## Abstract

The stability of a family of boundary-layer flows, which includes the von Kármán, Bödewadt and Ekman flows for a rotating incompressible fluid between a rotating disc and a stationary lid, is investigated. Numerical computations with the use of a spectral method are carried out to analyse absolute and convective instability. It is shown that the stability of the system is enhanced with a decrease in distance between the disc and the lid.

## 1. Introduction

Extensive studies have been carried out on fluid flow when a solid disc of infinite radius and the fluid over it are in a state of rigid-body rotation in the same direction. Some of the most interesting flows are those of von Kármán (1921), Bödewadt (1940) and Ekman (1905). In a von Kármán flow, the disc is rotating but the fluid is stationary; whereas in Bödewadt flow, the disc is stationary and the fluid is in rotation. The Ekman flow is characterized by the angular velocities of disc and the fluid being almost equal.

In this paper, we shall study a family of boundary-layer flows which occur when an incompressible fluid is trapped between a solid disc and a solid lid. The disc and the fluid are both undergoing rigid-body rotation, whereas the lid is stationary. The disc has infinite radius. The disc and the fluid are rotating about the same vertical axis with angular velocities *Ω*_{D} and *Ω*_{F}, respectively. There are four distinct flows: the Bödewadt layer is characterized by *Ω*_{D}=0 and *Ω*_{F}≠0; the von Kármán layer when *Ω*_{D}≠0 and *Ω*_{F}=0; the Ekman layer when *Ω*_{D}≈*Ω*_{F}; and the general case when *Ω*_{D} and *Ω*_{F} are unequal and non-vanishing. We shall call the distance between the disc and the lid *D*, and shall carry out a linear stability analysis of the system for different values of *D*. When *D* tends to infinity, our problem becomes identical to that of Lingwood (1997). More recently, Schouveiler *et al*. (2001) have investigated the stability of the flows confined between a rotating and a stationary disc enclosed by a stationary sidewall. Annular and spiral patterns in flows between rotating and stationary discs have been examined by Serre *et al*. (2001).

To begin with, we analyse the neutral stability characteristics of disturbances over a rotating-disc boundary-layer flow. The traditional sixth-order stability equations are derived. With the use of a spectral method and arc-length continuation, we have solved these equations numerically for different values of *D*. The criteria used for analysing absolute or convective instability involve solving the full sixth-order viscous equations. The branch points where the group velocity ∂*ω*/∂*α* tends to zero are searched in the *α* and *ω* planes. For velocity profiles of von Kármán, Bödewadt and Ekman flows, the stability characteristics are investigated and the absolute instability range is calculated for different values of *D*. The values we obtain show that the rotating-disc boundary-layer is absolutely unstable when *D* is large, confirming the results obtained by Lingwood (1995). However, the system's stability is increased with the decrease in the value of *D*.

## 2. The basic equations

### (a) Governing equation of the flow

We consider a family of boundary-layer flows caused by a differential rotation rate between a solid boundary, or disc, and an incompressible fluid in rigid-body rotation. Particular cases of this family are the Bödewadt, Ekman and von Kármán boundary-layer flows. The radius of the disc and the extent of the fluid above the disc are considered to be infinite, and the disc and fluid rotate about the same vertical axis with angular velocities *Ω*_{D}* k* and

*Ω*

_{F}

*.*

**k**The Navier–Stokes equations are non-dimensionalized by introducing the quantities *r*^{*}, *θ*^{*}, *z*^{*}, *t*^{*}, *u*^{*}, *v*^{*}, *w*^{*}, and *P*^{*}: *r*=*Lr*^{*}, *z*=*Lz*^{*}, *θ*=*θ*^{*}, , , *u*=*U*_{c}*u*^{*}, *v*=*U*_{c}*v*^{*}, *w*=*U*_{c}*w*^{*}, where *L* is a given length-scale and *U*_{c}=*LR*_{0}*Ω* is a given velocity scale. For convenience of writing, we shall suppress the asterisk on the non-dimensional variables. Here, (*r*,*θ*,*z*) denote cylindrical polar coordinates; *u*, *v*, and *w* are the non-dimensional radial, azimuthal and axial velocities; *p* is the fluid pressure; and is the Coriolis parameter. We have a global Reynolds number , where *R* is the Reynolds number based on the displacement thickness . *R*_{0} is a Rossby number (see also below), which measures the ratio between the convective and Coriolis effects.

