unsteady simulations of generated combustion-noise in...

Universidad Politecnica de Madrid - E.T.S.I. Aeronauticos

Institut Superieur de l’Aeronautique et de l’Espace - ENSICA

C.E.R.F.A.C.S.

Final Project

Unsteady simulations of generated

combustion-noise in aero-engines

by

Ignacio Duran Garcıa-Rama

Supervisor: Stephane Moreau

Toulouse, 2010

Reference: WN-CFD-10-78

Contents

Contents i

Company presentation 3

General introduction 5

1 Comparison of direct and indirect combustion noise 9

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 First order analytical model . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.1 Acoustic and wave generation in the combustion chamber . . 12

1.2.2 Compact nozzle analysis . . . . . . . . . . . . . . . . . . . . . 13

1.3 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Mesh convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.6 Convergent-divergent subsonic nozzle . . . . . . . . . . . . . . . . . . 22

1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 The Entropy Wave Generator experience 25

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Analytical Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.5 Semi-Analytical method using SuperNozzle . . . . . . . . . . . . . . 37

2.5.1 Study of the acoustic waves generated by the heating device . 40

2.5.2 Study of the inlet boundary condition . . . . . . . . . . . . . 44

2.5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.6 Fully analytical double nozzle analysis . . . . . . . . . . . . . . . . . 48

2.6.1 Double nozzle transfer function method . . . . . . . . . . . . 50

2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

i

ii CONTENTS

3 Simulation of the waves transmission through a blade row 55

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 Two dimensional modelization of the wave propagation . . . . . . . 56

3.2.1 Conservation equations . . . . . . . . . . . . . . . . . . . . . 56

3.2.2 Waves calculation . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.3.1 Post-treatment of the solution . . . . . . . . . . . . . . . . . 64

3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

General conclusion 69

Bibliography 73

A Linearised Euler equations 75

A.1 Isentropic relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

A.2 Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

A.3 Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

B Propagation equations through a rotor 79

B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

B.2 Stator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

B.3 Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

B.3.1 Inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

B.3.2 Outlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

C Double-Nozzle quasi transfer functions 85

1

Abstract

Combustion generated noise is becoming significantly important due to the

decrease of other noise sources in the airplane. Theories developed in the 70’s

showed that the combustion noise is generated by acoustic waves leaving the

combustion chamber (direct noise), and by entropy waves accelerated through

the turbine (indirect noise). Recent work using numerical simulations com-

pared direct and indirect combustion noise, concluding that the indirect noise

has to be considered in actual aero-engines. In this work analytical methods

are used combined with numerical simulations to study the validity of the hy-

pothesis made, identifying when they cannot be used. Alternative methods of

solving the equations have been proposed and discussed in those cases where

fully analytical methods cannot be used. Analytical methods and the numer-

ical simulations have been compared with experimental data to conclude that

first order methods predict correctly the noise transmision through a nozzle.

Two dimensional models of solving the blade rows have been also analyzed,

establishing the basics of a simple first order analytical method to predict the

total combustion noise at the engine outlet, and concluding that these methods

can be applied.

Keywords: Combustion noise, entropy noise.

Company presentation

CERFACS (Centre Europeen de Recherche et de Formation Avancee en Calcul

Scientifique) is a research organization that aims to develop advanced methods for

the numerical simulation and the algorithmic solution of large scientific and techno-

logical problems of interest for research as well as industry, and that requires access

to the most powerful computers presently available.

Approximately 115 people work at CERFACS, including more than 95 researchers

and engineers. They work on specific projects in nine main research areas: parallel

algorithms, code coupling, aerodynamics, gas turbines, combustion, climate, envi-

ronmental impact, data assimilation, and electromagnetism.

CERFACS develops all this research in collaboration with seven partners: CNES,

the French Space Agency; EADS France, European Aeronautic and Defense Space

Company; EDF, Electricite de France; Meteo-France, the French meteorological

service; ONERA, the French Aerospace Lab; SAFRAN, an international high-

technology group, TOTAL, a multinational energy company.

The objective of the Computational Fluid Dynamics (CFD) group at CERFACS

is to solve problems involving both CFD and High Performance Computing (HPC).

Despite the recent progresses observed in CFD, the solution of many flows of inter-

est is still beyond present capabilities and the challenge of HPC for CFD remains

as open and difficult as it ever was. In most CFD problems, brute force approaches

still fail and advances in this field rely on defining proper compromises between

physics and numerics.

This is especially true in the fields of CFD chosen at CERFACS: aerodynamics,

turbulence, combustion, unsteady flows, coupled phenomena between fluid mechan-

ics and other mechanisms (fluid structure interaction, optimization, two-phase flows,

radiation, etc).

In the last years, the requests of CERFACS partners as well as the general ori-

3

4 CONTENTS

entation of the CFD community have lead the CFD project to a deeper implication

in Direct Numerical Simulation tools, especially for reacting flows or for flows in

complex geometries as well as to the development of new aspects of CFD such as

multiphysics or active control. This has been done through an increase of the CFD

staff so that the classical expertise of the CFD team (aerodynamics, turbulence mod-

eling, optimization and parallelization, combustion) has been maintained or even

reinforced. An important new field of application for CERFACS is Large Eddy

Simulation (LES). The role of the CFD team and of its partners in the development

of Large Eddy Simulation is now significant through multiple collaborations, con-

tracts and dissemination of information and tools. The LES approach has emerged

as a prospective technique for problems associated with time dependent phenomena

and coherent eddy structures. This leading edge CFD technology can nowadays be

applied to geometries of reasonable complexity (such as a combustion chambers

in gas turbines but also in piston engines), which is the result of both constantly

increasing computer capacities along with improved underlying numerical methods

and grid techniques.

During the last years, Ph.D. students working for Snecma have developed tools

to predict and reduce combustion instabilities in the aero-engines. Recently, efforts

have been made in the prediction and reduction of combustion noise, a field in

which little research had been done since the 1970’s. The objective of this research

is to understand the fundamental mechanisms of combustion noise generation and

to build a tool able to predict the total combustion noise at the outlet of the aero-

engine. It is in this framework that this project will be done.

General introduction

Increasingly restrictive rules are being applied to reduce noise emissions of civil

aviation. This occurs due to the air traffic growth in the last decades and to the

fact that airports are placed near large urban areas. Noise reduction is therefore

becoming a great issue for aircraft and engines manufacturers who have worked in

common to reduce acoustic emissions significantly. Recently combustion noise has

been identified as a significant source for the global air plane noise: other acoustic

sources (such as jet noise and turbomachinery noise) have been reduced in the last

decades and therefore the relative impact of combustion in the global noise has in-

creased.

The measurement of combustion generated noise is of particular difficulty in

experimental facilities due mainly to three factors: Firstly, aero-engines are built

compactly, dificulting the introduction of microphones in the combustion chamber

to measure the sound level; secondly, high temperatures in the combustion chamber

limits the materials used to perform the measurements; and finally, as it will be seen,

combustion noise is strongly affected by the waves propagation through the turbine,

and therefore the measurement of combustion noise has to be done in association

with the turbine stages that follow, introducing noise in the measurement. For this

reason, combustion generated noise has been measured using the total engine noise,

and substracting the partial contribution of the compressor, the turbine, the jet

noise, etc. The long term objective is to study the combustion generated noise and

validate the analytical theories used to predict this noise generation.

Combustion noise was firstly studied in the 1970s by Marble and Candel [7],

showing that there are two mechanisms of combustion noise: the direct and the

indirect combustion noise (which will be explained in Chapter 1). Little progress

has been done since then as jet noise and turbomachinery noise had greater im-

portance and efforts were concentrated in reducing these sources. Recently Leyko

studied, during his Ph.D. [4], the different mechanisms of combustion noise shown

by Marble and Candel and created a tool to predict the combustion noise generated

by an aero-engine using LES simulations of the combustion chamber and analytical

methods to propagate the waves through the turbine stages.

5

6 CONTENTS

Objectives

The objective of this internship is to analyze the generation of combustion noise

and its propagation through the turbine to understand the mechanisms involved.

Analytic results will be combined with simulations and experimental data to study

the precision of first order models in the prediction of combustion-generated noise.

Firstly, in Chapter 1, one dimensional simplified analytical models and numeri-

cal simulations will be used to study the propagation of acoustic and entropy waves

through a nozzle. A first order comparison will be made to analyze the influence

of entropy noise in the global combustion noise. At the same time, the numerical

simulations will extend the validity of the analytical method and understand the

possible limitations of the method.

An existing code (SuperNozzle) will be used to solve the linearized equations of

the model and compare them to full Euler numerical simulations, in order to verify

if the calculated noise is actually created by first order mechanisms. The objective

is also to understand basic one dimensional mechanisms of noise generation and

propagation through the nozzle before considering more complex two dimensional

theories.

An experimental case will be analyzed with the developed tools in order to eval-

uate fist order theories with experimental data in Chapter 2. The data from the

Entropy Wave Generator Experiment performed by Bake et al. [2] will be used to

compare the experimental measurements and understand the mechanisms generat-

ing combustion noise.

The EWG experiment was already studied numerically and analytically by Leyko

et al. [5] for the supersonic case, showing that the noise generated is essentially in-

direct combustion noise and that it is correctly predicted using first order models.

The objective is therefore to understand the indirect noise generation in the sub-

sonic case and verify if the analytical methods can be used to predict this type of

noise.

To finish, simulations of the flow through a static turbine row will be performed

in Chapter 3 to consider the flow deviation and the 2D effects on the noise and en-

tropy propagation comparing the results to simplified models developed by Cump-

sty and Marble [3]. These simulations will help to validate the theoretical methods

that will be used to predict the combustion noise at the exit of the aero-engine.

Leyko [4] performed already some of these simulations, showing that the noise gen-

eration and propagation was correctly predicted for the downstream propagating

acoustic and entropy waves. The objective is to analyze the upstream propagating

acoustic wave. After this work, future simulations and analysis will be suggested to

continue in the understanding of combustion noise and the generation of indirect

7

combustion noise in the turbine stages.

Other objectives considered at the begining of the internship were to simulate

the wave transmison through a blade row when considering circumferential compo-

nent of the wave propagation, to study the influence of three dimensional effects

and to study the effect of rotating blade rows in turbines. These objectives will be

discussed through the project, though the simulations were not performed due both

to lack of time and computational resources, and to the concentration of efforts to

fully understand the fundamental physical phenomena of one dimensional problems

before jumping to complex simulations.

After the internship, a PH.D. will follow in the same subject. For this reason,

during the study presented here future work and objectives will be proposed.

Chapter 1

Comparison of direct and indirect

combustion noise

1.1 Introduction

Combustion noise is generated by a turbulent flame burning in a confined com-

bustion chamber. Turbulent flames are needed to achieve high burning rates using

small burners, but they generate combustion instabilities and noise which have to

be studied and reduced. Two main mechanisms have been identified by Marble and

Candel [7] in the generation of combustion noise by a turbulent flame:

• Direct noise: Acoustic perturbations generated by the unsteady heat re-

lease in the combustion chamber propagate upstream and downstream. These

acoustic waves interact with the turbine or the compressor stages and can be

attenuated, amplified or reflected.

• Indirect noise: Temperature fluctuations generated by the turbulent flame

propagate downstream and interact with the turbine stages. The acceleration

of these entropy waves generates acoustic waves as shown by Marble and Can-

del [7]. This is a first order effect which has to be considered to calculate the

total combustion noise.

In both types of noise the propagation of the associated wave through the turbine

is a key factor in the prediction of the overall acoustic noise at the exit of the

aeroengine.

9

10 CHAPTER 1. COMPARISON OF DIRECT AND INDIRECT COMBUSTION NOISE

1.1.1 Objectives

The objective of this chapter is to calculate the ratio between indirect and di-

rect combustion noise in a simple one dimensional combustion configuration. For

this purpose, an analytical method will be used to estimate the wave generation in

a simplified combustion chamber. At the same time, this analytical method will

be used to calculate the wave transmission through the nozzle. Simulations were

performed to validate the theoretical models, and at the same time to extend the

validity of the conclusions obtained.

This work is similar to the work done by Leyko et al. [6]. In this case a mesh

sensitivity analysis will be performed to validate the results, and a comparison with

SuperNozzle results will also be done.

1.2 First order analytical model

The direct to indirect combustion noise ratio is calculated as in Leyko et al. [6],

namely,

η =w+

2

ws1

[SA]

︸︷︷︸

Indirect noise

×ws

1

w+1

[CC]

︸︷︷︸

Wave Ratio

×[w+

2

w+1

[AA]

]−1

︸︷︷︸

Direct Noise

. (1.1)

Subscripts 1 and 2 make reference to the inlet and the outlet of the nozzle as

shown in Fig 1.1; w+, w− and ws are the acoustic waves propagating downstream,

upstream and the entropy wave respectively (Eq. 1.2-1.4)

Figure 1.1: Acoustic and entropy waves in the subsonic nozzle.

w+ =p′

γp+

u′

c, (1.2)

w− =p′

γp−

u′

c, (1.3)

ws =s′

Cp. (1.4)

1.2. FIRST ORDER ANALYTICAL MODEL 11

The first and the third terms in Eq. 1.1 characterize the outgoing acoustic wave

generated by incoming entropy wave (indirect noise) and by the incoming acoustic

wave (direct noise) respectively. The second term is the ratio between entropy and

acoustic waves generated in the combustion chamber.

