Gappy pod-based inverse design method for endwall contouring of a diffuser

Abstract

The application discloses a Gappy POD-based inverse design method for end wall shaping of a diffuser cascade, and belongs to the technical field of diffuser cascade design. The method comprises the following steps: setting a near-end wall speed inverse design target; parameterizing the end wall of the diffuser cascade; generating a sample database of the end wall shaping obtained through parameterization to fill the entire shaping design space; forming a missing snapshot according to the inverse design target; constructing a new database based on the sample database and the missing snapshot; and solving the POD (Proper Orthogonal Decomposition) base of the database by using the POD method. After the first p POD bases are selected, the missing data of the end wall shaping is solved through fitting. The missing data of the new end wall shaping is calculated iteratively, and whether the convergence condition is met is judged according to the difference between the parameters of the missing data before and after the iteration. If the convergence condition is met, the iteration is stopped, and the inverse design shaping result is output. If the convergence condition is not met, the solution of the missing part is replaced with the solution of the iteration in the previous step. The shaping result shows that the method has strong robustness and obvious flow control effect.

Description

Technical Field

[0001] This invention belongs to the field of diffuser blade design technology, specifically relating to a reverse design method for diffuser blade endwall shaping based on Gappy POD. Background Technology

[0002] As the performance requirements of aero-engines continuously evolve to meet societal demands, the pursuit of high performance presents numerous challenges. While increasing the thrust-to-weight ratio of aero-engines, the increased reverse pressure gradient within the axial compressor inevitably leads to aerodynamic problems. To mitigate the negative impacts of corner separation and corner stall, scholars both domestically and internationally have developed various flow control methods, including active and passive control.

[0003] Rose et al. (Rose, MGNon-Axisymmetric Endwall Profiling in the HP NGV's sofan Axial Flow Gas Turbine [R]. ASME Paper 1994-GT-249, 1994.) proposed a non-axisymmetric endwall profiling method in 1994. This method alters the fluid flow within the near-wall boundary layer by creating undulations on the endwall surface, thereby controlling and adjusting the flow structure within the compressor channel. After decades of research, a deeper understanding of the structural characteristics and flow control mechanisms of endwall profiling has been achieved. The endwall profiling inverse design method establishes a correspondence between design objectives and profiling schemes based on physical theory. It requires setting reasonable inverse design objectives and employing a margin correction iterative method for inverse design of the endwall profiling. Inverse design offers advantages such as short design and simulation times and universality.

[0004] In order to ensure that the endwall profile obtained by the reverse design method can effectively suppress corner separation and corner stall, and improve compressor performance, the following two key issues need to be addressed in the endwall profile reverse design (Wuli Chu, Xiangjun Li, Yanhui Wu. Reduction of end wall loss in axial compressor by using non-axisymmetric profiled end wall: A new design approach based on end wall velocity modification[J]. Aerospace Since and Technology, 55, 2016.): (1) reasonably specifying the target pressure or velocity distribution; (2) solving the many limitations of the margin correction iterative method in use and the large amount of time consumed by calling the solver. These two key issues determine the quality of the endwall profile reverse design effect, and the existing methods cannot solve these problems simultaneously. Summary of the Invention

[0005] The technical problem to be solved:

[0006] To overcome the shortcomings of existing technologies, this invention provides a reverse design method for the endwall design of a diffuser cascade based on Gappy POD. First, by analyzing the flow field inside the diffuser cascade, the near-endwall velocity distribution is set as the reverse design target. Then, based on the parameterized endwall, the endwall design parameters are reconstructed using flow field information. Design results show that this method has strong robustness and a significant effect on flow control.

[0007] The technical solution of this invention is: a reverse design method for the endwall shaping of a diffuser blade based on Gappy POD, characterized by the following specific steps:

[0008] Step 1: Set the inverse design target for the near-endwall velocity, denoted as V. Target ;

[0009] Step 2: Parameterize the diffuser blade endwall;

[0010] Step 3: Use Latin hypercube sampling to generate a sample database D from the end-wall shapes obtained in Step 2 through parameterization. i The obtained sample database D i It fills the entire design space;

[0011] Step 4: Based on the reverse design target V set in Step 1 Target To form a missing snapshot D Tj ;

[0012] Step 5: Based on the sample database D from Step 3 i Missing snapshot D from step 4 Tj Build a new database K i+1 =D i +D Tj Then, the POD method is used to obtain the database K. i+1 POD base Φ i+1 ;

[0013] Step 6: After selecting the first p POD bases, solve for the missing end-wall shape data through fitting.

