A Robust Calculation Method for Aerothermal Performance Parameters of High-Efficiency Turbine Blades

Abstract

An efficient method for calculating the robustness of aerothermal performance parameters of turbine blades mathematically models the response model of aerothermal performance parameters of gas turbine blades to generate a universal Kriging equation to be solved, uses the Latin hypercube method to obtain the coordinates of sampling points, calculates the aerothermal performance parameters of turbine blades under the working conditions represented by each sampling point, combines hyperparameters to solve the explicit equation of the universal Kriging equation, then generates a batch of new sample points and applies normal distribution perturbations to establish a polynomial chaos expansion to be solved; solves the statistical variance of the aerothermal performance parameters of each new sample point, outputs the optimal hyperparameters according to the convergence conditions, solves the explicit equation of the optimal universal Kriging equation, then obtains the uncertainty nodes, and solves the robustness parameters of the aerothermal performance parameters of gas turbine blades. The present invention is based on the universal Kriging method and the polynomial chaos method to quantify the uncertainty of the aerothermal performance parameters of turbine blades at a relatively low computational cost.

Description

Technical Field

[0001] The present invention belongs to the technical field of gas turbines, and relates to the uncertainty analysis and performance optimization of its turbines, and particularly relates to a method for calculating the robustness of aerothermal performance parameters of high-efficiency turbine blades. Background Art

[0002] In order to improve the output power and work efficiency, the inlet temperature and inlet pressure of modern gas turbines are continuously increased. However, the working conditions of high temperature and high pressure will lead to a significant reduction in the service life of gas turbine components. Among all the components of a gas turbine, the working conditions of turbine blades are the most severe, and thus the service life of turbine blades in actual operation is much lower than the designed life. Therefore, turbine researchers have developed uncertainty quantification methods to quantify the mean and variance of the actual performance parameters of turbine blades, that is, the robustness of the aerothermal performance parameters of turbine blades. However, as a new discipline, due to the lack of efficient mathematical tools, the calculation of uncertainty quantification of gas turbine turbine blades requires a large amount of computing resources and time costs. The mainstream Monte Carlo simulation requires hundreds of thousands of samples to calculate a three-dimensional uncertainty problem, which is almost impossible to achieve in engineering practice. And the relatively advanced polynomial chaos method also has very low performance in the uncertainty quantification of high-dimensional problems. Summary of the Invention

[0003] In order to overcome the above-mentioned disadvantages of the prior art, the purpose of the present invention is to provide a method for calculating the robustness of aerothermal performance parameters of high-efficiency turbine blades, so as to perform uncertainty quantification work on the aerothermal performance parameters of turbine blades at a lower calculation cost. It is mainly based on the universal Kriging method and the polynomial chaos method, which helps turbine designers to efficiently quantify the robustness of the aerothermal performance parameters of turbine blades to more deeply explore the uncertainty phenomenon of turbine blades in practical operation, thereby guiding the design work of strongly robust turbine blades.

[0004] In order to achieve the above purpose, the technical solution adopted by the present invention is:

[0005] A method for calculating the robustness of aerothermal performance parameters of high-efficiency turbine blades, characterized by comprising the following steps:

[0006] Step 1, based on the universal Kriging surrogate model theory, perform mathematical modeling on the aerothermal performance parameter response model of the gas turbine blade to generate a universal Kriging equation to be solved, where the aerothermal performance parameter is the output quantity whose uncertainty needs to be evaluated;

[0007] Step 2: Obtain the coordinates of the sampling points using the Latin hypercube method according to the research parameter range and the number of sampling points of the gas turbine blade; the research parameters are the preset research objects, which are set as the geometric or aerodynamic parameters with obvious uncertainties during the operation of the gas turbine; the sampling points refer to the operating conditions required for calculating the universal Kriging equation. The number of coordinate points of the sampling points is equal to the number of research parameters, and each sampling point represents a set of operating conditions that need to be calculated.

[0008] Step 3: Calculate the gas-thermal performance parameters of the turbine blade under the operating conditions represented by each sampling point using the open-source program library OPERNFORM according to the coordinates of the sampling points.

[0009] Step 4: Randomly generate a set of hyperparameters using the Random function.

[0010] Step 5: Solve the explicit equation of the universal Kriging equation by combining the hyperparameters according to the coordinates of the sampling points and the gas-thermal performance parameters of the turbine blade under the operating conditions represented by each sampling point.

[0011] Step 6: Generate a batch of new sample points using the Latin hypercube method within the sampling space determined by the research parameters of the gas turbine blade; the new sample points evaluate the operating conditions required for calculating the uncertainty of the established universal Kriging equation. The number of coordinate points of the new sample points is equal to the number of research parameters, and each new sample point represents a set of operating conditions that need to be calculated.

