Rate-dependent bifurcation dodging in a thermoacoustic system driven by colored noise
RRate-dependent bifurcation dodging in a thermoacoustic systemdriven by colored noise
Xiaoyu Zhang a , Yong Xu a,b, ∗ , Qi Liu a , J¨urgen Kurths c , Celso Grebogi d a School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710072, China b MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern PolytechnicalUniversity, Xian, 710072, China c Potsdam Institute for Climate Impact Research, Potsdam 14412, Germany d Institute for Complex Systems and Mathematical Biology, School of Natural and Computing Sciences,Kings College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
Abstract
Tipping in multistable systems occurs usually by varying the input slightly, resulting inthe output switching to an often unsatisfactory state. This phenomenon is manifested inthermoacoustic systems. This thermoacoustic instability may lead to the disintegration ofrocket engines, gas turbines and aeroengines, so it is necessary to design control measuresfor its suppression. It was speculated that such unwanted instability states may be dodgedby changing quickly enough the bifurcation parameters. Thus, in this work, based on afundamental mathematical model of thermoacoustic systems driven by colored noise, thecorresponding Fokker-Planck-Kolmogorov equation of the amplitude is derived by using astochastic averaging method. A transient dynamical behavior is identified through a prob-ability density analysis. We find that the rate of change of parameters and the correlationtime of the noise are helpful to dodge thermoacoustic instability, while a relatively largenoise intensity is a disadvantageous factor. In addition, power-law relationships between themaximum amplitude and the noise parameters are explored, and the probability of success-fully dodging a thermoacoustic instability is calculated. These results serve as a guide to thedesign of engines and to propose an effective control strategy, which is of great significanceto aerospace-related fields.
Keywords thermoacoustic system, colored noise, rate-dependent tipping, transient ∗ Corresponding author
Email addresses: [email protected] (Xiaoyu Zhang), [email protected] (Yong Xu), [email protected] (Qi Liu),
[email protected] (J¨urgen Kurths), [email protected] (Celso Grebogi) a r X i v : . [ n li n . AO ] S e p . Introduction In a multistable system, tipping usually refers to the event that a small change in inputresults in a sudden and disproportionate change in output [1, 2]. Tipping was first introducedinto sociology in the 1950s to describe the phenomenon of “white escape” in American urbancommunities, i.e., when the number of non-white residents in a community reaches a certainlevel, that quickly leads to the result in which the community is completely occupied by non-white people [3]. After that, the publication of the book [4] made the concept of “tipping”be generally accepted. It is currently widely used in climatology, ecology and several otherresearch fields [5, 6, 7, 8]. In general, one of the equilibrium points in multistable systemsis desirable, and the shift to another equilibrium point leads to disastrous consequences,often irreversible to some extent, which requires control strategies to avoid or slow down thesystem from drifting towards a tipping [9, 10, 11].In view of the catastrophic consequences of tipping, the control of tipping has alwaysbeen a timely research topic in various fields. Among them, the effective means is to identifyearly warning signals for a timely control [12, 13, 14, 15]. In addition, time-delay feedbackcontrol strategy is proposed to mitigate the tipping [16]. In a pollinator-plant mutualisticnetwork, it was shown that Gaussian white noise is an important factor to promote thestate recovery after tipping, so as to avoid the adverse consequences of tipping [17]. Due tothe complexity of multistable systems, a single control strategy may not be suitable for allsystems. Therefore, it is necessary to determine the main factors in a specific system, andthen put forward effective control strategies to dodge the undesirable states.Tipping in thermoacoustic systems was shown to result from thermoacoustic instabilityin combustion chambers [18]. When there is a positive feedback between the fluctuation ofsound pressure and unsteady heat release rate in the combustion chamber, a thermoacousticinstability occurs [19]. It may lead to a structural damage of rocket and aeroengine, compro-mising rocket launching missions due to engine disintegration, which is clearly an unwantedstate in thermoacoustic