Jonathan B. Freund

University of Illinois, Urbana-Champaign, Urbana, Illinois

Jet Noise Physics and Modeling Using

First-Principles Simulations

NASA/CR—2003-212123

February 2003

https://ntrs.nasa.gov/search.jsp?R=20030032418 2018-05-15T15:53:33+00:00Z
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Jonathan B. Freund

University of Illinois, Urbana-Champaign, Urbana, Illinois

Jet Noise Physics and Modeling Using

First-Principles Simulations

NASA/CR—2003-212123

February 2003

National Aeronautics and

Space Administration

Glenn Research Center

Prepared under Grant NAG3–2662
Acknowledgments

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National Technical Information Service

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This report contains preliminary

findings, subject to revision as

analysis proceeds.

The Propulsion and Power Program at

NASA Glenn Research Center sponsored this work.

Available electronically at http://gltrs.grc.nasa.gov
Jet Noise Physics and Modeling

Using First-Principles Simulations

Accomplishments of this year are in four categories, which are summarized here. Detailed appendices are attached as appropriate.

1 Noise and source statistics relevant to the MGBK framework

An extensive analysis of our jet DNS database has provided for the first time the complex correlations that are the core of many statistical jet noise models, including MGBK. We have also for

the first time explicitly computed the noise from different components of a commonly used noise

source as proposed in many modeling approaches. Key findings are:

1. While two-point (space and time) velocity statistics are well-fitted by decaying exponentials,

even for our low-Reynolds-number jet, spatially integrated fourth-order space/retarded-time

correlations which constitute the noise “source” in MGBK and are instead well-fitted by

Gaussians. The width of these Gaussians depends (by a factor of 2) on which components

are considered. This is to current modeling practice.

2. A standard decomposition of the Lighthill source is shown by direct evaluation to be somewhat

artificial since the noise from these nominally separate components is in fact highly correlated.

We anticipate that the same will be the case for the Lilley source.

3. The far-field sound is computed in a way that explicitly includes all quadrupole cancellations,

yet evaluating the Lighthill integral for only small part of the jet yields a far-field noise far

louder than that from the whole jet due to missing non-quadrupole cancellations.

Details of this study are discussed in a draft of a paper included as appendix A.

NASA/CR—2003-212123 1

Jonathan B. Freund

University of Illinois, Urbana-Champaign

Urbana, Illinois 62801

counter
2 Large-scale dynamics and large-eddy simulation

It seems that the noise from a jet is primarily due to the locally largest and most energetic scales,

making large-eddy simulation (LES) potentially effective for predicting jet noise. At the same

time the dynamics of the very large scales in a jet are of high interest from the perspective of

understanding the mechanics of noise generation. Our work in this area follows several related

paths, some in collaboration with Prof. Lele’s group at Stanford.

2.1 Large-scale structural dynamics – quantitative

We have designed spatial filters that remove small-scale, acoustically irrelevant components of the

jet’s turbulence. These filtered fields are being used in two ways. In the first, we use the DNS

to provide an “ideal” (determined a priori) sub-grid-scale model that is used to close linearized

equations for these large-scales,

∂

∂tρ(L) +

∂

∂xj



ρu(L)

j + ρ(L)

uj



= ∂

∂xj

(ρuj − ρuj )

∂

∂t



ρu(L)

i + ρ(L)

ui



+

∂

∂xj



ρuiu(L)

j + ρu(L)i uj + ρ(L)

uiuj + p(L)δij

= ∂

∂t (ρuj − ρuj ) + ∂∂xj



ρuiuj − ρuiuj



∂

∂t



ρE(L) + ρ(L)E



+

∂

∂xj



ρE + p



u(L)

j +

	

ρ(L)E + ρE(L) + p(L)




uj



= ∂

∂t



ρE − ρE+

∂

∂xj

	ρE + p


uj −



ρEu j + puj

.

This will allow us to explicitly study their dynamics. Figure 1 shows the effect of filters of various widths on the turbulence. Despite the dramatic effect of the filter on the appearance of

the turbulence, flow quantities remain essentially unaltered in the region of frequency/streamwisewavenumber (k–ω) space that is capable of radiating to the far field (figure 2). The evolution

equations are currently being solved to identify the effect of the sub-grid-scales on the evolution

of the structures capable of radiating to the far field and the dynamics of these structure as they

interact with the mean flow. This part of the work is in close collaboration with Sanjiva Lele and

his student Dan Bodony.

NASA/CR—2003-212123 2
10 15 20 25 30 −5

0

5

10 15 20 25 30 −5

0

5

10 15 20 25 30 −5

0

5

Figure 1: The vθ field. Left: unfiltered; middle: filtered with 
 = 2.5ro; and right: filtered with


 = 5.0ro. These filters define our very large scales.

(a) kro ωro/a

∞

(a) kro ωro/a

∞

(a) kro ωro/a

∞

Figure 2: Frequency/streamwise-wavenumber makeup of the data: (a) unfiltered, (b) filtered with


 = 2.5ro, (c) filtered with 
 = 5.0ro. The straight lines separate regions that can not radiate to

the far field in r (subsonic phase velocity in the right and left wedges) from those can can radiate

to the far field in r (supersonic phase velocity in the top and bottom wedges).

NASA/CR—2003-212123 3
2.2 Large-scale structural dynamics – qualitative

In the past, we have not been able to identify noisy events based on visualizations. However,

with much of the non-radiating turbulence removed by the selective filtering procedure described

above we are revisiting this approach. Many quantities (in fact all that we can think of) are being

investigated. Sample movies showing simultaneously the far- and near-field pressure for the full

field and a highly streamwise and azimuthally filtered near field are at

http://taylor.tam.uiuc.edu/∼jfreund/migall.gif and at

http://taylor.tam.uiuc.edu/∼jfreund/mig5.0.gif,

respectively. We have also started making visualizations that take into account the retarded time.

In this case flow quantities q(x, t) are plotted at q(x, τ ) = q(x, t − |xl − x|/a∞) for listener points

xl. We anticipate that this view will be more relevant for the far-field sound.

2.3 Direct evaluation of mode conversion

With the data in k–ω coordinates it is possible to explicitly compute the convolutions (products in

real coordinates) that put energy into modes that radiate. The entire machinery has been assembled

to do this and some initial convolutions was computed, but a subtle “bug” was found and the

procedure is being rerun. While this approach can precisely identify what types of interactions

convert turbulent energy to radiation components, it has the drawback that the spectral-space

data no longer have an immediate interpretation in physical coordinates. Thus, this method will

be used in close conjunction with the more “dynamical” approach above. Just as figures 1 and 2

are complementary, once the dynamic simulation for the very large scale modes are solved, similar

interpretations in k–ω space will deepen understanding.

2.4 Large-eddy simulation (LES)

We collaborated with Benjamin Rembold at the Center for Turbulence Research this summer to

compute the sound from a large-eddy simulation of a turbulent jet. In particular, we were looking

for errors due explicitly to the sub-grid-scale models in the LES. Figure 3 (a) shows that the α = 30◦

spectrum of this jet is well predicted and in good agreement with the corresponding DNS spectrum

shown in figure 3 (b) and in general agreement with other jet data at similar Reynolds numbers.

However, at upstream angles the spectra in 3 (c) and (d) show that the LES has created substantial

spurious noise. We are investigating the reasons for this. We see two possible causes. One is explicit

numerical errors due to “pushing” the resolution limit as is common practice in LES. The other is

NASA/CR—2003-212123 4
(a) (b)

(c) (d)

Figure 3: Far-field noise spectra: (a) DNS at α = 30◦; (b) DNS at α = 150◦; (c) LES at α = 30◦;

and (d) LES at α = 150◦. This is for a 5:1 rectangular jet with the major plane and the

minor plane. For reference, we also show the low-Reynolds-number round jet of Stromberg1

(experiment) and ◦ the low-Reynolds-round jet of Freund2 (DNS).

direct radiation by the sub-grid-scale model, which are universally designed to model turbulence

dynamics not noise.

3 Flow acoustic interactions

This past year we published a journal article (Appendix B) which showed qualitatively, based on ray

trajectories traced through unsteady jet turbulence, that jet turbulence can substantially scatter

noise. In collaboration with Tom Bewley and Laura Cerveno at UCSD we have extended the adjoint

based control procedure we use at UIUC to control jet noise to numerically compute an adjoint

Green’s function for the unsteady jet flow. This provides both the quantitative amplitudes that

were missing in the ray tracing study, and at the same time provides the effective forward Green’s

function for a single far-field direction with one computation, the same way Tam’s3 approach does

for a steady jet flow. Figure 4 shows preliminary computations of the adjoint Green’s function

NASA/CR—2003-212123 5
Figure 4: Adjoint Green’s function for an unsteady two-dimensional jet flow at several instances.

The vorticity is shown by the black contours.

for a two-dimensional jet. We see the substantial effect of the unsteady unsteady two-dimensional

turbulence structures.

4 Near-nozzle mixing layer DNS

We have designed a new algorithm to compute both a hot and cold near nozzle jet mixing layer.