[0014] Step 7: Repeat steps 5 and 6 to iteratively calculate the new missing endwall shape data. according to and The difference in parameters is used to determine if the convergence condition is met. If the convergence condition is met, the iteration stops, and the inverse design modeling result is output. If the convergence condition is not met, the missing solution is output. Replace the solution from the previous iteration Repeat steps 5 through 7.

[0015] A further technical solution of the present invention is as follows: In step 1, the near-endwall velocity flow field is analyzed through numerical simulation. At the design point, the endwall shape should increase the transverse tendency of the near-endwall flow in the suction angle region of the blade and suppress its longitudinal tendency; near the trailing edge suction angle region, it is necessary to promote longitudinal flow; according to the flow rules in the above different regions, an anti-design target is set as V. Target The inverse design target is located on the section of the near-end wall where y+ = 5.

[0016] A further technical solution of the present invention is: in step 2, the parameterization is to divide the end wall into 4 Besizer control lines along the circumferential direction, and the Besizer control line function is:

[0017]

[0018] Among them, P i t represents the control points on the Besizer curve, n is the order of the Besizer curve, and t is a parameter; each control line is controlled by 4 free control points, and a total of 16 free control points generate the endwall shaping surface.

[0019] A further technical solution of the present invention is: in step 3, the sample database includes the end wall shape geometric parameters constructed by Latin hypercube sampling and the velocity distribution on the cross section of the end wall y+=5 constructed by numerical simulation, the velocity distribution including the magnitude and direction of the velocity.

[0020] A further technical solution of the present invention is: in step 3, the sample database is configured as: D i =[H ij V ij ] T , i = 1, 2, ..., N; j = 1, 2, ..., 16, where H ij V represents the actual height of each control point on the section where y+ = 5 of the end wall. ij Let N represent the velocity distribution of the end wall section at y+=5, where N represents the number of end wall designs in the sample database, i represents the i-th end wall design, and j represents the j-th free control point.

[0021] A further technical solution of the present invention is: in step 4, the missing snapshot, i.e., D Tj =[H Tj V Target ] T , where H Tj Missing data for the endwall shape corresponding to the reverse design target; initial missing data H Tj Through the initial sample database D i H in ij The average value is used as a substitute, and the formula is as follows:

[0022]

[0023] A further technical solution of the present invention is: in step 5, the current database K i+1 It contains N+1 database samples, namely the initial database D. i +Missing snapshot D Tj Assume K i+1 The corresponding POD base is Φ i+1 Then K i+1 With Φ i+1 The relationship between them is represented as follows:

[0024]

[0025] The modal coefficient b in equation (3) i+1 The result is obtained using the least squares method:

[0026] F i+1 =M i+1,j+1 b i+1 (4)

[0027] Where F i+1 = < K i+1 ,Φ i+1 >, M i+1,j+1 =<Φ i+1 ,Φ j+1 >.

[0028] A further technical solution of the present invention is: in step 6, based on the POD base Φ obtained in step 5... i+1 and modal coefficient b i+1 The snapshot is approximated using the first p POD bases. The first p POD bases must satisfy the condition that the sum of their corresponding eigenvalues ​​accounts for 99.99% of the sum of the eigenvalues ​​of all POD bases, i.e.:

[0029]

[0030] Where λ represents the eigenvalue corresponding to the POD basis.

[0031] A further technical solution of the present invention is: in step 6, the end wall shape data is missing. The formula is as follows:

[0032]

[0033] The new result is obtained through steps 5 and 6.

[0034] A further technical solution of the present invention is: in step 7, the L2 norm is used to measure the difference between the two generations. and The difference between the parameters, i.e.:

[0035]

[0036] The convergence condition is met when Norm2 < 2.5; where m is the number of iterations.

[0037] Beneficial effects

[0038] The beneficial effects of this invention are as follows: It employs Gappy POD to reverse-design the endwall shape of the high-load diffuser cascade. Under near-stall conditions, the velocity distribution after iterative convergence is as follows: Figure 5 It is evident that in regions A and B, as iterations continue, the velocity distribution tends towards the target velocity direction. Application on a high-load planar blade cascade demonstrates that the reverse design endwall profile significantly reduces total pressure loss at the design point. Table 1 shows the difference between the reverse design endwall profile and the prototype at 40% C downstream of the trailing edge. x Percentage of total pressure loss. Figure 7 The images show the secondary flow intensity cloud maps and low-energy fluid isosurfaces at different cross-sections of the prototype and convergence results in this invention, where (a) is the end-wall shaping convergence result and (b) is the prototype. By comparing 50% C... x With 100% C x By examining the equivalence surfaces of the secondary flow energy and the low-energy fluid, it was found that the endwall design near the trailing edge can reduce the intensity of the transverse secondary flow at the endwall, alleviate the transport of the low-energy fluid from the pressure surface to the suction surface, and thus reduce the equivalence surface of the low-energy fluid.

