Rotating-disk Boundary Layer Research Articles

Transport enhancement in the laminar rotating disk boundary layer

Enhanced momentum and heat transport in a three-dimensional rotating disk boundary layer due to the consideration of a wavy disk surface is an already known fact in the literature. Such a transport enhancement is primarily associated with two fundamental attributes of the wavy disk: the increased surface area and the enhanced convection due to the surface undulations. However, there exists no further information about their individual role, mutual interaction, and individual share in enhancing the convective transport in the rotating disk boundary layer. This study aims to investigate these facts in detail by considering radially heterogeneous different wavy configurations on the disk surface. It is revealed that the said attributes contribute non-trivially to enhance the transport phenomena. Their impactful role is identified due to the strengths of circumferential and crossflow velocity components, the magnitude of flow swirl angle, and hence due to the coefficients of skin friction and their inter relationship. In this regard, a fundamental benchmark regarding the relative strength of the two coefficients of skin friction is identified from the flat disk case. Decisive observations regarding the impact of surface undulations and the increased surface area of the wavy disk on the enhanced convective transport are made with the aid of this benchmark. It is observed that the enhanced transport phenomena, especially the heat transfer, mainly depend upon the further dominance of the circumferential velocity component. In the situations of increased dominance of rotational flow Kw>Kf, the increased convection and surface area collectively contribute to enhance the thermal transport while in a reverse situation Kw<Kf the convection contributes adversely by even reducing the impact of the increased surface area. In the neutral situation where the dominance of rotational velocity is neither increased nor decreased (i.e., Kw≈Kf), any increase in the transport phenomena is wholly due to the increased surface area. Therefore, the presence of surface undulations or increase in surface area is not the only criterion for an optimum rise in the transport phenomena, rather the impact of surface undulations on the velocity profiles is important which is manipulated due to their positioning on the disk surface. It is observed that the presence of surface undulations in the central and intermediate regions on the disk surface causes to increase the gradients of the crossflow velocity while their presence in the near rim location causes to enhance the gradients of the circumferential velocity. Thus, the configurations that have higher undulations in the near rim region outperform the other configurations.

Onset of turbulence in rotor–stator cavity flows

Numerous studies have indicated that turbulence typically initiates along the boundary layer of the stationary disk within a rotor–stator cavity. To describe the transition process to turbulence on the stationary side of a closed rotor–stator cavity, a comprehensive approach combining global linear stability analysis with direct numerical simulation was adopted in the present study. The proposed model aligns with that of Yim et al. (J. Fluid Mech., vol. 848, 2018, pp. 631–647), who investigated the stability characteristics of the rotating-disk boundary layer in a rotor–stator cavity. In order to achieve a stable inflow for the stationary-disk boundary layer, we rotate the shroud together with the rotating disk. Through careful global stability analysis, the predominant spiral mode exhibiting the highest instability in the boundary layer of the stationary disk was discerned, corroborating observations from simulations. Initially, the spiral mode undergoes linear amplification, reaches a state of linear saturation and enters the nonlinear regime. Following nonlinear saturation in the flow field, a circular wave mode arises due to the influence of mean flow distortion. As the Reynolds number attained a sufficiently high level, the interplay between the downstream-propagating circular mode and spiral mode amplified disturbances in the boundary layer of the stationary disk, ultimately leading to the development of localised turbulence at the mid-radius of the rotor–stator cavity. Notably, the present study is the first to elucidate the coexistence of laminar–transitional–turbulent flow states in the stationary-disk boundary layer through direct numerical simulations.

Study on transition to turbulence of rotating-disk boundary layer in a rotor-stator cavity with temperature gradient

Study on transition to turbulence of rotating-disk boundary layer in a rotor-stator cavity with temperature gradient

Receptivity of the rotating disk boundary layer to traveling disturbances

An adjoint approach is developed to undertake a receptivity study of the rotating disk boundary layer. The adjoint linearized Navier–Stokes equations are first derived in cylindrical coordinates. A receptivity formula is then formulated that specifies the response of stationary and traveling linear perturbations to an external force, including sources of momenta and mass and unsteady wall motion. Using the parallel flow approximation, in which the radial dependence of the undisturbed flow is ignored, receptivity characteristics are computed for a broad range of temporal frequencies, radial wavenumbers, azimuthal mode numbers, and Reynolds numbers. The type-I crossflow instability attains a maximum amplitude for external forces fixed near the wall-normal location of the critical layer (i.e., α¯rF+βG=ω), and the type-II Coriolis instability achieves larger amplitudes when external forces are located in the vicinity of a vanishing effective shear stress (i.e., α¯rF′+βG′=0). Sources of radial momenta fixed about these wall-normal locations establish larger-sized disturbances than equivalent-sized sources of azimuthal momenta, wall-normal momenta, and mass. At the disk surface, motion along the wall-normal direction induces a stronger receptivity response than wall motions acting along the radial and azimuthal directions. In general, the crossflow instability achieves larger-sized amplitudes than the Coriolis instability, with the peak response realized for Reynolds numbers near the critical conditions for linear instability.

