Sensitivity of the Rayleigh criterion in thermoacoustics
TThis draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics JFM RAPIDS journals.cambridge.org/rapids
Sensitivity of the Rayleigh criterion inthermoacoustics
Luca Magri † , Matthew P. Juniper and Jonas P. Moeck Cambridge University Engineering Dept., Trumpington St, CB2 1PZ, Cambridge, UK Department of Energy and Process Engineering, NTNU, Trondheim, Norway(Received xx; revised xx; accepted xx)
Thermoacoustic instabilities are one of the most challenging problems faced by gasturbine and rocket motor manufacturers. The key instability mechanism is describedby the
Rayleigh criterion . The Rayleigh criterion does not directly show how to altera system to make it more stable. This is the objective of sensitivity analysis. Becausethermoacoustic systems have many design parameters, adjoint sensitivity analysis hasbeen proposed to obtain all the sensitivities with one extra calculation. Although adjointsensitivity analysis can be carried out in both the time and the frequency domain,the frequency domain is more natural for a linear analysis. Perhaps surprisingly, theRayleigh criterion has not yet been rigorously derived and comprehensively interpretedin the frequency domain. The contribution of this theoretical paper is threefold. First, theRayleigh criterion is interpreted in the frequency domain with integral formulae for thecomplex eigenvalue. Second, the first variation of the Rayleigh criterion is calculated bothin the time and frequency domain, both with and without Lagrange multipliers (adjointvariables). The Lagrange multipliers are physically related to the system’s observables.Third, an adjoint Rayleigh criterion is proposed. The paper also points out that theconclusions of
Juniper, M. P. (2018), Phys. Rev. Fluids, vol. 3, 110509 apply to the firstvariation of the Rayleigh criterion, not to the Rayleigh criterion itself. The mathematicalrelations of this paper can be used to compute sensitivities directly from measurablequantities to enable optimal design.
1. Introduction
The most commonly used criterion to assess the stability of a thermoacoustic system,such as a gas turbine combustor or a rocket motor, is the Rayleigh criterion (LordRayleigh 1878). This criterion states that the acoustic energy increases when the heatrelease rate is sufficiently in phase with the acoustic pressure that the work doneby gas expansion exceeds the energy loss over a cycle. The Rayleigh criterion hasbeen used in experiments, numerical simulations, and theory since the 1940’s (Juniper& Sujith 2018). The literature is thoroughly reviewed by Candel (2002); Lieuwen &Yang (2005); Poinsot (2017); Juniper & Sujith (2018) and here we describe only afew examples. In experiments (e.g., Lieuwen & Yang 2005), the Rayleigh criterion is † Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] O c t Luca Magri, Matthew P. Juniper and Jonas P. Moeck used as a diagnostic tool. The spatial product between the acoustic pressure and theheat release rate, p ˙ Q , also known as the Rayleigh index, is used to identify spatiallocations where the thermoacoustic instability is amplified or damped. In active controlof thermoacoustic oscillations (e.g., Dowling & Morgans 2005), the Rayleigh index isused to measure the influence of the heat release caused by a secondary fuel injection;if the Rayleigh index becomes negative, the secondary fuel damps out the instability.In large eddy simulations (e.g., Poinsot 2017), the Rayleigh criterion has been used toanalyse the thermoacoustic instability of staged and annular combustors. In low-ordermodels of annular combustors, the Rayleigh criterion provides conditions for the stabilityof spinning and standing waves (Ghirardo et al. et al. et al. et al. (2010); Luchini & Bottaro (2014); Camarri (2015). In sensitivity analysis inthe frequency domain, Magri & Juniper (2013) compared the Rayleigh index witheigenvalue sensitivities for passive control of the oscillations in a time-lag model, andMagri & Juniper (2014) compared the Rayleigh index of a diffusion flame with eigenvaluesensitivity to identify the most sensitive regions of the flame. The Rayleigh criterion wasuseful to gain insight into the active physical mechanisms leading to instability, whilesensitivity analysis was useful to make optimal changes to the system.Chu ( § III, 1965) examines the behaviour of acoustic disturbances subject to arbitraryinjections of mass, momentum, and heat. He asks the question “what is the contributionof an arbitrary injection of heat to the energy of an acoustic oscillation” and shows thatthe rate of change of energy is the integral of the injected heat