Editorial:Recent advances in hydrodynamic instability and transition to turbulence

Hydrodynamic instability and the mechanisms for turbulence onset and self-sustaining are two research questions that have puzzled the scientists for over a century，yet renewed interest and significant progress occurred in the last decade or so.We know only few exact analytical solutions of the Navier-Stokes equations，the equations governing the flow ofsimple fluids like waterand air，and most of these are never observed in applications or laboratory experiments since these configurations are prone to instabilities that quickly bring the flow to the turbulent regime，the most common state in environmental and industrial applications.The question of why and how a flow become turbulent has therefore interested scientists and engineers with very different background and applications in mind.

Thanks to the quick development of computational resources，of new computational algorithms and ways to deal with big amounts of experimental data，important progress has also been made in the area of hydrodynamic stability and turbulence，in addition to many other branches in science.Historically，the 1990s saw the appearance of non-modal approaches and transient growth analyses that enabled researchers to investigate the linear mechanisms of disturbance energy growth in flows that linear asymptotic theory predicted to be stable.In our opinion，two aspects are characterizing research in hydrodynamic instability and transition to turbulence in the last decade，something we believe will continue in the coming years.These are（1）dynamical systemapproach to investigate the phase-space portraitofthe flow and the onset of turbulence，（2）extension of stability，sensitivity，and receptivity analysis to complex flows，i.e.，the flow of complex and non-Newtonian fluids，fluids with a microstructure and with a more complex rheological behavior，and also flows in more complex three-dimensional geometries.These two lines are well represented by the selection presented in this special issue.

The idea of this dedicated issue originated at the Euromech Colloquium EC565:‘‘Subcritical Transition to Turbulence’’organized by Yohann Duguet，LIMSI-CNRS，F(xiàn)rance，JoséEduardo Wesfreid，ESPCI，PMMH-CNRS，F(xiàn)rance，and Bj?rn Hof，IST，Austria，and held in Cargese，F(xiàn)rance，May 6th-9th 2014.Not surprisingly，most of the investigations reported in this special issue were presented at the workshop in Corsica，to whom more than 80 researchers participated from all over the world.

As stated among the scopes of the colloquium，an especially challenging topic is the question of‘‘subcritical’’transition in wallbounded flows，i.e.，when the base flow is linearly stable and classical stability analysis fails at explaining the coherent structures commonly observed in experiments and in simulations.This concerns most flows in simple geometries such as pipes，ducts，channel，and also boundary layer flows.This special problem lies at the crossroad between hydrodynamics，chaos theory and statistical physics.Specific cutting-edge experimental and/or numerical strategies have been developed in the last decade.Indeed，scientists with background in theoretical and applied mathematics，physics，mechanical，aeronautical，and chemical engineering actively participate to the developmentofthis research area.In addition，as shown by the selection presented here，hydrodynamic stability and the turbulence onset are studied via large-scale numerical simulations and state-of-the-art experiments.

Understanding whether and how a flow becomes turbulent can provide useful insights on the possible strategies to manipulate the flow to favor/delay transition to turbulence.We indeed note that many of the research groups with strong tradition on hydrodynamic stability and transition to turbulence are now dedicating a larger effort to flow control.Although we decided not to include flow control here，different approaches are emerging and are worth mentioning here:active feedback control exploiting and adapting existing control and dynamical system theory to fluid flow systems characterized by a huge number of degrees of freedom（here knowledge of the potential to correctly estimate from realistic sensors becomes an important prerequisite），and passive control based on sensitivity and receptivity analysis as well as on the comprehension of the specific features of the flow at hand.

In this special issue the reader can find interesting examples of the two research directions mentioned above，which emerged clearly at the workshop in the spectacular scenario offered by Corsica，a beautiful island in the Mediterranean sea，just south of France.The rapid increase of available computer power has allowed scientists to perform numerical experiments of the full non-linear behavior of the flow and probe the flow systems as never done before；this is still the case in canonical configurations such as pipe and channel flow，where subcritical bifurcations reveal rich dynamics and unexpected behaviors.This has introduced a new picture of the transition to turbulence，based on the nonlinear system behaviorand the analysis oftrajectories in phase space，characterized by the approach to stable and unstable fixed points，such as traveling waves，orbits，and saddle points. Hand in hand with experiments，we are now able to explore the long time behavior of the flow and determine lifetimes of turbulence，just to mention an example.Thanks to large-scale simulations，we are also starting to understand and document the behavior of large systems and the occurrence of spatio-temporalchaos，examples being the appearance of localized traveling waves and turbulentstripes in large channelflows.These new discoveries call also for new theories and approaches and this is why the most recent efforts are often collaborations of fluid dynamicists，applied mathematicians and theoretical physicists.In addition to spatiotemporal chaos，we would like to mention as future challenges the extension of the current analysis to non-parallel nonhomogeneous flows where exact nonlinear solutions are difficult to compute and the analysis of the system stochastic behavior.

The second emerging development is in the direction of complex fluids.Non-Newtonian fluids，droplet，and particle suspensions as those found in many physiological flows are clearly the least explored by linear modal and non-modal instability analysis，not to mention a full nonlinear approach，despite the fact that they are probably the most often encountered in diverse fields and applications.Here，you will find examples of transition and turbulence ofa suspension ofrigid particles，the mixing and chaotic behavior induced by the elastic properties of dilute polymer suspensions，the so called elastic turbulence，and the instability of a plume induced by density and temperature differences between two fluids.New challenges arise in the field of complex fluids: the difficulty to write linearized stability equations as in the case of strong fluid-structure interactions，numerical stiffness of the governing equations（e.g.，polymer solutions），lack of reliable constitutive equations（e.g.，particle and bubble suspensions），and limitations of the optical and traditional measurement technique in some multiphase systems.Because of these exciting challenges，we expect great advances like novel methodologies and tools to emerge from applications and extensions of the current methods to complex fluids.A multidisciplinary approach based on both laboratory experiments and numerical experiments will be necessary to tackle this kind of problems.

We wish to take this opportunity to thank all the authors for their contributions and support to this special issue.Expert readers can find an overview of recent advances whereas readers with different background possibly appreciate the progresses and variety of tools developed during the years to investigate stability and transition to turbulence and hopefully benefit from them.

Luca Brandt?

Linné FLOW Centre，KTH Mechanics，SE 10044，Stockholm，Sweden

E-mail address：luca@mech.kth.se.

Jianjun Tao

CAPT，HEDPS，and IFSA Collaborative Innovation Center of MoE，

Department of Mechanics and Engineering Science，

College of Engineering，Peking University，Beijing 100871，China

Available online 29 April 2015

?Corresponding editor.

http://dx.doi.org/10.1016/j.taml.2015.03.007

2095-0349/

?2015 The Author.Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics.This is an open access article under the CC BY-NC-ND license（http://creativecommons.org/licenses/by-nc-nd/4.0/）.