Abstract
Direct numerical simulations are carried out with different disturbance forms introduced into the inlet of a flat plate boundary layer with the Mach number 4.5. According to the biorthogonal eigenfunction system of the linearized Navier-Stokes equations and the adjoint equations, the decomposition of the direct numerical simulation results into the discrete normal mode is easily realized. The decomposition coefficients can be solved by doing the inner product between the numerical results and the eigenfunctions of the adjoint equations. For the quadratic polynomial eigenvalue problem, the inner product operator is given in a simple form, and it is extended to an Nth-degree polynomial eigenvalue problem. The examples illustrate that the simplified mode decomposition is available to analyze direct numerical simulation results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Fedorov, A. Transition and stability of high-speed boundary layers. Annual Review of Fluid Mechanics, 43, 79–95 (2010)
Ma, Y. B. and Zhong, X. L. Receptivity of a supersonic boundary layer over a flat plate, part 2: receptivity to free-stream sound. Journal of Fluid Mechanics, 488, 79–121 (2003)
Salwen, H. and Grosch, C. E. The continuous spectrum of the Orr-Sommerfeld equation, part 2: eigenfunction expansions. Journal of Fluid Mechanics, 104, 445–465 (1981)
Balakumar, P. and Malik, M. R. Discrete modes and continuous spectra in supersonic boundary layers. Journal of Fluid Mechanics, 239, 631–656 (1992)
Tumin, A. The biorthogonal eigenfunction system of linear stability equations: a survey of applications to receptivity problems and to analysis of experimental and computational results. 41st AIAA Fluid Dynamics Conference and Exhibit, American Institute of Aeronautics and Astronautics, America (2011)
Fedorov, A. and Tumin, A. Initial-value problem for hypersonic boundary layer flows. AIAA Journal, 41(3), 379–389 (2003)
Tumin, A. Multimode decomposition of spatially growing perturbations in a two-dimensional boundary layer. Physics of Fluids, 15, 2525–2540 (2003)
Tumin, A. Three-dimensional spatial normal modes in compressible boundary layers. Journal of Fluid Mechanics, 586, 295–322 (2007)
Chiquete, C. and Tumin, A. Biorthogonal eigenfunction system for supersonic inviscid flow past a flat plate. 37th AIAA Fluid Dynamics Conference and Exhibit, American Institute of Aeronautics and Astronautics, America (2007)
Tumin, A., Wang, X. W., and Zhong, X. L. Direct numerical simulation and the theory of receptivity in a hypersonic boundary layer. Physics of Fluids, 19, 014101 (2007)
Zhou, H. and Li, L. The eigenvalue problem and expansion theorems associated with Orr-Sommerfeld equation. Applied Mathematics and Mechanics (English Edition), 2(3), 319–330 (1981) DOI 10.1007/BF01877398
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by the National Natural Science Foundation of China (Nos. 11332007, 11202147, and 9216111), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20120032120007), and the Open Fund from State Key Laboratory of Aerodynamics (Nos. SKLA201201 and SKLA201301)
Rights and permissions
About this article
Cite this article
Gao, J., Luo, Js. Mode decomposition of nonlinear eigenvalue problems and application in flow stability. Appl. Math. Mech.-Engl. Ed. 35, 667–674 (2014). https://doi.org/10.1007/s10483-014-1820-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-014-1820-6