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Register a new userTowards a model of large scale dynamics in transitional wall-bounded flows
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Paul Manneville
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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 process 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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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 wall-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.
Input-output analysis of transitional channel flows has proven to be a valuable analytical tool for identifying important flow structures and energetic motions. The traditional approach abstracts the nonlinear terms as forcing that is unstructured, in the sense that this forcing is not directly tied to the underlying nonlinearity in the dynamics. This paper instead employs a structured singular value-based approach that preserves certain input-output properties of the nonlinear forcing function in an effort to recover the larger range of key flow features identified through nonlinear analysis, experiments, and direct numerical simulation (DNS) of transitional channel flows. Application of this method to transitional plane Couette and plane Poiseuille flows leads to not only the identification of the streamwise coherent structures predicted through traditional input-output approaches, but also the characterization of the oblique flow structures as those requiring the least energy to induce transition in agreement with DNS studies, and nonlinear optimal perturbation analysis. The proposed approach also captures the recently observed oblique turbulent bands that have been linked to transition in experiments and DNS with very large channel size. The ability to identify the larger amplification of the streamwise varying structures predicted from DNS and nonlinear analysis in both flow regimes suggests that the structured approach allows one to maintain the nonlinear effects associated with weakening of the lift-up mechanism, which is known to dominate the linear operator. Capturing this key nonlinear effect enables the prediction of the wider range of known transitional flow structures within the analytical input-output modeling paradigm.
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.
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 flow (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.
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