A Shape Hessian-Based Boundary Roughness Analysis of Navier-Stokes Flow

The influence of boundary roughness characteristics on the rate of dissipation in a viscous fluid is analyzed using shape calculus from the theory of optimal control of systems governed by partial differential equations. To study the mapping $\mathscr{D}$ from surface roughness topography to the dissipation rate of a Navier–Stokes flow, expressions for the shape gradient and Hessian are determined using the velocity method. In the case of Couette and Poiseuille flows, a flat boundary is a local minimum of the dissipation rate functional. Thus, for small roughness heights the behavior of $\mathscr{D}$ is governed by the flat-wall shape Hessian operator, whose eigenfunctions are shown to be the Fourier modes. For Stokes flow, the shape Hessian is determined analytically and its eigenvalues are shown to grow linearly with the wavenumber of the shape perturbation. For Navier–Stokes flow, the shape Hessian is computed numerically, and the ratio of its eigenvalues to those of a Stokes flow depend only on the Re...