Location of Repository

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

### Abstract

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 Navier–Stokes equations obtained with the novel velocity–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

Topics: QC
Publisher: Cambridge University Press
Year: 2003
OAI identifier: oai:wrap.warwick.ac.uk:766

### Citations

1. (1991). A frequency selection criterion in spatially developing ﬂows.
2. (1962). A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability.
3. (2001). A novel velocity-vorticity formulation of the Navier–Stokes equations with applications to boundary-layer disturbance evolution.
4. (1966). A numerical study of the instability of the laminar Ekman boundary layer.
5. (1987). Absolute instability in the near-wake of two-dimensional bluﬀ bodies.
6. (1995). Absolute instability of the boundary layer on a rotating disc.
7. (1997). Absolute instability of the Ekman layer and related rotating ﬂows.
8. (1996). An experimental study of absolute instability of the rotating-disk boundarylayer ﬂow.
9. (1999). An experimental study of boundary-layer transition over a rotating, compliant disk.
10. (1988). Bifurcations to local and global modes in spatially-developing ﬂows.
11. (2000). Control of global instability in a non-parallel near wake.
12. (2000). Direct spatial resonance in the laminar boundary layer due to a rotating-disk. S¯ adhan¯ a-Acad.
13. E.&C r i g h t o n ,D .G .1991 Instability of ﬂows in spatially developing media.
14. (1964). Electron-Stream Interaction with Plasmas.M I TP r
15. (1996). Fully nonlinear global modes in spatially developing media.
16. (1993). Global linear stability analysis of weakly non-parallel shear ﬂows.
17. (1997). Global modes in falling capillary jets.
18. (1968). Growth of disturbances in both space and time.
19. (1991). Instability and transition of the disturbed ﬂow over a rotating disk.
20. (1996). Linear global modes in spatially developing media.
21. (1984). Local and global baroclinic instability of zonally varying ﬂow.
22. (1990). Local and global instabilities in spatially developing ﬂows.
23. (1985). Local instability characteristics and frequency determination of self-excited wake ﬂows.
24. (1989). Numerical simulation of the absolutely and convectively unstable wake.
25. (1997). On the application of the Briggs’ and steepest-descent methods to a boundary-layer ﬂow.
26. (1998). On the convective and absolute nature of instabilities in ﬁnite diﬀerence numerical simulations of open ﬂows.
27. (1991). On the cross-ﬂow instability near a rotating disk.
28. (1986). On the formation of vortex streets behind stationary cylinders.
29. (1965). On the generation of spatially growing waves in both space and time.
30. (2003). On the global stability of the boundary layer on rotating bodies.
31. (1995). On the spatial structure of global modes in wake ﬂow.
32. (1955). On the stability of three-dimensional boundary layers with application to the ﬂow due to a rotating disk.
33. (1989). S a r i c,
34. (1990). S r e e n i v a s a n ,K .R
35. (1993). Shallow-water wave ﬂow past isolated topography. Pt. II: Transition to vortex shedding.
36. (1980). Spiral vortices in boundary layer transition regime on a rotating disk.
37. (2003). Stability and transition of three-dimensional boundary layers.
38. (2001). Status of the use of wall compliance for laminar-ﬂow control.
39. (1998). Steep nonlinear global modes in spatially developing media.
40. (1988). The absolute and convective nature of instability in two-dimensional wakes at low Reynolds numbers.
41. (1984). The B´ enard-von K´ arm´ an instability: an experimental study near the threshold.
42. (1999). The ﬂickering candle: transition to a global oscillation in a thermal plume.
43. (2000). The hydrodynamics of compliant walls: Does the dolphin have a secret? Current Sci.
44. (1989). The interaction of two-dimensional ﬂows with a free surface at low Froude Numbers.
45. (1979). The role of negative energy waves in some instabilities of parallel ﬂows.
46. (1991). The stability of countercurrent mixing layers in circular jets.
47. (1997). The stability of rotating-disc boundary-layer ﬂow over a compliant wall. Part 1. Type I and II instabilities.
48. (1997). The stability of rotating-disc boundary-layer ﬂow over a compliant wall. Part 2. Absolute instability.
49. (1985). The wave pattern produced by a point source on a rotating disk.
50. (1921). Uber laminare und turbulente Reibung.
51. (1996). Unsteadiness and convective instabilities in two-dimensional ﬂow over a backward-facing step.
52. (1990). Wakes behind blunt bodies.

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.