JET NOISE MODELS FOR FORCED MIXER NOISE …lyrintzi/Thesis_Garrison.pdffor Forced Mixer Noise Predictions . Major Professor: Anastasios S. Lyrintzis and Gregory A. Blaisdell. The Four-Source

JET NOISE MODELS FOR FORCED MIXER NOISE PREDICTIONS

A Thesis

Submitted to the Faculty

of

Purdue University

by

Loren A. Garrison

In Partial Fulfillment of the

Requirements for the Degree

of

Masters of Science in Aeronautics and Astronautics

December 2003

ii

ACKNOWLEDGMENTS

I would like to thank Professor Tasos Lyrintzis and Professor Greg Blasidell for

giving me the opportunity to work on this project and for their leadership and

guidance. The work summarized in this thesis is part of a joint effort with the

Rolls-Royce Corporation, Indianapolis and has been sponsored by the Indiana 21st

Century Research and Technology Fund. I would also like to thank Bill Dalton at

the Rolls-Royce Corporation, Indianapolis for his many valuable discussions and for

providing the technical data and the experimental acoustic data used in this research.

I would like to thank Professor Stuart Bolton for serving on my advisory committee.

I would like to thank Dr. Rod Self, Dr. Brian Tester, and Prof. Mike Fisher at the

Institute of Sound and Vibration Research for both their guidance while I studied

there, and for their valuable advice and suggestions throughout my research. I would

like to thank my colleague Ali Uzun for his help and assistance.

iii

TABLE OF CONTENTS

Page

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Current Jet Noise Prediction Models . . . . . . . . . . . . . . . . . . 1

1.3 Goals of the Present Research . . . . . . . . . . . . . . . . . . . . . . 6

2 Coaxial Jet Noise Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Four-Source Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Practical Jet Configurations . . . . . . . . . . . . . . . . . . . . . . . 17

3 Experimental Acoustic Data . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Current Jet Noise Model Comparisons . . . . . . . . . . . . . . . . . . . . 27

4.1 Single Jet Noise Predictions . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Four-Source Single Jet Characteristic Parameters . . . . . . . . . . . 31

4.3 Confluent Mixer Comparisons . . . . . . . . . . . . . . . . . . . . . . 33

5 Forced Mixer Noise Predictions . . . . . . . . . . . . . . . . . . . . . . . . 37

5.1 Forced Mixer Jet Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2 Two-Source Forced Mixer Noise Models . . . . . . . . . . . . . . . . . 38

5.3 Two-Source Model Parameter Optimization . . . . . . . . . . . . . . 40

5.4 Two-Source Model Results . . . . . . . . . . . . . . . . . . . . . . . . 41

5.4.1 Model 1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.4.2 Model 2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 55

iv

Page

5.4.3 Parameter Correlations . . . . . . . . . . . . . . . . . . . . . . 62

5.4.4 Two-Source Model Performance . . . . . . . . . . . . . . . . . 68

6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

v

LIST OF TABLES

Table Page

3.1 Non-Dimensional Lobed Mixer Penetration . . . . . . . . . . . . . . . 22

3.2 Experimental Data Test Conditions . . . . . . . . . . . . . . . . . . . 22

3.3 Dual Flow Aerodynamic Test Conditions . . . . . . . . . . . . . . . . 23

4.1 ARP876C Input Parameters . . . . . . . . . . . . . . . . . . . . . . . 28

5.1 Model 1 Optimized Parameters for the Low Penetration Mixer . . . . 49

5.2 Model 1 Optimized Parameters for the Intermediate Penetration Mixer 51

5.3 Model 1 Optimized Parameters for the High Penetration Mixer . . . . 53

5.4 Model 2 Optimized Parameters for the Low Penetration Mixer . . . . 57

5.5 Model 2 Optimized Parameters for the Intermediate Penetration Mixer 59

5.6 Model 2 Optimized Parameters for the High Penetration Mixer . . . . 61

5.7 Final Optimized Parameters for Model 1 . . . . . . . . . . . . . . . . 63

5.8 Coefficients from the Linear Curve-fit of the Results from Model 1 . . 63

5.9 Final Optimized Parameters for Model 2 . . . . . . . . . . . . . . . . 65

5.10 Coefficients from the Linear Curve-fit of the Results from Model 2 . . 66

5.11 Average Weighted Errors in dB for Model 1 . . . . . . . . . . . . . . 70

5.12 Average Errors in dB for Model 1 . . . . . . . . . . . . . . . . . . . . 70

5.13 Maximum Errors in dB for Model 1 . . . . . . . . . . . . . . . . . . . 71

5.14 Average Weighted Errors in dB for Model 2 . . . . . . . . . . . . . . 82

5.15 Average Errors in dB for Model 2 . . . . . . . . . . . . . . . . . . . . 83

5.16 Maximum Errors in dB for Model 2 . . . . . . . . . . . . . . . . . . . 83

vi

LIST OF FIGURES

Figure Page

2.1 Coaxial Jet Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Single Stream Source Distribution Function for fc = 1000 Hz . . . . . 10

2.3 FU and FD Functions for fc = 1000 Hz . . . . . . . . . . . . . . . . . 11

2.4 Spectral Filter Functions for fc = 1000 Hz . . . . . . . . . . . . . . . 11

2.5 Effective Jet Source Reduction Function . . . . . . . . . . . . . . . . 16

2.6 Dual Flow Configurations (a) Coplanar, Coaxial Jet (b) InternallyMixed Jet with a Confluent Mixer (c) Internally Mixed Jet Configu-ration with a Forced Mixer . . . . . . . . . . . . . . . . . . . . . . . . 18

2.7 Internally Mixed Jet Configuration with a Forced Mixer . . . . . . . 19

2.8 Typical Lobed Mixer Geometry . . . . . . . . . . . . . . . . . . . . . 19

2.9 Lobed Mixer Vortex Strutcure . . . . . . . . . . . . . . . . . . . . . . 20

3.1 NASA Glenn Aero-Acoustic Propulsion Laboratory . . . . . . . . . . 22

3.2 Confluent and 12-Lobe Mixer Experimental Data at Set Point 1 . . . 24



4.1 OASPL Dependence on the Fully Expanded Mean Jet Velocity . . . . 29

4.2 OASPL Directivity Dependence on the Fully Expanded Mean Jet Ve-locity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 SPL Dependence on the Jet Total Temperature at 90 degrees . . . . . 30

4.4 SPL Dependence on the Jet Total Temperature at 150 degrees . . . . 30

4.5 Confluent Mixer Predictions for Set Point 1 . . . . . . . . . . . . . . 34



5.1 Forced Mixer Penetration . . . . . . . . . . . . . . . . . . . . . . . . 37

vii

Figure Page

5.2 Model 1 Parameter Optimization Error Results for the Low Penetra-tion Mixer at Set Point 1 . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.3 Model 1 Parameter Optimization Non-Dimensional Error Results forthe Low Penetration Mixer at Set Point 1 . . . . . . . . . . . . . . . 44

5.4 Model 1 Parameter Optimization Results for the Low PenetrationMixer at Set Point 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.5 Model 1 Optimized Predictions for the Low Penetration Mixer at SetPoint 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46





5.10 Model 1 Parameter Optimization Average Weighted Error Results forthe Low Penetration Mixer at Set Points 1, 2 and 3 . . . . . . . . . . 50

5.11 Model 1 Parameter Optimization Results for the Low PenetrationMixer at Set Points 1, 2 and 3 . . . . . . . . . . . . . . . . . . . . . . 50

5.12 Model 1 Parameter Optimization Average Weighted Error Results forthe Intermediate Penetration Mixer at Set Points 1, 2 and 3 . . . . . 52

5.13 Model 1 Parameter Optimization Results for the Intermediate Pene-tration Mixer at Set Points 1, 2 and 3 . . . . . . . . . . . . . . . . . . 52

5.14 Model 1 Parameter Optimization Average Weighted Error Results forthe High Penetration Mixer at Set Points 1, 2 and 3 . . . . . . . . . . 54

5.15 Model 1 Parameter Optimization Results for the High PenetrationMixer at Set Points 1, 2 and 3 . . . . . . . . . . . . . . . . . . . . . . 54

5.16 Model 2 Parameter Optimization Average Weighted Error Results forthe Low Penetration Mixer at Set Points 1, 2 and 3 . . . . . . . . . . 56

5.17 Model 2 Parameter Optimization Results for the Low PenetrationMixer at Set Points 1, 2 and 3 . . . . . . . . . . . . . . . . . . . . . . 56

5.18 Model 2 Parameter Optimization Average Weighted Error Results forthe Intermediate Penetration Mixer at Set Points 1, 2 and 3 . . . . . 58

viii

Figure Page

5.19 Model 2 Parameter Optimization Results for the Intermediate Pene-tration Mixer at Set Points 1, 2 and 3 . . . . . . . . . . . . . . . . . . 59

5.20 Model 2 Parameter Optimization Average Weighted Error Results forthe High Penetration Mixer at Set Points 1, 2 and 3 . . . . . . . . . . 60

5.21 Model 2 Parameter Optimization Results for the High PenetrationMixer at Set Points 1, 2 and 3 . . . . . . . . . . . . . . . . . . . . . . 61

5.22 Model 1 Optimized Parameter Correlation of the Source Strengths . . 63

5.23 Model 1 Optimized Parameter Correlation of the Cut-off StrouhalNumber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.24 Model 2 Optimized Parameter Correlation of the Source Strengths . . 66

5.25 Model 2 Optimized Parameter Correlation of the Cut-off StrouhalNumber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.26 Model 1 Predictions for the Low Penetration Mixer at Set Point 1 . . 72



5.29 Model 1 Predictions for the Intermediate Penetration Mixer at SetPoint 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75



5.32 Model 1 Predictions for the High Penetration Mixer at Set Point 1 . . 78








ix

Figure Page





x

NOMENCLATURE

V Velocity

T Total Temperature

P Total Pressure

ρ Density

D Diameter

A Area

f frequency

θ Far-Field Angle (Referenced from the Inlet Axis)

r Far-Field Radius

β Area Ratio As/Ap

λ Velocity Ratio Vs/Vp

δ Density Ratio ρs/ρp

τ Temperature Ratio TJ/To

FU Upstream Spectral Filter Function

FD Downstream Spectral Filter Function

∆dB Source Strength Augmentation

St Strouhal Number

Ew Error Weighting Function

α Ratio of Turbulence Intensities

Fturb Ratio of Turbulence Intensities

Id Dipole Source Intensity

Iq Quadrupole Source Intensity

Tij Lighthill Stress Tensor

xi

Subscripts

e Effective Jet

m Mixed Jet

p Primary Jet

s Secondary Jet

n Nozzle Exhaust Condition

o Ambient Condition

J Jet Condition

xii

ABBREVIATIONS

FAA Federal Aviation Administration

ISVR Institute of Sound and Vibration

SAE Society of Automotive Engineers

ESDU Engineering Sciences Data Unit

LES Large Eddy Simulation

RANS Reynolds Averaged Navier-Stokes

AAPL Aero-Acoustic Propulsion Laboratory

NPR Nozzle Pressure Ratio

NTR Nozzle Temperature Ratio

OASPL Overall Sound Pressure Level

SPL Sound Pressure Level

xiii

ABSTRACT

Garrison, Loren A. MSAE, Purdue University, December, 2003. Jet Noise Modelsfor Forced Mixer Noise Predictions . Major Professor: Anastasios S. Lyrintzis andGregory A. Blaisdell.