[0039] Table 1. Reverse design endwall shape at 40% C downstream of the trailing edge x Total pressure loss coefficient

[0040] Attached Figure Description

[0041] Figure 1 This is a roadmap of the Gappy POD endwall shape reverse design technology in this invention;

[0042] Figure 2 This is a schematic diagram of the internal flow field analysis of the diffuser cascade in this invention;

[0043] Figure 3 This is a velocity distribution diagram of the target flow field near the endwall in this invention;

[0044] Figure 4 This is a schematic diagram of the parameterization of the diffuser blade endwall in this invention;

[0045] Figure 5 This is a velocity distribution diagram during iterative convergence in this invention;

[0046] Figure 6 This is a schematic diagram of the endwall shape of the convergence result in this invention;

[0047] Figure 7 These are the secondary flow intensity cloud maps and low-energy fluid isosurfaces of different cross sections in the prototype and convergence results of this invention. Detailed Implementation

[0048] The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the invention, and should not be construed as limiting the invention.

[0049] This invention provides a reverse design method for the endwall design of a diffuser cascade based on Gappy POD. First, by analyzing the flow field inside the diffuser cascade, the near-endwall velocity distribution is set as the reverse design target. Based on the parameterized endwall, the endwall design parameters are reconstructed using flow field information. Design results show that this method has strong robustness and a significant effect on flow control.

[0050] Reference Figure 1 As shown in the figure, this embodiment presents a reverse design method for the endwall shape of a diffuser blade based on Gappy POD. The specific steps are as follows:

[0051] Step 1: Set the inverse design target for the near-endwall velocity, denoted as V. Target ;

[0052] Step 2: Parameterize the diffuser blade endwall;

[0053] Step 3: Use Latin hypercube sampling to generate a sample database D from the end-wall shapes obtained in Step 2 through parameterization. i The obtained sample database D i It fills the entire design space;

[0054] Step 4: Based on the reverse design target V set in Step 1 Target To form a missing snapshot D Tj ;

[0055] Step 5: Based on the sample database D from Step 3 i Missing snapshot D from step 4 Tj Build a new database K i+1 =D i +D Tj Then, the POD method is used to obtain the database K. i+1 POD base Φ i+1 ;

[0056] Step 6: After selecting the first p POD bases, solve for the missing end-wall shape data through fitting.

[0057] Step 7: Repeat steps 5 and 6 to iteratively calculate the new missing endwall shape data. according to and The difference in parameters is used to determine if the convergence condition is met. If the convergence condition is met, the iteration stops, and the inverse design modeling result is output. If the convergence condition is not met, the missing solution is output. Replace the solution from the previous iteration Repeat steps 5 through 7.

[0058] Gappy POD was used to reverse design the endwall shape of the high-load diffuser cascade. Under near-stall conditions, the velocity distribution after iterative convergence is as follows: Figure 5 It is evident that in regions A and B, as iterations continue, the velocity distribution tends towards the target velocity direction. Application on a high-load planar blade cascade demonstrates that the reverse design endwall profile significantly reduces total pressure loss at the design point. Table 1 shows the difference between the reverse design endwall profile and the prototype at 40% C downstream of the trailing edge. x Percentage of total pressure loss.

[0059] To further verify the detailed steps and advantages of this invention, the following description is provided in conjunction with the accompanying drawings and examples. The purpose of this invention is to introduce a novel method for reverse design of end-wall shapes.

[0060] Figure 1This is a roadmap for the Gappy POD-based reverse design technology for diffuser cascade endwalls. The following example uses a low-speed, high-load diffuser cascade (design parameters are shown in Table 1) and combines numerical simulation to verify that the Gappy POD-based reverse design method for endwalls can be effectively implemented and improve diffuser cascade performance.

[0061] Table 2 shows the design parameters for a high-load diffuser cascade.

[0062] Table 2 Design parameters of high-load diffuser cascade

[0063]

[0064] Taking this as an example, the specific implementation steps are as follows:

[0065] Step 1: Analyze the flow field through numerical simulation. At the design point, the endwall design should increase the lateral flow tendency near the endwall in the suction angle region of the blade, and suppress its directional flow tendency. For example... Figure 3 In the middle region B; near the trailing edge suction angle region, it is necessary to promote flow along the flow direction, such as... Figure 3 Region A. Therefore, according to... Figure 3 The flow rules within regions A and B are denoted as the inverse design objective (near-end wall y). + On the cross section of 5, it is denoted as V. Target .