[0012] Step 7: Apply a normal distribution perturbation to each new sample point.

[0013] Step 8: Establish the polynomial chaos expansion to be solved for each new sample point.

[0014] Step 9: Solve the explicit formula of the polynomial chaos expansion of each new sample point based on the polynomial chaos theory according to the explicit equation of the universal Kriging equation.

[0015] Step 10: Solve the statistical variance of the gas-thermal performance parameters of each new sample point according to the explicit formula of the polynomial chaos expansion of each new sample point.

[0016] Step 11: Solve the sum of the statistical variances of the gas-thermal performance parameters of each new sample point as the evaluation value of the hyperparameters generated in Step 4.

[0017] Step 12: Determine whether the convergence condition is satisfied. If not, execute Steps 4 - 11. If satisfied, output the optimal hyperparameters.

[0018] Step 13: Solve the explicit equation of the optimal universal Kriging equation by combining the optimal hyperparameters and the gas-thermal performance parameters of the turbine blade under the operating conditions represented by each sampling point.

[0019] Step 14: Obtain the coordinates of the uncertainty nodes using the Latin Hypercube method according to the research parameter range and the number of uncertainty nodes of the gas turbine blade; the uncertainty nodes refer to the working conditions required for solving the polynomial chaos expansion, the number of coordinates of the uncertainty nodes is equal to the number of research parameters, and each uncertainty node represents a set of working conditions to be calculated.

[0020] Step 15: Receive the coordinates of the uncertainty nodes and the explicit equation of the optimal universal Kriging equation, and solve the explicit formula of the polynomial chaos expansion of the gas turbine blade gas-thermal performance parameters based on the polynomial chaos theory.

[0021] Step 16: Receive the explicit formula of the polynomial chaos expansion of the gas turbine blade gas-thermal performance parameters, and solve the robustness parameters of the gas turbine blade gas-thermal performance parameters, where the robustness parameters include the statistical mean and the statistical variance.

[0022] In one embodiment, the gas turbine blade gas-thermal performance parameters are leakage flow rate, total pressure loss coefficient, or heat transfer amount; the research parameters include the inlet flow angle, tip clearance, mainstream inlet total pressure, and mainstream inlet total temperature.

[0023] In one embodiment, in Step 1, the basic equation of the universal Kriging equation is as follows:

[0024] M(θ) = f T (θ)β + z(θ)

[0025] In the formula, θ represents the research parameter, which is a random variable, f T (θ) is the regression function, z(θ) represents the approximation of the local deviation, β represents the coefficient of the regression function, and f T (θ) and β are characterized by polynomials as follows:

[0026] f T (θ)β = 8x 3 - 12x

[0027] z(θ) is expressed as follows:

[0028] E[(z(θ1)z(θ2))] = σ 2 R(γ, θ1, θ2)

[0029] In the formula, γ represents the set of hyperparameters, which contains n hyperparameters, n represents the order of the research problem, that is, the number of research parameters, θ1 and θ2 represent any two sample points in the sample space, σ represents the preset standard deviation of the research parameter, and R(γ, θ1, θ2) represents the spatial correlation function of θ1 and θ2, and its calculation method is as follows:

[0030]

[0031] where γ j , θ 1j and θ 2j represent γ, θ1, and θ2 in the j-th dimension; the correlation between any sample point θ x and the sample point θ s is expressed as follows:

[0032] r(θ) = R(γ, θ x , θ s ) T

[0033] In one embodiment, for step 2, if the number of sampling points is N, the steps to obtain the sampling point coordinates using the Latin hypercube method are as follows:

[0034] Step (1): Divide the interval (0, 1) into N segments evenly to obtain N small intervals;

[0035] Step (2): Use the random function Random in Python to randomly generate a value in each small interval and store them in the array P in sequence. Then the array P contains N elements;

[0036] Step (3): Map the elements in the array P to the coordinate values of the corresponding sampling points using the inverse function of the standard normal distribution in sequence.

[0037] In one embodiment, for step 4, use the random function Random in Python to give a set of random hyperparameters to obtain a hyperparameter set containing n elements, where n is the number of research parameters.

[0038] In one embodiment, for step 5, store the gas-thermal performance parameters of the turbine blade under the working conditions represented by each sampling point into the array Y. Then the universal Kriging model is calculated by the following formula:

[0039] M(θ) = f T (θ)β + r T (θ)R(γ, θ x , θ s ) -1 (Y - β)

[0040] The number of elements contained in the array Y is equal to the number of sampling points, and each element represents the gas-thermal performance parameter under the working conditions represented by a sampling point.