Effect of slip on the linear stability of the rotating disk boundary layer

The linear stability of the rotating disk boundary layer with surface roughness is investigated. Surface roughness is modeled using slip boundary conditions [M. Miklavčič and C. Y. Wang, Z. Angew. Math. Phys. 55, 235–246 (2004)], which establish concentric grooves, radial grooves, and isotropic roughness. The effect on the stationary crossflow and Coriolis instabilities is analyzed by applying slip conditions to the undisturbed flow and linear disturbances. This analysis builds on the work of Cooper et al. [Phys. Fluids 27, 014107 (2015)], who modeled slip effects on the base flow but applied the no-slip condition to the linear perturbations. Neutral stability curves and critical parameter settings for linearly unstable behavior are computed for several radial and azimuthal slip length settings. The application of slip on the linear disturbances has a significant impact on the flow stability. In particular, the Coriolis instability undergoes considerable destabilization in the instance of concentric grooves (i.e., radial slip) and radial grooves with sufficiently large azimuthal slip lengths. In addition, concentric grooves destabilize the crossflow instability when the radial slip length is small. Moreover, in the instance of isotropic roughness, the stabilizing effect is markedly less than the observations of Cooper et al. [Phys. Fluids 27, 014107 (2015)]. Finally, an energy analysis is undertaken to ascertain the physical mechanisms brought about by surface roughness.

Linear impulse response in three-dimensional oscillatory boundary layers formed by periodic modulation of a rotating disk

A numerical investigation is undertaken on the development of linear disturbances in the rotating disk boundary layer, in which a time-periodic modulation is applied to the disk rotation rate. The model gives a prototypical example of a three-dimensional oscillatory boundary layer, by adding a Stokes layer to the von Kármán flow that develops on a steady disk. The study extends the Floquet analysis of Morgan et al. (J. Fluid Mech., vol. 925, 2021, A20), who showed that disk modulation stabilises the stationary convective instabilities found on the steady rotating disk. Using a radial homogeneous flow approximation, whereby the radial dependence of the basic state is ignored, disturbance development is simulated for several modulation settings, with flow conditions matched to both convective and absolute forms of linear instability. Disturbances excited via a stationary periodic wall forcing display behaviour consistent with that found using Floquet theory; time-periodic modulation stabilises the cross-flow instability by reducing the radial growth rate. In addition, convective and absolute instabilities, generated by an impulsive wall forcing, are both stabilised by the introduction of modulation to the disk rotation rate. Modulation establishes a significant reduction in both the temporal growth rate and the disturbance amplitude as it propagates away from the impulse origin. Moreover, greater stabilising control benefits are realised as the modulation amplitude increases.

An adjoint approach for computing the receptivity of the rotating disc boundary layer to surface roughness

An adjoint approach is developed to compute the receptivity of the rotating disc boundary layer to surface roughness. The adjoint linearised Navier–Stokes equations, in cylindrical coordinates, are derived and receptivity characteristics are computed for a broad range of azimuthal mode numbers using a fully equivalent velocity–vorticity formulation. For each set of flow conditions (i.e. azimuthal mode number), the adjoint method only requires that the linear and adjoint solutions be computed once. Thus, the adjoint approach offers significant computational and time advantages over alternative receptivity schemes (i.e. direct linearised Navier–Stokes) as they can be used to instantaneously compute the receptivity of boundary layer disturbances to many environmental mechanisms. Stationary cross-flow disturbances are established by randomly distributed surface roughness that is periodic in the azimuthal direction and modelled via a linearisation of the no-slip condition on the disc surface. Each roughness distribution is scaled on its respective root-mean-square. A Monte-Carlo type uncertainty quantification analysis is performed, whereby mean receptivity amplitudes are computed by averaging over many thousands of roughness realisations with variable length and wavelength filters. The amplitude of the cross-flow instability is significantly larger for roughness distributions near the conditions for neutral linear instability, while roughness elements radially outboard have a negligible effect on the receptivity process. Furthermore, receptivity increases sharply for roughness distributions that encompass wavelength scales equivalent to that associated with the cross-flow instability. Finally, mean receptivity characteristics are used to predict the radial range that stationary cross-flow vortices achieve amplitudes sufficient to invalidate the linear stability assumptions.