The governing equations are the linearized Euler equations obtained by Mar-

ble and Candel [7]. They read,

[∂

∂t+ u

∂

∂x

](p′

γp

)

+ u∂

∂x

(u′

u

)

= 0 , (1.5)

[∂

∂t+ u

∂

∂x

](u′

u

)

+c2

u

∂

∂x

(p′

γp

)

+

(

2u′

u− (γ − 1)

p′

γp

)du

dx=

du

dx

s′

Cp. (1.6)

[∂

∂t+ u

∂

∂x

](s′

Cp

)

= 0 , (1.7)

The demonstration of these equations is shown in Appendix A. The last term of

Eq. 1.6 is the source term of the indirect combustion noise (or entropy noise): The

entropy waves s′/Cp generated in the combustion chamber are accelerated through

a nozzle, creating therefore a non-zero source term in the propagating equations.

The waves propagating through the nozzle can be obtained as a function of the

pulsating frequency. For that reason a dimensionless frequency has been defined as

Ω =fLn

c1, (1.8)

where Ln is the nozzle length, c1 is the mean sound speed at the inlet and f is

the pulsating frequency (in Hz). The dimensionless frequency Ω characterizes the

acoustic compactness of the nozzle. Eqs. 1.5-1.7 can be solved analytically for the

compact case (Ω = 0) as shown in [7]. Using the compact nozzle hypothesis, it

can be shown that any perturbation at the inlet of the nozzle is recovered at the

outlet with no time delay. To solve for the outgoing waves, the mass, enthalpy

and entropy conservation equations are therefore written for small time-dependent

perturbations as

m′

¯m=

1

M

u′

c+

p′

γp−

s′

Cp, (1.9)


T ′t

Tt=

1

1 + [(γ − 1)/2]M2

[

(γ − 1)Mu′

c+ (γ − 1)

p′

γp+

s′

Cp

]

, (1.10)

s′

Cp=

p′

γp−

ρ′

ρ. (1.11)

The conservation relations can be written between the inlet and the outlet using

the compact nozzle hypothesis, namely,

(

m′

¯m

)

1

=

(

m′

¯m

)

2

, (1.12)

(T ′

t

Tt

)

1

=

(T ′

t

Tt

)

2

, (1.13)

(s′

Cp

)

1

=

(s′

Cp

)

2

. (1.14)

The three terms of Eq.1.1 can be obtained analytically using these equations.

1.2.1 Acoustic and wave generation in the combustion chamber

Solving the generation of acoustic and entropy waves in a combustion chamber

requires expensive LES simulations and combustion models. For the simplified one

dimensional model developed here, the second term of Eq. 1.1 will be estimated

analytically as done by Leyko et al. [6] using Eq. 1.12-1.14 (considering a compact

flame) with a source term fluctuation q′, written as a function of the ingoing and

outgoing waves Eqs. 1.2-1.4,

(1 +1

M1)w+

0 + (1 −1

M1)w−

0 − 2ws0 = (1 +

1

M1)w+

1 + (1 −1

M1)w−

1 − 2ws1 , (1.15)

[

(1 + M1)w+0 + (1 − M1)w

−

0 +2(ws

0 + q′)

γ − 1

]

=

[

(1 + M1)w+1 + (1 − M1)w

−

1 +2ws

1

γ − 1

]

, (1.16)

ws0 + q′ = ws

1 , (1.17)

1.2. FIRST ORDER ANALYTICAL MODEL 13

where subscript 0 represents the waves upstream the flame, and 1 downstream. It

has been considered that the mean Mach number is the same at the inlet and at the

outlet of the flame as the mean heat release is supposed to be zero. The problem is

solved considering no incoming waves, and solving for the outcoming ones generated

by the heat release fluctuation. Even if the real heat release fluctuation of a real

chamber is unknown for the model the ratio of the acoustic to entropy waves can

be obtained as a function of mean flow characteristics as the problem is linearized.

It reads,

ws1

w+1

[CC] =1 + M1

M1. (1.18)

This ratio is large for most combustion chambers, as the Mach number is low.

1.2.2 Compact nozzle analysis

Eqs. 1.5-1.6 have to be solved numerically to obtain the noise ratio through the

nozzle as a function of frequency, but for small frequencies an analytical solution

can be obtained using the compact nozzle hypothesis. Eqs. 1.12-1.14 can be written

as a function of the three waves (w+, w− and ws) at the inlet and the outlet sections

using Eqs. 1.2-1.4 and Eqs. 1.9-1.11, namely,

(1 +1

M1)w+

1 + (1 −1

M1)w−

1 − 2ws1 = (1 +

1

M2)w+

2 + (1−1

M2)w−

2 − 2ws1 , (1.19)

1

1 + γ−12 M2

1

[

(1 + M1)w+1 + (1 − M1)w

−

1 +2ws

1

γ − 1

]

= ...

1

1 + γ−12 M2

2

[

(1 + M2)w+2 + (1 − M2)w

−

2 +2ws

2

γ − 1

]

, (1.20)

ws1 = ws

2 . (1.21)

Out of the six waves involved, three of them can be imposed for the unchoked

nozzle, and the problem can be solved directly (Fig. 1.1). For the choked nozzle

the wave w−

2 is propagating downstream, as the flow is supersonic, and it cannot

be imposed. An extra relation would be needed to close the problem, and therefore

both cases should be treated separately.


Unchoked Nozzle

The waves w−

2 and ws1 can be supposed to be zero to solve for the equation of direct

combustion noise. Solving Eqs. 1.19-1.21 gives the desired relation between w+2 and

w+1 , namely,

w+2

w+1

[AA] =2M2

1 + M2

1 + M1

M1 + M2

1 + [(γ − 1)/2]M22

1 + [(γ − 1)/2]M2M1. (1.22)

The indirect combustion noise relation can be obtained with w−

2 and w+1 equal

to zero,

w+2

ws1

[SA] =M2 − M1

1 + M2

M2

1 + [(γ − 1)/2]M2M1. (1.23)

Choked Nozzle

An independent equation should be written to solve the choked nozzle. This equa-

tion can be obtained from the choked-flow relation considering small perturbations

as

m′

¯m=

1

1 + [(γ − 1)/2]M2

[γ

2(1 − M2)

p′

γp+

γ + 1

2M

u′

c+

1 + γM2

2

ρ′

ρ

]

. (1.24)

With this new equation the two noise relations of Eq. 1.1 can be calculated

without imposing w−

2 , namely

w+2

w+1

[AA] =1 + [(γ − 1)/2]M2

1 + [(γ − 1)/2]M1, (1.25)

w+2

ws1

[SA] =(M2 − M1)/2

1 + [(γ − 1)/2]M1. (1.26)

1.3 Numerical simulations

The objective of the simulations is to calculate the noise ratio of Eq. 1.1 without

the compact nozzle hypothesis. Several simulations were performed with AVBP in

order to study the influence of all possible configurations of the nozzle: choked and

1.3. NUMERICAL SIMULATIONS 15

unchoked. In all cases the wave ratio of the combustion chamber (second term of

Eq. 1.1) is calculated analytically. The mesh size varied from one simulation to

another, but the following numerical parameters were kept constant:

• Euler equations

• Numerical Scheme: Lax-Wendroff (Second order)

• Artificial Viscosity: 0.05 for both second and fourth order.

• Relaxation Coefficients at boundaries: 0

• CFL: 0.7

The initial solution was obtained by converging the flow through the nozzle with

artificial viscosity of 0.1 and high relaxation coefficients to obtain a fully stationary

uniform flow.

Air at 1300K was used for the simulations, giving γ = 1.32.

Boundary conditions imposed are:

• Inlet: Imposed pressure and temperature (800kPa, 1300K)

• Outlet: Imposed Pressure

• Wall: Adiabatic and slip wall

The following nozzles were created with CFD-GEOM with a structured mesh

and converted to an unstructured grid using HIP (Table 1.1).

M1 M2 A1/Ac A2/Ac

0.1 1.2 5.873 1.032

0.1 0.8 5.651 1.000

0.025 0.8 22.482 1.000

Table 1.1: Geometric parameters for different Mach numbers


To obtain the results, two simulations were performed for each nozzle:

• Acoustic: In order to calculate the third term of Eq. 1.1:w+

2

w+

1

[AA] the nozzle

inlet was perturbed with normal propagating acoustic waves entering the do-

main.

• Entropic: Entropy waves were sent through the inlet to analyze the acoustic

response (indirect noise). The termw+

2

ws1

[SA] of Eq. 1.1 was then calculated.

White noise was introduced as a perturbation to obtain the response in all fre-

quencies using a cut-off frequency Ω = 1.7.

The simulations lasted long enough to capture at least 20 wave cycles at the

shortest frequency of interest (Ωmin = 0.01). This corresponded to 2.5 seconds of

simulation time.

Temporal solutions were stored at least each 2 · 10−5 seconds to obtain a good

resolution for high frequencies.

1.4 Mesh convergence

The influence of the mesh size in the results was studied at high frequencies,

when the numerical dissipation is larger since the wavelengths are shorter. The

same nozzle was simulated for the same frequency (Ω = 1.59) with different mesh-

ing. Results are shown in Fig. 1.2, where the indirect to direct noise ratio is plotted

against the number of nodes in the axial direction.

Theory shows that, to first order, the entropy wave should propagate undis-

turbed, as seen in Eq. 1.7. This means that the inlet-outlet entropy wave ratio

should be equal to 1 if no numerical dissipation exists. It is also known that the

effect of numerical dissipation in this wave is greater than in the acoustic wave,

since its associated wavelength is shorter. This inlet-outlet entropy ratio (the en-

tropy transfer function) will permit us to evaluate the mesh quality and it has

been therefore plotted as a function of frequency (Fig 1.3). It can be seen that, for

the frequency range of interest (0 < Ω < 1), numerical dissipation is not significant.

The same study has been done as a function of the mesh size (Fig. 1.4). It

can be seen that for coarsened meshes dissipation is strong and the entropy wave is

attenuated before it gets to the end of the nozzle.

1.4. MESH CONVERGENCE 17

Figure 1.2: ]

Indirect to direct noise ratio plotted as a function of the number of nodes forM1 = 0.1, M2 = 1.2 at Ω = 1.59

Figure 1.3: ]

Entropy transfer function for M1 = 0.1, M2 = 1.2 nozzle using 1000 points in theaxial direction.

For small inlet Mach number, the entropy wavelength is smaller. For that reason

the mesh has been refined in those cases.

The analysis of the entropy wave at the inlet and the outlet for the M1 = 0.025,

M2 = 0.8 case with 2000 nodes in X direction (Fig. 1.5, left) shows that the numer-

ical error made is too large. It can also be seen the effect of dispersion: there is a

zone where the ratio of the entropy wave is larger than one. This occurs because

high frequencies are dispersed, shifting the energy contents to lower frequencies.

The same mesh has been calculated with TTGC scheme (Fig. 1.5, right). Results


Figure 1.4: ]

Entropy transfer function for Ω = 1.59 as a function of mesh size for M1 = 0.1,M2 = 1.2.

show dissipation in the high frequency range, but significantly less dispersion of the

wave. Nevertheless, the mesh cannot be used for our purpose as the entropy wave

is not transported unaltered for the frequency range of interest. As the inlet Mach

number is four times smaller than the previous case, the mesh should be, as a first

estimate, four times more refined. In practice this has to be validated with a mesh

convergence study.

Figure 1.5: Entropy transfer function for M1 = 0.025, M2 = 0.8 nozzle with 2000nodes in the axial direction using Lax Wendroff scheme (left), and TTGC (right).

A mesh refinement was performed to achieve low numerical dissipation required

for our case. The entropy transfer function has been plotted in Fig. 1.6 and Fig. 1.7

for a mesh that was two and four times more refined in the the axial direction.

1.4. MESH CONVERGENCE 19

Figure 1.6: Entropy transfer function for M1 = 0.025, M2 = 0.8 nozzle with 4000nodes in the axial direction using Lax Wendroff scheme (left), and TTGC (right).

Figure 1.7: Entropy transfer function for M1 = 0.025, M2 = 0.8 nozzle with 8000nodes in the axial direction using Lax Wendroff scheme.

It can be seen that the refinement improves the dissipation characteristics of the

mesh, but the mesh should be still refined to achieve the low dissipation required

for these simulations.

A last simulation was performed using the Galerkin-Runge-Kutta scheme (GRK).

This scheme is of higher order, and therefore better dissipation and diffusion prop-

erties. Results are shown in Fig 1.8

GRK scheme is much better than the Lax Wendroff, but there is still too much

dissipation.


Figure 1.8: Entropy transfer function for M1 = 0.025, M2 = 0.8 nozzle with 8000nodes in the axial direction using Galerkin Runge-Kutta scheme.

1.5 Results

Results shown are compared to SuperNozzle code and to analytic solutions. Su-

perNozzle is a code that solves Eqs. 1.5-1.7 in the frequency domain. This gives a

third solution to compare to the analytical and numerical results. At the same time,

SupperNozzle gives only first order effects (as it solves the linearized equations), and

therefore it will show if combustion noise is a first order phenomena.

The ratio between direct and indirect noise has been plotted as a function of

the reduced frequency for the first nozzle (M1 = 0.1, M2 = 1.2) in Fig. 1.9.

Figure 1.9: Combustion noise ratio as a function of the reduced frequency Ω calcu-lated analytically (circle), with AVBP simulations (solid line) and with SuperNozzle(dotted line) for the M1 = 0.1, M2 = 1.2 nozzle

1.5. RESULTS 21

AVBP simulations agree with the analytical values. It can be seen that Su-

perNozzle results agree with the AVBP simulations for low frequency. At higher

frequencies, both become different and have a maximum error at the highest fre-

quency (Ω = 0.9, Error = 14%). This difference can be due to one of the following

factors:

• Numerical dissipation: Both AVBP calculations or SuperNozzle simulations

may have dissipation due to the mesh and/or the numerical scheme.

