Three-dimensional instabilities in the boundary-layer flow over a long rectangular plate

2011 ◽  
Vol 681 ◽  
pp. 411-433 ◽  
Author(s):  
HEMANT K. CHAURASIA ◽  
MARK C. THOMPSON
Keyword(s):  
Reynolds Number ◽  
Shear Layer ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Vortical Flow ◽  
Forced Flow ◽  
Recirculation Region ◽  
Two Dimensional ◽  
Optimal Perturbation ◽  
Dimensional Instability

A detailed numerical study of the separating and reattaching flow over a square leading-edge plate is presented, examining the instability modes governing transition from two- to three-dimensional flow. Under the influence of background noise, experiments show that the transition scenario typically is incompletely described by either global stability analysis or the transient growth of dominant optimal perturbation modes. Instead two-dimensional transition effectively can be triggered by the convective Kelvin–Helmholtz (KH) shear-layer instability; although it may be possible that this could be described alternatively in terms of higher-order optimal perturbation modes. At least in some experiments, observed transition occurs by either: (i) KH vortices shedding downstream directly and then almost immediately undergoing three-dimensional transition or (ii) at higher Reynolds numbers, larger vortical structures are shed that are also three-dimensionally unstable. These two paths lead to distinctly different three-dimensional arrangements of vortical flow structures. This paper focuses on the mechanisms underlying these three-dimensional transitions. Floquet analysis of weakly periodically forced flow, mimicking the observed two-dimensional quasi-periodic base flow, indicates that the two-dimensional vortex rollers shed from the recirculation region become globally three-dimensionally unstable at a Reynolds number of approximately 380. This transition Reynolds number and the predicted wavelength and flow symmetries match well with those of the experiments. The instability appears to be elliptical in nature with the perturbation field mainly restricted to the cores of the shed rollers and showing the spatial vorticity distribution expected for that instability type. Indeed an estimate of the theoretical predicted wavelength is also a good match to the prediction from Floquet analysis and theoretical estimates indicate the growth rate is positive. Fully three-dimensional simulations are also undertaken to explore the nonlinear development of the three-dimensional instability. These show the development of the characteristic upright hairpins observed in the experimental dye visualisations. The three-dimensional instability that manifests at lower Reynolds numbers is shown to be consistent with an elliptic instability of the KH shear-layer vortices in both symmetry and spanwise wavelength.

The wake behind a cylinder rolling on a wall at varying rotation rates

2010 ◽  
Vol 648 ◽  
pp. 225-256 ◽  
Author(s):  
B. E. STEWART ◽  
M. C. THOMPSON ◽  
T. LEWEKE ◽  
K. HOURIGAN
Keyword(s):  
Reynolds Number ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Wake Flow ◽  
Recirculation Region ◽  
Two Dimensional ◽  
Number Range ◽  
Rotation Rates ◽  
Transition Mechanism ◽  
Three Dimensional Simulations

A study investigating the flow around a cylinder rolling or sliding on a wall has been undertaken in two and three dimensions. The cylinder motion is specified from a set of five discrete rotation rates, ranging from prograde through to retrograde rolling. A Reynolds number range of 20–500 is considered. The effects of the nearby wall and the imposed body motion on the wake structure and dominant wake transitions have been determined. Prograde rolling is shown to destabilize the wake flow, while retrograde rotation delays the onset of unsteady flow to Reynolds numbers well above those observed for a cylinder in an unbounded flow.Two-dimensional simulations show the presence of two recirculation zones in the steady wake, the lengths of which increase approximately linearly with the Reynolds number. Values of the lift and drag coefficient are also reported for the steady flow regime. Results from a linear stability analysis show that the wake initially undergoes a regular bifurcation from a steady two-dimensional flow to a steady three-dimensional wake for all rotation rates. The critical Reynolds number Rec of transition and the spanwise wavelength of the dominant mode are shown to be highly dependent on, but smoothly varying with, the rotation rate of the cylinder. Varying the rotation from prograde to retrograde rolling acts to increase the value of Rec and decrease the preferred wavelength. The structure of the fully evolved wake mode is then established through three-dimensional simulations. In fact it is found that at Reynolds numbers only marginally (~5%) above critical, the three-dimensional simulations indicate that the saturated state becomes time dependent, although at least initially, this does not result in a significant change to the mode structure. It is only at higher Reynolds numbers that the wake undergoes a transition to vortex shedding.An analysis of the three-dimensional transition indicates that it is unlikely to be due to a centrifugal instability despite the superficial similarity to the flow over a backward-facing step, for which the transition mechanism has been speculated to be centrifugal. However, the attached elongated recirculation region and distribution of the spanwise perturbation vorticity field, and the similarity of these features with those of the flow through a partially blocked channel, suggest the possibility that the mechanism is elliptic in nature. Some analysis which supports this conjecture is undertaken.