The Four-Source method is a recently developed noise prediction tool applicable

to simple coaxial jets. Extensions to this noise prediction model are investigated

with the goal of developing a semi-empirical jet noise prediction method that would

be applicable to jet configurations with internal forced mixers. In the following

study, the noise signals resulting from an internally mixed jet are compared to both

a coplanar, coaxial and single jet prediction. It is shown that the current Four-Source

coaxial jet noise prediction method predicts with reasonable accuracy the noise from

an internally mixed jet for the case with a confluent mixer. However, the standard

Four-Source model does not have the capability to model the differences in the noise

spectrum that result from changes in the mixer geometry. It is shown that these

spectra can be modeled using a modified Two-Source model that has three variable

parameters. These parameters are optimized to best match the experimental data,

and they are then correlated back to the changes in the mixer geometry to yield a

jet noise prediction method for a specific family of forced mixers.

xiv

1

1. Introduction

1.1 Background

The subject of jet noise has been a topic of interest ever since the introduction

of the commercial jet aircraft in the early 1950’s. The problem of jet noise is still

prevalent today; a reality that is reinforced by the increased restrictions on aircraft

noise during take-off and landing that have been imposed by the Federal Aviation

Administration (FAA) in recent decades. Jet noise is a major component of the

overall aircraft noise during take-off. However, currently there are no industry design

tools for the prediction of the jet noise resulting from complex jet flows. As a result

the noise levels of modern turbofan jet engines can only be determined by expensive

experimental testing after they have been designed and built.

1.2 Current Jet Noise Prediction Models

Single Jet Models

The far-field noise spectrum of a simple, single stream jet is determined by three

characteristic parameters, the jet velocity, jet temperature, and jet diameter. Given

these parameters a similarity spectrum for the relative sound pressure level can be

determined for a given jet velocity and temperature ratio at a specified angular

location in the far field. These similarity spectra are functions of Strouhal number,

where the frequency is non-dimensionalized by the fully expanded jet velocity and

diameter. In addition, a similarity spectrum for the overall sound-pressure level

(OASPL) is determined based on the velocity of the jet. The single stream jet

noise is then found by appropriately scaling the similarity spectra using the jet area,

observer radius, and ambient pressure. This method for predicting single stream jet

2

noise is outlined in the Society of Automotive Engineers (SAE) standard ARP876:

Gas Turbine Jet Exhaust Noise Prediction [1].

A similar approach is used by the Engineering Sciences Data Unit (ESDU) in their

single stream jet noise prediction code, ESDU 98019 [2]. The jet noise prediction

method used in the ESDU 98019 code uses an experimental database with a test

matrix of various jet velocities and temperatures. The database of jet noise spectra

are normalized based on the jet area, observer distance, and ambient pressure, and

then interpolated/extrapolated based on the jet velocity and temperature at each

far-field angular location. These values are then scaled appropriately to yield a single

jet noise prediction.

Coaxial Jet Models

Although the aerodynamic process that leads to the generation of sound in a

coaxial jet is the same as that of a single stream jet, the aerodynamic structure of

a coaxial jet is greatly different. In addition, the coaxial jet structure is dependent

on a number of additional variables such as the velocity, temperature, and area

ratios between the two streams. Furthermore, the effects of various parameters are

not always separable. These additional complexities make it difficult to develop a

noise prediction method that is based solely on the interpolation of an experimental

database. Even so, there are a few coaxial jet noise prediction methods that are

based on interpolating an experimental database. In particular, the SAE standard,

AIR1905: Gas Turbine Coaxial Exhaust Flow Noise Prediction [3], and the ESDU

program ESDU 01004 [4], provide coaxial jet noise predictions based the interpolation

of an experimental database. However, there are two main limitations to these

methods. First, they require the interpolation over a multi-dimensional matrix of

experimental data. Second, the predictions are only valid within the bounds of the

matrix of jet conditions, thereby limiting the range of jet conditions which can be

predicted.

3

An alternative approach to predicting the noise from a coaxial jet, named the

Four-Source method, has recently been developed by Fisher et al. [5,6]. This method

is based on the observation that distinct regions can be identified in coaxial jets which

exhibit similarity relationships that are identical to those observed in simple single

stream jets. Based on this fact, it is then proposed that the noise of a simple coaxial

jets can be described as the combination of four noise producing regions each of

whose contribution to the total far field noise levels is the same as that produced

by a single stream jet with the appropriate characteristic velocity and length scales.

This allows existing experimental databases of single stream jet noise spectra to

be used as a foundation for determining the noise from a coaxial jet. A detailed

description of the Four-Source method is given in Chapter 2.

RANS Based Models

Traditionally, the noise resulting from the turbulent mixing in the shear layer of

a jet, referred to as jet mixing noise, is known to be the primary source of noise in

subsonic jets. Lightihill [7, 8] first derived an equation to describe the generation

of this type of aerodynamically generated noise by rearranging the Navier-Stokes

equations. His approach for modeling the noise generated by turbulent flow is now

referred to as the acoustic analogy. In particular, Lighthill derived the acoustic

analogy by combining the continuity and momentum equations. He then formed a

wave equation on the left hand side and moved all other terms to the right hand side

resulting in the following form

∂2ρ′

∂t2− c2

o∇2ρ′ =∂2

∂yi∂yj

Tij (1.1)

where the Lighthill stress tensor, Tij, given as

Tij = ρuiuj +(p− c2

oρ)δij (1.2)

contains all of the source terms responsible for the generation of the noise.

4

However, both the strength and the weakness of the acoustic analogy theory lies

in the simplicity of the model. For the case of a turbulent jet, to appropriately model

the sources in the Lighthill stress tensor it is necessary to have information regarding

the turbulence statistics. In particular, this method requires a model for the two-

point space-time cross correlation of turbulent sources [9,10]. Measurement of these

statistics is difficult at best and has been completed for only a small number of flow

fields. Based on the data that is available, a number of closure models have been

developed but none have proven universally acceptable. As a result, this predictive

method, which requires a detailed description of the turbulence, is not of sufficient

accuracy at this time to use for engine design purposes.

Further developments have been made to the standard acoustic analogy developed

by Lighthill to account for noise sources that are embedded in a mean flow. An

acoustic analogy was derived by Lilley [11, 12] in which the the governing equation

is linearized about a parallel sheared mean flow, which is representative of the mean

flow in a jet. An added benefit of this approach is that it accounts for the refraction

of sound waves by the jet’s mean flow.

Despite the drawbacks of the acoustic analogy approach, a number of jet noise

prediction methods have been developed based on this method. The most current

acoustic analogy based jet noise prediction methods commonly use a Reynolds aver-

aged Navier-Stokes (RANS) solution with a two-equation turbulence model to obtain

information about the turbulence in the jet [9]. The most common of these methods

is referred to as MGBK [13–15]. In this method the length and time scales of the

turbulence in each volume element are used in conjunction the Acoustic Analogy

theory to determine the characteristic frequency, spectrum and acoustic intensity of

each volume element. The total noise from the jet is then found by summing the

uncorrelated contributions from each volume element. A Similar method based on

the Acoustic Analogy has also been recently developed by Self [16]. In addition,

an Acoustic Analogy based noise prediction method currently being developed by

NASA [17] has been applied to full three-dimensional, non-axisymmetric flow fields.

5

An alternative RANS based noise model has been developed by Tam [10]. This

approach explicitly models the noise sources based on a modeled space-time cor-

relation function. The sound from these sources is then propagated to the far-field

through the use of the linearized Euler equations. The implementation of this method

is similar to the Acoustic Analogy models in that the turbulent flow field is deter-

mined from a RANS solution with a two-equation turbulence model. The turbulence

information from the two-equation model is used as inputs to the space-time corre-

lation function.

An important limitation of the RANS based noise models is the fact that good

quality RANS solutions are required to obtain accurate noise predictions. As a result,

for jets with complex geometries and flow fields, an accurate solution must first be

obtained before running the acoustic solver. For the application of the current study,

the jet has a strongly rotating flow field due to presence of stream-wise vorticies

produced by the forced mixers. Consequently, it may be fairly difficult to obtain a

reliable solution of the turbulent flow field using traditional two-equation turbulence

models.

LES/DNS Based Models

The use of Direct Numerical Simulation (DNS) has recently been used to find the

far-field noise of a low Reynolds number jet [18,19]. Through this approach the time

history of the flow field is determined from a DNS simulation. Direct Numerical

Simulation solves the time dependent Navier-Stokes equations and resolves all of

the relevant length scales in the turbulent flow field. The flow field data is then

post-processed using Lighthill’s acoustic analogy to determine the far-field sound.

The advantage of this approach is that no turbulence models are required for the

application of the acoustic analogy, since the entire turbulent flow field is known.

However, DNS simulations are limited to relatively low Reynolds numbers, on the

6

order of 3,000-4,000, due to the large range of length and time scales in a turbulent

flow. As a result, this approach is not feasible for the application at hand.

Another noise prediction approach currently being investigated involves the use

of a Large Eddy Simulation (LES). A Large Eddy Simulation also solves the time

dependent Navier-Stokes equations, however, a spatial filter is applied to remove the

small scales that are not resolved by the grid. Using this method, the large scale

motion is calculated directly, and a subgrid-scale model is used to model the effects of

the small scales. The LES solution provides the time history of the unsteady pressure

fluctuations on a surface that encloses the noise source mechanisms. These pressure

fluctuations are then extended to the far field by the use of Kirchoff’s method or

Ffowcs Williams-Hawkins method to determine the far-field noise characteristics [20–

22]. However, even with the use of the most advanced supercomputers, presently it is

not practical to perform LES calculations for Reynolds numbers that are consistent

with modern jet engines, especially if the internal mixed flow region is included.

Consequently, it is not feasible at this time to use DNS or LES as a design tool for

the application at hand.

1.3 Goals of the Present Research

The objective of the current study is to extend the Four-Source coaxial jet pre-

diction method to predict the noise from a jet with an internal forced mixer. First,

the Four-Source method formulation for coplanar, coaxial jets is evaluated for the

practical confluent mixer configuration considered in this study. Then a modified

Two-Source model is described in which the noise from an internally forced mixed jet

is matched using a combination of two modified single jet noise predictions. Three

free parameters in the Two-Source model are optimized to match the forced mixer

experimental data. These optimized parameters are then correlated to the changes

in the mixer geometry to yield a semi-empirical noise model for a given family of

forced mixers.

7

2. Coaxial Jet Noise Prediction

2.1 Four-Source Model

An novel approach to predicting the noise from a coaxial jet, referred to as the

Four-Source method, has been previously formulated by Fisher et al. [5, 6]. In this

method the total jet noise is found by adding the contributions of four representative

sources that are modeled as single stream jets. An experimental database of single

stream jet noise spectra is then used as a foundation for determining the noise from

a coaxial jet. Although, the Four-Source method is dependent on the magnitude

of the turbulent fluctuations in the jet, it uses experimental far field measurements

of single stream jets to determine the noise spectra. Therefore, this method is not

dependent on assumptions made about the nature of the turbulent statistics. As a

result, the Four-Source method has been shown to provide accurate predictions of

the noise spectra of coaxial jets.

The structure of a simple coaxial jet is shown in Figure 2.1. The coaxial jet

plume is divided into three regions, the initial region, the interaction region and the

mixed flow region. In the initial region there are two noise producing elements, the

secondary-ambient shear layer and the primary-secondary shear layer.

The heart of the Four-Source method relies on the fact that a simple coaxial jet

can be broken down into regions whose mean flow and turbulent properties resemble

a single stream jet. These properties of a simple coaxial jet were concluded based

on the analysis of the experimental coaxial jet data of Ko [23].

