This lecture is aimed at an audience that is technically conversant with fundamental fluid mechanics. It is intended to lay out (what the speaker thinks are) some outstanding open problems and opportunities for mathematical analysis regarding transport and mixing in turbulent incompressible fluid flows. The emphasis will be on challenges to derive rigorous estimates for physically important quantities. Questions include:
1) Boundary-driven flows. Can logarithmic (or power law) corrections to naive Kolmogorov scaling for bulk energy dissipation in turbulence Couette flow be rigorously derived from the Navier-Stokes equations without substantial secondary assumptions, e.g., closures? Can experimentally observed classical Kolmogorov scaling for rough boundaries be deduced directly? We will also introduce the perplexing 'Jacuzzi problem' for flux-driven flows.
2) Buoyancy-driven flows. Rayleigh-Bénard convection, notably thermal turbulence in a fluid layer heated from below and cooled from above, remains an outstanding problem for analysis, simulation, and experiment. A generally accepted mathematical model, the Boussinesq approximation to the Navier-Stokes equations, is in place but mysteries remain. Can mathematics resolve the ongoing controversies regarding the 'ultimate' state of turbulent convection?
3) Tracer-transport and mixing. The 'standard' model of passive scalar transport imposes a mean gradient and assumes spatially periodic flows and tracer distributions. We argue for the need for rigorous estimation of meaningful and useful definitions of effective diffusion in the context of this model. Can conclusions and predictions of direct numerical simulations be rigorously derived from the fundamental equations of motion?
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