Effective transition of steady flow over a square leading-edge plate

2012 ◽  
Vol 698 ◽  
pp. 335-357 ◽  
Author(s):  
Mark C. Thompson
Keyword(s):  
Reynolds Number ◽  
Shear Layer ◽  
Three Dimensional ◽  
Leading Edge ◽  
Maximum Energy ◽  
Transient Growth ◽  
Two Dimensional ◽  
Optimal Perturbation ◽  
Perturbation Mode ◽  
Optimal Mode

AbstractPrevious experimental studies have shown that the steady recirculation bubble that forms as the flow separates at the leading-edge corner of a long plate, becomes unsteady at relatively low Reynolds numbers of only a few hundreds. The reattaching shear layer irregularly releases two-dimensional vortices, which quickly undergo three-dimensional transition. Similar to the flow over a backward-facing step, this flow is globally stable at such Reynolds numbers, with transition to a steady three-dimensional flow as the first global instability to occur as the Reynolds number is increased to 393. Hence, it appears that the observed flow behaviour is governed by transient growth of optimal two-dimensional transiently growing perturbations (constructed from damped global modes) rather than a single three-dimensional unstable global mode. This paper quantifies the details of the transient growth of two- and three-dimensional optimal perturbations, and compares the predictions to other related cases examined recently. The optimal perturbation modes are shown to be highly concentrated in amplitude in the vicinity of the leading-edge corners and evolve to take the local shape of a Kelvin–Helmholtz shear-layer instability further downstream. However, the dominant mode reaches a maximum amplitude downstream of the position of the reattachment point of the shear layer. The maximum energy growth increases at 2.5 decades for each increment in Reynolds number of 100. Maximum energy growth of the optimal perturbation mode at a Reynolds number of 350 is greater than $1{0}^{4} $, which is typically an upper limit of the Reynolds number range over which it is possible to observe steady flow experimentally. While transient growth analysis concentrates on the evolution of wavepackets rather than continuous forcing, this appears consistent with longitudinal turbulence levels of up to 1 % for some water tunnels, and the fact that the optimal mode is highly concentrated close to the leading-edge corner so that an instantaneous projection of a perturbation field from a noisy inflow onto the optimal mode can be significant. Indeed, direct simulations with inflow noise reveal that a root-mean-square noise level of just 0.1 % is sufficient to trigger some unsteadiness at $\mathit{Re}= 350$, while a 0.5 % level results in sustained shedding. Three-dimensional optimal perturbation mode analysis was also performed showing that at $\mathit{Re}= 350$, the optimal mode has a spanwise wavelength of 11.7 plate thicknesses and is amplified 20 % more than the two-dimensional optimal disturbance. The evolved three-dimensional mode shows strong streamwise vortical structures aligned at a shallow angle to the plate top surface.

Bifurcation Characteristics of Flows in Rectangular Sudden Expansion Channels

2005 ◽  
Author(s):  
Francine Battaglia ◽  
George Papadopoulos
Keyword(s):  
Reynolds Number ◽  
Laminar Flow ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Expansion Ratio ◽  
Sudden Expansion ◽  
Two Dimensional ◽  
Effective Expansion ◽  
Three Dimensional Simulations

The effect of three-dimensionality on low Reynolds number flows past a symmetric sudden expansion in a channel was investigated. The geometric expansion ratio of in the current study was 2:1 and the aspect ratio was 6:1. Both experimental velocity measurements and two- and three-dimensional simulations for the flow along the centerplane of the rectangular duct are presented for Reynolds numbers in the range of 150 to 600. Comparison of the two-dimensional simulations with the experiments revealed that the simulations fail to capture completely the total expansion effect on the flow, which couples both geometric and hydrodynamic effects. To properly do so requires the definition of an effective expansion ratio, which is the ratio of the downstream and upstream hydraulic diameters and is therefore a function of both the expansion and aspect ratios. When the two-dimensional geometry was consistent with the effective expansion ratio, the new results agreed well with the three-dimensional simulations and the experiments. Furthermore, in the range of Reynolds numbers investigated, the laminar flow through the expansion underwent a symmetry-breaking bifurcation. The critical Reynolds number evaluated from the experiments and the simulations was compared to other values reported in the literature. Overall, side-wall proximity was found to enhance flow stability, helping to sustain laminar flow symmetry to higher Reynolds numbers in comparison to nominally two-dimensional double-expansion geometries. Lastly, and most importantly, when the logarithm of the critical Reynolds number from all these studies was plotted against the reciprocal of the effective expansion ratio, a linear trend emerged that uniquely captured the bifurcation dynamics of all symmetric double-sided planar expansions.

