This paper is concerned with the theoretical behaviour of the boundary-layer flow over a disk rotating in otherwise still fluid. The flow is excited impulsively at a certain radius at time t = 0. This paper analyses the inviscid stability of the flow and the stability with viscous, Coriolis and streamline curvature effects included. In both cases, within a specific range of the parameter space, it is shown that the flow is absolutely unstable, i.e. disturbances grow in time at every fixed point in space. Outside this range, the flow is convectively unstable or stable. The absolute or convective nature of the instabilities is determined by examining the branch-point singularities of the dispersion relation. Absolute instability is found for Reynolds numbers above 510. Experimentally observed values for the onset of transition from laminar to turbulent flow have an average value of 513. It is suggested that absolute instability may cause the onset of transition to turbulent flow. The results from the inviscid analysis show that the absolute instability is not caused by Coriolis effects nor by streamline curvature effects. This indicates that this mechanism may be possible on swept wings, where Coriolis effects are not present but the boundary layers are otherwise similar.

Absolute instability of the boundary layer on a rotating disk

In this paper, the results of experiments on unsteady disturbances in the boundary-layer flow over a disk rotating in otherwise still air are presented. The flow was perturbed impulsively at a point corresponding to a Reynolds number R below the value at which transition from laminar to turbulent flow is observed. Among the frequencies excited are convectively unstable modes, which form a three-dimensional wave packet that initially convects away from the source. The wave packet consists of two families of travelling convectively unstable waves that propagate together as one packet. These two families are predicted by linear-stability theory: branch-2 modes dominate close to the source but, as the packet moves outwards into regions with higher Reynolds numbers, branch-1 modes grow preferentially and this behaviour was found in the experiment. However, the radial propagation of the trailing edge of the wave packet was observed to tend towards zero as it approaches the critical Reynolds number (about 510) for the onset of radial absolute instability. The wave packet remains convectively unstable in the circumferential direction up to this critical Reynolds number, but it is suggested that the accumulation of energy at a well-defined radius, due to the flow becoming radially absolutely unstable, causes the onset of laminar–turbulent transition. The onset of transition has been consistently observed by previous authors at an average value of 513, with only a small scatter around this value. Here, transition is also observed at about this average value, with and without artificial excitation of the boundary layer. This lack of sensitivity to the exact form of the disturbance environment is characteristic of an absolutely unstable flow, because absolute growth of disturbances can start from either noise or artificial sources to reach the same final state, which is determined by nonlinear effects.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112096000365

An experimental study of absolute instability of the rotating-disk boundary-layer flow

Boundary-layer transition-to-turbulence studies are conducted in the Arizona State University Unsteady Wind Tunnel on a 45-deg swept airfoil. The pressure gradient is designed so that the initial stability characteristics are purely crossflow dominated. Flow-visualization and hot-wire measurements show that the development of the crossflow vortices is influenced by roughness near the attachment line. Comparisons of transition location are made between a painted surface (distributed 9-μm peaks and valleys on the surface), a machine-polished surface (0.5-μm rms finish), and a hand-polished surface (0.25-μm rms finish). Then isolated 6-μm roughness elements are placed near the attachment line on the airfoil surface under conditions of the final polish (0.25-μm rms). These elements create an enhanced packet of stationary crossflow waves, which results in localized early transition. The diameter, height, and location of these roughness elements are varied in a systematic manner. Spanwise hot-wire measurements are taken behind the roughness element to document the enhanced vortices. These scans are made at several different chord locations to examine vortex growth

Effect of Isolated Micron-Sized Roughness on Transition in Swept-Wing Flows

A study is carried out of the linear global behaviour corresponding to the absolute instability of the rotating-disc boundary layer. It is based on direct numerical simulations of the complete linearized Navierâ€“Stokes equations obtained with the novel velocityâ€“vorticity method described in Davies & Carpenter (2001). As the equations are linear, they become separable with respect to the azimuthal coordinate, $\theta$. This permits us to simulate a single azimuthal mode. Impulse-like excitation is used throughout. This creates disturbances that take the form of wavepackets, initially containing a wide range of frequencies. When the real spatially inhomogeneous flow is approximated by a spatially homogeneous flow (the so-called parallel-flow approximation) the results ofthe simulations are fully in accordance with the theory of Lingwood (1995). If the flow parameters are such that her theory indicates convective behaviour the simulations clearly exhibit the same behaviour. And behaviour fully consistent with absolute instability is always found when the flow parameters lie within the theoretical absolutely unstable region. The numerical simulations of the actual inhomogeneous flow reproduce the behaviour seen in the experimental study of Lingwood (1996). In particular, there is close agreement between simulation and experiment for the ray paths traced out by the leading and trailing edges of the wavepackets. In absolutely unstable regions the short-term behaviour of the simulated disturbances exhibits strong temporal growth and upstream propagation. This is not sustained for longer times, however. The study suggests that convective behaviour eventually dominates at all the Reynolds numbers investigated, even for strongly absolutely unstable regions. Thus the absolute instability of the rotating-disc boundary layer does not produce a linear amplified global mode as observed in many other flows. Instead the absolute instability seems to be associated with transient temporal growth, much like an algebraically growing disturbance. There is no evidence of the absolute instability giving rise to a global oscillator. The maximum growth rates found for the simulated disturbances in the spatially inhomogeneous flow are determined by the convective components and are little different in the absolutely unstable cases from the purely convectively unstable ones. In addition to the study of the global behaviour for the usual rigid-walled rotating disc, we also investigated the effect of replacing an annular region of the disc surface with a compliant wall. It was found that the compliant annulus had the effect of suppressing the transient temporal growth in the inboard (i.e. upstream) absolutely unstable region. As time progressed the upstream influence of the compliant region became more extensive.

/pdf/global-behaviour-corresponding-to-the-absolute-instability-499y0piv4d.pdf

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

Boundary-layer transition-to-turbulence studies are conducted in the Arizona State University Unsteady Wind Tunnel on a 45-degree swept airfoil. The pressure gradient is designed so that the initial stability characteristics are purely crossflow-dominated. Flow visualization and hot-wire measurements show that the development of the crossflow vortices is influenced by roughness near the attachment-line. Comparisons of transition location are made between a painted surface, a machine-polished surface, and a hand-polished surface. Then, isolated 6 micron roughness elements are placed near the attachment line on the airfoil surface under conditions of the final polish (0.25 micron rms). These elements amplify a centered stationary crossflow vortex and its neighbors, resulting in localized early transition. The diameter, height, and location of these roughness elements are varied in a systematic manner. Spanwise hot-wire measurements are taken behind the roughness element to document the enhanced vortices. These scans are made at several different chord locations to examine vortex growth.

Effect of micron-sized roughness on transition in swept-wing flows

It is pointed out that the boundary layer on a rotating disk is important in stability theory because it provides a particularly simple way to study the important phenomenon of crossflow instability. This type of instability is responsible for early transition on sweptback wings. Mack and Kendall (1983) have studied the wave patterns formed by harmonic point sources in a Blasius boundary layer on the basis that the source uniformly excites all oblique normal modes of the source frequency. The calculation procedure for planar boundary layers was modified to fit the different geometry of the rotating disk and the lack of an axis of symmetry. Calculations were performed of the wave pattern produced by a zero-frequency point source located at the Reynolds number of the artificial roughness element in an experiment conducted by Wilkinson and Malik (1983). The results provided in the present investigation confirm that the experimental wave pattern is a superposition of the complete azimuthal wavenumber spectrum of zero-frequency normal modes with uniform initial amplitude and phase.

L. M. Mack

Papers

The wave pattern produced by a point source on a rotating disk