A method for identifying combustion instability linear growth rate

Abstract

This invention discloses a method for identifying the linear growth rate of combustion instability, comprising: establishing a stochastic dynamics model of the combustion system; constructing an identification model based on the stochastic dynamics model of the combustion system; and identifying the linear growth rate, nonlinear saturation coefficient, and noise intensity of the combustion system based on the identification model. This invention aims to solve the problem in traditional methods where the changes in the linear growth rate, nonlinear saturation coefficient, and noise intensity cannot be identified from the data when the units of the experimental data are transformed.

Description

Technical Field

[0001] This invention relates to the field of aerospace engine development technology, and particularly to a method for identifying combustion instability linear growth rate. Background Technology

[0002] Combustion instability is a key concern in the development of aerospace engines. Extensive research has been conducted both domestically and internationally on its generation mechanisms, assessment and prediction methods, and control measures. Currently, the application of prediction methods in engineering is not yet mature, and reliable predictions of engine stability cannot be made before testing. Therefore, once combustion instability occurs, it needs to be addressed from two aspects: reducing excitation energy and increasing acoustic damping in the combustion chamber. The former requires considering the coupling process between combustion and the sound field, and due to the complexity of combustion phenomena, implementing improvement schemes is quite difficult. In comparison, the design of passive damping devices is easier.

[0003] From a linear perspective, the growth rate characterizes the rate of change of combustion chamber oscillation energy under small disturbances. If its value is greater than zero, it indicates that the system is unstable; conversely, it indicates that the system is stable. In this case, the growth rate is also called the decay rate. For combustion chambers exhibiting combustion instability, the decay rate of the newly designed damping device should be greater than the combustion chamber's growth rate while retaining a certain margin. That is, the growth rate of the unstable acoustic mode of the combustion chamber needs to be used as a design input parameter.

[0004] Scholars have proposed various methods to identify the growth rate. These are mainly divided into external excitation methods (input-output identification) and methods without external excitation (output identification only). In external excitation methods, for Hopf-type bifurcations, Lee utilizes the coherent resonance characteristics of noise before the bifurcation to identify the nonlinear combustion response of the system, extrapolating the location and type of the bifurcation point, the growth rate, the amplitude of the limiting cycle oscillation, etc., and further predicting the system stability boundary. Another type of method uses active control, applying and closing control at unstable operating points to obtain the process of the oscillation amplitude increasing from a small disturbance to the limiting cycle. Using the initial growth stage data of the oscillation amplitude (satisfying the linear assumption), an exponential function is fitted or other data processing methods (such as DMD) are used to identify the growth rate. In actual engine tests, it is difficult to add specific forms of external excitation or active control, therefore, external excitation methods are inconvenient in engineering applications. The methods without external excitation, in the first category, are the same as the processing method after closing control in active control methods, the difference being that they utilize the initial growth stage data of the spontaneous combustion unstable oscillation amplitude. Another type of method is based on the stochastic dynamics theory of the combustion chamber acoustic modes under noise excitation, extracting the linear growth rate from the limiting cycle oscillation data.

[0005] The existing methods for identifying the linear growth rate from limit cycle oscillation data have the following two shortcomings: (1) When the experimental data units are changed (e.g., pressure units include Pa, kPa, MPa, etc.), it is unclear how the linear growth rate, nonlinear saturation coefficient, and noise intensity identified from the data change; (2) When the experimental data length is finite, there are no reports on how to quantify the uncertainty of the identified linear growth rate, nonlinear saturation coefficient, and noise intensity. Summary of the Invention

[0006] The technical problem solved by this invention is to overcome the shortcomings of the prior art and provide a method for identifying the linear growth rate of combustion instability. This method aims to solve the problem in traditional schemes where the linear growth rate, nonlinear saturation coefficient, and noise intensity cannot be identified from the data when the experimental data units are changed.

[0007] To address the aforementioned technical problems, this invention discloses a method for identifying unstable linear growth rate of combustion, comprising:

[0008] Establish a stochastic dynamic model of the combustion system;

[0009] Based on the stochastic dynamics model of the combustion system, an identification model is constructed;

[0010] Based on the identification model, the linear growth rate, nonlinear saturation coefficient, and noise intensity of the combustion system are identified.

[0011] In the above method for identifying the linear growth rate of combustion instability, the stochastic dynamics model of the combustion system is expressed as follows:

[0012]

[0013] Where η represents pulsating pressure, α represents acoustic mode attenuation rate, ω0 represents acoustic mode angular frequency, ξ represents combustion noise, and f represents excitation source term generated by aerodynamic force and combustion process.

