[PPT] - Network Models in Thermoacoustics Ph.D. Camilo Silva Prof. PowerPoint Presentation

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Network Models in Thermoacoustics

Ph.D. Camilo Silva

Prof. Wolfgang Polifke

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SLIDE 2

Entropy-Acoustics coupling

How to study combustion instabilities? Why changing an injector position could make a flame unstable? Can entropy couple with the flame through acoustic waves?

u0 φ0 s0

Ac. wave

¯ ˙ Q

+ +

Ac. wave

Stable flame Acoustics flame coupling Unstable flame Unstable flame

Acoustic BC downstream Acoustic BC upstream Air supply impedance Fuel supply impedance

˙

Q0

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2

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Experiments High fidelity CFD Network models All together may be the best solution !! How to study combustion instabilities?

Flame dynamics from experiments or CFD Network Models 3D Acoustic Solvers For example …

Helicopter Engine

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4

All together may be the best solution !!

For example …

How to study combustion instabilities?

Network Models

r Network Models

Helicopter Engine

Flame dynamics from experiments or CFD

Experiments High fidelity CFD Network models

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5

Full System

Understand the System

What we want to study?

Of longitudingal, transversal, azimuthal or radial acoustic waves? Thermoacoustics X

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Full System

Understand the System

What we want to study?

Of longitudinal plane acoustic waves Thermoacoustics X X Of short or long wavelengths?

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Full System

Understand the System

What we want to study?

Of longitudinal plane acoustic waves Thermoacoustics X X Of long wavelengths X Main Assumption Acoustic compactness in most elements

f the thermoacoustic system

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Thermoacoustic Network models Full Thermoacoustic System

Decompose the System Understand the System

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Ducts Compact Flames Nozzles Joints

…

Gray Box Black Box White Box

Thermoacoustic Network models Full Thermoacoustic System Acoustic two-port element

White Box

Decompose the System Understand the System

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WHITE BOX

˙ Q0 ρ0 u0 p0 s0

Ducts Compact Flames Nozzles Joints

…

Gray Box Black Box White Box

Thermoacoustic Network models Full Thermoacoustic System Acoustic two-port element

Quasi 1D Conservation Equations

White Box ∂ ∂t (ρA) + ∂ ∂x (ρuA) = 0 ∂ ∂t (ρuA) + ∂ ∂x

ρu2A
= −A ∂p

∂x ∂ ∂t (ρsA) + ∂ ∂x (ρusA) = A T ˙ q ∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Decompose the System Understand the System Model Acoustic and entropy waves

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WHITE BOX

˙ Q0 ρ0 u0 p0 s0

Ducts Compact Flames Nozzles Joints

… Gather all elements in a single matrix and compute acoustic response of the ensemble.

Gray Box Black Box White Box

Thermoacoustic Network models Full Thermoacoustic System Acoustic two-port element

White Box ∂ ∂t (ρA) + ∂ ∂x (ρuA) = 0 ∂ ∂t (ρuA) + ∂ ∂x

ρu2A
= −A ∂p

∂x ∂ ∂t (ρsA) + ∂ ∂x (ρusA) = A T ˙ q ∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

2 6 6 4 1 −Rin −Rout 1 T11 T12 −1 T21 T22 −1 3 7 7 5 | {z }

M

2 6 6 4 f0 g0 f3 g3 3 7 7 5 = 2 6 6 4 3 7 7 5

Decompose the System Understand the System Quasi 1D Conservation Equations Model Acoustic and entropy waves Note that under a suitable treatment, tens of elements can reduce to a 4 x 4 matrix !

CONNEXIONS

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12

WHITE BOX

˙ Q0 ρ0 u0 p0 s0

Ducts Compact Flames Nozzles Joints

… Gather all elements in a single matrix and compute acoustic response of the ensemble.

STABILITY ANALYSIS.

Gray Box Black Box White Box

Thermoacoustic Network models Full Thermoacoustic System Acoustic two-port element

White Box ∂ ∂t (ρA) + ∂ ∂x (ρuA) = 0 ∂ ∂t (ρuA) + ∂ ∂x

ρu2A
= −A ∂p

∂x ∂ ∂t (ρsA) + ∂ ∂x (ρusA) = A T ˙ q ∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

2 6 6 4 1 −Rin −Rout 1 T11 T12 −1 T21 T22 −1 3 7 7 5 | {z }

M

2 6 6 4 f0 g0 f3 g3 3 7 7 5 = 2 6 6 4 3 7 7 5

Study stability of the system

Growth rate Freq. Neg. Pos. Unstable Region

Decompose the System Understand the System Quasi 1D Conservation Equations Model Acoustic and entropy waves

CONNEXIONS

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WHITE BOX

˙ Q0 ρ0 u0 p0 s0

∂ ∂t (ρA) + ∂ ∂x (ρuA) = 0 ∂ ∂t (ρuA) + ∂ ∂x

ρu2A
= −A ∂p

∂x ∂ ∂t (ρsA) + ∂ ∂x (ρusA) = A T ˙ q ∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Gather all elements in a single matrix and compute acoustic response of the ensemble.

2 6 6 4 1 −Rin −Rout 1 T11 T12 −1 T21 T22 −1 3 7 7 5 | {z }

M

2 6 6 4 f0 g0 f3 g3 3 7 7 5 = 2 6 6 4 3 7 7 5

STABILITY ANALYSIS.

Study stability of the system

Growth rate Freq. Neg. Pos. Unstable Region

OUTLINE

Quasi 1D Conservation Equations Model Acoustic and entropy waves Spatial integration and the compact assumption Linearization Further Assumptions Definition of waves Boundary conditions Isentropic ducts Modeling of Flame dynamics

f = B+e−iωx/¯

c(1+M) = 1

2 ✓ ˆ p ¯ ρ¯ c + ˆ u ◆ g = B−eiωx/¯

c(1−M) = 1

2 ✓ ˆ p ¯ ρ¯ c − ˆ u ◆ ¯ ˙ Q = ¯ ρ1¯ u1A1cp ¯ T1 ✓ ¯ T2 ¯ T1 − 1 ◆ = ¯ u1A1 γ¯ p1 (γ − 1) ✓ ¯ T2 ¯ T1 − 1 ◆

CONNEXIONS

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WHITE BOX

˙ Q0 ρ0 u0 p0 s0

∂ ∂t (ρA) + ∂ ∂x (ρuA) = 0 ∂ ∂t (ρuA) + ∂ ∂x

ρu2A
= −A ∂p

∂x ∂ ∂t (ρsA) + ∂ ∂x (ρusA) = A T ˙ q ∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Gather all elements in a single matrix and compute acoustic response of the ensemble.

2 6 6 4 1 −Rin −Rout 1 T11 T12 −1 T21 T22 −1 3 7 7 5 | {z }

M

2 6 6 4 f0 g0 f3 g3 3 7 7 5 = 2 6 6 4 3 7 7 5

STABILITY ANALYSIS.

