## Abstract

A linear stability theory is presented for the boundary-layer flow produced by an infinite disc rotating at constant angular velocity in otherwise undisturbed fluid. The theory is developed in the limit of long waves and when the effects of viscosity on the waves can be neglected. This is the parameter regime recently identified by the author in a numerical stability investigation where a curious new type of instability was found in which disturbances propagate and grow exponentially in the direction normal to the disc, (i.e. the growth takes place in a region of zero mean shear). The theory describes the mechanisms controlling the instability, the role and location of critical points, and presents a saddle-point analysis describing the large-time evolution of a wave packet in frames of reference moving normal to the disc. The theory also shows that the previously obtained numerical solutions for numerically large wavelengths do indeed lie in the asymptotic long-wave regime, and so the behaviour and mechanisms described here may apply to a number of cross-flow instability problems.

## 1. Introduction

Localized disturbances to an unstable shear flow can grow in two distinct ways. The flow is called convectively unstable if the wave packet produced by an impulsive disturbance propagates away from its source, for example, as shown in the experiment on the Blasius boundary-layer by Gaster & Grant (1975). In such flows, the neighbourhood of the impulsive source is eventually left undisturbed, and it is then usual to study the effects of periodic forcing, as in the experiment by Schubauer & Skramstad (1947). This results in growth in space, but not in time (i.e. once the start-up transients have propagated out of the system). Alternatively, the disturbance can grow in the reference frame of the source, eventually leading to nonlinear behaviour in the neighbourhood of the source. The flow is then called absolutely unstable.

The possibility that disturbances could evolve in these qualitatively different ways was first recognized by Twiss (1951) in the field of plasma physics. However, a mathematical method for identifying which is present in a particular stability problem was not developed until the 1960s (see Briggs 1964). These methods were first applied to a fluid mechanics problem by Tam (1971), and became widely used in fluid mechanics following the work of Huerre & Monkewitz (1985), though Gaster (1965) had already calculated spatial, as opposed to temporal, evolution of disturbances in a boundary layer, and independently made the distinction between absolute and convective instabilities.

Briggs' method consists of studying the behaviour of spatial branches. A spatial branch is the locus in the complex wavenumber plane of a root of the dispersion relation for a fixed imaginary part of the frequency, as the real part of the frequency is varied. When the imaginary part is positive and large enough, no spatial branches cross the real wavenumber axis (this result follows from a consideration of the principle of causality, and the finite propagation speeds of disturbances; see Briggs 1964). Spatial branches then confined to the upper half-plane correspond to downstream propagating waves, while spatial branches confined to the lower half-plane correspond to upstream propagating waves. If a spatial branch crosses the real axis when the imaginary part is reduced, then it represents an instability. If a spatial branch from the upper half-plane forms a branch-point with a spatial branch from the lower half-plane (such a branch-point is called a pinch-point) while the imaginary part of the frequency is positive, then the flow is absolutely unstable. Otherwise, the flow is convectively unstable, with the direction of spatial amplification determined by the origin of spatial branches that have crossed the real axis. Briggs' method corresponds to a saddle-point analysis for the case of a reference frame at rest, with the evolution of the spatial branches corresponding to establishing that the hills and valleys of the saddle are appropriately arranged so that the contour of integration of the inverse Fourier transform can lie entirely within the valleys of the saddle-point.

This paper presents a long-wave theory for a new type of convective instability in which a spatial branch that first crosses the real axis as the imaginary part of the frequency is reduced, then goes on to cross the imaginary axis while the imaginary part of the frequency is still positive. The significance of this behaviour lies in the fact that branch cuts arising from square roots in the eigenfunction outside the boundary layer are usually placed along the imaginary axis so that eigenfunctions decay exponentially outside the boundary layer. The branch cuts are moved to accommodate the spatial branches, but the part of the spatial branch lying on the other side of the imaginary axis of the complex wavenumber plane has eigenfunctions that diverge exponentially in the wall-normal direction. This apparently unphysical behaviour requires careful interpretation.

