The effects of surface mass flux on the instability of the BEK system of rotating boundary-layer flows

Abstract

We consider the effect of mass flux through the lower boundary of the general class of rotating BEK boundary-layer flows. This class includes the Bödewadt, Ekman and von Kármán flows as particular cases. A theoretical study is presented which considers the onset of convective instability modes (both stationary and travelling relative to the rotating system) and local absolute instability. Suction is found to be universally stabilising in terms of both the delayed onset and reduced amplification rates of both instability types. Furthermore, the radial span of convective instability preceding the onset of local absolute instability is extended with increased suction. Slowly travelling modes are predicted to be more dangerous than stationary modes within the convectively unstable region of each flow. Such modes are expected to be selected over highly polished lower disks. Extensive theoretical data is presented for future comparison to experiment.

Introduction

This paper details the stability properties of the family of boundary-layer flows caused by a differential rotation rate between a lower disk and an incompressible fluid in rigid-body rotation above. Particular cases are the Bödewadt, Ekman and von Kármán boundary layers, hence this family is referred to as the BEK system. The Bödewadt layer arises when the lower disk is stationary and the fluid rotates; the Ekman layer has the disk and fluid rotating at approximately the same rate; and the von Kármán layer has the disk rotating in otherwise still fluid. In each case the lower disk is permeable, allowing enforced mass flux (injection or suction) in the normal direction.

There has been considerable interest in the stability characteristics of three-dimensional rotating boundary-layer flows since Gregory et al. [1] first studied the stability properties of the von Kármán boundary layer in the mid 1950s. This benchmark rotating system still remains of interest today due to its similarity to the practically significant flow over a swept wing. However, a natural extension to this model flow is the introduction of an additional rotating disk in the far field above, this leads to the BEK class of flows with the differential rotation rate as a key system parameter. In addition to having a theoretical interest, this broad class of flows can occur in turbo-machinery and rotor–stator devices such as mixers. Their stability characteristics therefore have practical importance. Furthermore, the stability characteristics of the Ekman layer may have applications for geophysical flows as discussed by Lingwood [2] and the references contained therein. In this paper we are motivated by turbo-machinery applications where laminar flow should be maintained as far as possible and suction is a potential stabilising mechanism. However, we acknowledge that the encouragement of turbulence may be beneficial for heat diffusion and mixing problems, and we also present mass-injection results.

The similarities in the stability characteristics of all flows within the BEK class are well established. See, for example, the experimental studies of [3], [4] for the Ekman layer; [5], [6], [7] for the von Kármán flow; and [8], [9], [10], [11], [12] for the Bödewadt layer. In all systems, spiral waves associated with the convective instability of the flow and rotating with the lower disk have been observed in regions outside the turbulent regime; see for example [1], [4], [13], [2]. The systems are all convectively unstable within certain regions to disturbances stationary in the frame rotating with the disk. These disturbances are excited by roughnesses on the disk surface and, because these roughnesses are fixed in time in the rotating frame, the stationary disturbances are consistently excited and reinforced such that they are evident in flow-visualisation experiments and in quantitative experiments, e.g. using hot-wire anemometry, where measurements tend to be ensemble averaged over many periods of rotations to improve the signal-to-noise ratio. The systems are also all convectively unstable within certain regions to disturbances that are not stationary in the frame rotating with the disk, i.e., travelling disturbances. However, travelling disturbances, which are excited by, for example, freestream turbulence, have received less attention because unless they are intentionally and repeatably excited (as in Lingwood [6]) flow-visualisation and ensemble-averaging techniques average the signals away. Furthermore, it is known that the class of flows is locally absolutely unstable [2], [14] and it is travelling modes that play a particular role in this instability mechanism. Although the determination of a local absolute instability requires the so-called parallel-flow approximation to be made (although the boundary-layer flows are physically parallel this terminology is used to describe the approximation of ignoring variations in Reynolds number with radius needed to render the linearised perturbation equations separable; see [15], [16], [17], [18]), it is clear that the observed repeatable onset of transition to turbulence of many of these flows is due to a related global physical phenomenon. The exact mechanism involved remains an active research area (see [19], for example) and the study of local absolute instability continues to be of significance.

Suction typically acts as a stabilising mechanism in boundary-layer flows. For example, Gregory and Walker [20] discuss how the introduction of suction extends the laminar-flow region over a swept wing by reducing the thickness of the boundary layer and the magnitude of crossflow velocity. As before, insight into swept-wing flow arose from studies of the von Kármán boundary layer (see [21], [22]) and work continued on this model flow with suction using numerical and asymptotic approaches (see [23], [24], [25], [26], for example). The literature shows that suction has a stabilising effect on the von Kármán flow which results in an increase in critical Reynolds numbers, a narrowing in the range of unstable parameters and a decrease in the amplification rates of unstable convective modes. In addition, Lingwood [27] demonstrates that suction has a stabilising effect on local absolute instability. Furthermore, the recent paper of Culverhouse et al. uses numerical and asymptotic techniques to consider the stabilising effects of suction on convective modes within the Bödewadt layer. All studies have shown injection to have the converse effect by destabilising the boundary layer to both instability types.

In this paper we extend both Lingwood and Culverhouse et al. by considering the effects of suction and injection on the stability characteristics of the entire class of BEK flows. In Section 2 the problem is formulated and steady laminar-flow profiles are calculated. In Section 3 convective instability modes are considered and neutral curves and critical Reynolds numbers are presented for a variety of flow parameters. In Section 4 local absolute instability is considered and our conclusions are drawn in Section 5. We are unable to obtain experimental data for comparison for flows other than the von Kármán and Bödewadt cases previously considered by Lingwood and Culverhouse et al., we therefore present detailed predictions of measurable quantities in the Appendix. It is hoped that these can be referred to when experimental data becomes available.