## Abstract

The stability of the flow produced over an infinite stationary plane in a fluid rotating with uniform angular velocity at an infinite distance from the plane is considered. The basic flow is an exact solution of the Navier–Stokes equations making it amenable to theoretical study. An asymptotic investigation is presented in the limit of large Reynolds number. It is shown that the stationary spiral instabilities observed experimentally can be described by a linear inviscid stability analysis. The prediction obtained for the wave angle of the disturbances is found to agree well with the available experimental and numerical results.

## 1. Introduction

The stability of the boundary layer produced on a stationary plane in a fluid rotating far from the plane is relevant to many industrial devices. For instance, in rotor–stator systems it seems that the boundary layer on the stator is the first to become unstable (see Itoh *et al*. 1990). This instability is also relevant to cavity elements of a turbine engine (see Owen 1988).

The first theoretical investigation of this problem was carried out by Bödewadt (1940). The steady flow produced over an infinite stationary plane in a fluid rotating with uniform angular velocity at an infinite distance from the plane is an exact solution of the Navier–Stokes equations. The flow is characterized by the radial pressure gradient being balanced by the centrifugal forces. Fluid is drawn to the axis of rotation and swept upwards.

Batchelor (1951) suggested that for large Reynolds numbers, the flow between a rotating disc and a stationary disc would consist of boundary layers on each disc separated by a core of fluid rotating as a solid body. Thus, when the boundary layers are separated, the flow on the stator can be identified as Bödewadt flow.

The stability of Bödewadt flow was investigated by Savaş (1983, 1987) who conducted spin-down experiments in a cylindrical cavity. He observed spiral modes of instability (type I) similar to those appearing in the experiments of Gregory *et al*. (1955) on the boundary layer on a rotating disc. Savaş (1987) measured the angle between the normal to the radius vector and the tangent to the vortices to be *ϕ*=12–18°. He observed an additional instability in the form of circular waves which propagated towards the axis of rotation before dying out. It is thought that these instability waves are type II waves, owing their existence to Coriolis and viscous forces. The circular waves have also been observed by Gauthier *et al*. (1999) in their experiments on the flow between a rotating and a stationary disc. Spiral vortices and circular waves were observed together, with the flow becoming disordered at large Reynolds number. Schouveiler *et al*. (2001) investigated the flow between a rotating disc and a stationary disc. They showed experimentally that the boundary layer which forms on the stationary plane becomes unstable to spiral vortices, which can be stationary or travelling outwards. The stationary vortices were found to have *ϕ*=12–15°. Transition was observed to occur by the interaction of spiral rolls and circular rolls.

There have been several numerical studies of the stability of the Bödewadt layer. Lingwood (1997) used a parallel-flow approximation and presented results for stationary vortices. She found that for a large Reynolds number *ϕ*≈15° for inviscid neutrally stable modes. Fernandez-Feria (2002) performed a spatial stability analysis for axisymmetric modes and concluded that the circular waves observed experimentally correspond to an inertial instability mode.

Serre *et al*. (2001) carried out three-dimensional direct numerical simulations in an annular cavity and also a cylindrical cavity. Their results show that the Bödewadt layer is unstable to axisymmetric and spiral instabilities.

An asymptotic study of the flow owing to a rotating disc was carried out by Hall (1986). He investigated the inviscid (type I) instabilities and the viscous (type II) instabilities. The current study will follow his approach to investigate the inviscid instability of the Bödewadt layer.

## 2. Formulation

We consider the flow of an incompressible, viscous fluid of kinematic viscosity *ν* above an infinite stationary plane where the fluid far from the plane is rotating with uniform angular velocity *Ω*. We take cylindrical polar coordinates (*r*, *θ*, *z*), with *r* and *z* having been made dimensionless with respect to some reference length *l*. We consider the governing continuity and Navier–Stokes equations in a stationary frame of reference.

The axisymmetric non-dimensional basic flow is given by(2.1)(2.2)Here, velocities and pressure have been non-dimensionalized by *Ωl* and *ρΩ*^{2}*l*^{2}, respectively, where *ρ* is the fluid density and *p* is the fluid pressure. The variable *ζ*=*Re*^{1/2}*z*, where *Re* is the Reynolds number defined by *Re*=*Ω*^{2}*l*/*ν*.

