A Quantification Method for the Uncertainty of the Linear Growth Rate of Combustion Instability

Abstract

The present invention provides a method for quantifying the uncertainty of the linear growth rate of combustion instability, including: Step 1, constructing a kinetic process model of the combustion system; Step 2, identifying the linear growth rate; Step 3, identifying the noise intensity; Step 4, quantifying the uncertainty. Based on the linear stochastic dynamics Fokker-Planck equation, the present invention extracts the noise intensity from the limit cycle pressure oscillation data. The Monte Carlo simulation method is used to numerically solve the nonlinear stochastic differential equation to generate more than 100 sets of time series of the growth process from small perturbations to finite amplitudes. Different time intervals are selected, the mean values and standard deviations of the identification results are compared, and the interval with smaller deviation and variance is selected as the linear growth stage. The growth rates of each group in the linear growth stage are identified and the mean value and standard deviation are taken to quantify the uncertainty under this working condition.

Description

Technical Field

[0001] The present invention belongs to the technical field of aerospace, relates to a combustion system, and particularly relates to a method for quantifying the uncertainty of the linear growth rate of combustion instability. Background Art

[0002] During the development of combustion systems such as aerospace engines and gas turbines, combustion instability problems often occur, which are usually manifested as high-amplitude pressure oscillations near the acoustic modal frequency of the combustion chamber. Combustion instability will exacerbate the heat transfer on the combustion chamber wall and structural vibration, and then lead to serious consequences such as system performance degradation and component failure. Combustion instability involves complex coupling processes among combustion, flow, and sound fields, and its generation mechanism, active and passive control, margin assessment, and prediction methods have always been concerned by the academic and industrial communities. Once combustion instability occurs in a combustion system, there are usually two solutions: reducing the excitation energy and increasing the acoustic damping of the combustion chamber. Due to the complexity of combustion phenomena, it is difficult to reduce the excitation energy. In comparison, the design of increasing the acoustic damping of the combustion chamber is easier.

[0003] The linear growth rate characterizes the change rate of the oscillation energy in the combustion chamber under small perturbations. For a combustion chamber with combustion instability, the attenuation rate of the newly designed damping device should be greater than the growth rate of the combustion chamber and retain a certain margin, which requires the growth rate of the unstable acoustic mode of the combustion chamber as the design input parameter. Scholars have proposed various methods to identify the growth rate. They are mainly divided into external excitation methods (input-output identification) and non-external excitation methods (only output identification). In the external excitation method, the nonlinear combustion response of the system can be identified by using the noise coherence resonance characteristics before bifurcation, and the position and type of the system bifurcation point, the growth rate, the amplitude of the limit cycle oscillation, etc. can be extrapolated. It is difficult to add a specific form of external excitation in actual engine tests, so the external excitation method is not convenient for engineering applications.

[0004] In the non-external excitation method, considering the process of spontaneous combustion instability increasing from small perturbations to a finite amplitude (limit cycle), the oscillation amplitude initially grows exponentially in the growth stage (satisfying the linear assumption). Therefore, the pressure oscillation time series in this stage can be enveloped, and an exponential function can be used to fit it, or the logarithm of the envelope can be taken and a linear function can be used to fit it to identify the growth rate.

[0005] The existing non-external excitation identification methods have the following deficiencies: When there is noise in the experimental data, how to quantify the uncertainty of the identified linear growth rate of combustion instability, and how to determine the linear growth stage in the process of the oscillation amplitude increasing from small perturbations to a finite amplitude. These problems have not been well solved in the existing technology. Summary of the Invention

[0006] Aiming at the deficiencies of the existing technology, the purpose of the present invention is to provide a method for quantifying the uncertainty of the linear growth rate of combustion instability, so as to solve the problem of uncertainty quantification in the identification of the linear growth rate of combustion instability when there is noise in the experimental data, and the problem of determining the linear growth stage during the process of the oscillation amplitude increasing from small perturbations to a finite amplitude.