systems. As an important example, in the Apollo program of theUnited States, more than 2000 full-scale engine experiments have been conducted on theF-1 engine due to the thermoacoustic instability, consuming a lot of human and materialresources [20]. Hence, the most important and significant reason to study tipping in ther-moacoustic systems is to fully understand the observed thermoacoustic instability and tolearn how to influence and control it.There are two kinds of control strategies for thermoacoustic instability: passive and activecontrol [18]. Passive control reduces the instability mainly by decreasing the combustion2esponse of propellant and increasing the structural damping. The design process of passivecontrol consumes a lot of resources and time, and the control results obtained only have effecton specific engines in a certain working range. Thus passive control is mainly applicable toa situation where the working environment is detrimental and an active control method isdifficult to be enacted. The active control of thermoacoustic instability is based on a certaincontrol algorithm, e.g. by periodically adding energy to the combustion system to suppressoscillation. It requires a mathematical model to describe the dynamical behavior of thethermoacoustic system. Through the mathematical model, the dynamic characteristics ofthe combustion chamber under different working conditions is obtained, and then the activecontrol function of the thermoacoustic instability is inferred. It should be noted that via thetraditional theoretical methods it is very difficult to predict the thermoacoustic instabilityin thermoacoustic systems, and the classical linear stability method is only suitable to studyasymptotic instability. Moreover, the dynamics of the nonlinear thermoacoustic instabilitybelongs to the transient growth process, and the coupling between the sound field and thetransient burning rate of propellant needs to be considered, which makes the analysis evenmore difficult. The transient characteristics and the nonlinear behaviors are very importantand fundamental for understanding the thermoacoustic instability, which is the reason whywe mainly study the transient dynamical behaviors of the nonlinear mathematical model ofthe thermoacoustic system.When the combustion chamber produces a nonlinear thermoacoustic instability, exceptfor a few cases which leads to engine damage, the amplitude generally stays at a finite value,and the system has a bounded motion, usually showing periodic limit cycle oscillation. Forpractical applications, it is desirable to understand the magnitude of this finite amplitudemotion and how it is affected by system parameters. This helps to actively change theparameters to reduce the amplitude of the oscillation, so as to dodge the thermoacousticinstability. Considering the non-autonomous dynamical characteristics of the thermoacousticsystem, the rate of change of the parameter is likely to be an important factor affecting theamplitude of the thermoacoustic instability.A variety of rate-dependent bifurcations are available in the literature. From the per-spective of mathematical theory, Ashwin et al. [1], based on the concept of a pull-backattractor, gave the mathematical definition of rate-induced tipping for a class of nonau-tonomous systems. Kiers [21] put forward the definition for discrete dynamical systems, andgave the condition of rate-induced tipping. Sujith [22] showed experimental evidence of rate-dependent tipping in a typical turbulent afterburner model. In addition, through numerical3imulation, Vanselow et al. [23] confirmed that the Rosenzweig-MacArthur predator-preymodel experienced rate-dependent bifurcation, which led to the collapse of predator and preypopulations. Suchithra et al. [24] studied the rate-dependent tipping in power systems, andfound that the tipping is dependent on the initial conditions of the system. Those studies onthe aforesaid dynamics systems suggest that it is imperative to introduce the rate-dependentbifurcation in thermoacoustic systems.Crucially, it is not enough to consider only the dynamical behavior of the deterministicthermoacoustic systems. A large number of experiments clarified that the amplitude andphase of the limit cycle oscillation vary between cycles, and the parameters of the stabilityboundary in the combustion chamber also vary [25], which shows that the thermoacousticsystem is stochastic. Although the deterministic system can also exhibit as a seeminglyrandom motion when strange attractors exist, Curlick et al. [26] used a fractional dimensiontest to reject