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

2018 ◽  
Vol 857 ◽  
pp. 239-269 ◽  
Author(s):  
Christian Thomas ◽  
Christopher Davies
Keyword(s):  
Boundary Layer ◽  
Numerical Simulations ◽  
Linear Instability ◽  
Absolute Instability ◽  
Global Instability ◽  
Rotating Disc ◽  
Azimuthal Mode ◽  
Global Behaviour ◽  
Von Karman ◽  
Global Linear Instability

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.

Absolute instability of the von Kármán, Bödewadt and Ekman flows between a rotating disc and a stationary lid

2005 ◽  
Vol 363 (1830) ◽  
pp. 1131-1144 ◽  
Author(s):  
Hosne Ara Jasmine ◽  
Jitesh S.B Gajjar
Keyword(s):  
Boundary Layer ◽  
Incompressible Fluid ◽  
Spectral Method ◽  
Convective Instability ◽  
Absolute Instability ◽  
Boundary Layer Flows ◽  
Rotating Disc ◽  
Von Karman ◽  
The Stability ◽  
Von Kármán

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.

scholarly journals On linear instability mechanisms in incompressible open cavity flow

2014 ◽  
Vol 752 ◽  
pp. 219-236 ◽  
Author(s):  
F. Meseguer-Garrido ◽  
J. de Vicente ◽  
E. Valero ◽  
V. Theofilis
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Thickness ◽  
Theoretical Study ◽  
Three Dimensional ◽  
Theoretical Data ◽  
Cavity Flow ◽  
Linear Instability ◽  
Open Cavity ◽  
Global Instability ◽  
Neutral Curves

AbstractA theoretical study of linear global instability of incompressible flow over a rectangular spanwise-periodic open cavity in an unconfined domain is presented. Comparisons with the limited number of results available in the literature are shown. Subsequently, the parameter space is scanned in a systematic manner, varying Reynolds number, incoming boundary-layer thickness and length-to-depth aspect ratio. This permits documenting the neutral curves and leading eigenmode characteristics of this flow. Correlations constructed using the results obtained collapse all available theoretical data on the three-dimensional instabilities.

An experimental study of edge effects on rotating-disk transition

2013 ◽  
Vol 716 ◽  
pp. 638-657 ◽  
Author(s):  
Shintaro Imayama ◽  
P. Henrik Alfredsson ◽  
R. J. Lingwood
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Boundary Layer Flow ◽  
Rotating Disk ◽  
Reynolds Numbers ◽  
Absolute Instability ◽  
Global Instability ◽  
Global Mode ◽  
Turbulent Transition ◽  
Layer Flow

AbstractThe onset of transition for the rotating-disk flow was identified by Lingwood (J. Fluid. Mech., vol. 299, 1995, pp. 17–33) as being highly reproducible, which motivated her to look for absolute instability of the boundary-layer flow; the flow was found to be locally absolutely unstable above a Reynolds number of 507. Global instability, if associated with laminar–turbulent transition, implies that the onset of transition should be highly repeatable across different experimental facilities. While it has previously been shown that local absolute instability does not necessarily lead to linear global instability: Healey (J. Fluid. Mech., vol. 663, 2010, pp. 148–159) has shown, using the linearized complex Ginzburg–Landau equation, that if the finite nature of the flow domain is accounted for, then local absolute instability can give rise to linear global instability and lead directly to a nonlinear global mode. Healey (J. Fluid. Mech., vol. 663, 2010, pp. 148–159) also showed that there is a weak stabilizing effect as the steep front to the nonlinear global mode approaches the edge of the disk, and suggested that this might explain some reports of slightly higher transition Reynolds numbers, when located close to the edge. Here we look closely at the effects the edge of the disk have on laminar–turbulent transition of the rotating-disk boundary-layer flow. We present data for three different edge configurations and various edge Reynolds numbers, which show no obvious variation in the transition Reynolds number due to proximity to the edge of the disk. These data, together with the application (as far as possible) of a consistent definition for the onset of transition to others’ results, reduce the already relatively small scatter in reported transition Reynolds numbers, suggesting even greater reproducibility than previously thought for ‘clean’ disk experiments. The present results suggest that the finite nature of the disk, present in all real experiments, may indeed, as Healey (J. Fluid. Mech., vol. 663, 2010, pp. 148–159) suggests, lead to linear global instability as a first step in the onset of transition but we have not been able to verify a correlation between the transition Reynolds number and edge Reynolds number.