[0007] To solve the above technical problems, the present invention is implemented by adopting the following technical solutions:

[0008] A method for quantifying the uncertainty of the linear growth rate of combustion instability, the method comprising the following steps:

[0009] Step 1, constructing a dynamic process model of the combustion system:

[0010] The pressure oscillation in the combustion chamber of the combustion system is a single-order acoustic mode oscillation, and the pressure oscillation in the combustion chamber is described by a non-linear vibration equation:

[0011]

[0012] In the formula:

[0013] represents the second derivative of η;

[0014] represents the first derivative of η;

[0015] η represents the pulsating pressure, with the unit of Pa;

[0016] ω0 represents the acoustic mode angular frequency, with the unit of rad / s;

[0017] ν represents the linear growth rate, with the unit of rad / s;

[0018] κ represents the non-linear saturation coefficient, with the unit of 1;

[0019] ξ represents the combustion noise, with the unit of Pa;

[0020] Assuming that the combustion noise is Gaussian white noise, then <ξξ τ > = Γδ(τ);

[0021] In the formula:

[0022] Γ represents the noise intensity, with the unit of Pa 2 / Hz;

[0023] δ represents the Dirac function, with the unit of 1;

[0024] τ represents the time shift, with the unit of s;

[0025] Assuming that the pressure oscillation is a harmonic oscillation, the amplitude-frequency form of the pulsating pressure is:

[0026] η(t) = A(t)cos(ωt + σ(t)), Equation Ⅱ;

[0027] Where:

[0028] t represents time, with the unit of s;

[0029] A(t) represents the amplitude of the pressure oscillation, with the unit of Pa;

[0030] σ(t) represents the phase of the pressure oscillation, with the unit of rad;

[0031] ω represents the angular frequency, with the unit of rad / s;

[0032] Substituting Equation Ⅱ into Equation Ⅰ and using the stochastic averaging method, Equation Ⅰ can be written as a first-order stochastic differential equation for the amplitude A and the phase σ; since the amplitude A and the phase σ can be decoupled, only the equation satisfied by the amplitude A is listed

[0033]

[0034] Where:

[0035] represents the first derivative of A;

[0036] ζ represents white noise with an intensity of / Hz, with the unit of Pa 2 / Hz;

[0037] The amplitude A is decomposed into the sum of the average value and the fluctuation value A', that is:

[0038]

[0039] The fluctuation value A' is less than the average value Perform a linearization process on Equation Ⅲ;

[0040] The specific process of the linearization process is as follows: Substitute Equation Ⅳ into Equation Ⅲ, retain the first-order terms, and omit the higher-order terms to obtain the linearized first-order stochastic differential equation as:

[0041]

[0042] Where:

[0043] represents the first derivative of A';

[0044] Let the intermediate quantity The power spectrum of the fluctuation value A' is:

[0045]

[0046] Finally, a kinetic process model of the combustion system is obtained;

[0047] Step two, linear growth rate identification;

[0048] Step three, noise intensity identification;

[0049] Step four, uncertainty quantification.

[0050] The present invention also has the following technical features:

[0051] In step two, for the pressure time series with small perturbations to finite amplitude growth, the envelope curve is obtained through Hilbert transform; initially, the curves in the range of 15% - 50% of the limit cycle oscillation amplitude of the oscillation amplitude are selected to identify the growth rate, and the exponential function e νt is used to fit the envelope curve to obtain the linear growth rate ν.

[0052] In step three, the envelope of the limit cycle oscillation is obtained through Hilbert transform, and the modulus is taken to obtain the amplitude A; the average value of the amplitude A is taken to obtain the average amplitude value The fluctuation value A' is obtained using Equation IV in step one; the power spectrum of the time series of the fluctuation value A' is calculated, and non-linear fitting is performed using Equation VI in step one to obtain the noise intensity Γ.

[0053] In step four, using the equation the non-linear saturation coefficient κ is obtained; thus, the linear growth rate ν, the noise intensity Γ, and the non-linear saturation coefficient κ are obtained. Substitute the values of the three into Equation I in step one, and use the stochastic Runge-Kutta method to solve Equation I in step one to generate more than 100 time series of the process from small perturbations to finite amplitude growth; use the method of linear growth rate identification in step two to identify the linear growth rate of each group of data, and calculate the average value and standard deviation; adjust the selected data interval, compare the deviation and variance of the calculation results, and the interval with better results is used as the linear growth stage, and the identification result of this interval is used as the uncertainty under this working condition.

