Stationary travelling cross-flow mode interactions on a rotating disk

Abstract

This work involves the study of the development of Type 1 stationary and travelling cross-flow modes in the three-dimensional boundary layer over a rotating disk. In order to control the characteristics of the stationary modes, we utilized organized patterns of roughness which were applied to the disk surface. These consisted of ink dots which were equally spaced in the azimuthal direction at a fixed radius in order to enhance particular azimuthal wavenumbers. Logarithmic spiral patterns of dots were also used to enhance azimuthal wave angles. Velocity fluctuation time series were decomposed into the components corresponding to the stationary and travelling modes using the instantaneous disk position as a reference. Their development was documented through the linear and nonlinear stages leading to turbulence. The linear stage agreed well with linear stability predictions for both modes. In the nonlinear stage we documented a triad coupling between pairs of travelling modes and a stationary mode. The strongest of these was a difference interaction which led to the growth of a low-azimuthal-number, stationary mode. This mode had the largest amplitude and appeared to dominate transition. In retrospect, we can observe the signs of this mechanism in past flow visualization (Kobayashi, Kohama & Takamadate 1980), and it can account for the ‘jagged’ front normally associated with cross-flow-dominated transition on swept wings.

Boundary-Layer Transition on Broad Cones Rotating in an Imposed Axial Flow

“…are stationary) so that γ i = β, which is consistent with experimental observations discussed in §I. As discussed by Corke & Knasiak, 30 traveling modes can dominate on a highly polished rotating disk and this is also likely to be the case over a highly polished broad rotating cone. However, we are particularly interested in practical engineering applications where highly polished surfaces will not be found.…”

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

“…However, non-stationary modes (i.e., those where γ 0) can also exist, and some have lower critical Re and higher growth rates than the stationary modes. Corke and co-workers 18,28,29 have shown that non-stationary modes can be important in the transition process over smooth disks, but these tend not to be naturally excited where roughnesses are present. In order to safely predict a stabilization of the boundary layer by surface roughness though, some calculations for travelling modes are included.…”

“…However, 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. Watanabe et al 22 studied uniformly distributed roughness on a rotating cone.…”

An energy analysis of convective instabilities of the Bödewadt and Ekman boundary layers over rough surfaces

“…Many studies have been performed to reveal the similarities in the stability characteristics across the entire BEK class (see, for example, the theoretical and experimental studies discussed in papers [9][10][11][12][13][14] for the Bödewadt layer; papers [15,16] for the Ekman layer; and papers [17][18][19][20] for the von Kármán flow). These studies have shown the existence of two main flow instability mechanisms, commonly referred to as the Type I and the Type II modes.…”