The experimental data of Ko illustrates that the mean velocity and turbulent

intensity profiles of the secondary-ambient shear layer resemble that of a single jet

characterized by the secondary diameter and exit velocity. The noise in this region

will therefore be modeled as that from a single jet based on the secondary velocity,

8

V s

V s

V p

InitialRegion

InteractionRegion

Mixed FlowRegion

Secondary / AmbientShear Layer

Primary / SecondaryShear LayerV s

V s

V p

InitialRegion

InteractionRegion

Mixed FlowRegion

Secondary / AmbientShear Layer

Primary / SecondaryShear Layer

Figure 2.1. Coaxial Jet Structure

9

temperature, and diameter. However, only the portion of the shear layer upstream

of the end of the secondary potential core is modeled in the initial region. Since

the shear layer generally produces high frequency noise in the upstream portion

and low frequency noise in the portions downstream of the potential core, a low

frequency spectral filter is applied to the noise of the single stream jet that models

this source. The remainder of the secondary-ambient shear layer interacts with the

primary-secondary shear layer in the interaction region. The model for the spectral

filter is based on the single stream jet source distribution given by

S(x) = xm−1 exp(−mx

xc

)(2.1)

where S(x) is the source strength per unit length, x is the position on the jet axis

downstream of the nozzle exit, xc, which is a function of frequency, determines the

centroid of the distribution, and m is a shape parameter, which has a typical value

of 4. The fraction of energy, FU , that is radiated from upstream of a given position

x1, is then given by

FU(x1) =

x1∫0

S(x)dx

∞∫0

S(x)dx(2.2)

For the case when the shape parameter m is equal to 4, the fraction of energy that

is radiated from upstream of x1 can then be written as

FU(x1) = 1− exp(−mx

xc

) [1 +

mx1

xc

+1

2

(mx1

xc

)2

+1

6

(mx1

xc

)3]

(2.3)

This relation can then be formulated in terms of frequency, f , by assuming the

centroid positions varies inversely with frequency. Using this assumption, the fraction

of energy radiated from upstream of x1 can be written as

FU(x1, f) = 1− exp

(−mf

f1

) 1 +

mf

f1

+1

2

(mf

f1

)2

+1

6

(mf

f1

)3 (2.4)

where f1 is the frequency that corresponds to the position x1. In addition, the

fraction of energy, FD radiated downstream of a given position x1 is given by,

FD = 1− FU (2.5)

10

A spectral filter can be formulated based on the radiated energy by simply taking

ten times the base ten logarithm of the fraction of the radiated energy.

A plot of the single stream jet source distribution as a function of frequency for

a centroid frequency (fc) of 1000 Hz is shown in Figure 2.2. In addition, the corre-

sponding FU and FD functions and the resulting spectral filters for this sample case,

with cut-off frequency fc = 1000 Hz, are shown in Figure 2.3 and 2.4 respectively.

101

102

103

104

105

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency [Hz]

S(x

) S

ourc

e D

istr

ibut

ion

Fun

ctio

n

Figure 2.2. Single Stream Source Distribution Function for fc = 1000 Hz

The noise from the secondary-ambient shear layer, SPLs, as a function of observer

angle, θ, and frequency, f , is given by,

SPLs(θ, f) = SPL(Vs, Ts, Ds, θ, f) + 10 log10 FU(fs, f) (2.6)

where SPL denotes a single jet prediction using the characteristic jet properties

Vs, Ts, and Ds, which are the secondary exit velocity, temperature, and diameter

respectively. In addition, fs is the spectral filter cut off frequency defined by

fs =Vs

Ds

(2.7)

11

101

102

103

104

105

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Frequency [Hz]

FU

and

FD

FU: Upstream Radiated Energy

FD: Downstream Radiated Energy

Figure 2.3. FU and FD Functions for fc = 1000 Hz

101

102

103

104

105

−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

Frequency [Hz]

Spe

ctra

l Filt

er [d

B] 10log

10(F

U)

10log10

(FD)

Figure 2.4. Spectral Filter Functions for fc = 1000 Hz

12

The second noise producing area within the initial region is the primary-secondary

shear layer. It is observed from the experimental results of Ko that turbulence

intensities in this shear layer are much less than those in the other regions of the jet.

It is therefore determined that this component can be neglected since its noise levels

will have little effect on the overall noise of the jet.

The mixed flow region in the coaxial jet is modeled as a fully mixed jet. The

velocity, temperature and diameter of the fully mixed jet are based on conserving

mass, momentum and energy and are given by

Vm = Vp

(1 + λ2βδ

1 + λβδ

)(2.8)

Dm = Dp

((1 + λβ)(1 + λβδ)

1 + λ2βδ

) 12

(2.9)

Tm = Tp1 + λβ

(1 + λβδ)(2.10)

where Vm, Dm, and Tm are the mixed jet velocity, diameter, and temperature re-

spectively, and λ, β, and δ are the secondary to primary ratios of velocity, geometric

area, and density respectively. In addition, Vp, Dp, and Tp are the primary flow veloc-

ity, diameter, and temperature respectively. Similar to the secondary-ambient shear

layer source region, a high frequency spectral filter is applied to the single stream jet

data which models the mixed flow source region. This spectral filter is necessary due

to the fact that only the downstream portion of the mixed jet is present in the mixed

flow region and this is where the low frequency part of the noise is produced. The

fraction of energy that is radiated from the region of the jet downstream of position

x1 is given by Equation 2.5. The noise from the mixed jet region, SPLm, is then

given by

SPLm(θ, f) = SPL(Vm, Tm, Dm, θ, f) + 10 log10 FD(f1, f) (2.11)

where, Vm, Dm, and Tm are mixed jet velocity, temperature, and diameter respec-

tively, and f1 is the spectral filter cut off frequency defined by

f1 =Vm

Dm

(2.12)

13

In the interaction region there are no obvious flow characteristics by which to

model a single stream jet. It is noted from the work of Ko, however, that the

interaction region contains the largest volume of highly turbulent flow and it exhibits

characteristics of a single jet. It is determined through noise scaling analysis based on

experimental data that the noise from the interaction region scales with the primary

velocity to the eighth power. Therefore, the velocity of the effective jet, which models

the noise noise from the interaction region, is taken to be equal to the primary jet.

The diameter of the effective jet is determined by finding the diameter of a jet with

the given effective velocity that would provide the same amount of thrust as the

original coaxial jet configuration. Based on this model, the diameter of the effective

single jet whose noise will model that of the interaction region is found from

De = Dp

(1 + λ2β

)1/2(2.13)

where De is the diameter of the effective jet and λ and β are the previously defined

velocity and area ratios.

In order to account for differences in the quadrupole noise sources due to tur-

bulence intensity levels in noise producing regions which differ from those of a sin-

gle stream jet, a scaling analysis of the turbulence intensity is performed based on

Lighthill’s solution to the far-field pressure fluctuations at 90◦ to the jet axis. The

results of this analysis show that the far field pressure fluctuations scale as

p2(ro) ∼ α4ρ2oU

8j D2

r2oc

4o

(2.14)

where p2(ro) is the far-field mean square pressure, ro is the distance from the source

to the observer, UJ and D are the jet velocity and diameter, ρo and co are the ambient

density and speed of sound, and α is the turbulence intensity defined as

α ≡ u′

UJ

(2.15)

14

where u′ is the magnitude of the velocity fluctuations and UJ is the jet velocity. As

a result of Equation 2.14, a variation in the turbulence level in a noise producing

region of the coaxial cold jet will result in an attenuation effect given by

∆dB = 40 log(

α

αo

)(2.16)

where α is the peak turbulence intensity in the interaction region of the coaxial jet

and αo is the peak turbulence intensity of a single stream jet, which is approximately

equal to 15%.

However, for a heated jet the attenuation effect is slightly more complicated

due to the addition of a dipole source resulting from the mixing process of fluids of

different densities. A scaling law for the intensity of the quadrupoles, Iq, was derived

based on the expression for the far field pressure fluctuations given in equation 2.14.

This scaling law is given by

Iq ∼ α4ρ2sU

8JD2

ρoc5or

2o

(2.17)

where ρs is the density in the dominant source region and α is the turbulence inten-

sity. Similarly, a scaling law for the dipoles source intensity, Id, is given as

Id ∼ α2(ρs − ρo)2U6

JD2

ρoc3or

2o

(2.18)

which is derived based on the dipole source strength given by Morfey [24]. It is seen

from these scaling laws that the quadrupole sources scale with the fourth power of

the turbulence intensity, while the dipole sources scale with the second power of the

turbulence intensity. Using this information the attenuation of a heated jet is then

given by,

∆dB = 10 log10

(r2Id + r4Iq

Id + Iq

)(2.19)

where r is the ratio of turbulence intensities (α/αo). As a result, for a single jet

peak turbulence intensity, αo, of 15% and an interaction region peak turbulence

intensity, α, of 10%, if the quadrupole sources were dominant, then an attenuation

of 7 dB of the single stream jet noise would occur. Similarly, if the dipole sources

were dominant, then an attenuation of only 3.5 dB would occur.

15

In general, to evaluate the expression for the attenuation due to varying turbu-

lence intensities, information regarding the relative contributions of the quadrupole

and dipole source is determined based on their jet properties. The result of this

analysis isId

Iq

= K(

TJ − To

Ts

) (Ts

To

)M−2

J (2.20)

where K is a constant determined from a ’master spectra’ to have a value of 7, and

MJ is the jet mach number defined by jet velocity divided by the ambient speed of

sound (UJ/co). In addition, the temperature in the source region, Ts, is defined as

Ts = To + 0.65 (TJ − To) (2.21)

Equations 2.19 and 2.20 are then combined to yield the final representation of the

effective jet source reduction, given as

∆dB = 10 log10

(7r2y + r4

7y + 1

)(2.22)

where r is the previously defined ratio of the turbulence intensities and y is defined

as

y =(τ − 1)2

1 + 0.65 (τ − 1)M−2

J (2.23)

where τ is the jet temperature ratio (TJ/To). A graph showing the effective jet source

decibel reductions as a function of jet temperature ratio and Mach number is shown

in Figure 2.5

Given the previously described attenuation factor, the noise spectra from the

interaction region, SPLe, is determined from

SPLe(θ, f) = SPL(Vp, Tp, De, θ,f) + ∆dB (2.24)

where Vp and Tp are the velocity and temperature of the primary jet, De is the

effective diameter from equation 2.13 and the attenuation factor, ∆dB, is determined

from Equation 2.22.

The overall noise of the coaxial jet is then found by the incoherent sum the

contributions from each of the three source regions. The results of this method

16

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3−7.5

−7

−6.5

−6

−5.5

−5

−4.5

−4

−3.5

Temperature Ratio (τ)

Effe

ctiv

e Je

t Noi

se R

educ

tion

[dB

]

MJ = 0.2

MJ = 0.4

MJ = 0.6

MJ = 0.8

MJ = 1.0

MJ = 1.2

Figure 2.5. Effective Jet Source Reduction Function

17

provide noise predictions that are within the order of ±1 dB of experimental data

for a wide range of angles of observation and for a wide range of jet operating

conditions, including primary jet temperatures up to 980◦F (800 K).

2.2 Practical Jet Configurations

Dual Flow Configurations

The geometry of modern jet engines can greatly deviate from that of a simple

coaxial jet. This fact is particularly true for the case of engines with internal flow

mixers. For these configurations the flow will be influenced by both the presence

of a center body or tail cone and the nozzle wall contour. Schematics of a simple

coplanar, coaxial jet and the internally mixed, dual flow configurations examined in

this study are shown in Figure 2.6. In addition, a 3-D rendering of the forced mixer

configuration is shown in Figure 2.7.

Forced Mixers

The introduction of a forced, or lobed mixer, shown in Figure 2.8, increases the

mixing in a turbulent jet through a number of mechanisms. First, the convolution

of the lobed mixer increases the initial interface area between the primary and sec-

ondary flows as compared to a confluent splitter plate. A second mechanism that

creates increased mixing is the introduction of stream-wise vortices. These vortices

assist the mixing process in two ways. First, they further increase the interface area

due to the roll up of the counter rotating vortices. Second, the cross stream convec-

tion associated with the stream-wise vortices sharpens the interface gradients [25].

