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A proper orthogonal decomposition (POD) based hybrid surrogate model and the applications to transonic flow reconstructions are presented in the paper. In the implementations, the radial basis function (RBF) model response instead of the least-square linear regression is employed in order to improve the coefficients of POD basis modes; moreover, an adaptive sampling strategy with both the model response error and sample independence taken into account is studied to reduce the sample number, while maintaining sufficient response accuracy. Firstly, the POD-RBF surrogate model is studied and compared with the least-square-based POD through pressure reconstruction studies on the twodimensional blade surface. The results demonstrate that the non-linear model response method significantly improves the coefficients of the basis modes and thus the averaged description error. Meanwhile, the beneficial gains on the convergence performance of the response error versus the number of basis modes are obtained. Then by comparing with the uniform sampling and the resampling strategy with taking only the response error into account, the adaptive sampling method proposed in the paper obtains the best performance on reducing the averaged description error. Finally, the flow characteristics of the flow fields on the suction surface, at the blade tip, in the blade passage of the sampled three-dimensional transonic compressor rotor blades with different spanwise sweeps based on the baseline blade, NASA Rotor 67 are illustrated through the flow basis modes. Compared with the suction flow, the flow at the blade tip contains more intensive flow characteristics including shock, tip-leakage flow and shock-leakage interaction, resulting in a higher averaged description error. Besides, the missed flow fields in the passages of the test blades are reconstructed from the flow basis modes by using the adaptive POD-RBF hybrid model and the corresponding aerodynamic parameters are then predicted. The spanwise distributions of the circumferentially averaged aerodynamic parameters at the blade outlet reconstructed from POD-RBF model are consistent well with the numerical solutions. The results demonstrate that the adaptive POD-RBF hybrid surrogate model is effective and accurate enough for reconstructing the transonic flow. In order to further evaluate the response performance of the adaptive POD-RBF model, statistic analysis is carried out for a group of hybrid models with different sampling strategies and different numbers of samples. Generally, although the number of adaptive samples is much less, the mean value and standard deviation of the adaptive model are close enough to those of the static model with sufficient uniform samples. Besides, the standard deviations of a lot of aerodynamic parameters of interest exhibit significant peaks near the blade tip, further demonstrating that the flow at the blade tip is more intensive in the three-dimensional transonic rotor blade passage.
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Keywords:
- flow analysis /
- proper orthogonal decomposition /
- transonic /
- surrogate model
[1] Xie G F, He X H, Tong J J, Zheng Y H 2007 Acta Phys. Sin. 56 3193 (in Chinese) [谢国锋, 何旭洪, 童节娟, 郑艳华 2007 56 3193]
[2] Luo J Q, Liu F 2013 Acta Phys. Sin. 62 190201 (in Chinese) [罗佳奇, 刘锋 2013 62 190201]
[3] Jordan M I, Jacobs R A 1994 Neural Comput. 6 181
[4] Couplet M, Basdevant C, Sagut P 2005 J. Comput. Phys. 207 192
