[PPT] - A combustion instability model accounting for dynamic PowerPoint Presentation

SLIDE 1

A combustion instability model accounting for dynamic flame-flow-acoustic interactions

Rapha¨ el Assier 1 Xuesong Wu 2

1University of Manchester 2Imperial College London

Analytical methods in thermoacoustics, Keele, September 2014

Rapha¨ el Assier (University of Manchester) Modelling combustion instabilities Keele, Sept 16, 2014 1 / 37

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1 Introduction

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Combustion instabilities

[Photos from Lieuwen, 2005]

Applications:

Jet engines Rocket engines Gas turbine engines

Problems:

Vibrations Structural fatigue Increase in fuel consumption

Phenomena

Self-sustained, large amplitude pressure fluctuations and flame oscillations Generally occurs around the characteristic frequency of the combustor

Objective:

Understanding the fundamental mechanisms of combustion instabilities

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Physical mechanisms: Flame-Flow-Acoustic interactions

1

Flame excites the acoustic pressure

Acoustic pressure amplifies according to Rayleigh’s criterion, i.e when heat release and acoustic pressure are in phase.

2

Acoustic velocity and acoustic acceleration advect and modulate the flame front.

3

The flame influences the hydrodynamic flow, which in turn influences the flame. Multi-scale, Extremely challenging for DNS!

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Existing models and comparison

The G-equation [see e.g. Dowling, JFM, 1999]

Model the advection effect of acoustic on the flame Purely kinematic (no hydrodynamics) Ignores dynamic effect of acoustics

Hydrodynamic theory of flames

Flame-flow interaction model, no acoustic considerations

[Pelc´ e & Clavin, JFM 1982], [Matalon & Matkowsky, JFM 1982]

Reduces to Michelson-Sivashinsky equation (M-S)

Advection Hydrodynamics (D-L) Flame-acoustics coupling R-T G-equation ✓ ✗ ✓ ✗ M-S ✓ ✓ ✗ ✗ Aim: ✓ ✓ ✓ ✓

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The G-equation

G-equation

∂G ∂t = −u · ∇ G + Su|∇ G|, ∂F ∂t = uG − Su

1 +

∂F ∂y 2 G = G(x, y, t) : level set function representing the flame Su: normal flame speed propagation Write G(x, y, t) = x − F(y, t) and consider 1D velocity fluctuations uG = uG(t): “u gutter”, acoustic velocity at the flame (1D) Allows for wrinkling of anchored flames... ... but not for freely propagating flames.

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The Michelson-Sivashinsky equation

The M-S equation (2D)

    

∂ϕ ∂t = 1 2I(ϕ; y) + 1 γ ∂2ϕ ∂y2 + 1 2

∂ϕ

∂y

2 ,

I (ϕ; y)(k, t) = |k|

ϕ (k, t) , ϕ : flame shape γ : “free” parameter : Fourier transform in y direction Good for unconfined freely propagating flames Predicts Darrieus-Landau instability: i.e. flames tend to curve

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Searby’s experiment [Searby, Combust. Sci. Technol., 1992] Setup Results

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Flame-flow-acoustics in the literature

Externally imposed acoustics

For example [Markstein & Squire, 1955], [Searby & Rochwerger, JFM 1991], [Clanet & Searby, PRL 1998], [Bychkov, PoF 1999]

No consideration of back-action of the flame onto the acoustics Focused on secondary instability

Including spontaneous acoustic field

[Pelc´ e & Rochwerger, JFM 1992]

First mathematical treatment of flow-flame-acoustics interactions Ad-hoc modelling of flame profile as cosine function Focused on primary instability

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Presentation of the Problem Geometry

Burnt

h∗

Unburnt Θ−∞ ρ−∞ Θ∞ ρ∞

y z x = f(y, z, t) ℓ∗ x

Physical assumptions

One-step irreversible chemical reaction Fuel deficient reactant: lean combustion Mixture obey state equation for perfect gas Newtonian compressible fluid

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Equations to be solved

Equations

Conservation of mass Conservation of momentum Transport equation governing the diffusion of chemical species Energy conservation State equation

