A data-driven method for predicting unsteady flow with static interference in turbomachinery.

Abstract

This invention discloses a data-driven reduced-order model method for predicting unsteady flow in a turbomachinery under static-to-dynamic interference. The main steps include: Step 1, calculating the unsteady flow field at a typical operating point; Step 2, extracting the initial modes, corresponding eigenvalues, and initial time coefficients for the typical operating point; Step 3, normalizing the initial modes to obtain dimensionless initial modes; Step 4, calculating the compressor characteristic curves using a steady-state method to obtain the characteristic curves and corresponding steady flow fields; Step 5, solving for the predicted dynamic modes; and Step 6, predicting the unsteady flow field information at different operating points. This invention calculates the unsteady-to-static interference flow field under a typical operating point and uses a data-driven reduced-order model to predict the unsteady pulsation information of the target operating point. This achieves the consideration of unsteady time-averaged effects in steady-state calculations without iteratively solving unsteady control equations, effectively solving the problems of high computational cost and long calculation cycles in unsteady calculations.

Description

Technical Field

[0001] This invention relates to the field of computational fluid dynamics in engineering, and in particular to a data-driven method for predicting unsteady flow with static-rotation interference in turbomachinery. Background Technology

[0002] Aero-engines are a fundamental strategic industry for advanced countries. The turbine components within aero-engines operate under conditions of high temperature, high pressure, and high speed, exhibiting highly complex internal flow structures characterized by strong three-dimensionality, unsteadiness, and nonlinearity, posing significant challenges to the aerodynamic design of advanced aero-engines. Research on the aerodynamic performance of turbines primarily employs two methods: experimental and computational fluid dynamics (CFD). Experimental measurements are limited by the arrangement of measurement points, making it impossible to obtain global flow field data. In recent years, with the rapid development of computing power, CFD technology has gradually become an important technical tool for theoretical research and engineering applications in fields such as aerodynamics and turbine design.

[0003] In the complex flow of turbomachinery, there exists an inherent unsteady flow with rotational-static interference, which has a significant impact on turbomachinery performance and needs to be accurately predicted during the design process. For multi-stage turbomachinery, the commonly used steady mixing surface simulation method ignores the unsteady flow with rotational-static interference inside the turbomachinery, and the simulation error often increases with the number of turbomachinery stages; while traditional unsteady simulation methods are computationally too expensive and difficult to apply to routine engineering design.

[0004] Currently used methods for calculating unsteady turbomachinery, including the sliding mesh method, harmonic balance method, and nonlinear harmonic method, suffer from high computational costs, long calculation cycles, and poor convergence under near-stall conditions. Therefore, developing alternative methods for predicting unsteady turbomachinery rotation-to-static interference based on data-driven technology has significant engineering application value. By fully utilizing existing unsteady data and developing data-driven reduced-order model methods, computational speed can be effectively improved, computational costs reduced, and thus, multi-stage turbomachinery rotation-to-static interference unsteady simulations can be achieved efficiently. Summary of the Invention

[0005] (a) Technical problems to be solved

[0006] The purpose of this invention is to provide a data-driven reduced-order model method for predicting unsteady flow in turbomachinery with static-rotation interference. This method aims to achieve rapid prediction of unsteady flow fields under multiple operating conditions in multi-stage turbomachinery and rapid estimation of deterministic correlation terms, thereby reducing the cost of unsteady calculations and solving the problems of excessively long unsteady calculation cycles and the inability of steady calculations to consider unsteady effects in engineering applications.

[0007] (II) Technical Solution

[0008] To address the aforementioned technical problems, this invention provides a data-driven reduced-order model method for predicting unsteady flow with static-rotation interference in turbomachinery, comprising the following steps:

[0009] Step 1: Calculate the unsteady flow field at typical operating conditions;

[0010] Step 2: Extract the initial modes of typical operating conditions, as well as the corresponding eigenvalues ​​and initial time coefficients;

[0011] Step 3: Normalize the initial modes to obtain dimensionless initial modes;

[0012] Step 4: Calculate the characteristic curves of the compressor using the steady-state method to obtain the characteristic curves and the corresponding steady-state flow field;

[0013] Step 5: Solve for the predicted dynamic modes;

[0014] Step 6: Predict unsteady flow field information at different operating conditions;

[0015] ① The calculation of the unsteady flow field at typical operating points includes:

[0016] A single-stage compressor blade bank with static-to-static interference effect is selected as the calculation example. The unsteady flow field c1 of a typical operating point (denoted as c1 condition) is calculated using the unsteady Reynolds-averaged Navier-Stokes (URANS) method. The operating point with a flow rate slightly lower than the flow rate at the highest efficiency point is selected as the typical operating point. The calculation methods used to solve the unsteady instantaneous flow field in the unsteady Reynolds-averaged Navier-Stokes (URANS) method include the time-progressive sliding grid method, the nonlinear harmonic method, and the harmonic balance method.