The dimensionless functions and are determined, after substitution of (2.1) and (2.2) into the governing equations by(2.3)where the prime denotes differentiation with respect to *ζ*. The appropriate boundary conditions are(2.4)and(2.5)

The solutions of (2.3) subject to (2.4) and were obtained by following the method of Van de Vooren *et al*. (1987). The solutions are oscillatory as *ζ*→∞, so a shooting method was applied starting from large values of *ζ* and integrating to *ζ*=0. Our numerical results determined(2.6)The solutions for and are shown in figure 1.

## 3. The inviscid stability problem

We write * u*=

**u**_{B}+

*where |*

**U***/*

**U**

**u**_{B}|≪1 and consider the linear disturbance equations. Following Hall (1986), we consider disturbances to the basic flow proportional to(3.1)where

*ϵ*=

*Re*

^{−1/6}. Thus

*U*=

*u*(

*z*)

*E*, and similarly for

*V*,

*W*and

*P*and the wavelengths scale on the boundary-layer thickness. In the first instance, we seek neutrally stable disturbances with

*α*and

*β*real and expand them as(3.2)Then, in the inviscid zone of thickness

*O*(

*ϵ*

^{3}), the velocities and pressure expand as(3.3)

Substituting into the governing linearized disturbance equations we find that *w*_{0} satisfies(3.4)where is the ‘effective’ two-dimensional velocity profile and *γ*_{0} is the effective wavenumber defined by . Thus, as in Hall (1986), *w*_{0} satisfies Rayleigh's equation with boundary conditions(3.5)

The solution for *w*_{0} is determined for the situation when so that (3.4) is not singular at . This yields(3.6)The eigenvalue problem for *γ*_{0} was solved by using central differences to yield(3.7)The solution for *w*_{0} is shown in figure 2. This has been normalized with at *ζ*=0. Since we have(3.8)to leading order (3.6) yields *ϕ*≈15°, which agrees with the experimental observations.

The corrections to the ‘effective’ wavenumber and wave angle *ϕ* of the disturbance may be determined by considering the next order solutions in the inviscid layer. It is found that *w*_{1} satisfies(3.9)This equation has a singularity at , where . This can be removed by introducing a critical layer at , where for the path of integration is deformed above the singularity in the complex plane. (Note that this is the opposite situation from the analysis of Hall (1986) for the flow owing to a rotating disc, because is positive.)

As in Hall (1986) we find that *w*_{1} is given by(3.10)

The solutions in the inviscid layer do not satisfy the boundary conditions at *ζ*=0, so we require a wall layer of thickness *O*(*ϵ*^{4}). In the wall layer set *z*=*ϵ*^{4}*ξ* and the basic flow expands aswith similar expansions for and . In the wall layer, the disturbance velocities and pressure expand as(3.11)

The leading order solutions are given in terms of the Airy function Ai. For large *ξ* we find that(3.12)where This provides the matching condition for *w*_{1} as *ζ*→0, i.e.(3.13)Thus, from (3.10) we obtain the linear eigenrelation(3.14)Here,(3.15)and(3.16)

The eigenrelation (3.14) provides the *O*(*ϵ*) corrections to the effective wavenumber and the wave angle. We find that for the effective wavenumber(3.17)while the wave angle *ϕ* is determined from(3.18)

## 4. Conclusion

We have demonstrated that the upper-branch neutral modes of Bödewadt flow for large Reynolds number may be described by inviscid stability theory. The instability takes the form of stationary spiral rolls. The prediction for the angle of the orientation of the spirals agrees well with the available experimental observations.

There is much scope for future work on this problem. The lower-branch stationary modes could be studied asymptotically. Travelling modes have been observed experimentally, thus the importance of these modes should be investigated. In particular, it is hoped that the instability mechanism governing the travelling circular rolls in the experiments can be identified. Then the relevance of these modes of instability to practical situations can be judged.

## Footnotes

One contribution of 19 to a Theme ‘New developments and applications in rapid fluid flows’.

- © 2005 The Royal Society