Iitm10.Key

Technische Universität München

Workshop on Advanced Instability Methods
Jan. 18 - 21, 2010, IIT Madras, Chennai, India

Thermo-Acoustic
System Modelling and Stability Analysis:
Conventional Approaches

Wolfgang Polifke
Lehrstuhl für Thermodynamik
TU München


thanks to ...

Jakob J. Keller, Oliver Paschereit, Bruno Schuermans

Stephanie Evesque, Christoph Hirsch, Thomas Sattelmayer

Alexander Gentemann, Andreas Huber, Roland Kaess, Jan
Kopitz, Robert Leandro, Christian Pankiewitz

Matthew Juniper, Raman Sujith

W. Polifke - AIM Workshop @ IITM, Jan. 2010 2


Outline of Talk

Combustion Instabilities

Stability Analysis
Unsteady Analysis
Eigenmodes and Eigenfrequencies
Nyquist Plots
Energy Balance

System Models
CFD
Computational Acoustics
Galerkin Methods

Network Models (“Toy Models”)


A history of trouble
(from Culick, 2006)



Physics of Combustion Instabilities

a flame is a source of volume

a fluctuating flame is a (monopole) source of sound

combustion noise & combustion instability

Rayleighʼs Criterion:

˙
p Q dt > 0

Rayleigh Index



Thermodynamic interpretation of Rayleigh

Fluctuations produce acoustic energy, if Rayleigh Index > 0

p p’, u’

t
2'
3'
Q’ 2'
2"
1 3'

4' 1 4' t
3"
4" Q’
1 4"

v 2" 3" t



Thermodynamic interpretation of Rayleigh

Fluctuations produce acoustic energy, if Rayleigh Index > 0

p p’, u’

t
2'
3'
Q’ 2'
2"
1 3'

4' 1 4' t
3"
4" Q’
1 4"

v 2" 3" t

If production of energy > dissipation, instability occurs !


Thermo-Akustische Instabilität
Flame dynamics and system acoustics
Eingeschlossene Flamme
(p’, u’) Q’ (p’, u’)

Rückkopplung zwischen Fluktuationen
Premix ﬂames are velocity sensitive: ˙ ˙
Q = Q (u )
der Strömung (p’,u’) und der Wärmefreisetzung Q’
-> Selbsterregte Schwingungen ! p
System acoustics controls phase pʼ - uʼ: Z =
Stabilitätskriterium nach Rayleigh: ! d
Q!p! u" > 0. #
"
W . Polifke / divide et imp era — Ercoftac TecTag / 2
˙
p Q dt > 0


Heat release in sync with pressure
p', u'

Q'

p' Q'

most likely
unstable !



Heat release in sync with velocity
p', u'

Q'

p' Q'

stable !



Heat release lags velocity

p’, u’

Q’

p’ Q’

possibly
unstable !



Flame front kinematics
Heat release rate of a premix ﬂame:
˙
Q = ρu φSA ∆h



Modulation of Equivalence Ratio
˙
Q = ρu φSA ∆h

˙
Q ρu φ S
= + +
Q˙ ρu φ S
φ pI (t − τ ) uI (t − τ )
= − − ,
φ 2∆p uI
˙
Q pI uI
= −(1 + a) + exp(−iωτ )
Q˙ 2∆p uI


Flame / Acoustic Interactions
Fuel Air Flame Combustor
Supply Supply

Position and
Area of Flame

Burning p’, u’
u’ Q’
Velocity

Equivalence
p’ Ratio



Stability Analysis needs a System Model

“there are no unstable flames”

Rayleigh criterion is necessary, but not sufficient.

The system controls

Impedance at the flame (→ phase between velocity and pressure)

Losses of acoustic energy (dissipation and radiation).

Intensity, phase and dispersion of convective waves
(equivalence ratio, entropy).



