In this letter, it is shown numerically that in plane Poiseuille flow and before the threshold of equilibrium turbulence defined by the directed-percolation universality class, a sparse turbulent state in form of localized turbulent band can sustain either by continuous increase of the turbulence fraction due to band extension when the flow domain is large enough, or by a dynamic balance between the band extension and the band breaking and decay caused by the band interaction in a finite domain. The width and tilt angle of the band keep statistically invariant during its oblique extension, a process which is not sensitive to random disturbances. PACS numbers: 47.27.Cn; 47.27.nd; 47.52.+j The subcritical nature of the transition to turbulence in Poiseuille flows and several other shear flows gives rise to a number of phenomena that have been documented for some time but only recently been combined in a coherent picture of the transition. Since the laminar profiles are stable against sufficiently small perturbations, turbulence can only be triggered if certain thresholds are exceeded [1-3]. Studies of optimal perturbations based on stable manifolds of critical states (so-called edge states) have shown that localized patches of downstream vortices can extract enough energy from the laminar profile and trigger turbulence [4][5][6][7][8][9][10]. Localized turbulent puffs in pipe flow and turbulent bands in long but narrow channels [11,12] were found to either decay or split at low Reynolds numbers, and hence several dynamical system approaches were suggested to describe the onset of turbulence, e.g. the 1-D dynamic model [13][14][15], the directed percolation (DP) model [16,17], and the ecological predator-prey model [18]. Recently, the 1-D DP model was examined experimentally and numerically for Couette flows [19], where the flow is highly confined in two directions and hence the turbulent-laminar intermittency could occur only along one spatial dimension.A natural extension of this behaviour to two spatial dimensions would suggest that localized perturbations first grow to localized spots which then grow and fill all of space. Accordingly, one would expect the transition to fall into the universality class of two spatial and one temporal dimension or (2+1)-D directed percolation. Such an expectation is consistent with recent numerical simulations of the planar shear flow between stress-free boundaries, where the statistics of the turbulent structures satisfy the power-law scaling of the (2+1)-D DP [20]. However, for planar flow with no-slip boundaries in a large domain, such as the plane-Poiseuille flow (PPF), localized turbulent bands [21] were found to survive at Reynolds numbers much lower than 830 [22,23], the critical threshold defined by the (2+1)-D DP model [24]. Such a discrepancy is observed as well in experiments of PPF, where the turbulent fraction deviated from zero as Re<830 [24]. Exploring the underlying causes of the inconsistency between the DP model and the experimental and numerical results is t...
In this letter, we show via numerical simulations that the typical flow structures appearing in transitional channel flows at moderate Reynolds numbers are not spots but isolated turbulent bands, which have much longer lifetimes than the spots. Localized perturbations can evolve into isolated turbulent bands by continuously growing obliquely when the Reynolds number is larger than 660. However, interactions with other bands and local perturbations cause band breaking and decay. The competition between the band extension and breaking does not lead to a sustained turbulence until Re becomes larger than about 1000. Above this critical value, the bands split, providing an effective mechanism for turbulence spreading.
This paper investigates the spatio-temporal instability of the natural-convection boundary-layer flow adjacent to a vertical heated flat plate immersed in a thermally stratified ambient medium. The temperature on the plate surface is distributed linearly. By introducing a temperature gradient radio $a$ between the wall and the medium, we obtain a similarity solution which can describe in a smooth way the evolution between the states with isothermal and uniform-heat-flux boundary conditions. It is shown that the flow reversal in the basic flow vanishes when $a$ is larger than a critical value. A new absolute–convective instability transition of this flow is identified in the context of the coupled Orr–Sommerfeld and energy equations. Increasing $a$ decreases the domain of absolute instability, and when $a$ is large enough the absolute instability disappears. In particular, when $a\,{=}\,0$ (isothermal surface), the interval of absolute instability becomes narrower for fluids of larger Prandtl numbers, and the absolute instability does not occur for Prandtl numbers greater than 70; when $a\,{=}\,1$ (uniform-heat-flux surface) the instability remains convective in a wide Prandtl number range. Analysis of the Rayleigh equations for this problem reveals that the basic flows supporting this new instability transition have inviscid origin of convective instability. Based on the steep global mode theory, the effects of $a$ and Prandtl number on the global frequency are discussed as well.
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