1. Themes of the Workshop (transition and turbulence presentations split roughly equally: Kerswell and Davidson) ● What recent progress has been made in understanding transition to turbulence in wall-flows --- as a subject in its own right? ● What progress has been made in linking transition to turbulent wall flows at high Reynolds numbers (if such links are at all likely)? ● What are the outstanding issues in 'fully developed' wall flows? ● What is the outlook for the near future?

2. Dynamical systems approach [partially incorporates discussions in sessions on 'finite lifetime of turbulence' (Eckhardt) and 'global unstable modes...' (Gibson)] Hope: The structural features may be a ('random'?) walk among invariant solutions (periodic orbits, saddles, travelling waves, perhaps tori...); if so, one can get suitably weighted averages So far: View has been helpful in confined systems at low Re (N ~ 10 5 ) --- starting with Kawahara and Kida Questions for discussion ● How to determine the 'more prominent' periodic orbits from the 'less prominent'? ● How to scale the number of invariant solutions to higher Re (in particular, to fully turbulent flows)? ● How does the knowledge of invariant solutions on the 'critical surface' help? ● Is transition an attractor? Is turbulence a transient? How complex is the boundary if the two states coexist in phase space? ● Are the observed differences in lifetime data (e.g., finite or not?) due to lack of full understanding of initial perturbation? Does this result bear on the notion that turbulence is self-sustaining? ● Do such features have to be studied in spatially extended systems before they can be taken seriously? ● What role do these invariant solutions play for small-scale processes such as turbulent mixing and chemical reactions (which are influenced by the spectrum of fluctuations)? ● More broadly, what is their role in high-Reynolds-number turbulence? How to account for the many scales? ● How to compute them in that limit? Is the use eddy viscosity or LES solutions the way to go?

3. Spatially extended systems and pattern formation Qualitative progress that I saw Pattern formation studies used to be mostly based on complex G-L equation with little connection with NS equations, and most patterns studied were devoid of fluctuations. There were good examples to bridge this connection through the intermediary of numerical solutions of NS equations (oblique bands of alternating steady and random motion, spiral turbulence, etc) Some questions ● What do the solutions in highly confined systems have to do with patterns in extended systems? With transitional work? ● Is the spatio-temporal complexity in extended systems key to understanding transition? Do localized structures such as puffs and turbulent spots spread essentially through contamination, and decay in small systems but may not do so in larger systems: thermodynamic limit (or, size does matter!). ● How relevant is the outlook of critical phenomena to transition?

4. Transition to turbulence [partially incorporates the discussions in the sessions on 'streaks' (Kim) and the 'optimal growth' (Tuckerman)] ● Controlled transition vs transient growth due to non-normality effects ('optimal solution') ● Why should one worry about optimal solutions? ● Why does energy optimization lead to streaks? (Why shouldn't it have been the optimization of energy production, for instance? ● What is the role of nonlinearity (or self-sustenance through some linear mechanism possible)? ● How to determine their amplitudes? ● What is their relevance to high-Re turbulence? ● What relevance to control? ● Relation between streaks and vortices? ● Is it possible to create streaks by a combination of random forcing, shear and viscosity (without the need for longitudinal vortices)? ● If so, what is the best definition of streaks (e.g., the most persistent anisotropic features near the wall)?

5. Fully developed turbulence [partially incorporates discussions in sessions on 'scaling of Reynolds stress tensor' (Klewicki) and 'very large turbulent structures' (Nickels and Marusic)] A basic question Traditional description of the wall-bounded flows, in terms of inner and outer layer existing separately and passively, is passé. Dynamical interaction is the key. Large scales will impress their identity on the wall phenomena and vice versa. How to characterize them is the broad question? Examples of interaction effects ● The peak value of u' (occurring roughly at y+ of 12) is weakly dependent on Re, but what is the right way to take account of it? ● Ambiguous scaling of the wall-normal and spanwise intensities ● Does one need an outer velocity scale other than the friction velocity? ● There are 'very large' structures (>3  even in wall and log-regions, how to model the appropriate interaction dynamics to account for this? ● Do these interaction effects call for a flow-dependent Karman constant? (Do we understand enough about the Reynolds number dependence, development length, geometry, aspect ratio, measurement method, etc, etc, to be certain that this is indeed the case?) ● Do we need more than two distinguishing layers --- leading to other forms of the overlap region(s)?

6. Fully developed turbulence (contd): Last slide! Somewhat specialized questions ● Elusive nature of the -1 power-law in the spectral density ● Second peak in rms intensity ● Role of inlet conditions in obtaining the self-sustained state? (Nikita's simulations, Jimenez's prescription; experiments) ● Very large structures (>3  ● Shorter for boundary layers (because of spatial growth?) ● Origin: Optimal transient growth calculated via turbulent mean velocity profile using eddy viscosity? Chance concatenation of O(  ) structures, as in percolation, say, or with some minor kinematic constraints? ● Appropriate way to discern scale contributions to Reynolds stresses ● Memory of transition process in the Karman flow, leading to non-unique turbulent states with hysteresis? ● Roughness effects: Limitations of Townsend's hypothesis; retire Colebrook gracefully? ● LES methods, episodic description, elastic turbulence, drag reduction, viscosity stratification, sudden expansions, etc ● New facilities, new instrumentation, new measurement techniques, etc