This scenario was recently found by the author (‘A new convective instability with exponential growth normal to the wall’, hereafter H1, unpublished) in a numerical study of the stability of the rotating-disc boundary-layer, and the divergent eigenfunctions were interpreted physically in terms of a convective instability with spatial growth in the wall-normal direction. A saddle-point analysis, where the saddle-points themselves sometimes had divergent eigenfunctions, was presented that predicted that wave packets could propagate and grow in the wall-normal direction, although this is a region of basic flow with zero shear. In order to provide an independent confirmation of this anomalous propagation and growth, the inverse Fourier transforms for an impulsive excitation were evaluated numerically along paths where the eigenfunctions decay exponentially. Nonetheless, their collective behaviour produced exponential growth in the wall-normal direction in agreement with the predictions of the saddle-point analysis based on divergent eigenfunctions.

Divergent eigenfunctions also arise in the propagation of leaky waves on a fluid loaded elastic plate (see Crighton 1989). In that case, the growth extends only a finite distance above the plate, proportional to the downstream distance. An important difference is that, here, the distance over which wall-normal growth occurs increases proportionally to time. Therefore, this growth has the characteristics of a convective instability in the wall-normal direction.

The results just described were all based on numerical solutions of the stability equation for inviscid disturbances (the Rayleigh equation) for numerically small wavenumbers. This suggests that a long-wave analytical theory may be possible that can explain the instability mechanisms at work, provide a basis for future viscous extensions, and indicate whether the results are in some sense generic, or quirks of the particular numerical values of parameters utilized previously. In this paper, we develop a long-wave theory to answer these questions. In §2, we describe the basic flow and derive the stability equations. The long-wave theory, based on the method of matched asymptotic expansions, is presented in §3, together with comparisons with numerical solutions. Conclusions are given in §4.

## 2. Governing equations

An infinite disc rotates at constant angular velocity *Ω*_{*} in an otherwise still viscous incompressible fluid of kinematic viscosity *ν*_{*} (dimensional quantities have an asterisk subscript). Viscous stresses at the disc surface accelerate fluid elements near to the disc into almost circular paths, but there is no radial pressure gradient to counter the centrifugal forces acting on these fluid elements, and so fluid near the disc spirals outwards. The disc thus acts as a centrifugal fan, and the fluid thrown outwards is replaced by an axial flow directed towards the disc surface. von Kármán's (1921) similarity solution describes this basic flow, and the first experimental and theoretical investigation into its stability was carried out by Gregory *et al*. (1955), who discovered the inviscid cross-flow instability mechanism (it depends on there being basic flow components in orthogonal directions) which produces a pattern of stationary (with respect to the disc) vortices.

However, our work follows from Lingwood's (1995) discovery that this flow can become absolutely unstable. She also found absolute instability in the inviscid problem for numerically small wavenumbers, which led Turkyilmazoglu & Gajjar (2001) to derive a long-wave inviscid theory for the pinch-points, see also the author's recent work (‘Inviscid long-wave theory for the absolute instability of the rotating-disc boundary-layer’, hereafter H2, unpublished).

We work in a system of cylindrical coordinates rotating with the disc. The axial and radial coordinates are *z*_{*} and *r*_{*}, respectively, the azimuthal angle is *θ*, time is *t*_{*} and *ρ*_{*} is the density of the fluid. The radial, azimuthal and axial velocities are *u*_{*}, *v*_{*} and *w*_{*}, respectively, and the pressure is *p*_{*}, giving equations of motion(2.1)(2.2)(2.3)(2.4)where the differential operators are(2.5a,b)

Lengths are scaled by the characteristic viscous length-scale, and time by the angular velocity of the disc: *r*_{*}=(*ν*_{*}/*Ω*_{*})^{1/2}*r*, *z*_{*}=(*ν*_{*}/*Ω*_{*})^{1/2}*z*, *t*_{*}=*t*/*Ω*_{*}. Flow variables are separated into an axisymmetric steady basic flow, which respects von Kármán's similarity structure, and a more general unsteady part, whose amplitude is characterized by a small parameter *δ*≪1:(2.6)(2.7)(2.8)(2.9)

Reduction of the equations for the disturbances to an ordinary differential equation in and so on is only possible far from the axis of rotation, and this is the limit we shall work within. Therefore, let *R*_{*} be the dimensional position of interest on the disc, then a Reynolds number, *Re*, can be introduced that is the ratio of *R*_{*} to the characteristic viscous length-scale: *Re*=*R*_{*}(*Ω*_{*}/*ν*_{*})^{1/2}. A new radial coordinate, *ρ*, is introduced where *r*=*Re ρ*, *Re*≫1 and *ρ*=*O*(1) near the position of interest. A WKB formulation for the disturbances can now be adopted to exploit the separation of length-scales over which the basic flow evolves and characteristic wavelengths, i.e. *Re α*≫1 where *α* is the radial wavenumber. Disturbance terms are expanded in the form(2.10)where *Re β* is an integer. We assume *Re β*≫1 and so will neglect the discretization of *β* (which is discretized in units of *Re*^{−1}), the scaled azimuthal wavenumber.

