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Application of receptivity and sensitivity analysis tothermoacoustic instability
Matthew P. Juniper∗ & Luca MagriEngineering Department, University of Cambridge, CB2 1PZ, UK
6th June 2014
Contents
1 Introduction 31.1 Receptivity and sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . 31.2 Thermoacoustic instability . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Direct and adjoint governing equations 52.1 Definition of the direct governing equations . . . . . . . . . . . . . . . . . . 52.2 Definition of the adjoint governing equations . . . . . . . . . . . . . . . . . 52.3 Derivation of the adjoint equations from the direct equations . . . . . . . . 62.4 The bi-orthogonality condition . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Receptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 A pedagogical example 113.1 The direct governing equations . . . . . . . . . . . . . . . . . . . . . . . . 113.2 The adjoint governing equations . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Direct and adjoint eigenmodes . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Application to Thermoacoustics 134.1 Reduced order models in thermoacoustics . . . . . . . . . . . . . . . . . . . 134.2 Hot wire Rijke tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 Burke-Schumann flame Rijke tube . . . . . . . . . . . . . . . . . . . . . . . 18
5 Concluding remarks 23
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1 INTRODUCTION
1 Introduction
1.1 Receptivity and sensitivity analysis
Receptivity and sensitivity analysis is a branch of linear stability analysis. In stabilityanalysis, one typically calculates a linear system’s eigenmodes. These encapsulate thefrequency, growth rate, and mode shape of each natural mode of the system. Receptivityanalysis then quantifies the receptivity of each mode to external (open loop) forcing.Sensitivity analysis quantifies the sensitivity of each mode either to internal feedback,which is known as the structural sensitivity, or to changes in the base state, which is knownas the base state sensitivity. Sensitivity analysis can be performed by finite difference -e.g. by computing the system’s eigenvalues at two slightly different base states and thencalculating the gradient with respect to the change between the two states - but thisis computationally expensive and prone to numerical error. A more efficient and moreaccurate method is to use adjoint equations, which is the subject of this lecture.
The use of adjoint equations in flow instability dates back to the early 1990s (Hill,1992a,b; Chomaz, 1993; Hill, 1995a) but did not become widespread until the late 2000s(Giannetti and Luchini, 2007; Marquet et al., 2008). The concepts are explained in threereview articles (Chomaz, 2005; Sipp et al., 2010; Luchini and Bottaro, 2014), which areessential reading. Pedagogical examples with Matlab tutorials designed for graduate stu-dents who are new to the field can be found in Schmid and Brandt (2014).
1.2 Thermoacoustic instability
When a fluctuating source of heat interacts with acoustic waves, for example inside com-bustion chambers of aeroplane and rocket engines, thermoacoustic oscillations can occur.In the cyclic process created by the acoustic waves, mechanical energy is fed into oscilla-tions over one cycle if higher heat release occurs at moments of higher pressure and lowerheat release occurs at moments of lower pressure. This is because the extra heat release atthe moments of high pressure causes more work to be extracted during the decompressionphase than was required during the preceding compression phase. The amplitude of thethermo-acoustic oscillations therefore grows with time if the heat release fluctuations arewithin one quarter cycle of the pressure fluctuations. For a linear stability analysis, onetypically examines small acoustic perturbations around a steady flow. If these grow intime then this steady flow is thermoacoustically unstable.
Thermoacoustic oscillations were first documented in the 1880s (Rayleigh, 1880, 1878)but at that stage were little more than a curiosity discovered during the glass-blowingprocess. They became a serious research subject from the late 1930s, particularly dur-ing the US Apollo program in the 1960s, during which thermoacoustic oscillations wereone of the most important challenges facing the program. Research during this periodis extensively reviewed in the early chapters of Culick (2006). Recently, as NOx emis-sions targets for civil aircraft and power generation have become stricter, manufacturershave attempted to lower the fuel to air ratio in the combustion chambers of gas turbines.This makes flames more receptive to acoustic perturbations and thereby increases theirpropensity for thermoacoustic oscillations (Lieuwen, 2012). Despite over 60 years of re-search, thermoacoustic oscillations still present one of the biggest problems facing rocket
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1 INTRODUCTION 1.2 Thermoacoustics
and aircraft engine manufacturers. Our intention is that, by applying receptivity andsensitivity analysis in this field, new insights into control of thermacoustic oscillations canbe achieved.
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2 DIRECT AND ADJOINT GOVERNING EQUATIONS
2 Direct and adjoint governing equations
2.1 Definition of the direct governing equations
In a linear stability analysis, the direct equations are formed by considering the behaviourof small perturbations around a base state. These perturbations are expressed as a statevector q whose time evolution is governed by the direct equation:
A∂q
∂t− Lq = s exp (σst) , (1)
where the term on the right hand side is a forcing signal. In this forcing signal, s is thespatial distribution and σs is its growth rate / frequency. For a thermoacoustic system,the state vector, q, is equal to (F, u, p)T , where F contains the flame variables, such asthe mixture fraction for diffusion flames, u is the acoustic velocity, and p is the acousticpressure.
2.2 Definition of the adjoint governing equations
The adjoint state vector, q+, evolves according to the adjoint equation:
A+∂+q+
∂t− L+q+ = 0, (2)
If (1) and (2) represent the equations for continuous distributions q and q+ then A, L, A+,and L+ are operators. In this case, the adjoint operators and equations are analyticallyderived and then numerically discretized (CA, discretization of Continuous Adjoints). If(1) and (2) represent the equations for numerically discretized systems then A, L, A+,and L+ are matrices (in bold from now on) and q = (G,η,α)T , where G,η,α representthe discretization of the heat source, the acoustic velocity and the acoustic pressure,respectively. In this case, the adjoint matrices and functions are directly derived from thenumerically discretized direct system (DA, Discrete Adjoints).
When we follow the CA approach, the adjoint systems are defined through a bilinearform1 [·, ·], such that:[
q+,
(A∂
∂t− L
)q
]−[(
A+∂+
∂t− L+
)q+,q
]= constant, (3)
which, in this paper, defines an inner product2. For brevity, in this lecture we define thefollowing bracket operators to represent inner products:
〈a,b〉 =1
V
∫V
a∗ · b dV, (4)
〈〈a,b〉〉 =1
∂V
∫∂V
a∗ · b d∂V, (5)
[a,b] =1
T
1
V
∫ T
0
∫V
a∗ · b dV dt, (6)
1The calculation of the adjoint function depends on the choice of the bilinear form.2A discussion of possible definitions of the adjoint operator is reported in the supplementary material
of Luchini and Bottaro (2014).
