Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-12-01T02:33:23.477Z Has data issue: false hasContentIssue false
  • English
  • Français

DNS of wavepacket evolution in a Blasius boundary layer

Published online by Cambridge University Press:  19 May 2010

K. S. YEO ,
X. ZHAO ,
Z. Y. WANG  and
K. C. NG
K. S. YEO*
Affiliation:
Department of Mechanical Engineering, National University of Singapore, Singapore117576
X. ZHAO
Affiliation:
Department of Mechanical Engineering, National University of Singapore, Singapore117576
Z. Y. WANG
Affiliation:
Department of Mechanical Engineering, National University of Singapore, Singapore117576
K. C. NG
Affiliation:
Department of Mechanical Engineering, National University of Singapore, Singapore117576
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper presents the direct numerical simulation (DNS) of wavepacket evolution and breakdown in a Blasius boundary layer. The study covers the physical, spectral and structural aspects of the whole transition process, whereas previous studies have tended to focus on issues of a more limited scope. The simulations are modelled after the experiments of Cohen, Breuer & Haritonidis (J. Fluid Mech., vol. 225, 1991, p. 575). The disturbance wavepackets are initiated here by a u-velocity and a v-velocity delta pulse. They evolve through a quasi-linear growth stage, a subharmonic stage and a strongly nonlinear stage before breaking down into the nascent turbulent spots. Pulse-initiated wavepackets provide a plausible model for naturally occurring laminar–turbulent transition because they contain disturbances in a broadband of frequencies and wavenumbers, whose sum of interactions determines the spatio-temporal progress of the wavepackets. The early development of the wavepackets accords well with established linear results. The ensuing subharmonic evolution of the wavepackets appears to be underpinned by a critical-layer-based mechanism in which the x-phase speeds of the fundamental two-dimensional and dominant three-dimensional waves with compatible Squire wavenumbers are approximately matched. Spectral data over the bulk of the subharmonic stage demonstrate good consistency with the action of a phase-locked theory recently proposed by Wu, Stewart & Cowley (J. Fluid Mech., vol. 590, 2007, p. 265), strongly suggesting that the latter may be the dominant mechanism in the broadband nonlinear evolution of wavepackets. The dominant two-dimensional and three-dimensional waves are observed to be spontaneously evolving towards triad resonance in the late subharmonic stage. The simulations reproduce many key features in the experiments of Cohen et al. (1991) and Medeiros & Gaster (J. Fluid Mech., vol. 399, 1999b, p. 301). A plausible explanation is also offered for the apparently ‘deterministic’ subharmonic behaviour of wavepackets observed by Medeiros & Gaster. The strongly nonlinear stage is signified by the appearance of low-frequency streamwise-aligned u-velocity structures at twice the spanwise wavenumber of the dominant three-dimensional waves, distortion of the local base flow by the strengthening primary Λ-vortex and rapid expansion of the spanwise wavenumber (β) spectrum. These are in broad agreement with the experimental observations of Breuer, Cohen & Haritonidis (J. Fluid Mech., vol. 340, 1997, p. 395). The breakdown into incipient turbulent spots occurs at locations consistent with the experiments of Cohen et al. (1991). A visualization shows that the evolving wavepackets comprise very thin overlapping vorticity sheets of alternating signs, in stacks of two or three. Strong streamwise stretching of the flow at the centre of the wavepacket in the late subharmonic and strongly nonlinear stages promotes the roll-up and intensification of the vorticity sheets into longitudinal vortices, whose mutual induction precedes the breakdown of the wavepacket. The critical layer of the dominant two-dimensional and oblique wave modes reveals the progressive coalescence of a strong pair of vortices (associated with the Λ-vortex) during the subharmonic stage. Their coalescence culminates in a strong upward burst of velocity that transports lower momentum fluid from below the critical layer into the upper boundary layer to form a high shear layer in the post-subharmonic stage.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 652 , 10 June 2010 , pp. 333 - 372