scholarly journals Global stability of the rotating-disc boundary layer with an axial magnetic field

2013 ◽  
Vol 724 ◽  
pp. 510-526 ◽  
Author(s):  
Christian Thomas ◽  
Christopher Davies
Keyword(s):  
Magnetic Field ◽  
Boundary Layer ◽  
Linear Stability ◽  
Axial Magnetic Field ◽  
Numerical Study ◽  
Control Strategies ◽  
Base Flow ◽  
Global Instability ◽  
Rotating Disc ◽  
Radially Inhomogeneous

AbstractA numerical study is conducted to investigate the influence of a uniform axial magnetic field on the global linear stability of the rotating-disc boundary layer. Simulation results obtained using a radially homogenized base flow were found to be in excellent agreement with an earlier linear stability analysis, which indicated that an axial magnetic field can locally suppress both convective and absolute instabilities. However, the numerical results obtained for the genuine, radially inhomogeneous, flow indicate that a global form of instability develops for sufficiently large magnetic fields. The qualitative nature of the global instability is similar to that which was observed in a previous study, where mass suction was applied at the rotating disc surface. It is shown that, just as for the case with mass suction, it is possible to explain the promotion of global instability by considering a model that includes detuning effects, which are associated with the radial variation of locally defined absolute temporal frequencies. The recurrence of the same type of instability behaviour when two distinct flow control strategies are implemented, one using suction and the other an axial magnetic field, indicates that the phenomena described by the model may be considered generic.

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

2017 ◽  
Vol 836 ◽  
pp. 43-71 ◽  
Author(s):  
E. Appelquist ◽  
P. Schlatter ◽  
P. H. Alfredsson ◽  
R. J. Lingwood
Keyword(s):  
Boundary Layer ◽  
Stokes Equations ◽  
Low Frequency ◽  
Rotating Disk ◽  
High Amplitude ◽  
Transition To Turbulence ◽  
Navier Stokes ◽  
Absolute Instability ◽  
Global Instability ◽  
Roughness Elements

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.

scholarly journals The effects of mass transfer on the global stability of the rotating-disk boundary layer

2010 ◽  
Vol 663 ◽  
pp. 401-433 ◽  
Author(s):  
CHRISTIAN THOMAS ◽  
CHRISTOPHER DAVIES
Keyword(s):  
Boundary Layer ◽  
Mass Transfer ◽  
Rotating Disk ◽  
Disk Surface ◽  
Reynolds Numbers ◽  
Base Flow ◽  
Absolute Instability ◽  
Unstable Region ◽  
Global Behaviour ◽  
Simulation Results

Numerical simulations were conducted to investigate the effects of surface suction and injection on the global behaviour of linear disturbances in the rotating-disk boundary layer. This extends earlier work, which considered the case with no mass transfer. For disturbances in the genuine base flow, where radially inhomogeneity is retained, mass injection at the disk surface led to behaviour that remained qualitatively similar to that which was found when there was no mass transfer. The initial development of disturbances within the absolutely unstable region involved temporal growth and upstream propagation, as should be anticipated for an absolute instability. However, this did not persist indefinitely. Just as for the case without mass transfer, the simulation results suggested that convective behaviour would eventually dominate, for all the Reynolds numbers investigated. In marked contrast, the results obtained for flows with mass suction indicate a destabilization due to the effects of the base-flow radial inhomogeneity. It was possible to identify disturbances excited within the absolutely unstable region that grew continually, with a temporal growth rate that increased as the disturbance evolved. The strong locally stabilizing effect of suction on the absolute instability, which gives rise to large increases in critical Reynolds numbers, appears to be obtainable only at the expense of introducing a new form of global instability. Analogous forms of global behaviour can be found in impulse solutions of the linearized complex Ginzburg–Landau equation. These solutions were deployed to interpret and make comparisons with the numerical simulation results. They illustrate how the long-term behaviour of a disturbance can be determined by the precise balance between radial increases in temporal growth rates, corresponding shifts in temporal frequencies and diffusion/dispersion effects. This balance provides some insight into why disturbances that are absolutely unstable, for the homogenized version of the rotating-disk boundary-layer flow, may become, in the genuine radially inhomogeneous flow, either globally stable or globally unstable, depending on the level of mass transfer that is applied at the disk surface.