Substituting (2.6)–(2.9) and expressions such as (2.10) for and into (2.1)–(2.4) and equating terms of *O*(*δ*^{0}) gives the basic flow equations with boundary conditions *U*(0)=*V*(0)=*W*(0)=0 and *U*(∞)=0, *V*(∞)=−1, which are solved as in H2. Equating terms of *O*(*δ*) while neglecting terms of *O*(*Re*^{−1}), when *z*=*O*(1), gives the linearized inviscid disturbance equations:(2.11)(2.12)(2.13)(2.14)where primes denote partial differentiation with respect to *z*. These equations become invalid close to the disc and near critical points (where the phase velocity and basic flow velocity coincide) because the viscous terms must be reinstated in these thin regions. The viscous correction to the inviscid dispersion relation will be larger than *O*(*Re*^{−1}). Choudhari (1995) found viscous corrections at *O*(*Re*^{−1/4}) for stationary vortices, although other scalings may apply in the present unsteady case. However, viscous corrections will not be included here.

These equations show that in the inviscid limit, not only are the viscous terms neglected, but also the Coriolis terms, streamline curvature terms, nonparallel terms (those which involve ∂/∂*ρ*) and the axial basic flow component, *W* are neglected as well. The Rayleigh equation can be derived by eliminating *u*, *v* and *p*, but we shall utilise (2.11)–(2.14).

## 3. Long-wave theory

Normally, in an inviscid long-wave theory for boundary-layer flows (see Turkyilmazoglu & Gajjar 2001) or H2, the phase velocity is the same order of magnitude as the wavenumber, which gives a critical point close to the wall. This turns out to be the situation for the long-wave part of the spatial branch lying in the lower half of the complex wavenumber plane. However, the part lying in the upper half-plane has two critical points in the main part of the boundary layer, away from the wall. We can capture both parts of the spatial branch by assuming that the phase velocity is *O*(1), then look at appropriate limiting cases. Although an *O*(1) phase velocity might usually be associated with *O*(1) wavelengths, we will show that long waves are nonetheless possible when a pair of critical points approach a stationary point of the basic velocity profile, and this is the case here.

Let *ϵ*≪1 be a small parameter characterizing the smallness of the wavenumber in the long-wave limit. Therefore, let(3.1a,b,c)where higher order terms can be added to the series for the frequency *ω*, but are not needed for present purposes. Note that waves cannot be allowed to become so long that the WKB approximation of §2 fails, or other viscous effects become important, so we require *Re*^{−τ}≪*α*≪1 for some undetermined positive *τ*. The scalings in (3.1*a–c*) lead to a disturbance structure with an upper layer, which is outside the boundary layer, where the basic flow is uniform, and a main layer, which is the thickness of the boundary layer, and within which two critical points are embedded.

### (a) Upper layer

We define an upper-layer variable, *Z*, where *z*=*Z*/*ϵ* and *Z*=*O*(1) in the upper layer, and introduce upper-layer expansions(3.2a,b)(3.2c,d)where the subscript u denotes upper layer and 0 denotes the leading order term in an expansion. Substituting (3.1*a–c*) and (3.2*a–d*), and *U*=0 and *V*=−1 into (2.11)–(2.14), and then solving the resulting differential equations gives(3.3a,b)where and √ denote the root with positive real part so that solutions decay exponentially and satisfy the homogeneous outer boundary condition. The square root thus has branch points at *α*_{0}=±i, with branch cuts emanating from these points along the imaginary axes of the complex wavenumber plane away from the origin, as illustrated in figure 1 below.