(A previous simulation was found to be effectively forced by the inflow disturbance rendering it

less useful for studying certain statistical representations of noise sources.) We have a mapped

spectral code and a de-aliased mapped spectral code that are being developed by John Moreland,

a graduate student of the PI. He is working out inflow conditions and zonal boundary conditions

to do the large simulations. Figure 5 shows visualization of the mixing layer computed with the

de-aliased code. There is clearly significant work to do to with regard to the inflow conditions,

which have proved more difficult than anticipated. Any excitation that we impose will necessarily

force the flow. In the past we “fed” turbulence from a streamwise periodic simulation into the

domain for a jet simulation,4 but there remained a small artifact of the particular excitation far

downstream.5 Our Mach 0.9 jet, which has a laminar inflow region excited by very random, very

low amplitude perturbations seems to be our best approach so far. Unfortunately, we do not know

what the disturbances are in reality because precise nozzle exit conditions are seldom reported.

A Turbulent Jet Noise: Shear Noise, Self-Noise and Other Contributions

Abstract

Using a previously validated simulation database of a Mach 0.9 cold jet, we examine the

components of Lighthill’s analogous noise source that are linear (shear noise terms) and quadratic

NASA/CR—2003-212123 6
Figure 5: Preliminary results of de-aliased mapped spectral method code shoing vorticity of a

M = 0.9 mixing layer.

NASA/CR—2003-212123 7
(self-noise terms) in velocity fluctuations, as well as components that are deviations from p =

a2

∞ρ

. It is found that the shear noise is highly directional, with an angle of extinction near

α = 90◦, measured from the downstream axis, which is consistent with Ribner’s theory6 that

I ∼ cos4 α+cos2 α. Its spectrum broadens at larger α. In contrast, the self-noise is more uniform.

Is is fit by five inverse Doppler factors, (1 − Mc cos α)−5, but only provided that Mc is dropped

below its typically assumed values of Mc ≈ 0.6Uj/a∞ = 0.5 to Mc ≈ 0.3. In previous work it

was found that Mc ≈ 0.3 was indeed the dominant phase velocity for the Lighthill source in this

same jet. The spectral shape of the self-noise is relatively independent of angle, in contrast to the

shear noise. The shear noise and self-noise are correlated, especially at small angles where their

mutual correlation coefficient reaches as low as −0.4, casting doubt on models that treat these

terms as distinct. The p = a2

∞ρ contribution is relatively small, not negligible as expected for a

cold jet, but it is so well correlated with the shear noise (correlation coefficient of −0.6 at small

angles) that it should not be neglected. The total radiated power of the component quadratic

in the velocity fluctuations is nearly the same as that of all components combined. Examining

turbulence statistics relevant to jet noise, we see that two-point correlations statics are well fitted

by exponential functions, as is typical of turbulence at all but the lowest Reynolds numbers,

but integrated fourth-order space/retarded-time covariances, which are directed related to the

radiated acoustic intensity, are instead very well fitted by Gaussian functions of different widths

for different components, which is counter to conventional modeling practice.

A.1 Background and objectives

Several attempts to model jet noise6–11 have followed the standard Reynolds decomposition of

the flow variables: q =  ̄q + q

, where q is a flow quantity and  ̄q and q are its average and an

instantaneous perturbation. This study uses a numerical simulation database to examine this

decomposition as it is typically used in jet noise models utilizing Lighthill’s theory.

Lighthill’s equation12 can be written

∂2ρ

∂t2 − a2

∞

∂2ρ

∂xj∂xj

= ∂2Tij

∂xi∂xj

, (1)

where Tij = ρuiuj + (p − a2

∞ρ)δij − τij is the Lighthill stress tensor, a∞ is the ambient sound

speed, τij is the viscous stress, and ρ is a density fluctuation. The double divergence of Tij

appearing on the right hand side of (1) serves as a nominal acoustic source, but its use here

should not be interpreted in any way as an attempt to define the true noise source in the jet,

if there is such a thing. Such an interpretation is impractical for at least two reasons. The

first, which has been well understood starting with Lighthill’s original derivation of (1), is that

all effects aside from propagation in a homogeneous stationary medium, such as refraction,

are lumped into this nominal source. More sophisticated theories attempt to treat mean-flow

refraction explicitly,13–15 but are more analytically challenging. The second is that most of

Tij,ij does not radiate to the far acoustic field, which is also widely understood. What Tij,ij

does provide us is an exact connection between near-field turbulence statistics and their noise.

This is how we use it.

Applying the Reynolds decomposition to Tij gives

Tij = T ̄

ij + ρ( ̄uiu

j + uiu ̄j )    shear

+ ρu

iuj    self

+ (p − a2

∞ρ

)δij    entropy− τ 

ij



viscous

, (2)

where source terms that are linear in the fluctuating velocities have been labeled ‘shear’ to reflect

that this source component entails turbulent fluctuations interacting with the sheared mean

flow, and source terms that are quadratic in the fluctuating velocities have been labeled ‘self’

to reflect that this source component entails turbulent fluctuation interacting with themselves.

For clarity we have not yet decomposed ρ into  ̄ρ and ρ

. The so-called entropic contribution has

often been neglected to facilitate analysis,12, 16 but is thought by some to be significant for both

hot and cold jets, having an efficient dipole component.15, 17 Lighthill12 stated emphatically

NASA/CR—2003-212123 8
that the viscous component of Tij could be neglected with respect to the far-field sound, and

this view has held.15, 16 An implicit result of Colonius & Freund’s18 computation of jet noise

using Lighthill’s analogy was that τij does not contribute substantially even at Re = 2000.

Until now, there has been no direct measurements to verify the modeling of different components, only of the net result. A common modeling assumption is that the self-noise and shear

noise contributions are independent, the validity of which is untested. We use an existing wellvalidated the direct numerical simulation database to directly evaluate the role of the different

components. Turbulence statistics that constitute noise sources, but are also difficult to measure

experimentally, are also computed to evaluate common modeling practice and identify scalings.

A.2 Simulation Database

The database we use was reported on in detail by Freund.2 It is a Mach 0.9, Reynolds number

3600, constant stagnation temperature (Tj/T∞ = 0.86) jet. It has been validated against the

corresponding experimental results of Stromberg,1 matching the mean flow development, sound

pressure level directivity at 60ro (60 jet radii) from the nozzle, and sound spectrum at α = 30◦.

Because of its low Reynolds number, this jet has laminar shear layers that transition before

the end of the potential core. After transition, its turbulence has broad-banded energy spectra,

rapidly decaying two-point correlations, and viscous dissipation is a significant component in

the turbulent kinetic energy budget. Reynolds stresses downstream of the potential core match

those measured in higher-Reynolds-number experiments.

Despite this agreement with some aspects of high-Reynolds-number turbulence, we must

remain aware in interpreting our results that there are qualitative differences in the noise from a

low- versus high-Reynolds-number jet. Most notably, the spectrum of a high-Reynolds-number

jet is broader because of the greater range of turbulence scales in the flow. There are two general

factors that might cause this. The first is that broadening of local turbulence spectra leads to

higher frequency noise. However, these fine scales contain relatively little energy, making it more

likely that the high frequencies missing at low Reynolds numbers instead come from energetic

structures in the shear layers near the nozzle. These near-nozzle shear layer structures are

higher frequency by virtue of their smaller size. Statistical arguments support this. Lighthill’s19

arguments suggest that most of the noise comes from “eddies only slightly smaller than the

main energy bearing eddies.” Similarly, noise source localization experiments20 have show that

most of the high-frequency noise has an apparent origin in the thin shear layers near the nozzle.

Under this interpretation, a low-Reynolds-number jet can be viewed as a model for that portion

of a high-Reynolds-number jet near and beyond the close of the potential core. Some additional

Reynolds number sensitive turbulence correlations relevant to noise are discussed later in this

paper.

The flow simulation used Nx × Nr × Nθ = 640 × 250 × 160 points in the axial, radial

and azimuthal directions, respectively. Data were stored on every other mesh point in all

three coordinate directions every 20 numerical time steps of ∆t = 0.0085ro/a∞ each, which

corresponds to a Strouhal number St = 14.1, well above any of our frequencies of interest in the

sound field.

A.3 Source Decomposition

Neglecting viscosity, Lighthill’s noise source is the double divergence of Tij = ρuiuj + (p −

a2

∞ρ)δij , where ui are the instantaneous velocities, and p and ρ are the instantaneous pressure

and density. The source decomposition we consider splits Tij into components that are linear,

Tl

ij , and quadratic, T nij , in velocity fluctuations plus the so-named entropy component, T sij :

Tij = ρu ̄iu ̄j +( ̄p − a2

∞ρ ̄)    T m

ij

+ ρu ̄iu

j + ρu ̄jui   

Tl

ij

+ ρu

iuj    T n

ij

+ (p − a2

∞ρ

)δij    Ts

ij

. (3)

NASA/CR—2003-212123 9
α Tij (5) Tij (11) T ρ ̄

ij (5) T nij (5) T nij (11)

30.00 30.45 30.74 31.81 21.45 21.65

60.00 11.00 11.33 11.32 11.14 11.35

90.00 5.66 5.99 5.84 4.93 5.10

120.00 2.99 3.09 3.17 1.56 1.61

Table 1: Evaluation of use of ρuiuj for  ̄ρuiuj and truncation of azimuthal Fourier series expansion.