[0066] Step 2: Parameterize the diffuser blade endwall. Specifically, parameterization involves dividing the endwall into four Besizer control lines along the circumferential direction. The Besizer control line function is as follows:

[0067]

[0068] Where P i Here, n represents the control points on the Besizer curve, n is the order of the Besizer curve, and t is a parameter. Each control line is controlled by 4 free control points. A total of 16 free control points generate the endwall shaping surface. Figure 4 A schematic diagram of the parameterization of the diffuser blade endwall is shown. The endwall shape is controlled by 16 free control points to control the concavity and convexity of the endwall surface.

[0069] Step 3: Use Latin hypercube sampling to generate a sample database D from the end-wall shapes obtained in Step 2 through parameterization. i The obtained sample database D i The sample database fills the entire design space. It includes the end-wall geometry parameters constructed through Latin hypercube sampling and the velocity distribution on the end-wall section at y+=5 constructed through numerical simulation. The velocity distribution includes both the magnitude and direction of the velocity. That is, the initial sample database is structured as: D i =[Hij V ij ] T , (i=1,2,…,N; j=1,2,…,16), where H ij V represents the actual modeling height of each control point on the near-end wall (on the section where y+=5 on the near-end wall surface). ij For the proximal wall (proximal wall surface y) + Velocity distribution on the cross section with a radius of 5.

[0070] Step 4: Based on the reverse design target V set in Step 1 Target This forms a missing snapshot, namely D. Tj =[H Tj V Target ] T , where H Tj Missing data for the endwall shape corresponding to the reverse design target. Initial missing data H Tj Through the initial database D i H in ij Replace with the average value:

[0071]

[0072] Step 5: Based on the initial database D from Step 3 i Missing snapshot D from step 4 Tj Build a new database K i+1 =D i +D Tj The POD method is used to obtain the database K. i+1 The POD base. Currently, the database contains N+1 database samples (initial database D). i +Missing snapshot D Tj Assume K i+1 The corresponding POD base is Φ i+1 Then K i+1 With Φ i+1 The relationship between them can be represented as:

[0073]

[0074] The modal coefficient b in equation (3) i+1 It can be obtained using the least squares method:

[0075] F i+1 =M i+1,j+1 b i+1 (4)

[0076] , where F i+1 = < K i+1 ,Φ i+1 >, M i+1,j+1 =<Φi+1 ,Φ j+1 >.

[0077] Step 6: Based on the calculated POD base Φ i+1 and modal coefficient b i+1 The snapshot is approximated using the first p POD bases. The first p POD bases must satisfy the condition that the sum of their corresponding eigenvalues ​​accounts for 99.99% of the sum of the eigenvalues ​​of all POD bases, i.e.:

[0078]

[0079] After selecting the first p POD bases, the missing parts are solved by fitting.

[0080]

[0081] The new result is obtained through steps 5 and 6.

[0082] Step 7: According to and The difference between the parameters is used to determine whether the convergence condition is met. Here, the difference between the parameters is represented by the L2 norm, i.e.:

[0083]

[0084] When Norm2 < 2.5, the convergence condition is met, the iteration stops, and the reverse design styling result is output. A series of results are verified by numerical simulation and flow field analysis of the GappyPOD reverse design result and the styling characteristics analysis. Figure 6 The design results of the end wall after reverse design convergence are shown. In area A, the shape is mainly convex, while in area B, it is mainly concave. Figure 7 The changes in secondary flow energy intensity and low-energy fluid accumulation in the front and rear end regions of the model were compared. The results show that, compared to the prototype, at 100% C x The secondary flow energy decreases at the endwall and increases at high blade span, promoting mixing between the low-energy fluid and the mainstream, increasing the kinetic energy of the low-energy fluid, and thus reducing losses. At 50% C x At this point, the combined effect of the transverse secondary flow at the endwall and the enhanced secondary flow at high blade span reduces the accumulation of low-energy fluid at the junction of the blade and the endwall, thereby improving the performance of the diffuser cascade.

[0085] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention without departing from the principles and spirit of the present invention.