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2 DIRECT AND ADJOINT GOVERNING EQUATIONS 2.3 Adjoint derivation
where a and b are arbitrary functions in the function space in which the problem isdefined; V is the space domain and ∂V is its boundary; t is the time; and ∗ is the complexconjugate. Therefore, in this paper, the adjoint operator is defined through the followingrelation:∫
T
∫V
q+∗·(
A∂
∂t− L
)q dV dt −
∫T
∫V
(A+∂
+
∂t− L+
)∗q+∗·q dV dt = constant.
(7)
2.3 Derivation of the adjoint equations from the direct equa-tions
To find the adjoint operator with the CA approach we perform integration by parts of(7). The above relation is an elaboration of the generalized Green’s identity (Dennery andKrzywicky, 1996; Magri and Juniper, 2013c). The adjoint boundary / initial conditions,which arise from integration by parts of (7), are defined such that the constant on the RHSis zero. By integration by parts, we find the important result that −A∂/∂t = A+∂+/∂t.Setting A+ = A∗, then −∂/∂t = ∂+/∂t. In other words, the adjoint operator must evolvebackwards in time for a problem to be well-posed.
When we follow the DA approach, the adjoint matrix, L+ij, is defined through the
Euclidean product (in Einstein’s notation)
q+∗i Lijqj − qiL+∗ij q
+∗j = 0. (8)
The above terms are scalars, so the transposition does not change the equation. Therefore,we take the transpose of the second term and equate it to the first term:
q+∗i Lijqj = (qiL+∗ij q
+∗j )T = q+∗i L+∗
ji qj, (9)
=⇒ L+ij = L∗ji. (10)
This shows that the adjoint matrix is the conjugate transpose of the direct matrix. Fromnow on, when we use the DA method, we denote the direct state vector as χ and thecorresponding adjoint vector as ξ.
A comparison between the numerical truncation errors between the CA and DA meth-ods is illustrated in Magri and Juniper (2013c). Although the two formulations shouldconverge in principle, it has been shown that convergence is not guaranteed a priori (Vogeland Wade, 1995; Bewley, 2001; Pierce and Giles, 2004). For the thermo-acoustic systemconsidered in this paper, the DA method is more accurate and easier to implement. How-ever, we show the results obtained via the CA method in order to describe how the methodworks.
2.4 The bi-orthogonality condition
In stability/receptivity analysis, we perform a Laplace transform and consider the eigen-value problems of (1) and (2):
σAq− Lq = 0, (11)
σ+A+q+ − L+q+ = 0, (12)
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2.5 Receptivity 2 DIRECT AND ADJOINT GOVERNING EQUATIONS
where q and q+ are the eigenfunctions, and σ and σ+ are the eigenvalues. A very im-portant property of the adjoint and direct eigenpairs {σi, qi} and {σ+
j , q+j } is the bi-
orthogonality condition: (σi − σ+∗
j
) ⟨q+j ,Aqi
⟩= 0, (13)
which states that the inner product⟨q+j ,Aqi
⟩is zero for every pair of eigenfunctions except
when i = j, as long as σ+j = σ∗j , in accordance with Salwen and Grosch (1981). This means
that the adjoint operator’s spectrum is the complex conjugate of the direct operator’sspectrum. This information serves as good check when validating adjoint algorithms.
2.5 Receptivity
Here, we show that the adjoint eigenfunction quantifies the system’s receptivity to open-loop forcing. The receptivity of boundary layers has been calculated from the Orr-Sommerfeld equation by Salwen and Grosch (1981) and Hill (1995b). Another elegant for-mulation of the receptivity problem, based on the inverse Laplace transform and residuestheorem, is described by Giannetti and Luchini (2007, pp. 172–174). A more generalapproach to the receptivity problem via adjoint equations can be found, among others,in Marino and Luchini (2009, p. 42), Meliga et al. (2009, p. 605), Sipp et al. (2010, p.10), and Luchini and Bottaro (2014). These studies all concern flow stability. In this lec-ture, we extend these methods to be able to consider thermoacoustic instability using theformulation by Chandler (2010, pp. 63–68), which is sufficiently general for our purposes.
Let q be a time-dependent state vector defined in a suitable function space and L bea linear operator that also encapsulates the boundary conditions. We consider the con-tinuous inhomogeneous linear problem (1), with harmonic forcing at complex frequency,σs, and initial condition q(t = 0) = q0. The general solution of this problem is (in theCA approach):
q = qs exp (σst) + qd + qcs, (14)
where qs is the spatially varying part of the particular solution, qd =∑N
j βjqj exp (σjt)is the discrete-eigenmodes solution, and qcs is the continuous spectrum solution. Oden(1979) and Kato (1980) contain rigorous mathematical treatises of spectral decomposi-tion of linear operators. Note that an open loop forcing term, such as s exp(σst), does notchange the spectrum of the operator. Assuming that the discrete eigenmodes and contin-uous spectrum form a complete basis, then the particular and homogenous solutions canbe projected onto these spaces. Invoking the adjoint eigenfunction and taking advantageof the bi-orthogonality condition (13), we rearrange (14) as:
q =N∑j=1
⟨q+j ,q0 exp (σjt) + s
exp (σst)− exp (σjt)
σs − σj
⟩qj⟨
q+j ,Aqj
⟩ + proj[qs, qcs] exp (σst) + qcs,
(15)
where proj[qs, qcs] is the projection of the forcing term onto the continuous spectrum. Thesolution (15) is valid for a continuous operator (e.g. Orr-Sommerfeld) in an unboundedor semi-unbounded domain. In these notes we wish to consider reduced-order thermo-acoustic systems in which acoustic and combustion domains are bounded. In this case,there is no continuous spectrum, therefore qcs = 0.