Main variables of non-dimensional problem

u, ρ, p, θ M: Mach number q: heat release β: activation energy ρ−∞, θ−∞,UL δ: flame thickness q = (θ∞ − θ−∞)/θ−∞ Ma: Markstein number

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Change of coordinate, flame frame of reference

2π x = 0 x = 2πℓ∗

h∗

y

2πℓ∗ h∗

x = f(y, z, t) Non-dimensionalised coordinates Unburnt Burnt x y = π z y = −π

Change of variable (x, y, z, t) → (ξ, η, ζ, τ) ξ = x − f(y, z, t), η = y, ζ = z and τ = t

u = ueξ + v

ξ = 0 ζ η η = π η = −π 2π ξ Unburnt (ξ < 0) Burnt (ξ > 0)

∇ =

    ∂ ∂η ∂ ∂ζ     Rapha¨ el Assier (University of Manchester) Modelling combustion instabilities Keele, Sept 16, 2014 11 / 37

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2 Asymptotic analysis

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Different scalings of the problem

ξ = 0 ζ O(1) Reaction zone O(1/M) O(1/M) Acoustic zone η O (δ/β) Hydrodynamic zone O(δ) Preheat zone Acoustic zone ua, pa ua, pa U, V, P U, V, P η = π η = −π 2π ξ

Large-Activation-Energy : β ≫ 1 Low Mach Number: M ≪ 1 Thin flame : δ ≪ 1 δ β ≪ δ ≪ 1 ≪ 1 M

[Hydro. theo. flames] [Wu et al, JFM 2003]

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Acoustic zone Stretch of variable ξ = Mξ

Acoustic Equations

       ∂pa ∂τ + ∂ua ∂ ξ = R∂ua ∂τ + ∂pa ∂ ˜ ξ = ρ = R to first order in δ

Acoustic Jumps (weakly nonlinear)

   pa+

− = 0

ua+

− = Ja(τ) = q

2

∇F

2 f = F to first order in δ Acoustic-flame coupling

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Hydrodynamic zone 1/2: (u, v, p) = (U, V , P) + O(δ2) Linearising the hydrodynamic equations and the jumps to second

rder in δ leads to:

Hydrodynamic equations

               ∂U ∂ξ + ˜ ∇ · V = 0, R∂U ∂τ + ∂U ∂ξ = −∂P ∂ξ + δ Pr ∆U, R∂V ∂τ + ∂V ∂ξ = − ˜ ∇P + δ Pr ∆V ,

Jump conditions

         [U]+

−

= JU

F, V −

[V ]+

−

= JV

F, V −

[P]+

−

= JP

F, U −, V −, Ba(τ)
U − = U(0−, η, ζ, τ)

Weakly nonlinear Flame equation

∂F ∂τ =

U − − V − · ˜ ∇F − (∇ F)2/2 + δMa ˜ ∇2F + δEF

V −, F
Ba(τ) =

∂pa ∂ ξ +

−

=

−R∂ua

∂τ +

−

Acoustic-flow coupling Flow-flame coupling

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What now? Using Fourier analysis, this whole system can be simplified. Making some simplifications, the equations can be reduced to

The G-equation The M-S equation

Objective

Retaining key terms to allow for a simple model accounting for the three-way coupling physics of the problem.

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Hydrodynamic zone 2/2: (u, v, p) = (U, V , P) + O(δ) Considering only the leading order in δ, and partially linearising the flame equation leads to:

Hydrodynamic equations

               ∂U ∂ξ + ∇ · V = R∂U ∂τ + ∂U ∂ξ = −∂P ∂ξ R∂V ∂τ + ∂V ∂ξ = − ∇P

Jump conditions

         [U]+

−

= [V ]+

−

= −q

∇F
[P]+

−

= −

Ba(τ) +

qG 1 + q

F

Weakly nonlinear Flame equation

∂F ∂τ = U − − 1 2

∇F

2 + δMa ∇2F Ba(τ) = ∂pa ∂ ξ +

−

=

−R∂ua

∂τ +

−

Acoustic-flow coupling Flow-flame coupling

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3 Steady state solutions and linear instability

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Steady results Steady solution ↔ solution of steady Michelson-Sivashinsky Free parameter γ = q δMa Results agree with analytical results of the theory of N-pole solutions.

