Crow instability: nonlinear response to the linear optimal perturbation

2016 ◽  
Vol 795 ◽  
pp. 652-670 ◽  
Author(s):  
Holly G. Johnson ◽  
Vincent Brion ◽  
Laurent Jacquin
Keyword(s):  
Nonlinear Response ◽  
Linear Growth ◽  
Vortex Pair ◽  
Periodic Oscillation ◽  
Vortex Rings ◽  
Optimal Perturbation ◽  
Evolution Stage ◽  
Counter Rotating Vortex Pair ◽  
Crow Instability ◽  
Loss Of Coherence

The potential for anticipated destruction of a counter-rotating vortex pair using the linear optimal perturbation of the Crow instability is assessed. Direct numerical simulation is used to study the development of the Crow instability and the subsequent evolution of the flow up to 30 characteristic times at a circulation-based Reynolds number of 1000. The conventional development of the instability leads to multiple contortions of the vortices including the linear growth of sinusoidal deformation, vortex linking and the formation of vortex rings. A new evolution stage is identified, succeeding this well-established sequence: the vortex rings undergo periodic oscillation. Two complete periods are simulated during which the vortical system is hardly altered, thereby demonstrating the extraordinary resilience of the vortices. The possibility of preventing these dynamics using the linear optimal perturbation of the Crow instability, the adjoint mode, is analysed. By appropriately setting the forcing amplitude, the lifetime of the vortices until their loss of coherence is reduced to approximately 13 characteristic times, which is less than half that of the natural Crow behaviour observed with infinitesimal forcing. The dynamics of the flow induced by the linear optimal perturbation that enable this result are connected to processes already known to efficiently alter vortical flows, in particular transient growth and four-vortex dynamics.

Effect of velocity ratio on the interaction between plasma synthetic jets and turbulent cross-flow

2019 ◽  
Vol 865 ◽  
pp. 928-962 ◽  
Author(s):  
Haohua Zong ◽  
Marios Kotsonis
Keyword(s):  
Boundary Layer ◽  
High Speed ◽  
Velocity Ratio ◽  
Vortex Pair ◽  
Synthetic Jets ◽  
Low Energy ◽  
Vortex Rings ◽  
Wall Region ◽  
Mean Velocity ◽  
Counter Rotating Vortex Pair

Plasma synthetic jet actuators (PSJAs) are particularly suited for high-Reynolds-number, high-speed flow control due to their unique capability of generating supersonic pulsed jets at high frequency (${>}5$  kHz). Different from conventional synthetic jets driven by oscillating piezoelectric diaphragms, the exit-velocity variation of plasma synthetic jets (PSJs) within one period is significantly asymmetric, with ingestion being relatively weaker (less than $20~\text{m}~\text{s}^{-1}$) and longer than ejection. In this study, high-speed phase-locked particle image velocimetry is employed to investigate the interaction between PSJAs (round exit orifice, diameter 2 mm) and a turbulent boundary layer at constant Strouhal number (0.02) and increasing mean velocity ratio ($r$, defined as the ratio of the time-mean velocity over the ejection phase to the free-stream velocity). Two distinct operational regimes are identified for all the tested cases, separated by a transition velocity ratio, lying between $r=0.7$ and $r=1.0$. At large velocity and stroke ratios (first regime, representative case $r=1.6$), vortex rings are followed by a trailing jet column and tilt downstream initially. This downstream tilting is transformed into upstream tilting after the pinch-off of the trailing jet column. The moment of this transformation relative to the discharge advances with decreasing velocity ratio. Shear-layer vortices (SVs) and a hanging vortex pair (HVP) are identified in the windward and leeward sides of the jet body, respectively. The HVP is initially erect and evolves into an inclined primary counter-rotating vortex pair ($p$-CVP) which branches from the middle of the front vortex ring and extends to the near-wall region. The two legs of the $p$-CVP are bridged by SVs, and a secondary counter-rotating vortex pair ($s$-CVP) is induced underneath these two legs. At low velocity and stroke ratios (second regime, representative case $r=0.7$), the trailing jet column and $p$-CVP are absent. Vortex rings always tilt upstream, and the pitching angle increases monotonically with time. An $s$-CVP in the near-wall region is induced directly by the two longitudinal edges of the ring. Inspection of spanwise planes ($yz$-plane) reveals that boundary-layer energization is realized by the downwash effect of either vortex rings or $p$-CVP. In addition, in the streamwise symmetry plane, the increasing wall shear stress is attributed to the removal of low-energy flow by ingestion. The downwash effect of the $s$-CVP does not benefit boundary-layer energization, as the flow swept to the wall is of low energy.