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2 DIRECT AND ADJOINT GOVERNING EQUATIONS 2.5 Receptivity
The first term of (15) provides a physical interpretation of the adjoint eigenfunction.The response of the jth component of q in the long-time limit increases (i) as the forcingfrequency, σs, approaches the jth eigenvalue, σj, and (ii) as the spatial structure of theforcing, s, approaches the spatial structure of the adjoint eigenfunction, q+
j . For constantamplitude forcing (Re(σs) = 0) of a system with one unstable eigenfunction (Re(σ1)> 0)the linear response (15), in the limit t→∞, reduces to
q =
⟨q+1 ,q0 −
s
σs − σ1
⟩q1⟨
q+1 ,Aq1
⟩ exp (σ1t) . (16)
This shows that the linear response has the frequency/growth rate, σ1, and the spatialstructure, q1, of the most unstable direct eigenfunction. Furthermore, the magnitudeof this response is determined by the extent to which the spatial structure of the initialconditions, q0, and the spatial structure of the forcing, s, project onto the spatial structureof the adjoint eigenfunction, q+
1 . In other words, the flow behaves as an oscillator withan intrinsic frequency, growth rate, and shape (Huerre and Monkewitz, 1990) and thecorresponding adjoint shape quantifies the sensitivity of this oscillation to changes in thespatial structure of the forcing or initial condition. For constant amplitude forcing actingon a stable system (i.e. with no unstable eigenfunctions), the linear response in the limitt→∞ reduces to
q =N∑j=1
⟨q+j ,
s
σs − σj
⟩qj⟨
q+j ,Aqj
⟩ exp (σst) . (17)
This shows that the linear response is at the forcing frequency, σs, and that the spatialstructure contains contributions from all eigenfunctions, qj. Furthermore, the amplitudeof each eigenfunction’s contribution increases (i) as σs approaches one eigenvalue, σj and(ii) as the spatial structure of the forcing, s, approaches the spatial structure of thatadjoint eigenfunction, q+
1 . This shows that the sensitivity of the response of each modeto changes in the spatial structure of the forcing is quantified by each (corresponding)adjoint eigenfunction, q+
j . This is seen most clearly by considering the special case inwhich the forcing term has σs → σj. By applying l’Hopital’s rule to (15) for t → ∞ thesolution is
q =⟨q+j , s⟩ qj⟨
q+j ,Aqj
⟩t exp (σjt) . (18)
The sensitivity of the amplitude of the response to changes in the spatial distribution ofthe forcing term is the adjoint eigenfunction, q+
j . This is how we define receptivity.If we consider the discretized system in the inhomogeneous form (DA approach), seek-
ing solutions of the form χ exp(σst) then we obtain:
(Aσs − L)χ = g + Aχ(0), (19)
where g is the discretized source term s. In the discretized system, the spectrum consistsof a finite set of points. Assuming that the eigenvectors form a complete set, the solutionis decomposed as follows:
χ =N∑j=1
αjχj. (20)
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2.6 Sensitivity 2 DIRECT AND ADJOINT GOVERNING EQUATIONS
Substituting eq. (20) back into the discretized version of eq. (11) and premultiplying by
the conjugate adjoint eigenvector, ξ∗j , (which is the conjugate left eigenvector), we obtain
N∑j=1
ξ∗j ·(−L + Aσs)αjχj = ξ
∗j ·(g + Aχ(0)). (21)
Finally, recalling the bi-orthogonality condition for generalized eigenvalues problems (13),we obtain
χ =1
ξ∗·Aχ
N∑j=1
1
σs − σjξ∗j ·(g + Aχ(0))χj (22)
This results is analogous to (15) for discretized systems.An alternative approach is via constrained optimization, which is particularly suitable
to thermoacoustics. In this approach, we define a Lagrangian functional as
L(q,q0,q
+,q+0
)= J (q0,q∂V ,q)−
[q+,A∂/∂tq− Lq
]−⟨q+0 ,q(0)− q0
⟩+ . . .
. . .−⟨⟨
q+∂V ,q(∂V )− q∂V
⟩⟩, (23)
where J is the cost functional to optimize, which is the eigenvalue σ in this lecture, andq∂V is the boundary condition. The first variation of L, along the generic direction δq isdefined through the Gateaux derivative, as
δLδqδq = lim
ε→0
L (q + εδq)− L (q)
ε. (24)
By imposing the first variations of L with respect to the state vector, q, to be zero,we define the adjoint equations (12), whose eigenfunctions can be regarded as Lagrangemultipliers from a constrained optimization perspective (Gunzburger, 1997). In a ther-moacoustic system, u+ is the Lagrange multiplier of the acoustic momentum equation,revealing the locations where the thermo-acoustic system is most receptive to forcing (e.g.acoustic forcing); p+ is the Lagrange multiplier of the acoustic energy equation, revealingthe locations where the system is most receptive to heat injection; F+ is the Lagrangemultiplier of the flame equation. If the flame is a fast-chemistry diffusion flame, thenF+ reveals in which regions the flame is most receptive to species injection (Magri andJuniper, 2014). The adjoint boundary conditions can be interpreted similarly.
2.6 Sensitivity
In a sensitivity analysis, one calculates how much an eigenvalue changes when the operator,L, changes slightly. With the CA approach (Continuous Adjoint) we study the continuoussystem - i.e. before numerical discretization. The direct operator, L, is perturbed toL+εδL. Consequently, the eigenvalues are perturbed to σj+εδσj, the direct eigenfunctionsto qj + εδqj, and the adjoint eigenfunctions to q+
j + εδq+j . We substitute these into the
continuous eigenproblem (11) and examine the terms of order ε:
(σjA− L)εδqj + (εδσjA− εδL)qj = 0 (25)
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2 DIRECT AND ADJOINT GOVERNING EQUATIONS 2.6 Sensitivity
Then we pre-multiply by the corresponding adjoint eigenvector:
〈q+j , (σjA− L)εδqj〉+ 〈q+
j , (εδσjA− εδL)qj〉 = 0 (26)
The first term on the left hand side is zero because taking the inner products of Aδqj andLδqj with q+
j extracts only the components that are parallel to qj (see eqn. (13)), forwhich (σjA− L)qj = 0. This means that the eigenvalue drift, at order ε, is
δσj =〈q+
j , δLqj〉〈q+
j ,Aqj〉. (27)
Note that the denominator is always different from zero because the dimension of theadjoint space is equal to the original space’s dimension, under not restrictive conditionsMaddox (1988).
