## A shape Hessian-based analysis of roughness effects on fluid flows

##### Abstract

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.

##### Description

text