Outline of Talk


Stability Analysis
Unsteady Analysis
Eigenfrequencies
Nyquist Plots
Energy Balance

System Models
CFD
Galerkin Methods

Network Models

Q = 385 ± 7 W; vmean = 0.0218 ± 0.0002 m/s

Instability in a Rijke tube

Experiment by Lumens, Kopitz 2006


Stability Analysis by Unsteady Simulation
1D CFD Model of Rijke tube
˙
with source term for energy Q(t) = u(t − τ )

30
gauze [m/s]

20
Velocity at [m/s]

10
c v

0

-10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
t [s]

Polifke et al, JSV, 2001
Time [s]


Unsteady Simulation

+ Simulation of (turbulent, reacting), compressible ﬂow
captures all relevant phenomena

- Computationally expensive

- Only the dominant mode is identiﬁed

- Numerical vs. physical instability

- Results can depend on initial perturbation

- Boundary conditions (acoustic impedance !) are a problem.



Stability Analysis with
Eigenmodes and Eigenfrequencies
Mode - a pattern of vibration

Eigen - German: own, peculiar, characteristic

Eigenmode / Eigenfrequency
- a mode / frequency that is easily excited in the system
- once established, an eigenmode will persist for some time

Typically, a system has many eigenmodes.
several eigenmodes may be unstable
one mode will be most unstable (“dominant mode”)



Eigenmodes / frequencies of a Rijke tube
computed with a low-order model
.
p’=0 Q p’=0

i c h x

Acoustic waves travel between “i” and “c”, “h” and “x”:

fc e−ikl 0 fi ω
= , k= .
gc 0 eikl gi c
1 p 1 p
f= +u , g= −u .
2 ρc 2 ρc


Coupling relations at the heat source
.
p’=0 Q p’=0

i c h x

At the heat source:
• no pressure drop, ph = pc
• time-lagged heat release, uh (t) = uc (t) + nuc (t − τ ).

ρh ch
(fh + gh ) = (fc + gc ),
ρc cc
fh − gh = 1 + ne −iωτ
(fc − gc ),


Boundary conditions
.
p’=0 Q p’=0

i c h x

open / closed ends:

p = 0 → f + g = 0,
u = 0 → f − g = 0,



Rijke tube system matrix
.
u p’=00
= Q p’=0

i c h x

  fi
 
0

Matrix
of  .  =  . .
.   . 
. .
 
Coeﬃcients gx 0

Eigenfrequencies fulﬁll Det (S(ω)) = 0, which yields:

cos kc lc cos kh lh − ξ sin kc lc sin kh lh 1 + n e−iωτ = 0,


Eigenfrequency vs. time lag (n = 0.1)

Re w Im w
1.03 0.04
1.02
1.01 0.02
1 2 3 4 tau Pi 0.5 1 1.5 2 2.5 3 tau Pi
0.99
0.98 -0.02
0.97

exact solution (--------------) vs. weak coupling approximation (- - - -)



Eigenfrequency vs. time lag (n = 0.3)

Re w Im w

1.1 0.1
1.05 0.05
1 2 3 4 tau Pi 0.5 1 1.5 2 2.5 3 tau Pi
0.95 -0.05
0.9
-0.1

exact solution (--------------) vs. weak coupling approximation (- - - -)



Stability map Rijke tube:
.
p’=0 Q p’=0
u =0

i c h x

cos kc lc cos kh lh − ξ sin kc lc sin kh lh 1 + ne−iωt = 0,

m=3
Mode - #

m=2

m=1

m=0

0 1π 2π ω0τ



Remarks on dynamic stability analysis

+ Results as presented agree with Rayleigh - because losses
are neglected. Could be included easily!

+ Build system of equations in software → network model

- Closed-form expressions for the transfer matrices are known
only for the simplest conﬁgurations.

- Eigenfrequencies give only asymptotic, long-time behaviour
→ not adequate for non-normal analysis.

- Iterative search for eigenfrequencies in complex plane can
be tedious and incomplete !

- Matrix coefﬁcients must be known for complex-valued
frequencies → Problem for TFMs from experimental data