the possibility that the thermoacoustic system is a chaotic system. Poinsot etal. [27] carried out an experimental study on the dump combustor, and they confirmed theexistence of noise in the combustor through the spatial diagram of the coherence function. Liet al. [28] systematically studied the influence of background noise on the nonlinear dynamicsof a thermoacoustic system with a subcritical Hopf bifurcation. All of these show that thethermoacoustic system with noise excitation is more suitable to describe its response.Recently, Bonciolini and Noiray [29] considered a thermoacoustic system excited by Gaus-sian white noise. The system has a non-monotonic dynamical behavior, which at low andhigh values of the bifurcation parameter range, it has a low amplitude, and at intermediatevalues, it shows a high amplitude. That is to say, when the bifurcation parameter valuechanges from minimum to maximum value, the amplitude changes from the low state tothe high state and then goes back to the low one. In the thermoacoustic system, the highamplitude of the system is closely related to the thermoacoustic instability, which is notideal. Bonciolini and Noiray [29], through experiments and numerical simulations, foundthat when the rate of change of the parameter is large enough, it can dodge the unwantedstate in the intermediate parameter region. However, the power of the white noise withzero correlation considered in that study is infinite. But since there is no noise with infinitepower, the actual noise must be “colored”. This makes it necessary to consider the influenceof colored noise with non-zero correlation time on the dynamical behavior of the system.It is essential for a complete stochastic theory to study the influence of the correlationtime of noises on the properties of stochastic systems. In particular, under certain nonlinearconditions, the noise with a long correlation time makes the stochastic system behave com-4letely different from that under white noise only [30]. But, for simplicity, when the macrovariables and noises can be clearly divided into two time scales and the macro variableschange slowly, the disturbance can be considered as white noise [31]. However, when thesystem can not be distinguished by the two time scales clearly, it is necessary to includecolored noise. In a system with rate-dependent bifurcation, increasing the rate of parameterchange reduces the time scale of the macro variables and decrease the discrimination betweenboth time scales. For our system, colored noise is more suitable to describe the random force.This is the necessity of considering colored noise in the rate-dependent systems.The objective of this paper is to investigate whether the rate of change of parametersis helpful to dodge thermoacoustic instability in thermoacoustic systems excited by colorednoises. For a typical thermoacoustic system model, the reduced dimension Fokker-Planck-Kolmogorov (FPK) equation is obtained by using the stochastic averaging method. Throughthe evolution of the probability density of the amplitude, the effect of the rate-dependentof parameters on dodging thermoacoustic instability is analyzed. In addition, we also studythe influence of the correlation time and intensity of noises on rate-dependent bifurcation,and calculate the probability of successfully dodging thermoacoustic instability.This paper is organized as follows. In Section 2, thermoacoustic instability and a clas-sical mathematical model of thermoacoustic systems are introduced. The influence of theparameter varying rate on dodging thermoacoustic instability is examined in Section 3. Sec-tion 4 discusses the dodge probability of the rate-dependent bifurcation. Finally, severalconclusions are given to close this paper in Section 5.
2. Mathematical model of the thermoacoustic system
A thermoacoustic system is closely related to many combustion power devices, such asliquid rocket engines, solid rocket engines, gas turbines and aeroengines. These combustionpower units convert chemical energy stored in the molecular bonds into kinetic energy in theengines. The first stage of energy conversion is the combustion of oxidant and fuel in thecombustion chamber. At this time, the chemical energy is converted into heat energy. Thesecond stage is realized in the nozzle. The heat energy released by combustion is convertedinto kinetic energy at the engine through the expansion and acceleration of the nozzle. Inthe first stage described above, thermoacoustic instability is where it occurs.High performance is usually achieved by increasing the energy release rate per unit