[5] Gao Q, Yi S H, Jiang Z F, He L, Xie W K 2013 Chin. Phys. B 22 014202
[6] Wang W, Guan X L, Jiang N 2014 Chin. Phys. B 23 104703
[7] Sirovich L, Kirby M 1987 J. Opt. Soc. Am. A 4 519
[8] Dowell E, Hall K, Thomas J, Florea R, Heeg J 1999 AIAA Paper 1999 1261
[9] LeGresley P A, Alonso J J 2001 AIAA Paper 2001 0926
[10] Wilcox K, Peraire J 2002 AIAA J. 40 2323
[11] Bui-Thanh T, Damodaran M, Willcox K 2004 AIAA J. 42 1063
[12] Duan Y H, Cai J S, Li Y Z 2012 AIAA J. 50 968
[13] Kato H, Funazaki K I 2014 ASME Paper 2014 27229
[14] Luo J, Duan Y, Tang X, Liu F 2015 ASME Paper 2015 42876
[15] Krige D G 1951 J. Chem. Metal. Min. Soc. S. AFR. 52 119
[16] Ostrowski Z, Bialecki R A, Kassab A J 2008 Inverse Probl. Sci. En. 16 39
[17] Rogers C A, Kassab A J, Divo E A, Ostrowski Z, Bialecki R A 2012 Inverse Probl. Sci. En. 20 749
[18] Qiu Y S, Bai J Q, Hua J 2013 Acta Aeronau. Astronau. Sin. 34 1249 (in Chinese) [邱亚松, 白俊强, 华俊 2013 航空学报 34 1249]
[19] Braconnier T, Ferrier M, Jouhaud J C, Montagnac M, Sagaut P 2011 Comput. Fluids 40 195
[20] Gao S G, Dong H R, Sun X B, Ning B 2015 Chin. Phys. B 24 010501
[21] McKay M D, Beckman R J, Conover W J 2000 Technometrics 42 55
[22] Wang G G 2003 J. Mech. Design 125 210
[23] Thomson Q, Martins J R R A 2011 Eng. Optimiz. 43 615
[24] Sirovich L, Kirby M 1987 Q. Appl. Math. 45 561
[25] Finkel R, Bentley J L 1974 Acta Inform. 4 1
[26] Strazisar A J, Wood J R, Hathaway M D, Suder K L 1989 NASA TP 1989 2879
[27] Denton J D, Xu L 2002 ASME Paper 2002 30327
[28] Luo J, Zhou C, Liu F 2014 J. Turbomach. 136 051005
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[1] Xie G F, He X H, Tong J J, Zheng Y H 2007 Acta Phys. Sin. 56 3193 (in Chinese) [谢国锋, 何旭洪, 童节娟, 郑艳华 2007 56 3193]
[2] Luo J Q, Liu F 2013 Acta Phys. Sin. 62 190201 (in Chinese) [罗佳奇, 刘锋 2013 62 190201]
[3] Jordan M I, Jacobs R A 1994 Neural Comput. 6 181
[4] Couplet M, Basdevant C, Sagut P 2005 J. Comput. Phys. 207 192
[5] Gao Q, Yi S H, Jiang Z F, He L, Xie W K 2013 Chin. Phys. B 22 014202
[6] Wang W, Guan X L, Jiang N 2014 Chin. Phys. B 23 104703
[7] Sirovich L, Kirby M 1987 J. Opt. Soc. Am. A 4 519
[8] Dowell E, Hall K, Thomas J, Florea R, Heeg J 1999 AIAA Paper 1999 1261
[9] LeGresley P A, Alonso J J 2001 AIAA Paper 2001 0926
[10] Wilcox K, Peraire J 2002 AIAA J. 40 2323
[11] Bui-Thanh T, Damodaran M, Willcox K 2004 AIAA J. 42 1063
[12] Duan Y H, Cai J S, Li Y Z 2012 AIAA J. 50 968
[13] Kato H, Funazaki K I 2014 ASME Paper 2014 27229
[14] Luo J, Duan Y, Tang X, Liu F 2015 ASME Paper 2015 42876
[15] Krige D G 1951 J. Chem. Metal. Min. Soc. S. AFR. 52 119
[16] Ostrowski Z, Bialecki R A, Kassab A J 2008 Inverse Probl. Sci. En. 16 39
[17] Rogers C A, Kassab A J, Divo E A, Ostrowski Z, Bialecki R A 2012 Inverse Probl. Sci. En. 20 749
[18] Qiu Y S, Bai J Q, Hua J 2013 Acta Aeronau. Astronau. Sin. 34 1249 (in Chinese) [邱亚松, 白俊强, 华俊 2013 航空学报 34 1249]
[19] Braconnier T, Ferrier M, Jouhaud J C, Montagnac M, Sagaut P 2011 Comput. Fluids 40 195
[20] Gao S G, Dong H R, Sun X B, Ning B 2015 Chin. Phys. B 24 010501
[21] McKay M D, Beckman R J, Conover W J 2000 Technometrics 42 55
[22] Wang G G 2003 J. Mech. Design 125 210
[23] Thomson Q, Martins J R R A 2011 Eng. Optimiz. 43 615
[24] Sirovich L, Kirby M 1987 Q. Appl. Math. 45 561
[25] Finkel R, Bentley J L 1974 Acta Inform. 4 1
[26] Strazisar A J, Wood J R, Hathaway M D, Suder K L 1989 NASA TP 1989 2879
[27] Denton J D, Xu L 2002 ASME Paper 2002 30327
[28] Luo J, Zhou C, Liu F 2014 J. Turbomach. 136 051005
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