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تسجيل مستخدم جديدOn the generation of drift flows in wall-bounded flows transiting to turbulence
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Paul Manneville
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Despite recent progress, laminar-turbulent coexistence in transitional planar wall-bounded shear flows is still not well understood. Contrasting with the processes by which chaotic flow inside turbulent patches is sustained at the local (minimal flow unit) scale, the mechanisms controlling the obliqueness of laminar-turbulent interfaces typically observed all along the coexistence range are still mysterious. An extension of Waleffes approach [Phys. Fluids 9 (1997) 883--900] is used to show that, already at the local scale, drift flows breaking the problems spanwise symmetry are generated just by slightly detuning the modes involved in the self-sustainment process. This opens perspectives for theorizing the formation of laminar-turbulent patterns.
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In this essay, we recall the specificities of the transition to turbulence in wall-bounded flows and present recent achievements in the understanding of this problem. The transition is abrupt with laminar-turbulent coexistence over a finite range of
Reynolds numbers, the transitional range. The archetypical cases of Poiseuille pipe flow and plane Couette flow are first reviewed at the phenomenological level, together with a few other flow configurations. Theoretical approaches are then examined with particular emphasis on the existence of special nontrivial solutions to the Navier-Stokes equations at finite distance from laminar flow. Dynamical systems theory is most appropriate to analyze their role, in particular with respect to the transient character of turbulence in the lower transitional range. The extensions needed to deal with the prominent spatiotemporal features of the transition are then discussed. Turbulence growth/decay in terms of statistical physics of many-body systems and the relevance of directed percolation as a stochastic process able to account for it are next scrutinized. To conclude, we advocate the recourse to well-designed modeling able to provide us with a conceptually coherent picture of the full transitional range and put forward some open issues.
In wall-bounded flows, the laminar regime remain linearly stable up to large values of the Reynolds number while competing with nonlinear turbulent solutions issued from finite amplitude perturbations. The transition to turbulence of plane channel fl
ow (plane Poiseuille flow) is more specifically considered via numerical simulations. Previous conflicting observations are reconciled by noting that the two-dimensional directed percolation scenario expected for the decay of turbulence may be interrupted by a symmetry-breaking bifurcation favoring localized turbulent bands. At the other end of the transitional range, a preliminary study suggests that the laminar-turbulent pattern leaves room to a featureless regime beyond a well defined threshold to be determined with precision.
The main part of this contribution to the special issue of EJM-B/Fluids dedicated to Patrick Huerre outlines the problem of the subcritical transition to turbulence in wall-bounded flows in its historical perspective with emphasis on plane Couette fl
ow, the flow generated between counter-translating parallel planes. Subcritical here means discontinuous and direct, with strong hysteresis. This is due to the existence of nontrivial flow regimes between the global stability threshold Re_g, the upper bound for unconditional return to the base flow, and the linear instability threshold Re_c characterized by unconditional departure from the base flow. The transitional range around Re_g is first discussed from an empirical viewpoint ({S}1). The recent determination of Re_g for pipe flow by Avila et al. (2011) is recalled. Plane Couette flow is next examined. In laboratory conditions, its transitional range displays an oblique pattern made of alternately laminar and turbulent bands, up to a third threshold Re_t beyond which turbulence is uniform. Our current theoretical understanding of the problem is next reviewed ({S}2): linear theory and non-normal amplification of perturbations; nonlinear approaches and dynamical systems, basin boundaries and chaotic transients in minimal flow units; spatiotemporal chaos in extended systems and the use of concepts from statistical physics, spatiotemporal intermittency and directed percolation, large deviations and extreme values. Two appendices present some recent personal results obtained in plane Couette flow about patterning from numerical simulations and modeling attempts.
On its way to turbulence, plane Couette flow - the flow between counter-translating parallel plates - displays a puzzling steady oblique laminar-turbulent pattern. We approach this problem via Galerkin modelling of the Navier-Stokes equations. The wa
ll-normal dependence of the hydrodynamic field is treated by means of expansions on functional bases fitting the boundary conditions exactly. This yields a set of partial differential equations for the spatiotemporal dynamics in the plane of the flow. Truncating this set beyond lowest nontrivial order is numerically shown to produce the expected pattern, therefore improving over what was obtained at cruder effective wall-normal resolution. Perspectives opened by the approach are discussed.
A system of simplified equations is proposed to govern the feedback interactions of large-scale flows present in laminar-turbulent patterns of transitional wall-bounded flows, with small-scale Reynolds stresses generated by the self-sustainment proce
ss of turbulence itself modeled using an extension of Waleffes approach (Phys. Fluids 9 (1997) 883-900), the detailed expression of which is displayed as an annex to the main text.
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