Bifurcation and instability problems in vortex wakes

Journal of Fluid Mechanics, 2003

A rapidly growing instability is observed to develop between unequal-strength counter- rotating vortex pairs. The vortex pairs are generated in a towing tank in the wakes of wings with outboard triangular flaps. The vortices from the wing tip and the inboard tip of the flap form the counter-rotating vortex pair on each side of the wing. The flow fields are studied using flow visualization and particle image velocimetry. Both chord- based and circulation-based Reynolds numbers are of O(105). The circulation strength ratios of the flap- to tip-vortex pairs range from −0.4 to −0.7. The initial sinuous stage of the instability of the weaker flap vortex has a wavelength of order one wing span and becomes observable in about 15 wing spans downstream of the wing. The nearly straight vortex filaments first form loops around the stronger wing-tip vortices. The loops soon detach and form rings and move in the wake under self-induction. These vortex rings can move to the other side of the wake...

Journal of Fluids and Structures, 2011

We present a mathematical model of the vortex wake modes that appear behind neighboring and/or oscillating, flapping, and swimming bodies in which there are four vortices generated in an anti-symmetric pattern during each shedding cycle. The twodimensional potential flow model consists of four point vortices with strengths 7 G in a spatially periodic domain. The relative vortex positions are restricted by a discrete symmetry that is motivated by the spatial symmetry observed in experimental wakes. The strength restriction and the imposed symmetry result in the model system being an integrable Hamiltonian dynamical system. We find that the point vortex motion can be one of four distinct types based on the values of linear impulse and Hamiltonian. Two of these types correspond to 2P wakes and consist of two oppositely signed, counterrotating vortex pairs. One of these types corresponds to 2C wakes and consists of two like-signed, co-rotating vortex pairs. The fourth type is an exchanging mode in which the two vortices near the wake centerline translate faster than the outer two vortices. Scaled comparisons of the model with both a 2P and a 2C experimental wake show good representation of the experimentally observed vortex dynamics and lead to estimates of the experimental vortex strengths.

Journal of Fluid Mechanics, 2007

Thiria, Goujon-Durand & Wesfreid (J. Fluid Mech. vol. 560, 2006, p. 123), it was shown that vortex shedding from a rotationally oscillating cylinder at moderate Reynolds number can be characterized by the spatial coexistence of two distinct patterns, one of which is related to the forcing frequency in the near wake and the other to a frequency close to the natural one for the unforced case downstream of this locked region. The existence and the modification of these wake characteristics were found to be strongly affected by the frequency and the amplitude of the cylinder oscillation. In this paper, a linear stability analysis of these forced regimes is performed, and shows that the stability characteristics of such flows are governed by a strong mean flow correction which is a function of the oscillation parameters. We also present experiments on the spatial properties of the global mode and on the selection of the vortex shedding frequency as a function of the forcing conditions for Re = 150. Finally, we elucidate a diagram of locked and non-locked states, for a large range of frequencies and amplitudes of the oscillation.

2020

Wakes of upswept afterbodies are often characterized by a counter-rotating streamwise vortex pair. The unsteady dynamics of these vortices are examined with a spatiotemporally resolved Large-Eddy Simulation (LES) dataset on a representative configuration consisting of a cylinder with upswept basal surface. Emphasis is placed on understanding the meandering motion of the vortices in the pair, including vortex core displacement, spectral content, stability mechanisms and overall rank-behavior. The first two energy-ranked modes obtained through Proper orthogonal decomposition (POD) of the time-resolved vorticity field reveals a pair of vortex dipoles aligned relatively perpendicularly to each other. The dynamics is successfully mapped to a matched Batchelor vortex pair whose spatial and temporal stability analyses indicate similar dipole structures associated with an |m| = 1 elliptic mode pair. This short-wave elliptic instability dominates the meandering motion, with strain due to axi...