• Gas properties: The value of Cp is constant in the SuperNozzle calculation,

while in AVBP simulations enthalpy is tabulated as a function of temperature.

• Boundary Conditions: SuperNozzle and AVBP treat in different ways the

boundary conditions for supersonic flows. This can be the source of the dif-

ference found in the calculations.

To analyze the origin of the error, the first and third terms of Eq. 1.1 (direct and

indirect noise) were plotted separately in Fig. 1.10.

Figure 1.10: Direct (left) and indirect (right) combustion noise for the M1 = 0.1,M2 = 1.2 nozzle as a function of the reduced frequency Ω

It can be seen that the results for the indirect noise are coincident for SuperNoz-

zle and AVBP while the direct combustion noise shows a slight difference at high

frequencies. This is an unexpected result if we consider the numerical simulation to

be responsible for the error, as it is the entropy wave the one which is dissipated the

most (and therefore, disagreement between the two simulations should be larger for

the indirect combustion noise than for the direct noise).


Simulations carried out showed that the influence of a variable Cp is negligible:

AVBP simulations with variable and constant Cp agree.

For the subsonic nozzle (M1 = 0.1, M2 = 0.8) results are plotted in Fig. 1.11.

It can be seen that the performance as a function of Ω is similar to that seen for

the choked nozzle: the ratio between direct and indirect noise decreases for high

frequencies.

Figure 1.11: Combustion noise ratio as a function of the reduced frequency Ω cal-culated analytically (circle), AVBP (solid line) for the M1 = 0.1, M2 = 0.8 nozzle

A similar error at high frequencies can be seen when comparing AVBP to Super-

Nozzle simulations.

1.6 Convergent-divergent subsonic nozzle

Subsonic nozzles studied have been in all cases simply convergent nozzles. A

last case has been simulated with a convergent divergent subsonic nozzle, in which

the inlet Mach number is the same as in the outlet. The theory developed in Sec-

tion 1.2.2 shows that the expected indirect noise is zero. In fact, for low frequencies,

the nozzle should have the same response as a simple constant section tube. This is

true for low frequencies, but not for high ones. The direct noise generation at low

frequencies is zero, but this occurs because the acoustic wave generated in the con-

vergent is canceled by the one generated in the divergent. The objective is therefore

to analyze the response of the nozzle to non-zero frequencies, where the addition of

the waves generated in the convergent and in the divergent is not trivial due to the

phase shift.

Simulations were performed using a Mach number of M1 = M2 = 0.1 for the

inlet and the outlet, and a throat Mach number of Mt = 0.9. This configuration gen-

1.7. CONCLUSION 23

erates therefore strong acoustic waves, due to the strong acceleration/deceleration.

Results are plotted in Fig 1.12, where it can be seen that the analytical predic-

tion is correct for strictly zero frequencies, but increases rapidly with frequency.

Figure 1.12: Direct combustion noise as a function of the reduced frequency Ω cal-culated analytically (circle) and with AVBP (solid line) for the M1 = 0.1, M2 = 0.8nozzle

The explanation of this sudden increase of noise with frequency can be found

considering the waves generated inside the nozzle. In a subsonic nozzle the acoustic

waves generated in the convergent and in the divergent by the entropy wave are

in opposite phase, due to the ∂u/∂x term of Eq. 1.6. For this reason the acoustic

waves generated in the convergent (which are strong due to the strong accelera-

tion) are canceled by those generated in the divergent. This occurs only when the

compact nozzle hypothesis is considered. When considering a non-compact nozzle

and resolving numerically Eqs. 1.5-1.7 the waves involved will be phase-shifted for

non-zero frequencies, due to the finite non-zero nozzle length, and therefore they

will not cancel directly, incrementing the noise generated when compared to the

zero frequency case.

1.7 Conclusion

The comparison of the first order model developed by Marble and Candel [7]

with the numerical simulations show that the direct and indirect combustion noise

can be correctly predicted using first order linearized equations. This has been

verified for the compact nozzle case using the analytical solution and for the non-

compact case solving the linearized equations with SuperNozzle.


As Leyko et al. [6] did it has been shown that indirect noise can be signifi-

cant and must be studied to predict correctly the global combustion noise of an

aero-engine. At the same time the extension of the theory to non-zero frequencies

performing numerical simulations showed that the analytical result can be extrap-

olated to higher frequencies with small errors. This is true for nozzles in which

the flow is simply accelerated. Instead, if the flow is accelerated and decelerated in

a subsonic convergent divergent nozzle, the analytical result is verified for strictly

zero frequencies, but noise increases rapidly for non-zero frequencies.

The mesh sensibility study performed showed that, when low Mach numbers are

involved, the mesh must be small enough to capture the wavelengths of interest.

For the M1 = 0.025 case the entropy wave has a very small wavelength λ ∼ cM/f ,

and therefore the mesh has to be highly refined, and the schema order increased.

The mesh analysis performed will be used in Chapter 2, as well as the study of

the convergent-divergent nozzle of section 1.6

Chapter 2

The Entropy Wave Generator

experience

2.1 Introduction

The Entropy Wave Generator experiment performed by Bake et al. [2] showed

the importance of entropy-generated noise. Leyko et al. [5] analyzed the supersonic

case numerically and analytically finding that the noise generated could be correctly

predicted by first order models proposed by Marble and Candel [7] by treating

correctly the inlet and outlet boundary conditions. The subsonic case is here treated

using both analytical methods and numerical calculations to evaluate first order

theories in the prediction of entropy noise.

2.2 Experimental Setup

The experiment carried out consists of a convergent divergent nozzle with an

electric heating device situated in the settling chamber as it can be seen in Fig. 2.1.

Main geometrical parameters are shown in Table 2.1. Microphones were placed in

the outlet region to capture the acoustic pressure generated during the experiment.

The heating device generated the temperature signal shown in Fig. 2.2, with a

period of 1 second, measured using a vibrometer. All experimental pressure signals

of the EWG used in this analysis come from a microphone placed at the outlet duct,

at a distance of 1150mm from the nozzle throat.

Physical parameters of the subsonic case studied (Reference Test Case 2 in ref-

erence [2]) are summarized in Table 2.2.

25

26 CHAPTER 2. THE ENTROPY WAVE GENERATOR EXPERIENCE

Figure 2.1: Sketch of the EWG

Convergent Divergent Inlet Outlet ThroatLength Length Diameter Diameter Diameter

13mm 250mm 30mm 40mm 7.5mm

Table 2.1: Geometric characteristics of the experimental set-up

Figure 2.2: Experimental temperature pulse induced by the heating device. Left:spectrum. Right: signal.

Inlet Throat Outlet Inlet OutletMach number Mach Number Mach number Pressure Pressure

0.033 0.7 0.01861 105,640Pa 101,300Pa

Table 2.2: Physical parameters of the subsonic case studied

It should be noticed that even if the inlet and outlet Mach numbers are low the

flow is strongly accelerated in the nozzle throat and afterwards decelerated. This

acceleration/deceleration is a key factor in the mechanism generating entropy noise.

2.3. NUMERICAL SIMULATION 27

2.3 Numerical Simulation

The numerical simulation was done using the AVBP code developed at CERFACS.

The simulation was performed in a axisymmetric mesh of the experimental setup

shown in Fig. 2.1, including the settling chamber and the outlet, 2100mm from the

nozzle throat. As Leyko et al. [5] showed, 3D effects are negligible in the configu-

ration studied. The mesh has approximately 15 cells in a full cross-section, making

a total of 5000 cells. The mesh size is about 1mm, small enough to resolve the

smallest perturbations induced by the heating device, with a wavelength of about

120mm. Since only first order phenomena are studied, vortex dynamics and bound-

ary layers are neglected, and therefore the simulation is done using Euler equations

and imposing zero normal velocity at the walls.

Lax-Wendroff scheme is used for all simulations, with artificial viscosity of 0.01

for the second order and 0.04 for the fourth order.

As shown by Muhlbauer et al. [8] and Leyko et al. [5], the outlet of the exper-

imental setup is not perfectly anechoic. A reflecting boundary condition has to be

imposed in order to take into account reflected waves. Leyko et al. [5] showed that

the experimental reflection coefficient could be mimicked using a first order filter in

the outlet of the computational domain. This first order filter is written as

Rout =1

1 + ıω/κ, (2.1)

and it has to be imposed at the correct distance lout, in order to have the correct

phase. The values of κ and lout are tuned to have the correct modulus and argument,

giving κ = 80s−1 and lout = 2100mm, as it can be seen in Fig. 2.3.

Figure 2.3: Reflection coefficient imposed at the outlet, phase-shifted to be ex-pressed at the nozzle throat. Left: Modulus, Right: argument. Solid line: analyticalreflection coefficient, Dots: experimental reflection coefficient.


In the numerical simulation, the first order filter can be easily imposed defining

the entering wave as Poinsot and Lele [9]

L− = 2κp∆t(pref − p2)/(ρc) , (2.2)

using κp = 2κ for the relaxation coefficient at the outlet boundary condition.

For the inlet of the computational domain non-reflecting boundary conditions

were used at the settling chamber inlet.

The heating device is modeled as a source term in the energy equation with an

appropriate analytic equation as done by Leyko et al. [5],

φ(x, t) = Φ01

2

[

tanh

(x − x0 + lh/2

d

)

tanh

(

−x − x0 − lh/2

d

)

+ 1

]

φ(t) , (2.3)

where lh = 60mm is the length of the heating device, d = 3mm the spatial charac-

teristic slope length and x0 the location of the heating device in the experimental

setup. The temporal evolution of the temperature signal is modeled with the ana-

lytical function φ(t), given by

φ(t) =

1 − exp( t−t0τ ) if t ∈ [t0, t0 + Tp]

φ(t0 + Tp) exp(− t−t0τ ) if t > t0 + Tp

, (2.4)

where t0 is the triggering time, Tp = 100ms the pulse duration and τ the relaxation

time of the pulse, set to 7ms. The experimental pulse is compared in Fig. 2.4 with

the analytical function and the one obtained in the numerical simulations.

Results of the numerical simulations plotted in Fig. 2.5 compared to the experi-

mental data show that the general shape is captured. To obtain a perfect match in

the shape, the exact reflection coefficient should be used instead of a first order filter

modelization. It can also be seen that the peak value of the signal is overpredicted.

Though many factors could be the cause, the saturation limit of the microphones

placed at the outlet of the experimental set-up should be analyzed as a possible

cause.

2.4. ANALYTICAL STUDY 29

Figure 2.4: Experimental temperature pulse induced by the heating device. Analyt-ical function modeling the temperature signal compared to the EWG experimentaldata and to the numerical simulation temperature pulse induced by the source term.

Figure 2.5: Numerical simulation pressure signal compared to experimental datafrom the EWG

During the analytical study, several simulations were performed to compare with

theoretical methods and to validate the hypothesis made. The aim of these simula-

tions is to isolate the different physical effects in order to study them individually.

2.4 Analytical Study

To perform the analytical study of the EWG experiment the perturbation equa-

tions of continuity and momentum obtained by Marble and Candel [7]


[∂

∂t+ u

∂

∂x

](p′

γp

)

+ u∂

∂x

(u′

u

)

= 0 , (2.5)

[∂

∂t+ u

∂

∂x

](u′

u

)

+c2

u

∂

∂x

(p′

γp

)

+

(

2u′

u− (γ − 1)

p′

γp

)du

dx=

du

dx

s′

Cp, (2.6)

[∂

∂t+ u

∂

∂x

](s′

Cp

)

= 0 , (2.7)

will be used. Here primed variables represent the perturbations (acoustic and en-

tropy), and non primed variables represent the mean steady flow through the nozzle.

Eqs. 2.5-2.7 are the linearized form of the one dimensional Euler equations in

a nozzle (see Appendix A). The last two terms of Eq. 2.6 represent the reflection

of acoustic waves in the nozzle and the generation of entropy noise respectively,

which are first order effects and occur only when the mean flow is accelerated or

decelerated, due to the term du/dx.

Though Eq. 2.5-2.7 have to be solved numerically in their general form, they

can be solved analytically making the compact nozzle hypothesis: The wavelength

of the perturbations is large compared to the nozzle length (and therefore the time-

derivative term of Eqs. 2.5-2.7 is negligible). This hypothesis can be validated by

calculating the frequency of the signal spectrum (Fig 2.2). The reduced frequency

is defined as the ratio between the nozzle length and a characteristic wavelength,

Ω =Ln

λ=

fLn

c1. (2.8)

with c1 =√

γrT1 the sound speed at the nozzle inlet. The spectrum shows that

most of the energy is contained in frequencies below the cut-off frequency 100Hz,

and therefore, using this value we obtain a reduced frequency of Ω ≈ 0.077. This

means that the nozzle length is short compared to the wavelength, and we can make

the hypothesis that the perturbations are transmitted with no time delay.

Considering therefore the compact nozzle hypothesis, Eqs.2.5-2.6 can be simpli-

fied to obtain two invariants,

IA = 2

(p′

γp+

1

M

u′

c−

s′

Cp

)

, (2.9)

IB =2

1 + γ−12 M2

[

Mu′

c+

p′

γp+

1

γ − 1

s′

Cp

]

(2.10)


which are equivalent to the mass and the enthalpy conservation through the nozzle.

