round jet and the results compared with those from 

an existing, established, round jet code. Two non-axisymmetric test cases are then described in Section 4.0. 

These test cases are meant to be representative of two noise reduction concepts of interest to NASA. Results 

from these test cases are presented in Section 5.0. In Section 6.0 we summarize the results and provide 

recommendations for the application of this method to noise predictions in non-axisymmetric jets. 

2.0 Governing Equation

For a locally parallel mean flow, the components of the adjoint vector Green’s function of the 

acoustic analogy equations can be expressed in terms of a single scalar function that, for observer 

locations in the far field, satisfies (Refs. 7 and 8)
NASA/CR—2014-218107 3

 ( )

( )

( )

( ( ) )

( ) ( )

2

2

2 22

2

2

;,:

1 cos

/ cos

 1 ; , : 0; 2,3,

1 cos

T T

j j

T

T

c g

y y M

c c

g j

M

⊥

∞

⊥

∂ ∂ φθ ω

∂ ∂   − θ  

  θ   +ω −   φ θ ω = =

    − θ    

y y

y

y

y

y

(2.1)

where y = (y1, yT) are Cartesian coordinates, with y1 being in the direction of the mean flow and 

yT = (y2, y3) the cross flow coordinates,  ( ) 2

T c y is the mean sound speed profile, whose ambient value is 

denoted as 2 c∞ , M(yT) = U(yT)/c∞ is the acoustic Mach number profile corresponding to the mean axial 

velocity profile U(yT), ω is the radian frequency and θ,φ are the observer polar and azimuthal angles, 

defined as ( )

1 1

1 2 222

1 23

cos

x

xxx

−

 

θ =  

+ +  

and 1 3

2

tan x

x −   φ =     , respectively with x = (x1, x2, x3) the 

Cartesian coordinates of the observer. 

The solution is subject to the far-field boundary condition

( ) ( ) ( )

( )

0 2

2

sin cos /4

2 2 sin /

/

; , : outgoing waves,

2

ic y i

c

c

g

e e ∞ ⊥

∞

∞

⊥

− ω θ φ−φ π

π θω

ω

φθ ω → +

π

y (2.2)

as yT = yT → ∞, where φ0 = tan–1 (y3 /y2 ), together with appropriate boundedness and/or continuity 

conditions. 

We express (2.1) in terms of the polar coordinates (yT, φ0),

2 2

2

2 2 2

0 0

1 1 0,

T TT T T T

g g gg g g

y yy y y y

∂ ∂∂∂ ∂

+ + + + +ω =

∂ ∂ ∂φ ∂ ∂φ RF H (2.3)

where

( ) ( )

( )  ( )

 ( ) 2

0 2

2cos 1 , , 1 cos

T T

T

T T T T

M c

y M y y c

    θ ∂ ∂

φ +     − θ ∂ ∂      

R y y

y y = (2.4)

( ) ( )

( )  ( )

 ( ) 2

0 2 0 0

2cos 1 , , 1 cos

T T

T

T T T T

M c

y M y y c

    θ ∂ ∂

φ +     − θ ∂φ ∂φ      

F y y

y y = (2.5)

and

( ) ( )  ( )

2

2

0 2 2

1 cos cos , . T

T

T

M

y

c c∞

        − θ θ

φ −  

   

H y

y = (2.6)

For applications where the mean flow is not too far from axisymmetric, the coefficients, (2.4) to (2.6), of 

Equation (2.3) can be reasonably well approximated by a relatively small number of terms, say L, of their 

Fourier series expansions 
NASA/CR—2014-218107 4

( ) ( ) ( ) ( ) ( ) ( ) 00 0 0 00 , ; , ; ,

L LL

il il il

T l T T l T T l T

l L l L l L

y ye y ye y y e φφ φ

=− =− =−

R R F FH H φ = ∑∑∑ φ = φ = (2.7)

Using (2.7) in (2.3), the solution for the far-field scalar adjoint Green’s function can be sought in terms of 

the coefficients of its truncated Fourier series,

( ) ( ) 0 0 , ;,: ;,:

N

in n

n N

g y g y e φ

⊥ ⊥

=−

φ φθ ω ≈ φθ ω ∑ (2.8)

from

( )

2 2

2 2

2 -

1

 0 ; 

L

n n n l

n l

T TT T l L T

L L

l n l l n l

l L T l L

g g n g g

y yy y y

i n lg g N n N

y

−

=−

−

=− =−

∂∂ ∂

+ −+ +

∂ ∂ ∂

+ − +ω = − ≤ ≤

∑

∑ ∑

R

F H

, (2.9)

where L ≤ N.

The far-field boundary conditions on the Fourier modes, gn, are obtained from (2.2) as 

( ) ( )

( )

( )

2

2

/4

2

2 sin /

/

; , : outgoing waves as ,

4 2

n T

i

in

c

c

g y e y

e

∞

∞ ⊥

π − φ+π

π θω

ω

φ θ ω → + → ∞

π

(2.10)

where (2) Hn is the Hankel function of the second kind. 

To eliminate the outgoing wave component of the far-field boundary condition (2.10) (whose 

amplitude is unknown), a Robin-type condition is imposed

( )

( )

( ) ( ) ( )

2

2

2

/4 /2

2 sin /

as 

/

sin

4 2

n n n n n T n T

T T

T

i

c

dg d c in

g Hy y dy dy c

y

e

e

∞

∞

∞

π φ+π

π θω

→ ∞

  ω −   ω

+κ → +κ     θ =Γ

  π   , (2.11)

where 

( )

( )

1

1

sin

cos ,

sin

n T

n

n T

H y

c

c

H y

c

∞

∞

∞

  ω ′  θ

−ω θ   κ =   ω

  θ

 

(2.12)

(1) Hn is the Hankel function of the first kind of order n, and the prime denotes differentiation with respect 

to the argument. 

