Numerical multiscale methods for boundary layer problems in fluid dynamics




Carney, Sean Patrick

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Physical processes typically occur over a wide range of scales in space and time. In many instances it is computationally preferable to couple models with differing levels of physical description for different portions of the domain in space and/or time. Such multiscale, hybrid strategies allow for an accurate representation of important physical phenomena where necessary while ensuring a feasible overall computational cost. In general, every multiphysics problem is different, and for every coupling strategy there are nontrivial mathematical and algorithmic details that must be worked out. However, many problems share similar features, for instance, asymptotically thin boundary layers and nonlinear interactions across scales, and sometimes strategies developed for one problem can be successfully applied to a related, but different area. This thesis develops numerical strategies for the accurate and efficient simulation of multiscale, boundary layer problems arising in fluid dynamics. Considered are four different physical models, namely viscous laminar flow over a rough surface, high Reynolds number wall-bounded turbulent flow, electrokinetic flows over charge-conducting surfaces, and upscaling in porous media flow. The first three of these situations typically involve asymptotically small, physical boundary layers while the fourth can involve computational boundary layers as a result of modeling artifacts. The numerical strategies are presented in the context of the heterogeneous multiscale method (HMM), which generally involves coupling a coarse-scale, macroscopic solver for bulk dynamics to a fine-scale, microscopic solver for the small and/or fast scales in space and/or time. The first section contains a coupled method for rough-wall laminar flow. The second describes a reduced-order, microscale model for the near-wall eddies present in high Reynolds number wall bounded turbulence. The third section concerns the application of stochastic, multiscale partial differential equation model known as Fluctuating Hydrodynamics to the mesoscale dynamics of electrokinetic flows. Finally, the fourth contains a method for reducing the nonphysical, boundary or "resonance" error term that arises in the numerical homogenization of multiscale elliptic operators as in, for example, the modeling of porous media flows or steady state heat conduction. All sections contain a description of and motivations for the problem, numerical examples of the model presented, and a discussion of future research and open problems to be addressed.



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