Eq.2.7 shows that ws = s′/Cp (the entropy wave) is propagated unperturbed. In

this way, the third invariant is the isentropic equation,

IC =s′

Cp. (2.11)

Using the acoustic and entropy waves definition

w+ =p′

γp+

u′

c= ϕ + ν , (2.12)

w− =p′

γp−

u′

c= ϕ + ν , (2.13)

ws =s′

Cp= σ . (2.14)

the invariants can be written as

IA = (1 +1

M)w+ + (1 −

1

M)w− − 2ws , (2.15)

IB =1

1 + γ−12 M2

[

(1 + M)w+ + (1 − M)w− +2ws

γ − 1

]

, (2.16)

IC = ws . (2.17)

The values of IA, IB and IC are conserved through the nozzle (assuming the

compact nozzle hypothesis) and therefore the relations

(1 +1

M1)w+

1 + (1 −1

M1)w−

1 − 2ws1 = (1 +

1

M2)w+

2 + (1−1

M2)w−

2 − 2ws1 , (2.18)

1

1 + γ−12 M2

1

[

(1 + M1)w+1 + (1 − M1)w

−

1 +2ws

1

γ − 1

]

= ...

1

1 + γ−12 M2

2

[

(1 + M2)w+2 + (1 − M2)w

−

2 +2ws

2

γ − 1

]

, (2.19)


ws1 = ws

2 . (2.20)

can be written to relate the inlet (1) and the outlet (2) waves. In the subsonic

problem there are three imposed and three unknown waves as shown in Fig. 2.6,

and therefore the problem is closed.

Figure 2.6: Acoustic and entropy waves in the nozzle.

The only imposed wave considered is the entropy wave, generated by the heat-

ing device, and it is calculated using the analytical model of Eq. 2.4. As shown in

section 2.3, a specific treatment of the outlet boundary condition has to be done to

take into account the reflected waves. Fig. 2.6 shows that two acoustic waves have

to be calculated, w+1 and w−

2 , using the reflection coefficients, namely,

w+1 = R1w

−

1 , (2.21)

w−

2 = R2w+2 . (2.22)

As seen in section 2.3 the boundary condition at the outlet is considered as first

order filter (Rout, calculated in Eq. 2.1), and phase-shifted for it to be expressed at

the nozzle throat. Considering low Mach number in the outlet duct, the reflection

coefficient R2 is given by

R2 = Rout exp(−2ıloutω/c) . (2.23)

The values of κ and lout used are the same as for the numerical simulation:

κ = 80s−1 and lout = 2100mm.

The domain used for the calculation is plotted in Fig. 2.7. It can be seen that

it does not take into account the settling chamber, and therefore a different model


for the inlet reflection coefficient should be used.

Figure 2.7: Sketch of the EWG experimental setup, with a dotted rectangle showingthe domain for the analytical calculation.

The reflection coefficient at the inlet is calculated considering that the settling

chamber section is large compared to the inlet of the heating device tube, the re-

flection coefficient can be written as Rin = −1, and then phase-shifted using the

tube length lin = 250mm, giving finally

R1 = − exp(−2ılinω/c) . (2.24)

The system can be written in a matrix form, with two equations (from the con-

servation of IA and IB) and two unknowns (w+2 and w−

1 ),

ξ+1 R1 + ξ−1 −(ξ+

2 + ξ−2 R2)

ζ1(β+1 R1 + β−

1 ) −ζ2(β+2 + β−

2 R2)

w−

1

w+2

=

0

ζ2 − ζ1

ws , (2.25)

where ξ, β and ζ are a function of the Mach number only,


ξ± = 1 ±1

M, (2.26)

β± =γ − 1

2(1 ± M) , (2.27)

ζ =(

1 +γ − 1

2M2)−1

. (2.28)

Eq. 2.25 is frequency-dependent through coefficients R1 and R2, but the nozzle

is still supposed to be compact. The complete set of equations is solved using the

fast Fourier transform of the entropy wave signal and solving the system for each

individual frequency.

The pressure signal is recomposed by adding the upstream and downstream

propagating acoustic waves, with a phase shift according to the microphone location.

2.4.1 Results

Results obtained using this method are shown in Fig. 2.8 compared to the ex-

perimental data. It can be seen that both the shape and the amplitude of the signal

do not agree.

Figure 2.8: Pressure signal obtained with the analytical method at the outlet (mi-crophone located 1150mm downstream from the nozzle throat). Plane line: exper-imental data, dashed line: Analytical results multiplied by a factor 10


Another calculation was made considering non-reflecting boundary conditions.

It is plotted compared to the experimental pressure signal and to the experimental

temperature signal in Fig. 2.9.

Figure 2.9: Pressure signal obtained with the analytical method at the outlet usingnon-reflecting boundary conditions at the inlet and the outlet. Plane line: experi-mental pressure signal, dashed line: Analytical pressure signal multiplied by a factor10, dash-dot line: experimental temperature pulse at the nozzle inlet

Using non-reflecting boundary conditions, Eq. 2.25 can be solved analytically,

and as it is not a function of frequency, no Fourier transform of the heating pulse

is needed. In this way, the outgoing acoustic wave is written as

[w+2

ws1

]

=M2 − M1

1 + M2

M2

1 + [(γ − 1)/2]M2M1. (2.29)

The inversion of the pressure signal compared to the temperature one can be

explained using Eq. 2.29, as the value of M2 − M1 is negative in our case, and

therefore the ratio[

w+

2

ws1

]

will also be negative. At the same time, the values of M1

and M2 are very small, which gives a ratio of

[w+2

ws1

]

≈ −2.6 · 10−4 , (2.30)

and therefore, for a temperature pulse of T ′

T≈ 4.6 · 10−2 it gives p′ ≈ −8.7 · 10−1Pa,

as seen in Fig 2.9.


To analyze the reasons of this disagreement, simulations using SuperNozzle were

done. SuperNozzle is a solver in the frequency domain used to calculate the response

of a nozzle to acoustic and entropy perturbations. The equations solved are the

one-dimensional linearized Euler equations (Eqs. 2.5-2.7), and therefore only first

order effects are captured. These simulations use a smaller computational domain

than Fig. 2.7, reducing it to have only the convergent/divergent nozzle as shown

in Fig. 2.11, and considering non-reflecting boundary conditions at the inlet and at

the outlet. The indirect noise ratio[

w+

2

ws1

]

is calculated by introducing an entropy

wave of a given amplitude at a given frequency through the inlet; SuperNozzle then

calculates the outgoing acoustic and entropy waves. The ratio is later calculated by

dividing the involved waves at each frequency. The modulus of this ratio is plotted

in Fig. 2.10 as a function of the reduced frequency, showing that it is correctly

predicted by the analytical function for low frequencies, and at the same time that

the validity of this result is limited to very low frequencies. Considering that the

EWG experiment generates an entropy pulse of frequencies between Ω = 0 and

Ω = 0.08 it can be seen that the Marble and Candel method will underestimate the

pressure signal obtained at the nozzle outlet.

Figure 2.10: Indirect noise ratio. Dot: Marble and Candel solution. Solid line:SuperNozzle.

As seen in Chapter 1, the convergent-divergent nozzle generates little indirect

noise for zero frequency, but this noise increases rapidly with frequency. This oc-

curs because the acoustic waves generated in the convergent and in the divergent of

the nozzles (with opposite signs, due to the ∂u/∂x term of Eq. 2.6) do not cancel

directly for non-zero frequencies.

2.5. SEMI-ANALYTICAL METHOD USING SUPERNOZZLE 37

2.5 Semi-Analytical method using SuperNozzle

To solve the problem another method has been proposed: analytical methods

will be combined with SuperNozzle to solve the linearized equations through the

nozzle. Using SuperNozzle the response of the nozzle to a defined perturbation can

be calculated for each frequency, and in that way the transfer functions of the nozzle

will be calculated. This transfer functions are defined as the ratio between each of

the outgoing waves (w−

1 and w+2 ) and the incoming ones (w+

1 , ws1 and w−

2 ) as a

function of frequency. There are therefore six transfer functions, namely,

[w−

1

w+1

]

,[w−

1

ws1

]

,[w−

1

w−

2

]

, (2.31)

for the upstream propagating acoustic wave, and

[w+2

w+1

]

,[w+

2

ws1

]

,[w+

2

w−

2

]

, (2.32)

for the downstream propagating acoustic wave.

Figure 2.11: Sketch of the EWG experimental setup, with the domain for the ana-lytical calculation and for the SuperNozzle simulations

SuperNozzle will therefore calculate this transfer functions in the ’SuperNozzle

Domain’ plotted in Fig. 2.11 using non reflecting boundary conditions at the inlet

and at the outlet. The results will be used in the analytical domain, combined with

the modeling of inlet and outlet boundary conditions to simulate the experiment.


Three sets of SuperNozzle simulations will be done in which a single wave will be

introduced in each one to calculate the outcoming waves:

• Entropy: ws1

• Acoustic Inlet: w+1

• Acoustic Outlet: w−

2

For each set of simulations, a frequency sampling was done. In this way each

SuperNozzle simulation has a single unitary incoming wave at a single frequency

and the transfer functions of Eqs. 2.31-2.32 can be calculated.

Once the transfer functions of the nozzle have been calculated using SuperNozzle,

the whole experience can be analyzed using arithmetic equations. As the problem

is linearized, the global outgoing waves can be calculated as the sum of the three

contributions, namely

w−

1 =[w−

1

w+1

]

· w+1 +

[w−

1

ws1

]

· ws1 +

[w−

1

w−

2

]

· w−

2 , (2.33)

w+2 =

[w+2

w+1

]

· w+1 +

[w+2

ws1

]

· ws1 +

[w+2

w−

2

]

· w−

2 . (2.34)

The whole system is written using the reflection coefficient of Eq. 2.24 and

Eq. 2.23. It reads

R1 −1 0 0

0 0 −1 R2

0[

w+

2

w+

1

] [w+

2

w−

2

]

−1

−1[

w−

1

w+

1

] [w−

1

w−

2

]

0

w−

1

w+1

w−

2

w+2

=

0

0

−[

w+

2

ws1

]

−[

w−

1

ws1

]

ws . (2.35)

It should be noticed that Eq. 2.35 depends on the frequency, not only through

R1 and R2, but also through the transfer function terms.


Figure 2.12: Pressure signal solved with SuperNozzle and AVBP. Solid line: Exper-imental Data. Dashed line: Numerical Simulations. Dots: SuperNozzle calculation

Solving Eq. 2.35 as a function of the frequency gives the prediction of the pressure

signal plotted in Fig. 2.12.

It can be seen that the signal shape is well recovered, though the amplitude

of the wave is not exactly captured by the simulations. It is important to notice

that the AVBP simulations and the SuperNozzle calculation differ significantly. To

study the reason of this disagreement a comparison between AVBP simulations and

SuperNozzle has been done with non reflecting boundary conditions at the outlet

to isolate different physical phenomena. The results plotted in Fig. 2.13 show that

the SuperNozzle simulation does not capture all the physics of the EWG.

An order of magnitude analysis can be made to estimate the pressure fluctuation

generated by the entropy wave for the non reflecting boundary conditions case of

Fig. 2.13. The ratio[

w+

2

ws1

]

can be estimated from Fig. 2.10 as ≈ 4 · 10−3, and using

this value an estimate of p′ can be obtained as,

[w+2

ws1

]

≈2

(p′

γP2

)

(T ′

T1

) ⇒ p′ ≈1

2γP2

[w+2

ws1

]

·T ′

T1, (2.36)

giving p′ ≈ 13Pa, far from the 75Pa of the numerical simulation of Fig. 2.13. This

suggests that there are other physical phenomena which have to be taken into ac-

count.


Figure 2.13: Pressure signal at the outlet solved with SuperNozzle and AVBP withnon-reflecting boundary conditions at the outlet. Dashed line: Numerical Simula-tions with non-reflecting boundary conditions at the settling chamber inlet. Dots:SuperNozzle calculation using fully reflecting boundary conditions at the heatingtube inlet.

2.5.1 Study of the acoustic waves generated by the heating device

As explained in section 2.4, it has been considered that only entropy waves

generate noise in the analysis of the experiment. It is known that the heating device

generates, at the same time, small acoustic perturbations which are negligible in

comparison.

Figure 2.14: Waves definition in the heating device

An analysis will be done to see if those small acoustic waves generated by the

heating device should or should not be taken into account considering a first order

of magnitude estimate of the noise generated at the outlet. These acoustic waves

can be calculated with first order models as done by Leyko et al. [6]. Writing

the conservation equations (Eqs. 2.18-2.20) for a non isentropic flow through the


heating device (Fig. 2.14) with a source term,

(1 +1

M1)w+

0 + (1 −1

M1)w−

0 − 2ws0 = (1 +

1

M1)w+

1 + (1−1

M1)w−

1 − 2ws1 , (2.37)

[

(1+M1)w+0 +(1−M1)w

−

0 +2(ws

0 + q′)

γ − 1

]

=[

(1+M1)w+1 +(1−M1)w

−

1 +2ws

1

γ − 1

]

,

(2.38)

ws0 + q′ = ws

1 , (2.39)

with q′ a non-dimensional form of the heat fluctuation, and considering no inlet

waves, the acoustic and entropy waves generated by the device can be obtained,

giving,

ws1 = q′ , (2.40)

w+1 =

M

M + 1q′ , (2.41)

w−

0 =M

1 − Mq′ . (2.42)

Knowing that the inlet Mach number is small, one is tempted to neglect the

acoustic waves compared to the entropy waves, as the ratio between them is very

small (approximately 0.035). Doing an order of magnitude analysis the ratio be-

tween indirect and direct noise in the EWG can be estimated. It will be considered

that the nozzle transfer function has a constant value (which can be estimated from

Fig 2.10 and Fig. 2.15), as it is a first order of magnitude comparison.

