The effect of anisotropic and isotropic roughness on the convective stability of the rotating disk boundary layer

Abstract

A theoretical study investigating the effects of both anisotropic and isotropic surface roughness on the convective stability of the boundary-layer flow over a rotating disk is described. Surface roughness is modelled using a partial-slip approach, which yields steady-flow profiles for the relevant velocity components of the boundary-layer flow which are a departure from the classic von Kármán solution for a smooth disk. These are then subjected to a linear stability analysis to reveal how roughness affects the stability characteristics of the inviscid Type I (or cross-flow) instability and the viscous Type II instability that arise in the rotating disk boundary layer. Stationary modes are studied and both anisotropic (concentric grooves and radial grooves) and isotropic (general) roughness are shown to have a stabilizing effect on the Type I instability. For the viscous Type II instability, it was found that a disk with concentric grooves has a strongly destabilizing effect, whereas a disk with radial grooves or general isotropic roughness has a stabilizing effect on this mode. In order to extract possible underlying physical mechanisms behind the effects of roughness, and in order to reconfirm the results of the linear stability analysis, an integral energy equation for three-dimensional disturbances to the undisturbed three-dimensional boundary-layer flow is used. For anisotropic roughness, the stabilizing effect on the Type I mode is brought about by reductions in energy production in the boundary layer, whilst the destabilizing effect of concentric grooves on the Type II mode results from a reduction in energy dissipation. For isotropic roughness, both modes are stabilized by combinations of reduced energy production and increased dissipation.

On the stability of von Kármán rotating-disk boundary layers with radial anisotropic surface roughness

“…Despite resulting from fundamentally different steady-flow models, both collections of neutral curves display the same qualitative behaviour: the Type I lobe is diminished (both in terms of critical Re and width) with increased roughness, and the Type II mode exaggerated. This is entirely consistent with the results of our previous study of radial isotropic roughness 1 . The results of the YHP model appear much more sensitive to the increased roughness, however this merely reflects the much greater response of the steady flows (as reported in Table I) over the range of a used.…”

“…Our theoretical results discussed in Ref. [1] did predict such stabilising roughness effects on the rotating-disk boundary layer. However, these results were based on one particular theoretical approach of modelling roughness 2 which has, as yet, not been tested experimentally.…”

“…Here we summarize the results obtained from this alternative analysis and compare them to our original set of data from Ref. [1]. The goal is to highlight similarities and differences in the two data sets which we expect to become of significance in the context of interpreting future experimental data when they become available.…”

“…The dramatic reduction in P 2 in this case results from a strong reduction in the amplitude of the normal velocity, w, as roughness increases. As explained elsewhere 1,17 , the viscous dissipation term D is dominated by the term σ 32 ∂v/∂z so that D ≈ −(2/Re) |∂v/∂z| 2 dz. The broadening of the v velocity profile for the Type II mode has the effect of decreasing D. The effect of roughness on the distribution for D is less significant for the Type I mode.…”

“…In particular, the model assumes partial slip at the disk surface but is otherwise identical to the von Kármán formulation 3 ; full details can be found elsewhere 1,2 . The full MW model has two parameters η and λ (giving empirical measures of the roughness in the radial and azimuthal directions, respectively) that appear in the surface boundary conditions of the von Kármán ODEs,…”