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Instabilities and transition in the rotating-disk boundary layer

Motivation

The rotating-disk boundary layer is the archetype of a three-dimensional boundary layer. Such boundary layers, where all three velocity components vary with wall-normal distance, are observed in a variety of practical configurations whenever a body rotates in or is swept through a viscous fluid. Examples include centrifugal pumps, ventilator fans, compressor stages and flows over aircraft wings or ship hulls. These flows develop along a streamwise direction and are known to display a rapid transition to a turbulent régime. Despite intensive research, the general mechanism responsible for turbulence onset remains unclear and no low-cost control strategy has yet been developed.

Context

The flow due to an infinite disk rotating in otherwise still fluid has served as the ideal configuration to study three-dimensional boundary layers ever since von Kármán (1921) obtained the basic flow as an exact similarity solution of the Navier–Stokes equations. The fluid near the disk acquires, by viscous stresses, an azimuthal velocity which linearly increases with radial distance. This circular motion results in centrifugal forces pulling the fluid outwards. The radial outflow induces, by continuity, a weak axial flow component towards the disk. This axial flow reaches a constant value far from the disk surface and counteracts diffusion of vorticity away from the disk, thus maintaining a constant boundary layer thickness in the entire system.

Laminar rotating-disk boundary layer

Interest in this flow has been renewed by Lingwood's discovery that the nature of the impulse response changes at a critical radius: inside the critical radius perturbations are advected outwards (convective instability), whereas beyond the critical radius perturbations grow in situ (absolute instability). Moreover this critical radius closely corresponds to the station of experimentally observed turbulence onset. Previous theoretical work has established that spatially developing systems displaying an absolutely unstable region give rise to self-sustained finite-amplitude structures. The properties of the rotating-disk flow precisely meet the requirements of this theory. Thus, the present investigation has been undertaken to address the fully nonlinear régime. The objective was to analyse the naturally selected finite-amplitude state and its secondary stability properties in order to elucidate the process responsible for the sudden transition to turbulence.

Primary waves

The first stage consisted in a complete characterization of how the local properties of the flow evolve with radial distance, i.e., in the computation of the local linear and nonlinear dispersion relations and associated eigenfunctions pertaining to each radial station. Whereas linear properties applying to small-amplitude perturbations are well established, the nonlinear equivalent had not yet been obtained.

Under the assumption of slow radial development of the basic flow, local saturated nonlinear wavetrains are obtained via numerical simulations of temporal evolution problems. These fully nonlinear waves take the form of spiral cross-flow vortices. Here the (1) azimuthal and (2) radial velocity components of a nonlinear saturated wave are shown over two wavelengths: (a) isolines of perturbation velocity fields,u (b) isolines of total velocity fields, (c) comparison of basic (thin lines) and total (thick lines) velocity profiles.

azimuthal velocity component of nonlinear wavetrain radial velocity component of nonlinear wavetrain

These profiles display several inflection points in both velocity components. It is thus very likely that these saturated crossflow vortices are unstable with respect to secondary perturbations. A multiple-scales analysis then consistently yields the global solution made up of these local linear and nonlinear wavetrains.

Self-sustained spatially extended structure

Having obtained the local linear and nonlinear waves in the boundary layer at each radial location, a global solution developing over an extended radial interval may be sought in the form of wavetrains that are slowly modulated in the radial direction. This approach is set on firm theoretical ground using WKBJ asymptotic techniques.

As demonstrated in earlier investigations, spatially developing systems display a nonlinear self-sustained state whenever a region of absolute instability is present. These finite-amplitude solutions (also called elephant global modes) are characterized by a stationary front located at the upstream transition from local convective to absolute instability. The selection mechanism is the following: in the absolutely unstable region amplified perturbations develop and their envelope advances upstream against the basic flow. At the station of neutral absolute instability a balance between upstream perturbation growth and downstream advection is reached and perturbations pile up at that location. Nonlinearities lead to saturation of the fluctuating amplitude and a stationary front is formed. This front generates a downstream propagating fully nonlinear wavetrain and an upstream exponentially decaying tail. It thus connects linear and nonlinear regions, acts as a source and effectively tunes the entire system to its frequency. In the present configuration, the self-sustained global solution displays the following spatial structure, where the outer region is convered by nonlinear saturated outward-spiralling cross-flow vortices while an exponentially decaying tail covers the inner region.

spatial structure of self-sustained nonlinear global mode

Secondary absolute instability

The procedure outlined above guarantees the existence of a global time-harmonic solution but does not tell us whether or not it is stable with respect to secondary perturbations. The experimental observation of a rapid transition to turbulence near the critical radius suggests that it is not. In order to to understand this transition the secondary stability of the saturated waves that make up the global solution must be carefully examined.

Linear secondary stability of the primary crossflow vortices is governed by Floquet theory. The numerical resolution of the associated eigenproblems yields the complete local secondary dispersion relation.

Whether or not the primary finite-amplitude waves are permanently affected by secondary disturbances depends on the absolute or convective nature of the secondary instability. Indeed, for convectively unstable secondary instabilities, an external impulse may only trigger a transient perturbation that is eventually carried away radially outwards. Without external noise and for a perfectly smooth rotating disk, transition can only occur because of secondary absolute instability of the naturally selected primary cross-flow vortices.

By carrying out an absolute stability analysis based on the local secondary dispersion relation, it has been found that the the saturated crossflow vortices that are naturally selected near the critical radius are absolutely unstable with respect to secondary perturbations. This strong secondary absolute instability explains why the naturally selected spiral vortices are not observed experimentally: as soon as the primary nonlinear vortices are generated near the critical radius, secondary perturbations develop in situ and are exponentially amplified and transition to turbulence immediately occurs.

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