Values shown are p

p/ρ2

jU4j × 1010 on a circular arc at 240ro. The numbers in parenthesis, (5) and

(11), are Nnmax .

The mean component T m

ij by definition does not make noise. We have not decomposed the

density as one might into ρ =  ̄ρ+ρ in the velocity terms, because, as seen in table 1, the sound

from Tij is nearly the same as that from

T ρ ̄

ij =  ̄ρuiuj + (p − a2

∞ρ

)δij . (4)

Thus we do not discuss the explicit effect of density fluctuations in the ρuiuj terms.

In the spirit of Lighthill’s pioneering work,12, 21 Ribner6 made extensive use of this source

decomposition. More recently this decomposition has been coupled to k–ε turbulence models

as a predictive tool.22, 23 A similar decomposition into self-noise and shear noise components is

also used in modeling the source terms7, 8, 10, 24 in the linearized Lilley equation,14 which extends

the acoustic analogy approach to explicitly include refraction.

A.4 Computing the Far-Field Sound

Given Tij or one of its components, the following procedure was used to compute the far-field

sound. The data were first transformed in θ by

T ̃(n)

ij (x, r, t) = 1

Nθ

N

θ−1

k=0

Tij (x, r, θk, t)einθk for n = −Nnmax ,...,Nnmax , (5)

which was done as an efficient and accurate means of compressing the data. The stress tensor

Tij was rotated into cylindrical coordinates to facilitate the transform. We see in table 1 that

the noise eventually computed from Nnmax = 5 versus Nnmax = 11 modes differs little, so we

take Nnmax = 5 as sufficient.

Next, T ̃(n)

ij (x, r, t) was transformed in time following a procedure similar to that we used in

the past.2 The time series at each x and r mesh point and n-mode was first multiplied by

w(t) = 1

2



tanh 5

t − t1

t1 − t0



− tanh 5

tf − t

tf − t2

	, (6)

where t0 and tf in (6) are the times in the simulation when it was determined to be statistically

stationary and the final time, respectively. Times t1 and t2 are the 5 and 95 percent points in this

time series. This contaminates the resulting noise, but only for a finite length of time that can

be unambiguously identified in advance through straight forward retarded time considerations.

The time transform was then

Tˇ(n)

ij (x, r, ωm) = 1

Nt



Nt

l=1

w(tl)T ̃(n)

ij (x, r, tl)e−iωmtl for m = −Nt/2,...,Nt/2 − 1, (7)

where Nt = 2304, and the discrete angular frequencies were defined ωm = 2πm/T, where T is

the implied time period.

NASA/CR—2003-212123 10
500 600 700 800 900

-0.0004

-0.0002

0

0.0002

0.0004

0.0006

ta∞/ro

p/ρ

∞

a2

∞

unaffected times

0 0.5 1 1.5 2

10-4

10-3

St

(ˆpˆp∗)1/2/ρ

∞

a

∞ro

Figure 6: Sound data at α = 30◦, 60ro from the nozzle. (a) Time history shows region affected

by the windowing procedure. (b) Time spectra: present solution of the Lighthill equation,

from Freund,2 and from the experiment of Stromberg et al.1 Only a relative scale was

provided by Stromberg et al. so it has been adjusted vertically.

Finally, the data were inverse transformed in θ by

Tˆ

ij (x, r, θk, ωm) =

N

θmax

n=−Nθmax

Tˇ(n)

ij (x, r, ωm)einθk for k = 0,...,Nθ − 1, (8)

rotated into Cartesian coordinates, and convolved with the twice differentiated free-space Green’s

function for the Helmholtz equation to compute the acoustic pressure Fourier coefficients ˆp

(x, ωm)

at listener point x,

pˆ

(x, ωm) = 


Ω

Tˆ

ij (y, ωm) ∂2

∂xi∂xj



e−iωm|x−y|/a∞

4π|x − y|

	

dy for m = −Nωmax ,...,Nωmax , (9)

where we have used ˆp = a2

∞ρˆ

. Ω is the physically realistic region of the cylindrical simulation

domain, which extended x = 31ro downstream from the nozzle and out to r = 8ro. Taking

Nωmax = 327 was sufficient to compute all frequencies up to St = 2.0, a range which, as we shall

see, constitutes most of the noise from this jet. The time dependent pressure at point x is

p

(x, tl) =

N

ωmax

m=−Nωmax

pˆ(x, ωm)e−iωmtl for l = 0,...,Nt − 1. (10)

Figure 6 (a) shows p(x, t) at 30◦ from the jet axis and 60ro from the nozzle. The region

affected by the ‘windowing’ procedure is evident, but constitutes less than one-quarter of the

time series. There are sufficient unaffected data to converge statistics. Figure 6 (b) shows the

energy spectrum of the pressure fluctuations at this same point, comparing with corresponding

experimental data and the spectrum at the same position computed previously using a different

method.

NASA/CR—2003-212123 11
0 30 60 90 120 150 180 10-2

10-1

100

101

102

α (degrees)

p

p

/(ρ2

jU4

j ) × 1010

6 dB

Figure 7: Directivity on a circular arc 240ro from the nozzle: Tij ; Tij − Ts

ij ; using

Tl

ij ; T nij ; and Tsij , as defined in (3).

A.5 Directivity

Figure 7 shows that the sound from the shear noise source, T l

ij , is more directional than the

self-noise source, T n

ij , and has an angle of extinction near α = 90◦. The entropic term is highly

directional at small angles, but has nearly uniform directivity for α  90◦. While the Ts

ij term

is indeed insubstantial at large angles, near to the downstream jet axis it is significant even

though the temperature ratio of this jet is only Tj/T∞ = 0.86. Mitchell et al.25 also found this

term to be non-negligible for the noise from vortex pairing in an truly (not just in the mean)

axisymmetric uniform temperature jet. This is surprising since, as expected, (p − a2

∞ρ

)2 is

small compared to the Reynolds stresses: (p − a2

∞ρ

)2

max = 0.0062ρ2

jU4

j , whereas the Reynolds

stresses have peaks of u

xuxmax ≈ 0.040U2

j and u

rurmax ≈ 0.023U2

j , for example.

Lilley26 finds it instuctive to rewrite the p − a2

∞ρ term in terms of the fluctuating total

enthalpy and a kinetic energy component. Using the energy equation and the perfect gas

equation of state, it is equivalent to

p − a2

∞ρ = − γ − 1

2 ρu2

   term I

+ a2

∞


 ∂

∂xj



ρuj

h∞ − hs

h∞

	 dt

   term II

. (11)

Lilley shows that the second (II) term has a dipole form and is hence more acoustically efficient

and suggests that it explains the less-than-U8 sound power scaling of hot jets. Here we find

that it is much larger than term I and constitues most of the p − a2

∞ρ term’s contribution to

the far-field sound.

The net acoustic power (all angles) of the different components is tabulated in table 2. We see

that the power from just quadratic velocity fluctuations terms is 83 percent of the total, which

suggests that a large part of the apparent acoustic energy due to T l

ij can, in fact, potentially be

interpreted as redirection due by flow-acoustic interaction.

None of the profiles show a decrease in intensity near to the jet axis, as might be expected

due to refraction, but a zone of ‘silence’ is not expected for the total for the frequencies in this

jet (e.g. Lush27). If we divide the data into higher and lower frequency components, we see the

expected behavior (figure 9). (We cannot analyze band-limited spectral components like Lush

since we do not have sufficient date to converge statistics in frequency bands.) Low frequencies

NASA/CR—2003-212123 12
0 30 60 90 120 150 180

10-1

100

101

α (degrees)

p

p

/(ρ2

jU4

j ) × 1010

Figure 8: Directivity on a circular arc 240ro from the nozzle: p − a2

∞ρ; term I in 11;

and term II in 11.

Component Power/ρjUj

3Aj Power/Power Tij

Tij 8.3 ×10−5 1.00

Tl

ij 8.7 ×10−5 1.05

T n

ij 6.9 ×10−5 0.83

Ts

ij 2.0 ×10−5 0.25

Table 2: Net radiated acoustic power.

are more intense along the axis, which Goldstein28 suggests is a low-frequency flow acoustic

interaction, and high-frequency noise is reduced near the jet axis, as expected for refraction. So

there is significant evidence of refraction, but we note that since the source is only evaluated for

x < 31ro any refraction down stream of this point is omitted, though this omitted part is likely

to be less substantial since the Mach number is substantially decreased here.

It is clear that some individual components are more intense than the total, which means

that the noise from the different components is correlated, significantly so given the amount

of cancellation evident in figure 7. Taking Tij ≈ ρu ̄ iuj , Goldstein & Rosenbaum29 show that

a sufficient condition to decorrelate the shear noise and self-noise is that the jet turbulence be

locally homogeneous. To quantify the observed correlation, we define a correlation coefficient as

Cβ−γ = ρβργ

ρβ

rmsργrms

, (12)

where β and γ are n, l, or s to indicate the noise from different source contributions defined in

(3). Cl−s, Cl−n, and Cn−s are plotted in figure 10. All pairs are most correlated at small angles

and for the most part are canceling. At very small angles, the entropic contribution cancels the

noise due to Tl

ij with correlation coefficient Cl−s ≈ −0.6. Some degree of correlation has been

deduced from experimental observations of hot supersonic jets,30 but it does not appear to have

been anticipated to this degree for a nearly uniform temperature subsonic jet. The linear and

quadratic velocity fluctuation contributions are also correlated at small angles, though they are

often assumed to be decorrelated in models.6, 7, 22 The noise from source components that are

quadratic in the velocity fluctuations are relatively mildly correlated with the entropic noise.