release rate multiplied bythe temperature (with adjustments for dissipation, Chu (eq. 19a, 1965)). If the thermalconductivity is zero, the rate of change of energy becomes the integral of the injectedheat release rate multiplied by the pressure (with adjustments for dissipation, Chu (eq.23a, 1965)). This is the Rayleigh criterion. On the other hand, sensitivity analysis such asMagri & Juniper (2013, 2014); Juniper (2018) examines the behaviour of thermoacousticdisturbances subject to infinitesimal injections of mass, momentum, and heat. They askthe question “what is the contribution of a small injection of heat to the energy of a thermo acoustic oscillation” and show that the rate of change of energy is the integral ofthe injected heat release rate multiplied by the adjoint pressure. This is best describedas the first variation of the Rayleigh criterion. The Rayleigh criterion is formed withthe physical pressure, while the first variation of the Rayleigh criterion is formed withthe adjoint pressure. The conclusions of Juniper (2018) should have been more carefullyworded with regard to this point.In this paper, we first derive and interpret the Rayleigh criterion. The novelty is theanalysis in the frequency domain. Second, we derive and interpret the first variation ofthe Rayleigh criterion with and without Lagrange multipliers (adjoint variables). Thefunctional derivatives are calculated and interpreted as the sensitivities of the Rayleighcriterion. Crucially, we physically interpret the adjoint variables in terms of observable ensitivity of the Rayleigh criterion in thermoacoustics
2. Time domain analysis
The Rayleigh criterion
We consider a duct of length L with a zero-Mach-number base flow and ideal acousticboundary conditions, i.e., the acoustics do no work on the surroundings. The cut-offfrequency is sufficiently high that the acoustics are longitudinal in the direction x . Thedimensional equations, which are derived by linearizing the Euler and energy equationsfor ideal gases, are¯ ρ ∂u∂t + ∂p∂x = 0 , (2.1)1 γ ¯ p ∂p∂t + ∂u∂x = ( γ − γ ¯ p ˙ Q, (2.2)where γ is the heat-capacity ratio; ¯ ρ is the base-flow density, which can be a function of x ;and ˙ Q is the heat release rate from a source. To keep the theory as general as possible, weassume that ˙ Q is a prescribed function of time and space, i.e., ˙ Q = ˙ Q ( x, t ). This functioncan consist of both open-loop terms, which are not functions of the acoustic variables,and closed-loop terms, which are functions of the acoustic variables. The closed-loopforcing term can be obtained from flame transfer functions (e.g., Lieuwen & Yang 2005).The base-flow pressure, ¯ p , is uniform and constant because of the zero-Mach-numberassumption. Equations (2.1)-(2.2) define a model of a prototypical thermoacoustic systemwith a generic heat-source term. The operation (cid:82) T (cid:82) L ( u · (2.1) + p · (2.2)) dx dt , where[0 , T ] is an arbitrary interval of time, yields the governing equation for the change in theacoustic energy E := 12 (cid:34)(cid:90) L (cid:18) ¯ ρu + p γ ¯ p (cid:19) dx (cid:35) T = γ − γ ¯ p (cid:90) T (cid:90) L p ˙ Q dx dt, (2.3)where we considered that up = 0 at x = 0 and x = L because the boundary conditionsare ideal. The symbol := defines a quantity. For brevity, we refer to E as the acousticenergy. Equation (2.3), which has an arbitrary T , can be interpreted as an extension of theRayleigh criterion: If the heat release rate of a source, ˙ Q , is in phase with the acousticpressure, p , on average then the acoustic energy of the duct, E , increases in time onaverage. Although strictly speaking the original Rayleigh criterion (Lord Rayleigh 1878)takes T to be the period of an oscillation, we refer to (2.3) as the Rayleigh criterion andto the integrand p ˙ Q as the Rayleigh index.We work in Hilbert spaces with the following inner products (cid:104) a, b (cid:105) V := (cid:90) L a ∗ b dx, (cid:104) a, b (cid:105) V,T := (cid:90) T (cid:104) a, b (cid:105) V dt, (2.4)where ∗ denotes the complex conjugate, and a and b are generic functions. In this sectionthe variables are real, so the complex conjugate has no effect. We will, however, use thecomplex conjugate in the frequency domain ( § E [ p, ˙ Q ] := (cid:18) γ − γ ¯ p (cid:19) (cid:68) p, ˙ Q (cid:69) V,T , (2.5) Luca Magri, Matthew P. Juniper and Jonas P. Moeck which provides a mapping from the acoustic pressure and heat release rate functionspaces to a real number. We regard the heat release rate as the independent parameter.Physically, the acoustic pressure is a function of the heat release rate through theconstraints imposed by the momentum and energy equations (2.1)-(2.2).2.2.