### (b) Mean flow

The mean flow relative to the disc is given by the von Kármán (1921) exact similarity solution to the Navier–Stokes equations. The boundary-layer coordinate *Z*, which is of order O(1), is defined as *Z*=*zR*. We define Δ*Ω*=*Ω*_{F}−*Ω*_{D}, and the Rossby number *R*_{0}, which measures the ratio between convective and Coriolis effect, by , where

The mean flow velocities take the form(2.1)where the suffix denotes the basic flow velocities. The mean pressure has the form

Substituting the values of *u*, *v*, *w* and *p* from (2.1) into the Navier–Stokes equations gives the following non-dimensional ordinary differential equations for the mean flow(2.2a)(2.2b)(2.2c)(2.2d)where primes denote differentiation with respect to *Z* and is the Coriolis parameter. For the von Kármán layer *R*_{0}=−1 and *Ω*=*Ω*_{D}; for the Ekman layer *R*_{0}=0 and *Ω*=*Ω*_{F}=*Ω*_{D}; for the Bödewadt layer *R*_{0}=1 and *Ω*=*Ω*_{F}. *C*_{0} is equal to 2 for the von Kármán and the Ekman layers, but *C*_{0}=0 for the Bödewadt layer. The appropriate boundary conditions for the mean flow are given as(2.3a)

(2.3b)

The constant *D* is a parameter. The value of *h*_{D} is a constant vertical velocity for different values of *D*.

### (c) Linear stability equations

The basic velocity profiles are perturbed by imposing infinitesimally small disturbances. The instantaneous non-dimensional velocity and pressure are given bywhere , , and are small perturbation quantities and the local Reynolds number is *R*. We linearize the Navier–Stokes equations for small perturbation quantities by making use of various approximations (see Turkyilmazoglu & Gajjar 1998; Jasmine 2003) and assuming normal-mode disturbance of the formwhere *β* and *ω* are the wavenumber in the azimuthal direction and the scaled frequency of the wave propagating in the disturbance wave direction, respectively, and c.c. denotes the complex conjugate. We obtain a reduced sixth-order linear system of stability equations given by(2.4a)(2.4b)(2.4c)(2.4d)Here *λ*^{2}=*α*^{2}+*β*^{2}, . The boundary conditions for this set of equations are *f*=*g*=*h*=0 at (*Z*=0, *D*). This system of equations (2.4*a–d*) is an extension of similar equations derived by Balakumar & Malik (1990), Malik (1986), Lingwood (1997) and Turkyilmazoglu & Gajjar (1998).

## 3. Results

### (a) Mean flow results

The basic flow equations (2.2*a–d*) were solved numerically to obtain profiles for *F*(*Z*), *G*(*Z*) and *H*(*Z*). These solutions are calculated for the Rossby numbers −1, 0 and 1, and for different values of *D*. For the von Kármán flow, i.e *R*_{0}=−1, velocity profiles are plotted in figure 1 for values of *D* varying from 20 to 4. We observe that the axial velocity *H* decreases in magnitude with a decrease in the value of *D*. Only the velocity profile *F* in the radial direction is inflectional. The velocity profiles *F*, *G* and *H* are shown in figure 2*a–c*, respectively, for *R*_{0}=0; and in figure 3*a–c* for *R*_{0}=1. For the Bödewadt flow, i.e *R*_{0}=1, the velocity profiles *F*, *G* and *H* are all infectional. When *D*=20, the values of *F*′(0) and *G*′(0) coincide with the value obtained by Lingwood (1997) for *R*_{0}=−1, 0 and 1.