volumein the engine, where the energy density is very large. Such a large energy density may beaccompanied by relatively small fluctuations, and the amplitude of these small fluctuationsmay be only a small disturbance. It may, however, be shown as an unacceptable large5mplitude disturbance, that is, a strong pressure oscillation. Pressure oscillations are theresult of the interaction of acoustic vibrations, internal combustions and flows, which areusually called thermoacoustic instability. Thermoacoustic instability can lead to abnormalinterior ballistics, structural damage, engine disintegration, resulting in mission failure. Thisrequires us to predict the thermoacoustic instability as early and as accurately as possible,and to find measures to suppress the instability without affecting the overall performance.Fundamentally speaking, the thermoacoustic instability is essentially the coupling of theunsteady combustion and the structural acoustic characteristics of the combustion chamber.As the simplest example of thermoacoustic instability, Rijke invented and first studied theRijke tube in 1859. Then Rayleigh proposed the Rayleigh criterion, which is the most originaland scientific description of a thermoacoustic coupling effect [32]. With the development ofresearch, the Helmholtz equation (2.1) is often used to describe the dynamical behavior ofthermoacoustic systems in recent years, ∂ x∂t − c ∇ x = ( γ − ∂q∂t , (2.1)where x , c , γ and q denote the acoustic pressure, the speed of sound, the heat capacity ratioand the fluctuating component of the heat release rate, respectively. After Laplace transform,orthogonal basis projection and truncated Taylor expansion approximation of the nonlinearterm [33], we get the following reduced mathematical model of the thermoacoustic system¨ x − (cid:0) v + β x − β x (cid:1) ˙ x + ω x + β x = ξ ( t ) , (2.2)in which v is bound up with the linear growth rate, and with β , β , and β being realparameters. The ξ ( t ) is the exponential correlated Gaussian colored noise with zero mean,defined as, (cid:104) ξ ( t ) (cid:105) = 0 , (cid:104) ξ ( t ) ξ ( s ) (cid:105) = Dτ exp (cid:18) − | t − t | τ (cid:19) , where D and τ are the two most important parameters for characterizing noise: noise inten-sity and the correlation time of the noise.By means of the stochastic averaging method, it yields the stochastic differential equationfor the amplitude A of the system. Then we can derive the FPK equation (2.3) for theamplitude A with the help of stochastic dynamical theory, ∂P ( A, t ) ∂t = − ∂∂A [ m ( A ) P ( A, t )] + 12 · ∂ ∂A [ b ( A ) P ( A, t )] , (2.3)6ith the drift and diffusion coefficients m ( A ) = v A + β A − β A + D ω A (1 + ω τ ) ,b ( A ) = Dω (1 + ω τ ) , where P ( A, t ) is the probability density function (PDF) of the amplitude A and the ω isthe natural angular frequency. Later, we will use the PDF P ( A, t ) to describe the randomtransient dynamical behavior of the thermoacoustic system. The amplitude effective po-tential function of the model (2.2) is a double well, which corresponds to the two states ofhigh and low amplitudes in the dynamics, as seen in the time series (see Figure 1(b)). Thisshows that the model can well describe the intermittent switching between the two statesobserved in the experimental thermoacoustic system [33] (Figure 1(a)), and further explainsthe rationality of the mathematical model (2.2) used in this paper. (b) low highAmplitude effective potentiallow highAmplitude effective potential (a) (b) low highAmplitude effective potential (a)
Figure 1: Time series diagram of a thermoacoustic system. (a) The experimental data; (b) The data fromthe mathematical model (2.2). The two states of high and low amplitude correspond to the double well ofeffective potential.
We consider in this work the suppression of the thermoacoustic instability of a thermoa-coustic system when the linear growth rate v is no longer a constant but a function of time.Through the experiments of thermoacoustic systems and the parameter identification of theresults, an approximate relationship between the linear growth rate v and the air mass flowrate ˙ m air is obtained [29], v ( ˙ m air ) = n cos ( n ˙ m air ) + n , (2.4)7here n , n , and n are constant coefficients. In order to be close to real situations, the˙ m air ( t ) is assumed to be a piecewise linear function involving a ramp˙ m air ( t ) = ˙ m + Rt. (2.5) R is the ramp rate and the ˙ m is the initial value. In the following sections, the time-varyingparameter v ( t ) is the combination of the cosine function (2.4) and the linear function (2.5).