Three-dimensional dynamics and transition to turbulence in the wake of bluff objects

1992 ◽  
Vol 238 ◽  
pp. 1-30 ◽  
Author(s):  
George Em Karniadakis ◽  
George S. Triantafyllou
Keyword(s):  
Reynolds Number ◽  
Three Dimensional ◽  
Period Doubling ◽  
Reynolds Numbers ◽  
Transition To Turbulence ◽  
Two Dimensional ◽  
Near Wake ◽  
Vortex Filaments ◽  
Average Flow ◽  
Time Average

The wakes of bluff objects and in particular of circular cylinders are known to undergo a ‘fast’ transition, from a laminar two-dimensional state at Reynolds number 200 to a turbulent state at Reynolds number 400. The process has been documented in several experimental investigations, but the underlying physical mechanisms have remained largely unknown so far. In this paper, the transition process is investigated numerically, through direct simulation of the Navier—Stokes equations at representative Reynolds numbers, up to 500. A high-order time-accurate, mixed spectral/spectral element technique is used. It is shown that the wake first becomes three-dimensional, as a result of a secondary instability of the two-dimensional vortex street. This secondary instability appears at a Reynolds number close to 200. For slightly supercritical Reynolds numbers, a harmonic state develops, in which the flow oscillates at its fundamental frequency (Strouhal number) around a spanwise modulated time-average flow. In the near wake the modulation wavelength of the time-average flow is half of the spanwise wavelength of the perturbation flow, consistently with linear instability theory. The vortex filaments have a spanwise wavy shape in the near wake, and form rib-like structures further downstream. At higher Reynolds numbers the three-dimensional flow oscillation undergoes a period-doubling bifurcation, in which the flow alternates between two different states. Phase-space analysis of the flow shows that the basic limit cycle has branched into two connected limit cycles. In physical space the period doubling appears as the shedding of two distinct types of vortex filaments.Further increases of the Reynolds number result in a cascade of period-doubling bifurcations, which create a chaotic state in the flow at a Reynolds number of about 500. The flow is characterized by broadband power spectra, and the appearance of intermittent phenomena. It is concluded that the wake undergoes transition to turbulence following the period-doubling route.

Properties of the mean recirculation region in the wakes of two-dimensional bluff bodies

1997 ◽  
Vol 351 ◽  
pp. 167-199 ◽  
Author(s):  
S. BALACHANDAR ◽  
R. MITTAL ◽  
F. M. NAJJAR
Keyword(s):  
Reynolds Number ◽  
Shear Stress ◽  
Reynolds Stress ◽  
Vortex Shedding ◽  
Reynolds Numbers ◽  
Recirculation Region ◽  
Bluff Bodies ◽  
Two Dimensional ◽  
Stress Components ◽  
The Mean

The properties of the time- and span-averaged mean wake recirculation region are investigated in separated flows over several different two-dimensional bluff bodies. Ten different cases are considered and they divide into two groups: cylindrical geometries of circular, elliptic and square cross-sections and the normal plate. A wide Reynolds number range from 250 to 140000 is considered, but in all the cases the attached portion of the boundary layer remains laminar until separation. The lower Reynolds number data are from direct numerical simulations, while the data at the higher Reynolds number are obtained from large-eddy simulation and the experimental work of Cantwell & Coles (1983), Krothapalli (1996, personal communication), Leder (1991) and Lyn et al. (1995). Unlike supersonic and subsonic separations with a splitter plate in the wake, in all the cases considered here there is strong interaction between the shear layers resulting in Kármán vortex shedding. The impact of this fundamental difference on the distribution of Reynolds stress components and pressure in relation to the mean wake recirculation region (wake bubble) is considered. It is observed that in all cases the contribution from Reynolds normal stress to the force balance of the wake bubble is significant. In fact, in the cylinder geometries this contribution can outweigh the net force from the shear stress, so that the net pressure force tends to push the bubble away from the body. In contrast, in the case of normal plate, owing to the longer wake, the net contribution from shear stress outweighs that from the normal stress. At higher Reynolds numbers, separation of the Reynolds stress components into incoherent contributions provides more insight. The behaviour of the coherent contribution, arising from the dominant vortex shedding, is similar to that at lower Reynolds numbers. The incoherent contribution to Reynolds stress, arising from small-scale activity, is compared with that of a canonical free shear layer. Based on these observations a simple extension of the wake model (Sychev 1982; Roshko 1993a, b) is proposed.

