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Effects of phase difference between instability modes on boundary-layer transition

Published online by Cambridge University Press:  24 September 2021

Minwoo Kim ,
Seungtae Kim ,
Jiseop Lim ,
Ray-Sing Lin ,
Solkeun Jee Open the ORCID record for Solkeun Jee [Opens in a new window]  and
Donghun Park
Minwoo Kim
Affiliation:
School of Mechanical Engineering, Gwangju Institute of Science and Technology, Gwangju 61005, Korea
Seungtae Kim
Affiliation:
School of Mechanical Engineering, Gwangju Institute of Science and Technology, Gwangju 61005, Korea
Jiseop Lim
Affiliation:
School of Mechanical Engineering, Gwangju Institute of Science and Technology, Gwangju 61005, Korea
Ray-Sing Lin
Affiliation:
School of Mechanical Engineering, Gwangju Institute of Science and Technology, Gwangju 61005, Korea
Solkeun Jee*
Affiliation:
School of Mechanical Engineering, Gwangju Institute of Science and Technology, Gwangju 61005, Korea
Donghun Park
Affiliation:
Department of Aerospace Engineering, Pusan National University, Busan 46241, Korea
*
†Email address for correspondence: [email protected]
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Abstract

Phase effect on the modal interaction of flow instabilities is investigated for laminar-to-turbulent transition in a flat-plate boundary-layer flow. Primary and secondary three-dimensional (3-D) oblique waves at various initial phase differences between these two instability modes. Three numerical methods are used for a systematic approach for the entire transition process, i.e. before the onset of transition well into fully turbulent flow. Floquet analysis predicts the subharmonic resonance where a subharmonic mode locally resonates for a given basic flow composed of the steady laminar flow and the fundamental mode. Because Floquet analysis is limited to the resonating subharmonic mode, nonlinear parabolised stability equation analysis (PSE) is conducted with various phase shifts of the subharmonic mode with respect to the given fundamental mode. The application of PSE offers insights on the modal interaction affected by the phase difference up to the weakly nonlinear stage of transition. Large-eddy simulation (LES) is conducted for a complete transition to turbulent boundary layer because PSE becomes prohibitively expensive in the late nonlinear stage of transition. The modulation of the subharmonic resonance with the initial phase difference leads to a significant delay in the transition location up to $\Delta Re_{x, tr} \simeq 4\times 10^5$ as predicted by the current LES. Effects of the initial phase difference on the spatial evolution of the modal shape of the subharmonic mode are further investigated. The mechanism of the phase evolution is discussed, based on current numerical results and relevant literature data.

JFM classification

Boundary Layers: Boundary layer stability Instability: Nonlinear instability Instability: Transition to turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 927 , 25 November 2021 , A14
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

