Stability of the wakes of cylinders with triangular cross-sections

2018 ◽  
Vol 844 ◽  
pp. 721-745 ◽  
Author(s):  
Zhi Y. Ng ◽  
Tony Vo ◽  
Gregory J. Sheard
Keyword(s):  
Stability Analysis ◽  
Cross Sections ◽  
Reynolds Numbers ◽  
Base Flow ◽  
Half Period ◽  
Two Dimensional ◽  
Instability Modes ◽  
The Stability ◽  
Time Periodic ◽  
Mode A

The stability of the wakes of cylinders with triangular cross-sections at incidence is investigated using Floquet stability analysis to elucidate the effects of cylinder inclination on the dominant flow instability. The upper limit of the Reynolds numbers (scaled by the height projected by the cylinder in this study) at which the wake of the two-dimensional base flow is time periodic is$Re\approx 140$for most cylinder inclinations, exceeding which the flow becomes aperiodic, restricting the range of Reynolds numbers permitted for the stability analysis. Two different instability modes are predicted to manifest as the first-occurring mode at various cylinder inclinations – a regular mode possessing perturbation structures consistent with mode A dominates the wakes of cylinders at inclinations$\unicode[STIX]{x1D6FC}\lesssim 34.6^{\circ }$and$\unicode[STIX]{x1D6FC}\gtrsim 55.4^{\circ }$, with a subharmonic mode consistent with mode C emerging as the primary mode in the wakes of the cylinder at the intermediate range of inclinations. For all inclinations, the mode B branch is not detected within the range of Reynolds numbers examined. The peak instability growth rates corresponding to mode A for all cylinder inclinations describe a linear variation with$(Re-Re_{A})/Re_{A}$, where$Re_{A}$is the mode A transition Reynolds number, while those corresponding to mode C vary only approximately linearly. The generalized trend most pertinently shows mode C to develop more rapidly than mode A at inclinations which permit it. Examination of the near wake of the two-dimensional time-periodic base flow demonstrates the dependence of the development and intensity of mode C on imbalances in the flow solution over each shedding period, directly implying that the two-dimensional base flow solutions deviate from the half-period-flip map as the cylinder inclination is increased. The degree of asymmetry of the two-dimensional base flow relative to the ideal half-period-flip map is quantified using several measures. The results show distinctly different trends in these asymmetry measures between inclinations where modes A or C are dominant, agreeing with results from the stability analysis. The nature of the predicted instability modes at transition are also investigated by applying the Stuart–Landau equation, showing the transitions to be supercritical for all cylinder inclinations, with mode C being consistently more strongly supercritical than mode A.

Three-dimensional stability analysis for a salt-finger convecting layer

2018 ◽  
Vol 841 ◽  
pp. 636-653
Author(s):  
Ting-Yueh Chang ◽  
Falin Chen ◽  
Min-Hsing Chang
Keyword(s):  
Stability Analysis ◽  
Three Dimensional ◽  
Longitudinal Mode ◽  
Transverse Mode ◽  
Solute Diffusion ◽  
Two Dimensional ◽  
Grashof Number ◽  
Significant Difference ◽  
The Stability ◽  
Mode A

A three-dimensional linear stability analysis is carried out for a convecting layer in which both the temperature and solute distributions are linear in the horizontal direction. The three-dimensional results show that, for $Le=3$ and 100, the most unstable mode occurs invariably as the longitudinal mode, a vortex roll with its axis perpendicular to the longitudinal plane, suggesting that the two-dimensional results are sufficient to illustrate the stability characteristics of the convecting layer. Two-dimensional results show that the stability boundaries of the transverse mode (a vortex roll with its axis perpendicular to the transverse plane) and the longitudinal modes are virtually overlapped in the regime dominated by thermal diffusion and the regime dominated by solute diffusion, while these two modes hold a significant difference in the regime the salt-finger instability prevails. More precisely, the instability area in terms of thermal Grashof number $Gr$ and solute Grashof number $Gs$ is larger for the longitudinal mode than the transverse mode, implying that, under any circumstance, the longitudinal mode is always more unstable than the transverse mode.

scholarly journals Linear stability of confined flow around a 180-degree sharp bend

2017 ◽  
Vol 822 ◽  
pp. 813-847 ◽  
Author(s):  
Azan M. Sapardi ◽  
Wisam K. Hussam ◽  
Alban Pothérat ◽  
Gregory J. Sheard
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Base Flow ◽  
Two Dimensional ◽  
Two Dimensional Flow ◽  
The Stability

