Geodesic
Computation of evolution of Tollmein-Schlichten waves based on the global stability analysis
Matematičeskoe modelirovanie, Tome 35 (2023) no. 9, pp. 45-60.

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A computational methodology is presented for calculating spatial evolution of Tollmien-Schlichting waves and their amplitude growth-rate factor in substantially non-parallel compressible flows, based on a global stability analysis of stationary solutions of the full Navier-Stokes equations. Three stages of this methodology (obtaining a stationary solution, carrying out a global stability analysis and post-processing) are described, and results are presented of its validation based on the comparison of the results of calculating the characteristics of Tollmien-Schlichting waves on a flat plate with the corresponding results of the classical stability theory in the parallel approximation. An example of calculating the flow around a plate with a rectangular cavity is presented to illustrate the possibility of applying the proposed methodology to non-parallel flows.
Keywords: Tollmien-Schlichting waves, non-parallel flows, global stability analysis, full Navier-Stokes equations, amplification factor
Mots-clés : laminar-turbulent transition. 
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K. Belyaev; A. Garbaruk; V. Golubkov; M. Strelets. Computation of evolution of Tollmein-Schlichten waves based on the global stability analysis. Matematičeskoe modelirovanie, Tome 35 (2023) no. 9, pp. 45-60. http://geodesic.mathdoc.fr/item/MM_2023_35_9_a3/

[1] A. V. Boiko, A. V. Dovgal, G. R. Grek, V. V. Kozlov, Physics of Transitional Shear Flows: Instability and Laminar-Turbulent Transition in Incompressible Near-Wall Shear Layers, Springer, 2012, 300 pp. (ASIN: 9400724977) | MR

[2] P. J. Schmid, D. S. Henningson, Stability and Transition in Shear Flows, Applied Math. Sci., 142, Springer, 2011, 556 pp. | DOI | MR

[3] W. Tollmien, “Über die Entstehung der Turbulenz”, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1929, 21–44

[4] G. B. Schubauer, H. K. Skramstad, “Laminar Boundary Layer Oscillations and Stability of Laminar Flow”, J. Aeronautical Science, 14:2 (1947), 69 | DOI

[5] V. Theofilis, “Global Linear Instability”, Annual Review of Fluid Mechanics, 43 (2011), 319–352 | DOI | MR | Zbl

[6] U. Ehrenstein, F. F. Gallaire, “On two-dimensional temporal modes in spatially evolving open flows: The flat-plate boundary layer”, J. of Fluid Mechanic, 536 (2005), 209–218 | DOI | MR | Zbl

[7] E. Akervik, U. Ehrenstein, F. F. Gallaire, D. S. Henningson, “Global two-dimensional stabil-ity measures of the flat plate boundary-layer flow”, European J. of Mechanics B/Fluids, 7:5 (2008), 501–513 | DOI | MR

[8] R. Bhoraniya, V. Narayanan, “Global stability analysis of spatially developing boundary layer: Effect of streamwise pressure gradients”, Fluid Dynamics, 54:6 (2019), 821–834 | DOI | MR | Zbl

[9] R. Bhoraniya, V. Narayanan, “Global stability analysis of the axisymmetric boundary layer: Effect of axisymmetric forebody shapes on the helical global modes”, Pramana Journal of Physics, 95:19 (2021), 11 | DOI

[10] J. Garicano-Mena, E. Ferrer, S. Sanvido, E. Valero, “A stability analysis of the compressible boundary layer flow over indented surfaces”, Computers and Fluids, 160 (2018), 14–25 | DOI | MR | Zbl

[11] M. S. Mathias, M. A.F. Medeiros, Global instability analysis of a boundary layer flow over a small cavity, AIAA Paper 2019–3535 | DOI

[12] M. S. Kuester, Growth of Tollmien-Schlichting waves over three-dimensional roughness, AIAA Paper 2020–1580 | DOI

[13] M. Shur, M. Strelets, A. Travin, “High-Order Implicit Multi-Block Navier-Stokes Code: Ten-Year Experience of Application to RANS/DES/LES/DNS of Turbulence”, Invited lecture, 7th Symposium on Overset Composite Grids and Solution Technology (October 2004, Huntington Beach, USA), 5–7 https://cfd.spbstu.ru/agarbaruk/doc/NTS_code.pdf

[14] P. L. Roe, “Approximate Riemann solvers, parameter vectors, and difference schemes”, J. of Comp. Phys., 43:2 (1981), 357–372 | DOI | MR | Zbl

[15] J. D. Crouch, A. V. Garbaruk, D. Magidov, “Predicting the onset of flow unsteadiness based on global instability”, J. of Comp. Phys., 224:2 (2007), 924–940 | DOI | MR | Zbl

[16] J.D Crouc, A. Garbaruk, M. Strelets, “Global instability in the onset of transonic-wing buf-fet”, Journal of Fluid Mechanics, 881 (2019), 3–22 | DOI | MR | Zbl

[17] S. Timme, Global instability of wing shock buffet, 2108, arXiv: 1806.07299 [physics.flu-dyn] | DOI | MR

[18] S. Timme, Global shock buffet instability on NASA common research model, AIAA Paper No 2019-0037 | DOI | MR

[19] R. B. Lehoucq, D. C. Sorensen, C. Yang, ARPACK Users' Guide: Solution of Large-scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, 1988 | DOI | MR

[20] D. Golub, Ch. Van Loun, Matrix Computations, 4th ed, The John Hopkins University Press, Baltimore, 1989, 756 pp. | MR | Zbl

[21] W. Yang, I. Zurbenko, “Kolmogorov-Zurbenko filters”, Wiley Interdisciplinary Reviews: Computational Statistics, 2:3 (2010), 340–351 | DOI | MR

[22] J. D. Crouch, V. S. Kosorygin, “Surface step effects on boundary-layer transition dominated by Tollmien-Schlichting instability”, AIAA Journal, 58:7 (2020), 2943–2950 | DOI

[23] Y. X. Wang, M. Gaster, “Effect of surface steps on boundary layer transition”, Experiments in Fluids, 39 (2005), 679–686 | DOI