Study stability of the system

Growth rate Freq. Neg. Pos. Unstable Region

OUTLINE

Quasi 1D Conservation Equations Model Acoustic and entropy waves Spatial integration and the compact assumption Linearization Further Assumptions Definition of waves Boundary conditions Isentropic ducts Modeling of Flame dynamics

f = B+e−iωx/¯

c(1+M) = 1

2 ✓ ˆ p ¯ ρ¯ c + ˆ u ◆ g = B−eiωx/¯

c(1−M) = 1

2 ✓ ˆ p ¯ ρ¯ c − ˆ u ◆ ¯ ˙ Q = ¯ ρ1¯ u1A1cp ¯ T1 ✓ ¯ T2 ¯ T1 − 1 ◆ = ¯ u1A1 γ¯ p1 (γ − 1) ✓ ¯ T2 ¯ T1 − 1 ◆

CONNEXIONS

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Quasi 1D Conservation Equations

From full Navier-Stokes equations

∂ ∂t (ρA) + ∂ ∂x (ρuA) = 0 ∂ ∂t (ρuA) + ∂ ∂x

ρu2A
= −A ∂p

∂x ∂ ∂t (ρsA) + ∂ ∂x (ρusA) = A T ˙ q ∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Mass Assumptions No viscous terms Quasi-1D

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Quasi 1D Conservation Equations

∂ ∂t (ρA) + ∂ ∂x (ρuA) = 0 ∂ ∂t (ρuA) + ∂ ∂x

ρu2A
= −A ∂p

∂x ∂ ∂t (ρsA) + ∂ ∂x (ρusA) = A T ˙ q ∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Mass Momentum From full Navier-Stokes equations Assumptions No viscous terms Quasi-1D

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Quasi 1D Conservation Equations

∂ ∂t (ρA) + ∂ ∂x (ρuA) = 0 ∂ ∂t (ρuA) + ∂ ∂x

ρu2A
= −A ∂p

∂x ∂ ∂t (ρsA) + ∂ ∂x (ρusA) = A T ˙ q ∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Mass Momentum Entropy From full Navier-Stokes equations Assumptions No viscous terms Quasi-1D

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Quasi 1D Conservation Equations

∂ ∂t (ρA) + ∂ ∂x (ρuA) = 0 ∂ ∂t (ρuA) + ∂ ∂x

ρu2A
= −A ∂p

∂x ∂ ∂t (ρsA) + ∂ ∂x (ρusA) = A T ˙ q ∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Mass Momentum Entropy Total Enthalpy From full Navier-Stokes equations Assumptions No viscous terms Quasi-1D

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19

Quasi 1D Conservation Equations

∂ ∂t (ρA) + ∂ ∂x (ρuA) = 0 ∂ ∂t (ρuA) + ∂ ∂x

ρu2A
= −A ∂p

∂x ∂ ∂t (ρsA) + ∂ ∂x (ρusA) = A T ˙ q ∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

From full Navier-Stokes equations Total Enthalpy Assumptions No viscous terms Quasi-1D

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20

Quasi 1D Conservation Equations

Assumptions No viscous terms Quasi-1D

∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Total Enthalpy

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Quasi 1D Conservation Equations

∂ ∂t (ρA) + ∂ ∂x (ρuA) = 0 ∂ ∂t (ρuA) + ∂ ∂x

ρu2A
= −A ∂p

∂x ∂ ∂t (ρsA) + ∂ ∂x (ρusA) = A T ˙ q ∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Mass Momentum Entropy

1 ρ Dρ Dt = − 1 A ∂ ∂x (uA)

Assumptions No viscous terms Quasi-1D

∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Total Enthalpy

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22

Quasi 1D Conservation Equations

∂ ∂t (ρA) + ∂ ∂x (ρuA) = 0 ∂ ∂t (ρuA) + ∂ ∂x

ρu2A
= −A ∂p

∂x ∂ ∂t (ρsA) + ∂ ∂x (ρusA) = A T ˙ q ∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Mass Momentum Entropy

1 ρ Dρ Dt = − 1 A ∂ ∂x (uA)

Assumptions No viscous terms Quasi-1D

∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Total Enthalpy

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Quasi 1D Conservation Equations

Mass Momentum Entropy

1 ρ Dρ Dt = − 1 A ∂ ∂x (uA)

ρDu Dt = − ∂p ∂x ρT Ds Dt = ˙ q

Assumptions No viscous terms Quasi-1D

∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Total Enthalpy

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24

Quasi 1D Conservation Equations

Momentum

ρDu Dt = − ∂p ∂x

Assumptions No viscous terms Quasi-1D

∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Total Enthalpy

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Quasi 1D Conservation Equations

Momentum

ρDu Dt = − ∂p ∂x

Assumptions No viscous terms Quasi-1D

∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Total Enthalpy

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Quasi 1D Conservation Equations

Mass Entropy

1 ρ Dρ Dt = − 1 A ∂ ∂x (uA)

Dt −∂x ρT Ds Dt = ˙ q

Assumptions No viscous terms Quasi-1D

∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Total Enthalpy Momentum

ρDu Dt = − ∂p ∂x

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Quasi 1D Conservation Equations

Mass Entropy

1 ρ Dρ Dt = − 1 A ∂ ∂x (uA)

1 cp Ds Dt = 1 γp Dp Dt − 1 ρ Dρ Dt

2nd law thermodynamics

Dt −∂x ρT Ds Dt = ˙ q

Assumptions No viscous terms Quasi-1D

∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Total Enthalpy Momentum

ρDu Dt = − ∂p ∂x

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28

Quasi 1D Conservation Equations

Mass Entropy

1 ρ Dρ Dt = − 1 A ∂ ∂x (uA)

1 cp Ds Dt = 1 γp Dp Dt − 1 ρ Dρ Dt

2nd law thermodynamics

Dt −∂x ρT Ds Dt = ˙ q

Assumptions No viscous terms Quasi-1D

∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Total Enthalpy Momentum

ρDu Dt = − ∂p ∂x

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29

Quasi 1D Conservation Equations

Assumptions No viscous terms Quasi-1D

∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Total Enthalpy Mass Entropy 2nd law therm.

A γp Dp Dt + ∂ ∂x (uA) = (γ − 1) γp ˙ qA

Momentum

ρDu Dt = − ∂p ∂x

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30

Quasi 1D Conservation Equations

Assumptions No viscous terms Quasi-1D

∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Total Enthalpy Mass Entropy 2nd law therm.

A γp Dp Dt + ∂ ∂x (uA) = (γ − 1) γp ˙ qA

Momentum

ρDu Dt = − ∂p ∂x

Not convenient … we have to reorganize somehow

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31

Quasi 1D Conservation Equations

Assumptions No viscous terms Quasi-1D

∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Total Enthalpy Mass Entropy 2nd law therm.

A γp Dp Dt + ∂ ∂x (uA) = (γ − 1) γp ˙ qA

Momentum

ρDu Dt = − ∂p ∂x

A γp ∂p ∂t + Au γp ∂p ∂x + ∂ ∂x (uA) = (γ − 1) γp ˙ qA

A∂ ln(p1/γ) ∂t + 1 p1/γ ∂ ∂x ⇣ p1/γuA ⌘ = (γ − 1) γp ˙ qA

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32

Quasi 1D Conservation Equations

Assumptions No viscous terms Quasi-1D

∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Total Enthalpy Mass Entropy 2nd law therm. Momentum

ρDu Dt = − ∂p ∂x

Ap1/γ ∂ ln(p1/γ) ∂t + ∂ ∂x ⇣ p1/γuA ⌘ = (γ − 1) γp ˙ qAp1/γ

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33

WHITE BOX

˙ Q0 ρ0 u0 p0 s0

∂ ∂t (ρA) + ∂ ∂x (ρuA) = 0 ∂ ∂t (ρuA) + ∂ ∂x

ρu2A
= −A ∂p

∂x ∂ ∂t (ρsA) + ∂ ∂x (ρusA) = A T ˙ q ∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Gather all elements in a single matrix and compute acoustic response of the ensemble.