[0014] In the above method for identifying the unstable linear growth rate of combustion, when the combustion noise is Gaussian white noise, we have: <ξξ τ >=Γδ(τ); where, ξ τ Let ξ be the combustion noise time series after shifting ξ by time τ, δ(τ) represent the Dirac function, and Γ represent the noise intensity.

[0015] In the above method for identifying the linear growth rate of combustion instability, only the nonlinear combustion response function q is considered in f. c ,Right now

[0016] In the above method for identifying the linear growth rate of combustion instability, q c It can be expressed as a cubic polynomial of the pulsating pressure η: qc =βη-κ / 3η 3 Where β represents the linear response coefficient and κ represents the nonlinear saturation coefficient.

[0017] In the above method for identifying the linear growth rate of combustion instability, an identification model is constructed based on the stochastic dynamics model of the combustion system, including:

[0018] Equation (1) can be expressed as the following stochastic differential equation:

[0019]

[0020] Where v represents the linear growth rate;

[0021] Determine the amplitude-frequency representation of pulsating pressure:

[0022] η(t)=A(t)cos(ωt+σ(t))···(3)

[0023] Where A represents the amplitude of the pressure oscillation, ω represents the angular frequency of the pressure oscillation, and σ represents the phase of the pressure oscillation;

[0024] Substituting equation (3) into equation (2), and using the stochastic averaging method, equation (2) can be written as a first-order stochastic differential equation with amplitude A:

[0025]

[0026] Where ζ represents the intensity. White noise;

[0027] Based on equation (4), the probability density function P of amplitude A is determined. A The FK equations satisfied are:

[0028]

[0029] Based on equation (5), the drift coefficient S(A) and diffusion coefficient of the FK equation are determined. The solution formula is as follows:

[0030]

[0031] Where P(A,t1+Δt|a,t1) represents the probability that the amplitude is a at time t1, given that the amplitude is a at time t1, and the amplitude is A at time t1+Δt, and Δt represents the time shift.

[0032] In the above method for identifying the linear growth rate of combustion instability, the identification model is constructed based on the stochastic dynamics model of the combustion system, and further includes:

[0033] Let A(t) = kB(t), and substitute A(t) = kB(t) into equation (5), we get the following equation (7):

[0034]

[0035] Where k represents the scaling factor, i.e., the conversion relationship between units; B represents the amplitude after unit conversion; P B Let B be the probability density function representing the magnitude.

[0036] In the above method for identifying the linear growth rate of combustion instability, the linear growth rate, nonlinear saturation coefficient, and noise intensity of the combustion system are identified based on the identification model, including:

[0037] Step a, obtain the time series D of the pulsating pressure measured experimentally;

[0038] Step b, based on equation (7), after performing unit transformation on the pulsating pressure in the pulsating pressure time series D, substitute it into equation (6) to obtain discrete data points S whose drift coefficient and diffusion coefficient depend on the amplitude A.

[0039] Step c: According to the expression of the drift coefficient in equation (4), the obtained discrete data points S are fitted by the nonlinear data fitting method to obtain the identification results of the combustion system; wherein, the identification results include: the identification value of the linear growth rate, the identification value of the nonlinear saturation coefficient and the identification value of the noise intensity.

[0040] The above method for identifying the linear growth rate of combustion instability further includes: determining whether the identified values ​​of the linear growth rate, the nonlinear saturation coefficient, and the noise intensity obtained in step c meet the set accuracy requirements; if not, adjusting the scaling factor and repeating steps b to c until the identified values ​​of the linear growth rate, the nonlinear saturation coefficient, and the noise intensity meet the set accuracy requirements.

[0041] The above-mentioned method for identifying the linear growth rate of combustion instability also includes:

[0042] Substituting the identification result obtained from the experimentally measured pulsating pressure time series D into equation (2), and solving equation (2) numerically, at least N sets of pulsating pressure time series D obtained through numerical solution are obtained. N Based on the obtained N sets of pulsating pressure time series D N Using the same processing method as steps b and c, at least N sets of solution results are obtained, and the average value P and standard deviation Q of the N sets of solution results are determined; wherein, the solution results include: the solution value of the linear growth rate, the solution value of the nonlinear saturation coefficient, and the solution value of the noise intensity;

[0043] The identification result obtained in step c is compared with the average value P; if the deviation between the identification result obtained in step c and the average value P is less than a set threshold, then the standard deviation Q is used as the standard deviation of the identification result to determine the uncertainty of the identification result.