First variation without Lagrange multipliers
We are interested in calculating the effect that a variation to the heat release rate˙ Q ( x, t ) → ˙ Q ( x, t )+ δ ˙ Q ( x, t ) has on the acoustic energy. In particular, we wish to calculatethe effect of an arbitrary small variation δ ˙ Q ( x, t ) = (cid:15) ˙ Q p ( x, t ), where (cid:15) is infinitesimal and˙ Q p is an arbitrary function, known as the test function. (For clarity, the dependence on( x, t ) will be used only in some passages.) The symbol δ refers to an infinitesimal virtualchange, whereas the differentiation symbols d or ∂ refer to an infinitesimal actual change.The objective is to find the first variation of the acoustic energy, E , in the vicinity of p and ˙ Q for any arbitrary perturbation, (cid:15) ˙ Q p , to the heat release rate δE = (cid:15) (cid:90) T (cid:90) L (cid:18) δ ˙ Q Eδ ˙ Q ( x (cid:48) , t (cid:48) ) + δ p Eδp ( x (cid:48) , t (cid:48) ) dpd ˙ Q ( x (cid:48) , t (cid:48) ) (cid:19) ˙ Q p ( x (cid:48) , t (cid:48) ) dx (cid:48) dt (cid:48) , (2.6)where the term within round brackets is the functional derivative, which is the kernel ofthe integral (e.g., Greiner & Reinhardt 1996). The functional derivative of the acousticenergy consists of two terms. The first term is the sensitivity of the acoustic energy toan arbitrary perturbation to the heat release rate, δ ˙ Q E/δ ˙ Q ( x (cid:48) , t (cid:48) ). The second term isthe sensitivity to an arbitrary perturbation to the acoustic pressure, δ p E/δp ( x (cid:48) , t (cid:48) ). Theacoustic pressure, however, cannot arbitrarily change because it is a function of the heatrelease rate. The term dp/d ˙ Q is the sensitivity of the acoustic pressure to the virtualperturbation to the heat release rate dpd ˙ Q := lim (cid:15) → p ( ˙ Q + (cid:15) ˙ Q p ) − p ( ˙ Q ) (cid:15) ˙ Q p , (2.7)where p ( · ) signifies that the pressure is the solution of (2.1)-(2.2) when the heat-releaserate is provided by the argument ( · ). This constrains the pressure variation to be in theadmissible function space imposed by the momentum and energy equations. Workingon the assumption that the acoustic energy is Fr´echet differentiable and taking the firstvariation of (2.3) yields the two functional derivatives δ ˙ Q Eδ ˙ Q = (cid:18) γ − γ ¯ p (cid:19) p, δ p Eδp = (cid:18) γ − γ ¯ p (cid:19) ˙ Q, (2.8)whence the functional derivative of the acoustic energy is δEδ ˙ Q := (cid:18) δ ˙ Q Eδ ˙ Q ( x (cid:48) , t (cid:48) ) + δ p Eδp ( x (cid:48) , t (cid:48) ) dpd ˙ Q ( x (cid:48) , t (cid:48) ) (cid:19) = (cid:18) γ − γ ¯ p (cid:19) (cid:18) p + dpd ˙ Q ˙ Q (cid:19) . (2.9)Therefore, the first variation is δE = ( γ − / ( γ ¯ p ) (cid:104) p + ( dp/d ˙ Q ) ˙ Q, (cid:15) ˙ Q p (cid:105) V,T . The functionalderivative of the acoustic energy (2.9) is the superposition of two physical effects. Thefirst is a purely acoustic effect in the absence of a main heat source, i.e. ˙ Q = 0, inwhich the derivative is the acoustic pressure. The maximum change in the acousticenergy is obtained when