2 6 6 4 1 −Rin −Rout 1 T11 T12 −1 T21 T22 −1 3 7 7 5 | {z }

M

2 6 6 4 f0 g0 f3 g3 3 7 7 5 = 2 6 6 4 3 7 7 5

STABILITY ANALYSIS.

Study stability of the system

Growth rate Freq. Neg. Pos. Unstable Region

OUTLINE

Quasi 1D Conservation Equations Model Acoustic and entropy waves Spatial integration and the compact assumption Linearization Further Assumptions Definition of waves Boundary conditions Isentropic ducts Modeling of Flame dynamics

f = B+e−iωx/¯

c(1+M) = 1

2 ✓ ˆ p ¯ ρ¯ c + ˆ u ◆ g = B−eiωx/¯

c(1−M) = 1

2 ✓ ˆ p ¯ ρ¯ c − ˆ u ◆ ¯ ˙ Q = ¯ ρ1¯ u1A1cp ¯ T1 ✓ ¯ T2 ¯ T1 − 1 ◆ = ¯ u1A1 γ¯ p1 (γ − 1) ✓ ¯ T2 ¯ T1 − 1 ◆

X

CONNEXIONS

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34

Spatial integration and the compact assumption

Assumptions No viscous terms Quasi-1D

∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Total Enthalpy

Z x2

x1

∂ ∂t (ρhtA) dx + Z x2

x1

∂ ∂x (ρuhtA) dx = Z x2

x1

A ˙ q dx − Z x2

x1

A∂p ∂t dx

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35

Spatial integration and the compact assumption

Assumptions No viscous terms Quasi-1D

∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Total Enthalpy

Z x2

x1

∂ ∂t (ρhtA) dx + Z x2

x1

∂ ∂x (ρuhtA) dx = Z x2

x1

A ˙ q dx − Z x2

x1

A∂p ∂t dx

(x1 x2) = ∆x ⌧ λ Compact element if

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36

Its inside quantity is bounded

Spatial integration and the compact assumption

Assumptions No viscous terms Quasi-1D

∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Total Enthalpy

Z x2

x1

∂ ∂t (ρhtA) dx + Z x2

x1

∂ ∂x (ρuhtA) dx = Z x2

x1

A ˙ q dx − Z x2

x1

A∂p ∂t dx

Compact assumption means to neglect those integral terms for which (x1 x2) = ∆x ⌧ λ Compact element if

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37

Its inside quantity is bounded

Spatial integration and the compact assumption

Assumptions No viscous terms Quasi-1D

∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Total Enthalpy

Z x2

x1

∂ ∂t (ρhtA) dx + Z x2

x1

∂ ∂x (ρuhtA) dx = Z x2

x1

A ˙ q dx − Z x2

x1

A∂p ∂t dx

Compact assumption means to neglect those integral terms for which (x1 x2) = ∆x ⌧ λ Compact element if

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38

Spatial integration and the compact assumption

Assumptions No viscous terms Quasi-1D

∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Total Enthalpy

Z x2

x1

∂ ∂t (ρhtA) dx + Z x2

x1

∂ ∂x (ρuhtA) dx = Z x2

x1

A ˙ q dx − Z x2

x1

A∂p ∂t dx

[ρuhtA]x2

x1 = ˙

Q

˙ Q = Z x2

x1

˙ qAdx where

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39

Spatial integration and the compact assumption

Assumptions No viscous terms Quasi-1D Total Enthalpy

[ρuhtA]x2

x1 = ˙

Q

Compactness Mass Entropy 2nd law therm.

Ap1/γ ∂ ln(p1/γ) ∂t + ∂ ∂x ⇣ p1/γuA ⌘ = (γ − 1) γp ˙ qAp1/γ

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40

Spatial integration and the compact assumption

Assumptions No viscous terms Quasi-1D Total Enthalpy

[ρuhtA]x2

x1 = ˙

Q

Compactness Mass Entropy 2nd law therm.

Ap1/γ ∂ ln(p1/γ) ∂t + ∂ ∂x ⇣ p1/γuA ⌘ = (γ − 1) γp ˙ qAp1/γ

Z x2

x1

Ap1/γ ∂ ln(p1/γ) ∂t dx + Z x2

x1

∂ ∂x ⇣ p1/γuA ⌘ dx = Z x2

x1

(γ − 1) γp ˙ qAp1/γ dx

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41

Spatial integration and the compact assumption

Assumptions No viscous terms Quasi-1D Total Enthalpy

[ρuhtA]x2

x1 = ˙

Q

Compactness Mass Entropy 2nd law therm.

Ap1/γ ∂ ln(p1/γ) ∂t + ∂ ∂x ⇣ p1/γuA ⌘ = (γ − 1) γp ˙ qAp1/γ

Z x2

x1

Ap1/γ ∂ ln(p1/γ) ∂t dx + Z x2

x1

∂ ∂x ⇣ p1/γuA ⌘ dx = Z x2

x1

(γ − 1) γp ˙ qAp1/γ dx

h p1/γuA ix2

x1 =

Z x2

x1

(γ − 1) γp ˙ qAp1/γ dx

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42

Spatial integration and the compact assumption

Assumptions No viscous terms Quasi-1D Total Enthalpy

[ρuhtA]x2

x1 = ˙

Q

Compactness Mass Entropy 2nd law therm.

h p1/γuA ix2

x1 =

Z x2

x1

(γ − 1) γp ˙ qAp1/γ dx

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43

WHITE BOX

˙ Q0 ρ0 u0 p0 s0

∂ ∂t (ρA) + ∂ ∂x (ρuA) = 0 ∂ ∂t (ρuA) + ∂ ∂x

ρu2A
= −A ∂p

∂x ∂ ∂t (ρsA) + ∂ ∂x (ρusA) = A T ˙ q ∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

Gather all elements in a single matrix and compute acoustic response of the ensemble.

2 6 6 4 1 −Rin −Rout 1 T11 T12 −1 T21 T22 −1 3 7 7 5 | {z }

M

2 6 6 4 f0 g0 f3 g3 3 7 7 5 = 2 6 6 4 3 7 7 5

STABILITY ANALYSIS.

Study stability of the system

Growth rate Freq. Neg. Pos. Unstable Region

OUTLINE

Quasi 1D Conservation Equations Model Acoustic and entropy waves Spatial integration and the compact assumption Linearization Further Assumptions Definition of waves Boundary conditions Isentropic ducts Modeling of Flame dynamics

f = B+e−iωx/¯

c(1+M) = 1

2 ✓ ˆ p ¯ ρ¯ c + ˆ u ◆ g = B−eiωx/¯

c(1−M) = 1

2 ✓ ˆ p ¯ ρ¯ c − ˆ u ◆ ¯ ˙ Q = ¯ ρ1¯ u1A1cp ¯ T1 ✓ ¯ T2 ¯ T1 − 1 ◆ = ¯ u1A1 γ¯ p1 (γ − 1) ✓ ¯ T2 ¯ T1 − 1 ◆