All correlations decrease to near zero by α = 90◦.

Ribner6 predicted that five inverse Doppler factors31 would set the directivity of the velocity

components of the source, with an additional factor of cos4 α + cos2 α for the Tl

ij component.

NASA/CR—2003-212123 13
0 30 60 90 120 150 180 10-1

100

101

α (degrees)

p

p

/(ρ2

jU4

j ) × 1010

Figure 9: Directivity for high and low frequencies: ◦ St < 0.45(1 − Mc cos α)−1;  St > 0.45(1 −

Mc cos α)−1. The line is (1 − Mc cos α)−5 with Mc = 0.3 suggested as suggested by direct

analysis of Tij,ij .

2

0 30 60 90 120 150 180 -1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

α (degrees)

Correlation coefficient

Figure 10: Correlation coefficients defined by (12): Cl−n with smoothed profile ;  Cl−s

with smoothed profile ; and ◦ with smoothed profile .

NASA/CR—2003-212123 14
0 30 60 90 120 150 180 10-1

100

101

102

α (degrees)

p

p

/(ρ2

jU4

j ) × 1010

6 dB

(a)

0 30 60 90 120 150 180

10-2

10-1

100

101

102

103

α (degrees)

p

p

/(ρ2

jU4

j ) × 1010

6 dB

(b)

Figure 11: Directivity on a circular arc 240ro from the nozzle. (a) • T n

ij , 2.5(1 − 0.5 cos α)−5,

2.5(1 − 0.3 cos α)−5; (b) • Tl

ij , 2.5(cos4 β + cos2 β)(1 − 0.5 cos α)−5, 4.5(cos4 β +

cos2 β)(1 − 0.3 cos α)−5. Angle β is defined based on an origin at x = 20ro, which is presumably

closer the center of the apparent source location.

Figure 11 (a) shows the self-noise component. It is significantly less directional than five inverse

Doppler factors would have it with Mc ≡ Uc/a∞ = 0.6Uj/a∞ = 0.5. However, Freund2 showed

that the wave-number/frequency makeup of the full Lighthill source for this jet had a dominate

phase Mach number of Mc ≈ 0.3, which is indeed a better fit for the data. Accounting for

refraction by the mean flow in the high-frequency limit, Goldstein shows that three inverse

Doppler factors are actually to be expected, which is also closer to what is observed than five

Doppler factors, but since the mean velocity profile does not appear in the solution for the

source T n

ij interactions with the mean flow leading to refraction are not a viable explanation in

this particular case. A possibility is that the emitted directivity of five inverse Doppler factors

is made more uniform by interaction with turbulence rather than the mean, which would be

included as an artificial source in T n

ij . Such a change of directivity has been suggested based

on qualitative analysis of this database,32 but has not been demonstrated quantitatively for the

frequencies and source locations present. For the shear noise, the observed angle of extinction

matches well with Ribner’s theory if the effective source location is shifted to x = 20ro (figure

11 b). Again, Mc = 0.3 yields a better fit than Mc = 0.5. Ribner6 predicted that the shear

noise would be 3dB higher than the self-noise on axis, but we observe nearly 6dB difference.

Nonetheless, the agreement is encouraging.

A.6 Quadrupole character

The approach taken to solve for the far-field noise described in section A.4, specifically the

choice of convolving Tij with the twice-differentiated Green’s function G,ij for the free-space

homogeneous reduced wave equation, was made to reduce numerical errors. The Tij ∗ G,ij is

mathematically equivalent, of course, to a convolution of the double-divergence of the stress

with just G: Tij,ij ∗ G. However, based on Lighthill’s theory, we anticipated that the noise

sources should have quadrupole character. The approach taken implicity includes cancellation

that would lead to effectively quadrupole type source, whereas the Tij,ij ∗G method would have

challenged the fidelty of the differentiaion of Tij and the quadrature to accurately represent

these cancellation in the numerics. Indeed, we were not able to successfully compute the farNASA/CR—2003-212123 15
0 30 60 90 120 150 180

100

101

102

α (degrees)

p

p

/(ρ2

jU4

j ) × 1010

Figure 12: The effecive directivity of streamwise portions of the jet as defined by dividing (9) into

sub-inetegral in x: 0 ≤ x ≤ 5ro ; 5ro ≤ x < 10ro ; 10ro ≤ x < 15ro  ; 15ro ≤ x < 20ro ;

20ro ≤ x < 25ro; and 25ro ≤ x < 30ro. The line is the total.

field noise by summing up the monopoles directly. (The principal difficulty appeared to be in

the accuate computation double-divergence operation on the reduced every-other-mesh-point

mesh.) Taking this selected approach, we find negligible sensitivity to how the downstream and

upstream boundaries are treated, in contrast to the Lighthill solution of Mitchell et al.,

33 and

good agreement with other methods for computing the far-field noise from this simulation.2

This success suggests that quadrupole-like cancelation are important, but we can not conclude

directly that the noise is quadruopole, or even if it is well modeled by quadrupoles.

It is difficult to attach a particular noise source type label to jet turbulence because it is

a distributed source, whereas the term ‘quadrupole’ itself is best defined for a point source.

Lighthill himself points out34 that treatment of finite Mach number jets by his theory is an

extrapolation of a M → 0 theory and therefore subject to the standard caveats.

We can make an indirect test of whether the source is close to a distribution of point

quadrupoles by computing the noise from a part of the jet. If the noise depends only on

quadrupole cancellations, the noise from any part of the jet should be less than the total and

we could then add up the directivity of different components without worry of missing cancellation. This would be attractive from a modeling perspective because it would allow us to

define the noise from a part of the jet, for example a particular streamwise slice. However, if

non-quadrupole cancellation are substantial, we will be able within this approach not be able

to make such a definition.

Figure 12 show that the noise from downstream chunks of the jet is actually much louder

than the total. There are substantial non-quadrupole cancellation that are disrupted when the

integral (9) is split in x. The total of course matches exactly by the nature of the numerical

quadrature, but the mean-square pressure directivities are individually far too loud. This is

congruous with different observations made in a previous study of this database which suggested

a non-quadrupole character for the sources.2 Within Lighthill’s theory it does not seem possible

to define the noise from a downstream region in a jet, though this does not substantial the

potential facility of models build on such an assumption. [check Lilly JSV paper]

A.7 Time spectra

All the components contribute to the far-field sound over a range of frequencies, but their

spectral shapes differ for different components and, for some, are strongly dependent on α.

Figure 13 shows far-field noise spectra at four different angles. Closest to the jet at α = 30◦

NASA/CR—2003-212123 16
0.5 1 1.5 2 10-6

10-5

10-4

10-3

St

ˆpˆp∗/ρ2

∞

U2

j r2

o

(a) α = 30◦

0.5 1 1.5 2 10-6

10-5

10-4

10-3

St

ˆpˆp∗/ρ2

∞

U2

j r2

o

(b) α = 54◦

0.5 1 1.5 2 10-7

10-6

10-5

10-4

St

ˆpˆp∗/ρ2

∞

U2

j r2

o

(c) α = 78◦

0.5 1 1.5 2 10-7

10-6

10-5

10-4

St

ˆpˆp∗/ρ2

∞

U2

j r2

o

(d) α = 102◦

Figure 13: Pressure spectra: Tij total; Tl

ij + T nij all velocity terms; Tlij velocity

terms linear in fluctuations; T n

ij velocity terms nonlinear in fluctuations; T sij entropy

term.

(figure 13 a), the low frequency noise is primarily due to T l

ij . The noise from just T sij is small

here, but because it is so highly anti-correlated with that from T l

ij it cancels a portion of the

Tl

ij noise. By itself, the noise from T nij at higher frequencies is comparable to that just from

Tl

ij . At α = 54◦ (figure 13 b), the noise from T lij makes of the greater part of the total noise

for St < 0.3 (though still somewhat canceled by the noise from T s

ij ) whereas the noise from T nij

is more significant for St > 0.3. Both have similar levels for St  0.8. At α = 78◦ and 102◦

(figures 13 c and d), the noise from T n

ij and Tij are essentially the same.

Figure 14 illustrates how the spectral shape of the different components change with α.

The spectrum from the full source Tij clearly becomes broader with increasing α (figure 14 a).

However, the spectral shape of the noise from T n

ij is relatively unchanged (figure 14 c). It is the

change in shape of the spectrum from T l

ij and its rapidly decreasing significance near α = 90◦

(figure 14 b) that account for most of the change of the full spectrum. The spectral peak of the

noise from just T s

ij (figure 14 d) shifts to lower frequencies (from St ≈ 0.35 to St ≈ 0.15), but its

shape is relatively broad and otherwise insensitive to α. The shifts follows to the Doppler factor

(1 − Mc cos α) with Mc now equal to 0.6Uj/a∞ = 0.5 (figure 15). This is surprising since we

saw in the previous section that the directivity was better fitted using Mc = 0.3. It seems that

this most intense frequency comes from the structures convecting near the end of the potential

core, as suggested by visualizations.2 This is where Mc should be closest to 0.6Uj/a∞.