Claims

1. A reverse design method for the endwall shape of a diffuser cascade based on Gappy POD, characterized in that... The specific steps are as follows: Step 1: Set the inverse design target for the near-endwall velocity, denoted as V. Target ; Step 2: Parameterize the diffuser blade endwall; Step 3: Use Latin hypercube sampling to generate a sample database D from the end-wall shapes obtained in Step 2 through parameterization. i The obtained sample database D i It fills the entire design space; Step 4: Based on the reverse design target V set in Step 1 Target To form a missing snapshot D Tj ; Step 5: Based on the sample database D from Step 3 i Missing snapshot D from step 4 Tj Build a new database K i+1 =D i +D Tj Then, the POD method is used to obtain the database K. i+1 POD base Φ i+1 ; Step 6: After selecting the first p POD bases, solve for the missing end-wall shape data through fitting. Step 7: Repeat steps 5 and 6 to iteratively calculate the new missing endwall shape data. according to and The difference in parameters is used to determine whether the convergence condition is met; if the convergence condition is met, the iteration stops and the reverse design modeling result is output. If the convergence condition is not met, then the missing solution will be... Replace the solution from the previous iteration Repeat steps 5 through 7.

2. The inverse design method for the endwall shaping of the diffuser cascade based on Gappy POD as described in claim 1, characterized in that: In step 1, the near-endwall velocity flow field is analyzed through numerical simulation. At the design point, the endwall design should increase the lateral tendency of the near-endwall flow in the suction angle region of the blade and suppress its directional tendency; near the trailing edge suction angle region, it is necessary to promote directional flow. According to the flow rules in the above different regions, an anti-design target is set as V. Target The inverse design target is located on the section of the near-end wall where y+ = 5.

3. The inverse design method for the endwall shaping of the diffuser cascade based on Gappy POD as described in claim 1, characterized in that: In step 2, the parameterization involves dividing the end wall into four Besizer control lines along the circumferential direction. The Besizer control line function is as follows: Among them, P i t represents the control points on the Besizer curve, n is the order of the Besizer curve, and t is a parameter; each control line is controlled by 4 free control points, and a total of 16 free control points generate the endwall shaping surface.

4. The inverse design method for the endwall shaping of the diffuser cascade based on Gappy POD as described in claim 3, characterized in that: In step 3, the sample database includes the end wall shape geometric parameters constructed by Latin hypercube sampling and the velocity distribution on the cross section of the end wall y+=5 constructed by numerical simulation. The velocity distribution includes the magnitude and direction of the velocity.

5. The inverse design method for the endwall shaping of the diffuser cascade based on Gappy POD as described in claim 4, characterized in that: In step 3, the sample database is configured as follows: D i =[H ij V ij ] T , i = 1, 2, ..., N; j = 1, 2, ..., 16, where H ij V represents the actual height of each control point on the section where y+=5 of the end wall. ij Let N represent the velocity distribution of the end wall section at y+=5, where N represents the number of end wall designs in the sample database, i represents the i-th end wall design, and j represents the j-th free control point.

6. The inverse design method for the endwall shaping of the diffuser cascade based on Gappy POD as described in claim 5, characterized in that: In step 4, the missing snapshot is D. Tj =[H Tj V Target ] T , where H Tj Missing data for the endwall shape corresponding to the reverse design target; initial missing data H Tj Through the initial sample database D i H in ij The average value is used as a substitute, and the formula is as follows:

7. The inverse design method for the endwall shaping of a diffuser cascade based on Gappy POD as described in claim 6, characterized in that: In step 5, the current database K i+1 It contains N+1 database samples, namely the initial database D. i +Missing snapshot D Tj Assume K i+1 The corresponding POD base is Φ i+1 Then K i+1 With Φ i+1 The relationship between them is represented as follows: The modal coefficient b in equation (3) i+1 The result is obtained using the least squares method: F i+1 =M i+1,j+1 b i+1 , (4) among themF i+1 = <K i+1 ,F i+1 >,M i+1,j+1 =<Φ i+1 ,F j+1 >。 8. The inverse design method for the endwall shaping of the diffuser cascade based on Gappy POD as described in claim 7, characterized in that: In step 6, based on the POD base Φ obtained in step 5 i+1 and modal coefficient b i+1 The snapshot is approximated using the first p POD bases. The first p POD bases must satisfy the condition that the sum of their corresponding eigenvalues ​​accounts for 99.99% of the sum of the eigenvalues ​​of all POD bases, i.e.: Where λ represents the eigenvalue corresponding to the POD basis.

9. The inverse design method for the endwall shaping of a diffuser cascade based on Gappy POD as described in claim 8, characterized in that: In step 6, the end-wall shape data is missing. The formula is as follows: The new result is obtained through steps 5 and 6.

10. The inverse design method for the endwall shaping of a diffuser cascade based on Gappy POD as described in claim 9, characterized in that: In step 7, the L2 norm is used to measure the difference between the two generations. and The difference between the parameters, i.e.: The convergence condition is met when Norm2 < 2.5; where m is the number of iterations.

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