Near the origin, a Frobenius analysis shows that appropriate inner boundary conditions are 

( )

0 ( )

0; , : 0, 0

0; , : 0

n

T

g n

dg

dy

φθ ω = ≠

φθ ω = . (2.13)
NASA/CR—2014-218107 5

3.0 Numerical Methods

For numerical solution of the Fourier components of the far-field scalar adjoint Green’s function, the 

radial derivatives in the governing Equations (2.9), far-field boundary condition (2.11) and the second 

starting condition of (2.13) are replaced by second-order central differences, and a system of algebraic 

equations for the Fourier modes of the Green’s function at discrete grid points is formed.

The domain is discretized using a uniform grid size, ∆, so that 

1 ; 1, 1 j j

T T y y jJ + − =∆ = − , (3.1)

and the Fourier components of the solution at the discrete grid points are represented by j

n g . 

Then the initial (or starting) conditions (2.13) become 

1

2 1

0 0

0, 0

0

n g n

g g

= ≠

−+ = , (3.2)

the governing Equations (2.9) become

1, 1 0; 2 1, 

LL L

j j jn j j j

l nl l nl l nl

lL lL l L

g g g jJ NnN − +

− −−

=− =− =−

∑∑ ∑ BA C + + = ≤≤− − ≤≤ , (3.3)

where 

l,0 1

2 2

j j

l l

Ty

    ∆ ∆ =δ − −        

B R , (3.4)

( )

22 2 , 2 2

l,0 2 2 1 j n j j

l l l

T T

n i n

y y

    ∆ ∆ = −δ + + − + ω ∆    

   

A FH , (3.5)

l,0 1

2 2

j j

l l

Ty

    ∆ ∆ =δ + +        

C R , (3.6)

and the far-field boundary condition (2.11) becomes 

1 1

2 2 1

1 2 1 1 2 2

J J

J n n J J

n n n g g

− − − −

      ∆κ ∆κ

− + + + = ∆Γ

  

. (3.7)

The system of equations is written in matrix form as 

1 1 ; 1, j jj j jj j j J − + Bg Ag C g   ++ = = Γ (3.8)

where Aj, Bj,Cj are (2N + 1) × (2N + 1) matrices, with B1 = CJ = 0,
NASA/CR—2014-218107 6

1

.

.

.

.

.

.

j

N

j

N

j

j

n

j

N

g

g

g

g

−

− +

 

 

 

 

 

 

    =  

 

 

 

 

 

 

   

g , (3.9)

is a (2N + 1)-length vector. The system is of block-tridiagonal form, and can be efficiently solved using a 

sparse system algorithm.

4.0 Validation and Test Cases

As a validation case for the computer code written to implement this method, a round jet is 

considered and results are compared with those from an established noise prediction code for round jets 

(Refs. 7 and 8). Two non-circular test cases are then presented that will be used as applications of the 

method. They are meant to be representative of two noise reduction concepts of current interest: the offset 

jet and the fluid shield. 

4.1 Round Jet

A round jet with mean acoustic Mach number profile

( ) 2r Mr Me C −α = , (4.1)

where the radial coordinate is now denoted by r = yT, and uniform mean sound speed profile,  ( ) 2 2 cr c ≡ ∞ ,

was considered as the validation case for the code. Calculations were carried out for MC = 0.5 and α = 1.

Figure 1 shows comparisons of the results computed for this case using the new code with those of 

the established round jet code for a polar angle θ = 60° at a range of Strouhal numbers, St

2 J

D

U

ω = π

and 

several azimuthal observer angles at a radial source location of one-half. Figure 2 shows corresponding 

results for θ = 30°. Results at other radial source locations are generally similar. The results are in good 

agreement, although there begins to be some discrepancy as the Strouhal number increases. Increasing the 

number of Fourier modes used to compute the solution can be shown to ameliorate this discrepancy.
NASA/CR—2014-218107 7

 

(a) (b)

 (c) (d)

Figure 1.—Comparisons of results for the far-field adjoint Green's function versus azimuthal location in a round jet at 

θ = 60°, r = 0.5. (a) φ = 0, (b) φ = 90, (c) φ = 180, (d) φ = 270. Results from code of Reference 7 and 8 (Curves), 

from current code (Symbols). Re(g) solid lines and squares, Im(g) dashed lines and triangles.

φ0

R

e

al(g)

Im

a

g(g)

-1 0 1 2 3 4 5 6 7 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

St = 0.2

St = 0.4

St = 0.6

φ0

R

e

al(g)

Im

a

g(g)

-1 0 1 2 3 4 5 6 7 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

St = 0.2

St = 0.4

St = 0.6

φ0

R

e

al(g)

Im

a

g(g)

-1 0 1 2 3 4 5 6 7 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

St = 0.2

St = 0.4

St = 0.6

φ0

R

e

al(g)

Im

a

g(g)

0 2 4 6 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

St = 0.2

St = 0.4

St = 0.6
NASA/CR—2014-218107 8

 

(a) (b)

 

(c) (d)

Figure 2.—Comparisons of results for the far-field adjoint Green's function versus azimuthal location in a round jet at 

θ = 30°, r = 0.5. (a) φ = 0, (b) φ = 90, (c) φ = 180, (d) φ = 270. Results from code of References 7 and 8 (Curves), 

from current code (Symbols).Re(g) solid lines and squares, Im(g)dashed lines and triangles.

In the next two subsections, we present the non-circular jet test cases used as example applications of 

the code. 

4.2 The Offset Jet

The first non-circular geometry considered is that of the offset jet. As a model for the mean flow for 

this case, we take the mean acoustic Mach number profile to be

( ) ( ) 2 1 2 cos , C

r Mr Me− +α− α φ φ = . (4.2)

Figure 3 shows mean acoustic Mach number contours for this profile with MC = 0.5, α = 0.8. The mean 

flow consists of a thick and a thin side and experiments have found significant asymmetry in the sound 

field emitted from jets of this form. 