In addition to the enhancement of the mixing process, the introduction of the

stream-wise vortices substantially alters the flow field as compared to the simple

coaxial configuration. The structure of lobed mixer flows, which is summarized in

the subsequent text, is shown in Figure 2.9. In a lobed mixer, each lobe produces

18

Figure 2.6. Dual Flow Configurations (a) Coplanar, Coaxial Jet (b)Internally Mixed Jet with a Confluent Mixer (c) Internally MixedJet Configuration with a Forced Mixer

19

Figure 2.7. Internally Mixed Jet Configuration with a Forced Mixer

Figure 2.8. Typical Lobed Mixer Geometry

20

a pair of counter rotating vortices. As these vortices evolve they effectively twist

the hot core flow and cold bypass flow in a helical manner. As the vortices move

downstream they grow due to turbulent diffusion and eventually begin to interact

with their pairing vortex, the vortex produced by the adjacent lobe, and possibly

the nozzle wall.

Figure 2.9. Lobed Mixer Vortex Strutcure

21

3. Experimental Acoustic Data

The experimental acoustic data of the mixers used in this study was taken in the

Aero-Acoustic Propulsion Laboratory (AAPL) at NASA Glenn during the spring of

2003. The Aero-Acoustic Propulsion Laboratory, shown in Figure 3.1, is an anechoic

geodesic dome, which is 130 feet in diameter and 65 feet high. This facility houses the

Nozzle Acoustic Test Rig (NATR), which is a 53 inch diameter free-jet acoustic wind

tunnel. This rig is capable of producing jet flows in simulated flight conditions up to

Mach 0.30. The NATR rig is fed by the High Flow Jet Exit Rig (HFJER). This rig

can provide nozzle exit conditions up to 1425◦F (1050 K) with a nozzle pressure ratio

(NPR) of 4.5. In addition, it has the capability to provide dual flow configurations

with independent primary and secondary flow temperature and pressure ratios. The

AAPL facility has two far field microphone arrays located at approximately 50 feet

from a test model in the Nozzle Acoustic Test Rig.

Experimental far-field jet noise data was obtained for the four mixer configura-

tions that are evaluated in this study. These mixer configurations are the confluent

mixer (CFM), the low penetration 12-lobe mixer (12CL), the intermediate penetra-

tion 12-lobe mixer (12UM), and the high penetration 12-lobe mixer (12UH). The

amount of penetration in the three lobed mixers is shown in Table 3.1. The acoustic

data for the four mixers considered was evaluated at three different operating points.

The nozzle total pressure ratios (NPR) and total temperature ratios (NTR) of these

operating points is shown in Table 3.2. In addition, the corresponding velocity and

temperature ratios between the two coaxial streams are shown in Table 3.3. Further-

more, all of experimental jet noise data used in this study was taken in the acoustic

far field at a radius of approximately 80 jet diameters.

The aerodynamic properties of the flow were recorded at each data point. These

properties include the primary and secondary flow charging station total temper-

22

Figure 3.1. NASA Glenn Aero-Acoustic Propulsion Laboratory

Table 3.1 Non-Dimensional Lobed Mixer Penetration

Mixer Name Mixer ID (Penetration/Nozzle Diameter)

Low Penetration 12CL 0.199

Inetermediate Penetration 12UM 0.241

High Penetration 12UH 0.280

Table 3.2 Experimental Data Test Conditions

Operating Point NPRprimary NPRsecondary NTRprimary NTRsecondary

1 1.39 1.44 2.80 1.20

2 1.54 1.61 3.13 1.20

3 1.74 1.82 3.34 1.20

23

Table 3.3 Dual Flow Aerodynamic Test Conditions

Operating Velocity Ratio Temperature Ratio

Point λ δ

1 0.68 2.34

2 0.64 2.62

3 0.62 2.79

atures, total pressures, and static pressures. The charging station is located just

upstream of the mixer or splitter plate. In addition, the primary and secondary flow

mass flow rates are measured, along with the ambient conditions. This informa-

tion is later used to determine the characteristic properties of the flows (velocities,

temperatures and diameters) that are used in the single jet noise predictions.

The acoustic data is supplied in the form of 1/3 octave Sound Pressure Level

(SPL) spectra. These SPL spectra cover a frequency range of 158.5 Hz to 79432.8

Hz (1/3 octave bands 22 to 49), at angles from 55◦ to 165◦ in 5◦ deg increments, as

referenced from the intake axis. The acoustic data is recorded for frequencies up to

80kHz because the mixer/nozzle model is 1/4 scale. The 80kHz frequency limit for

the model scale data corresponds to 20kHz at full scale, which is the approximate

upper frequency limit of human hearing. The acoustic data is normalized to a 50

ft arc, which results in a far-field observer radius to jet diameter ratio of 82.8. The

acoustic data is corrected for microphone response and referenced to an acoustic

standard day (Tamb = 298.3 K, Pamb = 98.595 kPa, 70% relative humidity). A

sample of the acoustic data for all four mixers at the lower power setting is shown in

Figure 3.2. In addition, the acoustic data at the operating points 2 and 3 is shown

in Figures 3.3 and 3.4, respectively.

24

102

103

104

Frequency [Hz]

SP

L [d

B]

2 dB

90°

CFM12CL12UM12UH

102

103

104

Frequency [Hz]

SP

L [d

B]

2 dB

120°

CFM12CL12UM12UH

102

103

104

Frequency [Hz]

SP

L [d

B]

2 dB

150°

CFM12CL12UM12UH

Figure 3.2. Confluent and 12-Lobe Mixer Experimental Data at Set Point 1

25

102

103

104

Frequency [Hz]

SP

L [d

B]

2 dB

90°

CFM12CL12UM12UH

102

103

104

Frequency [Hz]

SP

L [d

B]

2 dB

120°

CFM12CL12UM12UH

102

103

104

Frequency [Hz]

SP

L [d

B]

2 dB

150°

CFM12CL12UM12UH


26

102

103

104

Frequency [Hz]

SP

L [d

B]

2 dB

90°

CFM12CL12UM12UH

102

103

104

Frequency [Hz]

SP

L [d

B]

2 dB

120°

CFM12CL12UM12UH

102

103

104

Frequency [Hz]

SP

L [d

B]

2 dB

150°

CFM12CL12UM12UH


27

4. Current Jet Noise Model Comparisons

In this chapter the ARP876C single jet prediction method and the influence of this

prediction method’s input parameters are discussed. In addition, the derivation

of the Four-Source single jet characteristc parameters are described. Finally, the

experimental data for the confluent mixer is compared to a single jet and a coaxial

jet noise prediction. The fully mixed flow conditions at the final nozzle exit are used

in the single jet prediction and the Four-Source method, as applied to an internally

mixed configuration, is used to make the coaxial jet prediction.

4.1 Single Jet Noise Predictions

In the present study all of the single jet predictions are made based on the SAE

ARP876C guidelines for predicting jet noise [1]. It should be noted that the SPL

spectra of these predictions are, in general, accurate to within approximately ±3 dB.

The ARP876C guidelines outline a method for predicting the noise from a simple

single stream jet. These guidelines are based on experimental data of jet engine

noise. The necessary input parameters that are used in the prediction are shown in

Table 4.1

The fully expanded mean jet velocity, VJ has the most influence on the single jet

noise prediction. This parameter scales the Overall Sound Pressure Level (OASPL)

and determines the shape of the OASPL spectrum as shown in Figures 4.1 and

4.2. In these figures lines with square markers indicate the maximum and minimum

velocity scales for the low power setting. The maximum velocity scale is the primary

velocity and the minimum velocity scale is the secondary velocity. The velocity of

the mixed jet will be somewhere between these two limiting velocities. Likewise the

lines with circle markers in Figures 4.1 and 4.2 indicate the maximum and minimum

28

Table 4.1 ARP876C Input Parameters

Parameter Description

VJ Fully Expanded Mean Jet Velocity

TJ Jet Total Temperature

DJ Exhaust Nozzle Diameter

AJ Cross Sectional Area of the Exhaust Nozzle

γ Ratio of Specific Heats

To Ambient Total Temperature

Po Ambient Total Pressure

RH Ambient Relative Humidity

r Radial Distance from Nozzle Exit to Observer

velocity scales for the high power settting. The jet velocity is also used to determine

the jet density exponent, ω, which also scales the OASPL spectrum. Furthermore,

at shallow angles to the jet axis the jet velocity influences the relative SPL spectrum

shape. Finally, the jet velocity is used to scale the relative SPL spectrum frequencies.

The jet total temperature, TJ , influences the relative SPL spectrum. An example

of this parameter’s influence is seen in Figures 4.3 and 4.4. The jet diameter, DJ ,

scales the frequencies of the relative SPL spectrum. The remaining parameters, the

jet exit area, AJ , the far-field radius, r, and the ambient total pressure, Po, all scale

the OASPL spectrum.

The ARP876C method produces noise predictions that correspond to a lossless

acoustic arena. As a result, to be consistent with the experimental data, an atmo-

spheric absorption correction is applied to the ARP876C noise prediction. In this

study the absorption model developed by Bass et. al [26] is used to correct for at-

mospheric absorption. This model uses the ambient pressure, Po, temperature, To,

and relative humidty, RH.

29

20 40 60 80 100 120 140 160100

110

120

130

140

150

160

170

180

190

Angle from the Inlet Axis [deg]

OA

SP

L [d

B]

OASPL for a Various Values of VJ/a

o

2.512.24 21.781.581.411.261.12 10.890.790.710.630.56 0.50.45 0.4

Figure 4.1. OASPL Dependence on the Fully Expanded Mean Jet Velocity

20 40 60 80 100 120 140 160−10

−5

0

5

10

15

20

Angle from the Inlet Axis [deg]

OA

SP

L [d

B]

OASPL − OASPL(90°) for a Various Values of VJ/a

o

2.512.24 21.781.581.411.261.12 10.890.790.710.630.56 0.50.45 0.4

Figure 4.2. OASPL Directivity Dependence on the Fully ExpandedMean Jet Velocity

30

10−2

10−1

100

101

102

−35

−30

−25

−20

−15

−10

log10

(Strouhal Number)

SP

Lrel

SPLrel at 90° for Various Values of TJ/T

o

1 22.5 33.5

Figure 4.3. SPL Dependence on the Jet Total Temperature at 90 degrees

10−2

10−1

100

101

102

−70

−60

−50

−40

−30

−20

−10

0

log10

(Strouhal Number)

SP

Lrel

SPLrel at 160° for Various Values of TJ/T

o

1 22.5 33.5

Figure 4.4. SPL Dependence on the Jet Total Temperature at 150 degrees

31

4.2 Four-Source Single Jet Characteristic Parameters

The Four-Source jet noise prediction method was developed to predict the noise

from simple coplanar, coaxial jets. An application of this method for the case of a jet

with a recessed, or buried, primary flow involves defining the equivalent primary and

secondary flow single jet properties at the final nozzle exit. The following describes a

method for determining these single jet properties based on flow properties measured

upstream of the coaxial flow splitter plate.

Jet Velocity (VJ)

The ARP876C noise prediction method is based on the fully expanded mean jet

velocity, calculated as

VJ =

√√√√√2γ

γ − 1RTJ

1−

(Po

PJ

) γ−1γ

(4.1)

where R is the ideal gas constant, γ is the ratio of specific heats, TJ is the jet

total temperature, PJ is the jet total pressure, and Po is the ambient total pressure.