X X

CONNEXIONS

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SLIDE 44

44

Linearization of Equations

[] = ¯ [] + []0 + O2

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SLIDE 45

45

Linearization of Equations

Assumptions No viscous terms Quasi-1D Total Enthalpy

[ρuhtA]x2

x1 = ˙

Q

Compactness Linear acoustics

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SLIDE 46

46

Linearization of Equations

Assumptions No viscous terms Quasi-1D

[ρuhtA]x2

x1 = ˙

Q

Compactness Linear acoustics Total Enthalpy

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˙ m h0

t|x2 x1 = ˙

Q0

r

ρ0

1

¯ ρ1 + u0

1

¯ u1 + T 0

t2 − T 0 t1

¯ Tt2 − ¯ Tt1 = ˙ Q0 ¯ Q

SLIDE 47

47

Linearization of Equations

Assumptions No viscous terms Quasi-1D

[ρuhtA]x2

x1 = ˙

Q

Compactness Linear acoustics Total Enthalpy

THERMODYNAMIK

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˙ m h0

t|x2 x1 = ˙

Q0

r

ρ0

1

¯ ρ1 + u0

1

¯ u1 + T 0

t2 − T 0 t1

¯ Tt2 − ¯ Tt1 = ˙ Q0 ¯ Q

λ (λχ1 − χ2) ✓ (γ − 1) ¯ M2 u0

2

¯ c2 + (γ − 1) p0

2

γ¯ p2 + s0

2

cp ◆ = ˙ Q0 ¯ Q + (λχ1 − χ2) ✓ (γ − 1) ¯ M1 u0

1

¯ c1 + (γ − 1) p0

1

γ¯ p1 + s0

1

cp ◆ − p0

1

γ¯ p1 − u0

1

¯ u1 + s1 cp

where

χ = ✓ 1 + γ − 1 2 ¯ M2 ◆ and λ = ¯ T2 ¯ T1

SLIDE 48

48

Linearization of Equations

Assumptions No viscous terms Quasi-1D Total Enthalpy Compactness Linear acoustics Mass Entropy 2nd law therm.

h p1/γuA ix2

x1 =

Z x2

x1

(γ − 1) γp ˙ qAp1/γ dx

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λ (λχ1 − χ2) ✓ (γ − 1) ¯ M2 u0

2

¯ c2 + (γ − 1) p0

2

γ¯ p2 + s0

2

cp ◆ = ˙ Q0 ¯ Q + (λχ1 − χ2) ✓ (γ − 1) ¯ M1 u0

1

¯ c1 + (γ − 1) p0

1

γ¯ p1 + s0

1

cp ◆ − p0

1

γ¯ p1 − u0

1

¯ u1 + s1 cp

SLIDE 49

49

Linearization of Equations

Assumptions No viscous terms Quasi-1D Compactness Linear acoustics Mass Entropy 2nd law therm.

h p1/γuA ix2

x1 =

Z x2

x1

(γ − 1) γp ˙ qAp1/γ dx

A lot of mathematical treats implemented so that after linearizing we get … A2πβ ✓ ¯ M2¯ c2p0

2

γ + ¯ p2u0

2

◆ = A1 ✓ ¯ M1¯ c1p0

1

γ + ¯ p1u0

1

◆ + β ˙ Q0 ✓ 1 + (1 − πβ) 2πβ ◆ − β2 ¯ p1 ¯ ˙ Q  1 + 1 (α + 1) ⇣ p0

2π(β+1) − p0 1

⌘

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π = ¯ p2/¯ p1

where Total Enthalpy

λ (λχ1 − χ2) ✓ (γ − 1) ¯ M2 u0

2

¯ c2 + (γ − 1) p0

2

γ¯ p2 + s0

2

cp ◆ = ˙ Q0 ¯ Q + (λχ1 − χ2) ✓ (γ − 1) ¯ M1 u0

1

¯ c1 + (γ − 1) p0

1

γ¯ p1 + s0

1

cp ◆ − p0

1

γ¯ p1 − u0

1

¯ u1 + s1 cp

SLIDE 50

50

Linearization of Equations

Assumptions No viscous terms Quasi-1D Total Enthalpy Compactness Linear acoustics Mass Entropy 2nd law therm. A2πβ ✓ ¯ M2¯ c2p0

2

γ + ¯ p2u0

2

◆ = A1 ✓ ¯ M1¯ c1p0

1

γ + ¯ p1u0

1

◆ + β ˙ Q0 ✓ 1 + (1 − πβ) 2πβ ◆ − β2 ¯ p1 ¯ ˙ Q  1 + 1 (α + 1) ⇣ p0

2π(β+1) − p0 1

⌘

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λ (λχ1 − χ2) ✓ (γ − 1) ¯ M2 u0

2

¯ c2 + (γ − 1) p0

2

γ¯ p2 + s0

2

cp ◆ = ˙ Q0 ¯ Q + (λχ1 − χ2) ✓ (γ − 1) ¯ M1 u0

1

¯ c1 + (γ − 1) p0

1

γ¯ p1 + s0

1

cp ◆ − p0

1

γ¯ p1 − u0

1

¯ u1 + s1 cp

Now we can start doing some simplifications

SLIDE 51

51

WHITE BOX

˙ Q0 ρ0 u0 p0 s0

∂ ∂t (ρA) + ∂ ∂x (ρuA) = 0 ∂ ∂t (ρuA) + ∂ ∂x

ρu2A
= −A ∂p

∂x ∂ ∂t (ρsA) + ∂ ∂x (ρusA) = A T ˙ q ∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

CONNEXIONS

Gather all elements in a single matrix and compute acoustic response of the ensemble.

2 6 6 4 1 −Rin −Rout 1 T11 T12 −1 T21 T22 −1 3 7 7 5 | {z }

M

2 6 6 4 f0 g0 f3 g3 3 7 7 5 = 2 6 6 4 3 7 7 5

STABILITY ANALYSIS.

Study stability of the system

Growth rate Freq. Neg. Pos. Unstable Region

OUTLINE

Quasi 1D Conservation Equations Model Acoustic and entropy waves Spatial integration and the compact assumption Linearization Further Assumptions Definition of waves Boundary conditions Isentropic ducts Modeling of Flame dynamics

f = B+e−iωx/¯

c(1+M) = 1

2 ✓ ˆ p ¯ ρ¯ c + ˆ u ◆ g = B−eiωx/¯

c(1−M) = 1

2 ✓ ˆ p ¯ ρ¯ c − ˆ u ◆ ¯ ˙ Q = ¯ ρ1¯ u1A1cp ¯ T1 ✓ ¯ T2 ¯ T1 − 1 ◆ = ¯ u1A1 γ¯ p1 (γ − 1) ✓ ¯ T2 ¯ T1 − 1 ◆

X X X

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SLIDE 52

52

Assumptions

Further assumptions No viscous terms Quasi-1D Compactness Linear acoustics Isentropic flow Total Enthalpy Mass Entropy 2nd law therm. A2πβ ✓ ¯ M2¯ c2p0

2

γ + ¯ p2u0

2

◆ = A1 ✓ ¯ M1¯ c1p0

1

γ + ¯ p1u0

1

◆ + β ˙ Q0 ✓ 1 + (1 − πβ) 2πβ ◆ − β2 ¯ p1 ¯ ˙ Q  1 + 1 (α + 1) ⇣ p0

2π(β+1) − p0 1

⌘ Total Enthalpy Mass Entropy 2nd law therm.