[0054] Compared with the prior art, the present invention has the following technical effects:

[0055] Based on the linear stochastic dynamics Fokker-Planck equation, the noise intensity is extracted from the limit cycle pressure oscillation data. The Monte Carlo simulation method is used to numerically solve the non-linear stochastic differential equation to generate more than 100 time series of the process from small perturbations to finite amplitude growth. Select different time intervals, compare the average value and standard deviation of the identification results, and select the interval with smaller deviation and variance as the linear growth stage. Identify the growth rate of each group in the linear growth stage and take the average value and standard deviation to quantify the uncertainty under this working condition. Description of the Drawings

[0056] Figures 1(a) and 1(b) are for the identification of the noise-free pressure time series and the growth rate.

[0057] Figures 2(a) and 2(b) are for the identification of the noisy pressure time series and the growth rate.

[0058] Figure 3 This is the influence of noise on the identification of the growth rate.

[0059] The following further elaborates on the specific content of the present invention in conjunction with embodiments. Specific implementation manners

[0060] It should be noted that all methods or units in the present invention, without special instructions, adopt the commonly known and used methods or units in the art. In each formula of the present invention, only the numerical part of the physical quantity is used for calculation, and the units of the physical quantities all adopt the commonly used standard units in the art.

[0061] It should be noted that in the present invention, the combustion system is the commonly known combustion systems such as aerospace engines and gas turbines.

[0062] The following gives specific embodiments of the present invention. It should be noted that the present invention is not limited to the following specific embodiments, and all equivalent transformations made on the basis of the technical solutions of the present application fall within the protection scope of the present invention.

[0063] Embodiment:

[0064] This embodiment provides a method for quantifying the uncertainty of the linear growth rate of combustion instability, and the method includes the following steps:

[0065] Step 1, constructing a dynamic process model of the combustion system:

[0066] When there is only a single-order acoustic mode oscillation in the combustion chamber of the combustion system, the coupling effect between modes can be ignored.

[0067] The pressure oscillation in the combustion chamber of the combustion system is a single-order acoustic mode oscillation, and the pressure oscillation in the combustion chamber is described by a nonlinear vibration equation:

[0068]

[0069] In the formula:

[0070] represents the second derivative of η;

[0071] represents the first derivative of η;

[0072] η represents the pulsating pressure, with the unit of Pa;

[0073] ω0 represents the acoustic mode angular frequency, with the unit of rad / s;

[0074] ν represents the linear growth rate, with the unit of rad / s;

[0075] κ represents the non-linear saturation coefficient, with the unit of 1;

[0076] ξ represents the combustion noise, with the unit of Pa;

[0077] If the combustion noise is set as Gaussian white noise, then <ξξ τ > = Γδ(τ);

[0078] In the formula:

[0079] Γ represents the noise intensity, with the unit of Pa 2 / Hz;

[0080] δ represents the Dirac function, with the unit of 1;

[0081] τ represents the time shift, with the unit of s.

[0082] If the pressure oscillation is set as a harmonic oscillation, then the amplitude-frequency form of the pulsating pressure is:

[0083] η(t) = A(t)cos(ωt + σ(t)) Equation II.

[0084] In the formula:

[0085] t represents time, with the unit of s;

[0086] A(t) represents the amplitude of the pressure oscillation, with the unit of Pa;

[0087] σ(t) represents the phase of the pressure oscillation, with the unit of rad;

[0088] ω represents the angular frequency, with the unit of rad / s.

[0089] Substitute Equation II into Equation I and use the stochastic averaging method. Equation I can be written as a first-order stochastic differential equation for the amplitude A and the phase σ; since the amplitude A and the phase σ can be decoupled, only the equation satisfied by the amplitude A is listed

[0090]

[0091] In the formula:

[0092] represents the first derivative of A;

[0093] ζ represents white noise with an intensity of in Pa 2 / Hz;

[0094] The amplitude A is decomposed into the average value and the fluctuation value A′, that is:

[0095]

[0096] The fluctuation value A′ is less than the average value Perform linearization on Equation Ⅲ.