This study seeks to characterise the breakdown of the steady two-dimensional solution in the flow around a 180-degree sharp bend to infinitesimal three-dimensional disturbances using a linear stability analysis. The stability analysis predicts that three-dimensional transition is via a synchronous instability of the steady flows. A highly accurate global linear stability analysis of the flow was conducted with Reynolds number $\mathit{Re}<1150$ and bend opening ratio (ratio of bend width to inlet height) $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 5$. This range of $\mathit{Re}$ and $\unicode[STIX]{x1D6FD}$ captures both steady-state two-dimensional flow solutions and the inception of unsteady two-dimensional flow. For $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 1$, the two-dimensional base flow transitions from steady to unsteady at higher Reynolds number as $\unicode[STIX]{x1D6FD}$ increases. The stability analysis shows that at the onset of instability, the base flow becomes three-dimensionally unstable in two different modes, namely a spanwise oscillating mode for $\unicode[STIX]{x1D6FD}=0.2$ and a spanwise synchronous mode for $\unicode[STIX]{x1D6FD}\geqslant 0.3$. The critical Reynolds number and the spanwise wavelength of perturbations increase as $\unicode[STIX]{x1D6FD}$ increases. For $1<\unicode[STIX]{x1D6FD}\leqslant 2$ both the critical Reynolds number for onset of unsteadiness and the spanwise wavelength decrease as $\unicode[STIX]{x1D6FD}$ increases. Finally, for $2<\unicode[STIX]{x1D6FD}\leqslant 5$, the critical Reynolds number and spanwise wavelength remain almost constant. The linear stability analysis also shows that the base flow becomes unstable to different three-dimensional modes depending on the opening ratio. The modes are found to be localised near the reattachment point of the first recirculation bubble.

scholarly journals Influence of confinement on a two-dimensional wake

2011 ◽  
Vol 688 ◽  
pp. 297-320 ◽  
Author(s):  
Luca Biancofiore ◽  
François Gallaire ◽  
Richard Pasquetti
Keyword(s):  
Stability Analysis ◽  
Velocity Ratio ◽  
Reynolds Numbers ◽  
Base Flow ◽  
Wake Flow ◽  
Two Dimensional ◽  
Dimensional Parameters ◽  
Spatio Temporal ◽  
Spatial Locations ◽  
Theoretical Results

AbstractThe spatio-temporal development of an incompressible two-dimensional viscous wake flow confined by two flat slipping plates is investigated by means of direct numerical simulation (DNS), using a spectral Chebyshev multi-domain method. The limit between unstable and stable configurations is determined with respect to several non-dimensional parameters: the confinement, the velocity ratio and two different Reynolds numbers, $100$ and $500$. The comparison of such limit curves with theoretical results obtained by Juniper (J. Fluid Mech., vol. 565, 2006, pp. 171–195) confirms the existence of a region at moderate confinement where the instability is maximal. Moreover, instabilities are also observed under sustained co-flow, in the form of a vacillating front. Using a direct computation of the two-dimensional base flow, we perform a local linear stability analysis for several velocity profiles prevailing at different spatial locations, so as to determine the local spatio-temporal nature of the flow: convectively unstable or absolutely unstable. Comparisons of the DNS and local stability analysis results are provided and discussed.

scholarly journals Two- and three-dimensional wake transitions of an impulsively started uniformly rolling circular cylinder

2017 ◽  
Vol 826 ◽  
pp. 32-59 ◽  
Author(s):  
F. Y. Houdroge ◽  
T. Leweke ◽  
K. Hourigan ◽  
M. C. Thompson
Keyword(s):  
Stability Analysis ◽  
Circular Cylinder ◽  
Rapid Development ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Two Dimensional ◽  
Dimensional Flow ◽  
Instability Modes ◽  
Two Dimensional Flow ◽  
Lift And Drag

