Nonlinear traveling waves as a framework for understanding turbulent drag reduction

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PROPOSAL NO.: CBET - 0730006 PRINCIPAL INVESTIGATOR: MICHAEL D. GRAHAMINSTITUTION: UNIVERSITY OF WISCONSIN-MADISONNONLINEAR TRAVELING WAVES AS A FRAMEWORK FOR UNDERSTANDING TURBULENT DRAG REDUCTIONThe level of energy dissipated in turbulent flow of a liquid can be dramatically reduced by low levels of certain polymer or surfactant additives. This rheological drag reduction effect is well-known, but not well understood even after many decades of active research. A detailed understanding of this phenomenon will have the direct benefit of allowing more rational design of additives and process systems to exploit the effect, and would elucidate one route by which turbulence is "controlled" to reduce drag, perhaps providing insight into general principles of flow control. Here, The PI continues the development of a unique theoretical and computational approach to drag reduction. This approach builds on new insights into transition and near-wall turbulence, especially the recently-discovered existence of a family of nonlinear three-dimensional traveling wave solutions to the Navier-Stokes equations in the pipe and channel flow geometries that provide, in outline at least, a unified solution to the puzzles of transition to turbulence and turbulent near-wall structure. These solutions are steady states in a traveling reference frame; they have been dubbed "exact coherent states" or ECS. They come into existence at a Reynolds number just below the observed transition to turbulence, and their flow structure in the channel flow geometry considered here consists of a mean shear and a pair of staggered streamwise-aligned counter-rotating vortices, as is found in the turbulent buffer layer. In prior work, the PI has shown that these flows are affected by viscoelasticity in the same qualitative manner as is full turbulence and begun the exploration of the relationship between these structures and the three main thresholds observed in drag-reduced flows as Reynolds number increases: (1) transition to turbulence, (2) onset of drag reduction and (3) approach to the maximum drag reduction (MDR) asymptote. The principal goal is to further study viscoelastic nonlinear traveling waves, as well as complementary direct numerical simulations (DNS) guided by the traveling wave studies, to gain a unified understanding of the important thresholds in drag reduction, and characterize flow and polymer dynamics in the MDR regime. The PI's prior work suggests that there may be other families of nonlinear traveling waves aside from the ECS family described above; one aim of the planned work is to seek such families, as evidence suggests that the ECS family vanishes in the MDR regime, and is replaced by some other self-sustaining flow structure. Preliminary DNS results in a regime suggested by the ECS studies already give clues to the dominant flow behavior in that regime. Models of both polymer and wormy micellar surfactant systems will be used; surfactant solutions show more drag reduction than polymers and have desirable properties for large scale heating and cooling applications. Broader impacts arising from this work include: involvement of undergraduate students in a project involving practical issues of implementing drag reducing fluids in a large scale flow system, the UW-Madison chilled water cooling system and providing background for turbulence control. Rigorous development of flow control strategies of all kinds will be enabled by a firm understanding of the dynamical framework of near-wall turbulence.