Influence of a wall on the three-dimensional dynamics of a vortex pair

2017 ◽  
Vol 817 ◽  
pp. 339-373 ◽  
Author(s):  
Daniel J. Asselin ◽  
C. H. K. Williamson
Keyword(s):  
Ground Plane ◽  
Rebound Effect ◽  
Three Dimensional ◽  
Vortex Pair ◽  
Vortex Rings ◽  
Wave Instability ◽  
Long Wave ◽  
Wall Interaction ◽  
Secondary Vortices ◽  
Crow Instability

In this paper, we are interested in perturbed vortices under the influence of a wall or ground plane. Such flows have relevance to aircraft wakes in ground effect, to ship hull junction flows, to fundamental studies of turbulent structures close to a ground plane and to vortex generator flows, among others. In particular, we study the vortex dynamics of a descending vortex pair, which is unstable to a long-wave instability (Crow, AIAA J., vol. 8 (12), 1970, pp. 2172–2179), as it interacts with a horizontal ground plane. Flow separation on the wall generates opposite-sign secondary vortices which in turn induce the ‘rebound’ effect, whereby the primary vortices rise up away from the wall. Even small perturbations in the vortices can cause significant topological changes in the flow, ultimately generating an array of vortex rings which rise up from the wall in a three-dimensional ‘rebound’ effect. The resulting vortex dynamics is almost unrecognizable when compared with the classical Crow instability. If the vortices are generated below a critical height over a horizontal ground plane, the long-wave instability is inhibited by the wall. We then observe two modes of vortex–wall interaction. For small initial heights, the primary vortices are close together, enabling the secondary vortices to interact with each other, forming vertically oriented vortex rings in what we call a ‘vertical rings mode’. In the ‘horizontal rings mode’, for larger initial heights, the Crow instability develops further before wall interaction; the peak locations are farther apart and the troughs closer together upon reaching the wall. The proximity of the troughs to each other and the wall increases vorticity cancellation, leading to a strong axial pressure gradient and axial flow. Ultimately, we find a series of small horizontal vortex rings which ‘rebound’ from the wall. Both modes comprise two small vortex rings in each instability wavelength, distinct from Crow instability vortex rings, only one of which is formed per wavelength. The phenomena observed here are not limited to the above perturbed vortex pairs. For example, remarkably similar phenomena are found where vortex rings impinge obliquely with a wall.

scholarly journals Optimal perturbation for two-dimensional vortex systems: route to non-axisymmetric state

2018 ◽  
Vol 855 ◽  
pp. 922-952 ◽  
Author(s):  
Navrose ◽  
H. G. Johnson ◽  
V. Brion ◽  
L. Jacquin ◽  
J. C. Robinet
Keyword(s):  
Steady State ◽  
Initial Perturbation ◽  
Vortex Pair ◽  
Vortex State ◽  
Quasi Steady State ◽  
Two Dimensional ◽  
Optimal Perturbation ◽  
Perturbation Energy ◽  
Counter Rotating Vortex Pair ◽  
Linear And Nonlinear

