Mathematical Problems in Fluid Dynamics

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NSF #DMS-9802611PI. P. ConstantinAbstract-------- We propose to extend work on variational bounds for bulkdissipation in boundary driven Navier-Stokes turbulence, with particularattention to Rayleigh-Benard convection. We intend to investigate the effect of rotation and of change in Prandtl number on high Rayleighnumber turbulence. We propose to investigate analytically and numericallythe scaling of generalized structure functions in Navier-Stokes turbulence and compare the results with those of simple stochastic models. We propose to study analytically and numerically the formation of singularitiesand dynamics of compressible dissipative active scalars and of quasi-geostrophic incompressible active scalars. We also propose to investigate models of granular dynamics and their relation to experimental data. The present proposal is aimed at the understandingof nonlinear phenomena in fluids. Understanding these phenomena is essentialto areas that range from weather prediction, climate and environmental studies, turbulent combustion (as for instance in engines) to the cosmological question of mass distribution in the universe. The objects of study in this proposal represent areas that challenge or lie beyond the capabilities of the most powerful computing systems available today; the proposal aim is to develop physical mathematicstheory and numerical models that will yield progress in our ability to perform computer simulations. NSF #DMS-9802611PI. P. ConstantinAbstract-------- We propose to extend work on variational bounds for bulkdissipation in boundary driven Navier-Stokes turbulence, with particularattention to Rayleigh-Benard convection. We intend to investigate the effect of rotation and of change in Prandtl number on high Rayleighnumber turbulence. We propose to investigate analytically and numericallythe scaling of generalized structure functions in Navier-Stokes turbulence and compare the results with those of simple stochastic models. We propose to study analytically and numerically the formation of singularitiesand dynamics of compressible dissipative active scalars and of quasi-geostrophic incompressible active scalars. We also propose to investigate models of granular dynamics and their relation to experimental data. The present proposal is aimed at the understandingof nonlinear phenomena in fluids. Understanding these phenomena is essentialto areas that range from weather prediction, climate and environmental studies, turbulent combustion (as for instance in engines) to the cosmological question of mass distribution in the universe. The objects of study in this proposal represent areas that challenge or lie beyond the capabilities of the most powerful computing systems available today; the proposal aim is to develop physical mathematicstheory and numerical models that will yield progress in our ability to perform computer simulations.