### (b) Main layer

The expansions in the main layer, where *z*=*O*(1), are(3.4a,b,c,d)where the subscript m denotes a main-layer variable and the order of magnitude of the pressure has been chosen to match the pressure in the upper layer; the magnitudes of the velocity components are then fixed by the requirement that they interact at leading order with the pressure. Substitute (3.1*a–c*) and (3.4*a–d*) into (2.11)–(2.14), then equating powers of *ϵ* at leading order leads to differential equations for the main-layer variables, and their solutions are:(3.5a,b)where *Q*=*U*+*V*/*α*_{0} is the basic flow resolved in the direction of the wave-vector and *c*_{0}=*ω*_{0}/*α*_{0} is the leading order phase velocity. However, there are two critical points *z*=*z*_{c1} and *z*=*z*_{c2} satisfying *Q*(*z*)=*c*_{0} (we take *Re*(*z*_{c1})<*Re*(*z*_{c2})), and it is convenient to extract these singularities from (3.5*b*) and instead use:(3.6)where the divergent part of the integral for large *z* has also been extracted (*Q*(∞)=−1/*α*_{0}), and subscripts c1 and c2 denote quantities evaluated at the critical points. In addition, the integration path is taken below *z*_{c1} and above *z*_{c2} since *Re*(*Q*′_{c1})>0 and *Re*(*Q*′_{c2})<0 so that solutions are the limiting case of the corresponding viscous problem as the viscosity tends to zero (see Lin 1955). The homogeneous wall boundary condition *w*_{m0}(*ρ*,0)=0 has been satisfied.

### (c) Matching between the layers

Matching the pressures gives *P*_{m0}=*P*_{u0}, and note that the relevant behaviour of (3.3*b*) for small *Z* is(3.7)while for large *z*, (3.6), expressed in terms of *Z*, behaves as(3.8)where the constant(3.9)The coefficients of *Z* in (3.7) and (3.8) match, but the constant terms only match if the square brackets in (3.8) are *O*(*ϵ*^{−1}). We now show that this can occur in two different ways.

### (d) Lower part of spatial branch

When *ω*_{0}=*ϵω*_{1}, *z*_{c1}=*ϵω*_{1}/(*α*_{0}*Q*′_{0}) and *Q*′_{c1}=*Q*′_{0}=*Q*′(0), and then matching the leading order constant term in (3.8) with the leading order constant term in (3.7) gives dispersion relation:(3.10)(which is the classic long-wave result that would have been obtained by assuming *ω*=*O*(*ϵ*^{2}) at the start). Figure 1*a* shows the spatial branch for Im(*ω*_{1})=0.7547. Spatial branches with Im(*ω*_{1})<0.7547 cross the imaginary axis, while those with Im(*ω*_{1})>0.7547 do not. Those parts that do cross the imaginary axis require the branch cuts to be moved, and have eigenfunctions proportional to in the upper layer. This is in good qualitative agreement with the numerical results in H1, where it was shown to produce convective instability in the *z*-direction.

The point marked by the circle corresponds to *Re*(*ω*_{1})=0, with *Re*(*ω*_{1})>0 and *Re*(*ω*_{1})<0 on this spatial branch below and above this point respectively. As Im(*ω*_{1}) increases, this point with purely imaginary frequency approaches *α*_{0}=−*V*′_{0}/*U*′_{0}, which gives the wave angle where *Q*′_{0}=0, and therefore corresponds to the long-wave stationary vortices of Hall (1986). As Im(*ω*_{1})→0, this point with purely imaginary frequency approaches the branch point at *α*_{0}=−i.

However, different scalings are needed to capture the upper part of this spatial branch, see §3*f*.

### (e) Group velocity in wall-normal direction

The propagation and growth of wave packets are obtained by considering the initial-value problem for an impulsive disturbance, which can be obtained by evaluating the appropriate inverse Fourier transforms. The impulse response for a particular azimuthal wavenumber, at the order of approximation where the basic flow is considered homogeneous in the radial direction, is of the form(3.11)where *Δ*=0 is the dispersion relation and the contours *A* and *F* are placed according to the principles of causality (see Briggs 1964). The *ω* integral is evaluated using the residue theorem, and outside the boundary layer the exponential behaviour of the eigenfunction in (3.3*a*,*b*) can be incorporated in the wavy term in (3.11), to give(3.12)where *ω*=*ω*(*α*) satisfies the dispersion relation, and *z*=Re*ζ* so that axial and radial coordinates are scaled in the same way. A similar step can be seen in (2.6) of Crighton (1989). The saddle-point of the exponent of (3.12) is now found from(3.13)and makes the dominant contribution to (3.12) at large times.