THERMODYNAMIK

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λ (λχ1 − χ2) ✓ (γ − 1) ¯ M2 u0

2

¯ c2 + (γ − 1) p0

2

γ¯ p2 + s0

2

cp ◆ = ˙ Q0 ¯ Q + (λχ1 − χ2) ✓ (γ − 1) ¯ M1 u0

1

¯ c1 + (γ − 1) p0

1

γ¯ p1 + s0

1

cp ◆ − p0

1

γ¯ p1 − u0

1

¯ u1 + s1 cp

SLIDE 53

53

1 1 + γ1

2

¯ M2

1

✓ ¯ M1 u0

1

¯ c1 + p0

1

γ¯ p1 + s0

1

cp ◆ = 1 1 + γ1

2

¯ M2

2

✓ ¯ M2 u0

2

¯ c2 + p0

2

γ¯ p2 + s0

2

cp ◆

Assumptions

Further assumptions No viscous terms Quasi-1D Compactness Linear acoustics Isentropic flow Total Enthalpy Mass Entropy 2nd law therm. Total Enthalpy Mass Entropy 2nd law therm. A1 ✓ ¯ ρ1u0

1 + ¯

M1 p0

1

¯ c1 ◆ = A2 ✓ ¯ ρ2u0

2 + ¯

M2 p0

2

¯ c2 ◆

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A2πβ ✓ ¯ M2¯ c2p0

2

γ + ¯ p2u0

2

◆ = A1 ✓ ¯ M1¯ c1p0

1

γ + ¯ p1u0

1

◆ + β ˙ Q0 ✓ 1 + (1 − πβ) 2πβ ◆ − β2 ¯ p1 ¯ ˙ Q  1 + 1 (α + 1) ⇣ p0

2π(β+1) − p0 1

⌘

λ (λχ1 − χ2) ✓ (γ − 1) ¯ M2 u0

2

¯ c2 + (γ − 1) p0

2

γ¯ p2 + s0

2

cp ◆ = ˙ Q0 ¯ Q + (λχ1 − χ2) ✓ (γ − 1) ¯ M1 u0

1

¯ c1 + (γ − 1) p0

1

γ¯ p1 + s0

1

cp ◆ − p0

1

γ¯ p1 − u0

1

¯ u1 + s1 cp

SLIDE 54

54

1 1 + γ1

2

¯ M2

1

✓ ¯ M1 u0

1

¯ c1 + p0

1

γ¯ p1 + s0

1

cp ◆ = 1 1 + γ1

2

¯ M2

2

✓ ¯ M2 u0

2

¯ c2 + p0

2

γ¯ p2 + s0

2

cp ◆

Assumptions

Further assumptions No viscous terms Quasi-1D Compactness Linear acoustics Isentropic flow Total Enthalpy Mass Entropy 2nd law therm. Total Enthalpy Mass Entropy 2nd law therm. A1 ✓ ¯ ρ1u0

1 + ¯

M1 p0

1

¯ c1 ◆ = A2 ✓ ¯ ρ2u0

2 + ¯

M2 p0

2

¯ c2 ◆

C. F. Silva, I. Duran, F. Nicoud and S. Moreau. Boundary conditions for the computation of

thermoacoustic modes in combustion chambers. AIAA Journal, 52(6): 1180–1193, 2014.

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A2πβ ✓ ¯ M2¯ c2p0

2

γ + ¯ p2u0

2

◆ = A1 ✓ ¯ M1¯ c1p0

1

γ + ¯ p1u0

1

◆ + β ˙ Q0 ✓ 1 + (1 − πβ) 2πβ ◆ − β2 ¯ p1 ¯ ˙ Q  1 + 1 (α + 1) ⇣ p0

2π(β+1) − p0 1

⌘

λ (λχ1 − χ2) ✓ (γ − 1) ¯ M2 u0

2

¯ c2 + (γ − 1) p0

2

γ¯ p2 + s0

2

cp ◆ = ˙ Q0 ¯ Q + (λχ1 − χ2) ✓ (γ − 1) ¯ M1 u0

1

¯ c1 + (γ − 1) p0

1

γ¯ p1 + s0

1

cp ◆ − p0

1

γ¯ p1 − u0

1

¯ u1 + s1 cp

SLIDE 55

55

Assumptions

Further assumptions No viscous terms Quasi-1D Compactness Linear acoustics Total Enthalpy Mass Entropy 2nd law therm. Total Enthalpy Mass Entropy 2nd law therm. (Non-Isentropic flow) Isobaric combustion Low Mach number

THERMODYNAMIK

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A2πβ ✓ ¯ M2¯ c2p0

2

γ + ¯ p2u0

2

◆ = A1 ✓ ¯ M1¯ c1p0

1

γ + ¯ p1u0

1

◆ + β ˙ Q0 ✓ 1 + (1 − πβ) 2πβ ◆ − β2 ¯ p1 ¯ ˙ Q  1 + 1 (α + 1) ⇣ p0

2π(β+1) − p0 1

⌘

λ (λχ1 − χ2) ✓ (γ − 1) ¯ M2 u0

2

¯ c2 + (γ − 1) p0

2

γ¯ p2 + s0

2

cp ◆ = ˙ Q0 ¯ Q + (λχ1 − χ2) ✓ (γ − 1) ¯ M1 u0

1

¯ c1 + (γ − 1) p0

1

γ¯ p1 + s0

1

cp ◆ − p0

1

γ¯ p1 − u0

1

¯ u1 + s1 cp

SLIDE 56

56

Assumptions

Further assumptions No viscous terms Quasi-1D Compactness Linear acoustics Total Enthalpy Mass Entropy 2nd law therm. Total Enthalpy Mass Entropy 2nd law therm. (Non-Isentropic flow) Isobaric combustion Low Mach number p0

2 = p0 1

A2u0

2 − A1u0 1 = (γ − 1)

γ¯ p ˙ Q0

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A2πβ ✓ ¯ M2¯ c2p0

2

γ + ¯ p2u0

2

◆ = A1 ✓ ¯ M1¯ c1p0

1

γ + ¯ p1u0

1

◆ + β ˙ Q0 ✓ 1 + (1 − πβ) 2πβ ◆ − β2 ¯ p1 ¯ ˙ Q  1 + 1 (α + 1) ⇣ p0

2π(β+1) − p0 1

⌘

λ (λχ1 − χ2) ✓ (γ − 1) ¯ M2 u0

2

¯ c2 + (γ − 1) p0

2

γ¯ p2 + s0

2

cp ◆ = ˙ Q0 ¯ Q + (λχ1 − χ2) ✓ (γ − 1) ¯ M1 u0

1

¯ c1 + (γ − 1) p0

1

γ¯ p1 + s0

1

cp ◆ − p0

1

γ¯ p1 − u0

1

¯ u1 + s1 cp

SLIDE 57

57

WHITE BOX

˙ Q0 ρ0 u0 p0 s0

∂ ∂t (ρA) + ∂ ∂x (ρuA) = 0 ∂ ∂t (ρuA) + ∂ ∂x

ρu2A
= −A ∂p

∂x ∂ ∂t (ρsA) + ∂ ∂x (ρusA) = A T ˙ q ∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

CONNEXIONS

Gather all elements in a single matrix and compute acoustic response of the ensemble.

2 6 6 4 1 −Rin −Rout 1 T11 T12 −1 T21 T22 −1 3 7 7 5 | {z }

M

2 6 6 4 f0 g0 f3 g3 3 7 7 5 = 2 6 6 4 3 7 7 5

STABILITY ANALYSIS.