N; ! < 4 ?*) 5)9'% ); #$ #* -"' (#+"- ",9;=59,*'B: -"'* -"' %1%-'6 #% %-,
" < 47
N; ! % 4: -"'* -"' 5)#*- !F 9#'% #*%#2' -"' G1H0#%- $)*-)0( ,*2 -"'(' 6
Nyquist Criterion in"=59,*'7 E#+0(' @7A #990%-(,-'% 5)%#-#.' ,*2 *'+,-#.' '*$#(
#* -"' (#+"-=",9; Control Theory

s
s G
HG H

−1 −1

N =1 N =0

Cauchyʼs argument principle: N = Z -); 5)%#-#.' ,*2 *'+,-#.' '*$#($9'6'*-%
!"#$%& '()( OP,659'% P )
N − # of anticlockG1H0#%- $)*-)0(7
clockwise encirclements of critical point (-1,i0)
Z − # of zeros of the open loop transfer function G(s)
P − # of poles in the right half plane
K% , 3*,9 ('6,(J: !' *)-' -",- -"' 5)(-#)* ); -"' G1H0#%- 5,-" ;)( "
Stable if N-"' P !
= #$=59,*'7 Q-"'(!#%': -"' #*#-#,9 .,90' -"')('6 +#.'% ;)( -"' ('%5)*%'
' ?& < 4MB < 9#6 "#$ ( ?"B


Open loop transfer function of a network model
(Polifke et al., 1997, Kopitz & Polifke, 2008)

Fuel Supply

Burner 1
Air Supply Combustor
& Flame

With fu (ω)
G(ω) ≡ − ,
fd
eigenfrequencies are mapped to the critical point -1



OLTF G(ω) as conformal mapping
(Polifke et al., 1997, Kopitz & Polifke, 2008)

Im(ω) ω Im(G(ω))
G(ω)

Re(ω) + 2i

Re(ω) + i
ωm
-1 Re(G(ω))
Re(ω)

Nyquist Criterion:
A mode is stable if the critical point “-1” lies to the left of
the image curve of the real axis.



Stability analysis with Nyquist plot

- what is “passage of the critical point” !?

- no rigorous proof !

+ this is not the “Barkhausen Criterion”
The Barkhausen Stability Criterion is simple, intuitive, and wrong.
(http://web.mit.edu/klund)

+ it is sufﬁcient to know transfer matrices / transfer functions
for real-valued frequencies ω ∈ |R.

+ no iterative searches for eigenmodes.

+ growth rate can be estimated from the OLTF



Eigenmodes of Rijke tube
o - Iterative search for eigenmodes
◆ - Nyquist plot

10

8

6

4
Growth Rate [%]

2

0

-2

-4

-6

-8

-10
0 1000 2000 3000
Frequency [Hz]



Outline of Talk


Stability Analysis
Unsteady Analysis
Eigenfrequencies
Nyquist Plots
Energy Balance

System Models
CFD
Galerkin Methods

Network Models


System Modelling

Time Domain Frequency Domain
Finite Element
Finite Volume,

CFD Computational Acoustics
nonlinear PDEs - linearized PDEs -
Navier-Stokes often extended Helmholtz
Mode-Based

Galerkin Methods Network Models
ODE algebraic equations

Nonlinear Linearized Equations


Computational Fluid Dynamics

Idea: use LES (or URANS) for Unsteady Analysis

+ conceptually straightforward !

- high computational cost !

- acoustic boundary conditions !?

- only dominant mode is detected.

- insight does not come easy.

Nota bene:
CFD can also be used to determine transfer functions /
matrices or to compute the OLTF !


Computational acoustics

Linearized Euler Equations:

∂ρ ∂ρ ∂ρ ∂ui ∂ui
+ ui + ui +ρ +ρ = 0,
∂t ∂xi ∂xi ∂xi ∂xi
∂ui ∂ui ∂ui 1 ∂p ρ ∂p
+ uj + uj + − 2 = 0,
∂t ∂xj ∂xj ρ ∂xi ρ ∂xi
∂p ∂p ∂p ∂ui ∂ui
+ ui + ui + γp + γp = (γ − 1) q .
˙
∂t ∂xi ∂xi ∂xi ∂xi

+ contain acoustic, entropy and vorticity modes (→APEs)

- numerically unstable



Starting point: equation for acoustic perturbations:

1 D2 p ∂ 1 ∂p γ − 1 Dq
˙
2 Dt2
−ρ = 2
.
c ∂xi ρ ∂xi c Dt

Add a model for the heat release ﬂuctuations:

q (x, t)
˙ ub (t − τ (x))
=n .
˙
q(x) ub

Apply FE-Solver (time-marching) for Unsteady Analysis !