The outgoing acoustic wave generated by the entropy wave is written as

w+2 )s =

[w+2

ws1

]

ws1 =

[w+2

ws1

]

q′ , (2.43)

and the one generated by the inlet acoustic wave is

w+2 )+ =

[w+2

w+1

]

w+1 =

[w+2

w+1

]

q′M

M + 1, (2.44)


Figure 2.15: Direct noise ratio. Dot: Analytic solution. Solid line: SuperNozzle.

which gives

η =w+

2 )s

w+2 )+

=M + 1

M

[w+

2

ws1

]

[w+

2

w+

1

] . (2.45)

In our case η ≈ 0.3, which shows that the effects of the acoustic wave are signif-

icant, due to the large value of the ratio between the transfer functions.

The same question should be posed for the supersonic case. Leyko et al. [5]

stated that the acoustic waves generated by the heating device were negligible in

the supersonic case, and did the analysis considering only indirect combustion noise,

obtaining acceptable results. The same analysis will be therefore carried out, con-

sidering the compact nozzle transfer functions for the choked case,

w+2

w+1

[AA] =1 + [(γ − 1)/2]M2

1 + [(γ − 1)/2]M1, (2.46)

for the direct combustion noise ratio and

w+2

ws1

[SA] =(M2 − M1)/2

1 + [(γ − 1)/2]M1. (2.47)


for the indirect noise.

The ratio of Eq. 2.45 can be calculated using the physical parameters of Ta-

ble 2.3 shown in Fig. 2.16.

Figure 2.16: Sketch of the EWG in the supersonic case

M1 M2 M3 M4

0.0365 1.34 0.7664 0.0225

Table 2.3: Physical parameters of the supersonic case

The direct combustion noise ratio calculated with Eq. 2.46 isw+

2

w+

1

[AA] ≈ 1.255

and the indirect noise ratio is thereforew+

2

ws1

[SA] ≈ 0.637 using Eq.2.47. The ratio

obtained for the supersonic case is η ≈ 15, showing that the indirect combustion

noise is much higher than the direct noise as it was supposed by Leyko et al. [5].

To take into account the acoustic waves generated in the heating device in the

subsonic case, an extra term has been added to Eqs. 2.33-2.34, giving

w−

1 =[w−

1

w+1

]

· (w+1 + w+

h ) +[w−

1

ws1

]

· ws1 +

[w−

1

w−

2

]

· w−

2 , (2.48)

w+2 =

[w+2

w+1

]

· (w+1 + w+

h ) +[w+

2

ws1

]

· ws1 +

[w+2

w−

2

]

· w−

2 , (2.49)

where w+h is the acoustic wave generated by the heating device entering the nozzle,

calculated as a combination of w+0 and the reflected part of w−

0 , namely,

w+h = w+

0 + R1 · w−

0 =

(M

M + 1+ R1

M

1 − M

)

ws . (2.50)


The matricial system of Eq. 2.35 can be rewritten as

R1 −1 0 0

0 0 −1 R2

0[

w+

2

w+

1

] [w+

2

w−

2

]

−1

−1[

w−

1

w+

1

] [w−

1

w−

2

]

0

w−

1

w+1

w−

2

w+2

=

0

0

−[

w+

2

ws1

]

−[

w−

1

ws1

]

ws +

0

0

−[

w+

2

w+

1

]

−[

w−

1

w+

1

]

w+h .

(2.51)

where the only difference is the extra source term of w+h . The equation is solved

like in section 2.4

2.5.2 Study of the inlet boundary condition

Another numerical simulation was made to study the influence of the inlet

boundary condition of the numerical simulation, imposed at the heating device

tube inlet as seen in Fig. 2.11. The objective was to validate the hypothesis made

to obtain Eq. 2.24, by which it was stated that the settling chamber was large

in comparison to the nozzle inlet and therefore the reflection coefficient could be

imposed as Rin = −1. A numerical simulation was made imposing fully reflecting

boundary conditions at the inlet of the settling chamber. If the hypothesis is reason-

able, this change should not affect the results obtained when using fully reflective

boundary conditions. As it can be seen in Fig. 2.17, the results of both simulations

differ significantly. This means that the boundary condition at the inlet of the noz-

zle should be revised to take into account the settling chamber.

Leyko et al. [5] showed that, in the supersonic case, the reflection coefficient at

the inlet had little impact in the solution at the outlet. For the subsonic case it

seems that the effect is significant and the settling chamber has to be considered.

This occurs because the upstream propagating acoustic waves in the inlet section

are of higher importance in the subsonic case than in the supersonic, for two main

reasons,

• For the subsonic case, the acoustic wave w−

0 generated by the heating device

is significant, while the one generated in the supersonic case is negligible.


Figure 2.17: Pressure signal solved with AVBP with non-reflecting boundary con-ditions at the outlet. Solid line: Fully reflective boundary conditions at the settlingchamber inlet. Dashed line: Non-reflecting boundary conditions at the settlingchamber inlet.

• In the supersonic case, waves reflected at the outlet cannot propagate up-

stream due to the supersonic region, and therefore have no effect in the inlet.

Instead, in the subsonic case, these waves propagate and are reflected at the

inlet boundary condition.

For this reasons the modelization of the reflection coefficient R1 has a stronger

effect in the subsonic case than in the supersonic. The inlet reflection coefficient R1

has to be modified to take into account the settling chamber. Using Fig. 2.18, the

conservation equations through a section change can be written as

Ssc · u′

sc = S0 · u′

0 , (2.52)

p′sc = p′sc (2.53)

using the low Mach number hypothesis. In this way, two relations can be written

between the waves at the settling chamber and at the heating device tube, namely,

w+sc − w−

sc = Γ (w+0 − w−

0 ) , (2.54)

w+sc + w−

sc = w+0 + w−

0 , (2.55)


Figure 2.18: Sketch of the settling chamber and the heating device tube.

with Γ = Ssc

S0the ratio between sections. Rin is defined as the relation between the

downstream propagating wave and the upstream propagating one,

Rin =w+

0

w−

0

, (2.56)

and using Eqs. 2.54-2.55 the reflection coefficient is written as,

Rin =(1 − Γ ) + Rsc(1 + Γ )

(1 + Γ ) + Rsc(1 − Γ ), (2.57)

where Rsc is the reflection coefficient at the inlet of the settling chamber, which is

calculated with u′ = 0 (no velocity fluctuations), to give

Rsc = exp(−2πfi(2L0)/c0) . (2.58)

Results obtained for the non-reflecting boundary conditions case, plotted in

Fig. 2.19, show the good agreement between both curves.

2.5.3 Results

Finally, the EWG experiment is calculated using SuperNozzle, taking into ac-

count the entropy and the acoustic wave generated by the heating device, the inlet

boundary condition extended to the settling chamber (Eq. 2.57) and the outlet


Figure 2.19: Pressure signal solved with AVBP and SuperNozzle using both theentropy and the acoustic wave generated by the heating device imposing boundaryconditions at the settling chamber inlet. Dashed line: Numerical Simulations. Dots:SuperNozzle calculation

boundary condition of Eq. 2.23 for the outlet. Results are shown in Fig. 2.20 com-

pared to the experimental data from the DLR and to numerical simulations.

Figure 2.20: Pressure signal solved with AVBP and SuperNozzle using both theentropy and the acoustic wave generated by the heating device. Dashed line: Nu-merical Simulations. Dots: SuperNozzle calculation. Solid line: Experimental data

It can be seen that there is a much better agreement between the numerical

simulations and the semi-analytical method.


2.6 Fully analytical double nozzle analysis

Another way of solving the problem is to calculate the convergent part of the

nozzle and the divergent part separately as shown in Fig 2.21.

Figure 2.21: Waves considering two nozzles

When doing so, the conservation of IA and IB of Eqs. 2.15-2.16 between the

inlet and the nozzle throat, and then from the throat to the outlet gives a set of

equations which can be reduced to the one shown in Eq. 2.25. This occurs because,

for zero frequency, solving Eq. 2.25 gives the correct solution (with the compact

nozzle hypothesis), and therefore solving it separately should give the same answer.

The key is therefore to introduce the frequency dependency in the equations in a

way in which, for zero frequency, the previous result is obtained. This can be done

by considering a finite non-zero distance between the convergent and the divergent.

The phase dependency is therefore introduced through the phase-shift in between

the nozzle parts as shown in Fig. 2.22

Figure 2.22: Waves in the two nozzles when considering a phase shift

Φ = exp(−ı2πfL/up) , (2.59)

where up is the propagating speed of the considered wave. It can be seen that the

analytical solution of section 2.4 will be obtained again when using f = 0Hz, as the

phase-shift will be zero.

The matricial system so solve can be obtained considering the compact nozzle

hypothesis in each part of the nozzle, and using Eqs. 2.18-2.20, it reads,

2.6. FULLY ANALYTICAL DOUBLE NOZZLE ANALYSIS 49

ξ+1 R1 + ξ−1 −ξ+

n −ξ−n 0

ζ1(β+1 R1 + β−

1 ) −ζnβ+n −ζnβ−

n 0

0 ξ+n Φ+ ξ−n Φ− −(ξ+

2 + ξ−2 R2)

0 ζnβ+n Φ+ ζnβ−

n Φ− −ζ2(β+2 + β−

2 R2)

w−

1

w+n

w−n

w+2

=

0

ζn − ζ1

0

−(ζn − ζ2)Φs

ws .

(2.60)

This system is solved as in Section 2.4.

To calculate the value of Φ using Eq. 2.59, a characteristic nozzle length L,

and a characteristic propagating speed up have to be defined. The nozzle length is

taken as characteristic length, and for the propagating speed, the inlet mean Mach

number and mean sound speed is used. In this way, the phase shift is given as a

function of the reduced frequency Ω, defined in Eq. 2.8,

Φ+ = exp(−ı2πΩ

M1 + 1) , (2.61)

Φ− = exp(−ı2πΩ

M1 − 1) , (2.62)

Φs = exp(−ı2πΩ

M1) , (2.63)

To analyze the ability of this method to predict entropy noise correctly, a

simple test was carried out: the noise predicted by SupperNozzle was compared to

the prediction of this analytic double nozzle method. Results illustrated in Fig. 2.23

show that the results do not agree with the SuperNozzle simulations.

Moreover, a test was be done to analyze the sensibility of the solution to the

characteristic length and speed choice. It was seen that the solution varies consider-

ably when changing the parameters. The best fit of the solution for low frequencies

(shown in Fig. 2.24) is obtained when considering the throat sound speed and Mach

number and the convergent length as characteristic parameters for the phase shift.

In any case, the double nozzle system is not able to capture the whole physical

phenomena of the non compact nozzle.


Figure 2.23: Indirect noise ratio. Comparison between SuperNozzle (Solid Line) andthe fully analytical method using two nozzles with the characteristic parameters: Lthe nozzle length, and c and M evaluated at the inlet (Dashed Line)

Figure 2.24: Indirect noise ratio. Comparison between SuperNozzle (Solid Line) andthe fully analytical method using two nozzles with the characteristic parameters: Lthe convergent length, and c and M evaluated at the nozzle throat (Dashed Line)

2.6.1 Double nozzle transfer function method

A method described by Bake et al. [1] has been also tested. The method calculates

three quasi transfer functions using the compact nozzle hypothesis and Eq. 2.18-2.20,

• For a sound pressure wave caused by entropy fluctuations:

2.6. FULLY ANALYTICAL DOUBLE NOZZLE ANALYSIS 51

p′2+ = S(1,2)q′

s1 =M2 − M1

1 + M2

12M2

1 + [(γ − 1)/2]M2M1a2

1Aq′s1 . (2.64)

• The transmission of entropy fluctuations:

q′s2q′s1

= Q(1,2) =

(

1 + [(γ − 1)/2]M21

1 + [(γ − 1)/2]M22

)( 1

γ−1)

. (2.65)

• For the transmission of an acoustic pressure wave:

p′2+p′1+

= T(1,2) =2M2

1 + M2

1 + M1

M1 + M2

1 + [(γ − 1)/2]M22

1 + [(γ − 1)/2]M2M1A . (2.66)

with

A =

(

1 + [(γ − 1)/2]M21

1 + [(γ − 1)/2]M22

)( γγ−1

)

. (2.67)

and the waves defined as,

2p′+γp

= w+ , (2.68)

q′sρ

= ws . (2.69)

The demonstration of these equations is shown in Appendix C.

As these equations are equivalent to the matrix system 2.25, the solution ob-

tained will be the same. The difference here comes from the way of solving the

equations: Bake et al. [1] divides the nozzle in the convergent and the divergent

part, solving Eqs. 2.64-2.66 for the convergent and for the divergent independently.

The three transfer functions are calculated for the convergent (S(1,n), Q(1,n) and

T(1,n)) and for the divergent (S(n,2), Q(n,2) and T(n,2)) separately. The total acoustic


wave at the nozzle exit is the sum of the one generated in the convergent multiplied

by the transmission factor of the divergent, and the one generated in the divergent,

namely,

p′2+ =[S(1,n) · T(n,2) + Q(1,n) · S(n,2)

]q′s1 . (2.70)

This method does not take into account the upstream propagating acoustic wave

generated by the divergent, which can eventually generate noise when interacting

with the convergent (this upstream propagating acoustic wave exists even when non

reflecting boundary conditions are considered). In fact, it can be proven that this

method of solving the double nozzle is the same as to solve Eq. 2.60 but considering

no phase-shift and that w−n does not affect the convergent (therefore replacing the

third term of the two first equations of 2.60 by zero).