[0017] ② The extraction of the initial mode, corresponding eigenvalues, and initial time coefficients for typical operating conditions includes:

[0018] Organize the unsteady flow field of condition c1 calculated by the unsteady Reynolds-averaged Navier-Stokes (URANS) method, including the unsteady instantaneous flow field at all times, to obtain the spatiotemporal matrix of the flow field variables under condition c1. The spatiotemporal matrix of the flow field variables Belonging to the m×n dimensional real number field The spatiotemporal matrix of the flow field variables Perform dynamic mode decomposition, the process of which includes:

[0019] Based on the spatiotemporal matrix of the flow field variables The sub-spacetime matrix X and sub-spacetime matrix Y are constructed as shown in the following equation:

[0020]

[0021] Perform singular value decomposition on the sub-spacetime matrix X.

[0022] X=UΣV T

[0023] Where ∑ is the singular value matrix, an n-order diagonal matrix; U is the left singular vector matrix, an m×m unitary matrix; V T Let X be a right singular vector matrix, which is an n×n unitary matrix; based on the singular value decomposition result of the sub-spacetime matrix X, the matrix is ​​further calculated.

[0024]

[0025] For the matrix Decompose to obtain the matrix Corresponding feature vector sum matrix The corresponding eigenvalue λ i According to the matrix Corresponding feature vector The matrix The corresponding eigenvalue λ i Calculate the dynamic mode matrix υ i As shown in the following formula:

[0026]

[0027] Let υ be the dynamic mode matrix under the c1 condition. i for The dynamic mode matrix under condition c1 Wherein, the dynamic mode matrix under the c1 condition is defined. The first eigenvector is the stable mode. Using the stable mode The reconstructed flow field is used as the time-averaged flow field under the c1 condition;

[0028] Based on the dynamic mode matrix under the c1 condition The left singular vector matrix under condition c1 Solving for the initial time coefficient vector As shown in the following formula:

[0029]

[0030] ③ The normalization process for the initial modes to obtain dimensionless initial modes includes:

[0031] Use the stable mode described in step two The dynamic mode matrix under condition c1 described in step two The column vectors in the matrix are dimensionless to obtain the dimensionless initial mode matrix. As shown in the following formula:

[0032]

[0033] ④ The method of calculating the compressor characteristic curves using a steady-state method to obtain the characteristic curves and corresponding steady-state flow fields includes:

[0034] The characteristic curves of the computational example are calculated using the steady Reynolds-averaged Navier-Stokes (RANS) method, which includes the mixing surface method or the channel-averaged equation method. The target operating condition c is obtained through the steady Reynolds-averaged Navier-Stokes (RANS) method. n steady flow field

[0035] ⑤ The predicted dynamic modes include:

[0036] The target working condition c obtained in step four is used. n steady flow field and the initial time coefficient vector obtained in step two Calculate the target operating condition c n Predicted stable modes As shown in the following formula:

[0037]

[0038] The dimensionless initial mode matrix obtained in step three is used. As the target working condition c n dimensionless mode matrix Immediately

[0039]

[0040] The target working condition c is calculated using the following formula. n The predicted mode matrix below

[0041]

[0042] ⑥ The unsteady flow field information for predicting different operating conditions includes:

[0043] Using the target operating condition c described in step five n The predicted mode matrix below Reconstruct the target operating condition c nUnsteady flow field As shown in the following formula:

[0044]

[0045] According to the reconstructed target working condition c n Unsteady flow field Calculate the target operating condition c n The following deterministic related terms As shown in the following formula:

[0046]

[0047] Among them, subscript ij For tensor subscripts and superscripts e Indicates ensemble average calculation, superscript t This indicates a time averaging calculation; the target operating condition c is... n The following deterministic related terms By coupling it into steady-state calculations, unsteady time-averaged effects can be considered in steady-state calculations, thereby improving the prediction accuracy of time-averaged flow fields and unsteady fluctuations.