[0044] This invention has the following advantages: Please add details.

[0045] (1) This invention discloses a method for identifying the linear growth rate of combustion instability and proposes an uncertainty quantification analysis method based on Monte Carlo simulation to solve the problem of uncertainty quantification of the identified linear growth rate, nonlinear saturation coefficient and noise intensity when the experimental data length is limited.

[0046] (2) This invention discloses a method for identifying the linear growth rate of combustion instability and proposes a transformation relationship (Equation (7)) to solve the problem of how the linear growth rate, nonlinear saturation coefficient and noise intensity are identified from the data when the experimental data unit is changed (such as pressure units such as Pa, kPa, MPa, etc.). Attached Figure Description

[0047] Figure 1 This is a flowchart illustrating the steps of a method for identifying an unstable linear growth rate of combustion according to an embodiment of the present invention.

[0048] Figure 2 This is a time series of pulsating pressures experimentally measured under four different operating conditions in an embodiment of the present invention; wherein, Figure 2 (a) to 2(d) correspond to four different working conditions;

[0049] Figure 3 This is a schematic diagram of the root mean square amplitude values ​​under four different operating conditions in an embodiment of the present invention;

[0050] Figure 4 This is a schematic diagram of the linear growth rate under four different operating conditions in an embodiment of the present invention;

[0051] Figure 5 This is a schematic diagram of the nonlinear saturation coefficient under four different working conditions in an embodiment of the present invention;

[0052] Figure 6 This is a schematic diagram of noise intensity under four different operating conditions in an embodiment of the present invention;

[0053] Figure 7 This is a schematic diagram comparing the amplitude probability density function of the experiment under four different working conditions with the identified amplitude probability density function in an embodiment of the present invention; wherein, Figure 7 (a) to 7(d) correspond to four different working conditions;

[0054] Figure 8This is a schematic diagram of uncertainty estimation of an identification result in an embodiment of the present invention. Detailed Implementation

[0055] To make the objectives, technical solutions, and advantages of the present invention clearer, the embodiments disclosed in the present invention will be described in further detail below with reference to the accompanying drawings.

[0056] One of the core ideas of this invention is to disclose a method for identifying the linear growth rate of combustion instability. This method includes: based on the nonlinear stochastic dynamics Fokker-Planck equation, using a proposed model transformation analysis method, establishing the transformation relationship between the linear growth rate, nonlinear saturation coefficient, and noise intensity when the pulsating pressure changes unit-wise. Based on Monte Carlo simulation, the identified values ​​under specified operating conditions are used as true values, and the corresponding stochastic differential equations are numerically solved to generate more than 100 sets of pressure-time series with equal time lengths. The growth rate is identified, and the average and standard deviation are taken to quantify the uncertainty of the identification results under that operating condition.

[0057] like Figure 1 In this embodiment, the method for identifying the unstable linear growth rate of combustion includes:

[0058] Step 1: Establish a stochastic dynamic model of the combustion system.

[0059] In this embodiment, when the combustion chamber only has single-order acoustic mode oscillations, the coupling effect between modes can be ignored, and the pressure oscillations can be described by nonlinear vibration equations. The stochastic dynamic model of the combustion system is then expressed as follows:

[0060]

[0061] Where η represents pulsating pressure, α represents acoustic mode attenuation rate, ω0 represents acoustic mode angular frequency, ξ represents combustion noise, and f represents excitation source term generated by aerodynamic force and combustion process.

[0062] Step 2: Based on the stochastic dynamics model of the combustion system, an identification model is constructed.

[0063] In this embodiment, when the combustion noise is Gaussian white noise, we have: <ξξ τ >=Γδ(τ). Only the nonlinear combustion response function q is considered in f. c ,Right now For a system with low pressure oscillation amplitude near the stability boundary, q c It can be assumed to be in the form of a cubic polynomial of the pulsating pressure η: q c =βη-κ / 3η 3 Where β represents the linear response coefficient, κ represents the nonlinear saturation coefficient, and ξ... τLet ξ be the combustion noise time series after shifting ξ by time τ, δ(τ) represent the Dirac function, and Γ represent the noise intensity.

[0064] Define the linear growth rate v as v = (β - α) / 2, and equation (1) can be expressed as the following stochastic differential equation:

[0065]

[0066] Assuming the pressure oscillation is close to resonance, the pulsating pressure can be written in the following amplitude-frequency form:

[0067] η(t)=A(t)cos(ωt+σ(t))···(3)

[0068] Where A represents the amplitude of the pressure oscillation, ω represents the angular frequency of the pressure oscillation, and σ represents the phase of the pressure oscillation.