3. Rate-dependent bifurcation dodge with the colored noise
In this section, we initially discuss the influence of parameter change rate R on avoidingthe thermoacoustic instability in a thermoacoustic system under the excitation of colorednoise. The system (2.2) can express its transient dynamical behavior through Monte Carlosimulations, but it needs a lot of calculation to collect the data. Here we use instead thereduced dimension FPK equation (2.3) and the more efficient Crank-Nicolson differencemethod whose accuracy was validated in [34], to get the evolution of the non-autonomousthermoacoustic system.Figure 2 displays the effect of different rates R on the transient dynamical behavior.The time-varying parameter v ( t ) is the combination of cosine and linear function mentionedabove. The contour map shows the probability density of the amplitude A changing with˙ m air . The probability of dark color is large, while that of light color is small. Here we dividethe probability density by its maximum value to normalize it, so its range of variation is 0to 1. As a reference, the yellow lines, which are the same in Figures 2(a) and (b), are for thecases when rates are not included. Figure 2(a) is the probability density of the amplitudeafter the introduction of a relatively small rate. The dark blue line in the middle is themean value. We find that after the introduction of the ramp rate, the amplitude is in factreduced, and the position of the highest amplitude point is delayed. This phenomenon ofrate-dependent tipping-delay has been discussed in detail in [34]. When the rate is relativelylarge (see Figure 2(b)), it can be seen from the dark red line representing the mean valuethat the amplitude reduction is stronger, and the phenomenon of bifurcation dodge is moreobvious. This shows that in a non-autonomous thermoacoustic system, the thermoacousticinstability can be avoided by properly controlling the changing rate R of the parameters.8 igure 2: The effect of different rates R of the time-varying parameter on the transient dynamical behavior.The contour map shows the transient probability density of the amplitude A with respect to the air mass flowrate ˙ m air , and the value is divided by the maximum probability to make the result normalized. The yellowline is the quasi-steady deterministic results, used as a reference. Dark blue and dark red lines represent themean value of the amplitude A under different R . (a) R = 10; (b) R = 75. Other parameters are β = − . β = 0 . ω = 100 × π , D = 8 . × , and τ = 0 .
9e have just analyzed that under the excitation of the colored noise, the effect of bi-furcation dodge becomes better with the increase of the rate R . Next, the influence of thecorrelation time τ and the intensity D of noise, which are important parameters to char-acterize the colored noise, on bifurcation dodge are studied. As shown in Figure 3(a), thelarger the noise correlation time τ is, the lower the amplitude becomes, and the better thedodging effect of thermoacoustic instability is. Considering that white noise is only an idealstate, the correlation time of noise always exists. The existence of the noise correlationtime τ is conducive to the avoidance of thermoacoustic instability in the thermoacousticsystem, which shows that the actual control effect is better than that of the idealized math-ematical model when the same R is selected. Figure 3(b) shows that the amplitude of thethermoacoustic system increases due to the noise intensity D , which intensifies the pressureoscillation. Therefore, it is necessary to pay attention to control the noise intensity in a realthermoacoustic system. A =0=0.001=0.01=0.02 (a) A D=10 D=10 D=10 D=10 (b) Figure 3: The influence of different correlation time τ and intensity D of noise on bifurcation dodge. (a)Different τ under the same R = 10 and D = 8 . × ; (b) Different D under the same R = 10 and τ = 0 . β = − . β = 0 . ω = 100 × π . Furthermore, we try to find a quantitative relationship between the amplitude of thesystem and the parameters characterizing the noise when the rate R is fixed, so as to better10mplement the control scheme. We consider the function of the maximum amplitude A max with respect to τ and D . It can be inferred from Figure 4 that the maximum amplitudedecreases rapidly with the increase of τ , and then tends to be flat. With the increase of D ,the rising speed of the maximum amplitude is also steep first and then slow. A m a x R=10R=50R=75 (a) D A m a x R=10R=50
R=75 (b)