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bech, K. H., Henningson, D. S. & Henkes, R. A. W. M. 1998 Linear and nonlinear development of localized disturbance in zero and adverse pressure gradient boundary layers. Phys. Fluids 10, (6), 1405–1418.Google Scholar
Borodulin, V. I., Kachanov, Y. S. & Koptsev, D. B. 2002 a Experimental study of resonant interactions of instability waves in a self-similar boundary layer with an adverse pressure gradient. Part I. Tuned resonance. J. Turbul. 3 (62), 1–38.Google Scholar
Borodulin, V. I., Kachanov, Y. S., Koptsev, D. B. & Roschektayev, A. P. 2002 b Experimental study of resonant interactions of instability waves in a self-similar boundary layer with an adverse pressure gradient. Part II. Detuned resonance. J. Turbul. 3 (63), 1–32.Google Scholar
Borodulin, V. I., Kachanov, Y. S. & Koptsev, D. B. 2002 c Experimental study of resonant interactions of instability waves in a self-similar boundary layer with an adverse pressure gradient. Part III. Broadband disturbances. J. Turbul. 3 (64), 1–19.Google Scholar
Borodulin, V. I., Kachanov, Y. S. & Roschektayev, A. P. 2006 Turbulence production in an APG-boundary-layer transition induced by randomized perturbations. J. Turbul. 7, (8), 1–30.CrossRefGoogle Scholar
Breuer, K. S., Cohen, J. & Haritonidis, J. H. 1997 The late stages of transition induced by a low-amplitude wavepacket in a laminar boundary layer. J. Fluid Mech. 340, 395–411.Google Scholar
Cantwell, B., Coles, D. & Dimotakis, P. 1978 Structure and entrainment in the plane of symmetry of a turbulent spot. J. Fluid Mech. 87, 641–672.Google Scholar
Cohen, J. 1994 The initial evolution of a wavepacket in a laminar boundary layer. Phys. Fluids 6, (3), 1133–1143.Google Scholar
Cohen, J., Breuer, K. S. & Haritonidis, J. H. 1991 On the evolution of a wave packet in a laminar boundary layer. J. Fluid Mech. 225, 575–606.Google Scholar
Corke, T. C. & Mangano, R. A. 1989 Resonant growth of three-dimensional modes in transitioning Blasius boundary layers. J. Fluid Mech. 209, 93–150.Google Scholar
Corke, T. C., Krull, J. D. & Ghassemi, M. 1992 Three-dimensional-mode resonance in far wakes. J. Fluid Mech. 239, 99–132.Google Scholar
Craik, A. D. D. 1971 Nonlinear resonant instability in boundary layers. J. Fluid Mech. 50, 393–413.CrossRefGoogle Scholar
Crow, S. C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8, 2172–2179.Google Scholar
Das, A. & Mathew, J. 2001 Direct numerical simulation of turbulent spots. Comput. Fluids 30, 533–541.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Fasel, H. F. 1976 Investigation of the stability of boundary layers by a finite-difference model of the Navier–Stokes equations. J. Fluid Mech. 78, 355–383.Google Scholar
Fasel, H. F., Rist, U. & Konzelmann, U. 1990 Numerical investigation of the three-dimensional development in boundary layer transition. AIAA J. 28 (1), 29.Google Scholar
Gaster, M. 1975 A theoretical model of a wave packet in the boundary layer on a flat plate. Proc. R. Soc. Lond. A 347, 271–289.Google Scholar
Gaster, M. & Grant, I. 1975 An experimental investigation of the formation and development of a wavepacket in a laminar boundary layer. Proc. R. Soc. Lond. A 347, 253–269.Google Scholar
Henningson, D. S. & Kim, J. 1991 On turbulent spots in plane Poiseuille flow. J. Fluid Mech. 228, 183–205.Google Scholar
Henningson, D. S., Lundbladh, A. & Johansson, A. V. 1993 A mechanism for bypass transition from localized disturbances in wall-bounded shear flows. J. Fluid Mech. 250, 169–207.CrossRefGoogle Scholar
Henningson, D. S., Spalart, P. & Kim, J. 1987 Numerical simulation of turbulent spots in plane Poiseuille and boundary layer flows. Phys. Fluids 30, 2914–2917.CrossRefGoogle Scholar
Herbert, T. 1983 Subharmonic three-dimensional disturbances in unstable plane shear flow. AIAA Paper 83-1759.Google Scholar
Herbert, T. 1984 Analysis of the subharmonic route to transition in boundary layers. AIAA Paper 84-0009.Google Scholar
Herbert, T. 1988 Secondary instability of boundary layers. Annu. Rev. Fluid Mech. 20, 487–526.Google Scholar
Holmes, P., Lumley, J. L. & Berkooz, G. 1996 Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press.Google Scholar
Jacobs, R. G. & Durbin, P. A. 2001 Simulations of bypass transition. J. Fluid Mech. 408, 185–212.Google Scholar
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 69–94.Google Scholar
Kachanov, Y. S. 1994 Physical mechanisms of laminar–boundary-layer transition. Annu. Rev. Fluid Mech. 26, 411–482.Google 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.Google Scholar
