Receptivity of three-dimensional boundary-layers to localized wall roughness and suction

Abstract

The receptivity of three-dimensional, incompressible, boundary-layer flows to localized roughness or suction is studied theoretically and numerically. Three-dimensional boundary-layer flows have a strong “nonparallel” component due to the curvature of the potential-flow streamlines. Our theoretical model extends the Fourier-transform methodology to nonparallel flows using a Taylor expansion of the laminar mean-flow at the location of the roughness. Both the near-field and the far-field solutions are contained in the model. Additionally, the use of MacLaurin expansions in the complex plane leads quickly to the receptivity factors of eigenmodes. The theoretical results are validated with solutions to the linearized Navier–Stokes equations efficiently obtained by replacing the actual wall forcing with a smoother, but equivalent, forcing. We find that the far-field response is proportional to both H̃(αe) and dH̃(αe)/dα, where H̃(α) is the Fourier transform of the forcing distribution, and αe is the eigenmode’s wave number. The receptivity coefficient for H̃(αe) decreases when nonparallelism is included in our models, while that for dH̃(αe)/dα increases.

The influence of the spatial frequency content of discrete roughness distributions on the development of the crossflow instability

“…From computational studies and Fourier theory analysis, it follows that the spectrum of the roughness distribution is strongly coupled to the response of the boundary layer 18,22,23 . Experimentally this has been verified with discrete roughness elements by Radeztsky et al 30 (who varied the diameter of the discrete roughness elements) and by Reibert et al 24 (who varied the spanwise spacing of the roughness elements).…”

“…In these studies, discrete roughness elements (D=O(mm), k=O(µm), where D is the diameter of the roughness and k the height) are placed close to the leading edge (near to the neutral stability point) at a certain spanwise spacing. Numerically, it has been found that, under the parallel-flow assumption, the response of the boundary layer on the small roughness elements can be computed using Fourier transform theory 18,22,23 . The response of the flow is dominated by the leaststable eigenmode having the same spanwise wavenumber, β, and frequency, ω, as the roughness distribution 23 .…”

“…Numerically, it has been found that, under the parallel-flow assumption, the response of the boundary layer on the small roughness elements can be computed using Fourier transform theory 18,22,23 . The response of the flow is dominated by the leaststable eigenmode having the same spanwise wavenumber, β, and frequency, ω, as the roughness distribution 23 . Experimentally, this has first been shown by Reibert et al 24,25 , who found that roughness elements spaced at spanwise spacing λ only excite the fundamental and sub-harmonic disturbances with wavelengths λ\n , whilst the disturbance with wavelengths greater than λ are not amplified.…”

Acoustic receptivity and transition modeling of Tollmien-Schlichting disturbances induced by distributed surface roughness

“…The inclusion of non-parallel effects in the finiteReynolds number theory was done approximately by Bertolotti 27 through a Taylor expansion of the baseflow treated in Fourier space (residue-based analysis). Previous approaches [28][29][30] restricted their analysis to a single slowly-varying perturbation ansatz.…”

On the effect of discrete roughness on crossflow instability in very low turbulence environment

“…An increased roughness height was found to advance transition, introducing mean-flow deformation in the velocity profiles and high-frequency spectral content in the fluctuations. Last but not least, many numerical simulations have also devoted effort to predict the stability of roughness induced crossflow (Bertolotti, 2000;Carpenter et al, 2010;Mughal and Ashworth, 2013;Tempelmann et al, 2012, amongst others).…”