With the DA approach (Discrete Adjoint) we study the discretized system, representedby the matrices A, L. From (10) we can infer that the adjoint eigenvector is the conjugate
left eigenvector of the system, i.e. ξ∗j ·(σjA−L) = 0. The bi-orthogonality property ensues
directly from definition of right and left eigenvectors
ξ∗j ·A · χi = δij. (28)
Now, let us consider a perturbation to the direct operator, as before:((σi + εδσi)A− (L + εδL)
)(χi + εδχi) = 0. (29)
At order ε this is:
(σiA− L)εδχi + (εδσiA− εδL)χi = 0. (30)
Now we pre-multiply by the ith adjoint eigenvector:
ξ∗i · (σiA− L)εδχi + ξ
∗i · (εδσiA− εδL)χi = 0. (31)
The first term is zero because of (28). The second term becomes:
δσiξ∗i ·A · χi = ξ
∗i · Lχi, (32)
which can be rearranged as:
δσi =ξ∗i · δLχiξ∗i ·A · χi
. (33)
Both eqns. (27) and (33) show that, once the perturbation operator/matrix is known,we can evaluate the first order eigenvalue drift exactly. To do this, we need to solve oneeigenvalue problem to obtain the direct eigenvalue and direct eigenfunction and then an-other eigenvalue problem to obtain the corresponding adjoint eigenfunction. (Althoughthe eigenvalue is already known for the second eigenvalue problem, it is usually quick-est to solve this eigenvalue problem from scratch.) This greatly reduces the number ofcomputations, compared with a finite difference calculation. Equations (27) and (33) are
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3 A PEDAGOGICAL EXAMPLE
well known results from perturbation methods (Stewart and Sun, 1990; Hinch, 1991).Although the adjoint equation depends on the choice of the bilinear form, as explainedpreviously, (27) and (33) do not.
When the perturbation operator, δL, represents a perturbation to the base state pa-rameters, we label this process a base-state sensitivity analysis. When δL represents aperturbation introduced by additional feedback between the direct variables and the lin-earized equations, e.g. by a passive feedback device, then we label it a structural sensitivityanalysis.
3 A pedagogical example
3.1 The direct governing equations
We will illustrate adjoint sensitivity analysis with a simple example, for which analyticalsolutions are available. This is a lightly-damped linear oscillator consisting of a mass-spring-damper system, whose displacement, x, obeys the governing equation:
d2x
dt2+ b
dx
dt+ cx = 0, (34)
given some initial conditions. This second order ODE can be written as two first orderODEs by introducing the velocity, y:
dx
dt= y, (35)
dy
dt= −by − cx. (36)
We define the state vector q and the operator L, which in this case is a matrix of constantcoefficients, such that (1) can be written as:
dq
dt− Lq = 0, (37)
where
q =
[xy
], Lq =
[0 1−c −b
] [xy
]. (38)
3.2 The adjoint governing equations
We define the adjoint operator, L+, through (3), which gives∫ T
0
x+∗(y) + y+
∗(−by − cx) dt =
∫ T
0
x+∗(y)(L+q+)∗xx+ (L+q+)∗yy dt,(39)
=⇒∫ T
0
x+∗(y)(−c∗y+)x∗ + (x+ − b∗y+)y∗ dt =
∫ T
0
x+∗(y)(L+q+)xx
∗ + (L+q+)yy∗ dt.(40)
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3 A PEDAGOGICAL EXAMPLE 3.3 Eigenmodes
Note that here the bilinear form does not involve spatial integration over V because theproblem is governed by ODEs. By inspection:
q+ =
[x+
y+
], L+q+ =
[0 −c∗1 −b∗
] [x+
y+
], (41)
so the adjoint governing equations are:
−dx+
dt= −c∗y+, (42)
−dy+
dt= −b∗y+ + x+. (43)
By comparing with (35),(36) we see that the two first order equations do not create aself-adjoint system3.
3.3 Direct and adjoint eigenmodes
The eigenvalues and eigenvectors of L and L+ can be found by hand calculations:
σ1 =−b+
√b2 − 4c
2, σ+
1 = σ∗1, (44)
q1 =
[xy
]1
=
[2
−b+√b2 − 4c
], q+
1 =
[x+
y+
]1
=
[−2c∗
−b∗ −√b∗2 − 4c∗
].
(45)
where the minus sign in front of the square root in the y component of q+1 in (45) arises
because the system is lightly damped and therefore b∗2 − 4c∗ is negative.
3.4 Sensitivity
We perturb the system (35),(36) with a small feedback mechanism that feeds from x intothe first governing equation:
dx
dt= εx+ y, (46)
dy
dt= −by − cx. (47)
Note that we considered b and c to be real, so b = b∗ and c = c∗. The perturbed state isnow:
(L + δL)q =
[ε 1−c −b
] [xy
], (48)
or in other words:
δL =
[ε 00 0
]. (49)
3Self-adjointness occurs when the direct operator is equal to its adjoint.
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4 APPLICATION TO THERMOACOUSTICS
We can work out the change in eigenvalue by using the formula derived with the aid ofthe adjoint eigenfunction (27):
δσ1 =〈q+
1 , δLq1〉〈q+
1 , q1〉(50)
=x+∗1 εx1〈q+
1 , q1〉(51)
= ε
(1
2+
b
2√b2 − 4c
)(52)
As a check, we can work out δσ1 by solving exactly the perturbed eigenproblem. We willuse the notation σ′j ≡ σj + δσj for convenience:
det[(L + δL)− σ′jI
]= 0, (53)
det
[ε− σ′j 1−c −b− σ′j
]= 0. (54)
(σ′j − ε)(σ′j + b) + c = 0. (55)
Therefore:
σ′1 =−(b− ε) +
√(b− ε)2 − 4(c− εb)
2(56)
σ′2 =−(b− ε)−
√(b− ε)2 − 4(c− εb)
2(57)
To calculate the sensitivity to the perturbation, we differentiate with respect to ε
d
dε
((b− ε)2 − 4(c− εb)
)1/2=
−(b− ε) + 2b((b− ε)2 − 4(c− εb)
)1/2 . (58)
So the Taylor expansion of (56) around ε = 0, at first order, gives:
σ′1 =−b+
√b2 − 4c
2+ ε
(1
2+−(b) + 2b
2(b2 − 4c
)1/2), (59)
and therefore the eigenvalue drift is
δσ1 = ε
(1
2+
b
2√b2 − 4c
), (60)
which is the same as (52), as we wished to show.
4 Application to Thermoacoustics
4.1 Reduced order models in thermoacoustics
Fully compressible reacting CFD codes can simulate thermoacoustic oscillations but thisis very computationally expensive. Instead, it is much more common to use reduced order
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4 APPLICATION TO THERMOACOUSTICS 4.2 Hot wire Rijke tube
models. In this lecture, we consider a reduced order thermoacoustic model that containstwo components. The first component is the acoustic network, which in this case is a tubethat is open at both ends. Here, the acoustics is assumed to be one-dimensional but ingeneral 3D acoustics can be modelled by using a Helmholtz solver. The second componentis the heat source. This is assumed to be very much smaller than an acoustic wavelengthand therefore its dilatation is treated as a compact monopole source of sound for theacoustics. Figure 1 shows a schematic of this reduced order thermoacoustic model. Themost important component of the model is the description of how the acoustics affects theheat release. In some models, the heat release is assumed to be an explicit function of thevelocity perturbation after a prescribed time delay. In other models, the time evolution ofthe flame is solved, with input from the acoustic variables, and the heat release is spatiallyintegrated in order to feed back into the acoustic energy equation. Either way, this createsa feedback loop between the acoustics and the flame.