The stability of rotating-disc boundary-layer flow over a compliant wall. Part 2. Absolute instability

1997 ◽  
Vol 350 ◽  
pp. 261-270 ◽  
Author(s):  
A. J. COOPER ◽  
PETER W. CARPENTER
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Flow ◽  
Transition To Turbulence ◽  
Control Technique ◽  
Absolute Instability ◽  
Rotating Disc ◽  
Compliant Wall ◽  
Wall Compliance ◽  
Layer Flow ◽  
The Stability

A numerical study has been undertaken of the influence of a compliant boundary on absolute instability. In a certain parameter range absolute instability occurs in the boundary layer on a rotating disc, thereby instigating rapid transition to turbulence. The conventional use of wall compliance as a laminar-flow control technique has been to lower growth rates of convective instabilities. This has the effect of reducing amplification of disturbances as they propagate downstream. For absolute instability, however, only the suppression of its onset would be a significant gain. This paper addresses the question of whether passive wall compliance can be advantageous when absolute instability exists in a boundary layer.A theoretical model of a single-layer viscoelastic compliant wall was used in conjunction with the sixth-order system of differential equations which govern the stability of the boundary-layer flow over a rotating disc. The absolute/convective nature of the flow was ascertained by using a spatio-temporal analysis. Pinch-point singularities of the dispersion relation and a point of zero group velocity identify the presence of absolute instability. It was found that only a low level of wall compliance was enough to delay the appearance of absolute instability to higher Reynolds numbers. Beyond a critical level of wall compliance results suggest that complete suppression of absolute instability is possible. This would then remove a major route to transition in the rotating-disc boundary layer.

scholarly journals Bifurcations in shock-wave/laminar-boundary-layer interaction: global instability approach

2007 ◽  
Vol 579 ◽  
pp. 85-112 ◽  
Author(s):  
J.-CH. ROBINET
Keyword(s):  
Boundary Layer ◽  
Shock Wave ◽  
Stability Analysis ◽  
Numerical Simulations ◽  
Laminar Boundary Layer ◽  
Three Dimensional ◽  
Oblique Shock Wave ◽  
Global Instability ◽  
Layer Interaction ◽  
Boundary Layer Interaction

The principal objective of this paper is to study some unsteady characteristics of an interaction between an incident oblique shock wave impinging on a laminar boundary layer developing on a flat plate. More precisely, this paper shows that some unsteadiness, in particular the low-frequency unsteadiness, originates in a supercritical Hopf bifurcation related to the dynamics of the separated boundary layer. Various direct numerical simulations were carried out of a shock-wave/laminar-boundary-layer interaction (SWBLI). Three-dimensional unsteady Navier–Stokes equations are numerically solved with an implicit dual time stepping for the temporal algorithm and high-order AUSMPW+ scheme for the spatial discretization. A parametric study on the oblique shock-wave angle has been performed to characterize the unsteady behaviour onset. These numerical simulations have shown that starting from the incident shock angle and the spanwise extension, the flow becomes three-dimensional and unsteady. A linearized global stability analysis is carried out in order to specify and to find some characteristics observed in the direct numerical simulation. This stability analysis permits us to show that the physical origin generating the three-dimensional characters of the flow results from the existence of a three-dimensional stationary global instability.

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

2021 ◽  
Vol 926 ◽  
Author(s):  
Christian Thomas ◽  
Christopher Davies
Keyword(s):  
Boundary Layer ◽  
Surface Roughness ◽  
Stokes Equations ◽  
Flow Instability ◽  
Cross Flow ◽  
Navier Stokes ◽  
Rotating Disc ◽  
Azimuthal Mode ◽  
The Cross ◽  
Adjoint Approach

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.

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