Tam35 suggests that jet noise spectra are composed of two distinct components: one due to

large-scale structures and one due to fine-grained turbulence. Here we see that the sound from

T n

ij is similar to Tam’s nominal fine-grained turbulence contributions, while that from T l

ij , being

NASA/CR—2003-212123 17
0.5 1 1.5 2 10-7

10-6

10-5

10-4

10-3

St

ˆpˆp∗/ρ2

∞

U2

j r2

o

(a)

0.5 1 1.5 2 10-7

10-6

10-5

10-4

10-3

St

ˆpˆp∗/ρ2

∞

U2

j r2

o

(b)

0.5 1 1.5 2 10-7

10-6

10-5

10-4

10-3

St

ˆpˆp∗/ρ2

∞

U2

j r2

o

(c)

0.5 1 1.5 2 10-7

10-6

10-5

10-4

10-3

St

ˆpˆp∗/ρ2

∞

U2

j r2

o

(d)

Figure 14: Pressure spectra: (a) Tij all components; (b) T l

ij components linear in velocity fluctuations; (c) T n

ij components quadratic in velocity fluctuations; (d) T sij ‘entropic’ component. Spectra

at every 6◦ form α = 18◦ (top curve) to α = 102◦ (bottom curve) are shown.

0 30 60 90 120 150 180 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

α (degrees)

Stn

peak

Figure 15: Peak Strouhal number due directly to T n

ij : ◦ from simulation; 0.175(1−Mc cos α)−1

with Mc = 0.6Uj/a∞. The peak was determined by fitting third-order polynomials to the spectra

in log-log coordinates. This removed statistical variations and provided an unambiguous recipe for

determining the peak.

NASA/CR—2003-212123 18
more peaked at lower frequencies, is qualitatively similar to his nominal large-scale contribution.

Unfortunately, the present jet has a Reynolds number dependent spectrum so direct comparison

with Tam’s empirical spectra is not instructive. However, the shear noise/self-noise decomposition is an artifact of the Reynolds average, which is made for modeling convenience, though it

does seem to also divide the radiated noise into a part whose spectrum varies substantially with

angle and a part that does not. But it does this without a formal splitting of the scales since all

velocity fluctuations participate in both the shear noise and self-noise sources. The empirical

predictive tool of Tam & Auriault36 for the noise near α = 90◦ might be assisted by the fact

the noise is, in a sense, simpler here, not depending on the mean flow through T l

ij and having

no substantial ‘entropic’ contribution.

A.8 Noise source statistics

Making a compact source assumption, the radiated acoustic intensity is directly related to the

volume integral of the of the space/retarded-time covariance of the Lighthill stress,

I(x) = xixjxkxl

16π2a5

∞|x|

5 ×




∞




∞

∂4

∂τ 4 Tij (y, t)Tkl(y + ξ, t + τ ) dξ dy, (13)

which follows after Lighthill12 and Ffowcs Williams31 and was used extensively by Ribner.6

Often in formulating statistical noise models, the stress tensors are approximated as6

Tij ≈ ρuiuj , (14)

which gives the sound due to a point in the flow as

Ip

ijkl ∝

∂4

∂τ 4




∞

uiuju

kul dξ, (15)

a formula that is the basis of some modern statistical jet noise models.7, 17 We have seen that

the (p − a2

∞ρ) term, which is neglected here, does indeed alter the directivity, but mostly near

the jet axis. It does not contribute much to the net radiated power (table 2) or for α  45◦.

The fourth-order integrated space/retarded-time correlation (15) that constitutes the core of

(13) is often simplified to facilitate the use of experimental data in tunning models. Assuming

that the turbulence has a normal joint probability distribution gives37

Rijkl = uiuju

kul = uiuj ukul + uiuk ujul + uiul ujuk. (16)

This is checked directly in figure 17 for the normal components at the points indicated in figure

16. Though convergence is relative poor for the fourth-order tensor, the agreement is seen to

be very good, justifying the approximation in (16). Other components are more difficult to

converge when computed as the fourth-order correlation because the τ -dependent component,

uiu

k ujul + uiul ujuk, is relatively smaller compared to the τ -independent component and are

not shown. Nevertheless, our results suggest that two-point correlation statistics, which are

relatively easy to measure at least in one direction, can be used in (16), a generalization of

which has been provided by Lighthill.19 Two-point correlations are known to be well-fitted

by exponential functions in high-Reynolds-number jets,38 but not necessarily in low-Reynoldsnumber simulations of homogeneous turbulence, as discussed in the context of noise by Lilley.39

Thus it is important to check their form here in our low-Reynolds-number jet. Figure 18 shows

vx(xo)vx(xo + ξ) for xo = 18.0ro and xo = 26.5ro, both at r = ro, and fitted by exponential

functions. The fits are good, essentially perfect at x = 26.5ro where the local Reynolds number

is, of course, higher. Likewise, the temporal two-point correlations are strongly peaked and also

well fitted by exponentials (figure 19).

NASA/CR—2003-212123 19
0 10 20 30 -6

-3

0

3

6

x = 15ro

x = 19ro

x = 23ro r/ro

x/ro

Figure 16: Schematic showing points used relative to mean velocity. All are coincident with the

nozzle lip at r = ro.

x/ro ijkl E2 τoa∞/ro

15

1111 0.015 3.851

1212 0.021 4.113

2222 0.042 2.595

3333 0.019 2.573

19

1111 0.028 4.379

1212 0.025 4.809

2222 0.032 2.866

3333 0.022 2.690

23

1111 0.048 6.131

1212 0.021 6.763

2222 0.024 3.461

3333 0.019 3.307

Table 3: Parameters for fitting with (18). E2 is the error norm for the fit: E2 = r−1 o a∞



(f −

P/Pmax)2 dτ .

Of course, the acoustic intensity (13) depends upon the integrated correlation,

Pijkl =




∞

uiuju

kul dξ, (17)

which, based on two-point correlations, is often assumed to also have an exponential form.7, 9

However, this is not the case as seen in figure (20) where it is instead well-fitted by Gaussian

functions,

f(τ ) = exp 

τ 2

τ 2

o

	

, (18)

despite the good exponential function fits of the one-dimensional two point in time correlations.

Data at x = 19ro, r = 1.1ro are plotted in figure 17 and the fitting coefficients for all points in

are given in table 3. As seen by the error norms given in this table, all the fits are very good.

We also see that the width of the Gaussian depends on the components, with the same relative

widths at the different x-points. To compute noise, this data is differenced in time four times,

so even the small deviation from Gaussian that we see in figure 17 will potentially be important.

Nevertheless, error for using an exponential fit for this flow would be much greater.

NASA/CR—2003-212123 20
18 20 22 24 26 28

0

0.2

0.4

0.6

0.8

1

1.2

ξx/ro

ˆ

Riiii/U4

j

(a)

012345

0

0.2

0.4

0.6

0.8

1

1.2

ξr/ro

ˆ

Riiii/U4

j

(b)

-2 -1 0 1 2

0

0.2

0.4

0.6

0.8

1

1.2

ξθ/ro

ˆ

Riiii/U4

j

(c)

-10 -5 0 5 10

0

0.2

0.4

0.6

0.8

1

1.2

τ a∞/ro

ˆ

Riiii/U4

j

(d)

Figure 17: Direct evaluation of (16) for normal components at x = 23ro and r = 1.1ro. The lines

are Rˆiiii = uiu

i uiui + uiui uiui and the symbols are Rˆiiii = uiuiuiui − uiui uiui: and  are

1111, and ◦ are 2222, and and are 3333. One dimensional offsets in (a) ξx, (b) ξr, (c)

ξθ, and (d) τ .

0 2 4 6 8 10 -0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ξ/ro

vx(xo)vx(xo + ξ)/vx(xo)vx(xo)

(a)

0 2 4 6 8 10 -0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ξ/ro

vx(xo)vx(xo + ξ)/vx(xo)vx(xo)

(b)

Figure 18: Axial two-point correlation for (a) xo = 19ro and (b) xo = 26.5ro with exponential fits

exp(−x/0.83) and exp(−x/1.34), respectively.

NASA/CR—2003-212123 21
-30 -20 -10 0 10 20 30 -0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τ

vxvx correlation

(a) vxvx

-30 -20 -10 0 10 20 30 -0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τ

vxvr correlation

(b) vxvr

-30 -20 -10 0 10 20 30 -0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τ

vrvr correlation

(c) vrvr

-30 -20 -10 0 10 20 30 -0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τ

vθvθ correlation

(d) vθvθ

Figure 19: Two point in time correlations, v(t)v(t + τ ), at r = 1.1ro and x = 19ro. The lines

are exponential fits by exp(−τ /τo): (a) vxvx with τo = 2.54, (b) vxvr with τo = 2.18, (c) vrvr with

τo = 1.32, (b) vθvθ with τo = 1.27.