Figure 1: Reduced-order thermo-acoustic model with acoustically compact flame andmean-flow temperature jump.
4.2 Hot wire Rijke tube
The first thermo-acoustic system we will use to demonstrate the adjoint framework isa constant-diameter tube heated by an electrical hot wire or gauze (Matveev, 2003).This is known as a Rijke tube. Full descriptions of such a system, with relevant non-dimensionalizations but no temperature jump across the wire, is given by Balasubrama-nian and Sujith (2008) and Juniper (2011). One-dimensional acoustic waves occur ontop of a mean flow, which undergoes a discontinuity of its uniform properties across theheat source (see figure 1). The mean-flow pressure does not undergo a discontinuity inthe low Mach number limit (Dowling, 1995; Nicoud and Wieczorek, 2009). The acousticmomentum, energy equations and heat-release law are, respectively:
ρ∂u
∂t+∂p
∂x= 0, (61)
∂p
∂t+∂u
∂x+ ζp− βqδf = −uc
1
γ
∂γ
∂xθc, (62)
q =
√3
2
[uf (t)− τ
(∂u(t)
∂t
)f
], (63)
where u and p are the non-dimensional acoustic velocity and pressure. The heat-transfercoefficient, β, is assumed to be constant and its complete expression, encapsulating the
VKI - 14 -
4.2 Hot wire Rijke tube 4 APPLICATION TO THERMOACOUSTICS
hot wire’s properties and ambient conditions, is reported in Juniper (2011). The acousticvelocity has been non-dimenensionalized with the mean flow velocity; the acoustic pressurewith κM1p, where κ is the heat capacity ratio, M1 is the cold-flow Mach number and pthe mean-flow pressure; the abscissa with the duct length, La; the time with La/c1, wherec1 is the cold-flow speed of sound. The heat-release rate, q, is the linearized version of thenonlinear time-delayed law proposed by Heckl (1990), in which the subscript f means thatthe variable is evaluated at the hot wire’s location x = xf (δf ≡ δ(x − xf ) is the Dirac delta(generalized) function). The time delay between the pressure and heat-release oscillationsis modelled by the constant coefficient, τ . This linearization has been performed both inamplitude and time. Eqn. (63) holds providing that |uf (t− τ)| � 1 and τ � 2/K, whereK is the number of Galerkin modes considered in the numerical discretization (Juniper,2011; Magri and Juniper, 2013c,b). The non-dimensional density is ρ = ρ1 when x < xfand ρ = ρ2 when x > xf . The positive mean-flow temperature jump, induced by the heattransferred to the mean-flow, makes the density ratio ρ2/ρ1 < 1 because of the ideal-gaslaw. γ ≡ A(x)/A0, where A(x) is the area at location x and A0 is a reference area; andθc is 1 at x = xc and zero elsewhere. If the duct is straight, then γ = 0. As shown byMagri and Juniper (2013b), a local smooth cross-sectional area variation, defined suchthat ∂γ/∂xθc is finite, can be regarded as a passive feedback mechanism. We assume thatthe area of the duct is constant except at location x = xc, where there is a small smoothchange in the area. At the ends of the tube, p and ∂u/∂x are both set to zero. Dissipationand end losses are modelled by the modal damping ζ = c1j
2 + c2j0.5 used by Matveev
and Culick (2003), based on models by Landau and Lifshitz (1987), where j is the jth
acoustic mode. The quadratic term represents the losses at the end of the tube, while thesquare-rooted term represents the losses in the viscous/thermal boundary layers.
The partial differential equations (61), (62) are discretized into a set of ordinary dif-ferential equations by choosing a basis that matches the boundary conditions and the dis-continuity condition at the flame. This Galerkin method, which is a weak-form method,ensures that in the subspace where the solution is discretized the error is orthogonal tothe chosen basis, so that the solution is an optimal weak-form solution. The pressure, p,and velocity, u, are expressed by separating the time and space dependence, as follows
p(x, t) =K∑j=1
{α(1)j (t)Ψ
(1)j (x), 0 ≤ x < xf ,
α(2)j (t)Ψ
(2)j (x), xf < x ≤ 1.
(64)
u(x, t) =K∑j=1
{η(1)j (t)Φ
(1)j (x), 0 ≤ x < xf ,
η(2)j (t)Φ
(2)j (x), xf < x ≤ 1.
(65)
The system (61), (62) reduces to the D’Alembert equation when ζ = 0 and βT = 0
∂2p
∂t2− 1
ρ
∂2p
∂x2= 0. (66)
The following procedure is applied to find the bases for u and p:
1. substitute the decomposition (64) into (66) to find the pressure eigenfunctions
Ψ(1)j (x), Ψ
(2)j (x);
VKI - 15 -
4 APPLICATION TO THERMOACOUSTICS 4.2 Hot wire Rijke tube
2. substitute the pressure eigenfunctions Ψ(1)j (x), Ψ
(2)j (x) into the momentum equation
(61) to find the velocity eigenfunctions Φ(1)j (x), Φ
(2)j (x);
3. impose the jump condition at the discontinuity, for which p(x→ x−f ) = p(x→ x+f )
and u(x→ x−f ) = u(x→ x+f ), to find the relations between α(1)j , α
(2)j , η
(1)j , η
(2)j .
Similarly to Zhao (2012), these steps give
p(x, t) =K∑j=1
{−αj(t) sin
(ωj√ρ1x), 0 ≤ x < xf ,
−αj(t)(
sin γjsinβj
)sin(ωj√ρ2(1− x)
), xf < x ≤ 1,
(67)
u(x, t) =K∑j=1
ηj(t)1√ρ1
cos(ωj√ρ1x), 0 ≤ x < xf ,
−ηj(t) 1√ρ2
(sin γjsinβj
)cos(ωj√ρ2(1− x)
), xf < x ≤ 1.