Study stability of the system

Growth rate Freq. Neg. Pos. Unstable Region

OUTLINE

Quasi 1D Conservation Equations Model Acoustic and entropy waves Spatial integration and the compact assumption Linearization Further Assumptions Definition of waves Boundary conditions Isentropic ducts Modeling of Flame dynamics

f = B+e−iωx/¯

c(1+M) = 1

2 ✓ ˆ p ¯ ρ¯ c + ˆ u ◆ g = B−eiωx/¯

c(1−M) = 1

2 ✓ ˆ p ¯ ρ¯ c − ˆ u ◆ ¯ ˙ Q = ¯ ρ1¯ u1A1cp ¯ T1 ✓ ¯ T2 ¯ T1 − 1 ◆ = ¯ u1A1 γ¯ p1 (γ − 1) ✓ ¯ T2 ¯ T1 − 1 ◆

X X X X

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SLIDE 58

58

Definition of acoustic waves

D2p0 Dt2 − ¯ c2 ∂2p0 ∂x2 = 0

1D Convective acoustic Wave equation Solution is the sum of two functions

f

g

and

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SLIDE 59

59

Definition of acoustic waves

D2p0 Dt2 − ¯ c2 ∂2p0 ∂x2 = 0

1D Convective acoustic Wave equation Solution is the sum of two functions

f

g

and are recognized as Riemann invariants

f

g

and

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SLIDE 60

60

Definition of acoustic waves

D2p0 Dt2 − ¯ c2 ∂2p0 ∂x2 = 0

1D Convective acoustic Wave equation Solution is the sum of two functions

f

g

and are recognized as Riemann invariants ()0 = ˆ ()eiωt and ˆ () = Beiφ

f

g

and Knowing that harmonic oscillations are defined as

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SLIDE 61

61

Definition of acoustic waves

D2p0 Dt2 − ¯ c2 ∂2p0 ∂x2 = 0

1D Convective acoustic Wave equation Solution is the sum of two functions

f

g

and are recognized as Riemann invariants ()0 = ˆ ()eiωt and ˆ () = Beiφ

f

g

and Knowing that harmonic oscillations are defined as can be defined as

f

g

and

f = B+e−iωx/¯

c(1+M) = 1

2 ✓ ˆ p ¯ ρ¯ c + ˆ u ◆

g = B−eiωx/¯

c(1−M) = 1

2 ✓ ˆ p ¯ ρ¯ c − ˆ u ◆

and

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SLIDE 62

62

Definition of acoustic waves

D2p0 Dt2 − ¯ c2 ∂2p0 ∂x2 = 0

1D Convective acoustic Wave equation Solution is the sum of two functions

f

g

and are recognized as Riemann invariants ()0 = ˆ ()eiωt and ˆ () = Beiφ

f

g

and Knowing that harmonic oscillations are defined as can be defined as

f

g

and

f = B+e−iωx/¯

c(1+M) = 1

2 ✓ ˆ p ¯ ρ¯ c + ˆ u ◆

g = B−eiωx/¯

c(1−M) = 1

2 ✓ ˆ p ¯ ρ¯ c − ˆ u ◆

and Downstream Travelling wave Upstream Travelling wave

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SLIDE 63

63

WHITE BOX

˙ Q0 ρ0 u0 p0 s0

∂ ∂t (ρA) + ∂ ∂x (ρuA) = 0 ∂ ∂t (ρuA) + ∂ ∂x

ρu2A
= −A ∂p

∂x ∂ ∂t (ρsA) + ∂ ∂x (ρusA) = A T ˙ q ∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

CONNEXIONS

Gather all elements in a single matrix and compute acoustic response of the ensemble.

2 6 6 4 1 −Rin −Rout 1 T11 T12 −1 T21 T22 −1 3 7 7 5 | {z }

M

2 6 6 4 f0 g0 f3 g3 3 7 7 5 = 2 6 6 4 3 7 7 5

STABILITY ANALYSIS.

Study stability of the system

Growth rate Freq. Neg. Pos. Unstable Region

OUTLINE

Quasi 1D Conservation Equations Model Acoustic and entropy waves Spatial integration and the compact assumption Linearization Further Assumptions Definition of waves Boundary conditions Isentropic ducts Modeling of Flame dynamics

f = B+e−iωx/¯

c(1+M) = 1

2 ✓ ˆ p ¯ ρ¯ c + ˆ u ◆ g = B−eiωx/¯

c(1−M) = 1

2 ✓ ˆ p ¯ ρ¯ c − ˆ u ◆ ¯ ˙ Q = ¯ ρ1¯ u1A1cp ¯ T1 ✓ ¯ T2 ¯ T1 − 1 ◆ = ¯ u1A1 γ¯ p1 (γ − 1) ✓ ¯ T2 ¯ T1 − 1 ◆

X X X X X

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SLIDE 64

64

f2 = f1e−iω(x2−x1)/¯

c(1+M)

f = B+e−iωx/¯

c(1+M) = 1

2 ✓ ˆ p ¯ ρ¯ c + ˆ u ◆

g = B−eiωx/¯

c(1−M) = 1

2 ✓ ˆ p ¯ ρ¯ c − ˆ u ◆

and Downstream Travelling wave Upstream Travelling wave

Isentropic ducts

x1 x2

B− = constant B+ = constant

Therefore

g2 = g1eiω(x2−x1)/¯

c(1−M)

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SLIDE 65

65

WHITE BOX

˙ Q0 ρ0 u0 p0 s0

∂ ∂t (ρA) + ∂ ∂x (ρuA) = 0 ∂ ∂t (ρuA) + ∂ ∂x

ρu2A
= −A ∂p

∂x ∂ ∂t (ρsA) + ∂ ∂x (ρusA) = A T ˙ q ∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

CONNEXIONS

Gather all elements in a single matrix and compute acoustic response of the ensemble.

2 6 6 4 1 −Rin −Rout 1 T11 T12 −1 T21 T22 −1 3 7 7 5 | {z }

M

2 6 6 4 f0 g0 f3 g3 3 7 7 5 = 2 6 6 4 3 7 7 5

STABILITY ANALYSIS.

Study stability of the system

Growth rate Freq. Neg. Pos. Unstable Region

OUTLINE

Quasi 1D Conservation Equations Model Acoustic and entropy waves Spatial integration and the compact assumption Linearization Further Assumptions Definition of waves Boundary conditions Isentropic ducts Modeling of Flame dynamics

f = B+e−iωx/¯

c(1+M) = 1

2 ✓ ˆ p ¯ ρ¯ c + ˆ u ◆ g = B−eiωx/¯

c(1−M) = 1

2 ✓ ˆ p ¯ ρ¯ c − ˆ u ◆ ¯ ˙ Q = ¯ ρ1¯ u1A1cp ¯ T1 ✓ ¯ T2 ¯ T1 − 1 ◆ = ¯ u1A1 γ¯ p1 (γ − 1) ✓ ¯ T2 ¯ T1 − 1 ◆

X X X X X X

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SLIDE 66

66

Boundary Conditions

Acoustic flux through a boundary is given by

˙ m0h0

t =

✓ p0 ¯ c2 ¯ u + ¯ ρu0 ◆ ✓ ¯ uu0 + p0 ¯ ρ ◆

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SLIDE 67

67

Boundary Conditions

Acoustic flux through a boundary is given by If acoustic energy it is not dissipated through the boundaries then

˙ m0h0

t =

✓ p0 ¯ c2 ¯ u + ¯ ρu0 ◆ ✓ ¯ uu0 + p0 ¯ ρ ◆

˙ m0h0

t = 0

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SLIDE 68

68

Boundary Conditions

Acoustic flux through a boundary is given by If acoustic energy it is not dissipated through the boundaries then

˙ m0h0

t =

✓ p0 ¯ c2 ¯ u + ¯ ρu0 ◆ ✓ ¯ uu0 + p0 ¯ ρ ◆

˙ m0h0

t = 0

f(1 + ¯ M) − g(1 − ¯ M) f(1 + ¯ M) + g(1 − ¯ M)

