Low-Reynolds-number wakes of elliptical cylinders: from the circular cylinder to the normal flat plate

2014 ◽  
Vol 751 ◽  
pp. 570-600 ◽  
Author(s):  
Mark C. Thompson ◽  
Alexander Radi ◽  
Anirudh Rao ◽  
John Sheridan ◽  
Kerry Hourigan
Keyword(s):  
Reynolds Number ◽  
Circular Cylinder ◽  
Flat Plate ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Near Wake ◽  
Aspect Ratios ◽  
Von Karman ◽  
Von Kármán ◽  
Mode A

AbstractWhile the wake of a circular cylinder and, to a lesser extent, the normal flat plate have been studied in considerable detail, the wakes of elliptic cylinders have not received similar attention. However, the wakes from the first two bodies have considerably different characteristics, in terms of three-dimensional transition modes, and near- and far-wake structure. This paper focuses on elliptic cylinders, which span these two disparate cases. The Strouhal number and drag coefficient variations with Reynolds number are documented for the two-dimensional shedding regime. There are considerable differences from the standard circular cylinder curve. The different three-dimensional transition modes are also examined using Floquet stability analysis based on computed two-dimensional periodic base flows. As the cylinder aspect ratio (major to minor axis) is decreased, mode A is no longer unstable for aspect ratios below 0.25, as the wake deviates further from the standard Bénard–von Kármán state. For still smaller aspect ratios, another three-dimensional quasi-periodic mode becomes unstable, leading to a different transition scenario. Interestingly, for the 0.25 aspect ratio case, mode A restabilises above a Reynolds number of approximately 125, allowing the wake to return to a two-dimensional state, at least in the near wake. For the flat plate, three-dimensional simulations show that the shift in the Strouhal number from the two-dimensional value is gradual with Reynolds number, unlike the situation for the circular cylinder wake once mode A shedding develops. Dynamic mode decomposition is used to characterise the spatially evolving character of the wake as it undergoes transition from the primary Bénard–von Kármán-like near wake into a two-layered wake, through to a secondary Bénard–von Kármán-like wake further downstream, which in turn develops an even longer wavelength unsteadiness. It is also used to examine the differences in the two- and three-dimensional near-wake state, showing the increasing distortion of the two-dimensional rollers as the Reynolds number is increased.

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 Linear biglobal analysis of Rayleigh–Bénard instabilities in binary fluids with and without throughflow

2012 ◽  
Vol 713 ◽  
pp. 216-242 ◽  
Author(s):  
Jun Hu ◽  
Daniel Henry ◽  
Xie-Yuan Yin ◽  
Hamda BenHadid
Keyword(s):  
Reynolds Number ◽  
Rayleigh Number ◽  
Aspect Ratio ◽  
Critical Rayleigh Number ◽  
Two Dimensional ◽  
Binary Fluids ◽  
Transverse Modes ◽  
Aspect Ratios ◽  
Rayleigh Benard ◽  
Mode A

AbstractThree-dimensional Rayleigh–Bénard instabilities in binary fluids with Soret effect are studied by linear biglobal stability analysis. The fluid is confined transversally in a duct and a longitudinal throughflow may exist or not. A negative separation factor $\psi = \ensuremath{-} 0. 01$, giving rise to oscillatory transitions, has been considered. The numerical dispersion relation associated with this stability problem is obtained with a two-dimensional Chebyshev collocation method. Symmetry considerations are used in the analysis of the results, which allow the classification of the perturbation modes as ${S}_{l} $ modes (those which keep the left–right symmetry) or ${R}_{x} $ modes (those which keep the symmetry of rotation of $\lrm{\pi} $ about the longitudinal mid-axis). Without throughflow, four dominant pairs of travelling transverse modes with finite wavenumbers $k$ have been found. Each pair corresponds to two symmetry degenerate left and right travelling modes which have the same critical Rayleigh number ${\mathit{Ra}}_{c} $. With the increase of the duct aspect ratio $A$, the critical Rayleigh numbers for these four pairs of modes decrease and closely approach the critical value ${\mathit{Ra}}_{c} = 1743. 894$ obtained in a two-dimensional situation, one of the mode (a ${S}_{l} $ mode called mode A) always remaining the dominant mode. Oscillatory longitudinal instabilities ($k\approx 0$) corresponding to either ${S}_{l} $ or ${R}_{x} $ modes have also been found. Their critical curves, globally decreasing, present oscillatory variations when the duct aspect ratio $A$ is increased, associated with an increasing number of longitudinal rolls. When a throughflow is applied, the symmetry degeneracy of the pairs of travelling transverse modes is broken, giving distinct upstream and downstream modes. For small and moderate aspect ratios $A$, the overall critical Rayleigh number in the small Reynolds number range studied is only determined by the upstream transverse mode A. In contrast, for larger aspect ratios as $A= 7$, different modes are successively dominant as the Reynolds number is increased, involving both upstream and downstream transverse modes A and even the longitudinal mode.