φ0

R

e

al(g)

mI a

g(g)

-1 0 1 2 3 4 5 6 7 0

0.002

0.004

0.006

0.008

0.01

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

St = 0.2

St = 0.4

St = 0.6

φ0

R

e

al(g)

Im

a

g(g)

-1 0 1 2 3 4 5 6 7 -0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

St = 0.2

St = 0.4

St = 0.6

φ0

R

eal(g)

Imag(g)

-1 0 1 2 3 4 5 6 7 -0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

St = 0.2

St = 0.4

St = 0.6

φ0

R

eal(g)

Imag(g)

-1 0 1 2 3 4 5 6 7 0

0.002

0.004

0.006

0.008

0.01

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

St = 0.2

St = 0.4

St = 0.6
NASA/CR—2014-218107 9

Figure 3.—Mean acoustic Mach number contours for the offset jet.

Figure 4.—Mean acoustic Mach number contours for the fluid shield.

4.3 Fluid Shield

The second non-circular case considered is the so-called fluid shield. In this geometry, a round core 

flow is partially surrounded by a secondary stream, with the latter intended to shield the region below it 

by refracting sound. A model for the mean flow for this configuration is 

( ) ( ) ( )

( )

4 2 2 1

2

,

0, 0

sin , 2

ar c r M r M e br e g C

g

  − − − φ= + φ    

 ≤φ<π

φ = 

 φ π≤φ< π

. (4.3)

Figure 4 shows the mean acoustic Mach number contours for this profile for MC = 0.5, a = 2.0, b = 0.3, 

c = 1.0.
NASA/CR—2014-218107 10

5.0 Results

In this section we present results for the two noncircular jet test cases considered as example

applications of the method. For each test case, we first show results demonstrating the Fourier series 

representation of the mean flow terms appearing in the governing Equation (2.3). The Fourier coefficients 

of the mean flow terms, (2.4) to (2.6), were computed at relatively high resolution (2048 points) to obtain 

accurate values of the coefficients using the routine FFTW. Subsets of these coefficients were then used 

in (2.7) to reconstruct these functions for different choices of L and the results compared with the full 

quantities. We then present results for the Green’s function using these Fourier series representations of 

the coefficients. For these calculations, we have taken the mean sound speed to be uniform (equal to its 

ambient value) and, therefore, for polar angles of θ = 90° the effect of the mean flow vanishes and the 

magnitude of the scalar adjoint Green’s function is an axisymmetric function of φ0 – φ. It has been 

verified that results from the code give this behavior for both test cases, but, since the emphasis here is on 

the asymmetry produced by the mean flow field, we only present results at polar angles of 30° and 60°

(relative to the downstream jet axis). Calculations were carried out using sixteen modes (N = 16) for the 

Green’s function, a step size ∆ = 0.005 and a maximum radial extent for the computational domain of 5.0. 

Testing was carried out to verify that these numerical parameters are sufficient for the two test cases of 

interest. 

Results are presented showing the effect of the number of Fourier modes used to represent the mean 

flow terms on the numerical solution for the Green’s function. In particular, we demonstrate that 

numerical results for the Green’s function converge (that is they are unchanged with increasing number of 

mean flow modes) for a reasonable number of mean flow modes for these two test cases. 

We note that the high-frequency asymptotic solution (Ref. 14) suggests that the scaled Green’s 

function, 3

2 g −

ω , should become independent of frequency as ω → ∞. As another check on the 

numerical results for the Green’s function, we plot this quantity for increasing Strouhal number and verify 

that it has this form.

Results are then presented showing the azimuthal directivity of the Green’s function as a function of 

the Strouhal number, at polar observation angles of 30° and 60° to the jet axis. 

In jet noise calculations based on an acoustic analogy formulation, prediction results are usually 

obtained by evaluating a formula derived for the far-field acoustic spectrum. This formula involves the 

Green’s function, and its derivatives, in combination with Doppler factors multiplying the various 

components of the Reynolds stress auto-covariance tensor, which are the source terms (Refs. 7 and 8). 

Prediction of the noise characteristics of a particular nozzle configuration requires the use of this full 

formula. However, as a rough estimate of the potential noise-shielding effects of non-axisymmetric mean 

flows, the directivity of the magnitude of the adjoint Green’s function itself can be used as a guide. 

Therefore, in the final set of results presented, we examine the computed results for directivity of the 

magnitude of the adjoint Green’s for a selected set of source locations in an attempt to gain some physical 

insight into the potential noise-shielding properties of the mean flows for these two test cases. 

5.1 Offset Jet

5.1.1 Fourier Representation of the Mean Flow Terms

In this section we present results for the Fourier series representation of the mean flow terms 

appearing in the governing equations for the case of the offset jet. Figure 5 shows contour plots in the 

cross flow, (y2, y3), plane of the mean flow terms, R, F, H, in Equation (2.3) for this case. 
NASA/CR—2014-218107 11

 

(a) (b)

(c)

Figure 5.—Mean flow dependent coefficients of the Greens function equation for the offset jet (4.2). (a) R, (b) F, and 

(c) H. θ = 30°.

In Figure 6 we show these quantities, and the acoustic Mach number profile, M, as functions of the 

azimuthal coordinate, φ, at a radial location of r = 0.5 and a polar angle of 30° from the jet centerline, 

along with their Fourier reconstructions using four, six and eight modes in (2.7). 

As seen in Figure 6, the mean flow Mach number profile, and the mean flow dependent coefficients 

of Equation (2.3), are well represented by six modes in their respective Fourier series for this case. 

Results computed for the Green’s function for this case, using these Fourier representations for the

coefficients, are presented in the next subsection. 
NASA/CR—2014-218107 12

(a) (b)

(c) (d)

Figure 6.—Fourier series reconstruction of the mean flow terms for the offset jet. (a) M; (b) R, (c) F, and (d) H.