[0069] Substituting equation (3) into equation (2), and using the stochastic averaging method, equation (2) can be rewritten as a first-order stochastic differential equation for amplitude A. Since A and σ can be decoupled, only the equation satisfied by amplitude A is listed:

[0070]

[0071] Where ζ represents the intensity. The white noise; A(t) satisfies a one-dimensional Markov process with probability density function P A The satisfied FK (Fokker–Planck) equation is as follows:

[0072]

[0073] If the pressure time series is known, then the drift coefficient S(A) and diffusion coefficient of the FK equation are... It can be calculated directly using the following formula:

[0074]

[0075] Where P(A,t1+Δt|a,t1) represents the probability that the amplitude is a at time t1, given that the amplitude is a at time t1, and the amplitude is A at time t1+Δt, and Δt represents the time shift.

[0076] Furthermore, the pressure data in the experiment has units, such as Pa, kPa, etc. Using different units for the pressure data will affect the identification results. Let A(t) = kB(t), and substitute A(t) = kB(t) into equation (5), we get the following equation (7):

[0077]

[0078] Where k represents the scaling factor, i.e., the conversion relationship between units; B represents the amplitude after unit conversion; P B Let B be the probability density function representing the magnitude.

[0079] Comparing equations (5) and (7), it can be seen that after the unit transformation, the linear growth rate remains unchanged, while the nonlinear saturation coefficient is the original k. 2 The noise intensity is 1 / k of the original. 2 times.

[0080] Step 3: Based on the identification model, the linear growth rate, nonlinear saturation coefficient, and noise intensity of the combustion system are identified.

[0081] In this embodiment, the identification process for the linear growth rate, nonlinear saturation coefficient, and noise intensity of the combustion system is as follows:

[0082] Step a: Obtain the time series D of the pulsating pressure measured experimentally.

[0083] Step b: Based on equation (7), after performing a unit transformation on the pulsating pressure in the pulsating pressure time series D, substitute it into equation (6) to obtain discrete data points S whose drift coefficient and diffusion coefficient depend on the amplitude A.

[0084] Step c: Based on the expression for the drift coefficient in equation (4), the obtained discrete data points S are fitted using a nonlinear data fitting method to obtain the identification results of the combustion system. The identification results include: the identification value of the linear growth rate, the identification value of the nonlinear saturation coefficient, and the identification value of the noise intensity.

[0085] Preferably, it can also be determined whether the identification values ​​of the linear growth rate, the nonlinear saturation coefficient, and the noise intensity obtained in step c meet the set accuracy requirements. If not, the scaling factor is adjusted, and steps b to c are repeated until the identification values ​​of the linear growth rate, the nonlinear saturation coefficient, and the noise intensity meet the set accuracy requirements.

[0086] Furthermore, experimental data for each operating condition is typically obtained only once, and the actual growth rate of the system is unknown. Therefore, to quantify the uncertainty of the identification results, the uncertainty can be determined as follows: Substitute the identification results obtained based on the experimentally measured pulsating pressure time series D into equation (2), and solve equation (2) numerically to obtain at least N sets of pulsating pressure time series D obtained through numerical solution. N Based on the obtained N sets of pulsating pressure time series D NUsing the same processing method as steps b and c, at least N sets of solution results are obtained, and the average value P and standard deviation Q of the N sets of solution results are determined. The solution results include: the solution value of the linear growth rate, the solution value of the nonlinear saturation coefficient, and the solution value of the noise intensity. The identification result obtained in step c is compared with the average value P. If the deviation between the identification result obtained in step c and the average value P is less than a set threshold, then the standard deviation Q is used as the standard deviation of the identification result to determine the uncertainty of the identification result.

[0087] Based on the above embodiments, the method for identifying the linear growth rate of combustion instability described in this invention is used to identify the linear growth rate, nonlinear saturation coefficient, and noise intensity of a coaxial centrifugal nozzle under four operating conditions using pulsating pressure time series, and the uncertainty of the results is obtained, such as... Figures 2-8 As shown. Among them, Figure 2 It refers to the data source (input conditions). Figures 3-6 It is the recognition result; Figure 7 It involves comparing the recognition results with the experimental results (probability density function) to qualitatively assess the quality of the recognition results. Figure 8 It is an uncertainty estimate of the identified linear growth rate (quantitatively assessing whether the identification result is good or not), and a quantitative assessment of whether the identification result is good or not.