If it is assumed that the flow from the charging station (upstream of the splitter

plate where the total pressure and total temperature measurements are taken) to

the final nozzle exit is isentropic, then the total pressure and total temperature at

the final nozzle exit will be the same as the total pressure and total temperature

at the charging station. Therefore, the primary and secondary fully expanded mean

velocities at the final nozzle are calculated with Equation 4.1 using the ambient

pressure measurement and the total temperature and total pressure measurements

taken at the charging station.

Jet Temperature (TJ)

The ARP876C noise prediction method is based on the jet total temperature.

If it is assumed that the flow from the charging station to the final nozzle is isen-

32

tropic, then the total temperature at the final nozzle will be the same as the total

temperature at the charging station.

Jet Area (AJ)

Given the areas of the ducts at the charging station, the final nozzle exit primary

and secondary areas are found by assuming that the flow is isentropic inside the

nozzle, the primary and secondary flows do not mix inside the nozzle, and that the

static pressures of the two flows at the nozzle exit are equal. The resulting problem

is then solved in an iterative manner using the following steps:

1. Guess a value for the primary flow area at the final nozzle exit (Ap)

2. Calculate the secondary flow area (As) using the equation

As = An − Ap (4.2)

3. Calculate the actual Mach number at the final nozzle exit for both the primary

and secondary flows using the isentropic area relation

Aexit

A=

M

Mexit

[1 + γ−1

2M2

exit

] γ+12(γ−1)

[1 + γ−1

2M2

] γ+12(γ−1)

(4.3)

where, for a given flow stream, A is the charging station area, M is the charging

station Mach number, Aexit is the final nozzle area, and Mexit is the unknown

final nozzle Mach number.

4. Calculate the static pressures of the two flows based on the calculated Mach

numbers using the isentropic relation

PJ

Pstatic

=(1 +

γ − 1

2M2

exit

) γγ−1

(4.4)

5. Adjust the core flow area until the static pressure of the two flows are equal.

A method for determining the charging station area for this particular application

is given in Appendix A.

33

Jet Diameter (DJ)

The diameter of the secondary flow, Ds, at the final nozzle will be equal to the

diameter of the final nozzle, Dn. The diameter of the primary flow, Dp at the final

nozzle is calculated based on the primary flow area, Ap at the final nozzle using the

geometric relation

Dp = 2

√Ap

π(4.5)

4.3 Confluent Mixer Comparisons

Using the primary and secondary flow properties at the final nozzle exit, which

were previously described, the standard Four-Source method is used to predict the

noise of the internally mixed configuration with a confluent mixer. In these predic-

tions a constant effective jet reduction of -7 dB is used. It was determined by Mike

Fisher [27] that based on previous experience this value of the effective jet reduc-

tion generally provides more accurate predictions for heated jets. The results of the

confluent mixer predictions are shown for the low power operating point in Figure

4.5. In addition, comparisons at operating points 2 and 3 are given in Figure 4.6

and Figure 4.7 respectively. From these comparisons it can be seen that the Four-

Source noise predictions agree well with the experimental data at angles close to

90◦. However, the predictions near the spectral peaks at angles near the jet axis are

slightly under-predicted by the Four-Source method. Furthermore, it is seen from

Figures 4.5-4.7 that the Four-Source predictions are more accurate than the single

jet predictions. This fact is particularly true at angles close the jet axis.

34

102

103

104

Frequency [Hz]

SP

L [d

B]

90°

2 dB

Experimental DataSingle Jet PredictionCoaxial Jet PredictionFS Sources

102

103

104

Frequency [Hz]

SP

L [d

B]

120°

2 dB


102

103

104

Frequency [Hz]

SP

L [d

B]

150°

2 dB


Figure 4.5. Confluent Mixer Predictions for Set Point 1

35

102

103

104

Frequency [Hz]

SP

L [d

B]

90°

2 dB


102

103

104

Frequency [Hz]

SP

L [d

B]

120°

2 dB


102

103

104

Frequency [Hz]

SP

L [d

B]

150°

2 dB



36

102

103

104

Frequency [Hz]

SP

L [d

B]

90°

2 dB


102

103

104

Frequency [Hz]

SP

L [d

B]

120°

2 dB


102

103

104

Frequency [Hz]

SP

L [d

B]

150°

2 dB



37

5. Forced Mixer Noise Predictions

5.1 Forced Mixer Jet Noise

In addition to the confluent mixer, three different forced mixers are evaluated in

this study. All three forced mixers have the same number of lobes, and are of similar

designs. The primary difference between them is their lobe heights, or penetration

(H). The penetration of a forced mixer is defined as the difference between the

maximum and minimum radii at the end of the mixer, as shown in Figure 5.1.

H

Figure 5.1. Forced Mixer Penetration

The effects of the differences in lobe penetration on the experimental far-field

noise were shown in Figures 3.2, 3.3, and 3.4. From these figures it is seen that

as the forced mixer penetration increases, the low frequency part of the spectrum

decreases, while the high frequency part of the spectrum increases. Based on the

experimental data shown in these figures, it is clear that additional noise generating

mechanisms will need to be accounted for in a forced mixer noise prediction method.

38

5.2 Two-Source Forced Mixer Noise Models

Based on observations of the changes in the forced mixer experimental noise data,

an alternate noise prediction model is proposed which uses portions of two corrected

single stream jet noise spectra. The low frequency region of the noise spectrum is

modeled using a reduced, filtered, fully mixed jet, given as

SPLmd(θ, f) = SPL(Vm, Tm, Dm, θ, f) + 10 log10 FD(fm, f) + ∆dBmd (5.1)

where SPLmd refers to the noise from the downstream fully mixed jet source, SPL

refers to a single jet prediction using the fully mixed jet values, Vm, Tm, and Dm. In

addition, the spectral filter, FD, filters out the high frequency part of the spectrum,

which corresponds to sources in the upstream portion of the fully mixed single stream

jet. This filter is a function of the filter cut-off frequency, fm. The form of this filter,

as given in Chapter 2, is

FD = exp

(−4

f

fc

) 1 +

(4

f

fc

)+

1

2

(4

f

fc

)2

+1

6

(4

f

fc

)3 (5.2)

The source reduction term, ∆dBmd, shifts the fully mixed jet noise spectra down.

This term can be related to differences in the turbulence intensities of a simple single

stream jet and of those which occur in the downstream portion of the actual jet

plume. This relationship, as given in Equation 2.16, is

∆dB = 40 log10 (Fturb) (5.3)

where Fturb is the ratio of peak turbulence intensities in the actual jet plume to the

peak turbulence intensities in a simple single stream jet. In this study, the source

strength parameter, ∆dB, is a free parameter whose value is determined empirically

through the parameter optimization process. Consequently, the resulting Fturb values

will be theoretical estimates of the magnitude of the turbulence intensities in the

actual jet plume. If this turbulence information is known, from experiments or

Computational Fluid Dynamics (RANS), then it can be compared to the optimized

values to provide a measurement of the validity of the assumptions in this model.

39

Two models are evaluated for the prediction of the high frequency region of

forced mixer noise spectra, which corresponds to the upstream portion of the actual

jet plume. The first model, which will be referred to as Model 1, uses an augmented,

filtered, fully mixed jet, given as

SPLmu(θ, f) = SPL(Vm, Tm, Dm, θ, f) + 10 log10 FU(fm, f) + ∆dBmu (5.4)

where SPLmu refers to the noise from the upstream fully mixed jet source, SPL

refers to a single jet prediction using the fully mixed jet values, Vm, Tm, and Dm.

The spectral filter, FU , filters out the low frequency part of the single jet noise

prediction, which corresponds to sources in the downstream region of the jet plume.

From Chapter 2, this filter is given as,

FU = 1− FD (5.5)

The source augmentation term, ∆dBmu, shifts the fully mixed jet noise spectra up.

This term is analogous to the differences that are seen in a single stream jet whose

turbulence intensities are increased.

The second model, which will be referred to as Model 2, uses an augmented,

filtered, secondary jet to predict the high frequency part of the forced mixer noise

spectrum. This source is given as

SPLsu(θ, f) = SPL(Vs, Ts, Ds, θ, f) + 10 log10 FU(fs, f) + ∆dBsu (5.6)

where SPLsu refers to the noise from the upstream secondary jet source and SPL

refers to a single jet prediction based on the secondary flow values, Vs, Ts, and

Ds. Similar to the previous source model, a spectral filter is applied to eliminate

the low frequency part of the single jet prediction, and the overall source levels are

augmented by the source strength term ∆dBsu.

For simplicity, it is assumed that the cut-off Strouhal numbers of the low fre-

quency and high frequency sources are equal. The cut-off frequency, fc, can be

calculated from the cut-off Strouhal number, Stc, through the relation

fc = StcV

D(5.7)

40

It is expected that Model 1, which uses a secondary jet to represent the upstream

portion of the actual jet plume, will produce better predictions for the low penetra-

tion mixer. Likewise, it is expected that Model 2, which uses a mixed jet to represent

the upstream portion of the actual jet plume, will produce better predictions for the

high penetration mixer. These trends are expected due to the fact that for the case

of the high penetration mixer the stronger stream wise vortices result in increased

mixing inside the nozzle. As a result as the penetration is increased, the flow at

the final nozzle exit will be more characteristic of a fully mixed. Similarly, as the

penetration decreases the impact of the stream wise vortices decreases resulting in a

flow that will be more characteristic of a secondary jet.

5.3 Two-Source Model Parameter Optimization

The proposed Two-Source model has three variable parameters, the low fre-

quency source reduction, ∆dBmd, the high frequency source augmentation, ∆dBmu

or ∆dBsu, and the common cut-off Strouhal number, Stc. The optimum values of

these variable parameters for a given mixer and nozzle geometry are determined em-

pirically through the use of a non-linear least squares optimization method. In this

method the best set of variable parameters are found which minimize the weighted

errors between the model prediction and the experimental data. This process essen-

tially curve-fits the experimental data using the Two-Source model. The optimized

parameters that result from this analysis can then be correlated with the changes in

the mixer geometry, namely the amount of penetration.

The non-linear least squares optimization is performed using MATLAB’s lsqnonlin

function. This routine uses a Levenberg-Marquardt method for minimizing the er-

rors between the model prediction and the experimental data. This non-linear least

squares optimization routine is used to find the optimum source strength parameters

for a given cut-off Strouhal number. This process is repeated for a range of cut-off

Strouhal numbers to find the set of optimized parameters which yields the lowest er-

41

ror. This exhaustive type of approach for determining the optimum cut-off Strouhal

number is necessary because of the non-linear nature of the filter functions and the

averaged weighted error, which cause difficulties due to both solution non-uniqueness

and the presence of local minima.

The errors between the model predictions and the experimental data are evalu-

ated for a range of angles from 90◦ to 150◦ from the intake axis, in 5◦ increments.

This process results in approximately 400 error values. At each angular location

these error values are weighted based on the experimental data spectra using the

weighting function

Ew (θi, f) = 10[0.1(SPLexp(θi,f)−[SPLexp(θi,f)]max)] (5.8)

This weighting function has a value of 1 at the peak of the experimental data, and

approaches 0 as the differences between a given experimental Sound Pressure Level

value and peak Sound Pressure Level in the experimental data spectrum approach

infinity. This weighting, which is similar to the one implicit in the calculation of the

Overall Sound Pressure Level, will weigh the errors in the predicted Sound Pressure

Level that are closer to the peak in the experimental data more heavily. A Perceived

Noise Level (PNL) type of weighting could also be used to weight the sound pressure

level errors. However, since the differences between these two weightings are expected

to be small, in this study the OASPL type will be used for simplicity.