NASA/CR—2003-212123 22
-10 -5 0 5 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

τ a∞/ro

P1111/Ppeak 1111

(a)

-10 -5 0 5 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

τ a∞/ro

P1212/Ppeak 1212

(b)

-10 -5 0 5 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

τ a∞/ro

P2222/Ppeak 2222

(c)

-10 -5 0 5 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

τ a∞/ro

P3333/Ppeak 3333

(d)

Figure 20: Integrated fourth-order correlation tensor as defined in (17) shown with open circles

◦ and Gaussian fits . Fitting parameters are given in table 3.

NASA/CR—2003-212123 23
A.9 Conclusions and closing

The implications for modeling using shear noise, self-noise, and ‘entropic’ source decompositions

are clear. These contributions are highly correlated at small angles to the jet, not statistically

independent as often assumed. Ribner’s theory, though qualitatively correct in predicting the

individual contributions of the shear noise and self-noise, especially if Mc = 0.3 is used to reflect

the computed dominant phase velocity of Tij for this jet, but will miss their substantial mutual

cancellation. Several more recent models, which have followed this same philosophy, also omit

shear noise/self-noise correlation. An implication of this conclusion is that locally homogeneous

turbulence, which is sufficient to decorrelate the shear noise and self-noise,29 may not be an

acceptable model for the noise source, at least at small angles to the downstream axis.

Even for this nearly uniform temperature jet, the entropic source’s contribution to the farfield noise is important at small angles, especially due to its high correlation with the shear

noise. Neglecting it leads to a substantial over prediction of the far-field sound. The present

jet with Uj/a∞ = 0.83 is close to the point where the experiments of Tanna40 show the least

sensitivity to jet temperature, so we can anticipate that this will become more significant at

substantially higher or lower jet velocities.

The sideline noise is dominated by terms in the Lighthill source that are independent of the

mean flow and entropic terms. This should greatly simplify modeling for this portion of the

radiated sound, perhaps making it more amenable to empirical fits.

We find that downstream of the potential core’s closing two-point correlations are well fitted

by exponential functions, which is characteristic of high-Reynolds-number turbulent flows. (In

the laminar/transitional shear layers they are more similar to the wavy correlations expected

for flows dominated by instability waves, as expected in this region at Re = 3600.2) The

fourth-order space/retarded-time covariance tensors, which are more directly related to the

far-field sound, are well approximated by the standard simplification into products of secondorder correlations, which assume a normal distribution for the turbulence. However, despite the

good exponential fits of the one-dimensional components, the volume integrated fourth-order

correlation tensor is very well fitted by Gaussian functions, of different widths for different

components. Both its form and dependence on components are counter to standard modeling

procedures.

It seems appropriate in closing to make a final comment concerning the Reynolds number of

the simulation used, which is low for a turbulent jet by every standard aside from modern direct

numerical simulation capabilities. It is conceivable that the correlations that we observe might

be an artifact of this low Reynolds number, but they are so significant that it seems unlikely

that they could disappear altogether at higher Reynolds numbers. Once turbulent, the Reynolds

stresses are in close agreement to jets of much higher Reynolds numbers and the two-point correlations decay rapidly, just as at higher Reynolds numbers.2 It does not seem feasible that

the energetic structures could change sufficiently with Reynolds number to display a behavior

much different than we observe. What will certainly change at higher Reynolds numbers is that

the near-nozzle shear layers will become turbulent. Since these eddies are relatively small, we

can anticipate that they will make the high-frequency noise missing from the present simulation. Unfortunately, there does not appear to be a definitive means of directly testing these

admittedly speculative assertions: we have pressed the limit of direct numerical simulation∗ and

experimentalists have pressed the limits of their measurement techniques. Further technological

advances in both might facilitate a better parameterization of Reynolds number effects in the

future and at the same time allow the details of high-Reynolds-number jet noise sources to be

probed in detail as we have done here in the low-Reynolds-number limit.

∗Large-eddy simulation is promising, but seems to have a resolution requirement for a high-Reynolds-number jet

that is greater than the present direct numerical simulation.41

NASA/CR—2003-212123 24
Acknowledgments

The author acknowledges the assistance of Devon Johnson in identifying the approach used to

compute the far-field sound. He also thanks Prof. Tim Colonius for making critical comments

on a early draft of this paper. Financial support from NASA is gratefully acknowledged.

B Ray Traces Through Unsteady Jet Turbulence

J. B. Freund

Theoretical and Applied Mechanics

University of Illinois at Urbana-Champaign

T. G. Fleischman

Mechanical and Aerospace Engineering

University of California, Los Angeles

B.1 Abstract

Results of an ongoing effort to quantify the role turbulence in scattering sound in jets are

reported. Using a direct numerical simulation database to provide the flow data, ray paths

traced through the mean flow are compared with those traced through the actual time evolving

turbulent flow. Significant scattering by the turbulence is observed. The most notable effect is

that upstream traveling waves that are trapped in the potential core by the mean flow, which

acts as a wave guide, easily escape in the turbulent flow. A crude statistical estimate based

on ray number density suggests that directivity is modified by the turbulence, but no rigorous

treatment of non-uniformities in the high-frequency approximation is attempted.

B.2 Nomenclature

a Sound speed

N Number of rays observed

Nr Number or rays released

M Mach number

p Pressure

Re Reynolds number

r Radial coordinate

ro Jet nozzle radius

s Entropy

t Time

ui Cartesian velocities (i = 1, 2, 3)

x Axial coordinate

xi Cartesian coordinates

α Directivity angle

φ Wave phase

ρ Density

θ Cylindrical polar coordinate

ω Angular frequency

Subscripts

j Jet exit

∞ Ambient

Accents

() Base flow (potentially time dependent)

() Acoustic perturbation

NASA/CR—2003-212123 25
r/ro

x/ro

0.2

0.1

0.8

0.3

0.5 0.4

0.7 0.6

0 10 20 30 -6

-4

-2

0

2

4

6

Figure 21: Contours of mean streamwise velocity: 8 evenly spaced contours from 0.1a∞ to 0.8a∞.

B.3 Introduction

Lighthill’s theory of aerodynamic noise is often criticized because it does not distinguish refraction from generation. Instead, these effects are grouped together in a nominal ‘source’:12

∂2ρ

∂t2 − a2

∞

∂2ρ

∂xj∂xj    sound propagation

= ∂2Tij

∂xi∂xj    ‘source’

, (19)

where Tij is the Lighthill stress tensor. Lighthill was, of course, aware of this but concluded that

refraction “may affect finer details, but it does not appear to be fundamental.”21 It has since

been argued that a distinction might not be necessary when developing predictive models,22 but

since generation and propagation are different physical processes, it is attractive to model them

separately. Lighthill was not faced with the stringent noise regulations that we are today. With

a great effort underway to achieve as little as a 3dB noise reduction, “finer details” are now

more important. More recent but more complex acoustic analogies, such Lilley’s equation,13, 14

D

DtD2ΠDt2 − ∂∂xj



a2 ∂Π

∂xj

 + 2∂uk

∂xj

∂

∂xk



a2 ∂Π

∂xj



   sound propagation

= −2

∂uj

∂xk

∂ui

∂xj

∂uk

∂xi

+ (viscous terms)

   source

, (20)

attempt to better separate propagation and generation.

Unfortunately, since (20) is nonlinear it must be linearized for implementation and interpretation.14 An estimate of the steady mean flow is typically chosen to linearize about and

many ongoing modeling endeavors take this approach.7, 24 Even modelers that circumvent the

exact governing equations as a starting point and designate a relatively ad hoc acoustic source36

choose to linearize about a steady mean flow. To linearize the propagation operator in (20), the

nonlinear terms can be omitted, which assumes they play no substantial role at all, or moved to

the right hand side, which is satisfying because the equation remains exact but once again blurs

the distinction between source and propagation. Though convenient, linearization about the

mean flow is well understood to be artificial since no individual sound wave actually encounters

the mean flow.42 Because local turbulence intensities can be over 100% in a jet, with large flow

structures on the scale of the local jet radius, scattering by the turbulence might indeed be

NASA/CR—2003-212123 26
r/ro

x/ro

0 10 20 30 -6

-4

-2

0

2

4

6

r/ro

x/ro

0 10 20 30 -6

-4

-2

0

2

4

6

Figure 22: (a) Instantaneous axial velocity: 9 evenly spaced contours from 0.1a∞ to 0.9a∞. (b)

Instantaneous radial velocity contours: -0.3a∞ to 0.3a∞ with 0.1 spacing. Negative contours are

dashed and the zero contour is omitted.

NASA/CR—2003-212123 27
significant. We investigate this possibility.

Refraction in jets has been investigated on many fronts. In Mach 0.5 and 0.9 jets it was

studied by MacGregor et al.43 by adding an artificial noise source into the jets. Suzuki & Lele44

used numerical methods and analysis to study both directional and frequency scattering of

sound by instability waves in a two-dimensional mixing layer and showed a significant influence

in some cases. While low frequency components of jet noise typically follow the Lighthill/Ffowcs

Williams I ∝ (1 − Mc cos θ)−5 law, higher frequencies are less directive,27 fit better by three

inverse Doppler factors.45 High-frequency solutions46 of Lilley’s equation for uni-directional

transversely sheared flow might explain this, but scattering by turbulence has been offered

as alternative explanation.45 Similar high-frequency formulations have been incorporated into

predictive tools47, 48 using a general high-frequency Green function derived by Durbin.49 Adjoint

Green functions have been proposed to simplify implementation of flow-acoustic interactions in

models.3

The purpose of this ongoing study is to estimate the role of scattering by the unsteady

turbulent fluctuations in a jet, and here we present a preliminary report on the effort. Since

turbulence is analytically intractable, we rely on an existing, well-validated direct numerical

simulation database to represent the turbulent jet flow. Flow-acoustic interaction is studied for

high-frequency noise using an unsteady geometrical acoustic formulation to identify ray paths.