Control of stationary convective instabilities in the rotating disk boundary layer via time-periodic modulation

The control of stationary convective instabilities in the rotating disk boundary layer via a time-periodic modulation of the disk rotation rate is investigated. The configuration provides an archetypal example of a three-dimensional temporally periodic boundary layer, encompassing both the von Kármán and Stokes boundary layers. A velocity–vorticity formulation of the governing perturbation equations is deployed, together with a numerical procedure that utilises the Chebyshev-tau method. Floquet theory is used to determine the linear stability properties of these time-periodic flows. The addition of a time-periodic modulation to the otherwise steady disk rotation rate establishes a stabilising effect. In particular, for a broad range of modulation frequencies, the growth of the stationary convective instabilities is suppressed and the critical Reynolds number for the onset of both the cross-flow and Coriolis instabilities is raised to larger values than that found for the steady disk without modulation. An energy analysis is undertaken, where it is demonstrated that time-periodic modulation induces a reduction in the Reynolds stress energy production and an increase in the viscous dissipation across the boundary layer. Comparisons are made with other control techniques, including distributed surface roughness and compliant walls.

Effects of partial slip on the local-global linear stability of the infinite rotating disk boundary layer

A numerical investigation is undertaken on the effect of small-scale surface roughness on the local absolute and global stability of the flow due to a rotating disk. Surface roughness is modeled via the imposition of the partial-slip wall boundary condition, with radial and concentric anisotropic roughnesses and isotropic roughness considered. The effect of the partial-slip parameters on the neutral characteristics for absolute instability is presented, while the azimuthal mode numbers required for global linear instability to occur are determined for the genuine inhomogeneous base flow. Predictions for the threshold values for the azimuthal mode numbers needed for globally unstable behavior are also computed by coupling solutions of the Ginzburg–Landau equation with the local linear stability properties obtained using the homogeneous flow approximation. These are found to be in excellent agreement with the exact values realized from the numerical simulations. In general, surface roughness is demonstrated to stabilize the absolute instability and the global linear instabilities.

Non-similar heat transfer phenomenon over non-isothermal rotating disk

Heat transfer phenomenon from a free rotating flat disk has been analyzed with the consideration of variable surface temperature cases. A new class of variable surface temperature of the disk has been considered for which the heat transfer phenomenon becomes non-similar in nature whereby the momentum transport continues to remain of self-similar nature. In view of the available literature, only those variable disk temperatures have so far been considered for which the self-similarity of the associated heat transport phenomenon does not break down; such a collection of variable disk temperature functions, in general, belongs to the power-law family. In this study, consideration has been given to those forms of variable disk temperature functions which do not belong to this family. Specifically, the sinusoidal, the exponentially increasing/decreasing and the polynomial-type (increasing/decreasing) surface temperatures of the rotating disk have been considered for which the self-similar solution is impossible and these have never been studied so far to the best of our knowledge. The motivation behind this study is twofold: first, to extend the study of heat transfer phenomenon in the rotating disk boundary layer from self-similar to non-similar nature, and second, to sort out the best possible situations for which an enhanced heat transfer process is guaranteed. Our analysis reveals the achievement of significant heat transfer enhancement for the cases of exponential and polynomial-type (increasing) specifications of the variable disk temperature function. For example, exponentially increasing disk temperature of a free rotating disk in the quiescent air yields 27% augmentation in heat transfer rate in comparison with the isothermal case. Similarly, an enhancement of 15% in the heat transfer rate is obtained for the case of (increasing) polynomial-type disk temperature when compared with the isothermal case whereby the other specifications show a reverse trend with some interesting findings highlighting the insulating role of rotating disk under certain circumstances. In addition to these conclusions, the considered cases have been investigated in detail and the results have been reported in several tables which would serve as a reference for the future experimental and theoretical studies in this direction.