5.4 Two-Source Model Results

In the following section the performance of two different the Two-Source models

are evaluated. The variable parameters in these models are optimized so that the pre-

dictions best match the experimental data. This optimization process is performed

at three different operating set points for each of the three forced mixer designs. The

resulting optimized parameters are then correlated back to the geometric differences

in the forced mixer designs.

42

5.4.1 Model 1 Results

The first Two-Source model that is evaluated is the Model 1. Using this model,

the upstream portion of the jet plume is modeled as a single stream fully mixed jet.

A spectral filter, which eliminates the low frequency region, is applied to the fully

mixed jet noise spectrum. The downstream portion of the jet plume is also modeled

as a single stream fully mixed jet. However, a spectral filter that eliminates the high

frequency part of the single stream noise spectrum is applied to this noise source.

The same cut-off Strouhal number is used in both spectral filters. In addition, each

of the two noise sources has a variable source strength term which shifts the entire

spectrum up or down.

Low Penetration Mixer

The results of the first step in the parameter optimization process for the low

penetration mixer at Set Point 1 are shown in Figures 5.2, 5.3, and 5.4. In Figure

5.2 the maximum error, average error, and average weighted errors are plotted as

a function of the cut-off Strouhal number. The circles on these plots show the

location of the minimum error for each type of error. It is seen here that each type

of error is minimized for different values of the cut-off Strouhal number. As a result

the optimum parameters for this configuration, as well as those for all subsequent

configurations, will be dependent on which metric is used to determine the optimum

criterion. For this study, the average weighted error is used as the metric to determine

the optimum criterion. This metric is chosen because it provides the best measure of

how well the prediction agrees with experimental data from an acoustics standpoint.

One of the difficulties of the parameter optimization problem is the existence of

non-unique solutions. This problem occurs in the parameter optimization process

when the errors in Figure 5.2 are relatively constant for a large range of cut-off

Strouhal numbers. When this condition occurs, there are multiple solutions to the

optimization process that yield roughly the same error. As a result, it is then not

43

obvious which set of parameters should be later used to correlate to the differences

in the mixer design. To overcome some of the non-linear behavior problems that

result from the optimization of the Two-Source model, an exhaustive type of search

is used to determine the optimum cut-off Strouhal number.

100

101

0

2

4

6

8

10

12

Stc [ ]

Err

or [d

B]

MaximumAverageWeighted

Figure 5.2. Model 1 Parameter Optimization Error Results for theLow Penetration Mixer at Set Point 1

In Figure 5.3 the non-dimensional errors resulting the parameter optimization

for the low penetration mixer at Set Point 1 are shown. These errors are the same

as those plotted in Figure 5.2, except they have been normalized by their respective

maximum values so that the behavior of the errors can be more clearly seen. The

problem of local minima is seen in Figure 5.3 for the averaged weighted error. The oc-

currence of local minima in the averaged weighted error is one reason that warranted

the need for the exhaustive type of search to determine the optimum cut-off Strouhal

number. Figure 5.4 shows the corresponding optimized source strengths for both the

upstream mixed jet and the downstream mixed jet sources. In addition, the vertical

dotted line signifies the optimum cut-off Strouhal number for this test case, which

44

100

101

0.4

0.5

0.6

0.7

0.8

0.9

1

Stc [ ]

Nor

mal

ized

Err

or


Figure 5.3. Model 1 Parameter Optimization Non-Dimensional ErrorResults for the Low Penetration Mixer at Set Point 1

100

101

0

5

10

15

20

25

Stc [ ]

∆ dB

[dB

]

Upstream JetDownstream Jet

Figure 5.4. Model 1 Parameter Optimization Results for the LowPenetration Mixer at Set Point 1

45

was determined from Figure 5.3. The optimum source strengths for this test case

are then taken to be those that resulted from the optimum cut-off Strouhal number.

An interesting point to note is that as the cut-off Strouhal number approaches the

lower bound, the noise prediction essentially consists of only the upstream single jet

source. Likewise, as the cut-off Strouhal number approaches the upper bound, the

noise prediction essentially consists of only the downstream single jet source.

The Two-Source model prediction using the optimized parameters for the low

penetration mixer at Set Point 1 is shown in Figure 5.5. It is seen from this figure

the optimized prediction agrees well with the experimental data. However, there are

some deviations present at angles close to the jet axis where the the predictions are

slightly under-predicted near the spectral peak. It is expected that these predictions

agree well since they were essentially curve-fit to match the experimental data. The

fact the optimized predictions are in agreement suggests that the Two-Source model

contains the necessary physics to model the noise from the forced mixer.

This optimization process was repeated for the low penetration mixer at the two

additional Set Points. The normalized error and optimized parameter results for Set

point 2 are shown in Figures 5.6 and 5.7. Similarly, the corresponding results for Set

point 3 are shown in Figures 5.8 and 5.9.

Once this process was completed for all three Set Points, the three corresponding

averaged weighted error curves for the low penetration mixer are combined onto

one graph, as shown in Figure 5.10. The averaged weighted errors from each Set

Point are then averaged again to yield an error that is representative of all three Set

Points. The final optimum cut-off Strouhal number for this mixer is then chosen

based on this metric. The corresponding optimized source strengths for all three Set

Points are shown in Figure 5.11. Once the final optimum cut-off Strouhal number is

determined, the final optimum source strength parameters are found by averaging the

source strength values for the three Set Points at the final optimum cut-off Strouhal

number.

46

102

103

104

Frequency [Hz]

SP

L [d

B]

90°

2 dBExperimentalMM Optimizied

102

103

104

Frequency [Hz]

SP

L [d

B]

120°


102

103

104

Frequency [Hz]

SP

L [d

B]

150°


Figure 5.5. Model 1 Optimized Predictions for the Low PenetrationMixer at Set Point 1

47

10−1

100

101

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Stc [ ]

Nor

mal

ized

Err

or



10−1

100

101

0

5

10

15

20

25

Stc [ ]

∆ dB

[dB

]



48

10−1

100

101

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Stc [ ]

Nor

mal

ized

Err

or



10−1

100

101

0

5

10

15

20

25

30

Stc [ ]

∆ dB

[dB

]



49

It is seen in Figure 5.11 that for this particular case there is a fairly large difference

in the upstream jet source strengths between the three Set Points at the optimum

Strouhal number. Ideally, the source strength parameters should collapse on one

another for all three Set Points. This result is expected since changes in aerodynamic

conditions are accounted for in the single jet predictions. The fact that there is a

discrepancy in the upstream jet source for the low penetration mixer suggests that

the Two-Source model does not contain all of the components necessary to model the

physics of the jet with the low penetration mixer. Consequently, an additional source

might be needed to model this case. Alternatively, this discrepancy could result if

the single jet characteristics are not representative of the actual flow field properties

in the jet plume. However, the only way to evaluate these differences would be to

analyze the experimental aerodynamic data in the full jet plume. Fortunately, the

optimized source strength results for the other two forced mixers do collapse fairly

well with respect to the three operating Set Points.

The final optimum parameters will later be used in the parameter correlation

process, described in Section 5.4.3. The optimized parameters for each Set Point

and the final set of optimized parameters for the low penetration mixer are given in

Table 5.1.

Table 5.1 Model 1 Optimized Parameters for the Low Penetration Mixer

Case ∆dBum ∆dBdm Stc

Set Pt 1 3.998 1.668 3.440

Set Pt 2 4.794 1.946 3.792

Set Pt 3 6.357 1.681 5.386

Final 5.050 1.765 4.331

50

10−1

100

101

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

Cut−off Strouhal Number []

Wei

ghte

d E

rror

[dB

]

High PowerMid PowerLow PowerAverage

Figure 5.10. Model 1 Parameter Optimization Average WeightedError Results for the Low Penetration Mixer at Set Points 1, 2 and3

10−1

100

101

0

5

10

15

20

25

30

Cut−Off Strouhal Number

Opt

imiz

ed S

ourc

e S

tren

gths

∆ d

B [d

B]

Upstream JetDownStream Jet

Figure 5.11. Model 1 Parameter Optimization Results for the LowPenetration Mixer at Set Points 1, 2 and 3

51

Intermediate Penetration Mixer

The optimization described in the previous section is repeated for the interme-

diate penetration mixer. The resulting averaged weighted error curves are shown in

Figure 5.12. Once again, the final optimum cut-off Strouhal number corresponds to

the location of the minimum of the averaged weighted error curve. The correspond-

ing optimized source strengths for the intermediate penetration mixer at all three

Set Points are shown in Figure 5.13. For this configuration, only a small variability

of the source strengths with respect to the operating condition is seen, which implies

that the flow physics are well represented by the two-source model. Once again,

after the final optimum cut-off Strouhal number is determined for this mixer, the

final optimum source strength parameters are found by averaging the source strength

values for the three Set Points at the final optimum cut-off Strouhal number. These

final parameters will later be used in the parameter correlation process. The opti-

mized parameters for each set point and the final set of optimized parameters for

the intermediate penetration mixer are given in Table 5.2.

Table 5.2 Model 1 Optimized Parameters for the Intermediate Penetration Mixer


Set Pt 1 7.251 0.284 3.372

Set Pt 2 7.643 0.379 3.706

Set Pt 3 7.911 0.515 5.245

Final 7.601 0.393 4.245

52

10−1

100

101

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Wei

ghte

d E

rror

[dB

]


Figure 5.12. Model 1 Parameter Optimization Average WeightedError Results for the Intermediate Penetration Mixer at Set Points1, 2 and 3

10−1

100

101

−5

0

5

10

15

20

25

30

35


Opt

imiz

ed S

ourc

e S

tren

gths

∆ d

B [d

B]


Figure 5.13. Model 1 Parameter Optimization Results for the Inter-mediate Penetration Mixer at Set Points 1, 2 and 3

53

High Penetration Mixer

The previously described optimization process is again repeated for the high pen-

etration mixer. The resulting averaged weighted error curves are shown in Figure

5.14. Once again, the final optimum cut-off Strouhal number corresponds to the

location of the minimum of the averaged weighted error curve. The corresponding

optimized source strengths for the high penetration mixer at all three Set Points are

shown in Figure 5.15. It is once again seen that there us only a small variability in

the source strengths with respect to the operating point for this configuration. After

the final optimum cut-off Strouhal number is determined for this mixer, the final op-

timum source strength parameters are found by averaging the source strength values

for the three Set Points at the final optimum cut-off Strouhal number. These final

parameters will later be used in the parameter correlation process. The optimized

parameters for each Set Point and the final set of optimized parameters for the high

penetration mixer are given in Table 5.3.

Table 5.3 Model 1 Optimized Parameters for the High Penetration Mixer


Set Pt 1 7.801 -0.283 3.443

Set Pt 2 8.101 0.123 3.795

Set Pt 3 8.105 0.200 4.283

Final 8.005 0.013 3.443

54

10−1

100

101

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Wei

ghte

d E

rror

[dB

]


Figure 5.14. Model 1 Parameter Optimization Average WeightedError Results for the High Penetration Mixer at Set Points 1, 2 and3

10−1

100

101

−5

0

5

10

15

20

25

30

35


Opt

imiz

ed S

ourc

e S

tren

gths

∆ d

B [d

B]


Figure 5.15. Model 1 Parameter Optimization Results for the HighPenetration Mixer at Set Points 1, 2 and 3

55

5.4.2 Model 2 Results

The second Two-Source model, Model 2, represents the upstream portion of the

jet plume using as a single stream secondary jet. A spectral filter, which eliminates

the low frequency region, is applied to the secondary jet noise spectrum. The down-

stream portion of the jet plume is modeled as a single stream fully mixed jet. A

spectral filter that eliminates the high frequency part of the single stream noise spec-

trum is applied to this noise source. The same cut-off Strouhal number is used in

both spectral filters. In addition, each of the two noise sources has a variable source

strength term which shifts the entire spectrum up or down.