[0048] (III) Beneficial Effects

[0049] The data-driven reduced-order model method for predicting unsteady flow under turbine rotational-stationary interference provided by this invention has the following advantages: By calculating the unsteady flow field under a typical operating condition, the predicted mode matrix is ​​obtained using dynamic mode decomposition. Using the steady-state calculation results and the predicted mode matrix of the target operating condition, the unsteady fluctuations and deterministic correlation terms of the target operating condition are predicted, thus considering the unsteady time-averaged effects in the steady-state calculation. When predicting unsteady fluctuations under the target operating condition using this method, it is not necessary to iteratively solve the unsteady control equations, effectively solving the problems of high computational cost and long calculation cycle in unsteady flow calculations.

[0050] The method of this invention does not involve modifying the physical model in CFD calculations, but only involves the processing of unsteady flow field data, feature extraction, and data-driven reduced-order model construction. Therefore, the code is relatively simple and easy to extend to existing design and analysis processes.

[0051] The method of this invention has shown good application results in the numerical test, and the results are in good agreement with the fully unsteady calculation results. Moreover, the calculation speed is fast, more than two orders of magnitude faster than the traditional unsteady solution. The reduced-order model involved in the method of this invention has a certain degree of universality and has good adaptability to different working states with different rotational speeds and boundary conditions. Attached Figure Description

[0052] Figure 1This is a flowchart of a data-driven reduced-order model method for predicting unsteady flow with rotational-stationary interference in turbomachinery, according to the present invention.

[0053] Figure 2 This is a schematic diagram of a three-dimensional mesh of a Stage-35 compressor, representing a specific embodiment of the data-driven reduced-order model method for predicting unsteady flow with turbine rotation-to-stationary interference according to the present invention.

[0054] Figure 3 This is a specific embodiment of the data-driven reduced-order model method for predicting unsteady flow in a turbine under static-rotation interference according to the present invention. The unsteady flow velocity fluctuations predicted by the data-driven reduced-order model method of the present invention for the Stage-35 compressor are compared with the original data.

[0055] Figure 4 This is a specific embodiment of the data-driven reduced-order model method for predicting unsteady flow with turbine rotation-to-station interference according to the present invention. The comparison between the deterministic correlation terms predicted by the data-driven reduced-order model method of the present invention and the unsteady prediction data of the Stage-35 compressor is shown. Detailed Implementation

[0056] The specific embodiments of the present invention will be further described in detail below with reference to the accompanying drawings and examples. The following examples are for illustrative purposes only and are not intended to limit the scope of the invention.

[0057] This invention provides a data-driven method for reducing the order of a model to predict unsteady flow with rotational-static interference in turbomachinery, comprising the following steps:

[0058] Step 1: Calculate the unsteady flow field at typical operating conditions;

[0059] In this step, a single-stage compressor blade bank with a static-to-rotation interference effect is selected as the calculation example. The unsteady flow field c1 of a typical operating point (denoted as c1 condition) is calculated using the unsteady Reynolds-averaged Navier-Stokes (URANS) method. A typical operating point is selected with a flow rate slightly lower than the flow rate at the highest efficiency point. The unsteady Reynolds-averaged Navier-Stokes (URANS) method includes, but is not limited to, time-progressive sliding grid method, nonlinear harmonic method, and harmonic balance method, which are calculation methods that can solve unsteady instantaneous flow fields.

[0060] Step 2: Extract the initial modes of typical operating conditions, as well as the corresponding eigenvalues ​​and initial time coefficients;

[0061] In this step, the unsteady flow field of condition c1 calculated by the unsteady Reynolds-averaged Navier-Stokes (URANS) method is organized, including the unsteady instantaneous flow field at all times, to obtain the spatiotemporal matrix of the flow field variables under condition c1. The spatiotemporal matrix of the flow field variables A real matrix of size m×n The spatiotemporal matrix of the flow field variables Perform dynamic mode decomposition, the process of which includes:

[0062] Based on the spatiotemporal matrix of the flow field variables The sub-spacetime matrix X and sub-spacetime matrix Y are constructed as shown in the following equation:

[0063]

[0064] Perform singular value decomposition on the sub-spacetime matrix X.