5.1.2 Results for the Green’s Function 

Figure 7 shows results computed for the real (a) and imaginary (b) parts of the far-field Green’s 

function at an observer location of 30° polar angle and 0° azimuthal angle for three radial locations in the 

jet. On each plot is shown results computed for the Green’s function using four, six and eight Fourier 

modes to represent the coefficients of the governing equation and five values of the Strouhal number. As 

Figure 7 shows, at low Strouhal number (black and red curves), results computed using only four modes 

for the mean flow terms appear to be sufficient for convergence, whereas as the Strouhal number 

increases (blue and orange curves), a larger number is needed. The results in Figure 7 (and those 

computed at other observation locations but not shown here) suggest that convergence is generally 

reached with eight mean flow modes. The results do not converge uniformly with radial location,

however; since the mean flow is relatively axisymmetric near the centerline, fewer modes are needed near 

the centerline (left most plots), whereas at intermediate radial locations (center plots), where there is 

greater asymmetry, convergence is slower.

φ -1 0 1 2 3 4 5 6 7 0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55 4 Modes

6 Modes

8 Modes

φ -1 0 1 2 3 4 5 6 7 -1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2 4 Modes

6 Modes

8 Modes

φ -1 0 1 2 3 4 5 6 7 -1

-0.5

0

0.5

1 4 Modes

6 Modes

8 Modes

φ -1 0 1 2 3 4 5 6 7 -0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0 4 Modes

6 Modes

8 Modes
NASA/CR—2014-218107 13

yT = 0.25 yT = 0.5 

(a) yT = 1.0 

Figure 7.—Results for the (a) real part and (b) imaginary part of the Green’s function versus azimuthal source 

location at three radial source locations using different numbers of Fourier modes for the mean flow dependent 

coefficients. Long-dashed L = 4; short-dashed L = 6; solid L = 8. θ = 30°, φ = 0°.

φ0

Real (g)

-1 0 1 2 3 4 5 6 7 -0.0015

-0.001

-0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

St = 0.2

St = 0.4

St = 0.6

St = 0.8

St = 1.0

φ0

Real(g)

-1 0 1 2 3 4 5 6 7 -0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004

0.005

St =0.2

St = 0.4

St = 0.6

St = 0.8

St = 1.0

φ0

Real ( g )

-1 0 1 2 3 4 5 6 7 -0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

St = 0.2

St = 0.4

St = 0.6

St = 0.8

St = 1.0
NASA/CR—2014-218107 14

(b)

Figure 7.—Concluded.

The plots in Figure 8 show results for the scaled Green’s function, 

3

2 g −

ω , versus observer 

azimuthal angle, at fixed source locations, for increasing Strouhal number. These results are shown at a 

polar angle of 60° from the jet axis, since, for this Mach number, a polar angle of 30° lies in the ‘zone of 

silence’ for the high-frequency solution. As required by the high-frequency asymptotic form of the 

solution, the curves are seen to eventually collapse and become independent of frequency by a Strouhal 

number of around two. 

Figure 9 and Figure 10 are polar plots of the magnitude of the scalar adjoint Green’s function versus 

φ0 – φ for observer polar angles of 60° and 30°, respectively, for a radial source location yT = 0.5, and 

Strouhal numbers ranging from 0.2 to 1.0. As noted previously, for an axisymmetric mean flow the 

solution depends only on the difference between source and observer azimuthal locations, and in this case 

the curves in these figures would coincide. For non-axisymmetric cases they do not, and plotting in this 

way gives an indication of the degree of asymmetry. At the lowest frequency shown, St =0.2, the Green’s 

function is fairly axisymmetric, although there is a slight bias toward larger values when the source is in 

line with the observer (i.e., at φ0 – φ = 0) at θ = 30°(Figure 10(a)) particularly for φ = π. As the frequency 

increases, however, there is significant asymmetry in the Green’s function. The largest asymmetry 

appears to occur at θ = 30°and for φ = 0°.

φ0

Imag ( g )

-1 0 1 2 3 4 5 6 7 -0.003

-0.0025

-0.002

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0.0015

St = 0.2

St = 0.4

St = 0.6

St = 0.8

St = 1.0

φ0

Imag( g )

-1 0 1 2 3 4 5 6 7 -0.004

-0.0035

-0.003

-0.0025

-0.002

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0.0015

0.002

St = 0.2

St = 0.4

St = 0.6

St = 0.8

St = 1.0

φ0

Imag( g )

-1 0 1 2 3 4 5 6 7 -0.007

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004

0.005

St = 0.2

St = 0.4

St = 0.6

St = 0.8

St = 1.0
NASA/CR—2014-218107 15

(a) (b)

(c) (d)

Figure 8.—Collapse of the scaled Green's function with increasing St – Offset jet. θ = 60°, yT = 0.5; (a) φ0 = 0°, 

(b) φ0 = 90°, (c) φ0 = 180°, and (d) φ0 = 270°.

φ

ω-3/2

|g|

-50 0 50 100 150 200 0.0008

0.0009

0.001

0.0011

0.0012

0.0013

0.0014

0.0015

0.0016

0.0017

0.0018

St = 0.5

St = 1.0

St = 1.5

St = 2.0

St = 2.5

St = 3.0

φ

ω-3/2

|g|

-50 0 50 100 150 200 0.0008

0.001

0.0012

0.0014

0.0016

0.0018

0.002

0.0022

0.0024

St = 0.5

St = 1.0

St = 1.5

St = 2.0

St = 2.5

St = 3.0

φ

ω-3/2

|g|

-50 0 50 100 150 200 0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.0018

0.002

0.0022

St = 0.5

St = 1.0

St = 1.5

St = 2.0

St = 2.5

St = 3.0

φ

ω-3/2

|g|

-50 0 50 100 150 200 0.0008

0.001

0.0012

0.0014

0.0016

0.0018

0.002

St = 0.5

St = 1.0

St = 1.5

St = 2.0

St = 2.5

St = 3.0
NASA/CR—2014-218107 16

(a) (b)

(c) (d)

(e)

Figure 9.—Polar plots of g(yT, φ0; φ, θ : ω) versus φ0 – φ at yT = 0.5 for θ =60° for the offset jet. (a) St = 0.2; 

(b) St = 0.4; (c) St = 0.6; (d) St = 0.8; and (e) St = 1.0.