We consider frames of reference at fixed radius *ρ*, that move away from the disc (i.e. *ρ*/*t*=0 and *ζ*/*t*≥0). As shown in H1, the frame of reference with the strongest growth rate has the temporal growth rate of the spatial branch that first touches the imaginary axis, such as the one shown in figure 1*a*, and then the saddle-point also lies on the imaginary axis of the *α*-plane. In the long-wave limit, using the dispersion relation (3.10) in (3.13) gives a wave packet whose peak amplitude grows at rate Im(*ω*/*ρ*)=0.7547(*β*/*ρ*)^{2} and which propagates into the freestream at velocity *ζ*/*t*=0.5270*β*/*ρ* when *ρ*/*t*=0.

### (f) Upper part of spatial branch

The resolved basic profile has a saddle-point at *z*_{s}, that is, *Q*′_{s}=*Q*′(*z*_{s})=0. The square brackets in (3.8) become *O*(*ϵ*^{−1}) when the critical points are close enough to *z*_{s}, so that *Q*′_{c1} and *Q*′_{c2} are small enough. Therefore, consider(3.14)The critical points are calculated from *Q*(*z*_{c})=*ω*_{0}/*α*_{0} by expanding about *z*_{s}:(3.15)and, hence, since *Re*(*Q*″_{s})<0,(3.16a,b)

The velocities in the main and upper layers can now be matched, giving at leading order the dispersion relation(3.17)Both critical points lie below the height of the inflexion point of *Q*, and so the phase jumps of each make a destabilizing contribution (the solutions can be continued from parameters where the critical points lie away from *z*_{s} and Lin's phase jump rule applies to where they approach *z*_{s}, keeping the correct phase jumps). The root of (3.17) that respects these destabilizing terms is:(3.18)since this gives Im(*ω*_{2})>0 when *α*_{0} is real and *Q*″_{s}<0 (the one-third power denotes the root with positive real part).

Points on a spatial branch for fixed values of *β*/*ρ*, Im(*ω*/*ρ*) and any given *Re*(*α*) are obtained by simultaneously solving *Q*′(*z*_{s})=0 with the imaginary part of the frequency given by (3.1*c*), (3.14) and (3.18) set equal to the given value of Im(*ω*/*ρ*), for *z*_{s} and Im(*α*). Figure 1*b* shows the results for *β*/*ρ*=0.004 and Im(*ω*/*ρ*)=0.000 015 compared with numerical solutions of the Rayleigh equation. There is good quantitative agreement between this long-wave theory and the numerical solutions in the upper half-plane. A distinctive feature of this part of the *α*-plane is that *z*_{s}, and hence the two critical points, lie far from the real axis of the complex *z*-plane. The solution must be obtained by taking an integration path that lies above one critical point and below the other. Therefore, the path passes between the two critical points, and these critical points become asymptotically close to one another as *β*/*ρ*→0. This makes the numerical solution highly ill-conditioned in this limit.

Cowley *et al*. (1985) have also considered the stability of profiles with a stationary point away from the wall, but they considered a single viscous critical point at the stationary point in the case where pressure and velocity fields do not interact. Our theory is inviscid, interactive and involves two critical points close to the stationary point.

## 4. Conclusions

A long-wave theory has been developed for the spatial branches that shows the same anomalous behaviour found previously in the numerical study of the Rayleigh equation for the rotating-disc boundary-layer in H1, in which spatial modes cross both the real and imaginary axes of the complex wavenumber plane. Unstable spatial branches that cross the imaginary axis correspond to unbounded exponentially divergent eigenfunctions. Nonetheless, the initial-value problem is always well behaved, with this wall-normal growth only extending over finite distances at finite times. The physical interpretation is of a new type of convective instability with propagation and growth in the wall-normal direction. For long waves, an impulsive disturbance of scaled azimuthal wavenumber *β*/*ρ* at the disc surface generates a wave packet that travels out of the boundary layer at vertical velocity *ζ*/*t*=0.5270*β*/*ρ* at fixed radii, with growth rate Im(*ω*/*ρ*)=0.7547(*β*/*ρ*)^{2}, which is greater than the growth rate of the absolute instability Im(*ω*/*ρ*)=1.910(*β*/*ρ*)^{9/4} found in H2. This behaviour is controlled by a critical point close to the wall, and these results depend only on the basic velocity gradient at the wall. Therefore, this new type of instability is ‘generic’ and could appear in a wide range of cross-flow instability problems.

## Footnotes

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

- © 2005 The Royal Society