(68)
whereγj ≡ ωj
√ρ1xf , βj ≡ ωj
√ρ2(1− xf ). (69)
Point 3 of the previous procedure provides the equation for the natural acoustic frequenciesωj
sin βj cos γj + cos βj sin γj
√ρ1ρ2
= 0. (70)
The full description and implementation of this method is available in Magri and Juniper(2014) based on the numerical model of Zhao (2012). The continuous adjoint equationsof the straight Rijke tube, derived via (7), are
ρ∂u+
∂t+∂p+
∂x+
√3
2β
(p+f + τ
(∂p+
∂t
)f
)δf = 0, (71)
∂u+
∂x+∂p+
∂t− ζp+ = 0. (72)
Note that (71) differs from the adjoint equations presented in previous work (Magri andJuniper, 2013c,a,b) because of the presence of ρ, which contains the information about themean-flow temperature jump. The localized smooth area variation term, −uc/γ∂γ/∂xθc,does not appear in the adjoint equations (71),(72). This is because this term is viewed as aforcing term of the energy equation (62) and the adjoint equations are defined with respectto the homogenous direct equations (see (1),(2),(3)). The direct and conjugate adjointeigenfunctions are arranged as column vectors [u, p]T and [u+∗, p+∗]T , respectively. Thestructural sensitivity tensor, defined in Magri and Juniper (2013c,b), is
S ≡ δσ
δC=
[u+∗, p+∗]T ⊗ [u, p]T∫ 1
0(uu+∗ + pp+∗)dx+ βτuf p
+∗f
, (73)
where ⊗ denotes the dyadic product and δC is a matrix of arbitrarily small perturbationcoefficients. Each component of this structural perturbation tensor quantifies the effectof a feedback mechanism between a variable and a governing equation, as explained in
VKI - 16 -
4.2 Hot wire Rijke tube 4 APPLICATION TO THERMOACOUSTICS
Magri and Juniper (2013c,b). Therefore we can identify the device, and the location, thatis most effective at changing the frequency or growth rate of the system just by inspectionof the components of the structural sensitivity tensor (73). Here, we discuss the two mostsignificant mechanisms, given by the components
Suu =u+∗u∫ 1
0(uu+∗ + pp+∗)dx+ βτuf p
+∗f
, (74)
Sup =p+∗u∫ 1
0(uu+∗ + pp+∗)dx+ βτuf p
+∗f
. (75)
The reader may refer to Magri and Juniper (2013c,b) for a detailed explanation of theremaining components of the structural sensitivity tensor. The components (74),(75) areshown in fig. 2 as a function of x , which is the location at which the passive device(structural perturbation) acts, both when ρ1/ρ2 = T2/T1 = 1 (solid line) and ρ1/ρ2 =T2/T1 = 2 (dash-dot line). The component Suu is the eigenvalue’s sensitivity to a feedbackmechanism proportional to the velocity at a given point and affecting the momentumequation. For example, this could be the (linearized) drag force about an obstacle in theflow, as proposed in Magri and Juniper (2013c,b). The real part of Suu (fig. 2a), beingthe sensitivity of the system’s growth rate, is positive for all values of x , which meansthat, whatever value of x is chosen, the growth rate will decrease if the forcing is in theopposite direction to the velocity, as it is in drag-exerting devices. This tells us that thedrag force of a mesh will always stabilize the thermo-acoustic oscillations but is mosteffective if placed at the ends of the duct. This effect is even stronger if the temperaturejump is considered. In summary, this type of feedback greatly affects the growth rate buthardly affects the frequency (fig. 2b), in agreement with Dowling (1995), who performedstability analysis via classical approaches.
The component Sup is the eigenvalue’s sensitivity to a feedback mechanism propor-tional to the velocity at a given point and affecting the energy equation. This type offeedback hardly affects the growth rate (fig. 2c) but greatly affects the frequency (fig. 2d).By inspection of the linearized heat law (63), we notice that a second hot wire with τ = 0causes this type of feedback, so this analysis shows that it will be relatively ineffective atstabilizing thermo-acoustic oscillations whereas it will be effective at changing the oscil-lation frequency. A detailed analysis and physical explanation of this finding is reportedin Magri and Juniper (2013c,b).
If γ 6= 0, the RHS of eqn. (62) shows that a change in the area can be interpretedas a forcing term, proportional to −uc, acting on the energy equation. In other words, apositive local smooth change of the cross-sectional area is a feedback mechanism acting likea second hot wire with negative β. Hence, the structural sensitivity is provided by −Sup.This means that where a control hot wire has a stabilizing effect, a positive change inarea in the same location has a destabilizing effect, and vice versa. It is worth mentioningthat the structural sensitivity coefficients depicted in figure 2 do not depend on the timedelay, τ , as long as it remains small compared with the oscillation period. We performedcalculations for time delays from 0 to 0.03 and observed negligible differences (results notshown).
VKI - 17 -
4 APPLICATION TO THERMOACOUSTICS 4.3 Burke-Schumann flame Rijke tube
−5
0
5
10
15x 10
−4
Re(S
uu)
(a)
−2
0
2
4x 10
−5
Im(S
uu)
(b )
T 2/T 1 = 1
T 2/T 1 = 2
0 0.2 0.4 0.6 0.8 1−2
0
2
4x 10
−5
Re(S
up)
( c )
x0 0.2 0.4 0.6 0.8 1
−5
0
5
10x 10
−4
Im(S
up)
(d )
x
Figure 2: Two significant components of the structural sensitivity tensor, which quantifythe effect of feedback mechanisms, placed at x, on the linear growth rate (left frames) andangular frequency (right frames). Solutions with no mean-flow temperature jump (solidlines) and with temperature jump (dash-dotted line). c1 = 0.01, c2 = 0.001, τ = 0.01,β = 0.433, xf = 0.25, 10 Galerkin modes are used for numerical discretization.
4.3 Burke-Schumann flame Rijke tube
The second thermoacoustic system is a Rijke tube heated by a diffusion flame with infinite-rate chemistry, known as a Burke-Schumann flame. The acoustic waves cause perturba-tions in the velocity field. In turn, these cause perturbations to the mixture fraction,which convect down the flame and cause perturbations in the heat-release rate and thedilatation rate at the flame. The dilatation described above provides a monopole sourceof sound, which feeds into the acoustic energy. As for the hot wire Rijke tube, we assumethat the flame is compact, meaning that the heat release is a point-wise impulsive forcingterm for the acoustics. The model and its governing equations are described fully in Magriand Juniper (2014). The governing equations are linearized and the direct and adjointeigenmodes are calculated.
The real and imaginary components of the least stable direct eigenfunction are shownin figure 3. The real and imaginary parts are in spatial quadrature, which shows that
!
(a) T 2/T 1 = 1
0
0.5
1(b) T 2/T 1 = 5
!