On the stability of a heated rotating-disk boundary layer in a temperature-dependent viscosity fluid

The paper presents a linear stability analysis of the temperature-dependent boundary-layer flow over a rotating disk. Gas- and liquid-type responses of the viscosity to temperature are considered, and the disk rotates in both a quiescent and an incident axial flow. Temperature-dependent-viscosity flows are typically found to be less stable than the temperature independent cases, with temperature dependences that produce high wall viscosities yielding the least stable flows. Conversely, increasing the incident axial flow strength produces greater flow stability. Transitional Reynolds numbers for these flows are then approximated through an eN-type analysis and are found to vary in approximate concordance with the critical Reynolds number. Examination of the component energy contributions shows that flow stability is affected exclusively through changes to the mean flow. The results are discussed in the context of chemical vapor deposition reactors.

Global linear instability of rotating-cone boundary layers in a quiescent medium

The global linear stability of the family of infinite rotating-cone boundary layers in an otherwise still fluid is investigated using a velocity-vorticity form of the linearized Navier-Stokes equations. The formulation is separable with respect to the azimuthal direction. Thus, disturbance development is simulated for a single azimuthal mode number. Numerical simulations are conducted for an extensive range of cone half-angles (ψ∈[20∘:80∘]) and stability parameters (Reynolds number, azimuthal mode number), where conditions are taken to be near those specifications necessary for the onset of absolute instability. A localized impulsive wall forcing is implemented that excites disturbances that form wave packets. This allows the disturbance evolution to be traced in the spatial-temporal plane. When a homogeneous flow approximation is utilised that neglects the spatial variation of the basic state, linear perturbations display characteristics consistent with local stability theory. For disturbances to the genuine spatially dependent inhomogeneous flow, global linear instability characterized by a faster than exponential temporal growth arises for azimuthal mode numbers greater than the conditions for critical absolute instability. Furthermore, a reasonable prediction for the azimuthal mode number needed to bring about a change in global behavior is achieved by coupling solutions of the Ginzburg-Landau equation with local stability properties. Thus, the local-global stability behavior is qualitatively similar to that found in the infinite rotating-disk boundary layer and many other globally unstable flows.

EXPERIMENTAL INVESTIGATION OF FLOW RESPONSE TO STATIONARY DISTURBANCE IN ROTATING-DISK FLOW

Three-dimensional boundary-layer flow is well known for its abrupt and sharp transition from laminar to turbulent regime. The presented study is a first attempt to achieve the target of delaying the natural transition to turbulence. The behaviour of two different shaped and sized stationary disturbances (in the laboratory frame) on the rotating-disk boundary layer flow is investigated. These disturbances are placed at dimensionless radial location (Rf = 340) which lies within the convectively unstable zone over a rotating-disk. Mean velocity profiles were measured using constant-temperature hot-wire anemometry. By careful analysis of experimental data, the instability of these disturbance wakes and its estimated orientation within the boundary-layer were investigated.

On the impulse response and global instability development of the infinite rotating-disc boundary layer

Linear disturbance development in the von Kármán boundary layer on an infinite rotating-disc is investigated for an extensive range of azimuthal mode numbers$n$. The study expands upon earlier investigations that were limited to those values of$n$located near the onset of absolute instability (Lingwood,J. Fluid Mech., vol. 299, 1995, pp. 17–33), where disturbances to the genuine inhomogeneous flow were shown to be globally stable (Davies & Carpenter,J. Fluid Mech., vol. 486, 2003, pp. 287–329). Numerical simulations corresponding to azimuthal mode numbers greater than the conditions for critical absolute instability display a form of global linear instability that is characterised by a faster than exponential temporal growth, similar in appearance to that found on the rotating-disc with mass suction (Thomas & Davies,J. Fluid Mech., vol. 724, 2010, pp. 510–526) and other globally unstable flows (Huerre & Monkewitz,Annu. Rev. Fluid Mech., vol. 22, 1990, pp. 473–537). Solutions indicate that a change in the global behaviour arises for $n\in [80:100]$that is marginally greater than those disturbances studied previously. Furthermore, the Reynolds number associated with the larger azimuthal mode numbers coincides with the upper bound of experimental predictions for transition. Thus, the local–global linear stability of the infinite rotating-disc is similar to the scenario outlined by Huerre & Monkewitz (1990) that states a region of local absolute instability is necessary but not sufficient for global instability to ensue. Conditions are derived to predict the azimuthal mode number needed to bring about a change in global behaviour, based on solutions of the linearised complex Ginzburg–Landau equation coupled with numerical simulations of disturbances to the radially homogeneous flow. The long term response is governed by a detuning effect, based on radial variations of the temporal frequency and matching shifts in temporal growth that increases for larger$n$, eventually attaining values sufficient to engineer global linear instability. The analysis is extended to include mass transfer through the disc surface, with similar conclusions drawn for disturbances to large enough azimuthal mode numbers. Finally, we conclude that the high$n$modes are unlikely to have a strong influence on disturbance development and transition in the von Kármán flow, as they will be unable to establish themselves across an extended radial range before nonlinear effects are triggered by the huge growth associated with the wavepacket maxima of the lower$n$-valued convective instabilities.