This paper presents the characteristics of the different stages in the evolution of the wake of a circular cylinder rolling without slipping along a wall at constant speed, acquired through numerical stability analysis and two- and three-dimensional numerical simulations. Reynolds numbers between 30 and 300 are considered. Of importance in this study is the transition to three-dimensionality from the underlying two-dimensional periodic flow and, in particular, the way that the associated transitions influence the fluid forces exerted on the cylinder and the development and the structure of the wake. It is found that the steady two-dimensional flow becomes unstable to three-dimensional perturbations at $Re_{c,3D}=37$, and that the transition to unsteady two-dimensional flow – or periodic vortex shedding – occurs at $Re_{c,2D}=88$, thus validating and refining the results of Stewart et al. (J. Fluid Mech. vol. 648, 2010, pp. 225–256). The main focus here is on Reynolds numbers beyond the transition to unsteady flow at $Re_{c,2D}=88$. From impulsive start up, the wake almost immediately undergoes transition to a periodic two-dimensional wake state, which, in turn, is three-dimensionally unstable. Thus, the previous three-dimensional stability analysis based on the two-dimensional steady flow provides only an element of the full story. Floquet analysis based on the periodic two-dimensional flow was undertaken and new three-dimensional instability modes were revealed. The results suggest that an impulsively started cylinder rolling along a surface at constant velocity for $Re\gtrsim 90$ will result in the rapid development of a periodic two-dimensional wake that will be maintained for a considerable time prior to the wake undergoing three-dimensional transition. Of interest, the mean lift and drag coefficients obtained from full three-dimensional simulations match predictions from two-dimensional simulations to within a few per cent.

scholarly journals Three-dimensional Floquet stability analysis of the wake of a circular cylinder

1996 ◽  
Vol 322 ◽  
pp. 215-241 ◽  
Author(s):  
Dwight Barkley ◽  
Ronald D. Henderson
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Circular Cylinder ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Neutral Stability ◽  
Two Dimensional ◽  
Numerical Stability Analysis ◽  
Mode A

Results are reported from a highly accurate, global numerical stability analysis of the periodic wake of a circular cylinder for Reynolds numbers between 140 and 300. The analysis shows that the two-dimensional wake becomes (absolutely) linearly unstable to three-dimensional perturbations at a critical Reynolds number of 188.5±1.0. The critical spanwise wavelength is 3.96 ± 0.02 diameters and the critical Floquet mode corresponds to a ‘Mode A’ instability. At Reynolds number 259 the two-dimensional wake becomes linearly unstable to a second branch of modes with wavelength 0.822 diameters at onset. Stability spectra and corresponding neutral stability curves are presented for Reynolds numbers up to 300.

scholarly journals Non-modal stability analysis of the boundary layer under solitary waves

2017 ◽  
Vol 836 ◽  
pp. 740-772 ◽  
Author(s):  
Joris C. G. Verschaeve ◽  
Geir K. Pedersen ◽  
Cameron Tropea
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Stability Analysis ◽  
Solitary Waves ◽  
Reynolds Numbers ◽  
Two Dimensional ◽  
Modal Theory ◽  
Critical Reynolds Numbers ◽  
Streamwise Streaks ◽  
The Stability

In the present work, a stability analysis of the bottom boundary layer under solitary waves based on energy bounds and non-modal theory is performed. The instability mechanism of this flow consists of a competition between streamwise streaks and two-dimensional perturbations. For lower Reynolds numbers and early times, streamwise streaks display larger amplification due to their quadratic dependence on the Reynolds number, whereas two-dimensional perturbations become dominant for larger Reynolds numbers and later times in the deceleration region of this flow, as the maximum amplification of two-dimensional perturbations grows exponentially with the Reynolds number. By means of the present findings, we can give some indications on the physical mechanism and on the interpretation of the results by direct numerical simulation in Vittori & Blondeaux (J. Fluid Mech., vol. 615, 2008, pp. 433–443) and Özdemir et al. (J. Fluid Mech., vol. 731, 2013, pp. 545–578) and by experiments in Sumer et al. (J. Fluid Mech., vol. 646, 2010, pp. 207–231). In addition, three critical Reynolds numbers can be defined for which the stability properties of the flow change. In particular, it is shown that this boundary layer changes from a monotonically stable to a non-monotonically stable flow at a Reynolds number of $Re_{\unicode[STIX]{x1D6FF}}=18$.

