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Spatial–temporal transformation for primary and secondary instabilities in weakly non-parallel shear flows

Published online by Cambridge University Press:  17 March 2023

Jiakuan Xu Open the ORCID record for Jiakuan Xu [Opens in a new window] ,
Jianxin Liu Open the ORCID record for Jianxin Liu [Opens in a new window] ,
Zhongyu Zhang Open the ORCID record for Zhongyu Zhang [Opens in a new window]  and
Xuesong Wu Open the ORCID record for Xuesong Wu [Opens in a new window]
Jiakuan Xu
Affiliation:
School of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, PR China
Jianxin Liu*
Affiliation:
Laboratory for High-Speed Aerodynamics, School of Mechanical Engineering, Tianjin University, Tianjin 300072, PR China
Zhongyu Zhang
Affiliation:
Laboratory for High-Speed Aerodynamics, School of Mechanical Engineering, Tianjin University, Tianjin 300072, PR China
Xuesong Wu*
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK
*
†Email addresses for correspondence: shookware@tju.edu.cn, x.wu@ic.ac.uk
†Email addresses for correspondence: shookware@tju.edu.cn, x.wu@ic.ac.uk
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Abstract

When studying instability of weakly non-parallel flows, it is often desirable to convert temporal growth rates of unstable modes, which can readily be computed, to physically more relevant spatial growth rates. This has been performed using the well-known Gaster's transformation for primary instability and Herbert's transformation for the secondary instability of a saturated primary mode. The issue of temporal–spatial transformation is revisited in the present paper to clarify/rectify the ambiguity/misunderstanding that appears to exist in the literature. A temporal mode and its spatial counterpart may be related by sharing either the real frequency or wavenumber, and the respective transformations between their growth rates are obtained by a simpler consistent derivation than the original one. These transformations, which consist of first- and second-order versions, are valid under conditions less restrictive than those for Gaster's and Herbert's transformations, and reduce to the latter under additional conditions, which are not always satisfied in practice. The transformations are applied to inviscid Rayleigh instability of a mixing layer and a jet, secondary instability of a streaky flow as well as general detuned secondary instability (including subharmonic and fundamental resonances) of primary Mack modes in a supersonic boundary layer. Comparison of the transformed growth rates with the directly calculated spatial growth rates shows that the transformations derived in this paper outperform Gaster's and Herbert's transformations consistently. The first-order transformation is accurate when the growth rates are small or moderate, while the second-order transformations are sufficiently accurate across the entire instability bands, and thus stand as a useful tool for obtaining spatial instability characteristics via temporal stability analysis.

JFM classification

Boundary Layers: Boundary layer stability Instability: Absolute/convective instability Instability: Shear-flow instability

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JFM Papers
Information
Journal of Fluid Mechanics , Volume 959 , 25 March 2023 , A21
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© The Author(s), 2023. Published by Cambridge University Press