the heat release rate perturbation is applied in phase with thepressure and where the acoustic pressure is maximum. The second is a purely thermaleffect when a main heat source is present, i.e., ˙ Q (cid:54) = 0, in which the derivative of theacoustic energy is the acoustic pressure sensitivity weighted by the unperturbed heatrelease rate ˙ Q . The maximum change in the acoustic energy is obtained when the heat ensitivity of the Rayleigh criterion in thermoacoustics First variation with Lagrange multipliers
The adjoint variables can be interpreted as the Lagrange multipliers of a constrainedoptimization problem (e.g, Sipp et al. § δE/δ ˙ Q subject to (2.1),(2.2). We define a Lagrangian L{ u, p, u + , p + , ˙ Q } := E − (cid:10) u + , (2.1) (cid:11) V,T − (cid:10) p + , (2.2) (cid:11) V,T , (2.10)and take the first variation with respect to virtual changes of its arguments. On integra-tion by parts, we obtain δ L = (cid:28) δ u Eδu , δu (cid:29)
V,T + (cid:28) δ p Eδp , δp (cid:29)
V,T − L ¯ ρ [ u + δu ] T − L γ ¯ p [ p + δp ] T . . . − T [ u + δp ] L − T [ p + δu ] L + (cid:42) ¯ ρ ∂u + ∂t + ∂p + ∂x + ( γ − γ ¯ p ∂ ˙ Q∂u p + , δu (cid:43) V,T . . . + (cid:42) γ ¯ p ∂p + ∂t + ∂u + ∂x + ( γ − γ ¯ p ∂ ˙ Q∂p p + , δp (cid:43) V,T + (cid:15) (cid:18) γ − γ ¯ p (cid:19) (cid:68) p + , ˙ Q p (cid:69) V,T . (2.11)From (2.1), (2.2) and (2.10), it becomes apparent that δ L = δE . To eliminate all exceptthe last term in (2.11), we define the adjoint problem¯ ρ ∂u + ∂t + ∂p + ∂x + ( γ − γ ¯ p ∂ ˙ Q∂u p + = 0 , (2.12)1 γ ¯ p ∂p + ∂t + ∂u + ∂x + ( γ − γ ¯ p ∂ ˙ Q∂p p + = 0 . (2.13)The adjoint initial and boundary conditions are defined such that they eliminate theboundary terms in (2.11). The partial derivatives ∂ ˙ Q/∂u and ∂ ˙ Q/∂p are zero if theheat-release rate has no closed-loop terms. Otherwise, these derivatives can be obtainedby differentiating the time-domain transformation of the flame transfer functions. Withthe Lagrange multiplier, the first variation of the acoustic energy is simply δE =( γ − / ( γ ¯ p ) (cid:104) p + , (cid:15) ˙ Q p (cid:105) V,T ; hence the functional derivative is δEδ ˙ Q = (cid:18) γ − γ ¯ p (cid:19) p + . (2.14)The Lagrange multiplier, p + (multiplied by ( γ − / ( γ ¯ p )), is the functional derivativeof the acoustic energy. Note that the calculation of dp/d ˙ Q in (2.9) is computationallyexpensive because it is the derivative of the pressure with respect to an arbitraryperturbation to the heat release rate. By solving for the adjoint problem (2.12)-(2.13),we obtain directly this sensitivity information. Luca Magri, Matthew P. Juniper and Jonas P. Moeck
Relation between first variations with and without Lagrange multipliers