The problem can be easily solved with non-reflecting boundary conditions as

there is no frequency dependency. The value of the direct noise ratio[

w+

2

ws1

]

is

compared with SuperNozzle results in Fig. 2.25

Figure 2.25: Indirect noise ratio. Comparison between SuperNozzle (Solid Line)and the Double nozzle transfer function method.

The value obtained is too high compared with the SuperNozzle results. This

means that the method is not able to predict the noise level at the outlet.

2.7. CONCLUSION 53

2.7 Conclusion

The Entropy Wave Generator experiment has been simulated using numerical

codes and analytical methods for the subsonic case. Leyko et al. [6] showed that

the compact nozzle hypothesis could be used to analyze the EWG experiment in

the supersonic case. For the subsonic case this hypothesis cannot be used, as the

outgoing acoustic wave increases rapidly when considering non-zero frequency. The

analysis has been therefore done using semi-analytical methods in which the nozzle

transfer function was calculated numerically solving the Euler first order pertur-

bation equations and was later implemented into an analytical model of the whole

experience.

The analysis of this experiment showed the importance of the reflection coef-

ficient at the inlet of the computational domain. This importance is due to the

fact that the subsonic nozzle generates strong upstream propagating waves, which

reflect in the inlet. An estimate of the generated acoustic waves at the outlet con-

cluded that the direct noise generated in the experiment was significant, while it

was negligible in the supersonic case. A method was therefore implemented to take

into account these acoustic waves, as well as the inlet reflection coefficient in the

semi-analytical method to prove that first order models are able to predict combus-

tion noise, even if there is a combination of both direct and indirect noise.

A method was already proposed by Leyko et al. [5] to reduce the impact of

the reflection coefficients in the solution by reducing the amplitude of the pulsed

wave. This method combined with a longer inlet tube may permit to separate the

entropy generated noise from the direct noise generated by the heating device: As

the acoustic waves generated by the heating device propagate much faster than the

entropy waves they would have generated their associated pressure pulse before the

entropy wave enters the nozzle inlet. An inlet tube of about 2m would be needed

to separate both mechanisms if a pulse of amplitude 5ms is used. Another way is

to change the nozzle in a way in which the indirect to direct noise ratio increases.

This can be done by having a simple convergent nozzle, with the same inlet Mach

number as in the EWG experience and an exit Mach number of 0.7. In this way

the nozzle would have η ≈ 6.5, which reduces the importance of the direct noise in

the results.

Finally other analytic methods have been tested to take into account the fre-

quency dependency of the transfer functions. These methods are based on the

division of the nozzle in the convergent and the divergent parts, each of which can

be considered compact. The dependency of the solution with the parameters chosen

difficults the application of this method.

Another possibility is to solve the original conservation equations (Eqs 2.5-2.7)

in a non-dimensional form, for small non-zero values of Ω. The invariants IA, IB


and IC would be written in the form,

I = I0 + ΩI1 + Ω2I2 + · · · + ΩnIn . (2.71)

where the equations for the zero order term are the conservation equations used

in Section 2.4 with the compact nozzle hypothesis. Equations for the following

terms should be written as a function of the mean flow variables using Eqs 2.5-2.7

which should be, in general, integrated numerically knowing the exact profile of the

nozzle.

Chapter 3

Simulation of the waves

transmission through a blade row

3.1 Introduction

As shown in Chapters 1 and 2, the acceleration of the mean flow is of great

relevance when studying the total noise generated by a combustion chamber. In

Chapter 1 it was seen that indirect combustion noise has to be considered to predict

correctly the total combustion noise and in Chapter 2 the Entropy Wave Generator

experiment performed by Bake et al. [2] was studied analytically and numerically,

verifying that first order theories predict correctly the generation and transmission

of noise in a nozzle. This is why, to study combustion noise, the propagation of

acoustic and entropy waves through the turbine stages should be considered. The

mean flow through a turbine stage is not only strongly accelerated, but also devi-

ated. This two dimensional configuration of the flow was studied analytically by

Cumpsty and Marble [3], developing a first order analytical method to calculate the

outgoing waves as a function of the incoming ones and the mean flow variables. As

the model is two dimensional, waves introduced in the blade row do not propagate

necessarily in the mean flow direction, having therefore circumferential components.

Leyko [4] compared this analytical method with numerical simulations, validating

the compact nozzle hypothesis and extending the results to higher frequencies. This

study was done for the entropy wave, the downstream propagating acoustic wave,

and the vorticity wave when propagating with no circumferential component. The

objective of this chapter is therefore to analyse the case of the upstream propagating

acoustic wave.

55

56 CHAPTER 3. SIMULATION OF THE WAVES TRANSMISSION THROUGH A BLADE ROW

3.2 Two dimensional modelization of the wave propagation

As done for the one dimensional nozzle, an analytical approach can be used

to calculate the outcoming waves as a function of the incoming ones. This theory

developed by Cumpsty and Marble [3] is based on the same compact nozzle hy-

pothesis of Marble and Candel [7], adding a flow deviation component. The flow

is assumed to be uniform at the inlet and the outlet and the blade row spacing is

small compared to the chord.

As the mean flow is two dimensional, there are two velocity components that can

be perturbed independently, introducing a fourth wave in the analysis: the vorticity

wave. It can be also seen that waves can now propagate in the circumferential

direction as well as in the mean flow direction. Cumpsty and Marble [3] analyzed

also the propagation direction, as it can induce attenuated solutions as well as waves.

3.2.1 Conservation equations

As done for the compact nozzle, the compact blade row can be analyzed by

considering inlet-outlet conservation relations of mass, enthalpy and entropy. The

four primitive variables used will be the pressure p, the velocity v, the entropy s

and the flow deviation θ. The conservation equations read,

IA =p′

γp+

1

M

u′

c−

s′

Cp− θ′ tan θ (3.1)

for the mass flow conservation,

IB =2

1 + γ−12 M2

[

Mu′

c+

p′

γp+

1

γ − 1

s′

Cp

]

(3.2)

for the total temperature conservation,

IC =s′

Cp, (3.3)

for the entropy conservation and a fourth equation relating the perturbation of the

flow deviation at the inlet and the outlet. This relation can be obtained using the

Kutta condition at the trailing edge of the blade row (θ′2 = 0), or a more general

relation proposed by Cumpsty and Marble,

3.2. TWO DIMENSIONAL MODELIZATION OF THE WAVE PROPAGATION 57

θ′2 = βθ′1 , (3.4)

This last relation is more general and will be therefore kept. Using relations 3.1-

3.4 between the inlet and the outlet section the primed variables can be calculated

using a matrix system. It reads,

[E1] ·

s′/Cp

u′/c

p′/γp

θ′

1

= [E2] ·

s′/Cp

u′/c

p′/γp

θ′

2

, (3.5)

with [E] the coefficients matrix,

[E1] =

1 0 0 0

−1 1M1

1 − tan θ1

µ11

γ−1 µ1M1 µ1 0

0 0 0 β

, (3.6)

and

[E2] =

1 0 0 0

−1 1M2

1 − tan θ2

µ21

γ−1 µ2M2 µ2 0

0 0 0 1

, (3.7)

where µ = 1/[1 + (γ − 1)M2/2].


3.2.2 Waves calculation

To solve the problem, Eq. 3.5 has to be written as a function of the upstream

and downstream propagating waves so that four of them can be imposed to close

the problem. The primitive variables can be expressed as a function of the waves

through a matrix relation,

s′/Cp

u′/c

p′/γp

θ′

= [M] ·

ws

wv

w+

w−

. (3.8)

Matrix [M] is calculated column by column, analyzing the influence of each wave

in the primitive variables. The linearized equations

D

Dt(ρ′) = −ρ

(∂u′

x

∂x+

∂u′y

∂y

)

, (3.9)

D

Dt(u′

x) = −1

ρ

∂p′

∂x, (3.10)

D

Dt(u′

y) = −1

ρ

∂p′

∂y, (3.11)

D

Dt(s′) = 0 , (3.12)

are used to obtain the dispersion equation of each wave. The dispersion equation

relates the angular frequency of the wave (ω) with the wave vector (−→k ), which de-

fines the propagation of a wave φ (shown in Fig. 3.1), therefore written as

wφ = Aφ exp[−ı(ωt −−→kφ · −→x )] . (3.13)

The waves do not propagate necessarily in the same direction as the mean flow,

and a specific study of the propagating characteristics of the wave should be done.

The wave vector can be expressed as the combination of the axial (kx,φ) and

circumferential (ky,φ) components, namely


Figure 3.1: Schema of the flow through the blade row upstream and downstream.

−→kφ · −→x = kx,φx + ky,φy , (3.14)

with

kx,φ = kφ cos νφ and ky,φ = kφ sin νφ , (3.15)

where ν is the angle of the propagating wave, and k the modulus of the wave vector.

The dispersion equation which will be obtained relates the two components of the

wave vector with the frequency. The frequency and the circumferential component

of the wave vector ky,φ are imposed by the combustion chamber which generates

the incoming waves. Using the dispersion equation for each type wave will give the

axial component of the wave vector kx,φ.

To simplify the relations the non-dimensional modulus of the wave vector will

be defined as Kφ = kφc/ω, as well as the azimuthal(Kx,φ = kx,φc/ω) and the

circumferential (Ky,φ = ky,φc/ω) components.

Acoustic waves

The definition of the acoustic waves is slightly different than in Chapter 2,

w± =

(p′

γp

)

±

, (3.16)

is used instead.


As the acoustic perturbations are independent from entropy ones, it can be

stated that s′ = 0, and therefore using Eq. 3.12 the density perturbation can be

written as ρ′ = p′/c2. Combining Eqs 3.9-3.11 an equation can be obtained for the

pressure perturbation, it reads,

(D

Dt

)2

(p′) − c2

(∂2

∂x2+

∂2

∂x2

)

(p′) = 0 . (3.17)

The dispersion relation can be obtained using Eq. 3.17, and writing the waves

w± as in Eq. 3.13, namely

(1 − K±M cos(ν± − θ))2 − K2± = 0 . (3.18)

The dispersion equation can be re-written as a function of the circumferential

and axial components of the wave vector, namely

(1 − Kx,±M cos θ − Ky,±M sin θ)2 − K2x,± − K2

y,± = 0 , (3.19)

to obtain the value of Kx,± imposing Ky,±.

Using Eq. 3.10 and 3.11 and considering u′x and u′

y written as a wave (Eq. 3.13)

the velocity perturbation can be written as a function of the wave,

u′x

c= −

K± cos ν±1 − K±M cos(ν± − θ)

, (3.20)

u′y

c= −

K± sin ν±1 − K±M cos(ν± − θ)

. (3.21)

The velocity modulus and direction perturbations, u′ and θ′, can be written as

a function of the x and y components of the velocity perturbation, namely,

u′

c=

u′x

ccos θ +

u′y

csin θ , (3.22)

θ′ =

(

−u′

x

csin θ +

u′y

ccos θ

)

/M . (3.23)


The acoustic waves generate therefore a fluctuation in the primitive variables

following the relation,

s′/Cp

u′/c

p′/γp

θ′

±

=

0

K± cos(ν± − θ)/(1 − K±M cos(ν± − θ))

1

K± sin(ν± − θ)/(M (1 − K±M cos(ν± − θ)))

· w± . (3.24)

Entropy wave

Defining the entropy wave as ws = s′/Cp, the dispersion relation can be obtained

using Eq. 3.12, namely,

KsM cos(νs − θ) − 1 = 0 . (3.25)

The entropy wave generates, by definition, only density fluctuations, giving

therefore

s′/Cp

u′/c

p′/γp

θ′

s

=

1

0

0

0

· ws . (3.26)

Vorticity wave

Due to the two dimensional configuration studied the vorticity wave has to be consid-

ered also. It is defined as ξ′ = ∂u′y/∂x−∂u′

x/∂y. Operating with Eq. 3.10 and 3.11,

the conservation equation for the vorticity,

D

Dt(ξ′) = 0 , (3.27)


can be obtained. This equation shows that the vorticity wave is convected with the

mean flow. The dispersion relation has the same form as for the entropy wave, it

reads

KvM cos(νv − θ)− 1 = 0 . (3.28)

The perturbation of the primitive variables created by this wave can be deduced

from Eq. 3.10 and 3.11,

u′x

c= −ı

ξ′

u

sin νv

Kv, (3.29)

u′y

c= ı

ξ′

u

cos νv

Kv, (3.30)

and using Eqs. 3.22 and 3.23 u′ and θ′ can be calculated to finally obtain the fluc-

tuation of the primitive variables

s′/Cp

u′/c

p′/γp

θ′

v

=

0

−ı sin(νv − θ)/Kv

0

ı cos(νv − θ)/(KvM)

· wv . (3.31)

The matrix [M] can be therefore written as

[M] =

1 0 0 0

0 −ısin(νv − θ)

Kv

K+ cos(ν+ − θ)

(1 − K+M cos(ν+ − θ))

K− cos(ν− − θ)

(1 − K−M cos(ν− − θ))

0 0 1 1

0 ıcos(νv − θ)

KvM

K+ sin(ν+ − θ)

M(1 − K+M cos(ν+ − θ))

K− sin(ν− − θ)

M(1 − K−M cos(ν− − θ))

.