Stability and dynamics of the laminar wake past a slender blunt-based axisymmetric body

2011 ◽  
Vol 676 ◽  
pp. 110-144 ◽  
Author(s):  
P. BOHORQUEZ ◽  
E. SANMIGUEL-ROJAS ◽  
A. SEVILLA ◽  
J. I. JIMÉNEZ-GONZÁLEZ ◽  
C. MARTÍNEZ-BAZÁN
Keyword(s):  
Stability Analysis ◽  
Numerical Simulations ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Velocity Ratio ◽  
Reynolds Numbers ◽  
Base Flow ◽  
Direct Numerical Simulations ◽  
The Body ◽  
Stream Velocity

We investigate the stability properties and flow regimes of laminar wakes behind slender cylindrical bodies, of diameter D and length L, with a blunt trailing edge at zero angle of attack, combining experiments, direct numerical simulations and local/global linear stability analyses. It has been found that the flow field is steady and axisymmetric for Reynolds numbers below a critical value, Recs (L/D), which depends on the length-to-diameter ratio of the body, L/D. However, in the range of Reynolds numbers Recs(L/D) < Re < Reco(L/D), although the flow is still steady, it is no longer axisymmetric but exhibits planar symmetry. Finally, for Re > Reco, the flow becomes unsteady due to a second oscillatory bifurcation which preserves the reflectional symmetry. In addition, as the Reynolds number increases, we report a new flow regime, characterized by the presence of a secondary, low frequency oscillation while keeping the reflectional symmetry. The results reported indicate that a global linear stability analysis is adequate to predict the first bifurcation, thereby providing values of Recs nearly identical to those given by the corresponding numerical simulations. On the other hand, experiments and direct numerical simulations give similar values of Reco for the second, oscillatory bifurcation, which are however overestimated by the linear stability analysis due to the use of an axisymmetric base flow. It is also shown that both bifurcations can be stabilized by injecting a certain amount of fluid through the base of the body, quantified here as the bleed-to-free-stream velocity ratio, Cb = Wb/W∞.

scholarly journals On the stability of plane Couette–Poiseuille flow with uniform crossflow

2010 ◽  
Vol 656 ◽  
pp. 417-447 ◽  
Author(s):  
ANIRBAN GUHA ◽  
IAN A. FRIGAARD
Keyword(s):  
Linear Stability ◽  
Poiseuille Flow ◽  
Energy Production ◽  
Base Flow ◽  
Two Dimensional ◽  
Wall Velocity ◽  
Resonant Interactions ◽  
Flow Stabilization ◽  
Tollmien Schlichting ◽  
The Stability

We present a detailed study of the linear stability of the plane Couette–Poiseuille flow in the presence of a crossflow. The base flow is characterized by the crossflow Reynolds number Rinj and the dimensionless wall velocity k. Squire's transformation may be applied to the linear stability equations and we therefore consider two-dimensional (spanwise-independent) perturbations. Corresponding to each dimensionless wall velocity, k ∈ [0, 1], two ranges of Rinj exist where unconditional stability is observed. In the lower range of Rinj, for modest k we have a stabilization of long wavelengths leading to a cutoff Rinj. This lower cutoff results from skewing of the velocity profile away from a Poiseuille profile, shifting of the critical layers and the gradual decrease of energy production. Crossflow stabilization and Couette stabilization appear to act via very similar mechanisms in this range, leading to the potential for a robust compensatory design of flow stabilization using either mechanism. As Rinj is increased, we see first destabilization and then stabilization at very large Rinj. The instability is again a long-wavelength mechanism. An analysis of the eigenspectrum suggests the cause of instability is due to resonant interactions of Tollmien–Schlichting waves. A linear energy analysis reveals that in this range the Reynolds stress becomes amplified, the critical layer is irrelevant and viscous dissipation is completely dominated by the energy production/negation, which approximately balances at criticality. The stabilization at very large Rinj appears to be due to decay in energy production, which diminishes like Rinj−1. Our study is limited to two-dimensional, spanwise-independent perturbations.

MHD Stability for a Class of Tokamak Equilibria with Fixed Boundaries

1978 ◽  
Vol 33 (7) ◽  
pp. 792-798
Author(s):  
W. Kerner
Keyword(s):  
Stability Analysis ◽  
Normal Mode ◽  
Cross Sections ◽  
Stability Limit ◽  
Stability Behavior ◽  
Mhd Stability ◽  
The Stability ◽  
High Beta ◽  
Fixed Boundaries ◽  
Flat Current

The stability behavior with respect to internal modes is discussed for a class of tokamak equilibria with non-circular cross-sections and essentially flat current profiles. The stability analysis is done by computer both symbolically and numerically with the help of a normal mode code, which extremizes the Lagrangian of the system . It is found that the stability limit agrees well with that of the Mercier criterion. There are stable high-beta equilibria in this model.

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