A shape hessian-based boundary roughness analysis of navier-stokes flow

Shan Yang, Georg Stadler, Robert Moser, Omar Ghattas

Research output: Contribution to journalArticle

Abstract

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 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 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 Reynolds number based on the wavelength of the perturbation. The consequences of these results on the analysis of the effects of roughness on fluid flows are discussed.

Original languageEnglish (US)
Pages (from-to)333-355
Number of pages23
JournalSIAM Journal on Applied Mathematics
Volume71
Issue number1
DOIs
StatePublished - 2011

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Stokes Flow
Navier-Stokes
Roughness
Surface roughness
Dissipation
Eigenvalues and eigenfunctions
Topography
Partial differential equations
Eigenvalue
Perturbation
Flow of fluids
Reynolds number
Couette Flow
Poiseuille Flow
Surface Roughness
Local Minima
Viscous Fluid
Wavelength
Fluids
Fluid Flow

Keywords

  • Navier-Stokes flow
  • Roughness
  • Shape gradient
  • Shape Hessian

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

A shape hessian-based boundary roughness analysis of navier-stokes flow. / Yang, Shan; Stadler, Georg; Moser, Robert; Ghattas, Omar.

In: SIAM Journal on Applied Mathematics, Vol. 71, No. 1, 2011, p. 333-355.

Research output: Contribution to journalArticle

Yang, Shan ; Stadler, Georg ; Moser, Robert ; Ghattas, Omar. / A shape hessian-based boundary roughness analysis of navier-stokes flow. In: SIAM Journal on Applied Mathematics. 2011 ; Vol. 71, No. 1. pp. 333-355.
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