[0041] In one embodiment, for step 7, the application method of the normal distribution is as follows:

[0042] Generate a mean value of ι i , and a standard deviation of N sam ι iFrom the normal distribution, obtain the normal distribution perturbation of each coordinate of each new sample point, N sam is a preset parameter. The larger this parameter is, the smoother the fitted Kriging model is, but the slower the convergence speed of the algorithm is.

[0043] In one embodiment, in step 8, the gas thermal performance parameter, i.e., the system output y, is expanded by polynomial chaos as the following formula:

[0044]

[0045] In the formula, a j is the coefficient of the j-th orthogonal basis, and Ψ j (ξ) is the j-th orthogonal basis in the discrete case;

[0046] In step 9, first input the coordinates of each new sample point into the explicit equation of the universal Kriging equation to solve the system output y of each new sample point; then, use the Galerkin projection method to solve the coefficient of the j-th orthogonal basis of the polynomial chaos expansion of each new sample point:

[0047]

[0048] In the formula, is the polynomial inner product, J(ξ) is the joint probability density function of the uncertainty input variables, and the obtained coefficients are combined with the orthogonal basis to obtain the explicit expression of the polynomial chaos expansion of each new sample point;

[0049] In steps 10 and 16, solve the statistical variance σ through the following formula 2 :

[0050]

[0051] In step 16, solve the statistical mean through the following formula

[0052] μ = a0

[0053] where a0 is the coefficient of the first term of the polynomial chaos expansion.

[0054] In one embodiment, in step 12, when the difference between the current evaluation value and the previous evaluation value is less than or equal to κ, stop the iteration and output the optimal hyperparameter; when the difference between the current evaluation value and the previous evaluation value is greater than κ, then run steps 4-11; where κ is a preset parameter, and the smaller κ is, the more accurate the Kriging model modeling is, but the slower the calculation convergence is.

[0055] In one embodiment, use the robustness parameter to quantify the uncertainty of the uncertainty input quantity on the system output of the gas turbine system.

[0056] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0057] (1) Based on the universal Kriging equation and polynomial chaos expansion, the present invention calculates the robustness of the gas turbine blade aerothermal performance parameters, which can significantly reduce the consumption of computing resources compared with the mainstream Monte Carlo method, and greatly improves the computing performance for high-dimensional uncertainty quantification problems compared with using the polynomial chaos expansion alone.

[0058] (2) Traditional Kriging-based algorithms use the maximum likelihood estimation method to solve hyperparameters. However, it has been proven that using the maximum likelihood estimation method to solve hyperparameters will overly focus on the vibration amplitude of the established Kriging model while ignoring the control of the complexity of the Kriging model. The current mainstream research problem lies in how to measure the complexity of the Kriging model. The present invention innovatively proposes that the Kriging model can be regarded as a system. From the perspective of cybernetics, the more complex a system is, the more obvious its initial value sensitivity is. Finally, the present invention mathematically quantifies the initial value sensitivity through steps 6-11. The complexity of the Kriging model generated by the present invention is greatly reduced.

[0059] (3) The defects of the Kriging model solved by the present invention can be applied not only to the universal Kriging equation but also to all Kriging models, such as the ordinary Kriging model and the multi-fidelity Kriging model, etc.

[0060] (4) The defects of the Kriging model solved by the present invention can be applied not only to uncertainty quantification problems. Considering that the intrinsic defects of the Kriging model exist in any problem, the solution method for the defects of the Kriging model proposed by the present invention can also be applied to related problems of using the Kriging model for optimization design. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] Figure 1 Schematic diagram of the blade geometric parameters of the GE-E3 gas turbine used in the embodiment.

[0062] Figure 2 Schematic diagram of the method of the present invention.

[0063] Figure 3 Axial distribution of the mean and variance of the heat transfer at the blade tip of the GE-E3 gas turbine calculated in the embodiment. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0064] The embodiments of the present invention will be described in detail below with reference to the drawings and embodiments.

[0065] The present invention relates to a method for calculating the robustness of aerothermal performance parameters of an efficient turbine blade. Among them, the aerothermal performance parameters are output quantities that need to evaluate uncertainty, such as parameters like leakage flow rate, total pressure loss coefficient, or heat transfer amount. The corresponding research parameters, that is, the preset research objects, are generally set as geometric or aerodynamic parameters with obvious uncertainties during the operation of a gas turbine.

[0066] In an embodiment of the present invention, the blade profile of GE-E3 which has been widely studied (Kwak J S, Han J C. Heat-transfer coefficients of a turbine blade-tip and near-tip regions[J]. Journalof thermophysics and heat transfer, 2003, 17(3):297-303.) is used as the research object. Its geometric parameters are shown in Table 1. The geometric model of GE-E3 and the meanings of each geometric parameter in Table 1 are shown in Figure 1 .