Directivity is estimated statistically based on the ray paths, but no attempt has yet been made

to construct instantaneous intensity profiles.

B.4 Simulation Database

Details of the direct numerical simulation database used in this study are reported in full elsewhere.2, 50, 51 In summary, it is of a Reynolds number 3600, Mach number 0.9, temperature ratio

Tj/T∞ = 0.86 turbulent jet, which matches the experimental conditions studied by Stromberg et

al.1 Contours of mean streamwise velocity are shown in figure 21 and contours of instantaneous

streamwise and radial velocity are shown in 22. At this Reynolds number, the initial shear

layers are laminar as expected and thus qualitatively different from a high-Reynolds-number

jet, which would have turbulent shear layers. However, after transition, which occurs a little

before the potential core closes at x ≈ 14ro, the jet’s development agrees well with those at

a much higher Reynolds number data. Downstream of the potential core, the jet’s spreading

rate and Reynolds stresses agree with those of much higher Reynolds number jets.2, 52, 53 This

is important because it suggests that the energetic large scales are similar to those in jets at

higher Reynolds numbers which should generalize the present results.

Other points of validation are reported elsewhere.2 In these references it was shown that the

mean flow, the noise directivity, and far-field noise spectrum are all in excellent agreement with

the data of Stromberg et al.1

B.5 High-Frequency Approximation

High-frequency asymptotics will be used to study the interactions of sound with the flow. This

approach is both convenient because it leads to tractable formulations, and important because

the high frequencies are a particularly annoying component of the noise. Of course, the noise

from a jet at Re = 3600 is relatively narrow banded compared to jets at typical engineering

Reynolds numbers. However, if we accept that the large, energy carrying turbulence scales

are realistic, high-frequency noise sources can be artificially added to the flow in order study

flow-acoustic interactions.

It is often found that high-frequency approximations give reasonable estimates for Helmholtz

numbers as low as unity,54–56 and we will use this to estimate a lower bound on the Strouhal

numbers that might be accurately represented by our procedure. We take as our length scale

the 50 percent two-point velocity correlation width of the turbulent eddies, which, depending

on the location within the jet, is as low as 4 ≈ 0.5ro.

2 This is smaller than the scale over which

the mean flow varies (figure 21) except in the initial shear layers. Setting the Helmholtz number

NASA/CR—2003-212123 28
He ≡ ω4/a = 1 gives ω ≈ 2a/ro or St ≡ fD/Uj = ω2ro/2πUj  1.0. Of course, the motion

of the turbulent structures might decrease or increase their effective size for a particular sound

wave they encounter. Assuming Mc ≈ 0.5, this would potentially increase the Strouhal number

limit by about one-third to 1.3, which is high but still relevant for many applications.

We develop an unsteady ray tracing formulation similar to that used by Colonius et al.57

We start with the Euler equations in Cartesian coordinates and three space dimensions, with

the energy equation written in terms of entropy,

ρ

∂ui

∂t + uj

∂ui

∂xj



+ ∂p

∂xi

= 0 (21)

∂ρ

∂t + uj

∂ρ

∂xj

+ ρ

∂uj

∂xj

= 0 (22)

∂s

∂t + uj

∂s

∂xj

= 0, (23)

and decompose the dependent variables as









ρ

ui

p

s









(xi,t)

=









ρ ̄

u ̄i

p ̄

s ̄









(xi,t)

+









ρ

u

i

p

s









(xi,t)

, (24)

where () terms will be obtained from the simulation database and in general are functions of

all space coordinates and time. The () terms are perturbations, but not necessarily acoustic at

this point. Retaining only linear terms in the perturbations gives

∂u

i

∂t +  ̄uj

∂u

i

∂xj

+ u

j

∂u ̄i

∂xj

+ ρ

ρ ̄

∂u ̄i

∂t +  ̄uj

∂u ̄i

∂xj



+

1

ρ ̄

∂p

∂xi

= 0 (25)

∂ρ

∂t +  ̄uj

∂ρ

∂xj

+  ̄ρ

∂u

j

∂xj

+ ρ ∂u ̄j

∂xj

+ u

j

∂ρ ̄

∂xj

= 0 (26)

∂s

∂t +  ̄uj

∂s

∂xj

+ u

j

∂s ̄

∂xj

= 0. (27)

We next assume harmonic fluctuations









u

i

ρ

p

s







 =









Ui

R

P

S









eiωφ(xi,t)

, (28)

where Ui, R, P and S are complex amplitudes that are functions of (xi, t), φ is a real-valued

phase function, and ω is the (large) angular frequency of the disturbances. Substituting and

retaining only the highest order terms in ω yields

∂φ

∂t +  ̄uj

∂φ

∂xj



Ui + p ̄

ρ ̄

∂φ

∂xi

S +

a ̄2

ρ ̄

∂φ

∂xi

R = 0 (29)

∂φ

∂t +  ̄uj

∂φ

∂xj



R +  ̄ρ

∂φ

∂xj

Uj = 0 (30)

∂φ

∂t +  ̄uj

∂φ

∂xj



S = 0. (31)

This leaves 5 equations and 5 unknowns. For a non-trivial solution the determinant of this

NASA/CR—2003-212123 29
α

θ

r

x1

∆α

ro

source

x2

x3

Figure 23: Coordinate system schematic.

system must be zero:

G3



G2 − a ̄2

 ∂φ

∂xj

∂φ

∂xj

	 = 0, (32)

where G = ∂φ

∂t +  ̄uj ∂φ∂xj . It is obvious that G3 = 0 roots correspond to vorticity and entropy

modes and are therefore not of concern to us. The remaining equation can be solved by method

of characteristics to yield a system of ordinary differential equations for the ray paths,

dxi

dt

=  ̄ui − a ̄2φxi

φt +  ̄ujφxj

(33)

dφxi

dt = −∂u ̄j∂xi

φxj +

∂a ̄2

∂xi

φxjφxj

φt +  ̄ujφxj

(34)

dφt

dt = −∂u ̄j∂t φxj +

∂a ̄2

∂t

φxjφxj

φt +  ̄ujφxj

(35)

dφ

dt = 0, (36)

where subscripts on φ indicate partial differentiation.

Once an initial coordinate is chosen, initial conditions for φxi are determined by solving (33)

with φt = ro/a∞ and

d

dt





x1

x2

x3



 =





a ̄ cos α +  ̄u1

a ̄ sin α cos θ +  ̄u2

a ̄ sin α sin θ +  ̄u3



. (37)

The results are insensitive to the choice of φt and, as we shall see, (35) in general.

B.6 Procedure

Equations (33) through (36) were integrated using a fourth-order Runge–Kutta algorithm with

∆t = 0.01ro/a∞. Coefficients  ̄ui and  ̄a were taken from the direct numerical simulation database.

This data was available every ∆t = 0.17ro/a∞ and at every other mesh point of the original

computation. It was interpolated in space using B-splines and in time using a linear method.

Differences were computed using second-order centered finite differences with δ = 0.001ro. Results were insensitive to this value and did not change if a fourth-order finite difference was used

instead. Results were also insensitive to the order of the B-spline used. We traced Nr = 1000

rays through the mean-flow and Nr = 500 rays through the unsteady flow. They were initially

directed at evenly spaced α-angles in the θ = 0, π and θ = ±π/2 planes (see figure 23). Results

shown here were insensitive to the number of rays traced. Except when noted, rays were traced

for a total time of 25.5ro/a∞.

Focusing by the turbulence will cause ray path to cross at points in their trajectories. At

these caustics, the high-frequency approximation for amplitude, which can be obtained from the

next highest terms in ω, fails because the wave fronts develop cusps, a feature that is always small

NASA/CR—2003-212123 30
0 10 20 30 -6

-3

0

3

6

A C B E

D

r/ro

x/ro

Figure 24: Schematic showing source points used. See also table 4.

Case x/ro r/ro

A 70

B 17 0

C 11 1

D 17 1

E 23 1

Table 4: Source points (see figure 24). All have θ = 0.

with respect to a wavelength of the sound. In addition, behind the caustics multiple rays pass

through each point and potentially interfere with each other. Even in a steady case with simply

defined caustics, a rigorous solution in regions of multiple rays adds considerable complexity.

In the present case where caustics are numerous, transitory and impossible to anticipate, we

have not undertaken the task of developing a uniformly valid amplitude procedure. Instead,

to estimate the effect of refraction on directivity we simply count the number of rays passing

through different regions on a sphere surrounding the source. The resulting ray number density,

N, can be shown inversely related to ray tube area, A, and thus proportional to tube intensity,

I, which is valid where there are no caustics. We assume that the volume of the causticaffected region crossing the observation sphere is negligible and the positions of the caustics

ever changing, as is indeed observed, and can therefore neglect them in the statistical average.