[Vaynblat & Matalon, SIAM J. Appl. Math., 2000]

−0.8 −0.6 −0.4 −0.2 0.2 0.4 −4 −π −2 −1 1 2 π 4 η

ne-pole

numerical numerical shift −0.8 −0.4 0.4 0.8 1.2 −4 −π −2 −1 1 2 π 4 η

ne-pole

two-pole numerical shift 1 shift 2

γ = 2.1 γ = 6.2

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Linear stability analysis No acoustic considerations [Vaynblat & Matalon, SIAM J. Apl. Math., 2000]

Flat ϕ1 γ3 γM γM+1 ϕM γ ϕ2 ϕ1 Steady unstable Steady stable ϕ1 ϕ2 Flat Flat Flat γ1 γ2 2.1 6.2

Considering acoustics : linear instability!

0.2 0.4 0.6 0.8 1 1.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −Im(ω) σ Growth Rate for γ = 2.1 2 4 6 8 10 12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −Im(ω) σ Growth Rate for γ = 6.2

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4 The one-equation model

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The spectral flame equation Using Fourier transforms in the η and ζ direction, we can solve the hydrodynamic system analytically. We can then reduce the hydro equations, hydro jumps and flame equation to one equation in the spectral space A∂2 F ∂τ 2 + B(k)∂ F ∂τ + C(k, Ba(τ)) F = −|k|

∇

F ⋆

∇

F

(k)

− A  

∇

F ⋆ ∂

∇

F ∂τ   (k), where A, B and C are functions known explicitly and ⋆ represents the convolution.

The whole problem reduces to two subproblems:

1

The acoustic system

2

The spectral flame equation

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5 2D unsteady numerical approach

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Acoustic resolution

System of PDEs with discontinuous coefficient

(pa)τ + (ua)˜

ξ

= (ua)τ + c2(pa)˜

ξ

= 0 , where c = c− if ˜ ξ < 0 c+ if ˜ ξ > 0

Boundary and jump conditions

˜ ξ ∈ [L−, L+] τ 0 , ua(L−, τ) = 0 pa(L+, τ) = 0 , ua+

− = Ja(τ)

pa+

− = 0

Solved by semi-analytical method of characteristics

L− C A D B L+ x

So if we know Ja(τ), we can solve the acoustic problem...

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Hydrodynamic-flame resolution

The spectral flame equation

A∂2 F ∂τ 2 + B(k)∂ F ∂τ + C(k, Ba(τ)) F = −|k|

∇

F ⋆

∇

F

(k)

− A  

∇

F ⋆ ∂

∇

F ∂τ   (k),

F(k, 0) =

F0(k) and ∂ F ∂τ (k, 0) = 0 Use FFT and IFFT to evaluate the convolutions March in time using e.g. 4th-order Adams-Bashforth So in theory if we know Ba(τ), the flame equation can be solved...

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Coupling The whole system can be solved by coupling the two methods

Solver Acoustic Solver Spectral flame Ba Ja

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6 Numerical results

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Parameters Constants of the problem Parameters σ M UL q ℓ∗ h∗ G Values 0.5 0.0007 0.24 m · s−1 5.25 1.2 m 0.1 m Results presented for two values of the “free” parameter γ: γ = 2.1 γ = 6.2

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Acoustic pressure and flame shape γ = 2.1

−4 −2 2 4 2 4 6 8 10 12 14 16 pa(L−, τ) τ ∆τ = 2.4 × 10−4

−1.5 −1 −0.5 0.5 1 1.5 2 −π −2 −1 1 2 π F(η, τ) η τ = 0.00 τ = 4.00 τ = 5.33 τ = 5.81 τ = 6.40 τ = 7.11 τ = 8.00 τ = 9.14 τ = 16.0

