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.

Numerical Integration of the Navier-Stokes Equations for a Disk Rotating in a Housing

1985 ◽  
Vol 40 (8) ◽  
pp. 789-799 ◽  
Author(s):  
A. F. Borghesani
Keyword(s):  
Boundary Layer ◽  
Cylindrical Cavity ◽  
Boundary Layer Thickness ◽  
Stokes Equations ◽  
Fluid Motion ◽  
Rotating Disk ◽  
Infinite Medium ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Viscous Torque

The Navier-Stokes equations for the fluid motion induced by a disk rotating inside a cylindrical cavity have been integrated for several values of the boundary layer thickness d. The equivalence of such a device to a rotating disk immersed in an infinite medium has been shown in the limit as d → 0. From that solution and taking into account edge effect corrections an equation for the viscous torque acting on the disk has been derived, which depends only on d. Moreover, these results justify the use of a rotating disk to perform accurate viscosity measurements.

Two-dimensional global low-frequency oscillations in a separating boundary-layer flow

2008 ◽  
Vol 614 ◽  
pp. 315-327 ◽  
Author(s):  
UWE EHRENSTEIN ◽  
FRANÇOIS GALLAIRE
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Perturbation Analysis ◽  
Boundary Layer Flow ◽  
Stokes Equations ◽  
Low Frequency ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Optimal Perturbation ◽  
Layer Flow

A separated boundary-layer flow at the rear of a bump is considered. Two-dimensional equilibrium stationary states of the Navier–Stokes equations are determined using a nonlinear continuation procedure varying the bump height as well as the Reynolds number. A global instability analysis of the steady states is performed by computing two-dimensional temporal modes. The onset of instability is shown to be characterized by a family of modes with localized structures around the reattachment point becoming almost simultaneously unstable. The optimal perturbation analysis, by projecting the initial disturbance on the set of temporal eigenmodes, reveals that the non-normal modes are able to describe localized initial perturbations associated with the large transient energy growth. At larger time a global low-frequency oscillation is found, accompanied by a periodic regeneration of the flow perturbation inside the bubble, as the consequence of non-normal cancellation of modes. The initial condition provided by the optimal perturbation analysis is applied to Navier–Stokes time integration and is shown to trigger the nonlinear ‘flapping’ typical of separation bubbles. It is possible to follow the stationary equilibrium state on increasing the Reynolds number far beyond instability, ruling out for the present flow case the hypothesis of some authors that topological flow changes are responsible for the ‘flapping’.

scholarly journals Non-axisymmetric rotating-disk flows: nonlinear travelling-wave states

2000 ◽  
Vol 413 ◽  
pp. 287-316 ◽  
Author(s):  
R. E. HEWITT ◽  
P. W. DUCK
Keyword(s):  
Boundary Layer ◽  
Stokes Equations ◽  
Rotating Disk ◽  
Travelling Wave ◽  
Numerical Results ◽  
Navier Stokes ◽  
Rotating Fluid ◽  
Navier Stokes Equations ◽  
Axisymmetric Solution ◽  
Wave States

We consider the classical problem of the laminar flow of an incompressible rotating fluid above a rotating, impermeable, infinite disk. There is a well-known class of solutions to this configuration in the form of an exact axisymmetric solution to the Navier–Stokes equations. However, the radial self-similarity that leads to the ‘rotating- disk equations’ can also be used to obtain solutions that are non-axisymmetric in nature, although (in general) this requires a boundary-layer approximation. In this manner, we locate several new solution branches, which are non-axisymmetric travelling-wave states that satisfy axisymmetric boundary conditions at infinity and at the disk. These states are shown to appear as symmetry-breaking bifurcations of the well-known axisymmetric solution branches of the rotating-disk equations. Numerical results are presented, which suggest that an infinity of such travelling states exist in some parameter regimes. The numerical results are also presented in a manner that allows their application to the analogous flow in a conical geometry.Two of the many states described are of particular interest. The first is an exact, nonlinear, non-axisymmetric, stationary state for a rotating disk in a counter-rotating fluid; this solution was first presented by Hewitt, Duck & Foster (1999) and here we provide further details. The second state corresponds to a new boundary-layer-type approximation to the Navier–Stokes equations in the form of azimuthally propagating waves in a rotating fluid above a stationary disk. This second state is a new non-axisymmetric alternative to the classical axisymmetric Bödewadt solution.