[0067] Table 1 Geometric parameters of the blade profile of GE_E3 gas turbine

[0068] Geometric parameter name Value (mm) Tip clearance (S) 0.4 Blade height (H) 122 Shoulder wall thickness (G) 2.29 Groove depth (D) 5.08

[0069] According to the geometric parameters in Table 1, the research parameters in this embodiment include the inlet flow angle, tip clearance, mainstream inlet total pressure, and mainstream inlet total temperature.

[0070] Refer to Figure 2 , the method for calculating the robustness of aerothermal performance parameters of an efficient turbine blade in an embodiment of the present invention is implemented based on the following functional modules and corresponding sequence:

[0071] 1. Universal Kriging equation modeling module: Mathematically model the response model of aerothermal performance parameters of a gas turbine blade based on the theory of the universal Kriging surrogate model to generate the universal Kriging equation to be solved. The basic equation of the universal Kriging equation is as follows:

[0072] M(θ)=f T (θ)β+z(θ) (1)

[0073] In the formula, θ represents the research parameter, which is a random variable, f T (θ) is the regression function, z(θ) represents the approximation of the local deviation, β represents the coefficient of the regression function, f T (θ) and β can be characterized by common polynomials. In this embodiment, a fourth-order Hermite polynomial is used as the regression function, so f T (θ) and β can be expressed as in formula (2):

[0074] fT (θ)β = 8x 3 -12x (2)

[0075] The key to solving the universal Kriging model lies in obtaining the value of the local deviation. In the present invention, the value of z(θ) can be expressed as in Equation (3)

[0076] E[(z(θ1)z(θ2))] = σ 2 R(γ, θ1, θ2) (3)

[0077] In the formula, γ represents a set of hyperparameters, which includes n hyperparameters. n represents the order of the research problem, that is, the number of preset research parameters. Obviously, in this example, n is taken as 4. θ1 and θ2 represent any two sample points in the sample space. σ represents the standard deviation of the preset research parameters, and R(γ, θ1, θ2) represents the spatial correlation function of θ1 and θ2. Its calculation method is as follows:

[0078]

[0079] In the formula, γ j , θ 1j and θ 2j represent γ, θ1 and θ2 in the j-th dimension; the correlation between any sample point θ x and the sample point θ s is expressed as follows:

[0080] r(θ) = R(γ, θ x , θ s ) T (5)

[0081] 2. Sampling point coordinate generation module. According to the research parameter range of the gas turbine blade and the number of sampling points N, the Latin hypercube method is used to obtain the sampling point coordinates. Among them, the sampling points are sampled from the sampling space represented by the research parameter range, which refers to the working conditions that need to be calculated to solve the universal Kriging equation. The number of coordinates of the sampling points is equal to the number of research parameters, and each sampling point represents a set of working conditions that need to be calculated. The research parameters and the research parameter range selected in this embodiment are shown in Table 2. The number of sampling points N is a preset value. The more sampling points there are, the more accurate the modeling of the same Kriging model will be, but the calculation time will be longer. In this embodiment, the number of sampling points N is selected as 100.

[0082] Table 2 Probability density distribution of uncertainty input quantities

[0083] Research parameter name Mean value Standard deviation Inlet airflow angle (H) 0.0° 0.67° Tip clearance (S) 0.4 mm 0.08 mm Mainstream inlet total pressure (G) 126900 Pa 7480 Pa Mainstream inlet total temperature (D) 709.0K 17.24K

[0084] The steps of calculating the sampling point coordinates using the Latin hypercube method based on the research parameter range and the number of sampling points N are as follows:

[0085] (1) Divide the interval (0, 1) into N equal segments to obtain N small intervals;

[0086] (2) Use the random function Random in Python to randomly generate a value in each small interval and store it in the array P in sequence. Then the array P contains N elements.

[0087] (3) Map the elements in the array P to the coordinate values of the sampling points corresponding to these elements in sequence using the inverse function of the standard normal distribution. The specific calculation process is as follows:

[0088] Let the coordinates of the η-th sampling point be (η1, η2, η3, η4). Then according to the Latin hypercube sampling theory:

[0089]

[0090]

[0091]

[0092]

[0093] According to formulas (6)-(9), the coordinates (η1, η2, η3, η4) of the η-th sampling point can be calculated. Perform the operation in step (3) for all sampling points to obtain the coordinates of all sampling points.

[0094] Among them, (P[η1], P[η2], P[η3], P[η4]) represents the η-th variable of the array P, represents the mean of the standard normal distribution, and its value is 0.5; represents the variance of the standard normal distribution, and its value is 1.