[0088] In summary, this invention discloses a method for identifying the linear growth rate of combustion instability, which solves the problem in traditional methods that cannot identify how the linear growth rate, nonlinear saturation coefficient, and noise intensity change from the data when the experimental data units are transformed. At the same time, it provides a method for quantifying and analyzing uncertainty based on Monte Carlo simulation to solve the problem of how to quantify the uncertainty of the identified linear growth rate, nonlinear saturation coefficient, and noise intensity when the experimental data length is finite.

[0089] Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make possible changes and modifications to the technical solutions of the present invention by utilizing the methods and techniques disclosed above without departing from the spirit and scope of the present invention. Therefore, any simple modifications, equivalent changes and alterations made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solutions of the present invention shall fall within the protection scope of the technical solutions of the present invention.

[0090] The contents not described in detail in this specification are common knowledge to those skilled in the art.

Claims

1. A method of identifying a combustion instability linear growth rate, characterized in that, The application relates to a method for identifying a combustion system, comprising the following steps: A stochastic dynamics model of the combustion system is established: （1） in, Indicates pulsating pressure. Indicates the acoustic modal attenuation rate. Indicates the angular frequency of the acoustic mode. Indicates combustion noise; This represents the excitation source terms generated by aerodynamic forces and the combustion process, considering only the nonlinear combustion response function. ,Right now ; Represented as pulsating pressure cubic polynomial form: , Represents the linear response coefficient. This represents the nonlinear saturation coefficient; when the combustion noise is Gaussian white noise, we have: , Indicates will Translation time The subsequent combustion noise time series, Represents the Dirac function, Indicates noise intensity; According to the stochastic dynamics model of the combustion system, an identification model is constructed; comprising: The formula (1) is expressed as the following stochastic differential equation: （2） wherein, represents the linear growth rate; The amplitude-frequency expression form of the pulsating pressure is determined: （3） wherein denotes the amplitude of the pressure oscillation, denotes the angular frequency of the pressure oscillation, denotes the phase of the pressure oscillation; Substituting equation (3) into equation (2), and using the stochastic averaging method, equation (2) can be written as a first-order stochastic differential equation: ​ （4） wherein represents white noise with intensity of 1. Based on equation (4), the probability density function of the amplitude is determined to satisfy the FK equation: （5） According to equation (5), the drift coefficient of the FK equation is determined and the diffusion coefficient is calculated from the following equation: （6） wherein represents the amplitude at time under the condition that the amplitude at time the probability that represents the time shift make and will Substituting into equation (5), we obtain the following equation (7): （7） wherein, denotes a scaling factor, i.e. a conversion relationship between units; denotes the amplitude after unit conversion; denotes the amplitude of the probability density function; According to the identification model, the linear growth rate, the nonlinear saturation coefficient and the noise intensity of the combustion system are identified; comprising: Step a, obtaining the pulsating pressure time sequence D measured through experiments; Step b, based on equation (7), the pulsation pressure in the pulsation pressure time series D is unit transformed, substituted into equation (6), and the discrete data points S of the drift coefficient and the diffusion coefficient dependent on the amplitude are obtained ; Step c, according to the expression of the drift coefficient in formula (4), the obtained discrete data points S are fitted through a nonlinear data fitting method to obtain the identification result of the combustion system; wherein the identification result comprises: the identified value of the linear growth rate, the identified value of the nonlinear saturation coefficient and the identified value of the noise intensity; Step d, judging whether the identified value of the linear growth rate, the identified value of the nonlinear saturation coefficient and the identified value of the noise intensity obtained through step c meet the set accuracy requirement, if not, adjusting the scale reduction coefficient, and repeating steps b-c until the identified value of the linear growth rate, the identified value of the nonlinear saturation coefficient and the identified value of the noise intensity meet the set accuracy requirement. Step e, the identification result obtained based on the experimentally measured pulsating pressure time series D is substituted into formula (2), and formula (2) is solved in a numerical manner to obtain at least N groups of pulsating pressure time series D obtained by numerical solution N Step f, based on the obtained N groups of pulsating pressure time series D N , the same processing mode as steps b and c is used to obtain at least N groups of calculation results, and the average P and the standard deviation Q of the N groups of calculation results are determined; wherein the calculation results include: the calculation value of the linear growth rate, the calculation value of the nonlinear saturation coefficient and the calculation value of the noise intensity; the identification result obtained by step c is compared with the average P; wherein the deviation between the identification result obtained by step c and the average P is less than the set threshold value, and the standard deviation Q is taken as the standard deviation of the identification result, and the uncertainty of the identification result is determined.

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