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.

Numerical investigation of the interaction of the Klebanoff-mode with a Tollmien–Schlichting wave

2002 ◽  
Vol 450 ◽  
pp. 1-33 ◽  
Author(s):  
HERMANN F. FASEL
Keyword(s):  
Boundary Layer ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Low Frequency ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Free Stream Turbulence ◽  
Turbulent Transition ◽  
Tollmien Schlichting

Direct numerical simulations (DNS) of the Navier–Stokes equations are used to investigate the role of the Klebanoff-mode in laminar–turbulent transition in a flatplate boundary layer. To model the effects of free-stream turbulence, volume forces are used to generate low-frequency streamwise vortices outside the boundary layer. A suction/blowing slot at the wall is used to generate a two-dimensional Tollmien–Schlichting (TS) wave inside the boundary layer. The characteristics of the fluctuations inside the boundary layer agree very well with those measured in experiments. It is shown how the interaction of the Klebanoff-mode with the two-dimensional TS-wave leads to the formation of three-dimensional TS-wavepackets. When the disturbance amplitudes reach a critical level, a fundamental resonance-type secondary instability causes the breakdown of the TS-wavepackets into turbulent spots.

scholarly journals The interaction of Blasius boundary-layer flow with a compliant panel: global, local and transient analyses

2017 ◽  
Vol 827 ◽  
pp. 155-193 ◽  
Author(s):  
Konstantinos Tsigklifis ◽  
Anthony D. Lucey
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Flow ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Travelling Wave ◽  
Navier Stokes ◽  
Global Instability ◽  
Blasius Boundary Layer ◽  
Layer Flow ◽  
High Level

We study the fluid–structure interaction (FSI) of a compliant panel with developing Blasius boundary-layer flow. The linearised Navier–Stokes equations in velocity–vorticity form are solved using a Helmholtz decomposition coupled with the dynamics of a plate-spring compliant panel couched in finite-difference form. The FSI system is written as an eigenvalue problem and the various flow- and wall-based instabilities are analysed. It is shown that global temporal instability can occur through the interaction of travelling wave flutter (TWF) with a structural mode or as a resonance between Tollmien–Schlichting wave (TSW) instability and discrete structural modes of the compliant panel. The former is independent of compliant panel length and upstream inflow disturbances while the specific behaviour arising from the latter phenomenon is dependent upon the frequency of a disturbance introduced upstream of the compliant panel. The inclusion of axial displacements in the wall model does not lead to any further global instabilities. The dependence of instability-onset Reynolds numbers with structural stiffness and damping for the global modes is quantified. It is also shown that the TWF-based global instability is stabilised as the boundary layer progresses downstream while the TSW-based global instability exhibits discrete resonance-type behaviour as Reynolds number increases. At sufficiently high Reynolds numbers, a globally unstable divergence instability is identified when the wavelength of its wall-based mode is longer than that of the least stable TSW mode. Finally, a non-modal analysis reveals a high level of transient growth when the flow interacts with a compliant panel which has structural properties capable of reducing TSW growth but which is prone to global instability through wall-based modes.

On the laminar–turbulent transition of the rotating-disk flow: the role of absolute instability

2014 ◽  
Vol 745 ◽  
pp. 132-163 ◽  
Author(s):  
Shintaro Imayama ◽  
P. Henrik Alfredsson ◽  
R. J. Lingwood
Keyword(s):  
Boundary Layer ◽  
Rotating Disk ◽  
Reynolds Numbers ◽  
Secondary Instability ◽  
Transition To Turbulence ◽  
Absolute Instability ◽  
Nonlinear Saturation ◽  
Global Mode ◽  
Turbulent Transition ◽  
Laminar Turbulent Transition