Open inlet Closed inlet

h0

t ≤

˙ m0 f g

Inlet

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= 0 = 0 = 0 = 0

SLIDE 69

69

Boundary Conditions

Acoustic flux through a boundary is given by If acoustic energy is not dissipated through the boundaries then

˙ m0h0

t =

✓ p0 ¯ c2 ¯ u + ¯ ρu0 ◆ ✓ ¯ uu0 + p0 ¯ ρ ◆

˙ m0h0

t = 0

f g f g

Inlet Outlet

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f(1 + ¯ M) − g(1 − ¯ M) f(1 + ¯ M) + g(1 − ¯ M)

Open inlet/outlet Closed inlet/outlet

h0

t ≤

˙ m0 = 0 = 0 = 0 = 0

SLIDE 70

70

Boundary Conditions

Acoustic flux through a boundary is given by If acoustic energy it is not dissipated through the boundaries then

˙ m0h0

t =

✓ p0 ¯ c2 ¯ u + ¯ ρu0 ◆ ✓ ¯ uu0 + p0 ¯ ρ ◆

˙ m0h0

t = 0

If acoustic energy it is no acoustic energy enters the system then

f g f g

Inlet Outlet

Rin = f g f g Rout = g f f g

THERMODYNAMIK

Lehrstuhl für

Rin = (1 − ¯ M) (1 + ¯ M) Rin = −(1 − ¯ M) (1 + ¯ M) Rout = −(1 + ¯ M) (1 − ¯ M) Rout = (1 + ¯ M) (1 − ¯ M)

Close end Open end

SLIDE 71

71

WHITE BOX

˙ Q0 ρ0 u0 p0 s0

∂ ∂t (ρA) + ∂ ∂x (ρuA) = 0 ∂ ∂t (ρuA) + ∂ ∂x

ρu2A
= −A ∂p

∂x ∂ ∂t (ρsA) + ∂ ∂x (ρusA) = A T ˙ q ∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

CONNEXIONS

Gather all elements in a single matrix and compute acoustic response of the ensemble.

2 6 6 4 1 −Rin −Rout 1 T11 T12 −1 T21 T22 −1 3 7 7 5 | {z }

M

2 6 6 4 f0 g0 f3 g3 3 7 7 5 = 2 6 6 4 3 7 7 5

STABILITY ANALYSIS.

Study stability of the system

Growth rate Freq. Neg. Pos. Unstable Region

OUTLINE

Quasi 1D Conservation Equations Model Acoustic and entropy waves Spatial integration and the compact assumption Linearization Further Assumptions Definition of waves Boundary conditions Isentropic ducts Modeling of Flame dynamics

f = B+e−iωx/¯

c(1+M) = 1

2 ✓ ˆ p ¯ ρ¯ c + ˆ u ◆ g = B−eiωx/¯

c(1−M) = 1

2 ✓ ˆ p ¯ ρ¯ c − ˆ u ◆ ¯ ˙ Q = ¯ ρ1¯ u1A1cp ¯ T1 ✓ ¯ T2 ¯ T1 − 1 ◆ = ¯ u1A1 γ¯ p1 (γ − 1) ✓ ¯ T2 ¯ T1 − 1 ◆

X X X X X X X

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SLIDE 72

72

ˆ ˙ Q ¯ ˙ Q = F(ω) ˆ u1 ¯ u1

Modeling Flame Dynamics

where

F(ω) = G(ω)eiϕ(ω)

CFD or Experiments

?

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SLIDE 73

73

WHITE BOX

˙ Q0 ρ0 u0 p0 s0

∂ ∂t (ρA) + ∂ ∂x (ρuA) = 0 ∂ ∂t (ρuA) + ∂ ∂x

ρu2A
= −A ∂p

∂x ∂ ∂t (ρsA) + ∂ ∂x (ρusA) = A T ˙ q ∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

CONNEXIONS

Gather all elements in a single matrix and compute acoustic response of the ensemble.

2 6 6 4 1 −Rin −Rout 1 T11 T12 −1 T21 T22 −1 3 7 7 5 | {z }

M

2 6 6 4 f0 g0 f3 g3 3 7 7 5 = 2 6 6 4 3 7 7 5

STABILITY ANALYSIS.

Study stability of the system

Growth rate Freq. Neg. Pos. Unstable Region

OUTLINE

Quasi 1D Conservation Equations Model Acoustic and entropy waves Spatial integration and the compact assumption Linearization Further Assumptions Definition of waves Boundary conditions Isentropic ducts Modeling of Flame dynamics

f = B+e−iωx/¯

c(1+M) = 1

2 ✓ ˆ p ¯ ρ¯ c + ˆ u ◆ g = B−eiωx/¯

c(1−M) = 1

2 ✓ ˆ p ¯ ρ¯ c − ˆ u ◆ ¯ ˙ Q = ¯ ρ1¯ u1A1cp ¯ T1 ✓ ¯ T2 ¯ T1 − 1 ◆ = ¯ u1A1 γ¯ p1 (γ − 1) ✓ ¯ T2 ¯ T1 − 1 ◆

X X X X X X X X

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SLIDE 74

74

In this case, the Mach number is considered zero.

Connexions

Turbulent combustion chamber

C. F. Silva, F. Nicoud, T. Schuller, D. Durox, and S. Candel. Combining a Helmholtz solver

with the flame describing function to assess combustion instability un a premixed swirled

combustion. Combust. Flame, 160(9): 1743-1754, 2013.
P. Palies, D. Durox, T. Schuller and S. Candel. Nonlinear combustion instabilities analysis

based on the Flame Describing Function applied to turbulent premixed swirling flames.

Combust. Flame, 158: 1980-1991, 2011.

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SLIDE 75

75

In this case, the Mach number is considered zero.

Connexions

Turbulent combustion chamber

CFD

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SLIDE 76

76

In this case, the Mach number is considered zero.

Connexions

Boundary Condition: Outlet Turbulent combustion chamber

CFD

Network Models

1 2 3 4 5

Boundary Condition: Inlet Cross section change Cross section change + Flame Duct (cold) Duct (cold) Duct (hot)

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SLIDE 77

77

In this case, the Mach number is considered zero.

Connexions

Boundary Condition: Open outlet Network Models

1 2 3 4 5

Boundary Condition: Closed Inlet Cross section change Cross section change + Flame Duct (cold) Duct (cold) Duct (hot)

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SLIDE 78

78

In this case, the Mach number is considered zero.

Connexions

Boundary Condition: Open outlet Network Models

1 2 3 4 5

Boundary Condition: Closed Inlet Cross section change Cross section change + Flame Duct (cold) Duct (cold) Duct (hot)

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Rin = f0 g0

Rout = g5 f5

SLIDE 79

79

In this case, the Mach number is considered zero.

Connexions

Boundary Condition: Open outlet Network Models

1 2 3 4 5

Boundary Condition: Closed Inlet Cross section change Cross section change + Flame Duct (cold) Duct (cold) Duct (hot)

f1 = f0e−iω(x1−x0)/¯

c

g1 = g0eiω(x1−x0)/¯

c

f5 = f4e−iω(x5−x4)/¯

c

g5 = g4eiω(x5−x4)/¯

c

¯ c ¯ c

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SLIDE 80

80

In this case, the Mach number is considered zero.