[0097] In most cases, the fluctuation value A′ is less than the average value The present invention only considers this case. For the case where the fluctuation value A′ is greater than or equal to the average value is not within the scope of consideration of the present invention.

[0098] The specific process of the linearization is as follows: Substitute Equation Ⅳ into Equation Ⅲ, retain the first-order term, and omit the high-order terms to obtain the linearized first-order stochastic differential equation as follows:

[0099]

[0100] In the formula:

[0101] represents the first derivative of A′.

[0102] Let the intermediate quantity The power spectrum of the fluctuation value A′ is:

[0103]

[0104] Finally, obtain the kinetic process model of the combustion system.

[0105] Step 2, linear growth rate identification:

[0106] For the pressure time series with small perturbations growing to a finite amplitude, obtain the envelope curve through the Hilbert transform; initially select the curve with the oscillation amplitude in the range of 15% - 50% of the limit cycle oscillation amplitude to identify the growth rate, and use the exponential function e νt to fit the envelope curve to obtain the linear growth rate ν.

[0107] Step 3, noise intensity identification:

[0108] Use the Hilbert transform to obtain the envelope of the limit cycle oscillation, take the modulus to obtain the amplitude A; take the average of the amplitude A to obtain the average amplitude value Use Equation Ⅳ in Step 1 to obtain the fluctuation value A′; calculate the power spectrum of the time series of the fluctuation value A′, and perform nonlinear fitting using Equation Ⅵ in Step 1 to obtain the noise intensity Γ.

[0109] Step 4, uncertainty quantification:

[0110] Use Equation The non - linear saturation coefficient κ is obtained. Thus, the linear growth rate ν, the noise intensity Γ, and the non - linear saturation coefficient κ are obtained. Substitute the values of the three into Equation Ⅰ in Step 1, and use the stochastic Runge - Kutta method to solve Equation Ⅰ in Step 1 to generate more than 100 time series of the growth process from small perturbations to finite amplitudes. Use the method for identifying the linear growth rate in Step 2 to identify the linear growth rate of each group of data, and calculate the mean value and standard deviation. Adjust the selected data interval, compare the deviation and variance of the calculation results, and use the interval with better results as the linear growth stage, and use the identification result of this interval as the uncertainty under this working condition.

[0111] Application example:

[0112] This application example presents a method for quantifying the uncertainty of the linear growth rate of combustion instability based on the above - mentioned embodiments. The linear growth rate is identified using the time series in the linear growth stage of the pulsating pressure, and the uncertainty of the result is obtained.

[0113] In the case of no noise (noise intensity is 0), a time series is generated through Equation Ⅰ, and the oscillation amplitude gradually increases from small perturbations and finally reaches saturation (the amplitude of the limit cycle oscillation), as shown in Figure 1(a) specifically. When the oscillation amplitude is less than 30% of the amplitude of the limit cycle oscillation, the logarithmic curve of the amplitude has a good linearity, and the fitting result is relatively accurate, as shown in Figure 1(b) specifically.

[0114] When the noise intensity is 5 Use Equation Ⅰ to generate 100 groups of pressure time series respectively, as shown in Figure 2(a) specifically (5 groups are listed in Figure 2(a)). Use the method in the embodiment to identify the linear growth rate of each group of data, and it is found that when selecting the curve in the interval where the oscillation amplitude is 15% - 50% of the amplitude of the limit cycle oscillation, as shown in Figure 2(b) specifically, the deviation between the mean value of the identified growth rate and the set value is relatively small.

[0115] From Figure 3 it can be seen that under three noise intensities ( 10, 15), as the noise intensity increases, the deviation between the mean value of the identified growth rate and the set value becomes larger, and the standard deviation also becomes larger. Therefore, when identifying the growth rate from the linear growth stage of the pulsating pressure, the deviation and standard deviation caused by combustion noise should be considered to improve the reliability of the analysis.