(3.32)

The matrix system is now written


[E1] · [M1]︸︷︷︸

[B]1

·

ws

wv

w+

w−

1

= [E2] · [M2]︸︷︷︸

[B]2

·

ws

wv

w+

w−

2

. (3.33)

To solve for the unknown waves, the system should be grouped, leaving a vector

of outgoing waves at one side and of incoming waves at the other. In the subsonic

case considered here, the upstream propagating acoustic wave in the inlet section,

w−

1 is unknown and should be grouped with the downstream propagating waves at

the outlet. The matrix system should be written as

[Aout] ·

ws2

wv2

w+2

w−

1

= [Ain] ·

ws1

wv1

w+1

w−

2

. (3.34)

Matrix [Ain] and [Aout] are obtained by permuting the last columns of [B1] and

[B2] changing its sign to obtain the relation of Eq. 3.34.

3.3 Numerical Simulation

A two dimensional blade row is considered to perform the numerical simulations.

As it can be seen in Fig.3.2, the mesh is extended in the inlet and the outlet regions

to perform the post-treatment. Only one blade is considered and periodic boundary

conditions are imposed. In order to solve correctly all the frequencies involved, the

physical time is of 66 periods of the smallest frequency, and the sampling period of

10−5s to have at least 20 samples per wavelength.

Figure 3.2: Schema of the computational domain with the blade simulated, the inletand outlet


The flow studied here is subsonic, with M1 = 0.12, θ1 = 0 and M2 = 0.66,

θ2 = 76.

The numerical simulation was performed using AVBP code developed at CER-

FACS, using TTGC4A discretization. The simulation is not done as in Chapter 1,

where Euler equations were used because they will not solve the mixing and the real

mean flow characteristics. The idea is to mimic the real blade mean flow configura-

tion and at the same time permit the unsteady simulation, but not to solve the real

turbulence of the turbine, or the boundary layer of the blade. A full LES is unfea-

sible due to the high meshing requirements, the small time step, three dimensional

configuration required and memory needed. This is why a two dimensional simu-

lation is performed using turbulent modeling. The modeling used is not intended

to solve the boundary layer or the vortex shedding at the trailing edge, but it is

included to add viscosity in order to stabilize the simulation. The mesh used is of

about 115000 triangles in total, making it impossible to solve the LES equations,

but enough to propagate the entropy wave without significant dispersion. Second

and fourth order artificial viscosity is used (0.1 and 0.001) to perform the simula-

tion and a turbulent Prandtl number of Prt = 100 large to reduce the heat diffusion

(mostly in the post-processing regions).

At the outlet boundary condition, a single normal upstream propagating acoustic

wave was introduced. This wave is a periodic function, built as a sum of sinusoidal

waves with frequencies from 100Hz to 5000Hz, namely,

f(t) =N∑

k=1

sin(2πf0kt) (3.35)

where in our case N = 50 and f0 = 100Hz. The signal is plotted in Fig. 3.3 as a

function of t/τ0, where τ0 = 1/f0.

3.3.1 Post-treatment of the solution

To obtain the waves a post-treatment must be done. In Chapter 1 the signal

was obtained from one point in the outlet region, performing a Fast Fourier Trans-

form (FFT). In this case a FFT cannot be performed, as the wave introduced is

not a white noise, but a combination of discrete frequencies. For this reason the

Fourier-Transform has to be done at each frequency. Moreover, an averaging of the

waves is needed to smooth the solution, as the turbulent field and mixing perturbs

the wave.


Figure 3.3: Wave perturbation imposed at the outlet of the computational domain

The post-treatment is performed in 5 steps:

1. Steady state calculation: The mean flow variables are calculated in the

domain.

2. Waves calculation: Primitive variables (s′/Cp, u′/c, p′/γp, θ′) are calcu-

lated for each point as a function of time. Using matrix [M] (Eq. 3.32) the

four waves ws, wv, w+, w− are calculated as a function of x, y and t.

3. Integration along the transversal direction: The waves are averaged

along the transversal direction by integration,

wφx(x, t) =

1

Ly

∫ Ly

0wφ(x, y, t)dy (3.36)

4. Fourier-Transform: The Fourier-Transform is performed before the inte-

gration along the propagating direction,

wφx(x, k) =

∣∣∣∣

1

tf

∫ tf

0wφ

x(x, t) exp(2πıf0kt)dt

∣∣∣∣

(3.37)


5. Integration along the propagating direction: Once the Fourier-Transform

has been done, the integration of the solution along the x axis can be done to

average the solution,

wφ(k) =

√

1

Lx

∫ Lx

0

[

wφx(x, k)

]2dx (3.38)

3.4 Results

Results are plotted in Fig. 3.4 (for the downstream propagating acoustic wave

at the outlet) and in Fig. 3.5 (for the upstream propagating acoustic wave at the

inlet), compared to the analytical method. The value of the frequency for which

the reduced frequency is Ω = 0.1 is also shown to delimitate the compact nozzle

hypothesis.

Figure 3.4: Download propagating acoustic wave at the outlet induced by an up-stream propagating acoustic wave at the outlet. Solid line: Analytical method.Dots: Numerical simulation

It can be seen that in both cases the wave is correctly predicted by the analyti-

cal method. The effects of non compact blade row show that for Ω = 0.1 the error

made is about ≈ 15%.

3.5. CONCLUSION 67

Figure 3.5: Upstream propagating acoustic wave at the inlet induced by an upstreampropagating acoustic wave at the outlet

3.5 Conclusion

Performing numerical simulations, the Cumpsty and Marble [3] analytical method

to calculate the wave propagation through a turbine stator has been validated for

the low frequency range. For higher frequencies the error made diverges, as the

compact blade row hypothesis is no longer validated. The error made when using

the theory for large frequencies is also estimated. Knowing that the combustion

noise is concentrated in a low frequency spectrum, the analytical model can be used

to create a first order method that calculates the acoustic waves at the outlet of a

whole aero-engine. The method, proposed by Cumpsty and Marble [3] and imple-

mented by Leyko [4] uses the matrices of Eq. 3.5 and 3.8 to solve several blade rows

of a turbine and therefore obtain the total acoustic power at the exit.

A lot of work has to be done to completely validate the analytical method

proposed. Firstly, as shown by Leyko [4], the entropy wave is perturbed when

propagated through a blade row. The effect of this perturbation in the following

stages should be analyzed to build a tool able to predict the combustion noise at

the outlet of the aero-engine. Rotating blades have to be analyzed numerically and

analytically. Actually, the method developed using Eq. 3.5 has to be changed when

considering moving blades: Eq. 3.2 has to be modified as the total temperature is no

longer conserved through a rotor. The equation has to be replaced with the rotalpy

equation: This modification is performed in Appendix B. The same numerical anal-

ysis as the one performed in this chapter should be performed using a stator and

a rotor. A third effect should be analyzed: the circumferential waves propagating

through the turbine. In this chapter only normal waves have been considered, and


the effect of the circumferential component of the wave has to be taken into account.

To do so, no periodic conditions can be assumed in the computational domain, and

therefore a two dimensional 360o computation should be performed, with the whole

set of blades. Perfectly non reflecting boundary conditions should be studied in the

case of oblique waves, as there might be an undesired reflection.

General conclusion

Combustion noise is, as explained in the introduction, difficult to measure. The

two main mechanisms of combustion noise generation (Direct and indirect, as ex-

plained in Chapter 1) have to be considered, and the noise at the outlet of an

engine is strongly influenced by the mean flow variables. Theoretical models and

first order propagation equations have been analyzed using both numerical methods

and comparisons to experimental data, showing that these models predict correctly

the propagation of waves through a nozzle (Chapters 1 and 2) in the supersonic

and subsonic case and through a static blade row (Chapter 3) when considering

the compact nozzle hypothesis, and studying the validity of this hypothesis and the

error made in the non compact case. Specific conclusions were discussed at the end

of each chapter, in sections 1.7, 2.7 and 3.5. A general overview of these conclusions

and a discussion about the continuation of this research project will be done in this

section.

The basic theory was shown in Chapter 1, where the importance of indirect

combustion noise (and therefore the wave propagation through the turbine) was

proven. It was also shown that, for small nonzero frequencies, the compact nozzle

hypothesis predicts the direct and indirect combustion noise with little error, though

a case was shown in which the validity domain was limited to very small frequen-

cies. This was the case of the Entropy Wave Generation, as it was seen in Chapter 2.

In Chapter 2 the Entropy Wave Generator (EWG) was studied. It was seen

that the compact nozzle hypothesis could not be used in this subsonic case. A

fully analytical double-nozzle method was proposed to predict the waves propaga-

tion through the nozzle for the non compact case, though it was proved that this

method depended too much on the parameters chosen, and therefore it could not

be used. Another method was proposed, though it was not implemented nor tested,

to solve the linear equations for non zero frequencies. This work can be continued

during the Ph.D. that follows. Using the SuperNozzle code, the linearized equations

were solved, showing that the linear model is still valid, and first order equations

apply and predict correctly the combustion noise in the EWG experience. The

analysis showed that, opposite to what happened in the supersonic case, the direct

69


combustion noise is not negligible and it should be considered in the study. The

boundary conditions of the problem were also reviewed: Leyko et al. [5] showed

that in the supersonic case the inlet boundary condition had a negligible effect in

the whole study; but in the subsonic case the waves leaving through the inlet of

the experimental device are significant, and therefore a detailed study of the inlet

boundary condition imposed had to be done.

The subsonic blade row of a turbine has been analyzed using numerical tools

in order to validate the analytical methods developed by Cumpsty and Marble [3].

It was seen that the numerical simulations converge well to the analytical solution

when the frequency tends to zero. The error made when considering non-compact

blade rows was also analyzed.

No 3D simulations were performed during the project, though at the beginning

it was considered. The lack of time, and mainly the interest in the physical phe-

nomena that could be analyzed using two dimensional simulations were the reasons

for not performing them. In any case, Leyko et al. [5] showed that, for the Entropy

Wave Generator experiment, three dimensional effects were negligible, and one di-

mensional theories could be used.

Future work

The results obtained during this work (and those from Leyko’s Ph.D, [4]) permit

the development of a linear matricial method to solve a complete turbine in order

to obtain the total acoustic power at the outlet of the global engine. This matricial

system, developed by Leyko during his Ph.D, takes the waves at the outlet of the

combustion chamber as an input, and solves the linearized propagation equations

using the Cumpsty and Marble [3] method with the compact blade row hypothesis.

The method has been already validated for a simple stage, but has to be compared

to numerical simulations in the case of multiple blade rows and of rotating blade

rows. In particular, Leyko showed that the entropy wave was strongly perturbed

at the blade row outlet due to the non-uniform mean flow in the blade spacing and

to the turbulence in the trailing edge. The coupling of blade rows can be prob-

lematic as the entropy wave is supposed to stay planar and unaltered through the

blade rows. Another validation should be performed regarding the circumferential

components of the inlet waves: At the outlet of the combustion chamber, the out-

going entropy and acoustic waves propagate both in the axial direction and in the

circumferential one. For the moment only longitudinal waves have been considered,

but the theoretical method has to be validated in the general case.

The rotating blade rows can be calculated analytically using the equations de-

veloped in Appendix B. A numerical simulation similar to the one performed in

3.5. CONCLUSION 71

Chapter 3 should be performed. To do so, a moving mesh has to be considered,

knowing that this induces dissipation of the acoustic and entropy waves.

During this project efforts have been concentrated to study the propagation of

the entropy and of the acoustic waves through the turbine, but little has been said

about the generation of the waves in the combustion chamber itself. The prediction

of these waves require LES simulations using combustion models. The precision of

these predictions are strongly dependent on the numerical scheme, the mesh refine-

ment and the combustion models used. It is for this reason that the work done

concentrates on the propagation of the waves, where simplified models obtained

permit the reduction of the numerical analysis, and therefore the computational

resources needed. Nevertheless, an analysis of the combustion chamber has to be

done in the future (during the Ph.D) to compare results with experimental data,

and to validate the combustion models involved in the LES.

This work will be continued during the Ph.D that follows in collaboration with

Snecma and CERFACS.

Bibliography

[1] F. Bake, N. Kings, A. Fischer, and Rohle I. Experimental investigation of the en-

tropy noise mechanism in aero-engines. International Journal of Aeroacoustics,

8(1-2):125–142, 2008. [cited at p. 50, 51]

[2] F. Bake, C. Richter, B. Muhlbauer, N. Kings, I.Rohle, F.Thiele, and B.Noll.

The entropy wave generator (ewg): a reference case on entropy noise. J. Sound

Vib. , pages 574–598, 2009. [cited at p. 6, 25, 55]

[3] N. A. Cumpsty and F. E. Marble. The interaction of entropy fluctuations with

turbine blade rows; a mechanism of turbojet engine noise. Proc. R. Soc. Lond.

A , 357:323–344, 1977. [cited at p. 6, 55, 56, 67, 70, 79, 80]

[4] M. Leyko. Mise en oeuvre et analyse de calculs aeroacoustiques de type SGE

pour la prevision du bruit de chambres de combustion aeronautiques. PhD thesis,

Institut National Polytechnique de Toulouse, 2010. [cited at p. 5, 6, 55, 67, 70]

[5] M. Leyko, F. Nicoud, S. Moreau, and T. Poinsot. Numerical and analytical

investigation of the indirect noise in a nozzle. In Proc. of the Summer Pro-

gram , pages 343–354, Center for Turbulence Research, NASA AMES, Stanford

University, USA, 2008. [cited at p. 6, 25, 27, 28, 42, 43, 44, 53, 70]

[6] M. Leyko, F. Nicoud, and T. Poinsot. Comparison of direct and indirect combus-

tion noise mechanisms in a model combustor. AIAA Journal, 47(11):2709–2716,

November 2009. [cited at p. 10, 12, 24, 40, 53]

[7] F. E. Marble and S. Candel. Acoustic disturbances from gas nonuniformities

convected through a nozzle. J. Sound Vib. , 55:225–243, 1977. [cited at p. 5, 9, 11,

23, 25, 29, 56]

[8] B. Muhlbauer, B. Noll, and M. Aigner. Numerical investigation of the funda-

mental mechanism for entropy noise generation in aero-engines. Acta Acustica

united with Acustica 95, pages 470–478, 2009. [cited at p. 27]

[9] T. Poinsot and S. Lele. Boundary conditions for direct simulations of compress-

ible viscous flows. J. Comput. Phys. , 101(1):104–129, 1992. [cited at p. 28]

73

Appendix A

Linearised Euler equations

In this appendix the linearized Euler equations used in Chapters 1 and 2 are

obtained for a one dimensional uniform flow.

