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The bypass transition mechanism of the Stokes boundary layer in the intermittently turbulent regime

Published online by Cambridge University Press:  27 May 2020

Chengwang Xiong Open the ORCID record for Chengwang Xiong [Opens in a new window] ,
Xiang Qi ,
Ankang Gao Open the ORCID record for Ankang Gao [Opens in a new window] ,
Hui Xu Open the ORCID record for Hui Xu [Opens in a new window] ,
Chengjiao Ren Open the ORCID record for Chengjiao Ren [Opens in a new window]  and
Liang Cheng
Chengwang Xiong
Affiliation:
College of Shipbuilding Engineering, Harbin Engineering University, 150001Harbin, PR China School of Civil Engineering, Hebei University of Technology, 300401Tianjin, PR China
Xiang Qi
Affiliation:
School of Civil Engineering, Hebei University of Technology, 300401Tianjin, PR China
Ankang Gao
Affiliation:
State Key Laboratory for Turbulence and Complex Systems (LTCS), Peking University, 100871Beijing, PR China
Hui Xu
Affiliation:
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai, PR China
Chengjiao Ren
Affiliation:
State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, 116024Dalian, PR China
Liang Cheng*
Affiliation:
State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, 116024Dalian, PR China School of Civil, Environmental and Mining Engineering, University of Western Australia, 35 Stirling Highway, Crawley, WA6009, Australia
*
†Email address for correspondence: [email protected]
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Abstract

This numerical study focuses on the coherent structures and bypass transition mechanism of the Stokes boundary layer in the intermittently turbulent regime. In particular, the initial disturbance is produced by a temporary roughness element that is removed immediately after triggering a two-dimensional vortex tube under an inflection-point instability. The present study reveals a complete scenario of self-induced motion of a vortex tube after rollup from the boundary layer. The trajectory of the vortex tube is reasonably described based on the Helmholtz point-vortex equation. The three-dimensional transition of the vortex tube is attributed to the Crow instability, which leads to a sinusoidal disturbance that eventually evolves into a ring-like structure, especially for the weaker vortex. Further investigation demonstrates that three-dimensional or quasi-three-dimensional vortex perturbations in the free stream play a critical role in the boundary layer transition through a bypass mechanism, which is featured by the non-modal and explosive transient growth of the subsequent boundary layer instabilities. This transition scenario is found to be analogous to the oblique transition in the steady boundary layer, both of which are characterised by the formation of streaks, rollup of hairpin-like vortices and burst into turbulent spots. In addition, the streamwise propagation of turbulent spots is discussed in detail. To shed more light on the nature of the intermittently turbulent Stokes boundary layer, a conceptual model is proposed for the periodically self-sustaining mechanism of the turbulent spots based on the present numerical results and experimental evidence reported in the literature.