φ0 - φ

|g|

0

_1_ π 6

_1_ π 3

_1_ π 2

_2_ π 3

_5_ π 6

π

_7_ π 6

_4_ π 3

_3_ π 2

_5_ π 3

_1_1_ π 6

0 0.001 0.002

φ = 0

φ = π/2

φ = π

φ = 3π/2

φ0 - φ

|g|

0

_1_ π 6

_1_ π 3

_1_ π 2

_2_ π 3

_5_ π 6

π

_7_ π 6

_4_ π 3

_3_ π 2

_5_ π 3

_1_1_ π 6

0 0.005

φ = 0

φ = π/2

φ = π

φ = 3π/2

φ0 - φ

|g|

0

_1_ π 6

_1_ π 3

_1_ π 2

_2_ π 3

_5_ π 6

π

_7_ π 6

_4_ π 3

_3_ π 2

_5_ π 3

_1_1_ π 6

0 0.005 0.01 0.015

φ = 0

φ = π/2

φ = π

φ = 3π/2

φ0 - φ

|g|

0

_1_ π 6

_1_ π 3

_1_ π 2

_2_ π 3

_5_ π 6

π

_7_ π 6

_4_ π 3

_3_ π 2

_5_ π 3

_1_1_ π 6

0 0.01 0.02

φ = 0

φ = π/2

φ = π

φ = 3π/2

φ0 - φ

|g|

0

_1_ π 6

_1_ π 3

_1_ π 2

_2_ π 3

_5_ π 6

π

_7_ π 6

_4_ π 3

_3_ π 2

_5_ π 3

_1_1_ π 6

0 0.01 0.02

φ = 0

φ = π/2

φ = π

φ = 3π/2
NASA/CR—2014-218107 17

(a) (b)

(c) (d)

(e)

Figure 10.—Polar plots of g(yT, φ0; φ, θ : ω) versus φ0 – φ at yT = 0.5 for θ =30°. (a) St = 0.2; (b) St = 0.4; 

(c) St = 0.6; (d) St = 0.8; and (e) St = 1.0. 

φ0 - φ

|g|

0

_1_ π 6

_1_ π 3

_1_ π 2

_2_ π 3

_5_ π 6

π

_7_ π 6

_4_ π 3

_3_ π 2

_5_ π 3

_1_1_ π 6

0 0.001 0.002

φ = 0

φ = π/2

φ = π

φ = 3π/2

φ0 - φ

|g|

0

_1_ π 6

_1_ π 3

_1_ π 2

_2_ π 3

_5_ π 6

π

_7_ π 6

_4_ π 3

_3_ π 2

_5_ π 3

_1_1_ π 6

0 0.002 0.004 0.006 0.008

φ = 0

φ = π/2

φ = π

φ = 3π/2

φ0 - φ

|g|

0

_1_ π 6

_1_ π 3

_1_ π 2

_2_ π 3

_5_ π 6

π

_7_ π 6

_4_ π 3

_3_ π 2

_5_ π 3

_1_1_ π 6

0 0.005 0.01 0.015

φ = 0

φ = π/2

φ = π

φ = 3π/2

φ0 - φ

|g|

0

_1_ π 6

_1_ π 3

_1_ π 2

_2_ π 3

_5_ π 6

π

_7_ π 6

_4_ π 3

_3_ π 2

_5_ π 3

_1_1_ π 6

0 0.01 0.02

φ = 0

φ = π/2

φ = π

φ = 3π/2

φ0 - φ

|g|

0

_1_ π 6

_1_ π 3

_1_ π 2

_2_ π 3

_5_ π 6

π

_7_ π 6

_4_ π 3

_3_ π 2

_5_ π 3

_1_1_ π 6

0 0.01 0.02

φ = 0

φ = π/2

φ = π

φ = 3π/2
NASA/CR—2014-218107 18

To try to get some idea of the noise shielding that may occur due to this non-axisymmetric mean 

flow, in Figure 11 we have selected two azimuthal observer locations, φ = 0 and φ= π, from the results of 

Figure 10 to examine in more detail. The curves shown correspond to radial source locations of yT = 1.0 

(b) and yT = 0.5 (c). The relatively low value for the Green’s function at φ – φ0 = 0 for φ = 0 (red curve),

relative to that for φ = π (blue curve) suggests that the observer at φ = 0 (right side) is shielded from the 

source at φ0 = 0 more than the observer at φ = π (left side) is from the source at φ0 = π. The asymmetry is 

larger at yT = 1.0 but also quite significant at yT = 0.5. Based on these results, the sound levels radiated to 

observer locations at φ = 0° from a source located at (yT, φ0) = (0.5, 0) could potentially be 5 dB lower 

than that radiated to φ = π from a source located at (yT, φ0) = (0.5, π). The differences at a radial location 

of yT = 1.0 is more like 15 dB, but it should be noted that the turbulence levels will probably be quite low 

at such a relatively distant radial location from the jet centerline, and the importance of these source 

locations would therefore be diminished. It is emphasized again that the magnitude of the Green’s 

function itself is not the only factor in determining the azimuthal directivity of the sound field in a nonaxisymmetric jet and that these results are illustrative only. 

(a)

(b) (c)

Figure 11.—Shielding of sound by the mean flow to two observer locations from two source locations—Offset Jet.

