Transition to turbulence in wall-bounded flows: Where do we stand?

Abstract

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.

Theoretical perspective on the route to turbulence in a pipe

“…At fixed r, puffs decay at rate 1/τ D independently of time and hence independently of their history. The exponential form of survival functions is well documented in numerous studies, not only of pipe flow, but also several other wall-bounded shear flows (Faisst & Eckhardt 2004;Peixinho & Mullin 2006;Avila et al 2010;Manneville 2015Manneville , 2016, and references therein). On a practical level, memoryless decay is absolutely essential to the study of puff dynamics, particularly in experiments, since it implies that all time intervals of a given size are equivalent.…”

“…As we will see, this is only one piece in the story of how turbulence arises, but it illustrates how only recently has it been possible to definitely answer some of the most basic questions about this flow and to obtain a more-or-less clear understanding of the route to turbulence in pipe flow. Manneville (2015Manneville ( , 2016 gives excellent reviews of the field in the broader context of wall-bounded shear flows.…”

Numerical methods for hydraulic fracture propagation: A review of recent trends

“…A number of models exist for the evolution of the Fanning factor from laminar to turbulent flow in pipes, most are based on a combination of theory and phenomenological relations to capture experimental observations (Nikuradse, 1950). One should always treat those turbulent models with great care and recognize the spatio-temporal statistics associated with the transition to turbulence (Manneville, 2016). It is also extremely important to bear in mind that in industrial practice, friction reducers (Virk, 1975) are always added to low viscosity fluid in order to reduce drag under turbulent conditions.…”

Laminar-Turbulent Patterning in Transitional Flows

“…The transition from laminar to turbulent dynamics has been an important field of study, in view of deep theoretical issues relating to the nature of stochasticity and its important consequences on macroscopic transfer properties in applications (consult [2] for an introduction). Basically two transition scenarios can be distinguished upon varying R [1].…”

“…This active field of research is rapidly evolving, and important results have been obtained recently. To set the frame, in Section 1, I will summarize a recent paper reviewing the subject from a more general standpoint [1], enabling me to focus on a specific feature of this transition: the existence of a statistically well-organized laminar-turbulent patterning of flows along planar walls in some intermediate range of Reynolds numbers [R g , R t ]. The Reynolds number is the main control parameter of the problem.…”