A.1 Isentropic relation

The isentropic flow relation is written as,

[∂

∂t+ u

∂

∂x

]

s = 0 . (A.1)

Considering small perturbations of s, s = s + s′,

[∂

∂t+ u

∂

∂x

](s′

Cp

)

= 0 . (A.2)

A.2 Mass conservation

The mass conservation equation is written as

∂(ρA)

∂t+

∂(uρA)

∂x= 0 , (A.3)

75

76 APPENDIX A. LINEARISED EULER EQUATIONS

where ρ is the density of the fluid, A is the nozzle section, and u is the speed in the

x direction. Developing the equation, it can be written

1

ρ

[∂ρ

∂t+ u

∂ρ

∂x

]

+∂u

∂x= −u

1

A

∂A

∂x. (A.4)

Considering small perturbations of the variables,

ρ = ρ

(

1 +ρ′

ρ

)

, (A.5)

u = u

(

1 +u′

u

)

, (A.6)

the equation can be rewritten,

1

ρ

∂ρ

∂t+

1

1 +ρ′

ρ

∂

∂t

(ρ′

ρ

)

+1

ρu

(

1 +u′

u

)∂ρ

∂x+

1

1 +ρ′

ρ

(

1 +u′

u

)

u∂

∂x

(ρ′

ρ

)

+

+u∂

∂x

(u′

u

)

+

(

1 +u′

u

)∂u

∂x= −u

(

1 +u′

u

)1

A

∂A

∂x. (A.7)

As the mean flow is steady, the temporal derivatives of the mean variables are

equal to zero. It can also be seen that, as the mean flow variables follows Eq. A.3,

all the terms including a mean flow variable cancel out. Using 1/(1+ ǫ) ≈ 1− ǫ and

neglecting second order terms, Eq.A.8 can be simplified, namely,

∂

∂t

(ρ′

ρ

)

+ u∂

∂x

(ρ′

ρ

)

+ u∂

∂x

(u′

u

)

= 0 . (A.8)

Using the result shown in Eq. A.2, writing s′/Cp = p′/(γp − ρ′/ρ) the equation

can be written as a function of p′,

[∂

∂t+ u

∂

∂x

](p′

γp

)

+ u∂

∂x

(u′

u

)

= 0 . (A.9)

A.3. MOMENTUM 77

A.3 Momentum

The momentum conservation is written,

[∂

∂t+ u

∂

∂x

]

(u) +1

ρ

∂p

∂x= 0 , (A.10)

and considering small perturbations as done before, cancelling the temporal deriva-

tives of the mean flow and second order terms, the equation is,

[∂

∂t+ u

∂

∂x

](u′

u

)

+2u′

u

∂u

∂x+

1

ρu

[(p′

p

)

−(

ρ′

ρ

)]∂p

∂x+

p

ρu

∂

∂x

(p′

p

)

= 0 . (A.11)

Knowing that ρ′/ρ = p′/(γp − s′/Cp) and using Eq. A.10 to re-write the mean

pressure gradient, the final equation is obtained,

[∂

∂t+ u

∂

∂x

](u′

u

)

+c2

u

∂

∂x

(p′

γp

)

+

[

2u′

u− (γ − 1)

(p′

γp

)]∂u

∂x=

s′

Cp

∂u

∂x. (A.12)

Appendix B

Propagation equations through a

rotor

B.1 Introduction

As it was seen in Chapter 3, four equations have to be written to study wave

propagation through a turbine in the subsonic case: Mass, entropy and energy

conservation, and a fourth closing condition (Kutta condition). The first two are

the same for the stator and the rotor, as well as the Kutta condition, but the energy

equation must be calculated differently. Four non-dimensional variables will be used

to describe the waves propagating through the flow, namely,

p′

γp,

s′

Cp,

v′

aand θ′ . (B.1)

B.2 Stator

The perturbed equations through a stator

s′1Cp

=s′2Cp

, (B.2)

p′1γp1

+1

M1

v′1a1

− θ′1 tan θ1 =p′2γp2

+1

M2

v′2a2

− θ′2 tan θ2 , (B.3)

θ′2 = 0 , (B.4)

T ′t1

Tt1=

T ′t2

Tt2. (B.5)

were obtained by Cumpsty and Marble [3] in a 2D configuration using the compact-

nozzle assumption. Eq. B.4 can be changed by any other condition involving the

79

80 APPENDIX B. PROPAGATION EQUATIONS THROUGH A ROTOR

exit angle (as for example θ′2 = Kθ′1). Eq. B.5 can be written as a function of the

dimensionless variables and the mean flow characteristics using

T ′t

Tt=

1

1 +(γ − 1)

2M2

[

(γ − 1)p′

γp+

s′

Cp+ (γ − 1)M

v′

a

]

. (B.6)

The unknown waves can be calculated using Eq. B.2-B.5, with for example the

matrix method described by Cumpsty and Marble [3].

B.3 Rotor

The energy equation must be re-written for the rotor. The total enthalpy is not con-

served through moving blades, and therefore Eq. B.5 cannot be used. Instead the

rothalpy equation (which is constant through a rotating machines) can be written as

I = ht1 − U1vθ1= ht2 − U2vθ2

, (B.7)

where ht is the total enthalpy, U the blade speed at the considered plane and vθ

is the tangential component of the speed in the fixed frame as it can be seen in

Fig. B.1.

Figure B.1: Velocity triangle of a turbine rotor.

For axial turbomachines, one can use U1 = U2 = U . It can be seen that making

U = 0 the total enthalpy conservation for a stator is recovered.

B.3. ROTOR 81

The conservation of rothalpy can be demonstrated using the energy equation in

fixed frame

∆ht = τ , (B.8)

and Euler’s turbine equation

τ = ∆(Uvθ) = U2 · vθ2 − U1 · vθ1 , (B.9)

Where τ is the specific energy introduced by the rotating blades (in kJ/kg). It can

be deduced that I = ht − Uvθ is conserved through a blade row.

Considering small perturbations in Eq. B.7 leads to the generalized form of the

energy conservation equation through a turbine row,

I ′

I=

h′t − Uv′θ

ht − Uvθ=

h′t

ht−

Uvθ

ht

v′θvθ

1 −Uvθ

ht

. (B.10)

For the compact-nozzle assumption, the perturbations are equal at both sides

of the blade row, and the relation

I ′1I1

=I ′2I2

, (B.11)

can be written.

Defining the parameter ζ as

ζ =Uvθ

ht, (B.12)

and writing the tangential speed perturbation as

v′θvθ

=1

M

v′

a+

θ′

tan θ, (B.13)

82 APPENDIX B. PROPAGATION EQUATIONS THROUGH A ROTOR

Eq. B.10 can be written in the form

I ′

I=

1

1 − ζ

[T ′

t

Tt− ζ

(1

M

v′

a+

θ′

tan θ

)]

, (B.14)

where it can be seen that for ζ = 0 Eq. B.14 becomes the enthalpy conservation

equation of a stator (Eq. B.5).

The parameter ζ needs to be calculated differently for the inlet and the outlet

of each row. It should be written as a function of the mean flow at the inlet and

the outlet of the row and the turbine characteristics: the row load factor Ψ and the

pressure ratio Π, defined as

Ψ = −∆ht

U2=

cp(Tt1 − Tt2)

U2, (B.15)

Π =Pt2

Pt1. (B.16)

B.3.1 Inlet

Using the Euler equation (Eq. B.9) with the definition of Ψ (Eq.B.15), U can be

eliminated from Eq B.12,

ζ1 =v2θ1

− vθ1vθ2

CpTt1Ψ. (B.17)

Using the relations

Tt1 = T1(1 +γ − 1

2M2

1 ) , T1 =a2

1

γrand vθ = aM sin θ (B.18)

Eq. B.17 can be written as

B.3. ROTOR 83

ζ1 =

(γ − 1

Ψ

)

M1 sin θ1

1 +γ − 1

2M2

1

(

M1 sin θ1 −a2

a1M2 sin θ2

)

. (B.19)

The relation between the inlet and outlet sound speeds ( a2

a1) can be written as

a function of both the inlet and outlet Mach numbers and the pressure ratio Π,

a2

a1=

√

T2

T1=

√√√√√√

Tt2

Tt1

1 +γ − 1

2M2

1

1 +γ − 1

2M2

2

= Π

γ−1

2γ

1 +γ − 1

2M2

1

1 +γ − 1

2M2

2

1/2

, (B.20)

to write the final expression of ζ1,

ζ1 =

(γ − 1

Ψ

)

M1 sin θ1

1 +γ − 1

2M2

1

M1 sin θ1 − Πγ−1

2γ

1 +γ − 1

2M2

1

1 +γ − 1

2M2

2

1/2

M2 sin θ2

.

(B.21)

B.3.2 Outlet

The expression of ζ2 is obtained using Euler’s equation (Eq. B.9) and the same

relations as for ζ1 (Eq. B.18) expressed using the output variables. ζ2 is written in

the form

ζ2 =

(γ − 1

Ψ

)

M2 sin θ2

1 +γ − 1

2M2

2

(a1

a2M1 sin θ1 − M2 sin θ2

)

. (B.22)

With Eq. B.20 ζ2 can be written as

ζ2 =

(γ − 1

Ψ

)

M2 sin θ2

1 +γ − 1

2M2

2

Π−

γ−1

2γ

1 +γ − 1

2M2

2

1 +γ − 1

2M2

1

1/2

M1 sin θ1 − M2 sin θ2

.

(B.23)

Appendix C

Double-Nozzle quasi transfer

functions

In this appendix, the double-nozzle transfer functions used in section 2.6.1 are ob-

tained from the compact nozzle hypothesis and the linearized Euler equations of

Appendix A.

For small perturbations of, u′, p′ et ρ′ the mass, energy and entropy conservation

equations read,

m′

¯m=

1

M

u′

c+

p′

γp−

s′

Cp, (C.1)

T ′t

Tt=

1

1 + [(γ − 1)/2]M2

[

(γ − 1)Mu′

c+ (γ − 1)

p′

γp+

s′

Cp

]

, (C.2)

s′

Cp=

p′

γp−

ρ′

ρ. (C.3)

Using the compact nozzle hypothesis, the outcoming waves can be calculated

(for the subsonic case),

w+2

w+1

[AA] =2M2

1 + M2

1 + M1

M1 + M2

1 + [(γ − 1)/2]M22

1 + [(γ − 1)/2]M2M1, (C.4)

w+2

ws1

[SA] =M2 − M1

1 + M2

M2

1 + [(γ − 1)/2]M2M1, (C.5)

85

86 APPENDIX C. DOUBLE-NOZZLE QUASI TRANSFER FUNCTIONS

and

ws2

ws1

[SA,AA] = 1 . (C.6)

Using the usual waves definition,

w+ =p′

γp+

u′

c, (C.7)

w− =p′

γp−

u′

c, (C.8)

ws =p′

γp−

ρ′

ρ. (C.9)

Defining the density entropy wave as,

q′sρ

= ws , (C.10)

and using Eq. C.6, the conservation of q′s can be written between the inlet and the

outlet,

q′s1ρ1

=q′s2ρ2

. (C.11)

Writing the mean density in the inlet and the outlet as a function of the Mach

number (with the isentropic relations), the transmission of entropy fluctuations is

obtained, namely,

q′s2q′s1

=

(

1 + [(γ − 1)/2]M21

1 + [(γ − 1)/2]M22

)( 1

γ−1)

. (C.12)

87

Using Eq. C.5, and defining p′2+ as,

p′2+γp2

=1

2w+

2 , (C.13)

the sound pressure wave generated by the entropy perturbations is,

p′2+γp2

=1

2

M2 − M1

1 + M2

M2

1 + [(γ − 1)/2]M2M1ws

1 . (C.14)

Using ws1 =

q′s1ρ1

, and knowing that a21 = γp1

ρ1the final relation can be obtained,

p′2+ =M2 − M1

1 + M2

12M2

1 + [(γ − 1)/2]M2M1a2

1Aq′s1 . (C.15)

where A = p2

p1is obtained from the isentropic relation,

A =

(

1 + [(γ − 1)/2]M21

1 + [(γ − 1)/2]M22

)( γγ−1

)

. (C.16)

Doing the same with Eq. C.4 the transmission of an acoustic pressure wave can

be calculated,

p′2+γp2

=2M2

1 + M2

1 + M1

M1 + M2

1 + [(γ − 1)/2]M22

1 + [(γ − 1)/2]M2M1

p′1+γp1

, (C.17)

giving finally,

p′2+p′1+

=2M2

1 + M2

1 + M1

M1 + M2

1 + [(γ − 1)/2]M22

1 + [(γ − 1)/2]M2M1A . (C.18)

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