(Pankiewitz et al, ʼ02 - ʼ04)


Eigenmodes of annular combustor
(1,0,0) (1,1,0) (1,0,0) cies and
0.8
agreeme
0.7

0.6 CONCL
We
0.5 coustic
f

especial
0.4 simulati
bitrary g
0.3 the prop
indicate
0.2 and hav
0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4
τ

FE: triangles, low-order: circles REFER
Figure 5. FREQUENCIES AND
Unstable: ﬁlled symbols. CORRESPONDING MODE TYPES
[1] S. H
FOR DIFFERENT DELAY TIMES. ( , ) LOW ORDER MODEL, (•,◦)
W. Polifke - AIM Workshop @ IITM, Jan. 2010
TIME DOMAIN SIMULATION. ( ,•) UNSTABLE MODE, (◦, ) STABLE 39 mix


FE / iterative subspace method for eigenmodes
(Benoit and Nicoud ʼ05, Sensiau et al. 2008)

Discretize eqn. for pressure perturbation (with source term)

[A][P ] + ω[B][P ] + ω 2 [P ] = [D(ω)][P ].

1
a) expand in thermo-acoustic coupling strength ≡ n(x) dV.
V V

ωm = ωm + ωm + O( 2 ),
(0) (1)

pm = pm + pm + O( 2 ).
(0) (1)

b) solve iteratively for sequence ω (k)

[A] − [D(ω (k−1) )] [P ] + ω (k) [B][P ] + (ω (k) )2 [P ] = 0.

Table I. E ect of reference position value and the grid resolution for the ÿrst eigen
Technische Universität München = 0:003,
frequency; = 10−4 s. Cross ‘X’ indicates unfeasible calculation.

Results for “Rijke Tube” 0:24 m
x ref = x ref = 0:249 m x ref = 0:25 m
Benoit and Nicoud ʻ05
Coarse mesh (561 nodes) 270:4 − 0:087i X X
Reÿned mesh (5231 nodes) 271:3 − 0:093i 271:4 − 0:088i X
Theoretical 271:5 − 0:098i 271:6 − 0:088i 271:6 − 0:088i
using the weak-coupling expansion

ε=0.003 ε=0.33 ε=1.6
0.4 40 200
expansion expansion expansion
theoretical theoretical 100 theoretical
0.2 20
Growth rate

Growth rate

Growth rate
0
0 0

-100
-0.2 -20

-200
-0.4 -40
-300
0 500 1000 1500 2000 0 500 1000 1500 2000 0 500 1000 1500 2000
Frequency Frequency Frequency

Figure 2. Representation in the complex plane of the theoretical and computed eigen frequencies.

available in Figure 2 and displayed in the complex plane. As expected, the computed


Computational acoustics

+ time domain: straightforward

+ fq domain: identify both stable and unstable eigenmodes

+ modest computational cost.

- acoustic boundary conditions, mean ﬂow effects, losses !?

- needs input on ﬂame dynamics.

Stay tuned !



System Modelling

Finite Element
Finite Volume,

Mode-Based




Galerkin method

∂ 2 ψm
1D Helmholtz-equation without sources: 2
+ km ψm = 0
2
∂x

Eigenmodes ψm (x) = sin(km x) are orthogonal
L
ψm ψn dx = δmn .
0

“Project” eigenmodes on PDE with source:
2
d ηm γ−1 L

2
+ ωm ηm =
2
2
q (x) ψm (x) dx.
˙
dt Em 0



Galerkin

+ very efﬁcient

+ can handle non-linearities

+ for complicated geometries, expansion functions ψ can be
computed with FE

+ eigenmodes of full problem need not be close to the ψʼs of
the homogeneous problem

- non-normal modes for non-trival boundary conditions

- input on ﬂame dynamics is needed, determination of source
term may be non-trivial.



Outline of Talk


Stability Analysis
Unsteady Analysis
Eigenfrequencies
Nyquist Plots
Energy Balance

System Models
CFD
Galerkin Methods

Network Models


Network models

Fuel Supply

Air Supply Burner Flame Combustor

  f1

0
 
Matrix
of  .  =  . .
.   . 
. .
 