‡

M. Kim and S. Kim contributed equally to this work.

References

REFERENCES

Bertolotti, F.P., Herbert, T. & Spalart, P.R. 1992 Linear and nonlinear stability of the Blasius boundary layer. J. Fluid Mech. 242, 441–474.CrossRefGoogle Scholar
Borodulin, V.I., Kachanov, Y.S. & Koptsev, D.B. 2002 Experimental study of resonant interactions of instability waves in a self-similar boundary layer with an adverse pressure gradient: I. Tuned resonances. J. Turbul. 3, N62.CrossRefGoogle Scholar
Borodulin, V.I., Kachanov, Y.S. & Roschektayev, A.P. 2011 Experimental detection of deterministic turbulence. J. Turbul. 12, N23.CrossRefGoogle Scholar
Chang, C.-L., Malik, M.R., Erlebacher, G. & Hussaini, M.Y. 1993 Linear and nonlinear PSE for compressible boundary layers. Technical Report NASA-CR-191537.Google Scholar
Corke, T.C. & Mangano, R.A. 1989 Resonant growth of three-dimensional modes in trnsitioning Blasius boundary layers. J. Fluid Mech. 209, 93–150.CrossRefGoogle Scholar
Craik, A.D.D. 1971 Non-linear resonant instability in boundary layers. J. Fluid Mech. 50 (2), 393–413.CrossRefGoogle Scholar
Durbin, P. & Wu, X. 2007 Transition beneath vortical disturbances. Annu. Rev. Fluid Mech. 39 (1), 107–128.CrossRefGoogle Scholar
El-Hady, N.M. 1988 Secondary subharmonic instability of boundary layers with pressure gradient and suction. Technical Report NASA-CR-4112.Google Scholar
Gao, B., Park, D. & Park, S.O. 2011 Stability analysis of a boundary layer over a hump using parabolized stability equations. Fluid Dyn. Res. 43 (5), 055503.CrossRefGoogle Scholar
Herbert, T. 1984 Analysis of the subharmonic route to transition in boundary layers. In AIAA 22nd Aerospace Sciences Meeting, January 09–12, Reno, NV. https://arc.aiaa.org/doi/pdf/10.2514/6.1984-9.Google 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
Higham, D.J. & Higham, N.J. 2016 MATLAB Guide, 3rd edn. SIAM. https://epubs.siam.org/doi/pdf/10.1137/1.9781611974669.ch9.Google Scholar
Issa, R.I. 1986 Solution of the implicitly discretised fluid flow equations by operator-splitting. J. Comput. Phys. 62 (1), 40–65.CrossRefGoogle Scholar
Jee, S., Joo, J. & Lin, R.-S. 2018 Toward cost-effective boundary layer transition computations with large-eddy simulation. Trans. ASME J. Fluids Engng 140 (11), 111201.CrossRefGoogle Scholar
Joslin, R.D., Streett, C.L. & Chang, C.-L. 1993 Spatial direct numerical simulation of boundary-layer transition mechanisms: validation of PSE theory. Theor. Comput. Fluid Dyn. 4 (6), 271–288.CrossRefGoogle Scholar
Kachanov, Y.S. 1994 Physical mechanisms of laminar-boundary-layer transition. Annu. Rev. Fluid Mech. 26 (1), 411–482.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
Kim, S. 2020 Effect of phase difference between instability modes on boundary layer transition. Master's thesis, Gwangju Institute of Science and Technology, Gwangju, Korea.Google Scholar
Kim, M., Lim, J., Kim, S., Jee, S. & Park, D. 2020 Assessment of the wall-adapting local eddy-viscosity model in transitional boundary layer. Comput. Meth. Appl. Mech. Engng 371, 113287.CrossRefGoogle Scholar
Kim, M., Lim, J., Kim, S., Jee, S., Park, J. & Park, D. 2019 Large-eddy simulation with parabolized stability equations for turbulent transition using OpenFOAM. Comput. Fluids 189, 108–117.CrossRefGoogle Scholar
Klebanoff, P.S., Tidstrom, K.D. & Sargent, L.M. 1962 The three-dimensional nature of boundary-layer instability. J. Fluid Mech. 12 (1), 1–34.CrossRefGoogle Scholar
Kovacic, I., Rand, R. & Sah, S.M. 2018 Mathieu's equation and its generalizations: overview of stability charts and their features. Appl. Mech. Rev. 70 (2), 020802.CrossRefGoogle Scholar
Lim, J., Kim, M., Kim, S., Jee, S. & Park, D. 2021 Cost-effective and high-fidelity method for turbulent transition in compressible boundary layer. Aerosp. Sci. Technol. 108, 106367.CrossRefGoogle Scholar
Morkovin, M.V. 1969 On the many faces of transition. In Viscous Drag Reduction (ed. C. Sinclair Wells), pp. 1–31. Springer.CrossRefGoogle Scholar
Nayfeh, A.H. & Masad, J.A. 1990 Recent advances in secondary instabilities in boundary layers. Comput. Syst. Engng 1 (2), 401–414.CrossRefGoogle Scholar
Nicoud, F. & Ducros, F. 1999 Subgrid-scale stress modelling based on the square of the velocity gradient tensor. Flow Turbul. Combust. 62 (3), 183–200.CrossRefGoogle Scholar
Park, D. & Park, S.O. 2013 Linear and non-linear stability analysis of incompressible boundary layer over a two-dimensional hump. Comput. Fluids 73, 80–96.CrossRefGoogle Scholar
Park, D. & Park, S.O. 2016 Study of effect of a smooth hump on hypersonic boundary layer instability. Theor. Comput. Fluid Dyn. 30 (6), 543–563.CrossRefGoogle Scholar
Park, D., Park, J., Kim, M., Lim, J., Kim, S. & Jee, S. 2021 Influence of initial phase on subharmonic resonance in an incompressible boundary layer. Phys. Fluids 33 (4), 044101.CrossRefGoogle Scholar
Saric, W.S., Reed, H.L. & White, E.B. 2003 Stability and transition of three-dimensional boundary layers. Annu. Rev. Fluid Mech. 35 (1), 413–440.CrossRefGoogle Scholar
Sayadi, T., Hamman, C.W. & Moin, P. 2013 Direct numerical simulation of complete H-type and K-type transitions with implications for the dynamics of turbulent boundary layers. J. Fluid Mech. 724, 480–509.CrossRefGoogle Scholar
Schmid, P.J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39 (1), 129–162.CrossRefGoogle Scholar
Schmid, P.J. & Henningson, D.S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Würz, W., Sartorius, D., Kloker, M., Borodulin, V.I., Kachanov, Y.S. & Smorodsky, B.V. 2012 a Nonlinear instabilities of a non-self-similar boundary layer on an airfoil: experiments, DNS, and theory. Eur. J. Mech. B/Fluids 31, 102–128.CrossRefGoogle Scholar
Würz, W., Sartorius, D., Kloker, M., Borodulin, V.I., Kachanov, Y.S. & Smorodsky, B.V. 2012 b Detuned resonances of Tollmien–Schlichting waves in an airfoil boundary layer: experiment, theory, and direct numerical simulation. Phys. Fluids 24 (9), 094103.CrossRefGoogle Scholar
Wu, X. 2019 Nonlinear theories for shear flow instabilities: physical insights and practical implications. Annu. Rev. Fluid Mech. 51 (1), 451–485.CrossRefGoogle Scholar
Xu, H., Lombard, J.W. & Sherwin, S.J. 2017 Influence of localised smooth steps on the instability of a boundary layer. J. Fluid Mech. 817, 138–170.CrossRefGoogle Scholar