Claims

1. A method for quantifying the uncertainty of the linear growth rate of combustion instability, characterized in that The method includes the following steps: Step 1, constructing a kinetic process model of the combustion system: The pressure oscillation in the combustion chamber of the combustion system is a single-order acoustic mode oscillation, and the pressure oscillation in the combustion chamber is described by a nonlinear vibration equation: In the formula: Denotes the second derivative of η; Denotes the first derivative of η; η represents the pulsating pressure, with the unit of Pa; ω0 represents the acoustic mode angular frequency, with the unit of rad / s; ν represents the linear growth rate, with the unit of rad / s; κ represents the nonlinear saturation coefficient, with the unit of 1; ξ represents the combustion noise, with the unit of Pa; Assume that the combustion noise is Gaussian white noise, then <ξξ τ > = Γδ(τ); In the formula: Γ represents the noise intensity, with the unit of Pa 2 / Hz; δ represents the Dirac function, with the unit of 1; τ represents the time translation, with the unit of s; Assume that the pressure oscillation is a resonance, then the amplitude-frequency form of the pulsating pressure is: η(t) = A(t)cos(ωt + σ(t)) Equation Ⅱ; In the formula: t represents time, with the unit of s; A(t) represents the amplitude of the pressure oscillation, with the unit of Pa; σ(t) represents the phase of the pressure oscillation, with the unit of rad; ω represents the angular frequency, with the unit of rad / s; Substitute Equation Ⅱ into Equation Ⅰ, and using the stochastic averaging method, Equation Ⅰ can be written as a first-order stochastic differential equation of the amplitude A and the phase σ; since the amplitude A and the phase σ can be decoupled, only the equation satisfied by the amplitude A is listed In the formula: Denotes the first derivative of A; ζ represents white noise with an intensity of , in Pa 2 / Hz; The amplitude A is decomposed into the sum of the average value and the fluctuation value A′, that is: The fluctuation value A' is less than the average value Perform linearization on Equation Ⅲ; The specific process of the linearization treatment is: substitute Equation Ⅳ into Equation Ⅲ, retain the first-order terms, and omit the higher-order terms, and the linearized first-order stochastic differential equation is obtained as: In the formula: Denote the first derivative of A′; Let the intermediate quantity The power spectrum of the fluctuation value A' is as follows: Finally, a kinetic process model of the combustion system is obtained; Step 2, identifying the linear growth rate; Step 3, identifying the noise intensity; Step 4, quantifying the uncertainty.

2. The quantization method for the uncertainty of the combustion instability linear growth rate according to claim 1, characterized in that, In Step 2, for the pressure time series with small perturbations growing to a finite amplitude, the envelope curve is obtained through Hilbert transform; initially, curves with an oscillation amplitude in the range of 15% - 50% of the limit cycle oscillation amplitude are selected to identify the growth rate, and the exponential function e νt is used to fit the envelope curve to obtain the linear growth rate ν.

3. The quantization method for the uncertainty of the linear growth rate of combustion instability according to claim 2, characterized in that In Step 3, the envelope of the limit cycle oscillation is obtained by using the Hilbert transform, and the modulus is taken to obtain the amplitude A; the average value of the amplitude A is taken to obtain the average amplitude value. The fluctuation value A′ is obtained by using Equation Ⅳ in Step 1; the power spectrum of the time series of the fluctuation value A′ is calculated, and nonlinear fitting is performed by using Equation Ⅵ in Step 1 to obtain the noise intensity Γ.

4. The quantization method for the uncertainty of the combustion instability linear growth rate according to claim 3, characterized in that In Step 4, use Equation to obtain the non-linear saturation coefficient κ. Thus, the linear growth rate ν, the noise intensity Γ, and the non-linear saturation coefficient κ are obtained. Substitute the values of the three into Equation I in Step 1, and use the stochastic Runge-Kutta method to solve Equation I in Step 1 to generate more than 100 time series of the growth process from small perturbations to finite amplitudes; use the method for identifying the linear growth rate in Step 2 to identify the linear growth rate of each group of data, and calculate the mean value and the standard deviation; adjust the selected data interval, compare the deviation and variance of the calculation results, and use the interval with better results as the linear growth stage, and use the identification result of this interval as the uncertainty.

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