[0095] 3. Sampling point gas-thermal parameter calculation module, which accepts the coordinates of the sampling points and inputs the geometric parameters of the turbine blade given in Table 1 into the open-source program library OPERNFORM to calculate all the gas-thermal performance parameters of the turbine blade under the working conditions represented by each sampling point. In this embodiment, the heat transfer amount at the blade tip is used as the main gas-thermal parameter to be studied.

[0096] 4. Random hyperparameter generation module, which receives the set of unknown hyperparameters in the universal Kriging equation modeling module and gives a set of random hyperparameters through the random function Random in Python to obtain a hyperparameter set containing n elements, denoted as the initial random hyperparameter set.

[0097] 5. Explicit equation solving module for the universal Kriging equation, which receives the sampling point coordinates and the gas-thermal performance parameters of the turbine blade under the working conditions represented by each sampling point, and combines the initial random hyperparameter set to solve the explicit equation of the universal Kriging equation.

[0098] Store the aerothermal performance parameters of the turbine blade under the working conditions characterized by each sampling point into the array Y, then the Kriging model can be calculated by the following formula:

[0099] M(θ) = f T (θ)β + r T (θ)R(γ, θ x , θ s ) -1 (Y - β)(10)

[0100] The number of elements contained in the array Y is equal to the number of sampling points, and each element represents the heat exchange amount under the working conditions represented by each sampling point. In this embodiment, the Y for calculating the first universal Kriging equation is [110064.984375, 120002.8359375, 99956.765625, 110487.8984375, 104221.8671875, 105730.53125, 114676.625, 96688.7421875, 127677.578125, 114589.171875, 91216.96875, 120471.390625, 114407.296875, 101247.359375, 108632.5078125, 109684.0390625, 113932.265625, 96684.046875, 112027.734375, 104842.8046875, 145045.234375, 119462.3125, 99111.7890625, 99771.8046875, 90754.65625, 108289.9921875, 122578.9453125, 93221.65625, 122380.5546875, 120467.7890625, 91172.1015625, 95812.421875, 99067.1015625, 104515.765625, 118573.265625, 97453.15625, 126621.1171875, 137919.9375, 107030.2734375, 108709.6640625, 107723.140625, 95927.2578125, 114564.1796875, 104999.296875, 141989.359375, 115197.5703125, 92053.21875, 89585.0078125, 123238.6484375, 115476.7578125, 87211.8125, 122325.9296875, 118206.5625, 97557.6875, 109024.53125, 85914.421875, 120471.9921875, 124863.7890625, 104296.515625, 105612.09375, 104137.5078125, 108914.5625, 108800.65625, 89308.296875, 136510.765625, 117307.421875, 97415.1484375, 111767.[5703125, 110499.7890625, 93693.1328125, 106856.8515625, 89662.7890625]。.

[0101] 6. Sample point coordinate generation module. In the sampling space determined by the research parameters of the gas turbine blade, a new batch of sample points is newly generated using the Latin hypercube method. The new sample points are sampled from the sampling space characterized by the research parameter range, and the working conditions for calculating the uncertainty required to evaluate the established ordinary Kriging equation are evaluated. The number of coordinates of the new sample points is equal to the number of research parameters, and each new sample point represents a set of working conditions to be calculated. Let the number of new sample points be N test , N test The larger N is, the more accurate the calculated hyperparameter set is, but the consumption of computing resources increases. In this embodiment, N test is set to 1000; the method for generating the sample point coordinates is exactly the same as in step 2.

[0102] 7. Normal distribution perturbation application module. A normal distribution perturbation is applied to each new sample point. The application method of the normal distribution is as follows:

[0103] Let the coordinates of the ι-th new sample point be (ι1, ι2, ι3, ι4), then a normal distribution with a mean of ι i (i = 1, 2, 3, 4) and a standard deviation of (N sam ι i ) can be generated for each coordinate of this new sample point. N sam is a preset parameter. The larger this parameter is, the smoother the fitted Kriging model is, but the slower the convergence speed of the algorithm. In this embodiment, N sam is set to 0.3. The above operation is performed on all samples, and then the normal distribution perturbations of each coordinate of each new sample point are obtained.