AbstractThis paper describes a detailed experimental study using hot-wire anemometry of the laminar–turbulent transition region of a rotating-disk boundary-layer flow without any imposed excitation of the boundary layer. The measured data are separated into stationary and unsteady disturbance fields in order to elaborate on the roles that the stationary and the travelling modes have in the transition process. We show the onset of nonlinearity consistently at Reynolds numbers, $R$, of $\sim $510, i.e. at the onset of Lingwood’s (J. Fluid Mech., vol. 299, 1995, pp. 17–33) local absolute instability, and the growth of stationary vortices saturates at a Reynolds number of $\sim $550. The nonlinear saturation and subsequent turbulent breakdown of individual stationary vortices independently of their amplitudes, which vary azimuthally, seem to be determined by well-defined Reynolds numbers. We identify unstable travelling disturbances in our power spectra, which continue to grow, saturating at around $R=585$, whereupon turbulent breakdown of the boundary layer ensues. The nonlinear saturation amplitude of the total disturbance field is approximately constant for all considered cases, i.e. different rotation rates and edge Reynolds numbers. We also identify a travelling secondary instability. Our results suggest that it is the travelling disturbances that are fundamentally important to the transition to turbulence for a clean disk, rather than the stationary vortices. Here, the results appear to show a primary nonlinear steep-fronted (travelling) global mode at the boundary between the local convectively and absolutely unstable regions, which develops nonlinearly interacting with the stationary vortices and which saturates and is unstable to a secondary instability. This leads to a rapid transition to turbulence outward of the primary front from approximately $R=565$ to 590 and to a fully turbulent boundary layer above 650.

Global linear instability of the rotating-disk flow investigated through simulations

2015 ◽  
Vol 765 ◽  
pp. 612-631 ◽  
Author(s):  
E. Appelquist ◽  
P. Schlatter ◽  
P. H. Alfredsson ◽  
R. J. Lingwood
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Rotating Disk ◽  
Linear Instability ◽  
Navier Stokes ◽  
Absolute Instability ◽  
Ginzburg Landau ◽  
Navier Stokes Equations ◽  
Rotating Disk Flow ◽  
Layer Flow

AbstractNumerical simulations of the flow developing on the surface of a rotating disk are presented based on the linearized incompressible Navier–Stokes equations. The boundary-layer flow is perturbed by an impulsive disturbance within a linear global framework, and the effect of downstream turbulence is modelled by a damping region further downstream. In addition to the outward-travelling modes, inward-travelling disturbances excited at the radial end of the simulated linear region, $r_{end}$, by the modelled turbulence are included within the simulations, potentially allowing absolute instability to develop. During early times the flow shows traditional convective behaviour, with the total energy slowly decaying in time. However, after the disturbances have reached $r_{end}$, the energy evolution reaches a turning point and, if the location of $r_{end}$ is at a Reynolds number larger than approximately $R=594$ (radius non-dimensionalized by $\sqrt{{\it\nu}/{\rm\Omega}^{\ast }}$, where ${\it\nu}$ is the kinematic viscosity and ${\rm\Omega}^{\ast }$ is the rotation rate of the disk), there will be global temporal growth. The global frequency and mode shape are clearly imposed by the conditions at $r_{end}$. Our results suggest that the linearized Ginzburg–Landau model by Healey (J. Fluid Mech., vol. 663, 2010, pp. 148–159) captures the (linear) physics of the developing rotating-disk flow, showing that there is linear global instability provided the Reynolds number of $r_{end}$ is sufficiently larger than the critical Reynolds number for the onset of absolute instability.

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

2019 ◽  
Vol XVI (2) ◽  
pp. 13-22
Author(s):  
Muhammad Ehtisham Siddiqui
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Flow ◽  
Three Dimensional ◽  
Rotating Disk ◽  
Careful Analysis ◽  
Laboratory Frame ◽  
Transition To Turbulence ◽  
Turbulent Regime ◽  
Mean Velocity ◽  
Layer 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.

Direct simulation of transition in an oscillatory boundary layer

1998 ◽  
Vol 371 ◽  
pp. 207-232 ◽  
Author(s):  
G. VITTORI ◽  
R. VERZICCO
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Turbulent Regime ◽  
Two Dimensional ◽  
Temporal Development ◽  
Direct Simulation ◽  
Navier Stokes Equations ◽  
Oscillatory Boundary Layer

Numerical simulations of Navier–Stokes equations are performed to study the flow originated by an oscillating pressure gradient close to a wall characterized by small imperfections. The scenario of transition from the laminar to the turbulent regime is investigated and the results are interpreted in the light of existing analytical theories. The ‘disturbed-laminar’ and the ‘intermittently turbulent’ regimes detected experimentally are reproduced by the present simulations. Moreover it is found that imperfections of the wall are of fundamental importance in causing the growth of two-dimensional disturbances which in turn trigger turbulence in the Stokes boundary layer. Finally, in the intermittently turbulent regime, a description is given of the temporal development of turbulence characteristics.

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