### (b) Stability results

#### (i) Stationary waves

We first consider stationary waves, i.e disturbances with *ω*=0. We solve the system of equations (2.4*a–d*) by employing a spectral method (more details of the method may be found in Turkyilmazoglu & Gajjar 1998). The wave angle *ϵ* of disturbance is defined as . Two types of instability waves exist in the rotating-disc boundary-layer flow. The first one is the inviscid type or the upper branch, and the other one is the viscous-type or the lower branch. The cups between the two branches mark the neutral branch-points. Figures 4–6 show the marginal curves for convective instability of stationary modes for the Rossby numbers −1, 0 and 1 for different values of *D*. The marginal curves for the von Kármán flow, i.e *R*_{0}=−1, consist of two branches. The two convective unstable branches exist outwards from the source (*r*>*r*_{a}) for *D*=20. With decreasing *D*, there are no longer two minima. The lower minimum disappears from the lower branch. The figures show that decreasing *D* leads to an increase in the values of the critical Reynolds number *R* and eigenvalues *α*. However, the corresponding critical eigenvalues *β* increase, as *D* decreases, from 20 to 6, and start decreasing beyond 6. For other Rossby numbers, the curves also consist of two branches. Results for the Ekman layer with *R*_{0}=0 are shown in figure 5. For *D*=20, the data shows that the critical Reynolds number *R* is 117 and wave angle 14.3° for the onset of (upper branch) stationary waves. It has been shown by Lilly (1966) and Lingwood (1997) that a critical Reynolds number *R* of about 115 and 116, respectively, is necessary for the onset of (upper branch) stationary waves. The critical Reynolds number for the convective instability of stationary waves in the Ekman layer agrees quite closely with an experimentally observed Reynolds number of about 125 and wave angle of about 14° by Faller & Kaylor (1966). Tatro & Mollö-Christensen (1967) observed upper branch modes in the Ekman layer with an almost constant wave angle of 14.6°. As *D* decreases, the critical value of *R* increases. However, the critical values of *α*, *β* and *ϵ* increase as *D* decreases from values 20 to 3.5, and start decreasing for values of *D* beyond 3.5. For Bödewadt flows, i.e *R*_{0}=1, the two convective unstable branches exist inwards from the source (*r*<*r*_{a}). In figure 6, we show the marginal curves for the convective instability of stationary modes for different values of *D*. Boundary-layer transition over a stationary disc in a rotating flow is studied experimentally by Savaş (1987). He determined the critical Reynolds number from frequency and wavelength measurements to be about 25. In this study, the critical Reynolds number *R*=27.36 and *β*≈0.1145 for *D*=20. The critical Reynolds number for convective instability waves in the Bödewadt layer agrees quite closely with the Reynolds number of about 27.4 and wavenumber *β* of about 0.1152, observed by Lingwood (1997). With the decrease in the value of *D*, the system becomes increasingly stable for all cases.

#### (ii) Travelling waves

Travelling waves are defined for non-zero values of frequency *ω*. The second minima instability is analogous to an instability mode reported experimentally and theoretically in the Ekman layer by Lilly (1966), Tatro & Mollö-Christensen (1967), Melander (1983) and Lingwood (1997). The existence of such a second minima was also observed in the experiments of Faller & Kaylor (1966). They showed that both viscous-type, as well as inviscid-type, disturbances were possible in the Ekman boundary layer in the rotating-disc flow. It was therefore suggested that the travelling modes are more important since they have a smaller critical Reynolds number. We have considered several positive non-dimensional frequencies and several values of *D*. For *ω*=7.9, figure 7 shows the neutral stability curves for different values of *D*. It is interesting to observe that the wavenumber *β* in the azimuthal direction and wave-angle *ϵ* are both negative until *D* is greater than or equal to 10, but change to positive values when *D* is less than 10. Additional results for travelling waves may be found in Jasmine (2003).

### (c) Absolute instability results

Using the Newton-Raphson search procedure, we have solved the sixth-order system of equations to find branch points for the von Kármán flow, Bödewadt and Ekman flows. The Briggs (1964) criterion has been applied, with fixed *β*, to distinguish between convectively and absolutely unstable time-asymptotic responses to the initial boundary-value perturbation. Marginal curves for absolute instability are given in figures 8–10 for Rossby numbers −1, 0 and 1, and different values of *D*. For *D*=20, the eigenvalues we obtain agree well with those obtained earlier by Lingwood (1997).

## 5. Conclusions

A family of boundary-layer flows, which include von Kármán, Bödewadt and Ekman flows for a rotating incompressible fluid between a rotating disc and a stationary lid, is investigated using linear stability theory for different values of distance between a rotating disc and a stationary lid *D*. The stability parameters for the stationary and non-stationary waves are computed. Using a spectral method, stability diagrams are produced. Neutral stability curves are given for various real frequencies and several values of *D*. The region enclosed by these curves is shown to be convectively unstable. In all cases these flows have been found to be absolutely unstable above a particular Reynolds number and certain frequencies and wavenumbers of disturbance. As *D* decreases, the flow becomes increasingly absolutely stable. It is expected that the parallel-flow approximation will have some small numerical effect on the stability calculations, but that the general absolute instability characteristics discussed in this paper are still relevant (for small *D*) to the physical behaviour of the flows, as has been shown for the particular case of the von Kármán flow by Lingwood (1995, 1996), where the onset of transition and absolute instability coincide when *D* is infinite.

The absolute instability always occurs for travelling (non-zero frequency) waves for the von Kármán boundary-layer. As *D* is decreased, the stationary waves also become absolutely stable. The stationary waves are of particular importance because they are excited by unavoidable roughnesses on the surface of the disc, and are therefore often observed in experiments as so-called crossflow vortices. With decreasing *D*, the flows become increasingly stable in both the convective and absolute senses. For the Bödewadt layer, the onset of convective and absolute instability occurs almost simultaneously at very low Reynolds numbers. Decrease in the value of *D* leads to increase in the critical Reynolds number, and to the flows becoming much more absolutely stable.

## Footnotes

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

- © 2005 The Royal Society