scholarly journals Modification of three-dimensional transition in the wake of a rotationally oscillating cylinder

2009 ◽  
Vol 643 ◽  
pp. 349-362 ◽  
Author(s):  
DAVID LO JACONO ◽  
JUSTIN S. LEONTINI ◽  
MARK C. THOMPSON ◽  
JOHN SHERIDAN
Keyword(s):  
Reynolds Number ◽  
Three Dimensional ◽  
Oscillation Mode ◽  
Critical Amplitude ◽  
Near Wake ◽  
Three Dimensional Flow ◽  
Double Row ◽  
Dimensional Flow ◽  
Spatio Temporal ◽  
Mode A

A study of the flow past an oscillatory rotating cylinder has been conducted, where the frequency of oscillation has been matched to the natural frequency of the vortex street generated in the wake of a stationary cylinder, at Reynolds number 300. The focus is on the wake transition to three-dimensional flow and, in particular, the changes induced in this transition by the addition of the oscillatory rotation. Using Floquet stability analysis, it is found that the fine-scale three-dimensional mode that typically dominates the wake at a Reynolds number beyond that at the second transition to three-dimensional flow (referred to as mode B) is suppressed for amplitudes of rotation beyond a critical amplitude, in agreement with past studies. However, the rotation does not suppress the development of three-dimensionality completely, as other modes are discovered that would lead to three-dimensional flow. In particular, the longer-wavelength mode that leads the three-dimensional transition in the wake of a stationary cylinder (referred to as mode A) is left essentially unaffected at low amplitudes of rotation. At higher amplitudes of oscillation, mode A is also suppressed as the two-dimensional near wake changes in character from a single- to a double-row wake; however, another mode is predicted to render the flow three-dimensional, dubbed mode D (for double row). This mode has the same spatio-temporal symmetries as mode A.

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.

scholarly journals Strouhal–Reynolds number relationship for flow past a circular cylinder

2017 ◽  
Vol 832 ◽  
pp. 170-188 ◽  
Author(s):  
Hongyi Jiang ◽  
Liang Cheng
Keyword(s):  
Reynolds Number ◽  
Circular Cylinder ◽  
Three Dimensional ◽  
Wake Flow ◽  
Pure Mode ◽  
Laminar Regime ◽  
3D Flow ◽  
Flow Past ◽  
3D Flows ◽  
Mode A

The Strouhal–Reynolds number ($St{-}Re$) relationship for flow past a circular cylinder in the low $Re$ range of $Re\leqslant 1000$ is investigated through two- (2D) and three-dimensional (3D) direct numerical simulations (DNS). An improved method is proposed for the determination of the separating velocity and the wake width to allow for a better estimation of the wake Strouhal number $St^{\ast }$. For $Re$ in the extended laminar regime calculated by 2D DNS, the $St^{\ast }$ values are found to be more uniform than the original $St$ for the 2D flow. It is also found that the $St^{\ast }$ values for the 2D and 3D flows agree well in the laminar regime of $Re$ up to approximately 270. In addition, uniform $St^{\ast }$ values are also obtained for different mode A and mode B flow structures triggered artificially by using different cylinder span lengths in DNS. It is demonstrated that the drop in $St$ (with respect to its 2D counterpart) with the development of different 3D wake structures is due to the decrease in the separating velocity and the increase in the wake width for a 3D flow, rather than the existence of a particular wake structure such as pure mode A or vortex dislocation. However, as the wake flow becomes increasingly turbulent with further increase in $Re$, the $St^{\ast }$ value for the 3D flow increases gradually and deviates from its 2D counterpart, since for turbulent 3D flows the vortex shedding frequency scales on a length smaller than the wake width.

