DNS of wavepacket evolution in a Blasius boundary layer

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.