[0104] 8. Sample point polynomial chaos expansion modeling module. Establish the polynomial chaos expansion to be solved for each new sample point. The system output y (i.e., the gas thermal performance parameter) can be expanded by polynomial chaos as the following formula. In this embodiment, the system output is selected as the heat transfer at the blade tip:

[0105]

[0106] In the formula, a j is the coefficient of the j-th orthogonal basis, and Ψ j (ξ) is the j-th orthogonal basis in the discrete case. In this embodiment, the orthogonal basis is selected as the fourth-order Hermite polynomial, and its form is exactly the same as formula (2);

[0107] 9. Sample point polynomial chaos expansion solving module, which receives the explicit equation of the universal Kriging equation and solves the explicit formula of the polynomial chaos expansion for each new sample point based on the polynomial chaos theory. First, input the coordinates of each sample point into the explicit equation of the universal Kriging equation to solve the system output y of each new sample point. Then, the Galerkin projection method can be used to solve the coefficient of the j-th orthogonal basis of the polynomial chaos expansion for each new sample point:

[0108]

[0109] In the formula, is the polynomial inner product, J(ξ) is the joint probability density function of the uncertainty input variables, that is, the corresponding normal distribution function obtained according to the means and variances of the four normal distributions shown in Table 2 combined with the formula of the normal distribution. The explicit expression of the polynomial chaos expansion of each new sample point can be obtained by combining the coefficients of the polynomial chaos expansion with the orthogonal basis in the formula.

[0110] 10. Sample point statistical variance calculation module, which receives the explicit formula of the polynomial chaos expansion of each new sample point; solves the statistical variance σ of the gas thermal performance parameters of each new sample point through formula (13) 2 ;

[0111]

[0112] 11. Hyperparameter evaluation module, which solves the sum of the statistical variances of the gas thermal performance parameters of each new sample point as the evaluation value of the random initial hyperparameters generated in step 4.

[0113] 12. Optimal hyperparameter output module, which determines whether the convergence condition is met, that is, when the difference between the current evaluation value and the previous evaluation value is less than or equal to κ, stop the iteration and output the optimal hyperparameters; when the difference between the current evaluation value and the previous evaluation value is greater than κ, then run steps 4-11; where κ is a preset parameter, the smaller κ is, the more accurate the Kriging model is, but the slower the calculation converges.

[0114] 13. Optimal universal Kriging equation explicit equation solving module, which receives the optimal hyperparameters and solves the explicit equation of the optimal universal Kriging equation by combining the gas thermal performance parameters of the turbine blade under the working conditions represented by each sampling point calculated in step 3. The solving formula is exactly the same as formula (10).

[0115] 14. Uncertainty node generation module, which uses the Latin hypercube method to obtain the coordinates of the uncertainty nodes according to the research parameter range of the gas turbine blade and the number of uncertainty nodes N un

[0116] ​In the present invention, the uncertainty nodes are sampled from the sampling space characterized by the research parameter range, which refers to the working conditions required for solving the polynomial chaos expansion. The number of coordinate of the uncertainty nodes is equal to the number of research parameters, and each uncertainty node represents a set of working conditions to be calculated.

[0117] In this embodiment, the selection of the research parameter range is consistent with Table 2, and the number of uncertainty nodes N un The larger the value, the higher the calculation accuracy of the uncertainty quantification calculation, but the slower the calculation convergence. In this embodiment, N un is selected as 4000. The process of calculating the uncertainty coordinates by the Latin hypercube method in step 14 is exactly the same as that in step 2.

[0118] 15. The explicit formula solving module for the polynomial chaos expansion of the gas thermal parameters of the turbine blade receives the coordinates of the uncertainty nodes and the explicit equation of the optimal universal Kriging equation, and solves the explicit formula of the polynomial chaos expansion of the gas thermal performance parameters of the gas turbine blade based on the polynomial chaos theory. The solving process is exactly the same as that in steps 8-9.

[0119] 16. The robust parameter calculation module for the gas thermal performance of the gas turbine blade receives the explicit formula of the polynomial chaos expansion of the gas thermal performance parameters of the gas turbine blade, and solves the robust parameters (including statistical mean and statistical variance) of the gas thermal performance parameters of the gas turbine blade. The statistical variance is solved by formula (12). The statistical mean is solved by formula (14).

[0120] μ = a0 (14)

[0121] where a0 is the coefficient of the first term of the polynomial chaos expansion.

[0122] The calculated robust parameters can quantify the uncertainty of the system output of the gas turbine system with strong nonlinearity and chaos degree caused by the uncertainty input quantity, reveal the internal mechanism of the uncertainty phenomenon in the actual operation of the gas turbine, and provide data and technical support for the research of gas turbine blades with high life and strong robustness.