The connection between (2.9) and (2.14), which is guaranteed by the Riesz represen-tation theorem, is p + = p + dpd ˙ Q ˙ Q. (2.15)This is a key equation in the time domain. It is the link between the adjoint pressureand the observable quantities, which gives physical meaning to the adjoint pressure.Relation (2.15) shows how to express the first variation of the Rayleigh criterion in thetime domain in a non-self-adjoint system ( ˙ Q (cid:54) = 0); non-self-adjointness manifests itselfvia the thermal sensitivity ( dp/d ˙ Q ) ˙ Q , which is a nonlocal effect. If the heat sources arelocalized as ˙ Q ( x, t ) = Q f ( t ) δ ( x − x f ) and ˙ Q p ( x, t ) = Q p ( t ) δ ( x − x p ), where δ ( x − x f,p ) isthe Dirac delta centred at x f,p , the adjoint pressure is p + p = p p + dp f Q f / ( (cid:15)Q p ). (A shortderivation can be found in § f and p signify that the variable is evaluated at x f and x p , respectively. To achieve the maximumfirst variation in the acoustic energy, the phase between Q p and the unperturbed pressure p p is such that p f shifts to be more in phase with Q f . Using the Lagrange multiplierinstead of the observables offers a more compact interpretation of the first variation ofthe Rayleigh criterion. To achieve the maximum first variation in the acoustic energy,the heat source, Q f , should be perturbed at the location x p where the adjoint pressure, p + , is maximum.
3. Frequency domain analysis
To switch to the frequency domain, we use modal decomposition u ( x, t ) =ˆ u ( x ) exp( σt ) + c.c. and p ( x, t ) = ˆ p ( x ) exp( σt ) + c.c. for the acoustic velocity andpressure, respectively. “c.c.” stands for complex conjugate of the preceding term andˆ u ( x ), ˆ p ( x ) and σ are complex. The real part of σ , denoted σ r , is the growth rate andthe imaginary part, denoted σ i , is the angular frequency. By substituting the modaldecomposition into the governing equations (2.1)-(2.2), we obtain¯ ρσ ˆ u + ∂ ˆ p∂x = 0 , (3.1)1 γ ¯ p σ ˆ p + ∂ ˆ u∂x = γ − γ ¯ p ˆ˙ Q. (3.2)To keep the theory as general as possible, we assume that ˆ˙ Q is a prescribed spatialfunction, i.e., ˆ˙ Q = ˆ˙ Q ( x ), which is in a linear closed loop with ˆ p and ˆ u (but usually it isa nonlinear function of σ ). Typically, the heat-release rate is specified by flame transferfunctions (e.g., Lieuwen & Yang 2005). Equations (3.1)-(3.2) define an eigenvalue prob-lem where ˆ u and ˆ p are the acoustic velocity and pressure eigenfunctions associated withthe eigenvalue σ . These quantities characterize the natural response of the thermoacousticsystem. 3.1. The Rayleigh criterion
In the frequency domain, the complex instantaneous acoustic energy varies as E t =(1 /
2) exp(2 σt ) (cid:82) L (¯ ρ ˆ u + ˆ p / ( γ ¯ p )) dx . The eigenvalue σ provides half the growth rate andangular frequency of the complex acoustic energy. As a consequence, interpreting theRayleigh criterion in the frequency domain is equivalent to interpreting the eigenvalue.We achieve this by deriving three integral formulae for the eigenvalue in § ensitivity of the Rayleigh criterion in thermoacoustics ∂E t /∂t over an arbitrary interval [0 , T ]. Weset C := (cid:82) L (¯ ρ ˆ u + ˆ p / ( γ ¯ p )) dx as a constant normalization factor. It is straightforwardto show that C is a nontrivial complex number in a thermoacoustic system ( § δE = ( C/ δ (exp(2 σT ) −