(c )
0
0.5
1(d)
0 1 2 3 40
1
2(e )
c
"0 1 2 3 4
( f )
"
!
(a) T 2/T 1 = 1
0
0.5
1(b) T 2/T 1 = 5
!
(c )
0
0.5
1(d)
0 1 2 3 40
1
2(e )
c
"0 1 2 3 4
( f )
"
Figure 3: Real (left) and imaginary (right) components of the mixture fraction per-turbation of the least stable eigenfunction for the thermoacoustic system containing aBurke-Schumann flame. The red/blue colour corresponds to positive/negative values.The dashed line is the flame position and only the top half of the flame domain is shown.The acoustic component of the eigenfunction is not shown. The real and imaginary com-ponents are in spatial quadrature, showing that this mode contains z-perturbations thatconvect down the flame at almost constant speed.
VKI - 18 -
4.3 Burke-Schumann flame Rijke tube 4 APPLICATION TO THERMOACOUSTICS
the mixture fraction perturbation, z, takes the form of a travelling wave that movesdown the flame in the streamwise direction. This shows that a simple model of theflame, in which mixture fraction perturbations convect down the flame at the mean-flowspeed and causes heat-release fluctuations when they reach the flame is reasonable. Toa first approximation, therefore, the time delay between the velocity perturbation andthe subsequent heat-release perturbation scales with Lf/U , where Lf is the length ofthe steady flame and U is the mean-flow speed (which is 1 in this model). The phasedelay between the velocity perturbation and the subsequent heat release perturbationtherefore scales with Lfσi/U , where σi is the dominant eigenvalue’s imaginary part, i.e.the linear-oscillation angular frequency.
The absolute value of the corresponding adjoint eigenfunction is shown in figure 4. Thisis a map, in the flame domain, of the first eigenfunction’s receptivity to species injection.In other words, it shows the most effective regions at which to place an open-loop activedevice to excite the dominant thermo-acoustic mode. The adjoint eigenfunction has highmagnitude around the flame. This is because species injection affects the heat releaseonly if it changes the gradient of z at the flame itself, which is achieved by injectingspecies around the flame. Its magnitude increases towards the tip of the flame, where∇z is weakest. This is because mixture fraction fluctuations diffuse out as they convectdown the flame, which means that open-loop forcing has a proportionately large influenceon the mixture fraction towards the tip. From a practical point of view, this shows thatopen-loop control of the mixture fraction has little influence at the injection plane butgreat influence at the flame tip. In this case, this could be achieved by injecting speciesat the wall.
Figure 5 shows two of the base state sensitivities for this model: to changes in thestoichiometric mixture fraction (top) and to changes in the flame aspect ratio (bottom).These results, obtained by the adjoint-based approach, have been checked against thesolutions obtained via finite difference and agree to within ∼ O(10−9). These figures areuseful from a design point of view. For example, they reveal that, at Zsto = 0.15 andα = 0.35, changes in Zsto strongly influence the growth rate but that, at Zsto = 0.15 andα = 0.30, changes in Zsto strongly influence the frequency instead. This demonstratesan inconvenient feature of thermo-acoustic instability: it is exceedingly sensitive to smallchanges in the operating point. In this analysis, the length of the unperturbed flame,Lf , emerges as a key parameter. This is defined here as the distance between the inletand the tip of the steady flame and is shown by the black lines in figure 5. Lines ofconstant δσ/δZsto and δσ/δα very nearly follow the lines of constant Lf . This can beexplained physically by considering the physical mechanism behind thermoacoustic insta-
!
(a) T 2/T 1 = 1
0
0.5
1(b) T 2/T 1 = 5
!
(c )
"0 1 2 3 40
0.5
1(d)
"0 1 2 3 4
Figure 4: Absolute value of the adjoint eigenfunction that corresponds to the directeigenfunction in figure 3. This shows the region of the flame domain that are mostreceptive to forcing of the mixture fraction - i.e. to periodic species injection.
VKI - 19 -
4 APPLICATION TO THERMOACOUSTICS 4.3 Burke-Schumann flame Rijke tube
Zsto
T 2/T 1 = 1 (a) ! "r/!Zsto
23
4
5
6
23
4
5
6
0.050.1
0.15
0.02
0.04
0.06
Zsto
(b) ! " i/!Zsto
23
4
5
6
0.050.1
0.15
0.04
0.06
Zsto
(c ) ! "r/! #
23
4
5
6
0.050.1
0.15
0.01
0
0.01
#
Zsto
(d) ! " i/! #
23
4
5
6
0.25 0.3 0.35 0.4
0.050.1
0.15
0.01
0
0.01
T 2/T 1 = 5 (e ) ! "r/!Zsto
23
4
5
6
0.0140.0160.0180.020.0220.024
( f ) ! " i/!Zsto
23
4
5
6
0.0220.0240.0260.0280.030.0320.034
(g) ! "r/! #
23
4
5
6
2
0
2x 10 3
#
(h) ! " i/! #
23
4
5
6
0.25 0.3 0.35 0.42
0
2x 10 3
Zsto
T 2/T 1 = 1 (a) ! "r/!Zsto
23
4
5
6
23
4
5
6
0.050.1
0.15
0.02
0.04
0.06
Zsto
(b) ! " i/!Zsto
23
4
5
6
0.050.1
0.15
0.04
0.06
Zsto
(c ) ! "r/! #
23
4
5
6
0.050.1
0.15
0.01
0
0.01
#
Zsto
(d) ! " i/! #
23
4
5
6
0.25 0.3 0.35 0.4
0.050.1
0.15
0.01
0
0.01
T 2/T 1 = 5 (e ) ! "r/!Zsto
23
4
5
6
0.0140.0160.0180.020.0220.024
( f ) ! " i/!Zsto
23
4
5
6
0.0220.0240.0260.0280.030.0320.034
(g) ! "r/! #
23
4
5
6
2
0
2x 10 3
#
(h) ! " i/! #
23
4
5
6
0.25 0.3 0.35 0.42
0
2x 10 3
Zsto
T 2/T 1 = 1 (a) ! "r/!Zsto
23
4
5
6
23
4
5
6
0.050.1
0.15
0.02
0.04
0.06
Zsto
(b) ! " i/!Zsto
23
4
5
6
0.050.1
0.15
0.04
0.06Z
sto
(c ) ! "r/! #
23
4
5
6
0.050.1
0.15
0.01
0
0.01
#
Zsto
(d) ! " i/! #