Figure 4: Variation of the maximum amplitude A max with respect to (a) noise correlation time τ , and (b)noise intensity D for different rates R . Other parameters are β = − . β = 0 . ω = 100 × π . In order to determine more accurately the functional relationship, we draw the log-logplot of the maximum amplitude A max and both the noise correlation time τ and D in Figure5. We then clearly see that the relationship between the ln ( A max ) and ln ( τ ) is a linearfunction. Let ln ( A max ) = − p · ln ( τ ) + a , where p > a is a constant. Then it can bededuced that ln ( A max ) = ln ( τ − p ) + ln ( e a ) = ln ( e a · τ − p ) . With logarithms removed from both sides of the equation, we get that A max and τ have thefollowing power-law relationship A max ( τ ) ∼ τ − p , (3.1)where p > A max and the noise intensity DA max ( D ) ∼ D q , (3.2)where q > τ is very small, i.e., the first few points in Figure 5(a), the power-law relationship is notwell expressed. In this case, the colored noise is close to the white noise, and the randomnessis relatively strong, so it is easy to destroy the power-law relationship. However, when R
11s large, the rate becomes the major cause of the system qualitative change, which weakensthe destructive noise impact caused by a small correlation time, as shown by the power-lawrelationship. But the power-law relationship is still relatively good when R is large and thecorrelation time is small. -6 -5 -4 -3 ln( ) -2024 l n ( A m a x ) R=10R=50R=75 (a)
16 17 18 ln(D) l n ( A m a x ) R=10R=50
R=75 (b)
Figure 5: (a) The log-log plot of the Eq. (3.1), between the maximum amplitude A max and the noisecorrelation time; (b) The log-log plot of the Eq. (3.2), between the maximum amplitude A max and the noiseintensity D . The solid lines are fittings with linear functions. For the sake of getting a more detailed relationship, we can estimate the values of thepower-law exponents p and q under different rates R by data fitting. We infer from the Figure6 that the power-law exponent p of the noise correlation time remains virtually unchangedin the process of R change. That is to say, p is a global constant and the rate of decreaseis constant for all rates R . In addition, the power law exponent q of noise intensity is alsostable at a constant value when R >
30, but it decreases with the decrease of R when theramp rate is small. When the parameters change a little faster, the change speed of themaximum amplitude A max with the noise intensity D is no longer affected by the rate R .This power-law relationship quantifies the influence of the correlation time of the noise andthe noise intensity on the larger amplitudes in the thermoacoustic system, which is of highestsignificance for the control strategy to dodge the thermoacoustic instability.
4. Probability of the rate-dependent bifurcation dodge
For a practical thermoacoustic system, the threshold of thermoacoustic instability isusually defined according to its actual utility. So in this section we present calculations ofthe probability of a successful bifurcation dodge for a given threshold in the thermoacousticsystem. 12
20 40 60 80 100 R p q Figure 6: Variation of the power exponents p and q of the Eqs. (3.1) and (3.2) calculated numerically fordifferent parameter change rate R . In the thermoacoustic system excited by additive colored noise, we consider that thethreshold value of the thermoacoustic instability A th is between the maximum of the ampli-tude of the stationary state A max and the minimum value A min . In this work, we consider thethreshold A th = ( A max + A min ) / T d , the probability of abifurcation dodge is the probability of the amplitude A in the threshold range [0 , A th ] beforetime T d . Thus, we can use the following formula to calculate the probability of bifurcationdodge P by the FPK equation (2.3) P ( T d ) = (cid:90) A th P ( A, T d ) dA. (4.1)Figure 7 shows the influence of different rates R of the time-varying parameter on the tran-sient dynamical behavior of the thermoacoustic system with a given threshold A th . Figures7(a) and (b) are the results corresponding to Figures 2(a) and (b) after the threshold value A th is being set. It can be seen that when R = 10 (see Figure 7(a)), most of the amplitudesbeyond ˙ m air ( t ) = 16 exceed the threshold range, and the probability density is absorbed bythe threshold boundary. However, when the rate is large (see Figure 7(b)), some of the am-plitudes remain in the threshold range, which makes the probability of successfully dodgingto reach 70%. Figure 7(c) shows the probability of successful dodging at different rates R .For the abscissa, we divide the time by the duration time of parameter change T d , unifyingthem in the same time scale. It is not difficult for us to understand that the larger R is, thehigher the probability of successfully dodging thermoacoustic instability is.Next, we consider the probability of the thermoacoustic system to successfully dodgethe thermoacoustic instability threshold A th excited by additive colored noise under thecondition of a fixed rate R with different noise correlation time τ and noise density D .Figure 8(a) shows that the larger τ is, higher the probability of successfully dodging the13 t/T d P R=10R=50R=75 (c)
