The stability of rotating-disc boundary-layer flow over a compliant wall. Part 1. Type I and II instabilities

Abstract

A theoretical study into the effects of wall compliance on the stability of the rotating-disc boundary layer is described. A single-layer viscoelastic wall model is coupled to a sixth-order system of fluid stability equations which take into account the effects of viscosity, Coriolis acceleration, and streamline curvature. The coupled system of equations is integrated numerically by a spectral Chebyshev-tau technique.Travelling and stationary modes are studied and wall compliance is found to greatly increase the complexity of the eigenmode spectrum. It is effective in stabilizing the inviscid Type I (or cross-flow) instability. The effect on the viscous (Type II) eigenmode is more complex and can be strongly destabilizing. An analysis of the energy flux indicates that this destabilization arises as a result of a large degree of energy production by viscous stresses at the wall/flow interface.The Type I and II instabilities are shown to be negative and positive energy waves respectively. The co-existence of eigenmodes of opposite energy type indicates the possibility of modal interaction and coalescence. It is found that, compared with the rigid disc, wall compliance promotes the interaction and coalescence of the Type I and II eigenmodes. There is an associated strong instability which appears to be characterized by marked horizontal motion of the compliant surface. Modal coalescence is interpreted physically as producing local algebraic growth which could advance the onset of nonlinear effects.

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

“…The resulting steady flows are directly related to the von Kármán flow, and, importantly, previous stability analyses for the rotating-disk flow 17,26,27 are directly applicable to this current study. Full details of the governing perturbation equations and the codes used can be found in those references; here, it is sufficient to understand that a normal-mode analysis is conducted with perturbations of the form…”

“…It is noted that the behaviour of the Type II mode under anisotropic roughness with concentric grooves is identified as being similar to the effect of wall compliance on this mode, 27 where the disk boundary was comprised of a single layer of viscoelastic material free to move under the influence of disturbances in the boundary layer. For certain levels of wall compliance, the Type II lobe of the neutral curve was augmented and the critical Re reduced significantly, in exactly the same way as exhibited in this study.…”

Global behaviour corresponding to the absolute instability of the rotating-disc boundary layer

“…Energy considerations suggest that Types I and II also could coalesce and form a self-excited instability. In fact, as shown by Lingwood (1995) and Cooper & Carpenter (1997a), they do coalesce, but the result is an algebraically growing disturbance rather than an absolute instability. This phenomenon was investigated in I and also in a recent paper by Turkyilmazoglu & Gajjar (2000).…”

“…Nevertheless, as she herself pointed out, there were clues to be found in the previous experimental studies. (We will not attempt to review the relevant literature here, but instead refer the reader to the reviews by Reed & Saric (1989) and Saric, Reed & White (2003) and the introduction in Cooper & Carpenter (1997a).) For example, in the surface flow visualizations of Gregory, Stuart & Walker (1955), using the china-clay technique, the transitional radius is very sharp.…”

“…Accordingly, one might expect that, owing to their opposite response to energy transfer, when a PEW and a NEW coalesce they become capable of being a true self-excited instability, rather than a noise amplifier like a convective instability. Cooper & Carpenter (1997a) showed that Types I and II are NEW and PEW respectively. It is not clear whether or not Type III is a PEW, but presumably it is.…”