2004

The wakes behind open rings of two different aspect ratios are computed using a spectral-element method, and the stability of the solutions are analysed by the application of a linear stability analysis. The predicted instabilities are compared with the computed three-dimensional flows and with experimentally obtained dye visualisations. The flows computed at the chosen aspect ratios show that the bifurcations from steady axisymmetric flow produce remarkably different wakes. This phenomenon is related to the existence of a transition aspect ratio that delineates annular and cylinder-like wakes at small and large aspect ratios, respectively. Background: The Stability of Bluff-Body Flows The subject of instability and the bifurcation of fluid flows is as rich as it is diverse. The motion of a fluid around even the simplest of geometries is governed by the mathematically formidable Navier-Stokes equations; a set of non-linear partial differential equations for which analytical solutions to practical flows are rare.

Lecture Notes in Physics, 2000

Airplane trailing vortices have a destabilizing e ect on ensuing aircrafts. Security spacings, related to the trailing vortices \lifetime" are actually enforced between take-o s and landings. This spacing limits the maximum take-o and landing frequency in saturated airports 19]. A number of studies have been devoted to the understanding of vortex wake dynamics, usually modeled by a pair of counterrotating vortices. Two types of vortex pair three-dimensional instabilities have been identi ed in the past: a long-wave instability (of the order of the spacing b between the two vortices) and a short-wave instability (of the order of the vortex core radius a) have been rst considered respectively by Crow 4] and by Moore & Sa man 15] and Tsai & Widnall 20]. These two mechanisms, which are thought to participate in the vortex wake dissipation, have been observed in recent experiments 12]. One possible technique to accelerate the dissipation of aircraft wakes is to force these instabilities by on-board control devices 2,21,22]. Until now only temporal vortex pair instability analyses are available. If one wants to force these instabilities, however, it would be more appropriate to analyze their spatial stability in the airplane reference frame. As the spatial analysis makes sense only when the instabilities are convective, an absolute/convective stability analysis is required. The results presented consist of the absolute/convective and spatial stability analyses of both long{ and short{wave instabilities.

1975

X see Eq (5.48) a) circular frequency Q angular rate of rotation TABLE OF CONTENTS Introduction The Roll-Up of Trailed Vorticity 1.1 Point Vortex Computations of the Roll-Up Phenomenon 1.2 The Methods of Prandtl and Betz 1.3 Comparison with Experimental Measurements Aircraft Wake Geometry 2.1 Lumped Vorticity and the Approximations Involved 2.2 The Wake Geometry of Multiple Pair Wakes Sinusoidal Instability and Vortex Breakdown 3.

Physics of Fluids, 2007

In this paper, we focus on the three-dimensional growth of the vortical structures behind two square cylinders when placed in a side-by-side arrangement and examine different possible modes of flow bifurcation along their spanwise extended corelines. For this, unsteady three-dimensional flow simulations are conducted with five different values of gap/diameter ratios ͑g * ͒, namely, 2.1, 1.7, 0.7, 0.5, and 0.2, that cover important known phases, i.e., coshedding ͑g * = 2.1 and 1.7͒, asymmetric ͑g * = 0.7 and 0.5͒, and single-body type ͑g * = 0.2͒ of wake evolution. In order to exploit flow physics within the transition range, the Reynolds number of the flow is taken as 100. Notably, for all the investigated cases, parallel vortex shedding has been observed behind the two cylinders. However, with the decrease of gap ratio g * , the downstream flow gradually lost stability and the corelines of the shedded vortices appeared in a wavy fashion. While for g * = 2.1 and 1.7, the simulated streamwise flows exhibit antiphase and in-phase synchronization of the wake, respectively, there is also observed a notable difference in the structural growth of corresponding spanwise flows. For the antiphase flow ͑at g * = 2.1͒ the downstream wake evolved through a symmetry-breaking mode-I Hopf bifurcation along various spanwise extended vortex corelines. However, the presence of an additional mode-II Hopf bifurcation was detected in the wake of an in-phase flow ͑at g * = 1.7͒. The development of a number of local pressure maxima over different spanwise extended vortex corelines, and the gradual decrease of pressure along their left and right, respectively, are noted to be responsible for initiating such flow bifurcations. In the biased ͑asymmetric͒ regime, with g * = 0.7 and 0.5, the flow behind the cylinders showed visible signs of growing instability and there occurred significant spanwise pressure fluctuations, which in turn inflicted large three dimensionality into the downstream wake and local velocity irregularities. The corresponding wake evolution patterns are thereby noted to exhibit predominantly unsteady and transitional features. In this biased flow regime, the gap flow flip-flopped randomly and became deflected alternately towards the top and bottom cylinders. On the one hand, clear dominance of both mode-I and mode-II flow bifurcations was observed with both of these initially locked biased flows. However, due to increased flow asymmetry the spanwise length scales of such bifurcations changed significantly. On the other hand, for the single-body regime flow with g * = 0.2, the occurrence of only mode-I bifurcation was detected in the wake of the cylinders.