[0065] X=UΣV T

[0066] Where ∑ is the singular value matrix, an n-order diagonal matrix; U is the left singular vector matrix, an m×m unitary matrix; V T The right singular vector matrix is ​​an n×n unitary matrix. Based on the singular value decomposition of the sub-spacetime matrix X, the matrix is ​​further calculated.

[0067]

[0068] For the matrix Decompose to obtain the matrix Corresponding feature vector sum matrix The corresponding eigenvalue λ i According to the matrix Corresponding feature vector The matrix The corresponding eigenvalue λ i Calculate the dynamic mode matrix υ i As shown in the following formula:

[0069]

[0070] Let υ be the dynamic mode matrix under the c1 condition. i for The dynamic mode matrix under condition c1 Wherein, the dynamic mode matrix under the c1 condition is defined. The first eigenvector is the stable mode. Using the stable mode The reconstructed flow field is used as the time-averaged flow field under the c1 condition;

[0071] Based on the dynamic mode matrix under the c1 condition The left singular vector matrix under condition c1 Solving for the initial time coefficient vector As shown in the following formula:

[0072]

[0073] Step 3: Normalize the initial modes to obtain dimensionless initial modes;

[0074] In this step, the stable mode described in step two is used. The dynamic mode matrix under condition c1 described in step two The column vectors in the matrix are dimensionless to obtain the dimensionless initial mode matrix. As shown in the following formula:

[0075]

[0076] Step 4: Calculate the characteristic curves of the compressor using the steady-state method to obtain the characteristic curves and the corresponding steady-state flow field;

[0077] In this step, the characteristic curves of the computational example are calculated using the steady Reynolds-averaged Navier-Stokes (RANS) method, which includes the mixing surface method or the channel-averaged equation method. The target operating condition c is obtained through the steady Reynolds-averaged Navier-Stokes (RANS) method. n steady flow field

[0078] Step 5: Solve for the predicted dynamic modes;

[0079] In this step, the target operating condition c obtained in step four is adopted. n steady flow field and the initial time coefficient vector obtained in step two Calculate the target operating condition c n Predicted stable modes As shown in the following formula:

[0080]

[0081] The dimensionless initial mode matrix obtained in step three is used. As the target working condition c n dimensionless mode matrix Immediately

[0082]

[0083] The target working condition c is calculated using the following formula. n The predicted mode matrix below

[0084]

[0085] Step 6: Predict unsteady flow field information at different operating conditions;

[0086] In this step, the target operating condition c described in step five is adopted. n The predicted mode matrix below Reconstruct the target operating condition c n Unsteady flow field As shown in the following formula:

[0087]

[0088] According to the reconstructed target working condition c n Unsteady flow field Calculate the target operating condition c n The following deterministic related terms As shown in the following formula:

[0089]

[0090] Among them, subscript ij For tensor subscripts and superscripts e Indicates ensemble average calculation, superscript t This indicates a time averaging calculation; the target operating condition c is... n The following deterministic related terms By coupling it into steady-state calculations, unsteady time-averaged effects can be considered in steady-state calculations, thereby improving the prediction accuracy of time-averaged flow fields and unsteady fluctuations.

[0091] like Figure 1 The diagram shows a flowchart of a data-driven reduced-order model method for predicting unsteady flow in a turbomachinery under static-to-rotation interference, according to the present invention. Compared to traditional computational methods that iteratively solve unsteady control equations point-by-point, this invention uses data-driven technology and applies the concept of initial modes. It only needs to calculate the unsteady flow field under a typical operating condition, and obtains the initial dynamic modes that are basically applicable to the entire characteristic curve through mode decomposition. Subsequently, a less computationally expensive steady-state method is used to solve for all time-averaged flow fields along the entire characteristic curve. Then, using the relationship between the time-averaged flow field and the mode sequence, all corresponding predicted modes at that point are obtained, further predicting the unsteady flow field and corresponding deterministic correlation terms at the target operating point.

[0092] The code for a data-driven order reduction model method for predicting unsteady flow in turbomachinery with static-to-rotation interference, as proposed in this invention, has been implemented using MATLAB software. Here, the Stage-35 compressor is used as an example to verify this method. To better compare and demonstrate the advantages of the method, two methods were simultaneously used for calculation at the target operating point: one was the unsteady time-progression method, and the other was the data-driven order reduction model method proposed in this invention. The calculation speeds of the different methods were compared.