Transition to turbulence in the rotating disk boundary layer of a rotor–stator cavity

The transition to turbulence in the rotating disk boundary layer is investigated in a closed cylindrical rotor–stator cavity via direct numerical simulation (DNS) and linear stability analysis (LSA). The mean flow in the rotor boundary layer is qualitatively similar to the von Kármán self-similarity solution. The mean velocity profiles, however, slightly depart from theory as the rotor edge is approached. Shear and centrifugal effects lead to a locally more unstable mean flow than the self-similarity solution, which acts as a strong source of perturbations. Fluctuations start rising there, as the Reynolds number is increased, eventually leading to an edge-driven global mode, characterized by spiral arms rotating counter-clockwise with respect to the rotor. At larger Reynolds numbers, fluctuations form a steep front, no longer driven by the edge, and followed downstream by a saturated spiral wave, eventually leading to incipient turbulence. Numerical results show that this front results from the superposition of several elephant front-forming global modes, corresponding to unstable azimuthal wavenumbers $m$, in the range $m\in [32,78]$. The spatial growth along the radial direction of the energy of these fluctuations is quantitatively similar to that observed experimentally. This superposition of elephant modes could thus provide an explanation for the discrepancy observed in the single disk configuration, between the corresponding spatial growth rates values measured by experiments on the one hand, and predicted by LSA and DNS performed in an azimuthal sector, on the other hand.

Turbulence in the rotating-disk boundary layer investigated through direct numerical simulations

Direct numerical simulations (DNS) are reported for the turbulent rotating-disk boundary layer for the first time. Two turbulent simulations are presented with overlapping small and large Reynolds numbers, where the largest corresponds to a momentum-loss Reynolds number of almost 2000. Simulation data are compared with experimental data from the same flow case reported by Imayama et al. (2014), and also a comparison is made with a numerical simulation of a two-dimensional turbulent boundary layer (2DTBL) over a flat plate reported by Schlatter and Örlü (2010). The agreement of the turbulent statistics between experiments and simulations is in general very good, as well as the findings of a missing wake region and a lower shape factor compared to the 2DTBL. The simulations also show rms-levels in the inner region similar to the 2DTBL. The simulations validate Imayama et al.’s results showing that the rotating-disk turbulent boundary layer in the near-wall region contains shorter streamwise (azimuthal) wavelengths than the 2DTBL, probably due to the outward inclination of the low-speed streaks. Moreover, all velocity components are available from the simulations, and hence the local flow angle, Reynolds stresses and all terms in the turbulent kinetic energy equation are also discussed. However there are in general no large differences compared to the 2DTBL, hence the three-dimensional effects seem to have only a small influence on the turbulence.

Heat transfer enhancement in rotating disk boundary layer

A generally admitted fact about the nanofluids is the expedition of heat transfer process in comparison to pure fluids. The calculation of enhanced rate of heat transfer depends strongly upon the nanofluid modeling. Following the experimental evidence most of the researchers assume the nanofluid to be a homogeneous mixture. However, this is a severe condition that results in under-prediction of heat transfer rates. Due to the ongoing convection phenomena the nanoparticle concentration is actually non-homogeneous within the boundary-layer because of the presence of concentration gradients. The objective of this study is to calculate the heat transfer enhancement in 3-D boundary-layer when the working fluid is a nanofluid. The rotating disk geometry, which perhaps serves as the bench mark for the 3-D boundary-layers, have been chosen for the purpose here. The non-homogeneous nanofluid modeling has been utilized and a percent increase in Nusselt number has been calculated. Detailed analyses of flow and heat transfer phenomena for nanofluids have been conducted under the influence of several physical parameters.