Konzelmann, U. & Fasel, H. 1991 Numerical simulation of a three-dimensional wavepacket in a growing flat-plate boundary layer. In Boundary Layer Transition and Control, pp. 1–24. The Royal Aeronautical Society.Google Scholar
Liu, C. & Liu, Z. 1994 Multigrid mapping box relaxation for simulation of the whole process of transition in 3D boundary layer. J. Comput. Phys. 119, 325–341.CrossRefGoogle Scholar
Lundbaldh, A., Schmidt, P., Berlin, S. & Henningson, D. 1994 Simulations of bypass transition for spatially evolving disturbances. In Application of Direct and Large Eddy Simulation to Transition and Turbulence. AGARD CP 551 (ed. Cantwell, B., Jimenez, J. & Lekoudis, S.), vol. 18, pp. 1–3.Google Scholar
Mankbadi, R. R., Wu, X. & Lee, S. S. 1993 A critical-layer analysis of the resonant triad in boundary-layer transition: nonlinear interactions. J. Fluid Mech. 256, 85–106.Google Scholar
Medeiros, M. A. F. & Gaster, M. 1999 a The influence of phase on the nonlinear evolution of wavepackets in boundary layers. J. Fluid Mech. 397, 259–283.Google Scholar
Medeiros, M. A. F. & Gaster, M. 1999 b The production of subharmonic waves in the nonlinear evolution of wavepackets in boundary layers. J. Fluid Mech. 399, 301–318.Google Scholar
Moin, P., Leonard, A. & Kim, J. 1986 Evolution of a curved vortex filament into a vortex ring. Phys Fluids. 29, 955–963.Google Scholar
Morkovin, M. V. 1969 The many faces of transition. In Viscous Drag Reduction (ed. Wells, C.). Plenum.Google Scholar
Orszag, S. A. & Patera, A. T. 1983 Secondary instabilities of wall-bounded shear flows. J. Fluid Mech. 128, 347–385.Google Scholar
Perry, A. E., Lim, T. T. & Teh, E. W. 1981 A visual study of turbulent spot. J. Fluid Mech. 104, 387–405.Google Scholar
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238.Google Scholar
Rist, U., Muller, K. & Wagner, S. 1998 Visualization of late-stage transitional structures in numerical data using vortex identification and feature extraction. Paper 103. In Eighth International Symposium on Flow Visualization, Sorrento, Italy.Google Scholar
Sartorius, D., Wuerz, W., Ries, T., Kloker, M., Wagner, S., Borodulin, V. I. & Kachanov, Y. S. 2006 Experimental study of resonant interactions of instability waves in an airfoil boundary layer. In Sixth IUTAM Symposium on Laminar–Turbulent Transition. Fluid Mech. Appl. 78, 159–166.Google Scholar
Schmid, P. J. & Henningson, D. S. 2000 Stability and Transition in Shear Flows. Springer.Google Scholar
Shaikh, F. N. 1997 The generation of turbulent spots in a laminar boundary layer. Eur. J. Mech. B: Fluids 16, 349–385.Google Scholar
Shaikh, F. N. & Gaster, M. 1994 The nonlinear evolution of modulated waves in a boundary layer. J. Engng Math. 28, 55–71.Google Scholar
Singer, B. A. 1996 Characteristics of a young turbulent spot. Phys. Fluids 8, (2), 509–521.Google Scholar
Spalart, P. R. & Yang, K. S. 1987 Numerical study of ribbon-induced transition in Blasius flow. J. Fluid Mech. 178, 345–365.Google Scholar
Vasudeva, B. R. 1967 Boundary-layer instability experiment with localized disturbance. J. Fluid Mech. 29, (4), 745–763.Google Scholar
Wang, Z. 2003 Computational simulation of unsteady boundary layer over compliant surfaces. PhD thesis, National University of Singapore.Google Scholar
Wang, Z., Yeo, K. S. & Khoo, B. C. 2005 Spatial direct numerical simulation of transitional boundary layer over compliant surfaces. Comput. Fluids 34, 1062–1095.Google Scholar
Wang, Z., Yeo, K. S. & Khoo, B. C. 2006 DNS of low Reynolds number turbulent flows in dimpled channels. J. Turbul. 7, (37), 1–31.Google Scholar
White, F. M. 1991 Viscous Fluid Flow, 2nd edn. McGraw-Hill.Google Scholar
Williamson, C. H. K. & Prasad, A. 1993 a A new mechanism for oblique wave resonance in ‘natural’ far wake. J. Fluid Mech. 256, 269–313.Google Scholar
Williamson, C. H. K. & Prasad, A. 1993 b Acoustic forcing of oblique wave resonance in the far wake. J. Fluid Mech. 256, 315–341.Google Scholar
Wu, X. & Stewart, P. A. 1996 Interaction of phase-locked modes: a new mechanism for rapid growth of three-dimensional disturbances. J. Fluid Mech. 316, 335–372.Google Scholar
Wu, X., Stewart, P. A. & Cowley, S. J. 2007 On the catalytic role of the phase-locked interaction of Tollmien–Schlichting waves in boundary-layer transition. J. Fluid Mech. 590, 265–294.Google Scholar
Wygnanski, I., Sokolov, M. & Friedman, D. 1976 On a turbulent spot in a laminar boundary layer. J. Fluid Mech. 78, 785–819.Google Scholar
Zhao, X. 2007 Computational simulation of wavepacket evolution over compliant surfaces. PhD thesis, National University of Singapore.Google Scholar