1) =
CT δσ . Thecalculation of the first variation of the acoustic energy is one-to-one related to the firstvariation of the eigenvalue, δσ . In light of this, we focus only on the calculation of δσ in §§ Eigenvalue integral formulae
Performing the operation (cid:104) ˆ u ∗ , (3.1) (cid:105) V + (cid:104) ˆ p ∗ , (3.2) (cid:105) V and dividing by C provides aneigenvalue functional σ [ˆ p ∗ , ˆ˙ Q ] := 1 C (cid:18) γ − γ ¯ p (cid:19) (cid:68) ˆ p ∗ , ˆ˙ Q (cid:69) V , (3.3)which was used in Magri & Juniper (2013) without any derivation. This relation isneeded to define the eigenvalue functional derivative in § C , which de-phases the eigenvalue with respect to the numerator. To enable physical interpretationof the Rayleigh criterion in the frequency domain, we derive two other integral for-mulae with real denominators, which do not affect the phase. Performing the opera-tion Re ( (cid:104) ˆ u, (3.1) (cid:105) V + (cid:104) ˆ p, (3.2) (cid:105) V ), after including the fact that the real part of (cid:82) L (ˆ u ∗ ∂ ˆ p∂x +ˆ p ∗ ∂ ˆ u∂x ) dx is zero (but the imaginary part is not) because of the ideal boundary conditions,offers a convenient functional for the growth rate σ r [ˆ p, ˆ˙ Q ] = 1 F (cid:18) γ − γ ¯ p (cid:19) Re (cid:68) ˆ p, ˆ˙ Q (cid:69) V , (3.4)where F := (cid:82) L (cid:0) ¯ ρ | ˆ u | + | ˆ p | / ( γ ¯ p ) (cid:1) dx . On the other hand, for the angular frequency,performing the operation Im ( (cid:104) ˆ u, (3.1) (cid:105) V − (cid:104) ˆ p, (3.2) (cid:105) V ), after including the fact that theimaginary part of (cid:82) L (ˆ u ∗ ∂ ˆ p∂x − ˆ p ∗ ∂ ˆ u∂x ) dx is zero (but the real part is not) because of theideal boundary conditions, offers a convenient functional for the angular frequency σ i [ˆ p, ˆ˙ Q ] = − G (cid:18) γ − γ ¯ p (cid:19) Im (cid:68) ˆ p, ˆ˙ Q (cid:69) V , (3.5)where G := (cid:82) L (¯ ρ | ˆ u | − | ˆ p | / ( γ ¯ p )) dx (cid:54) = 0 (when ˆ˙ Q (cid:54) = 0) following the argument of § / G/ σ i >
0, so the sign of the right-hand side ispositive. Hence the acoustic potential energy is smaller (larger) than the acoustic kineticenergy when the imaginary part of the Rayeligh integral (3.5) is negative (positive).Equations (3.4) and (3.5) represent the Rayleigh criterion in the frequency domain,which states that if the heat release rate is in phase with the pressure, the growthrate is maximized. We also gain an extra piece of information with respect to thetime domain analysis: if the heat release rate is in quadrature with the pressure, thefrequency is maximized. The latter is the mathematical formalization and generalizationof a statement made in Lord Rayleigh (1878) regarding the pitch of the oscillation. Theintegral formulae are showcased analytically in § Luca Magri, Matthew P. Juniper and Jonas P. Moeck
First variation without Lagrange multipliers