St = 0.6. 
NASA/CR—2014-218107 19

5.2 Fluid Shield

5.2.1 Fourier Representation of the Mean Flow Terms

Results for the Fourier series representation of the mean flow terms appearing in the governing 

equations for the case of the fluid shield are presented in this subsection. For this geometry, the mean 

acoustic Mach number profile is given by (4.3), and again a uniform mean sound speed is used. Contour 

plots in the cross flow, (y2, y3), plane for, R, F, H, in Equation (2.3) are shown in Figure 12 for this case. 

The mean flow for this case is somewhat more asymmetric than the offset jet considered above, and 

we might expect that it would require more Fourier modes to adequately represent these quantities in this 

case. In Figure 13 we show these quantities, and the acoustic Mach number profile, M, as functions of the 

azimuthal coordinate, φ, at a radial location of r = 0.5 and a polar angle of 30° from the jet centerline, 

along with their Fourier reconstructions using four, eight and twelve modes in (2.7). 

a) (b)

(c)

Figure 12.—Mean Flow Quantities—Fluid Shield. (a) R; (b) F and (c) H. θ =30°
NASA/CR—2014-218107 20

(a) (b)

(c) (d)

Figure 13.—Fourier reconstruction of the mean flow terms for the fluid shield. (a) M; (b) R; (c) F; and (d) H.

As expected, the results in Figure 13 show that for this (more asymmetric) case, a larger number of 

Fourier modes, generally eight, is needed to adaquately represent the mean flow terms.

5.2.2 Green’s Function

In this section we present results for the Green’s function for the fluid shield. Figure 14 shows results 

computed for the real (a) and imaginary (b) parts of the far-field Green’s function at an observer location 

of 30° polar angle and 0° azimuthal angle for three radial locations. On each plot is shown results 

computed using four, six and eight Fourier modes to represent the coefficients of the governing equation 

and five values of the Strouhal number. For this case, even though the mean flow is more asymmetric 

than the offset jet, it appears that using only four modes for the mean flow terms is sufficient for 

convergence of the Green’s function, up to a Strouhal number of one. 

φ -1 0 1 2 3 4 5 6 7 0

0.05

0.1

0.15

0.2

0.25 4 Modes

8 Modes

12 Modes

φ -1 0 1 2 3 4 5 6 7 -0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2 4 Modes

8 Modes

12 Modes

φ -1 0 1 2 3 4 5 6 7 -0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4 4 Modes

8 Modes

12 Modes

φ -1 0 1 2 3 4 5 6 7 -0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05 4 Modes

8 Modes

12 Modes
NASA/CR—2014-218107 21

yT = 0.25 yT = 0.5 

(a) 

yT = 1.0

Figure 14.—Results for the (a) real part and (b) imaginary part of the Green’s function versus azimuthal source 

location at three radial source locations using different numbers of Fourier modes for the mean flow dependent 

coefficients. Long-dashed L = 4; Short-dashed L = 6; solid L = 8. θ = 30°, φ = 0°.

φ0

Real( g )

-1 0 1 2 3 4 5 6 7 0

0.002

0.004

0.006

0.008

0.01

St = 0.2

St = 0.4

St = 0.6

St = 0.8

St = 1.0

φ0

Real( g )

-1 0 1 2 3 4 5 6 7 -0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

St = 0.2

St = 0.4

St = 0.6

St = 0.8

St = 1.0

φ0

Real( g )

-1 0 1 2 3 4 5 6 7 -0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

St = 0.2

St = 0.4

St = 0.6

St = 0.8

St = 1.0
NASA/CR—2014-218107 22

(b)

φ0 φ0

φ0

Figure 14.—Concluded.

Figure 15 shows results for the scaled Green’s function, 3

2 g −

ω , at a fixed source location as the 

Strouhal number is increased. A slightly higher Strouhal number (around 2.5) is needed for these curves 

to collapse compared with the offset jet case, but the required frequency independence is also achieved in 

this case.

Figure 16 and Figure 17 are polar plots of the magnitude of the scalar adjoint Green’s function versus 

φ0 – φ for observer polar angles of 60° and 30°, respectively, corresponding to radial source location, 

yT = 0.5 and Strouhal numbers ranging from 0.2 to 1.0 for the fluid shield case. As was found in the offset 

jet case, at the lowest frequency shown, St =0.2, the Green’s function is fairly axisymmetric, and there is 

greater asymmetry at θ = 30° relative to that at θ = 60°. Also, as the frequency increases there is again 

significant asymmetry in the Green’s function.

Imag( g )

-1 0 1 2 3 4 5 6 7 -0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

St = 0.2

St = 0.4

St = 0.6

St = 0.8

St = 1.0

Imag( g )

-1 0 1 2 3 4 5 6 7 -0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

St = 0.2

St = 0.4

St = 0.6

St = 0.8

St = 1.0

Imag( g )

-1 0 1 2 3 4 5 6 7 -0.055

-0.05

-0.045

-0.04

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

St = 0.2

St = 0.4

St = 0.6

St = 0.8

St = 1.0
NASA/CR—2014-218107 23

(a) (b)

(c) (d)

Figure 15.—Collapse of scaled Green's function with increasing St—Fluid Shield. θ = 60°, yT = 0.5; (a) φ0 = 0°, 

(b) φ0 = 90°, (c) φ0 = 180°, and (d) φ0 = 270°.