γ = 6.2

−15 −10 −5 5 10 15 0.5 1 1.5 2 2.5 3 pa(L−, τ) τ ∆τ = 6.1 × 10−5

−2 −1 1 2 3 4 5 −π −2 −1 1 2 π F(η, τ) η τ = 0 τ = 1.14 τ = 1.37 τ = 1.52 τ = 2.75 τ = 3.40 τ = 3.42

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Validation 1: growth rate γ = 2.1

0.2 0.4 0.6 0.8 1 1.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Growth Rate σ

1

1 −π π η Feig stability analysis numerical using Feig numerical Npert = 10

γ = 6.2

2 4 6 8 10 12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Growth Rate σ

1

1 −π π η Feig stability analysis numerical using Feig

Agreement between numerics and linear stability analysis

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Validation 2: nonlinear behaviour 1/2 γ = 2.1

−200 −150 −100 −50 50 100 150 2 4 6 8 10 12 14 16 Ba(τ) τ 130 ∆τ = 2.4 × 10−4 Amplitude estimate

γ = 6.2

−800 −600 −400 −200 200 400 600 800 1000 1200 0.5 1 1.5 2 2.5 3 Ba(τ) τ 430 450 ∆τ = 6.1 × 10−5 Amplitude estimate

ω1 dominant + pressure saturation → Ba(τ) ≈ Aa cos(ω1τ) flattening of the flame → linearisation of the flame equation around flat state

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Validation 2: nonlinear behaviour 2/2

Simplified spectral flame equation: damped Mathieu Equation

∂2 ˆ F ∂τ 2 + ν∗(k)∂ ˆ F ∂τ + [δ∗(k) + ǫ∗(k) cos(ω1τ)] ˆ F = γ = 2.1

Amplitude Aa k Unstable region Amplitude estimate stable modes 50 100 150 200 250 300 5 10

γ = 6.2

Amplitude Aa k Unstable regions Amplitude estimate unstable modes stable modes 100 200 300 400 500 600 700 5 10 15 20 Rapha¨ el Assier (University of Manchester) Modelling combustion instabilities Keele, Sept 16, 2014 27 / 37

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Physical interpretation The curved steady states are linearly unstable. For reasonably low values of γ, a flat flame (intrinsically unstable in silent environment) can survive in a noisy (spontaneous) environment. For larger values of γ, a cellular flame is forming, corresponding to a weakly-nonlinear instability (subharmonic parametric). Both cases correspond qualitatively to experimental observations

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Propagating flame with gravity Experiment [Searby, 1992] Numerics

−0.01 −0.005 0.005 0.01 0.015 0.02 0.025 −0.5 0.5 1 1.5 2 2.5 3 −0.4 −0.2 0.2 0.4 0.6 0.8 1 Acoustic pressure (bar) Relative flame position τ Acoustic pressure and flame position Curved Flat Flat

0.05

0.05 0.1 −0.5 0.5 1 1.5 −0.5 0.5 1 Acoustic pressure (bar) Relative flame position Time (secs) Acoustic pressure and flame position Flat Curved Flat Cellular

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Towards better agreement? Hint from steady states Current

−2 −1 1 2 3 4 5 −π −2 −1 1 2 π F(η) η γ = 2.1 γ = 6.2 γ = 30 γ = 45 γ = 55

Full model

−0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −π −2 −1 1 2 π F(η) η γ = 13 γ = 15 γ = 20 γ = 30 γ = 45 γ = 55

For similar values of γ, the steady states of the full model are less cusped and more compact.