23
4
5
6
0.25 0.3 0.35 0.4
0.050.1
0.15
0.01
0
0.01
T 2/T 1 = 5 (e ) ! "r/!Zsto
23
4
5
6
0.0140.0160.0180.020.0220.024
( f ) ! " i/!Zsto
23
4
5
6
0.0220.0240.0260.0280.030.0320.034
(g) ! "r/! #
23
4
5
6
2
0
2x 10 3
#
(h) ! " i/! #
23
4
5
6
0.25 0.3 0.35 0.42
0
2x 10 3
Zsto
T 2/T 1 = 1 (a) ! "r/!Zsto
23
4
5
6
23
4
5
6
0.050.1
0.15
0.02
0.04
0.06
Zsto
(b) ! " i/!Zsto
23
4
5
6
0.050.1
0.15
0.04
0.06
Zsto
(c ) ! "r/! #
23
4
5
6
0.050.1
0.15
0.01
0
0.01
#
Zsto
(d) ! " i/! #
23
4
5
6
0.25 0.3 0.35 0.4
0.050.1
0.15
0.01
0
0.01
T 2/T 1 = 5 (e ) ! "r/!Zsto
23
4
5
6
0.0140.0160.0180.020.0220.024
( f ) ! " i/!Zsto
23
4
5
6
0.0220.0240.0260.0280.030.0320.034
(g) ! "r/! #
23
4
5
6
2
0
2x 10 3
#
(h) ! " i/! #
23
4
5
6
0.25 0.3 0.35 0.42
0
2x 10 3
Figure 5: Sensitivity of the growth rate, σr, (left frames) and frequency, σi (right frames)to changes in the stoichiometric mixture fraction, Zsto, (top frames) and flame aspectratio, α, (bottom frames). These sensitivities are shown as functions of Zsto (verticalaxis, from 0.025 to 0.20) and α (horizontal axis, from 0.25 to 0.4). The contour levelsshow: Real(∂σ/∂Zsto) on the top left, Imag(∂σ/∂Zsto) on the top right, Real(∂σ/∂α)on the bottom left, Imag(∂σ/∂α) on the bottom right. These are two of the base statesensitivities for this model. The sensitivities to the other flame parameters and to theacoustics are also calculated but are not shown. The numbered black lines are contoursof constant flame length.
bility: acoustic velocity perturbations cause mixture fraction perturbations at the base ofthe flame. These are convected downstream and cause a heat-release perturbation sometime later. The time delay between acoustic velocity and subsequent heat release, τ ,scales with Lf/U , where Lf is the length of the flame. The influence of this heat-releaseperturbation on the growth rate or the frequency of the acoustic wave depends on thephase of the heat release relative to the phase of the velocity or pressure, which are intemporal quadrature. This is why the base-state sensitivity plots are in spatial quadra-ture in parameter space. The phase delay, ψ, is given by τ/T , where T = 2π/σi. In thissimple model, δσ depends only on ψ, which means that the eigenvalue drifts in figure 5should collapse onto a single line when plotted as a function of Lfσi/U . This is shown infigure 6 for δσ/δZsto (top) and δσ/δα (bottom). The data collapse reasonably closely to aline, showing that the simple model provides a physical understanding of the sensitivities,while figure 5 provides their exact values. Although not shown here, once the direct andadjoint eigenfunctions have been calculated, all the other base state sensitivities follow atvery little extra cost, demonstrating the utility of these techniques.
VKI - 20 -
4.3 Burke-Schumann flame Rijke tube 4 APPLICATION TO THERMOACOUSTICS
0
0.05
0.1
!"r/!Z
sto
(a) T 2/T 1 = 1
0
0.05
0.1
!"i/!Z
sto
0.02
0
0.02
!"r/!#
2 3 4 50.02
0
0.02
$
!"i/!#
0.01
0.02
0.03(b) T 2/T 1 = 5
0.02
0.03
0.04
5
0
5x 10 3
2 3 4 55
0
5x 10 3
$
0
0.05
0.1
!"r/!Z
sto
(a) T 2/T 1 = 1
0
0.05
0.1
!"i/!Z
sto
0.02
0
0.02
!"r/!#
2 3 4 50.02
0
0.02
$
!"i/!#
0.01
0.02
0.03(b) T 2/T 1 = 5
0.02
0.03
0.04
5
0
5x 10 3
2 3 4 55
0
5x 10 3
$
0
0.05
0.1
!"r/!Z
sto
(a) T 2/T 1 = 1
0
0.05
0.1
!"i/!Z
sto
0.02
0
0.02
!"r/!#
2 3 4 50.02
0
0.02
$
!"i/!#
0.01
0.02
0.03(b) T 2/T 1 = 5
0.02
0.03
0.04
5
0
5x 10 3
2 3 4 55
0
5x 10 3
$
0
0.05
0.1
!"r/!Z
sto
(a) T 2/T 1 = 1
0
0.05
0.1
!"i/!Z
sto
0.02
0
0.02!"r/!#
2 3 4 50.02
0
0.02
$
!"i/!#
0.01
0.02
0.03(b) T 2/T 1 = 5
0.02
0.03
0.04
5
0
5x 10 3
2 3 4 55
0
5x 10 3
$Figure 6: Sensitivity of the growth rate, σr, (left frames) and frequency, σi (right frames)to changes in the stoichiometric mixture fraction, Zsto, (top frames) and flame aspect ratio,α, (bottom frames). These sensitivities are shown as functions of Lfσi/U (horizontal axis),which is plotted from 1.6 to 5.6. The data, which is the same as that in figure 5, nearlycollapses to a single line in each figure.
VKI - 21 -
REFERENCES
5 Concluding remarks
The aim of this lecture is to show how adjoint sensitivity analysis can be applied tothermoacoustics. We describe the physical meaning of the adjoint eigenfunction in termsof the system’s receptivity to open-loop forcing and show how to combine the direct andadjoint eigenfunctions in order to obtain an analytical formula for the first-order eigenvaluedrift.
The results in this paper are for a simple thermo-acoustic model and are, of course,only as accurate as the model itself. The adjoint-based techniques, however, can readilybe applied to more realistic models, as long as they can be linearized. This could quicklyreveal, for example, the best position for an acoustic damper in a complex acoustic net-work, the optimal change in the flame shape and the best strategies for open loop control.The usefulness of adjoint techniques applied to thermo-acoustics is that, in a few calcula-tions, one can predict accurately how the growth rate and frequency of thermo-acousticoscillations are affected either by all possible passive control elements in the system or byall possible changes to its base state.
This work is supported by the European Research Council through Project ALORS2590620 and by the FP7 Marie Curie ITN Programme ANADE 289428.
References
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