JFM classification

Boundary Layers: Boundary layer stability Instability: Transition to turbulence Turbulent Flows: Shear layer turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 896 , 10 August 2020 , A4
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Akhavan, R., Kamm, R. D. & Shapiro, A. H. 1991a An investigation of transition to turbulence in bounded oscillatory Stokes flows. Part 1. Experiments. J. Fluid Mech. 225, 395–422.CrossRefGoogle Scholar
Akhavan, R., Kamm, R. D. & Shapiro, A. H. 1991b An investigation of transition to turbulence in bounded oscillatory Stokes flows. Part 2. Numerical simulations. J. Fluid Mech. 225, 423–444.CrossRefGoogle Scholar
Berlin, S.1998 Oblique waves in boundary layer transition. PhD thesis, Royal Institute of Technology, Stockholm.Google Scholar
Berlin, S. & Henningson, D. S. 1999 A nonlinear mechanism for receptivity of free-stream disturbances. Phys. Fluids 11 (12), 3749–3760.CrossRefGoogle Scholar
Berlin, S., Wiegel, M. & Henningson, D. S. 1999 Numerical and experimental investigations of oblique boundary layer transition. J. Fluid Mech. 393, 23–57.CrossRefGoogle Scholar
Biau, D. 2016 Transient growth of perturbations in stokes oscillatory flows. J. Fluid Mech. 794, R4.CrossRefGoogle Scholar
Blackburn, H. M. & Henderson, R. D. 1999 A study of two-dimensional flow past an oscillating cylinder. J. Fluid Mech. 385, 255–286.CrossRefGoogle Scholar
Blennerhassett, P. J. & Bassom, A. P. 2002 The linear stability of flat Stokes layers. J. Fluid Mech. 464, 393–410.CrossRefGoogle Scholar
Blennerhassett, P. J. & Bassom, A. P. 2006 The linear stability of high-frequency oscillatory flow in a channel. J. Fluid Mech. 556, 1–25.CrossRefGoogle Scholar
Blondeaux, P. & Vittori, G. 1994 Wall imperfections as a triggering mechanism for Stokes-layer transition. J. Fluid Mech. 264 (264), 107–135.CrossRefGoogle Scholar
Bolis, A.2013 Fourier spectral/$hp$ element method: investigation of time-stepping and parallelisation strategies. PhD thesis, Imperial College London.Google Scholar
Brandt, L., Schlatter, P. & Dan, S. H. 2004 Transition in boundary layers subject to free-stream turbulence. J. Fluid Mech. 517 (517), 167–198.CrossRefGoogle Scholar
Bristol, R. L., Ortega, J. M., Marcus, P. S. & Savas, O. 2004 On cooperative instabilities of parallel vortex pairs. J. Fluid Mech. 517, 331–358.CrossRefGoogle Scholar
Cantwell, C. D., Moxey, D., Comerford, A., Bolis, A., Rocco, G., Mengaldo, G., De Grazia, D., Yakovlev, S., Lombard, J.-E., Ekelschot, D. et al. 2015 Nektar + +: an open-source spectral/hp element framework. Comput. Phys. Commun. 192, 205–219.CrossRefGoogle Scholar
Carstensen, S., Sumer, B. M. & Fredsøe, J. 2010 Coherent structures in wave boundary layers. Part 1. Oscillatory motion. J. Fluid Mech. 646, 169–206.CrossRefGoogle Scholar
Cook, A. W. & Cabot, W. H. 2004 A high-wavenumber viscosity for high-resolution numerical methods. J. Comput. Phys. 195 (2), 594–601.CrossRefGoogle Scholar
Costamagna, P., Vittori, G. & Blondeaux, P. 2003 Coherent structures in oscillatory boundary layers. J. Fluid Mech. 474, 1–33.CrossRefGoogle Scholar
Crow, S. C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8 (12), 2172–2179.CrossRefGoogle Scholar
Dipankar, A., Sengupta, T. K. & Talla, S. B. 2007 Suppression of vortex shedding behind a circular cylinder by another control cylinder at low Reynolds numbers. J. Fluid Mech. 573, 171–190.CrossRefGoogle Scholar
Duguet, Y., Schlatter, P., Henningson, D. S. & Eckhardt, B. 2012 Self-sustained localized structures in a boundary-layer flow. Phys. Rev. Lett. 108 (4), 1–4.CrossRefGoogle Scholar
Hall, P. 1978 The linear stability of flat Stokes layers. Proc. R. Soc. Lond. A 359, 151–166.Google Scholar
Hino, M., Sawamoto, M. & Takasu, S. 1976 Experiments on transition to turbulence in an oscillatory pipe flow. J. Fluid Mech. 75 (2), 193–207.CrossRefGoogle Scholar
Jensen, B. L., Sumer, B. M. & Fredsoe, J. 1989 Turbulent oscillatory boundary layers at high Reynolds numbers. J. Fluid Mech. 206, 265–297.CrossRefGoogle Scholar
Jeong, J. & Hussain, F. 1992 On the identification of a vortex. J. Fluid Mech. 285, 69–94.CrossRefGoogle Scholar
Jiménez, J. 2018 Coherent structures in wall-bounded turbulence. J. Fluid Mech. 842, P1.CrossRefGoogle Scholar
Johnson, A. A. & Tezduyar, T. E. 1994 Mesh update strategies in parallel finite element computations of flow problems with moving boundaries and interfaces. Comput. Meth. Appl. Mech. Engng 119 (1-2), 73–94.CrossRefGoogle Scholar
Karniadakis, G. E. 1990 Spectral element-Fourier methods for incompressible turbulent flows. Comput. Meth. Appl. Mech. Engng 80 (1–3), 367–380.CrossRefGoogle Scholar
Kerczek, C. von & Davis, S. H. 1974 Linear stability theory of oscillatory Stokes layers. J. Fluid Mech. 62 (4), 753–773.CrossRefGoogle Scholar
Kirby, R. M. & Sherwin, S. J. 2006 Stabilisation of spectral/hp element methods through spectral vanishing viscosity: application to fluid mechanics modelling. Comput. Meth. Appl. Mech. Engng 195 (2324), 3128–3144.CrossRefGoogle Scholar
Lamb, G. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Landahl, M. T. 1975 Wave breakdown and turbulence. SIAM J. Appl. Maths 28 (4), 735–756.CrossRefGoogle Scholar