Low Penetration Mixer

The same optimization process that was used with Model 1 is also used here with

the Model 2. The resulting averaged weighted error curves for the low penetration

mixer are shown in Figure 5.16. The final optimum cut-off Strouhal number corre-

sponds to the location of the minimum of the averaged weighted error curve. The

corresponding optimized source strengths for the low penetration mixer at all three

Set Points is shown in Figure 5.17. It is seen from Figure 5.17 that the variability

in the optimized source strength terms with respect to the Set Points using Model 2

are similar to those obtained with Model 1. After the final optimum cut-off Strouhal

number is determined for this mixer, the final optimum source strength parameters

are found by averaging the source strength values for the three Set Points at the

final optimum cut-off Strouhal number. These final parameters will later be used in

the parameter correlation process. The optimized parameters for each Set Point and

the final set of optimized parameters for the low penetration mixer based on Model

2 are given in Table 5.4.

56

10−1

100

101

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7


Wei

ghte

d E

rror

[dB

]


Figure 5.16. Model 2 Parameter Optimization Average WeightedError Results for the Low Penetration Mixer at Set Points 1, 2 and3

10−1

100

101

0

5

10

15

20

25

30

35


Opt

imiz

ed S

ourc

e S

tren

gths

∆ d

B [d

B]


Figure 5.17. Model 2 Parameter Optimization Results for the LowPenetration Mixer at Set Points 1, 2 and 3

57

Table 5.4 Model 2 Optimized Parameters for the Low Penetration Mixer

Case ∆dBus ∆dBdm Stc

Set Pt 1 7.788 1.671 4.331

Set Pt 2 8.560 1.983 3.792

Set Pt 3 10.474 1.707 5.386

Final 8.941 1.787 5.452

58

Intermediate Penetration Mixer

The averaged weighted error curves for the intermediate penetration mixer that

resulted from the parameter optimization process are shown in Figure 5.18. In addi-

tion, the corresponding optimized source strengths for the intermediate penetration

mixer at all three Set Points is shown in Figure 5.19. It is seen from Figure 5.19

that much like the results from Model 1, the optimum source strength curves at all

three Set Points for the intermediate penetration mixer collapse on one another. The

optimized parameters for each Set Point and the final set of optimized parameters

for the intermediate penetration mixer are given in Table 5.5.

10−1

100

101

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


Wei

ghte

d E

rror

[dB

]


Figure 5.18. Model 2 Parameter Optimization Average WeightedError Results for the Intermediate Penetration Mixer at Set Points1, 2 and 3

59

10−1

100

101

−5

0

5

10

15

20

25

30

35

40


Opt

imiz

ed S

ourc

e S

tren

gths

∆ d

B [d

B]


Figure 5.19. Model 2 Parameter Optimization Results for the Inter-mediate Penetration Mixer at Set Points 1, 2 and 3

Table 5.5 Model 2 Optimized Parameters for the Intermediate Penetration Mixer


Set Pt 1 10.508 0.291 3.372

Set Pt 2 10.969 0.360 3.706

Set Pt 3 11.089 0.525 5.245

Final 10.855 0.392 4.245

60

High Penetration Mixer

The resulting averaged weighted error curves for the intermediate penetration

mixer are shown in Figure 5.20. In addition, the corresponding optimized source

strengths for this mixer at all three Set Points is shown in Figure 5.21. Similar to

the results from Model 1, with Model 2 the optimized source strengths terms at all

three Set Points collapse for the high penetration mixer. The optimized parameters

for each Set Point and the final set of optimized parameters for the high penetration

mixer are given in Table 5.6.

10−1

100

101

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Wei

ghte

d E

rror

[dB

]


Figure 5.20. Model 2 Parameter Optimization Average WeightedError Results for the High Penetration Mixer at Set Points 1, 2 and3

61

10−1

100

101

−5

0

5

10

15

20

25

30

35

40


Opt

imiz

ed S

ourc

e S

tren

gths

∆ d

B [d

B]


Figure 5.21. Model 2 Parameter Optimization Results for the HighPenetration Mixer at Set Points 1, 2 and 3

Table 5.6 Model 2 Optimized Parameters for the High Penetration Mixer


Set Pt 1 11.256 -0.370 3.443

Set Pt 2 11.399 0.115 3.795

Set Pt 3 11.491 0.197 4.283

Final 11.382 -0.020 3.443

62

5.4.3 Parameter Correlations

Once a fixed set of optimized parameters are determined for each forced mixer

geometry, these parameters can then be correlated to the geometric differences in

the force mixer designs. For this particular family of forced mixers, the primary

geometric difference is the lobe height, or amount of penetration (H). This parameter

is non-dimensionalized by the final nozzle diameter (Dnozzle), and is then correlated

to the optimized parameters. This parameter correlation process is applied to the

results from each of the Two-Source models.

Model 1 Correlations

The optimized parameters for each mixer geometry using Model 1 are shown

in Table 5.7. These optimized parameters are plotted versus the non-dimensional

mixer penetration values in Figures 5.22 and 5.23. It is seen from Figure 5.22 that the

source strength for the single stream fully mixed jet that represents the downstream

portion of the actual jet plume exhibits a linear behavior. In addition, the source

strength for the upstream jet source exhibits approximately a linear behavior. Based

on this result, a linear curve-fit can be applied to each of the source strength curves.

The resulting coefficients for this linear curve-fit are given in Table 5.8.

The variation in the optimum cut-off Strouhal number, Stc, in Figure 5.23 is

fairly small when compared the full range of Strouhal numbers in the experimental

data. In fact the difference between a Strouhal number of 3.4 and 4.4 approximately

corresponds to a span of one 1/3 Octave band. As a result, the variation in the

Strouhal number from 3.4 to 4.4 will have a little effect on the overall noise prediction.

Consequently, a constant cut-off Strouhal number can be used. This final cut-off

Strouhal number is found by simply averaging the values for the three mixer designs.

For this Two-Source model, the cut-off Strouhal number will have a value of 4.01.

63

Table 5.7 Final Optimized Parameters for Model 1

Mixer ID HDnozzle

∆dBum ∆dBdm Stc

12CL 0.1994 5.050 1.765 4.331

12UM 0.2602 7.601 0.393 4.245

12UH 0.2801 8.005 0.013 3.443

0.18 0.2 0.22 0.24 0.26 0.28 0.3−1

0

1

2

3

4

5

6

7

8

9

10

Sou

rce

Str

engt

h ∆d

B [d

B]


Figure 5.22. Model 1 Optimized Parameter Correlation of the Source Strengths

Table 5.8 Coefficients from the Linear Curve-fit of the Results from Model 1

Source Strength Slope (A) Intercept (b)

∆dBum 37.887 -2.457

∆dBdm -21.917 6.128

64

0.18 0.2 0.22 0.24 0.26 0.28 0.310

−1

100

101

Lobe Penetration / Nozzle Diameter

Cut

−O

ff S

trou

hal N

umbe

r

Figure 5.23. Model 1 Optimized Parameter Correlation of the Cut-offStrouhal Number

65

Model 2 Correlations

The optimized parameters for each mixer geometry using Model 2 are shown in

Table 5.9. These optimized parameters are plotted versus the non-dimensional mixer

penetration values in Figures 5.24 and 5.25. It is seen from Figure 5.24 that the both

source strengths exhibit a linear behavior. Based on this result a linear curve-fit can

be applied to each of the source strength curves. The resulting coefficients for this

linear curve-fit are given in Table 5.10.

Once again it is seen from Figure 5.25 that the cut-off Strouhal number varies over

a relatively small range for all three mixer designs. For this case, the cut-off Strouhal

number ranges from 3.5 to 5.5, which approximately corresponds to two 1/3 Octave

bands. Once again, the effects of this variation on the noise prediction will be fairly

small. As a result, constant cut-off Strouhal number is also assumed for this Two-

Source model. This final cut-off Strouhal number, which is the average of the values

from the three mixers, has a value of 4.38 for this model. As an alternative approach,

since the behavior of the cut-off Strouhal number for this model is approximately

linear, a linear curve-fit could also be used to find the cut-off Strouhal number.

Table 5.9 Final Optimized Parameters for Model 2

Mixer ID HDnozzle

∆dBus ∆dBdm Stc

12CL 0.1994 8.941 1.787 5.452

12UM 0.2602 10.855 0.392 4.245

12UH 0.2801 11.382 -0.019 3.443

66

Table 5.10 Coefficients from the Linear Curve-fit of the Results from Model 2

Source Strength Slope (A) Intercept (b)

∆dBus 30.550 2.859

∆dBdm -22.522 6.274

0.18 0.2 0.22 0.24 0.26 0.28 0.3

0

2

4

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8

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12

Sou

rce

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engt

h ∆d

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Figure 5.24. Model 2 Optimized Parameter Correlation of the Source Strengths

67

0.18 0.2 0.22 0.24 0.26 0.28 0.310

−1

100

101

Lobe Penetration / Nozzle Diameter

Cut

−O

ff S

trou

hal N

umbe

r

Figure 5.25. Model 2 Optimized Parameter Correlation of the Cut-offStrouhal Number

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5.4.4 Two-Source Model Performance

In the following section the final optimized parameter correlations are evaluated

for the nine data points from which they were developed. It is important to note

that through the optimization process, a number of intermediate steps required av-

eraging of various optimized results. As a result of this process, the final prediction

method will not agree with the experimental data as well as some of the results from

the intermediate steps. In this section errors from predictions using the final fixed

parameters will be compared to errors from the optimized Two-Source solutions for

each data point. In addition, these errors will also be compared to the errors that

would result from using both a coaxial jet and a single jet prediction.

Model 1 Performance

The average weighted errors for four different prediction methods are given in

Table 5.11. The Two-Source Optimized prediction corresponds to the Two-Source

(Model 1) prediction which was optimized to best match the particular data point.

The Fixed Parameters prediction corresponds to a Two-Source (Model 1) predic-

tion made using the parameters which result from the parameter correlation linear

curve-fits described in Section 5.4.3. The Coaxial Jet prediction corresponds to a

prediction made using the Four-Source method with the jet properties based ’Equiv-

alent Coaxial’ jet. The Single Jet prediction corresponds to a prediction made using

a single stream fully mixed jet with the final nozzle exit diameter. In addition to

the averaged weighted errors, the average errors and maximum errors using these

predictions are given in Tables 5.12 and 5.13 respectively. The most important of

these errors are the average weighted errors since they best describe how well the pre-

dictions match the experimental data from an acoustics standpoint. It is seen from

Table 5.11 that the Two-Source model predictions best match the experimental data

for all three forced mixers at all three Set Points. In addition, it is noted that only

a small amount of error is introduced to the Two-Source model predictions as a re-

69

sult of the parameter optimization process. Through this process a number of steps

required either averaging a set of quantities or curve fitting quantities. As a result,

the Two-Source model predictions made using the final correlated parameters will

not match the experimental data as well as they could if the model was optimized

for just one specific data point. Fortunately, it is seen that errors introduced by

the optimization process are small compared to the difference in the errors of other

current jet noise prediction methods. In general, it is noted that the Two-Source

model predictions produce the best match to the experimental data, followed by the

Four-Source prediction, and finally the single jet prediction.

The Sound Pressure Level spectra predictions for the three forced mixers at all

three data points using Model 1 are given in Figures 5.26 through 5.34. It is seen

from these Figures that for all nine data points the predictions at angles close to

90◦ are in strong agreement with the experimental data. However, similar to the

confluent mixer predictions, the spectrum peak is often slightly under-predicted at

angles close to the jet axis. In addition, for all three forced mixers some deficiencies

in the predictions are seen at the high power Set Point at angles close to the jet

axis. In these cases it appears as if there is an additional high frequency noise source

that is not modeled by the Two-Source formulation. It has been suggested that this

excess noise may be due to sources other than those related to jet mixing. As a

result, at this time no efforts have been made to account for this additional source

in the current forced mixer noise models.