Coeﬃcients gN 0



Contoured duct

A(x)

1 2 3 4
(x)

e−ikx+ l1 0 1 0 e−ikx+ l2 0 1 0
M= −ikx− l1 −ikx− l2 ···
0 e 0 α1 0 e 0 αN



Non-plane modes in thin annular duct

fd e−ikx+ l 0 fu
= −ikx− l .
gd 0 e gu
 
2
ω/c  k⊥ m
kx± = 2
−M ± 1− (1 − M 2 ) , k⊥ ≡ .
1−M ω/c R



Compact element ( l λ )

l

Au
Ad

xu xd

   
p p
 ρc  = 1 −i k leﬀ − ζM  ρc  .
−i k lred α
u d
u u

xd
Au
leﬀ ≈ dx.
xu A(x)


Transfer matrix of (compact) ﬂame

Linearize Rankine-Hugoniot relations
(conservation of mass, momentum, energy across discontinuity)

p p TH uc ˙
Q
ξ = − − 1 u c Mc + ,
ρc h ρc c TC uc Q˙

TH ˙
Q pc
uh = uc + − 1 uc − .
TC Q˙ pc

closure with ﬂame frequency response,
˙
Q u
= F (ω)
Q˙ u



Example network calculations

Acoustics in duct system with low-Mach-# ﬂow

x
L

open end - duct - area change - duct - open end



Eigenfrequencies
with reﬂection coefﬁcient r = -1

$"$ $

$
#"*
/012345267238029:4-4!

#"*

/0123450637
#"( #"(

#"&
#"&
#"$

#
#"$
!"*

!"( #
! !"# !"$ !"% !"& !"' !"( !") !"* !"+ # ! !"# !"$ !"% !"& !"' !"( !") !"* !"+ #
,-. ,-.

How is instability possible in a system without energy source?



Eigenfrequencies
with energy-conserving boundary conditions

$"$ #

$
!"+'
/012345267238029:4-4!

#"*

/0123450637
#"( !"+
#"&
!"*'
#"$

#
!"*
!"*

!"( !")'
! !"# !"$ !"% !"& !"' !"( !") !"* !"+ # ! !"# !"$ !"% !"& !"' !"( !") !"* !"+ #
,-. ,-.

1 2
p+ ρ = p∞ ,
2
p
+M = 0,
ρc
ƒ (1 + M) + g(1 − M) = 0.


Riemann Twist
The Riemann Twist

!$
!"
#%

&
!% #$

&' #"
&"
&$

Messung von Transfermatrixen (3)
Prof. Wolfgang Polifke


Network Models

+ Fast & Flexible, low computational effort

+ Great for qualitative / exploratory studies

- Not suitable for many geometries of applied interest

- Only in frequency domain ?

- Non-linear phenomena ?

- Non-normal phenomena ?

→ n3l Workshop in Munich


Summary

Finite Element
Finite Volume,

Mode-Based




My questions on non-normality ...

Real-world conﬁgurations:
how typical / important are n-n effects?
which methods / tools are adequate for the study of n-n effects?

How to adopt exisiting modelling approaches:
how to formulate appropriate (time-domain) network models?
what is the proper norm?
what are the physically realistic / permissible initial states ?

How to describe / identify the ﬂame dynamics ?


Announcement and Call for Papers

Summer School and Workshop on
Non-Normal and Nonlinear Effects in
Aero- and Thermoacoustics
In aero-acoustics, nonlinear effects play an important role in generation as well as
dissipation of sound. Stability limits and limit cycle amplitudes of self-excited aero- or
thermoacoustic instabilities are influenced by nonlinearities. For thermoacoustic
interactions, standard linear modal analysis can in general not predict the response
of the system to finite amplitude perturbations due to the non-normality of the
corresponding evolution operator and nonorthogonality of eigenmodes.

At TU München, a Summer School / Workshop on non-normality and nonlinearity in
aero- and thermoacoustics will be held in May 2010.
During the Summer School (May 17 and 18), a series of invited lectures will give an
introduction to the workshop topics and present the state of the art. Expected
http://www.td.mw.tum.de/n3l-conf-2010
audience are doctoral students with some background in fluid mechanics, flow
instabilities, aero- or thermoacoustics, or combustion. Of course, more experienced
researchers interested in the workshop topics are also welcome.
Tentative list of speakers:
W. Polifke - AIM Workshop (École Jan. 2010 Lyon)
C. Bailly @ IITM, Centrale 59

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