Experimental Study on the Stability of the Flow Behind an Accelerating Circular Cylinder

2007 ◽  
Author(s):  
A. Inasawa ◽  
K. Toda ◽  
M. Asai
Keyword(s):  
Reynolds Number ◽  
Circular Cylinder ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Cylinder Wake ◽  
Global Instability ◽  
The Stability ◽  
Dimensional Instability ◽  
Wake Oscillation

Disturbance growth in the wake of a circular cylinder moving at a constant acceleration is examined experimentally. The cylinder is installed on a carriage moving in the still air. The results show that the critical Reynolds number for the onset of the global instability leading to a self-sustained wake oscillation increases with the magnitude of acceleration, while the Strouhal number of the growing disturbance at the critical Reynolds number is not strongly dependent on the magnitude of acceleration. It is also found that with increasing the acceleration, the Ka´rma´n vortex street remains two-dimensional even at the Reynolds numbers around 200 where the three-dimensional instability occurs to lead to the vortex dislocation in the case of cylinder moving at constant velocity or in the case of cylinder wake in the steady oncoming flow.

scholarly journals Localised turbulence in a circular pipe flow with gradual expansion

2015 ◽  
Vol 771 ◽  
Author(s):  
Kamal Selvam ◽  
Jorge Peixinho ◽  
Ashley P. Willis
Keyword(s):  
Reynolds Number ◽  
Pipe Flow ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Finite Amplitude ◽  
Transition To Turbulence ◽  
Circular Pipe ◽  
Recirculation Region ◽  
Circular Pipe Flow ◽  
Gradual Expansion

We report the results of three-dimensional direct numerical simulations for incompressible viscous fluid in a circular pipe flow with a gradual expansion. At the inlet, a parabolic velocity profile is applied together with a constant finite-amplitude perturbation to represent experimental imperfections. Initially, at low Reynolds number, the solution is steady. As the Reynolds number is increased, the length of the recirculation region near the wall grows linearly. Then, at a critical Reynolds number, a symmetry-breaking bifurcation occurs, where linear growth of asymmetry is observed. Near the point of transition to turbulence, the flow experiences oscillations due to a shear layer instability for a narrow range of Reynolds numbers. At higher Reynolds numbers, the recirculation region breaks into a turbulent state which remains spatially localised and unchanged when the perturbation is removed from the flow. Spatial correlation analysis suggests that the localised turbulence in the gradual expansion possesses a different flow structure from the turbulent puff of uniform pipe flow.

Three-dimensional instability of a plane free shear layer: an experimental study of the formation and evolution of streamwise vortices

1988 ◽  
Vol 189 ◽  
pp. 53-86 ◽  
Author(s):  
J. C. Lasheras ◽  
H. Choi
Keyword(s):  
Shear Layer ◽  
Strain Field ◽  
Principal Direction ◽  
Three Dimensional ◽  
Streamwise Vortex ◽  
Shear Instability ◽  
Two Dimensional ◽  
Free Shear Layer ◽  
Dimensional Instability ◽  
Vortex Tubes

The three-dimensional development of a plane free shear layer subjected to small sinusoidal perturbations periodically placed along the span is experimentally studied. Both laser induced fluorescence and direct interface visualization are used to monitor the interface between the two fluids. The development of the different flow stabilities is obtained through analysis of the temporal and spatial evolution of the interface separating the two streams. It is shown that the characteristic time of growth of the two-dimensional shear instability is much shorter than that of the three-dimensional instability. The primary Kelvin-Helmholtz instability develops first, leading to the formation of an almost two-dimensional array of spanwise vortex tubes. Under the effect of the strain field created by the evolving spanwise vortices, the perturbed vorticity existing on the braids undergoes axial stretching, resulting in the formation of vortex tubes whose axes are aligned with the principal direction of the positive strain field. During the formation of these streamwise vortex tubes, the spanwise vortices maintain, to a great extent, their two-dimensionality, suggesting an almost uncoupled development of both instabilities. The vortex tubes formed through the three-dimensional instability of the braids further undergo nonlinear interactions with the spanwise vortices inducing on their cores a wavy undulation of the same wavelength, but 180° phase shifted with respect to the perturbation. In addition, it is shown that owing to the nature of the three-dimensional instability, the effect of vertical and axial perturbations are coupled. Finally, the influence of the amplitude and wavelength of the perturbation on the development of the two- and three-dimensional instabilities is described.