[0093] like Figure 2 As shown, the Stage-35 compressor is selected as the research case, and a three-dimensional mesh with an O4H topology is used for the calculation. The unsteady calculation uses a rotor-to-stator channel ratio of 3:4 and a total of 24 million meshes; the steady calculation uses a rotor-to-stator channel ratio of 1:1 and a total of 6 million meshes. The physical time step is set to 180 steps per cycle, with 35 virtual time steps per physical time step. This invention's data-driven reduced-order model method for predicting unsteady flow with rotor-to-stator interference in turbomachinery does not have special requirements for the turbulence model and can use any turbulence model. In this embodiment, the SA turbulence model is used. The rotor speed in this example is 17189 rpm. Two operating conditions were calculated using the unsteady time-progression method, with condition 1 used as the dataset to extract the initial modes and condition 2 used for model validation. For operating condition 2, a steady mixing surface method was first used for simulation to obtain a steady flow field. Then, a data-driven reduced-order model method for predicting unsteady flow with turbine rotation-static interference, proposed in this invention, was used to predict unsteady fluctuations across the entire field by inputting steady results. Data validation was performed at several sampling points to verify the model's accuracy. For example... Figure 3 As shown in the figure, "Unsteady" represents the applied unsteady method, "ROM" represents the prediction result of the data-driven reduced-order model method for predicting unsteady flow with turbine rotation-stationary interference according to the present invention, and "at WP-" following "Unsteady" or "ROM" indicates the operating condition number. Among them, point WP1 is the operating condition used for feature extraction, located to the left of the highest efficiency point of the characteristic line, while WP2 is the target operating condition used for prediction, located near the near stall point of the characteristic line.

[0094] like Figure 3The figure shows a comparison of flow velocity fluctuations at sampling points using the data-driven reduced-order model method for predicting unsteady flow with turbine-to-station interference, as described in this invention, and using fully unsteady calculations. The sampling points in the figure are located near 50% of the stator blade height at the leading edge. Under WP1 conditions, the results predicted by the data-driven reduced-order model method for predicting unsteady flow with turbine-to-station interference at the sampling points are very close to the results of unsteady calculations, only slightly higher near the first trough. In terms of computational speed, fully unsteady calculations at a single operating point require at least 15 cycles to achieve good periodicity, and the total computational cost is nearly two orders of magnitude higher than steady-state calculations. However, if the data-driven reduced-order model method for predicting unsteady flow of turbomachines with static-rotation interference is used, it is only necessary to calculate a steady result on the characteristic line and then import it into the data-driven reduced-order model to realize the consideration of unsteady time-averaged effects in steady calculations without iteratively solving unsteady control equations. Therefore, the data-driven reduced-order model method for predicting unsteady flow of turbomachines with static-rotation interference is used to predict the unsteady flow field of a working condition, which only requires about 1 / 90 of the traditional unsteady calculation. It can be seen that the present invention has great advantages in terms of computational efficiency and computational cost.

[0095] like Figure 4 The image shows a comparison between the circumferential average of the deterministic correlation terms for 50% blade height and the unsteady predicted data. The deterministic correlation terms predicted by the data-driven reduced-order model method for predicting unsteady flow with turbine rotation-stationary interference, as proposed in this invention, are very close to the calculated unsteady values. The fluctuations of the deterministic correlation terms near the blade leading edge can be accurately reproduced, demonstrating the accuracy of the data-driven reduced-order model method for predicting unsteady flow with turbine rotation-stationary interference, as proposed in this invention.

[0096] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

[0097] In summary, this invention extracts the unsteady-to-static interference flow field under typical operating conditions, obtains the predicted mode matrix using dynamic mode decomposition, and predicts the unsteady fluctuations and deterministic correlation terms of the target operating condition using the steady-state calculation results and the predicted mode matrix, thus realizing the consideration of unsteady time-averaged effects in steady-state calculations. It effectively solves the problems of high computational cost and long computation cycle of the unsteady methods used, achieves good agreement with the fully unsteady calculation results, and is approximately two orders of magnitude faster than traditional unsteady solutions. It also has a certain degree of versatility, showing good adaptability to different operating states with varying speeds and boundary conditions, providing a new and highly efficient method for predicting turbomachinery flow fields in complex engineering fields.