Physics of Fluids, 1998

The effect of swirl on jets and wakes is investigated by analyzing the inviscid spatiotemporal instability of the Rankine vortex with superimposed plug flow axial velocity profile. The linear dispersion relation is derived analytically as a function of two nondimensional control parameters: the swirl ratio S and the external axial flow parameter a ͑aϾϪ0.5 for jets, aϽϪ0.5 for wakes͒. For each azimuthal wave number m, there exists a single unstable Kelvin-Helmholtz mode and an infinite number of neutrally stable inertial waveguide modes. Swirl decreases the temporal growth rate of the axisymmetric Kelvin-Helmholtz mode (mϭ0), which nonetheless remains unstable for all axial wave numbers. For helical modes (m 0), small amounts of swirl lead to the widespread occurrence of direct resonances between the unstable Kelvin-Helmholtz mode and the inertial waveguide modes. Such interactions generate, in the low wave number range, neutrally stable wave number bands separated by bubbles of instability. As S increases above a critical value, all bubbles merge to give rise to unstable wave numbers throughout, but the growth rate envelope decreases continuously with increasing swirl. In the high wave number range, negative helical mode growth rates are enhanced for small swirl and decrease continuously for large swirl, while positive helical mode growth rates monotonically decrease with increasing swirl. For a given positive swirl, negative modes are more unstable than their positive counterparts, although their growth rates may not necessarily be larger than in the nonrotating case. The absolute/convective nature of the instability in swirling jets and wakes is determined in an aϪS control parameter plane by numerical implementation of the Briggs-Bers criterion. In the absence of swirl, jets (aϾϪ0.5) become absolutely unstable ͑AI͒ to the axisymmetric mode mϭ0 only for a sufficiently large external axial counterflow aϽϪ0.15. AI is found to be significantly enhanced by swirl: for SϾ1.61, AI occurs, even for coflowing jets (aϾ0). As S is gradually increased, the transitional mode to AI successively becomes mϭ0, Ϫ1, Ϫ2, Ϫ3, etc. In the absence of swirl, wakes (aϽϪ0.5) become absolutely unstable to the bending modes mϭϮ1 only for sufficiently large counterflow 1ϩaϾ0.091. For SϾ0.47, AI occurs even for coflowing wakes (aϽϪ1) and, as S increases, the transitional mode to AI successively becomes mϭϪ1, Ϫ2, Ϫ3, etc. This instability analysis is found to provide a preliminary estimate of the critical Rossby number for the onset of vortex breakdown: for zero external axial flow jets (aϭ0), absolute/convective transition first takes place at a Rossby number RoϵS Ϫ1 ϳ0.62, which very favorably compares with available experimental and numerical threshold values for vortex breakdown onset.

International Journal for Numerical Methods in Fluids, 2004

A computational code EZ-vortex is developed for the motion of slender vortex ÿlaments of closed or open shape. The integro-di erential equations governing the motion of the vortex centrelines are either the Callegari and Ting equations, which are the leading order solution of a matched asymptotic analysis, or equivalent forms of these equations. They include large axial velocity and nonsimilar proÿles in the vortical cores. The uid may be viscous or inviscid. This code is validated both against known solutions of these equations and results from linear stability analyses. The linear and non-linear stages of a perturbed two-vortex wake and of a four-vortex wake model are then computed.