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References

REFERENCES

Bertolotti, F.P. 1985 Temporal and spatial growth of subharmonic disturbances in Falkner–Skan flows. MSc thesis, Virginia Polytechnic Institute and State University, VA.Google Scholar
Brevdo, L. 1992 A note on the Gaster transformation. Z. Angew. Math. Mech. 72 (7), 305–306.CrossRefGoogle Scholar
Brevdo, L. & Bridges, T.J. 1996 Absolute and convective instabilities of spatially periodic flows. Phil. Trans. R. Soc. Lond. A 354 (1710), 1027–1064.Google Scholar
Drazin, P.G. & Howard, L.N. 1962 The instability to long waves of unbounded parallel inviscid flow. J. Fluid Mech. 14 (2), 257–283.CrossRefGoogle Scholar
Gaster, M. 1962 A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability. J. Fluid Mech. 14 (2), 222–224.CrossRefGoogle Scholar
Herbert, T. 1983 Secondary instability of plane channel flow to subharmonic three-dimensional disturbances. Phys. Fluids 26 (4), 871–874.CrossRefGoogle Scholar
Herbert, T. 1984 Analysis of the subharmonic route to transition in boundary layers. AIAA Paper 1984-0009.CrossRefGoogle Scholar
Herbert, T. 1988 Secondary instability of boundary layers. Annu. Rev. Fluid Mech. 20 (1), 487–526.CrossRefGoogle Scholar
Herbert, T., Bertolotti, F.P. & Santos, G.R. 1987 Floquet analysis of secondary instability in shear flows. In Stability of Time Dependent and Spatially Varying Flows (ed. D.L. Dwoyer & M.Y. Hussaini), pp. 43–57. Springer.CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P.A. 1985 Absolute and convective instabilities in free shear layers. J. Fluid Mech. 159, 151–168.CrossRefGoogle Scholar
Kachanov, Y.S. & Levchenko, V.Y. 1984 The resonant interaction of disturbances at laminar-turbulent transition in a boundary layer. J. Fluid Mech. 138, 209–247.CrossRefGoogle Scholar
Koch, W., Bertolotti, F.P., Stolte, A. & Hein, S. 2000 Nonlinear equilibrium solutions in a three-dimensional boundary layer and their secondary instability. J. Fluid Mech. 406, 131–174.CrossRefGoogle Scholar
Lehoucq, R.B., Sorensen, D.C. & Yang, C. 1998 Computational routines. In ARPACK Users’ Guide – Solution of Large-scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, chap. 5, pp. 67–77. DBLP. https://doi.org/10.1137/1.9780898719628.CrossRefGoogle Scholar
Li, F. & Choudhari, M.M. 2011 Spatially developing secondary instabilities in compressible swept airfoil boundary layers. Theor. Comput. Fluid Dyn. 25 (1), 65–84.CrossRefGoogle Scholar
Li, F., Choudhari, M., Chang, C.-L. & White, J. 2012 Secondary instability of second mode disturbances in hypersonic boundary layers. NASA Tech. Rep. NF1676L-13407.Google Scholar
Li, F. & Malik, M.R. 1995 Fundamental and subharmonic secondary instabilities of Görtler vortices. J. Fluid Mech. 297, 77–100.CrossRefGoogle Scholar
Malik, M.R., Li, F., Choudhari, M.M. & Chang, C.-L. 1999 Secondary instability of crossflow vortices and swept-wing boundary-layer transition. J. Fluid Mech. 399, 85–115.CrossRefGoogle Scholar
Michalke, A. 1965 On spatially growing disturbances in an inviscid shear layer. J. Fluid Mech. 23 (3), 521–544.CrossRefGoogle Scholar
Náraigh, L.O. & Spelt, P.D. 2013 An analytical connection between temporal and spatio-temporal growth rates in linear stability analysis. Proc. R. Soc. A 469, 20130171.CrossRefGoogle Scholar
Nayfeh, A.H. & Padhye, A. 1979 Relation between temporal and spatial stability in three-dimensional flows. AIAA J. 17 (10), 1084–1090.CrossRefGoogle Scholar
Ng, L.L. & Erlebacher, G. 1992 Secondary instabilities in compressible boundary layers. Phys. Fluids A 4 (4), 710–726.CrossRefGoogle Scholar
Peng, M. & Williams, R. 1987 On the transformations between temporal and spatial growth rates. J. Atmos. Sci. 44 (18), 2668–2673.2.0.CO;2>CrossRefGoogle Scholar
Ricco, P., Luo, J. & Wu, X. 2011 Evolution and instability of unsteady nonlinear streaks generated by free-stream vortical disturbances. J. Fluid Mech. 677, 1–38.CrossRefGoogle Scholar
Roychowdhury, A. & Sreedhar, B. 1992 Gaster's transform. AIAA J. 30 (11), 2776–2778.CrossRefGoogle Scholar
Swearingen, J. & Blackwelder, R. 1987 The growth and breakdown of streamwise vortices in the presence of a wall. J. Fluid Mech. 182, 255–290.CrossRefGoogle Scholar
Wu, X. & Stewart, P. 1996 Interaction of phase-locked modes: a new mechanism for the rapid growth of three-dimensional disturbances. J. Fluid Mech. 316, 335–372.CrossRefGoogle Scholar
Wu, X., Stewart, P. & Cowley, S. 1996 On the weakly nonlinear development of Tollmien–Schlichting wavetrains in boundary layers. J. Fluid Mech. 323, 133–171.CrossRefGoogle Scholar
Wu, X. & Tian, F. 2012 Spectral broadening and flow randomization in free shear layers. J. Fluid Mech. 706, 431–469.CrossRefGoogle Scholar
Xu, J. & Liu, J. 2022 Wall-cooling effects on secondary instabilities of Mack mode disturbances at Mach 6. Phys. Fluids 34 (4), 044105.CrossRefGoogle Scholar
Xu, J., Liu, J., Mughal, S., Yu, P. & Bai, J. 2020 Secondary instability of Mack mode disturbances in hypersonic boundary layers over micro-porous surface. Phys. Fluids 32 (4), 044105.CrossRefGoogle Scholar
Xu, D., Zhang, Y. & Wu, X. 2017 Nonlinear evolution and secondary instability of steady and unsteady Görtler vortices induced by free-stream vortical disturbances. J. Fluid Mech. 829, 681–730.CrossRefGoogle Scholar