Claims

1. A data-driven reduced-order model method for predicting unsteady flow with rotational-stationary interference in turbomachinery, characterized in that, Includes the following steps: Step 1: Calculate the unsteady flow field at typical operating conditions; Step 2: Extract the initial modes of typical operating conditions, as well as the corresponding eigenvalues ​​and initial time coefficients; Step 3: Normalize the initial modes to obtain dimensionless initial modes; Step 4: Calculate the characteristic curves of the compressor using the steady-state method to obtain the characteristic curves and the corresponding steady-state flow field; Step 5: Solve for the predicted dynamic modes; Step 6: Predict unsteady flow field information at different operating conditions; ① The calculation of the unsteady flow field at typical operating points includes: A single-stage compressor blade bank with a static-to-static interference effect is selected as the calculation example. The unsteady flow field at a typical operating point is calculated using the unsteady Reynolds-averaged Navier-Stokes method, denoted as operating condition c1. The operating condition with a flow rate slightly lower than the flow rate at the highest efficiency point is selected as the typical operating point. The calculation methods used in the unsteady Reynolds-averaged Navier-Stokes method for solving unsteady instantaneous flow fields include the time-progressive sliding grid method, the nonlinear harmonic method, and the harmonic balance method. ② The extraction of the initial mode, corresponding eigenvalues, and initial time coefficients for typical operating conditions includes: Organize the unsteady flow field of condition c1 calculated by the unsteady Reynolds-averaged Navier-Stokes method, including the unsteady instantaneous flow field at all times, to obtain the spatiotemporal matrix of the flow field variables under condition c1. The spatiotemporal matrix of the flow field variables Belonging to the m×n dimensional real number field The spatiotemporal matrix of the flow field variables Perform dynamic mode decomposition, the process of which includes: Based on the spatiotemporal matrix of the flow field variables The sub-spacetime matrix X and sub-spacetime matrix Y are constructed as shown in the following equation: Perform singular value decomposition on the sub-spacetime matrix X. X=UΣV T Where ∑ is the singular value matrix, an n-order diagonal matrix; U is the left singular vector matrix, an m×m unitary matrix; V T Let X be a right singular vector matrix, which is an n×n unitary matrix; based on the singular value decomposition result of the sub-spacetime matrix X, the matrix is ​​further calculated. For the matrix Decompose to obtain the matrix Corresponding feature vector sum matrix The corresponding eigenvalue λ i According to the matrix corresponding feature vector The matrix The corresponding eigenvalue λ i Calculate the dynamic mode matrix υ i As shown in the following formula: Let υ be the dynamic mode matrix under the c1 condition. i for The dynamic mode matrix under condition c1 Wherein, the dynamic mode matrix under the c1 working condition is defined. The first eigenvector is the stable mode. Using the stable mode The reconstructed flow field is used as the time-averaged flow field under the c1 condition. Based on the dynamic mode matrix under the c1 condition The left singular vector matrix under condition c1 Solving for the initial time coefficient vector As shown in the following formula: ③ The normalization process for the initial modes to obtain dimensionless initial modes includes: Use the stable mode described in step two The dynamic mode matrix under condition c1 described in step two The column vectors in the matrix are dimensionless to obtain the dimensionless initial mode matrix. As shown in the following formula: ④ The method of calculating the compressor characteristic curves using a steady-state method to obtain the characteristic curves and corresponding steady-state flow fields includes: The characteristic curves of the computational example are calculated using the steady Reynolds-averaged Navier-Stokes method, which includes the mixing surface method or the channel-averaged equation method. The target operating condition c is obtained through the steady Reynolds-averaged Navier-Stokes method. n steady flow field ⑤ The predicted dynamic modes include: The target working condition c obtained in step four is used. n steady flow field and the initial time coefficient vector obtained in step two Calculate the target operating condition c n Predicted stable modes As shown in the following formula: The dimensionless initial mode matrix obtained in step three is used. As the target working condition c n The dimensionless mode matrix under Immediately The target working condition c is calculated using the following formula. n The predicted mode matrix below ⑥ The unsteady flow field information for predicting different operating conditions includes: Using the target operating condition c described in step five n The predicted mode matrix below Reconstruct the target operating condition c n Unsteady flow field As shown in the following formula: According to the reconstructed target working condition c n Unsteady flow field Calculate the target operating condition c n The following deterministic related terms As shown in the following formula: Wherein, the subscript ij is a tensor subscript, the superscript e indicates ensemble averaging, and the superscript t indicates time averaging; the target working condition c n The following deterministic related terms By coupling it into steady-state calculations, unsteady time-averaged effects can be considered in steady-state calculations.

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