Figure 7: The influence of different rates R of the time-varying parameter on the transient dynamicalbehavior of the thermoacoustic system with a given threshold A th . The yellow lines indicate the quasi-steady deterministic results as a reference. Dark blue and dark red lines represent the mean value of theamplitude A under different R . (a) R = 10; (b) R = 75. (c) The probability of successful bifurcation dodgewith different rates R under the excitation of additive colored noise. For comparison, the abscissa is dividedby the duration time of parameter change T d to make it unified in the same time scale. Other parametersare β = − . β = 0 . ω = 100 × π , D = 8 . × , and τ = 0 . τ increases to a certain value, the system can quench the thermoacousticinstability with a probability close to 1. This shows that the noise correlation time τ is afavorable factor to dodge the thermoacoustic instability. On the contrary, the noise intensity D is not conducive to avoiding thermoacoustic instability. The higher D , the lower theprobability of successfully avoiding thermoacoustic instability, as shown in Figure 8(b). t/T d P =0.001=0.005=0.01=0.02 (a) t/T d P D=10 D=10 D=10 D=10 (b) Figure 8: The probability of a successful bifurcation dodging under the excitation of additive colored noisewith (a) different noise correlation time τ , and (b) different noise intensity D . Other parameters are β = − . β = 0 . ω = 100 × π , and R = 10. When the rate R and τ vary at the same time, the probability of a bifurcation dodgingthe instability is discussed in Figure 9(a). This gives that moderately accelerating R andincreasing τ can make the probability of the thermoacoustic system dodging the high am-plitude to reach more than 90%, so as to avoid the occurrence of thermoacoustic instability.Combined with R , we find that when R is larger and D is smaller, it is helpful to avoidthermoacoustic instability, as shown in Figure 9(b). When D is relatively small or large,the parameter change rate has little effect on avoiding the thermoacoustic instability. Thatis to say, when the thermoacoustic system is affected by strong noise, the system almost15nevitably produces thermoacoustic instability. At this time, it is impossible to keep thesystem in a safe state by changing the rate of parameters. Hence, the noise intensity mustbe controlled first. Figure 9: The probability of the thermoacoustic system to successfully dodge the thermoacoustic instabilitywhen combined with the two factors under the excitation of additive colored noise. (a) The noise correlationtime τ versus the ramp rate R ; (b) the noise intensity D versus the ramp rate R .
5. Conclusions
In this work, the avoidance of the undesirable state of a thermoacoustic system excitedby colored noise is studied. Here we argue that there is a strong reason for assuming thatthe changing rate of the parameters affects the dodging of thermoacoustic instability, andcolored noise is the most suitable form to describe the random forces in such rate-dependentsystem. We take into account the stochastic averaging method to overcome the non-Markovproperty of colored noise and to simplify the mathematical model. By means of the transientdynamical behavior shown by the evolution of the probability density, we find that thechanging rate of parameter is conducive to dodge the thermoacoustic instability. Besides,we uncover that the increase of the noise correlation time makes the dodging effect better,but an increase of the noise intensity brings an adverse effect. Furthermore, the functionalrelationship between the rate of parameters, the correlation time of the noise, the noiseintensity, and the maximum amplitude of the system are quantified in this work. The power-law relationship obtained is of utmost importance for understanding the internal mechanismof thermoacoustic systems and controlling thermoacoustic instability. Finally, the probabilityof successfully dodging the thermoacoustic instability under various conditions is calculatedto validate the effectiveness of the control. 16ur research is crucial to ensure a safe operation of thermoacoustic engines, becausereal nonlinear thermoacoustic systems are non-autonomous, the parameters do change withtime and the correlation time of the noise can not be ignored. Considering that rocketengine, gas turbine and aeroengine are the main thermoacoustic systems, the cost of full-scale experiments is huge. Our theoretical research is helpful to save human and materialresources. The rate of parameters, correlation time of the noise and noise intensity are usedtogether to better regulate the thermoacoustic instability, which has a guiding role in thedesign of the engines and the control of the combustion process, and it is also of significancefor the control of other systems with similar bifurcation properties.
Acknowledgments
This paper was supported by the National Natural Science Foundation of China un-der Grant No.11772255, the Fundamental Research Funds for the Central Universities, theResearch Funds for Interdisciplinary Subject of Northwestern Polytechnical University, theShaanxi Project for Distinguished Young Scholars,and Shaanxi Provincial Key R&D Pro-gram 2020KW-013 and 2019TD-010.
Conflict of interest
The authors declare that they have no conflict of interest.
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