70

Table 5.11 Average Weighted Errors in dB for Model 1

Mixer Set Two-Source Fixed Coaxial Single

ID Point Optimized Parameters Jet Jet

12CL 1 0.31 0.35 0.41 0.64

12CL 2 0.38 0.38 0.56 0.86

12CL 3 0.48 0.53 0.82 0.96

12UM 1 0.26 0.29 0.93 0.68

12UM 2 0.35 0.37 0.98 0.82

12UM 3 0.46 0.52 1.04 0.98

12UH 1 0.27 0.28 1.24 0.96

12UH 2 0.36 0.41 1.30 1.09

12UH 3 0.45 0.55 1.29 1.12

Table 5.12 Average Errors in dB for Model 1



12CL 1 1.35 1.42 2.56 2.53

12CL 2 1.32 1.35 2.54 2.60

12CL 3 1.98 2.02 3.16 3.40

12UM 1 1.09 1.37 3.61 2.85

12UM 2 1.41 1.38 3.65 3.02

12UM 3 1.75 2.00 4.33 4.41

12UH 1 1.08 1.12 4.87 4.06

12UH 2 1.24 1.28 4.97 4.18

12UH 3 1.52 1.81 4.81 4.66

71

Table 5.13 Maximum Errors in dB for Model 1



12CL 1 9.30 7.86 13.18 12.02

12CL 2 6.20 5.90 10.74 9.87

12CL 3 7.09 10.12 16.70 14.31

12UM 1 8.35 7.12 14.48 13.56

12UM 2 5.67 6.15 11.32 11.25

12UM 3 7.46 9.51 18.46 16.22

12UH 1 6.14 5.78 14.13 13.00

12UH 2 7.38 6.84 13.04 12.45

12UH 3 5.73 8.24 17.88 15.58

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Model 2 Performance

The average weighted errors for the four different prediction methods are given

in Table 5.14. The Two-Source Optimized prediction corresponds to the Two-Source

(Model 2) prediction which was optimized to best match the particular data point.

The Fixed Parameters prediction corresponds to a Two-Source (Model 2) prediction

made using the parameters which result from the parameter correlation linear curve-

fits described in Section 5.4.3. The Coaxial Jet prediction corresponds to a prediction

made using the Four-Source method. The Single Jet prediction corresponds to a

prediction made using a single stream fully mixed jet. In addition to the averaged

weighted errors, the average errors and maximum errors using these predictions are

given in Table 5.15 and Table 5.16 respectively. Similar to the Model 1 performance,

it is seen from Table 5.14 that the Two-Source model predictions best match the

experimental data for all three forced mixers at all three Set Points. In addition, it

is noted that once again only a small amount of error is introduced to the Two-Source

model predictions as a result of the parameter optimization process. It is seen that

errors introduced by the optimization process of Model 2 are small compared to the

difference in the errors of other current jet noise prediction methods. In general it

is again noted for Model 2 that the Two-Source model predictions produce the best

match to the experimental data, followed by the Four-Source prediction, and finally

the single jet prediction.

The Sound Pressure Level spectra predictions for the three forced mixers at all

three data points using Model 2 are given in Figures 5.35 thru 5.43. It is seen from

these Figures that for all nine data points the predictions at angles close to 90◦ are

in strong agreement with the experimental data. However, similar to the confluent

mixer and Model 1 predictions, the spectrum peak is often slightly under-predicted

at angles close to the jet axis. In addition, similar to the results from Model 1,

for all three forced mixers some deficiencies in the predictions are seen at the high

power Set Point at angles close to the jet axis. In these cases it appears as if there

82

is an additional high frequency noise source that is not modeled by the Two-Source

formulation.

Table 5.14 Average Weighted Errors in dB for Model 2



12CL 1 0.30 0.36 0.41 0.64

12CL 2 0.38 0.38 0.56 0.86

12CL 3 0.48 0.52 0.82 0.96

12UM 1 0.26 0.29 0.93 0.68

12UM 2 0.35 0.38 0.98 0.82

12UM 3 0.46 0.52 1.04 0.98

12UH 1 0.27 0.31 1.24 0.96

12UH 2 0.35 0.45 1.30 1.09

12UH 3 0.45 0.57 1.29 1.12

83

Table 5.15 Average Errors in dB for Model 2



12CL 1 1.29 1.55 2.56 2.53

12CL 2 1.37 1.47 2.54 2.60

12CL 3 2.08 1.97 3.16 3.40

12UM 1 1.16 1.48 3.61 2.85

12UM 2 1.45 1.43 3.65 3.02

12UM 3 1.77 1.89 4.33 4.41

12UH 1 1.03 1.13 4.87 4.06

12UH 2 1.24 1.31 4.97 4.18

12UH 3 1.57 1.75 4.81 4.66

Table 5.16 Maximum Errors in dB for Model 2



12CL 1 8.35 6.81 13.18 12.02

12CL 2 4.78 4.75 10.74 9.87

12CL 3 7.54 10.36 16.70 14.31

12UM 1 5.40 5.44 14.48 13.56

12UM 2 5.93 5.77 11.32 11.25

12UM 3 7.34 10.00 18.46 16.22

12UH 1 4.24 4.08 14.13 13.00

12UH 2 7.36 6.82 13.04 12.45

12UH 3 6.55 9.04 17.88 15.58

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6. Conclusions

It has been shown that the current Four-Source coaxial jet prediction method ac-

curately predicts the noise from an internally mixed jet with a confluent mixer.

However, neither a standard coaxial jet nor a single jet prediction are capable of ac-

curately predicting the noise from an internally mixed jet with a forced mixer. The

forced mixer noise spectra can, however, be predicted using a Two-Source model.

The three variable parameters in this Two-Source model are determined for a given

mixer geometry through a multi-step optimization process. These parameters have

then been curve-fit to the differences in the mixer geometry. As a result, for the

family of forced mixers studied here, given the mixer penetration and the aerody-

namic conditions of the co-flowing jet, a noise prediction can be made based on a

Two-Source model.

The fact that a fully mixed jet and a secondary jet can be used to model the

noise from a forced mixer suggests that the differences in the structure of a forced

mixer jet plume essentially eliminates the effective jet component of the Four-Source

model. This hypothesis, which was originally proposed by Mike Fisher and Brian

Tester [27] based on the analysis of the forced mixer experimental acoustic data, is

supported by results found in this study.

A notable deviation in the forced mixer noise predictions is seen near the spectrum

peak at angles close the jet axis. However, it is noted that the same deviations are

present in the confluent mixer predictions using the Four-Source method. Since the

basic components that make up the Two-Source model are taken from the Four-

Source method it is logical that any limitation in the predictions of the Four-Source

method for a coaxial jet prediction would be inherited by the Two-Source model. It

is likely that these deviations in the predictions could result from the differences in

the geometric configurations, such as the presence of the center body or the nozzle

94

wall. In addition, these deviations could also be attributed to the quality of the

single jet predictions, which are common to both the Four-Source and Two-Source

models.

It is seen in the predictions at the high power set point that there appears to be an

additional noise source mechanism that is not modeled by the two single jet sources

in the Two-Source model. This additional noise source mechanism may result from

the stream-wise vortices interacting with the nozzle wall. In addition, this noise

source could also be generated by the test rig in the experimental facility. Since the

origin of this noise source is not yet known, no efforts have yet been made to account

for this noise in the current prediction models.

Two Two-Source models were evaluated in this study, a mixed jet - mixed jet

model (Model 1) and a mixed jet - secondary jet model (Model 2). In general,

the results from Model 2 appear to correlate with geometric differences in a more

linear fashion. However, it is possible that the type of Two-Source model which best

represents the actual flow field could be dependent on the forced mixer geometry.

This case could result due to the fact that as the forced mixer penetration increases,

the flow at the final nozzle exit becomes more like a fully mixed jet. Furthermore, as

the forced mixer penetration decreases, the flow at the final nozzle exit will resemble

more that of a secondary jet. The validity of this hypothesis could be determined

through the analysis of the aerodynamic data of the flow field at the final nozzle

exit. In practice, it is possible that a CFD solution may aide in determining which

form of the Two-Source model is most applicable. Based on the performance of

the two Two-Source models in this study, it is difficult to support this hypothesis

due to the limited range of velocity ratios in the current data set. For the current

set of operating points the velocity ratio between the secondary and primary flows

varies little (from 0.62 to 0.68). In addition, at these velocity ratios the secondary

jet and mixed jet have similar jet velocities. As a result, at this time it is difficult to

definitively determine which Two-Source model is best for a given mixer geometry.

95

Future work on this research topic could include relating information from the

experimental aerodynamic data to the source strength terms in the Two-Source

models. In addition, information from the experimental aerodynamic data may also

be used to assist in the identification of the additional noise source at the high power

set point. A similar effort could also made using the results from a CFD (RANS)

analysis to construct a predictive noise tool for evaluating the noise from jets with

forced mixers. In this approach, information about the predicted turbulent flow field

could be used to determine the source strengths in the Two-Source models.

LIST OF REFERENCES

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LIST OF REFERENCES

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[10] C. K. W. Tam and L. Auriault. Jet mixing noise from fine-scale turbulence.AIAA Journal, 37(2):145–153, 1999.

[11] G. M. Lilley. AFAPL-TR-72-53 Volume IV, 1972.

[12] R. Mani. The influence of jet flow on jet noise. Journal of Fluid Mechanics,73(4):753–793, 1976.

[13] T. F. Balsa and P. R. Gliebe. Aerodynamics and noise from fine-scale turbulence.AIAA Journal, 15(11):1550–1558, 1977.

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[18] J. B. Freund, S. K. Lele, and P. Moin. Direct numerical simulation of a mach1.92 turbulent jet and its sound field. AIAA Journal, 38(11):2023–2031, 2000.

[19] J. B. Freund. Noise sources in a low-reynolds number turbulent jet at mach 0.9.Journal of Fluid Mechanics, 438:277–305, 2001.

[20] A. Uzun, G. A. Blaisdell, and A. S. Lyrintzis. 3-d large eddy simulation for jetaeroacoustics. AIAA Paper No. 2003-3322, May 2003.

[21] W. Zhao, S. H. Frankel, and L. Mongeau. Large eddy simulations of soundradiated from subsonic turbulent jets. AIAA Journal, 39(8):1469–1477, 2001.

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[24] C. L. Morfey, V. M. Szewczyk, and B. J. Tester. New scaling laws for hot andcold jet mixing noise based on a geometric acoustic model. Journal of Soundand Vibration, 61:255–292, 1978.

[25] I. A. Waitz, Y. J. Qui, T. A. Manning, A. K. S. Fung, J. K. Elliot, J. M. Kerwin,J. K. Krasnodebski, M. N. O’Sullivan, D. E. Tew, E. M. Greitzer, F. E. Marble,C. S. Tan, and T. G. Tillman. Enhanced mixing with streamwise vorticity.Progress in Aerospace Sciences, 33:323–351, 1997.

[26] H. E. Bass, L. C. Sutherland, A. J. Zuckerwar, D. T. Blackstone, and D. M.Hester. Atmospheric absorption of sound: Further developments. Journal ofthe Acoustical Society of America, 97(1):680–683, 1995.

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Documents

JET NOISE MODELS FOR FORCED MIXER NOISE …lyrintzi/Thesis_Garrison.pdffor Forced Mixer Noise Predictions . Major Professor: Anastasios S. Lyrintzis and Gregory A. Blaisdell. The Four-Source