Luo, J. & Wu, X. 2010 On the linear instability of a finite Stokes layer: instantaneous versus Floquet modes. Phys. Fluids 22, 054106.CrossRefGoogle Scholar
Mazzuoli, M., Vittori, G. & Blondeaux, P. 2011 Turbulent spots in oscillatory boundary layers. J. Fluid Mech. 685, 365–376.CrossRefGoogle Scholar
McKeon, B. J. 2017 The engine behind (wall) turbulence: perspectives on scale interactions. J. Fluid Mech. 817, P1.CrossRefGoogle Scholar
Ozdemir, C. E., Hsu, T. & Balachandar, S. 2013 Direct numerical simulations of instability and boundary layer turbulence under a solitary wave. J. Fluid Mech. 731, 545–578.CrossRefGoogle Scholar
Ozdemir, C. E., Hsu, T.-J. & Balachandar, S. 2014 Direct numerical simulations of transition and turbulence in smooth-walled Stokes boundary layer. Phys. Fluids 26 (4), 1–25.CrossRefGoogle Scholar
Perry, A. E. & Chong, M. S. 1982 On the mechanism of wall turbulence. J. Fluid Mech. 119, 173–217.CrossRefGoogle Scholar
Rayleigh, L. 1880 On the stability, or instability, of certain fluid motions. Proc. Lond. Math. Soc. 9, 57–70.Google Scholar
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23 (1), 601–639.CrossRefGoogle Scholar
Salon, S., Armenio, V. & Crise, A. 2007 A numerical investigation of the stokes boundary layer in the turbulent regime. J. Fluid Mech. 570 (570), 253–296.CrossRefGoogle Scholar
Sarpkaya, T. 1993 Coherent structures in oscillatory boundary layers. J. Fluid Mech. 253, 105–140.CrossRefGoogle Scholar
Scandura, P. 2013 Two-dimensional vortex structures in the bottom boundary layer of progressive and solitary waves. J. Fluid Mech. 728, 340–361.CrossRefGoogle Scholar
Schlichting, H. 1979 Boundary Layer Theory, 7th edn. McGraw-Hill.Google Scholar
Sengupta, T. K., De, S. & Sarkar, S. 2003 Vortex-induced instability of an incompressible wall-bounded shear layer. J. Fluid Mech. 493, 277–286.CrossRefGoogle Scholar
Sengupta, T. K. & Dipankar, A. 2005 Subcritical instability on the attachment-line of an infinite swept wing. J. Fluid Mech. 529, 147–171.CrossRefGoogle Scholar
Smith, C. R. 1984 A synthesized model of the near-wall behavior in turbulent boundary layers. In Proceedings of the 8th Symposium on Turbulence (ed. Patterson, G. K. & Zakin, J. K.). University of Missouri-Rolla.Google Scholar
Stokes, G. G. 1851 On the effect of the internal friction of fluids on the motion of pendulums. Trans. Camb. Phil. Soc. 9, 8–106.Google Scholar
Sumer, B. M., Jensen, P. M., Sørensen, L. B., Fredsøe, J., Liu, P. L. F. & Carstensen, S. 2010 Coherent structures in wave boundary layers. Part 2. Solitary motion. J. Fluid Mech. 646, 207–231.CrossRefGoogle Scholar
Thomas, C., Blennerhassett, P. J., Bassom, A. P. & Davies, C. 2015 The linear stability of a stokes layer subjected to high-frequency perturbations. J. Fluid Mech. 764, 193–218.CrossRefGoogle Scholar
Thomas, C., Davies, C., Bassom, A. P. & Blennerhassett, P. J. 2014 Evolution of disturbance wavepackets in an oscillatory Stokes layer. J. Fluid Mech. 752, 543–571.CrossRefGoogle Scholar
Traykovski, P., Wiberg, P. L. & Geyer, W. R. 2007 Observations and modeling of wave-supported sediment gravity flows on the Po prodelta and comparison to prior observations from the Eel shelf. Cont. Shelf Res. 27 (3), 375–399.CrossRefGoogle Scholar
Vittori, G. & Verzicco, R. 1998 Direct simulation of transition in an oscillatory boundary layer. J. Fluid Mech. 371, 207–232.CrossRefGoogle Scholar
Womersley, J. R. 1955 Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known. J. Physiol. 127 (3), 553–563.CrossRefGoogle ScholarPubMed
Wu, X. 1992 The nonlinear evolution of high-frequency resonant-triad waves in an oscillatory stokes layer at high Reynolds number. J. Fluid Mech. 245, 553–597.CrossRefGoogle Scholar
Xiong, C., Cheng, L., Tong, F. & An, H. 2018a Influence of plane boundary proximity on the Honji instability. J. Fluid Mech. 852, 226–256.CrossRefGoogle Scholar
Xiong, C., Cheng, L., Tong, F. & An, H. 2018b On regime c flow around an oscillating circular cylinder. J. Fluid Mech. 849, 968–1008.CrossRefGoogle Scholar
Xu, C. & Pasquetti, R. 2004 Stabilized spectral element computations of high Reynolds number incompressible flows. J. Comput. Phys. 196 (2), 680–704.CrossRefGoogle Scholar
Xu, H., Sherwin, S. J., Hall, P. & Wu, X. 2016 The behaviour of Tollmien–Schlichting waves undergoing small-scale localised distortions. J. Fluid Mech. 792, 499–525.CrossRefGoogle Scholar

Xiong et al. supplementary movie 1

(fig. 18) The video illustrating the propagation of turbulent spots in the moving reference frame $\hat{S}$. Case 16, $Re = 600$ and $h=0.5$.

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Video 2.3 MB

Xiong et al. supplementary movie 2

Experimental visualizations are reproduced from supplementary movie 3 of Carstensen et al. (2010) to demonstrate the inception, growth and breakdown of turbulent spots in accordance with figure 21.

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Xiong et al. supplementary movie 3

The comparison of the trajectory of the vortex pair and the inception of turbulent spots between the ones generated by a fixed roughness element and the counterparts by a temporary roughness element.

Download Xiong et al. supplementary movie 3(Video)
Video 203 KB