We also assume that behind the caustics the rays are decorrelated due to the stochastic action

of the turbulence so we also neglect interference. The results of these assumptions is a statistical

beam-like method which provides an estimate of directivity.

Ray position statistics were accumulated as they exited a sphere of radius r = 6.0ro centered

on the jet axis closest to the source. Monitoring a sphere at r = 10ro instead of r = 6.0ro caused

a negligible change in our results. These data were collected in 20 bins of width ∆α (see figure

23). Some rays were trapped in the jet, usually in the initial shear layers that act as a wave

guide, and did not reach this spherical surface in the run time. These rays were not counted on

the assumption they would have dissipated or entered the nozzle. Using a run time of 10ro/a∞

did not change the directivity estimates for α 150◦. For the unsteady case, rays were traced

through 39 separate time series that were separated in initial times by at least one large-eddy

turnover time. The turbulence evolved as the rays progressed.

B.7 Results and Conclusions

The five point-source positions studied are labeled in figure 24 and summarized in table 4. The

ray number-density directivities for all cases are plotted in figure 25 (a-e). In figure 25 (a),

where the source is in the laminar potential core region (x = 7ro, r = 0), we see little difference

NASA/CR—2003-212123 31
N/

Nr

α

0 30 60 90 120 150 180

0

0.02

0.04

0.06

0.08

0.1

0.12

(a) Case A

N/

Nr

α

0 30 60 90 120 150 180

0

0.02

0.04

0.06

0.08

0.1

0.12

(b) Case B

N/

Nr

α

0 30 60 90 120 150 180

0

0.02

0.04

0.06

0.08

0.1

0.12

(c) Case C

Figure 25: Figure continues on the following page.

NASA/CR—2003-212123 32
N/

Nr

α

0 30 60 90 120 150 180

0

0.02

0.04

0.06

0.08

0.1

0.12

(d) Case D

N/

Nr

α

0 30 60 90 120 150 180

0

0.02

0.04

0.06

0.08

0.1

0.12

(e) Case E

Figure 25: (a-e) Bin number density for rays traced through mean and turbulent jet

flows. See table 4 for source points.

NASA/CR—2003-212123 33
r/ro

x/ro

0 5 10 15 20 25 30

-10

-5

0

5

10

(a) Case A

r/ro

x/ro

0 5 10 15 20 25 30

-10

-5

0

5

10

(b) Case B

r/ro

x/ro

0 5 10 15 20 25 30

-10

-5

0

5

10

(c) Case C

r/ro

x/ro

0 5 10 15 20 25 30

-10

-5

0

5

10

(d) Case D

r/ro

x/ro

0 5 10 15 20 25 30

-10

-5

0

5

10

(e) Case E

Figure 26: Every tenth ray path released in the x–y plane in the mean flow for labeled cases (see

table 4).

NASA/CR—2003-212123 34
r/ro

x/ro

0 5 10 15 20 25 30

-10

-5

0

5

10

(a) Case A

r/ro

x/ro

0 5 10 15 20 25 30

-10

-5

0

5

10

(b) Case B

r/ro

x/ro

0 5 10 15 20 25 30

-10

-5

0

5

10

(c) Case C

r/ro

x/ro

0 5 10 15 20 25 30

-10

-5

0

5

10

(d) Case D

r/ro

x/ro

0 5 10 15 20 25 30

-10

-5

0

5

10

(e) Case E

Figure 27: Every fifth ray released in the x–y plane in the unsteady flow for the labeled cases (see

table 4).

NASA/CR—2003-212123 35
r/ro

x/ro

0 30 60 90 120 150 180

0

0.02

0.04

0.06

0.08

0.1

0.12

Figure 28: Mean flow ray number-density directivity for trace times of 10ro/a∞ and 25ro/a∞

. Case E is shown.

between the steady and unsteady ray number-density directivity. However, for rays released

on the jet’s axis just past the end of the potential core at x = 17ro (figure 25 b), there is a

significant difference between the two cases. In the mean-flow case, substantially fewer rays

leave the observation sphere at angles greater than α = 70◦. The reason for this is clear in

figure 26 (b): the upstream traveling waves are trapped by the jet’s shear layers. However, in

the unsteady case the rays escape and radiate as seen in figure 27 (b). Other source points are

seen to give similar results.

The freeing of trapped rays appears to be the principle difference between the steady and

unsteady cases for all the source points considered. In figure 25 (e), the upstream velocity is slow

enough that trapped upstream traveling waves appear as a spike in the profile near α = 180◦ at

the observation time. The trapped rays must eventually escape or enter the nozzle. Whether

they will eventually affect the directivity is unclear, but the rate of release is slow: the steadyflow number-density directivity is unchanged for ray trace times of 10ro/a∞ versus 25ro/a∞ for

α 150◦ (see figure 28). If they do eventually leave the jet, they will do so differently than the

rays in a realistic turbulent jet.

So far we have assumed that the source itself has no inherent directivity. However, if the

sources are convecting quadrupoles, this is not true. In figures 29 (a-e) the rays have been given

an initial weighting so that they would have a three Doppler factor, W −3 =(1 − Mc cos α)−3,

directivity in absence of additional refraction. The convection Mach number was taken to be

Mc = 0.5, which corresponds to a convection velocity of Uc = 0.6Uj . We again see significant

influence of the turbulence on the directivity estimates.

It is interesting to estimate how much of the scattering is because of the presences of the

turbulence at any instance in time versus its change in time. To this end we have plotted a

directivity for case B with the time derivative terms of the background flow in (36) set to zero.

We compare with the correct equation results in figure 30, and see that there is an effect for

α  60◦, but not a large one. It appears that accounting for the turbulence as a succession of

steady states would provide most of the scattering.

In summary, we have made a crude estimate of the effect of turbulence on the propagation

of high-frequency noise in turbulent jets and found that turbulence does affect the ray number

density on a spherical surface outside the jet, increasing it in most cases. The most significant

mechanism observed was that the turbulence frees rays that would otherwise be trapped traveling

upstream in the potential core if only refraction by the mean flow were considered. The spatial

NASA/CR—2003-212123 36
10 log(W

−3N/

Nr)

α

0 30 60 90 120 150 180

-30

-25

-20

-15

-10

-5

0

(a) Case A

10 log(W

−3N/

Nr)

α

0 30 60 90 120 150 180

-30

-25

-20

-15

-10

-5

0

(b) Case B

Figure 29: Figure continues on page 33.

NASA/CR—2003-212123 37
10 log(W

−3N/

Nr)

α

0 30 60 90 120 150 180

-30

-25

-20

-15

-10

-5

0

(c) Case C

10 log(W

−3N/

Nr)

α

0 30 60 90 120 150 180

-30

-25

-20

-15

-10

-5

0

(d) Case D

10 log(W

−3N/

Nr)

α

0 30 60 90 120 150 180

-30

-25

-20

-15

-10

-5

0

(e) Case E

Figure 29: (a-e) Directivity when rays are given an effective three Doppler factors weighting.

NASA/CR—2003-212123 38
N/

Nr

α

0 30 60 90 120 150 180

0

0.02

0.04

0.06

0.08

0.1

0.12

Figure 30: Directivity for Case B, x = 17ro, r = 0, with time derivatives in (35) and without

.

variation of the turbulence was found to be more an influence on scattering than its time

evolution.

This work is ongoing and a rigorous means of computing directivity as well as more realistic

sources are being considered.

B.8 Acknowledgments

Support for this work provided by NASA is very gratefully acknowledged.

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NASA/CR—2003-212123 39
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NASA/CR—2003-212123 42
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NAG3–2662

48

Jet Noise Physics and Modeling Using First-Principles Simulations

Jonathan B. Freund

Jet noise; CFD; Jet; Acoustic; Noise prediction

Unclassified -Unlimited

Subject Categories: 07 and 34 Distribution: Nonstandard

University of Illinois, Urbana-Champaign

Department of Theoretical and Applied Mechanics

Urbana, Illinois 62801

Project Manager, James E. Bridges, Structures and Acoustics Division, NASA Glenn Research Center, organization code

5940, 216–433–2963.

An extensive analysis of our jet DNS database has provided for the first time the complex correlations that are the core

of many statistical jet noise models, including MGBK. We have also for the first time explicitly computed the noise

from different components of a commonly used noise source as proposed in many modeling approaches. Key findings

are: (1) While two-point (space and time) velocity statistics are well-fitted by decaying exponentials, even for our lowReynolds-number jet, spatially integrated fourth-order space/retarded-time correlations, which constitute the noise

"source" in MGBK, are instead well-fitted by Gaussians. The width of these Gaussians depends (by a factor of 2) on

which components are considered. This is counter to current modeling practice, (2) A standard decomposition of the

Lighthill source is shown by direct evaluation to be somewhat artificial since the noise from these nominally separate

components is in fact highly correlated. We anticipate that the same will be the case for the Lilley source, and (3) The

far-field sound is computed in a way that explicitly includes all quadrupole cancellations, yet evaluating the Lighthill

integral for only a small part of the jet yields a far-field noise far louder than that from the whole jet due to missing nonquadrupole cancellations. Details of this study are discussed in a draft of a paper included as appendix A.