We follow the same procedure as that of § Q ( x ) → ˆ˙ Q ( x ) + (cid:15) ˆ˙ Q p ( x ) where ˆ˙ Q p ( x ) is the test function in thefrequency domain. The objective is to find the first variation of the eigenvalue, δσ , inthe vicinity of ˆ p and ˆ˙ Q . The eigenvalue functional derivative we wish to calculate is thekernel of the eigenvalue first variation δσ = (cid:15) (cid:90) L (cid:32) δ ˆ˙ Q σδ ˆ˙ Q ( x (cid:48) ) + δ ˆ p σδ ˆ p ( x (cid:48) ) d ˆ p ∗ d ˆ˙ Q ( x (cid:48) ) ∗ (cid:33) ∗ ˆ˙ Q p ( x (cid:48) ) dx (cid:48) . (3.6)A similar physical explanation to that given in § δ ˆ˙ Q σδ ˆ˙ Q = 1 C ∗ (cid:18) γ − γ ¯ p (cid:19) ˆ p ∗ , δ ˆ p σδ ˆ p = 1 C ∗ (cid:18) γ − γ ¯ p (cid:19) ˆ˙ Q ∗ , (3.7)whence the eigenvalue functional derivative is δσδ ˆ˙ Q := (cid:32) δ ˆ˙ Q σδ ˆ˙ Q ( x (cid:48) ) + δ ˆ p σδ ˆ p ( x (cid:48) ) d ˆ p ∗ d ˆ˙ Q ( x (cid:48) ) ∗ (cid:33) = 1 C ∗ (cid:18) γ − γ ¯ p (cid:19) (cid:32) ˆ p + d ˆ pd ˆ˙ Q ˆ˙ Q (cid:33) ∗ , (3.8)such that the eigenvalue first variation is provided by δσ = (cid:104) ( δσ/δ ˆ˙ Q ) , (cid:15) ˆ˙ Q p (cid:105) V .3.3. First variation with Lagrange multipliers
The objective of this section is the same as that of § δσ/δ ˆ˙ Q subject to (3.1), (3.2). We define a Lagrangian L ( σ, ˆ u, ˆ p, ˆ u + , ˆ p + , ˆ˙ Q ) := σ − (cid:10) ˆ u + , (3.1) (cid:11) V − (cid:10) ˆ p + , (3.2) (cid:11) V . (3.9)After integrating by parts, the first variation of the Lagrangian reads δ L = δσ (cid:32) − (cid:90) L (cid:32) ¯ ρ ˆ u + ∗ ˆ u + 1 γ ¯ p ˆ p + ∗ ˆ p − γ − γ ¯ p ∂ ˆ˙ Q∂σ ˆ p + ∗ (cid:33) dx (cid:33) . . . − [ˆ u + ∗ δ ˆ p ] L − [ˆ p + ∗ δ ˆ u ] L + (cid:42) − σ ¯ ρ ˆ u + + ∂ ˆ p + ∂x + γ − γ ¯ p (cid:32) ∂ ˆ˙ Q∂ ˆ u (cid:33) ∗ ˆ p + , δ ˆ u (cid:43) V . . . + (cid:42) − σ ˆ p + γ ¯ p + ∂ ˆ u + ∂x + γ − γ ¯ p (cid:32) ∂ ˆ˙ Q∂ ˆ p (cid:33) ∗ ˆ p + , δ ˆ p (cid:43) V + (cid:15) γ − γ ¯ p (cid:68) ˆ p + , ˆ˙ Q p (cid:69) V . (3.10)We define the adjoint eigenvalue problem − σ ¯ ρ ˆ u + + ∂ ˆ p + ∂x + γ − γ ¯ p (cid:32) ∂ ˆ˙ Q∂ ˆ u (cid:33) ∗ ˆ p + = 0 , (3.11) − σ ˆ p + γ ¯ p + ∂ ˆ u + ∂x + γ − γ ¯ p (cid:32) ∂ ˆ˙ Q∂ ˆ p (cid:33) ∗ ˆ p + = 0 , (3.12) ensitivity of the Rayleigh criterion in thermoacoustics u + ∗ δ ˆ p ] L = 0 and [ˆ p + ∗ δ ˆ u ] L = 0. Hence, the firstvariation of the eigenvalue can be calculated as δσ = δ L = (1 /D )( γ − / ( γ ¯ p ) (cid:104) ˆ p + , (cid:15) ˆ˙ Q p (cid:105) V ,where D := (cid:82) L (¯ ρ ˆ u + ∗ ˆ u + γ ¯ p ˆ p + ∗ ˆ p − γ − γ ¯ p ˆ p + ∗ ∂ ˆ˙ Q∂σ ) dx . Finally, the eigenvalue functionalderivative is δσδ ˆ˙ Q = 1 D ∗ (cid:18) γ − γ ¯ p (cid:19) ˆ p + . (3.13)3.4. Relation between first variations with and without Lagrange multipliers
The relations (3.8) and (3.13) imply thatˆ p + ∗ = DC (cid:32) ˆ p + d ˆ pd ˆ˙ Q ˆ˙ Q (cid:33) . (3.14)This is a key equation in the frequency domain. It is the link between the Lagrangemultiplier (adjoint variable) and the system’s observables. Physically, the adjoint eigen-function is the sum of two terms multiplied by a constant D/C . Starting from the firstterm on the right-hand side they are: (i) The acoustic pressure at the location where theperturbation is applied (acoustic sensitivity), and (ii) the acoustic pressure sensitivityat the main source spatial support weighted