φ

ω-3/2 |g|

-50 0 50 100 150 200

0.0012

0.0013

0.0014

0.0015

0.0016

0.0017

0.0018

0.0019

St = 0.5

St = 1.0

St = 1.5

St = 2.0

St = 2.5

St = 3.0

φ

ω-3/2 |g|

-50 0 50 100 150 200 0.0016

0.00165

0.0017

0.00175

0.0018

0.00185

0.0019

St = 0.5

St = 1.0

St = 1.5

St = 2.0

St = 2.5

St = 3.0

φ

ω-3/2 |g|

-50 0 50 100 150 200

0.0012

0.0013

0.0014

0.0015

0.0016

0.0017

0.0018

0.0019

St = 0.5

St = 1.0

St = 1.5

St = 2.0

St = 2.5

St = 3.0

φ

ω-3/2 |g|

-50 0 50 100 150 200 0.001

0.0011

0.0012

0.0013

0.0014

0.0015

0.0016

0.0017

0.0018

0.0019

0.002

0.0021

0.0022

St = 0.5

St = 1.0

St = 1.5

St = 2.0

St = 2.5

St = 3.0
NASA/CR—2014-218107 24

(a) (b) 

(c) (d)

(e)

Figure 16.—Polar plots of g(yT, φ0; φ, θ : ω) versus φ0 – φ at yT = 0.5 for θ = 60° for the fluid shield. (a) St = 0.2; 

(b) St = 0.4; (c) St = 0.6; (d) St = 0.8; and (e) St = 1.0.

φ0 - φ

|g|

0

_1_ π 6

_1_ π 3

_1_ π 2

_2_ π 3

_5_ π 6

π

_7_ π 6

_4_ π 3

_3_ π 2

_5_ π 3

_1_1_ π 6

0 0.001 0.002 0.003 0.004

φ = 0

φ = π/2

φ = π

φ = 3π/2

φ0 - φ

|g|

0

_1_ π 6

_1_ π 3

_1_ π 2

_2_ π 3

_5_ π 6

π

_7_ π 6

_4_ π 3

_3_ π 2

_5_ π 3

_1_1_ π 6

0 0.0025 0.005 0.0075

φ = 0

φ = π/2

φ = π

φ = 3π/2

φ0 - φ

|g

0

_1_ π 6

_1_ π 3

_1_ π 2

_2_ π 3

_5_ π 6

π

_7_ π 6

_4_ π 3

_3_ π 2

_5_ π 3

_1_1_ π 6

0 0.005 0.01 0.015

φ = 0

φ = π/2

φ = π

φ = 3π/2

φ0 - φ

|g|

0

_1_ π 6

_1_ π 3

_1_ π 2

_2_ π 3

_5_ π 6

π

_7_ π 6

_4_ π 3

_3_ π 2

_5_ π 3

_1_1_ π 6

0 0.005 0.01 0.015 0.02

φ = 0

φ = π/2

φ = π

φ = 3π/2

φ0 - φ

|g|

0

_1_ π 6

_1_ π 3

_1_ π 2

_2_ π 3

_5_ π 6

π

_7_ π 6

_4_ π 3

_3_ π 2

_5_ π 3

_1_1_ π 6

0 0.01 0.02

φ = 0

φ = π/2

φ = π

φ = 3π/2
NASA/CR—2014-218107 25

(a) (b)

(c) (d)

(e)

Figure 17.—Polar plots of g(yT, φ0; φ, θ : ω) versus φ – φ0 at yT = 0.5 for θ = 30° for the fluid shield. (a) St = 0.2; 

(b) St = 0.4; (c) St = 0.6; (d) St = 0.8; (e) St = 1.0.

φ0 - φ

|g|

0

_1_ π 6

_1_ π 3

_1_ π 2

_2_ π 3

_5_ π 6

π

_7_ π 6

_4_ π 3

_3_ π 2

_5_ π 3

_1_1_ π 6

0 0.001 0.002 0.003 0.004

φ = 0

φ = π/2

φ = π

φ = 3π/2

φ0 - φ

|g|

0

_1_ π 6

_1_ π 3

_1_ π 2

_2_ π 3

_5_ π 6

π

_7_ π 6

_4_ π 3

_3_ π 2

_5_ π 3

_1_1_ π 6

0 0.002 0.004 0.006 0.008

φ = 0

φ = π/2

φ = π

φ = 3π/2

φ0 - φ

|g|

0

_1_ π 6

_1_ π 3

_1_ π 2

_2_ π 3

_5_ π 6

π

_7_ π 6

_4_ π 3

_3_ π 2

_5_ π 3

_1_1_ π 6

0 0.005 0.01 0.015

φ = 0

φ = π/2

φ = π

φ = 3π/2

φ0 - φ

|g|

0

_1_ π 6

_1_ π 3

_1_ π 2

_2_ π 3

_5_ π 6

π

_7_ π 6

_4_ π 3

_3_ π 2

_5_ π 3

_1_1_ π 6

0 0.005 0.01 0.015

φ = 0

φ = π/2

φ = π

φ = 3π/2

φ0 - φ

|g|

0

_1_ π 6

_1_ π 3

_1_ π 2

_2_ π 3

_5_ π 6

π

_7_ π 6

_4_ π 3

_3_ π 2

_5_ π 3

_1_1_ π 6

0 0.005 0.01 0.015 0.02 0.025

φ = 0

φ = π/2

φ = π

φ = 3π/2
NASA/CR—2014-218107 26

(a)

Figure 18.—Shielding of sound by the mean flow to two observer locations from two source locations—Fluid Shield. 

St = 0.6.

As for the case of the fluid shield, we have selected two azimuthal observer locations, 2 φ = π and 

3

2 φ = π here, from the results of Figure 17 to examine in more detail. The curves shown in Figure 18

again correspond to radial source locations of yT = 1.0 (b) and yT = 0.5 (c). The relatively low value for the 

Green’s function magnitude at φ – φ0 = 0 for 3

2 φ = π (yellow curve), relative to that for 2 φ = π (green

curve) suggests that the observer at 3

2 φ = π (bottom) is shielded from the source at 0 3

2 φ = π more than 

the observer at 2 φ = π (top) is from the source at 0 2 φ = π , suggesting that the partial secondary stream 

does in fact act like a noise shield for source locations below the core stream. The asymmetry is again 

more pronounced at yT = 1.0 than at yT = 0.5, but to a somewhat lesser extent than for the offset jet case. 