scholarly journals Nonlinear dynamics and pattern formation in turbulent wake transition

1997 ◽  
Vol 352 ◽  
pp. 65-112 ◽  
Author(s):  
RONALD D. HENDERSON
Keyword(s):  
Reynolds Number ◽  
Large Scale ◽  
Three Dimensional ◽  
Spatiotemporal Chaos ◽  
Secondary Instability ◽  
Transition To Turbulence ◽  
Two Dimensional ◽  
Instability Modes ◽  
Mode A ◽  
Near Threshold

Results are reported on direct numerical simulations of transition from two-dimensional to three-dimensional states due to secondary instability in the wake of a circular cylinder. These calculations quantify the nonlinear response of the system to three-dimensional perturbations near threshold for the two separate linear instabilities of the wake: mode A and mode B. The objectives are to classify the nonlinear form of the bifurcation to mode A and mode B and to identify the conditions under which the wake evolves to periodic, quasi-periodic, or chaotic states with respect to changes in spanwise dimension and Reynolds number. The onset of mode A is shown to occur through a subcritical bifurcation that causes a reduction in the primary oscillation frequency of the wake at saturation. In contrast, the onset of mode B occurs through a supercritical bifurcation with no frequency shift near threshold. Simulations of the three-dimensional wake for fixed Reynolds number and increasing spanwise dimension show that large systems evolve to a state of spatiotemporal chaos, and suggest that three-dimensionality in the wake leads to irregular states and fast transition to turbulence at Reynolds numbers just beyond the onset of the secondary instability. A key feature of these ‘turbulent’ states is the competition between self-excited, three-dimensional instability modes (global modes) in the mode A wavenumber band. These instability modes produce irregular spatiotemporal patterns and large-scale ‘spot-like’ disturbances in the wake during the breakdown of the regular mode A pattern. Simulations at higher Reynolds number show that long-wavelength interactions modulate fluctuating forces and cause variations in phase along the span of the cylinder that reduce the fluctuating amplitude of lift and drag. Results of both two-dimensional and three-dimensional simulations are presented for a range of Reynolds number from about 10 up to 1000.

Experimental and Theoretical Study of Three-Dimensional Effects on Vertical Wave-in-Deck Forces

2009 ◽  
Author(s):  
Rolf Baarholm
Keyword(s):  
Model Test ◽  
Three Dimensional ◽  
Water Entry ◽  
Scale Model ◽  
Two Dimensional ◽  
Wave Impact ◽  
Dimensional Effects ◽  
Von Karman ◽  
Von Kármán ◽  
Water Exit

In order to validate theory for computing wave-in-deck loads of offshore platforms, a small scale model test campaign of wave impact on an idealized platform deck is performed at Towing Tank no. II at MARINTEK. The main objectives of the tests were to assess three-dimensional effects and to better understand the effect transverse and longitudinal structural members have on the fluid flow. The emphasis in the present paper is to demonstrate the three-dimensional effects. Model tests of the same structure were performed for both two-dimensional and three-dimensional flow conditions. The model test results show that three-dimensional effects significantly reduce the wave-in-deck loads. In particular, for the water exit phase, the vertical force is almost halved due to three-dimensional effects. Two different two-dimensional methods are used to study water impact on the deck: one method is based on a generalization of Wagner’s impact theory while the latter is a simple von Karman approach. Moreover, a three-dimensional correction is introduced. Comparisons show that the Wagner based method yields good results for the water entry phase, but it overestimates the water exit force and underestimates the duration of the wave-in-deck event. The von Karman type approach underestimates the water entry force.

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