[0123] Figure 3 is the axial distribution of the mean and variance of the tip heat transfer of the GE-E3 gas turbine obtained in the embodiment. In the figure, Q represents the tip heat transfer, μ represents the mean of the heat transfer, and σ represents the variance of the heat transfer. It can be found that Figure 3 gives the detailed axial distribution of the robust performance (mean and variance) of the tip heat transfer under the influence of the uncertainty input quantity. There are two peak regions of the tip heat transfer mean, which are located at the positions of 0.18 axial chord length and 0.83 axial chord length respectively. Therefore, in the actual operation of the gas turbine, special attention should be paid to the protection of these two regions, and the thermal barrier coatings in these two regions should be thickened to a certain extent. In addition,Figure 3 It also shows that in the region with a larger mean heat transfer amount, the variance of the heat transfer amount is also larger, indicating that in actual operation, thermal corrosion (high mean heat transfer amount) and thermal fatigue (high variance of heat transfer amount) are often coupled with each other. This internal interaction mechanism of the heat transfer field revealed by the present invention can guide turbine designers to more deeply understand the uncertainty mechanism of gas turbine blades during actual operation.

Claims

1. A robust calculation method for the gas thermal performance parameters of high-efficiency turbine blades, characterized in that, It includes the following steps: Step 1, perform mathematical modeling on the gas-thermal performance parameter response model of the gas turbine blade based on the universal Kriging surrogate model theory to generate the universal Kriging equation to be solved, where the gas-thermal performance parameter is the output quantity for which uncertainty needs to be evaluated; Step 2, obtain the sampling point coordinates using the Latin hypercube method according to the research parameter range and the number of sampling points of the gas turbine blade; the research parameters are the preset research objects, which are set as the geometric or aerodynamic parameters with obvious uncertainty during the operation of the gas turbine; the sampling points refer to the working conditions required for calculating the universal Kriging equation, and the number of coordinates of the sampling points is equal to the number of research parameters, and each sampling point represents a set of working conditions that need to be calculated; Step 3, calculate the gas-thermal performance parameters of the turbine blade under the working conditions represented by each sampling point using the open-source program library OPERNFORM according to the coordinates of the sampling points; Step 4, randomly generate a set of hyperparameters using the Random function; Step 5, solve the explicit equation of the universal Kriging equation by combining the sampling point coordinates, the gas-thermal performance parameters of the turbine blade under the working conditions represented by each sampling point, and the hyperparameters; Step 6, generate a batch of new sample points using the Latin hypercube method within the sampling space determined by the research parameters of the gas turbine blade; The new sample points evaluate the working conditions for calculating the uncertainty required by the established universal Kriging equation. The number of coordinates of the new sample points is equal to the number of research parameters, and each new sample point represents a set of working conditions that need to be calculated; Step 8, apply a normal distribution perturbation to each new sample point; Step 9, establish the polynomial chaos expansion to be solved for each new sample point; Step 10, solve the explicit formula of the polynomial chaos expansion of each new sample point based on the polynomial chaos theory according to the explicit equation of the universal Kriging equation; Step 11, solve the statistical variance of the gas-thermal performance parameters of each new sample point according to the explicit formula of the polynomial chaos expansion of each new sample point; Step 12, solve the cumulative sum of the statistical variances of the gas-thermal performance parameters of each new sample point as the evaluation value of the hyperparameters generated in Step 4; Step 13, determine whether the convergence condition is satisfied. If not, execute Steps 4 - 11. If satisfied, output the optimal hyperparameters; Step 14, solve the explicit equation of the optimal universal Kriging equation by combining the optimal hyperparameters and the gas-thermal performance parameters of the turbine blade under the working conditions represented by each sampling point; Step 15, obtain the uncertainty node coordinates using the Latin hypercube method according to the research parameter range and the number of uncertainty nodes of the gas turbine blade; the uncertainty nodes refer to the working conditions required for calculating the polynomial chaos expansion. The number of coordinates of the uncertainty nodes is equal to the number of research parameters, and each uncertainty node represents a set of working conditions that need to be calculated; Step 16, receive the uncertainty node coordinates and the explicit equation of the optimal universal Kriging equation, and solve the explicit formula of the polynomial chaos expansion of the gas-thermal performance parameters of the gas turbine blade based on the polynomial chaos theory; Step 16: Receive the explicit formula of the polynomial chaos expansion of the gas turbine blade gas-thermal performance parameters, and solve the robustness parameters of the gas turbine blade gas-thermal performance parameters, where the robustness parameters include statistical mean and statistical variance.

2. The robust calculation method of the gas thermal performance parameters of the high-efficiency turbine blade according to claim 1, characterized in that The gas turbine blade gas-thermal performance parameters are leakage flow rate, total pressure loss coefficient, or heat transfer quantity; the research parameters include inlet gas flow angle, tip clearance, mainstream inlet total pressure, and mainstream inlet total temperature.

