Improved algorithm for solving nonlinear parabolized stability equations

doi:10.1088/1674-1056/25/8/084701

Abstract Due to its high computational efficiency and ability to consider nonparallel and nonlinear effects, nonlinear parabolized stability equations (NPSE) approach has been widely used to study the stability and transition mechanisms. However, it often diverges in hypersonic boundary layers when the amplitude of disturbance reaches a certain level. In this study, an improved algorithm for solving NPSE is developed. In this algorithm, the mean flow distortion is included into the linear operator instead of into the nonlinear forcing terms in NPSE. An under-relaxation factor for computing the nonlinear terms is introduced during the iteration process to guarantee the robustness of the algorithm. Two case studies, the nonlinear development of stationary crossflow vortices and the fundamental resonance of the second mode disturbance in hypersonic boundary layers, are presented to validate the proposed algorithm for NPSE. Results from direct numerical simulation (DNS) are regarded as the baseline for comparison. Good agreement can be found between the proposed algorithm and DNS, which indicates the great potential of the proposed method on studying the crossflow and streamwise instability in hypersonic boundary layers.

Keywords: improved NPSE approach stationary crossflow vortices fundamental resonance hypersonic boundary layers

Received: 29 January 2016
Revised: 28 March 2016
Accepted manuscript online:

PACS:	47.20.Ib	(Instability of boundary layers; separation)
	47.40.Ki	(Supersonic and hypersonic flows)
	47.20.Lz	(Secondary instabilities)

Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11332007 and 11402167).

Corresponding Authors: Jian-xin Liu
E-mail: shookware@tju.edu.cn

Cite this article:

Lei Zhao(赵磊), Cun-bo Zhang(张存波), Jian-xin Liu(刘建新), Ji-sheng Luo(罗纪生) Improved algorithm for solving nonlinear parabolized stability equations 2016 Chin. Phys. B 25 084701

[1]	Mack L M 1984 "Spec. Course Stab. Transit. Laminar Flows", AGARD Rep. 709, pp. 3.1-81
[2]	Reed H L, Saric W S and Arnal D 1996 Ann. Rev. Fluid Mech. 28 389
[3]	Bertolotti F P 1991 "Linear and nonlinear stability of boundary layers with streamwise varying properties", Ph. D. Dissertation (Columbus:Ohio State University)
[4]	Bertolotti F P, Herbert T and Spalart P R 1992 J. Fluid Mech. 242 441
[5]	Herbert T 1997 Ann. Rev. Fluid Mech. 29 245
[6]	Wang M J, Herbert T and Stuckert G 1994 m25th AIAA Fluid Dynamics Conference, 1994, Colorado Springs, USA
[7]	Haynes T S and Reed H L 2000 J. Fluid Mech. 405 325
[8]	Reibert M S, Saric W S, Carrillo R B and Chapman K L 1996 34th AIAA Aerospace Sciences Meeting and Exhibit, January 15-18, 1996, Reno, NV, USA, p. 184
[9]	Malik M R, Li F, Choudhari M M and Chang C L 1999 J. Fluid Mech. 399 85
[10]	Xu G L, Xiao Z X and Fu S 2011 Sci. China-Phys. Mech. Astron. 54 724
[11]	Xu G L, Xiao Z X and Fu S 2011 Sci. China-Phys. Mech. Astron. 54 2040
[12]	Li F and Choudhari M M 2011 Theor. Comput. Fluid Dyn. 25 65
[13]	Li F, Choudhari M M, Duan L and Chang C L 1994 Phys. Fluids 26 064104
[14]	Zhang Y M and Zhou H 2008 Appl. Math. Mech.-Engl. Ed. 29 833
[15]	Li N 2007 "Investigation on transition of laminar-turbulent in a boundary layer on a flat plate based on the spatial model", Ph. D. Dissertation (Tianjin:Tianjin University) (in Chinese)
[16]	Mayer C S J, Fasel H F, Choudhari M M and Chang C L 2014 AIAA J. 52 882
[17]	Mayer C S J, Dominic A and Fasel H F 2011 J. Fluid Mech. 674 5
[18]	Zhang Y M and Zhou H 2008 Appl. Math. Mech.-Engl. Ed. 29 1
[19]	Chang C L 2004 NASA Langley Research Center, NASA TM 213233
[20]	Ren J and Fu S 2015 J. Fluid Mech. 781 388
[21]	Choudhari M M, Chang C L, Streett C and Balakumar P 2003 AIAA m41st Aerospace Sciences Meeting and Exhibit, January 6-9, 2003, Reno, Nevada, USA, p. 973
[22]	Kuehl J, Perez E and Reed H L 2012 m50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, January 9-12, 2012, Nashville, Tennessee, USA, p. 921
[23]	Dong M and Luo J S 2007 Appl. Math. Mech.-Engl. Ed. 28 1019
[24]	Cao W and Huang Z F 2006 Appl. Math. Mech.-Engl. Ed. 27 425
[25]	Yu M and Luo J S 2014 Sci. China-Phys. Mech. Astron. 57 2141

[1]	Improved nonlinear parabolized stability equations approach for hypersonic boundary layers Shaoxian Ma(马绍贤), Yi Duan(段毅), Zhangfeng Huang(黄章峰), and Shiyong Yao(姚世勇). Chin. Phys. B, 2021, 30(5): 054701.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Improved algorithm for solving nonlinear parabolized stability equations

Cite this article:

Online attention