A shape Hessian-based analysis of roughness effects on fluid flows
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The flow of fluids over solid surfaces is an integral part of many technologies, and the analysis of such flows is important to the design and operation of these technologies. Solid surfaces, however, are generally rough at some scale, and analyzing the effects of such roughness on fluid flows represents a significant challenge. There are two fluid flow situations in which roughness is particularly important, because the fluid shear layers they create can be very thin, of order the height of the roughness. These are very high Reynolds number turbulent wall-bounded flows (the viscous wall layer is very thin), and very low Reynolds number lubrication flows (the lubrication layer between moving surfaces is very thin). Analysis in both of these flow domains has long accounted for roughness through empirical adjustments to the smooth-wall analysis, with empirical parameters describing the fluid dynamic roughness effects. The ability to determine these effects from a topographic description of the roughness is limited (lubrication) or non-existent (turbulence). The commonly used parameter, the equivalent sand grain roughness, can be determined in terms of the change in the rate of viscous energy dissipation caused by the roughness and is generally obtained by measuring the effects on a fluid flow. However, determining fluid dynamic effects from roughness characteristics is critical to effective engineering analysis. Characterization of this mapping from roughness topography to fluid dynamic impact is the main topic of the dissertation. Using the mathematical tools of shape calculus, we construct this mapping by defining the roughness functional and derive its first- and second- order shape derivatives, i.e., the derivatives of the roughness functional with respect to the roughness topography. The results of the shape gradient and complete spectrum of the shape Hessian are presented for the low Reynolds number lubrication flows. Flow predictions based on this derivative information is shown to be very accurate for small roughness. However, for the study of high Reynolds number turbulent flows, the direct extension of the current approach fails due to the chaotic nature of turbulent flows. Challenges and possible approaches are discussed for the turbulence problem as well as a model problem, the sensitivity analysis of the Lorenz system.