3. The robust calculation method for the gas thermal performance parameters of the high-efficiency turbine blade according to claim 1 or 2, characterized in that In Step 1, the basic equation of the universal Kriging equation is as follows: M(θ) = f T (θ)β + z(θ) where θ represents a research parameter, which is a random variable, and f T (θ) is the regression function, z(θ) represents the approximation of the local deviation, β represents the coefficient of the regression function, and f T (θ) and β are represented by polynomials as follows: f T (θ)β = 8x 3 -12x z(θ) is expressed as follows: E[(z(θ1)z(θ2))] = σ 2 R(γ, θ1, θ2) In the formula, γ represents a set of hyperparameters, including n hyperparameters, n represents the order of the research problem, that is, the number of research parameters, θ1 and θ2 represent any two sample points in the sample space, σ represents the preset standard deviation of the research parameters, and R(γ, θ1, θ2) represents the spatial correlation function of θ1 and θ2, and its calculation method is as follows: where γ j , θ 1j and θ 2j represent γ, θ1, and θ2 of the j-th dimension; the correlation between any sample point θ x and the sample point θ s is expressed as follows: r(θ) = R(γ, θ x , θ s ) T 。 4. The robust calculation method for the gas thermal performance parameters of the high-efficiency turbine blade according to claim 3, wherein In Step 2, if the number of sampling points is N, the steps to obtain the sampling point coordinates using the Latin hypercube method are as follows: Step (1): Divide the interval (0, 1) into N segments evenly to obtain N small intervals; Step (2): Use the random function Random in Python to randomly generate a value in each small interval and store it in the array P in sequence, then the array P contains N elements; Step (3): Map the elements in the array P to the coordinate values of the corresponding sampling points using the inverse function of the standard normal distribution in sequence.

5. The robust calculation method for the gas thermal performance parameters of the high-efficiency turbine blade according to claim 3, characterized in that, In Step 4, use the random function Random in Python to give a set of random hyperparameters to obtain a set of hyperparameters containing n elements, where n is the number of research parameters.

6. The robust calculation method for the gas thermal performance parameters of the high-efficiency turbine blade according to claim 3, characterized in that, In Step 5, store the gas-thermal performance parameters of the turbine blade under the working conditions represented by each sampling point in the array Y, then the universal Kriging model is calculated by the following formula: M(θ) = f T (θ)β + r T (θ)R(γ, θ x , θ s ) -1 (Y - β) The number of elements contained in the array Y is equal to the number of sampling points, and each element represents the gas-thermal performance parameter under the working conditions represented by a sampling point.

7. The robust calculation method for the gas thermal performance parameters of the high-efficiency turbine blade according to claim 3, wherein In Step 7, the application method of the normal distribution is as follows: Generate a normal distribution with a mean of ι i and a standard deviation of N sam ι i for each coordinate of each new sample point, and obtain the normal distribution perturbation of each coordinate of each new sample point. N sam is a preset parameter. The larger this parameter is, the smoother the fitted Kriging model is, but the slower the convergence rate of the algorithm is.

8. The robust calculation method for the gas thermal performance parameters of the high-efficiency turbine blade according to claim 3, characterized in that In Step 8, for the gas-thermal performance parameter, that is, the system output y, the polynomial chaos expansion is as follows: where a j is the coefficient of the j-th orthogonal basis, and Ψ j (ξ) is the j-th orthogonal basis in the discrete case; In Step 9, first input the coordinates of each new sample point into the explicit equation of the universal Kriging equation to solve the system output y of each new sample point; then, use the Galerkin projection method to solve the coefficient of the jth orthogonal basis of the polynomial chaos expansion of each new sample point: wherein, is the polynomial inner product, J(ξ) is the joint probability density function of the uncertain input variables, and the explicit expression of the polynomial expansion of each new sample point can be obtained by matching the obtained coefficients with the orthogonal basis; In the said step 10 and step 16, the statistical variance σ is solved by the following formula 2 :[[]]END]] In Step 16, solve the statistical mean through the following formula μ = a0 where a0 is the coefficient of the first term of the polynomial chaos expansion.

9. The robust calculation method for the gas thermal performance parameters of the high-efficiency turbine blade according to claim 8, characterized in that, In Step 12, when the difference between the current evaluation value and the previous evaluation value is less than or equal to κ, stop the iteration and output the optimal hyperparameters; when the difference between the current evaluation value and the previous evaluation value is greater than κ, then run Steps 4 - 11; where κ is a preset parameter, the smaller κ is, the more accurate the Kriging model modeling is, but the slower the calculation convergence is.

10. The robust calculation method for the gas thermal performance parameters of the high-efficiency turbine blade according to claim 1, characterized in that, Use the robustness parameters to quantify the uncertainty of the system output of the gas turbine system caused by the uncertain input quantities.

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