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

A J Cooper,P J Thomas,J H Harris,S J Garrett,M Özkan

doi:10.1063/1.4906091

A J Cooper, P J Thomas + Show 3 more

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https://doi.org/10.1063/1.4906091

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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.

Highlights

It was believed that surface roughness of bodies inevitably increases their skinfriction drag as they move through fluids
These are subjected to a linear stability analysis to reveal how roughness affects the stability characteristics of the inviscid Type I instability and the viscous Type II instability that arise in the rotating disk boundary layer
The effects of surface roughness on the absolute instability are the focus of a separate study, but preliminary results have shown that roughness can act in the same way as wall compliance and restrict the onset of absolute instability to significantly higher Re compared to the smooth disk

Summary

INTRODUCTION

It was believed that surface roughness of bodies inevitably increases their skinfriction drag as they move through fluids. The existing studies focussed almost exclusively on the effects of a small number of roughness elements arranged to form certain roughness patterns, the goal being to excite particular disturbance modes to test their characteristics for the development of specific control mechanisms.[9,10,18,19,20] Work with more general roughness was briefly considered by Zoueshtiagh et al.[21] for a rotating disk and, in more depth, by Watanabe et al.[22] Zoueshtiagh et al.[21] documented a short set of experimental data for velocity profiles over quartz-grain roughened disks and these will be revisited here after the presentation of the current results.

THE STEADY FLOW

CONVECTIVE INSTABILITY

ENERGY ANALYSIS

TRAVELLING MODES AND ABSOLUTE INSTABILITY

SUMMARY AND CONCLUSION