Transition to turbulence in the rotating-disk boundary-layer flow with stationary vortices

This paper proposes a resolution to the conundrum of the roles of convective and absolute instability in transition of the rotating-disk boundary layer. It also draws some comparison with swept-wing flows. Direct numerical simulations based on the incompressible Navier–Stokes equations of the flow over the surface of a rotating disk with modelled roughness elements are presented. The rotating-disk flow has been of particular interest for stability and transition research since the work by Lingwood (J. Fluid Mech., vol. 299, 1995, pp. 17–33) where an absolute instability was found. Here stationary disturbances develop from roughness elements on the disk and are followed from the linear stage, growing to saturation and finally transitioning to turbulence. Several simulations are presented with varying disturbance amplitudes. The lowest amplitude corresponds approximately to the experiment by Imayama et al. (J. Fluid Mech., vol. 745, 2014a, pp. 132–163). For all cases, the primary instability was found to be convectively unstable, and secondary modes were found to be triggered spontaneously while the flow was developing. The secondary modes further stayed within the domain, and an explanation for this is a proposed globally unstable secondary instability. For the low-amplitude roughness cases, the disturbances propagate beyond the threshold for secondary global instability before becoming turbulent, and for the high-amplitude roughness cases the transition scenario gives a turbulent flow directly at the critical Reynolds number for the secondary global instability. These results correspond to the theory of Pier (J. Engng Maths, vol. 57, 2007, pp. 237–251) predicting a secondary absolute instability. In our simulations, high temporal frequencies were found to grow with a large amplification rate where the secondary global instability occurred. For smaller radial positions, low-frequency secondary instabilities were observed, tripped by the global instability.

Large scale motions of multiple limit-cycle high Reynolds number annular and toroidal rotor/stator cavities

Rotating cavity flows are essential components of industrial applications but their dynamics are still not fully understood when it comes to the relation between the fluid organization and monitored pressure fluctuations. From computer hard-drives to turbo-pumps of space launchers, designed devices often produce flow oscillations that can either destroy the component prematurely or produce too much noise. In such a context, large scale dynamics of high Reynolds number rotor/stator cavities need better understanding especially at the flow limit-cycle or associated statistically stationary state. In particular, the influence of curvature as well as cavity aspect ratio on the large scale organization and flow stability at a fixed rotating disc Reynolds number is fundamental. To probe such flows, wall-resolved large eddy simulation is applied to two different rotor/stator cylindrical cavities and one annular cavity. Validation of the predictions proves the method to be suited and to capture the disc boundary layer patterns reported in the literature. It is then shown that in complement to these disc boundary layer analyses, at the limit-cycle the rotating flows exhibit characteristic patterns at mid-height in the homogeneous core pointing the importance of large scale features. Indeed, dynamic modal decomposition reveals that the entire flow dynamics are driven by only a handful of atomic modes whose combination links the oscillatory patterns observed in the boundary layers as well as in the core of the cavity. These fluctuations form macro-structures, born in the unstable stator boundary layer and extending through the homogeneous inviscid core to the rotating disc boundary layer, causing its instability under some conditions. More importantly, the macro-structures significantly differ depending on the configuration pointing the need for deeper understanding of the influence of geometrical parameters as well as operating conditions.

Global stability behaviour for the BEK family of rotating boundary layers

Numerical simulations were conducted to investigate the linear global stability behaviour of the Bödewadt, Ekman, von Kármán (BEK) family of flows, for cases where a disc rotates beneath an incompressible fluid that is also rotating. This extends the work reported in recent studies that only considered the rotating-disc boundary layer with a von Kármán configuration, where the fluid that lies above the boundary layer remains stationary. When a homogeneous flow approximation is made, neglecting the radial variation of the basic state, it can be shown that linearised disturbances are susceptible to absolute instability. We shall demonstrate that, despite this prediction of absolute instability, the disturbance development exhibits globally stable behaviour in the BEK boundary layers with a genuine radial inhomogeneity. For configurations where the disc rotation rate is greater than that of the overlying fluid, disturbances propagate radially outwards and there is only a convective form of instability. This replicates the behaviour that had previously been documented when the fluid did not rotate beyond the boundary layer. However, if the fluid rotation rate is taken to exceed that of the disc, then the propagation direction reverses and disturbances grow while convecting radially inwards. Eventually, as they approach regions of smaller radii, where stability is predicted according to the homogeneous flow approximation, the growth rates reduce until decay takes over. Given sufficient time, such disturbances can begin to diminish at every radial location, even those which are positioned outwards from the radius associated with the onset of absolute instability. This leads to the confinement of the disturbance development within a finitely bounded region of the spatial–temporal plane.