Connexions

Boundary Condition: Open outlet Network Models

1 2 3 4 5

Boundary Condition: Closed Inlet Cross section change Cross section change + Flame Duct (cold) Duct (cold) Duct (hot) f1 g1

=

e−iω(x1−x0)/¯

c

eiω(x1−x0)/¯

c

f0 g0

f5

g5

=

e−iω(x5−x4)/¯

c

eiω(x5−x4)/¯

c

f4 g4

¯

c ¯ c

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SLIDE 81

81

In this case, the Mach number is considered zero.

Connexions

Boundary Condition: Open outlet Network Models

1 2 3 4 5

Boundary Condition: Closed Inlet Cross section change Duct (cold) Duct (cold) Duct (hot)

(f4 + g4) = (f3 + g3) ξ (f4 − g4) = (f3 − g3) α [1 + θF(ω)]

Cross section change + Flame

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SLIDE 82

82

In this case, the Mach number is considered zero.

Connexions

Boundary Condition: Open outlet Network Models

1 2 3 4 5

Boundary Condition: Closed Inlet Cross section change Duct (cold) Duct (cold) Duct (hot)

f4 g4

= 1

2 ξ + α + αθF(ω) ξ − α − αθF(ω) ξ − α − αθF(ω) ξ + α + αθF(ω) f3 g3

Cross section change +

Flame

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SLIDE 83

83

In this case, the Mach number is considered zero.

Connexions

Boundary Condition: Open outlet Network Models

1 2 3 4 5

Boundary Condition: Closed Inlet Cross section change Duct (cold) Duct (cold) Duct (hot)

f4 g4

= F

f3 g3

f1

g1

= D1

f0 g0

f5

g5

= D3

f4 g4

f3

g3

= D2

f2 g2

f2

g2

= C

f1 g1

Cross section change +

Flame

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SLIDE 84

84

In this case, the Mach number is considered zero.

Connexions

Network Models

1 2 3 4 5 f5 g5

= T

f0 g0

where

T = D3FD2CD1

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SLIDE 85

85

In this case, the Mach number is considered zero.

Connexions

Network Models

f5 g5

= T

f0 g0

where

T = D3FD2CD1

Final matrix

2 6 6 4 1 −Rin −Rout 1 T11 T12 −1 T21 T22 −1 3 7 7 5 | {z }

M

2 6 6 4 f0 g0 f5 g5 3 7 7 5 = 2 6 6 4 3 7 7 5

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SLIDE 86

86

In this case, the Mach number is considered zero.

Connexions

Network Models

f5 g5

= T

f0 g0

where

T = D3FD2CD1

Final matrix

det(M) = 0 ⇒ T22 − RoutT12 + RinT21 − RinRoutT11 = 0

Solution comes by solving the characteristic equation

2 6 6 4 1 −Rin −Rout 1 T11 T12 −1 T21 T22 −1 3 7 7 5 | {z }

M

2 6 6 4 f0 g0 f5 g5 3 7 7 5 = 2 6 6 4 3 7 7 5

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SLIDE 87

87

WHITE BOX

˙ Q0 ρ0 u0 p0 s0

∂ ∂t (ρA) + ∂ ∂x (ρuA) = 0 ∂ ∂t (ρuA) + ∂ ∂x

ρu2A
= −A ∂p

∂x ∂ ∂t (ρsA) + ∂ ∂x (ρusA) = A T ˙ q ∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

CONNEXIONS

Gather all elements in a single matrix and compute acoustic response of the ensemble.

2 6 6 4 1 −Rin −Rout 1 T11 T12 −1 T21 T22 −1 3 7 7 5 | {z }

M

2 6 6 4 f0 g0 f3 g3 3 7 7 5 = 2 6 6 4 3 7 7 5

STABILITY ANALYSIS.

Study stability of the system

Growth rate Freq. Neg. Pos. Unstable Region

OUTLINE

Quasi 1D Conservation Equations Model Acoustic and entropy waves Spatial integration and the compact assumption Linearization Further Assumptions Definition of waves Boundary conditions Isentropic ducts Modeling of Flame dynamics

f = B+e−iωx/¯

c(1+M) = 1

2 ✓ ˆ p ¯ ρ¯ c + ˆ u ◆ g = B−eiωx/¯

c(1−M) = 1

2 ✓ ˆ p ¯ ρ¯ c − ˆ u ◆ ¯ ˙ Q = ¯ ρ1¯ u1A1cp ¯ T1 ✓ ¯ T2 ¯ T1 − 1 ◆ = ¯ u1A1 γ¯ p1 (γ − 1) ✓ ¯ T2 ¯ T1 − 1 ◆

X X X X X X X X

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SLIDE 88

88

Stability analysis

T22 − RoutT12 + RinT21 − RinRoutT11 = 0

Solving the characteristic equation we obtain a value for ω (complex number) Resonance frequency Growth rate (Stable or unstable)

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ω = ωr + iωi

SLIDE 89

89

Stability analysis

T22 − RoutT12 + RinT21 − RinRoutT11 = 0

Solving the characteristic equation we obtain a value for ω (complex number) Resonance frequency Growth rate (Stable or unstable)

Growth rate

Freq.

Neg.

Pos.

U n s t a b l e

R

e g i

n

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ω = ωr + iωi

SLIDE 90

90

Stability analysis

ω = ωr − iωi

Resonance frequency Growth rate

Growth rate

Freq.

Neg.

Pos.

U n s t a b l e

R

e g i

n

a0 = ˆ aeiωt

ωi > 0 stability ωi < 0 instability

Harmonic Oscillations a fluctuating quantity is expressed as

a0 = ˆ aeωiteiωrt.

Therefore

⇒

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SLIDE 91

91

WHITE BOX

˙ Q0 ρ0 u0 p0 s0

∂ ∂t (ρA) + ∂ ∂x (ρuA) = 0 ∂ ∂t (ρuA) + ∂ ∂x

ρu2A
= −A ∂p

∂x ∂ ∂t (ρsA) + ∂ ∂x (ρusA) = A T ˙ q ∂ ∂t (ρhtA) + ∂ ∂x (ρuhtA) = A ˙ q − A∂p ∂t

CONNEXIONS

Gather all elements in a single matrix and compute acoustic response of the ensemble.

2 6 6 4 1 −Rin −Rout 1 T11 T12 −1 T21 T22 −1 3 7 7 5 | {z }

M

2 6 6 4 f0 g0 f3 g3 3 7 7 5 = 2 6 6 4 3 7 7 5

STABILITY ANALYSIS.

Study stability of the system

Growth rate Freq. Neg. Pos. Unstable Region

OUTLINE

Quasi 1D Conservation Equations Model Acoustic and entropy waves Spatial integration and the compact assumption Linearization Further Assumptions Definition of waves Boundary conditions Isentropic ducts Modeling of Flame dynamics

f = B+e−iωx/¯

c(1+M) = 1

2 ✓ ˆ p ¯ ρ¯ c + ˆ u ◆ g = B−eiωx/¯

c(1−M) = 1

2 ✓ ˆ p ¯ ρ¯ c − ˆ u ◆ ¯ ˙ Q = ¯ ρ1¯ u1A1cp ¯ T1 ✓ ¯ T2 ¯ T1 − 1 ◆ = ¯ u1A1 γ¯ p1 (γ − 1) ✓ ¯ T2 ¯ T1 − 1 ◆

X X X X X X X X

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SLIDE 92

92

The End

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SLIDE 93

93

Exercises

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Boundary Condition: Outlet

1 2 3 4 5

Boundary Condition: Inlet Cross section change Cross section change + Flame Duct (cold) Duct (cold) Duct (hot)

l1 l3 l2 d1 d2 d3

Open end Closed end