A numerical study of the motion of drops in Poiseuille flow. Part 1. Lateral migration of one drop

2000 ◽  
Vol 411 ◽  
pp. 325-350 ◽  
Author(s):  
SAEED MORTAZAVI ◽  
GRÉTAR TRYGGVASON
Keyword(s):  
Reynolds Number ◽  
Steady State ◽  
Poiseuille Flow ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Lateral Position ◽  
Physical Parameters ◽  
Computational Domain ◽  
Two Dimensional ◽  
Lateral Migration

The cross-stream migration of a deformable drop in two-dimensional Hagen–Poiseuille flow at finite Reynolds numbers is studied numerically. In the limit of a small Reynolds number (< 1), the motion of the drop depends strongly on the ratio of the viscosity of the drop fluid to the viscosity of the suspending fluid. For viscosity ratio 0.125 a drop moves toward the centre of the channel, while for ratio 1.0 it moves away from the centre until halted by wall repulsion. The rate of migration increases with the deformability of the drop. At higher Reynolds numbers (5–50), the drop either moves to an equilibrium lateral position about halfway between the centreline and the wall – according to the so-called Segre–Silberberg effect or it undergoes oscillatory motion. The steady-state position depends only weakly on the various physical parameters of the flow, but the length of the transient oscillations increases as the Reynolds number is raised, or the density of the drop is increased, or the viscosity of the drop is decreased. Once the Reynolds number is high enough, the oscillations appear to persist forever and no steady state is observed. The numerical results are in good agreement with experimental observations, especially for drops that reach a steady-state lateral position. Most of the simulations assume that the flow is two-dimensional. A few simulations of three-dimensional flows for a modest Reynolds number (Re = 10), and a small computational domain, confirm the behaviour seen in two dimensions. The equilibrium position of the three-dimensional drop is close to that predicted in the simulations of two-dimensional flow.

Dynamics and stability of the wake behind tandem cylinders sliding along a wall

2013 ◽  
Vol 722 ◽  
pp. 291-316 ◽  
Author(s):  
A. Rao ◽  
M. C. Thompson ◽  
T. Leweke ◽  
K. Hourigan
Keyword(s):  
Three Dimensional ◽  
Flow Characteristics ◽  
Reynolds Numbers ◽  
Two Dimensional ◽  
Feedback Cycle ◽  
Tandem Cylinders ◽  
Two Dimensional Flow ◽  
Shedding Frequency ◽  
Dimensional Instability ◽  
Unsteady Regimes

AbstractThe dynamics and stability of the flow past two cylinders sliding along a wall in a tandem configuration is studied numerically for Reynolds numbers ($\mathit{Re}$) between 20 and 200, and streamwise separation distances between 0.1 and 10 cylinder diameters. For cylinders at close separations, the onset of unsteady two-dimensional flow is delayed to higher $\mathit{Re}$ compared with the case of a single sliding cylinder, while at larger separations, this transition occurs earlier. For Reynolds numbers above the threshold, shedding from both cylinders is periodic and locked. At intermediate separation distances, the wake frequency shifts to the subharmonic of the leading-cylinder shedding frequency, which appears to be due to a feedback cycle, whereby shed leading-cylinder vortices interact strongly with the downstream cylinder to influence subsequent leading-cylinder shedding two cycles later. In addition to the shedding frequency, the drag coefficients for the two cylinders are determined for both the steady and unsteady regimes. The three-dimensional stability of the flow is also investigated. It is found that, when increasing the Reynolds number at intermediate separations, an initial three-dimensional instability develops, which disappears at higher $\mathit{Re}$. The new two-dimensional steady flow again becomes unstable, but with a different three-dimensional instability mode. At very close spacings, when the two cylinders are effectively seen by the flow as a single body, and at very large spacings, when the cylinders form independent wakes, the flow characteristics are similar to those of a single cylinder sliding along a wall.

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