by the main heat release rate (thermalsensitivity). Relation (3.14) shows how to express the first variation of the Rayleighcriterion in the frequency domain in a non-self-adjoint system ( ˆ˙ Q (cid:54) = 0); as in § d ˆ p/d ˆ˙ Q ) ˆ˙ Q , which is a nonlocaleffect. For a localized source, similarly to § p + ∗ p = ( D/C )(ˆ p p + d ˆ p f ˆ Q f / ( (cid:15) ˆ Q p )).Relation (3.14) can be used in two ways. On the one hand, we can obtain an approxima-tion of the adjoint variable by interpolating experimental measurements of ˆ p +( d ˆ p/d ˆ˙ Q ) ˆ˙ Q .On the other hand, we can cheaply estimate the sensitivity function from a low-ordermodel as ˆ p + ∗ − ( D/C )ˆ p . This indicates which areas of the flow should be sampled toreconstruct an accurate sensitivity function. In other words, this relation gives a prior toguide the measurements in experiments and sampling in high-fidelity simulations.3.5. The adjoint Rayleigh criterion
The adjoint Rayleigh criterion answers the question: “Under which conditions is thefirst variation of the eigenvalue positive, negative, or zero, when the heat source isperturbed?”. This was numerically investigated in Magri & Juniper (2013, 2014); Juniper(2018). Here, we formalize the adjoint Rayleigh criterion following the argument of Magri(Appendix C, 2019). The factor D can be set to a real number (e.g., Eq. (28), Aguilar et al. D = ( γ − / ( γ ¯ p ) in this section without loss of generality.For localized sources, the first variation of the growth rate is δσ r = | ˆ p ∗ + p || ˆ˙ Q p | cos( ∆θ )where ∆θ is the difference between the arguments of the heat release rate and adjointpressure. This means that δσ r is maximum when ∆θ = ± k − π , where k is a positiveinteger, i.e., when a perturbation to the heat release rate is in phase with the adjointpressure, the system is most destabilized. Second, the first variation of the growthrate, δσ r , is minimum when ∆θ = ± (2 k − π . i.e., when a perturbation to the heatrelease rate is in antiphase with the adjoint pressure, the system is most stabilized.Third, the first variation of the growth rate δσ r is zero when ∆θ = ± (2 k + 1) π/ Luca Magri, Matthew P. Juniper and Jonas P. Moeck is δσ i = | ˆ p ∗ + p || ˆ˙ Q p | sin( ∆θ ). In contrast to the growth rate, δσ i is maximum when theadjoint pressure is in quadrature with the heat release rate. Note that we could have usedthe adjoint without the complex conjugation in (2.4). The results discussed would stillbe valid by taking into account that, without complex conjugate, the adjoint eigenvectorrotates in time as exp( − σt ), which is to say that the angular frequency is negative.
4. Conclusions
Thermoacoustic instabilities are a major challenge for the gas turbine and rocketmotor industries. The Rayleigh criterion is widely used to establish whether or not athermoacoustic system is unstable. One result of this paper is to formalize and interpretthe Rayleigh criterion in the frequency domain ( § References
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