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7 Instability triggering by vortical disturbances

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Vortical disturbances: another instability trigger Periodic forcing of hydrodynamic velocity U

Spectral flame equation

A∂2 ˆ F ∂τ 2 + B(k)∂ ˆ F ∂τ + C(k, τ) ˆ F = −|k|

iu ˆ

F(u)

⋆
iu ˆ

F(u)

(k)

− A

iu ˆ

F(u)

⋆
iu∂ ˆ

F ∂τ (u)

(k)

+ N0(ω, k, k0, τ, ε) ω: frequency of the disturbance k0: wavenumber of the disturbance ε: amplitude of the disturbance

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γ = 2.1 (was “stable” wihtout vortical disturbances)

−20 −15 −10 −5 5 10 15 20 10 20 30 40 50 60 70 80 Envelope of pa(L−, τ) τ

ε = 0 ε = 0.2 ε = 0.4 ε = 0.6 ε = 0.8 ε = 1 ε = 2

ε = 0

τ0 τ1 τ3 τ4 τ5 τ6 τ7 τ8 −π −2 1 2 π −1 1 2 3 4 F(η, τ) − F(π, τ) η F(η, τ) − F(π, τ) τ1 τ2 −2 −1

τ = . τ

1

= 4 . τ

2

= 5 . 3 3 τ

3

= 5 . 8 1 τ

4

= 6 . 4 τ

5

= 7 . 1 1 τ

6

= 8 . τ

7

= 9 . 1 4 τ

8

= 1 6 .

ε = 0.6

τ0 τ3 τ4 τ6 τ7 τ8 −π 1 2 π −2 −1 1 2 3 4 F(η, τ) − F(π, τ) η F(η, τ) − F(π, τ) τ1 τ2 τ3 τ4 τ5 −2 −1 0

τ = . τ

1

= 4 . 2 τ

2

= 5 . 2 τ

3

= 5 . 7 4 τ

4

= 8 . 4 τ

5

= 2 3 . τ

6

= 2 6 . 8 τ

7

= 3 2 . 2 τ

8

= 4 . 2

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8 Feedback control of combustion instabilities

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Feedback control implementation

Loud speaker Pref K(s) Burnt Unburnt

i(t) + − e(t) vc(t) G(s) K(s) pref(t)

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Adaptive control: STR algorithm [see e.g. Morgans 2009]

The controller: 1st order phase compensator

K (s) = K (s, t) = k1 (t) s + zc s + zc + k2 (t)

Update rules

k1 (t) = −γ1 t (pref (τ))2 dτ k2 (t) = +γ2 t pref

t′

k1

t′

J

t′

dt′ J (t) = t pref (τ) exp {− [zc + k2 (t − τ)] (t − τ)} dτ

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Reduction of instability effect 1/2 Without vortical disturbances

−4 −2 2 4 0 10 20 40 60 80 100 120 140 160 Envelope of pa(L−, τ) τ

1.6

0.2 0 10 50 100 150 τ

without control with control control start

103 × k2 103 × k1

With vortical disturbances

−15 −10 −5 5 10 15 0 10 20 40 60 80 100 120 140 160 180 200 Envelope of pa(L−, τ) τ

3.5

0.5 0 10 50 100 150 200 τ

with control without control control start

ε = 0.4

103 × k2 103 × k1

−20 −15 −10 −5 5 10 15 20 0 10 20 40 60 80 100 120 140 160 180 Envelope of pa(L−, τ) τ

13

2.5 10 50 100 150 200 τ

with control without control control start

ε = 1

103 × k2 103 × k1

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Reduction of instability effect 2/2 New strategy to “kill” the signal faster: sequential control Changing the convergence rate when gradient is “calm” γ1,2 → 10 × γ1,2

−20 −15 −10 −5 5 10 15 20 0 10 20 40 60 80 100 120 140 160 180 Envelope of pa(L−, τ) τ

60

10 10 50 100 150 180 τ

without control sequential control normal control control start multiply γ1,2 by 10

ε = 1

102 × k2 102 × k1

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Conclusion and perspectives

Summary

[RCA & Wu, JFM, 2014]

Implementation of “complete” flame model Analytical linear stability analysis of curved flames Unsteady coupled numerical scheme

Other things that have been done

[RCA & Wu, AIAA, 2014]

Modelling effect of weak turbulence in fresh mixture Use adaptive feedback control to suppress instabilities Implementation of O(δ) model

Future work and challenges

Refining model to get quantitative agreement with experiments Analyse the effect of not making a weak nonlinear assumption Implementation of acoustic loss in the system

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