6.0 Summary and Conclusions

In this report we have documented a numerical method for computing the far-field scalar adjoint 

Green’s function of the acoustic analogy equations for a locally parallel, non-axisymmetric, mean flow. 

A computer code was written to implement this method and we have reported here on its validation and 

on the results for a set of sample test cases chosen for relevance to certain noise-reduction concepts of 

current interest. The method relies on a Fourier series representation of the azimuthal variation of the 

mean flow depenedent coefficients appearing in the governing equation for the Green’s function, and 

solves for the latter in terms of its own azimuhal Fourier components.

(b)

(c)
NASA/CR—2014-218107 27

The method was initially validated by using it to compute the Green’s function for a round jet and 

comparing the results with those obtained from an exisitng round jet noise prediction code. 

It was demonstrated, for the two non-axisymmetric test cases examined, that the mean flow 

dependent coefficients appearing in the governing equation for the Green’s function can be represented by 

a reasonably small number of Fourier modes—an important result for the viability of the method and its 

potential use in jet noise predictions. The effect of the number of Fourier components used to represent 

the mean flow on the numerical solution for the Green’s function was tested and the number of modes 

needed for convergence of the results was also found to be relatively small for the two test cases

considered. It was also verified that the appropriately scaled Green’s function becomes independent of 

frequency as the Strouhal number becomes large, as the high-frequency asymptotic solution suggests. 

The azimuthal directivity characteristics of the Green’s function for the two test cases considered here 

were examined and results for a pair of azimuthal observer locations for each case were considered in a 

bit more detail to try to glean some information about the noise shielding potential of the respective nonaxisymmetric mean flows.

For a complete noise prediction, of course, the Green’s function solver described in this report must 

be combined with appropriate models for the source terms, with the mean flow and turbulence quantities 

being supplied by a Reynolds-averaged Navier-Stokes (RANS) solution for the flow. Based on the results 

obtained for the two test cases considered in this report, it appears that the numerical method developed 

here could be used in more realistic mean flow profiles and incorporated into a noise prediction scheme 

for non-axisymmetric jets. The next steps in the development of a noise prediction method based on this 

Green’s function solver include extending the routines that carry out the Fourier series decomposition of 

the mean flow terms to handle profiles from a RANS solution and its integration with an appropriate 

source model.

References

1. Lighthill, M.J., “On Sound Generated Aerodynamically: I. General Theory,” Proc. R Soc. Lond., A211, 

pp. 546-587, 1952.

2. Lighthill, M.J., “On Sound Generated Aerodynamically: II. Turbulence as a Source of Sound,” Proc. R 

Soc. Lond., A222, pp. 1-32, 1954.

3. Lilley, G.M., “On the Noise From Jets,” AGARD-CP-131, pp. 131-132, 1974.

4. Khavaran, A., Bridges, J. and Georgiadis, N., “Prediction of Turbulence-Generating Noise in Jets,” 

NASA/TM—2002-211696. 

5. Dahl, M.D. (ed.) “Assessment of NASA’s Aircraft Noise Prediction Capability,” NASA/TP—2012-

215653, Bridges, J.E., Khavaran, A. and Hunter, C.A., Chapter 8—Jet Noise Prediction. 

6. Goldstein, M.E., “A Generalized Acoustic Analogy,” Journal of Fluid Mechanics, Vol. 488, pp.315-

333, 2003.

7. Goldstein, M.E. and Leib, S.J., “The Aeroacoustics of Slowly Diverging Supersonic Jets,” Journal of 

Fluid Mechanics, Vol. 600, pp. 291-337, 2008.

8. Leib, S.J. and Goldstein, M.E. “Hybrid Source Model for Predicting High-Speed Jet Noise,” AIAA 

Journal, Vol. 49, No. 7, 2011.

9. Leib, S.J. “Noise Predictions for Rectangular Jets Using a Conformal Mapping Method, “AIAA Journal, 

Vol. 51, No. 3, pp. 721-737.

10. Frate, F.C. and Bridges, J.E., “Extensible Rectangular Nozzle Model System,” AIAA paper 2011-975. 

11. Zaman, K.B.M.Q, “Flow-Field Surveys for Rectangular Jets,” NASA/TM—2012-217410, 2012.

12. Bridges, J.E. and Wernet, M.P., “Turbulence Measurements of Rectangular Nozzles with Bevel,” 

Extended abstract for 19th AIAA/CEAS Aeroacoustics Conference, May 27-29 2013.

13. Bridges, J.E., “Acoustic Measurements of Rectangular Nozzles with Bevel,” AIAA paper 2012-2252.

14. Goldstein, M.E. 1982 “High Frequency Sound Emission from Moving Point Multipole Sources 

Embedded in Arbitrary Transversely Sheared Mean Flows.” Journal of Sound and Vibration, Vol. 80, 

No. 4, pp. 499-522.
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Modeling Sound Propagation Through Non-Axisymmetric Jets 

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13. SUPPLEMENTARY NOTES

14. ABSTRACT

A method for computing the far-field adjoint Green’s function of the generalized acoustic analogy equations under a locally parallel mean 

flow approximation is presented. The method is based on expanding the mean-flow-dependent coefficients in the governing equation and 

the scalar Green’s function in truncated Fourier series in the azimuthal direction and a finite difference approximation in the radial direction 

in circular cylindrical coordinates. The combined spectral/finite difference method yields a highly banded system of algebraic equations that 

can be efficiently solved using a standard sparse system solver. The method is applied to test cases, with mean flow specified by analytical 

functions, corresponding to two noise reduction concepts of current interest: the offset jet and the fluid shield. Sample results for the Green’s 

function are given for these two test cases and recommendations made as to the use of the method as part of a RANS-based jet noise 

prediction code. 

15. SUBJECT TERMS

Aeronautics; Aeroacoustics 

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