We investigate perturbations that maximize the gain of disturbance energy in a two-dimensional isolated vortex and a counter-rotating vortex pair. The optimization is carried out using the method of Lagrange multipliers. For low initial energy of the perturbation ( $E(0)$ ), the nonlinear optimal perturbation/gain is found to be the same as the linear optimal perturbation/gain. Beyond a certain threshold $E(0)$ , the optimal perturbation/gain obtained from linear and nonlinear computations are different. There exists a range of $E(0)$ for which the nonlinear optimal gain is higher than the linear optimal gain. For an isolated vortex, the higher value of nonlinear optimal gain is attributed to interaction among different azimuthal components, which is otherwise absent in a linearized system. Spiral dislocations are found in the nonlinear optimal perturbation at the radial location where the most dominant wavenumber changes. Long-time nonlinear evolution of linear and nonlinear optimal perturbations is studied. The evolution shows that, after the initial increment of perturbation energy, the vortex attains a quasi-steady state where the mean perturbation energy decreases on a slow time scale. The quasi-steady vortex state is non-axisymmetric and its shape depends on the initial perturbation. It is observed that the lifetime of a quasi-steady vortex state obtained using the nonlinear optimal perturbation is longer than that obtained using the linear optimal perturbation. For a counter-rotating vortex pair, the mechanism that maximizes the energy gain is found to be similar to that of the isolated vortex. Within the linear framework, the optimal perturbation for a vortex pair can be either symmetric or antisymmetric, whereas the structure of the nonlinear optimal perturbation, beyond the threshold $E(0)$ , is always asymmetric. No quasi-steady state for a counter-rotating vortex pair is observed.

scholarly journals On the formation of the counter-rotating vortex pair in transverse jets

2001 ◽  
Vol 446 ◽  
pp. 347-373 ◽  
Author(s):  
L. CORTELEZZI ◽  
A. R. KARAGOZIAN
Keyword(s):  
Flow Field ◽  
Computational Study ◽  
Near Field ◽  
Three Dimensional ◽  
Vortex Pair ◽  
Transverse Jets ◽  
Vortical Structures ◽  
Jet In Crossflow ◽  
Dynamical Generation ◽  
Counter Rotating Vortex Pair

Among the important physical phenomena associated with the jet in crossflow is the formation and evolution of vortical structures in the flow field, in particular the counter-rotating vortex pair (CVP) associated with the jet cross-section. The present computational study focuses on the mechanisms for the dynamical generation and evolution of these vortical structures. Transient numerical simulations of the flow field are performed using three-dimensional vortex elements. Vortex ring rollup, interactions, tilting, and folding are observed in the near field, consistent with the ideas described in the experimental work of Kelso, Lim & Perry (1996), for example. The time-averaged effect of these jet shear layer vortices, even over a single period of their evolution, is seen to result in initiation of the CVP. Further insight into the topology of the flow field, the formation of wake vortices, the entrainment of crossflow, and the effect of upstream boundary layer thickness is also provided in this study.

Formation of a Counter-Rotating Vortex Pair in a Plume in a Cross-Flow

2010 ◽  
Author(s):  
Masahiko Shinohara
Keyword(s):  
Vortex Ring ◽  
Three Dimensional ◽  
Cross Flow ◽  
Vortex Pair ◽  
Boussinesq Approximation ◽  
Streamwise Vorticity ◽  
Three Dimensional Flow ◽  
Flow Feature ◽  
Heated Area ◽  
Counter Rotating Vortex Pair

Numerical simulations are performed to study the formation of a counter-rotating vortex pair (CVP), a dominant flow feature in plumes inclined in a cross-flow. The unsteady three-dimensional flow fields are calculated by a finite difference method using the Boussinesq approximation. A plume rises from an isothermally heated square surface facing upward in air. Calculations show that the CVP originates not from horizontal spanwise vorticity in the velocity boundary layer on the bottom wall around the heated area, but from horizontal streamwise vorticity just above each side of the heated area. When the cross-flow begins after a plume forms a vortex ring in the cap above the heated area in a still environment, the vortex ring does not form a CVP. However, as the cap and the stem of the plume move downwind, a rotation about the streamwise axis appears just above each side edge of the heated area and grows into the CVP. We discuss the effect of entrainment into the stem and cap on the formation of the streamwise rotation that causes the CVP.

scholarly journals Three-dimensional instabilities and transient growth of a counter-rotating vortex pair

2009 ◽  
Vol 21 (9) ◽  
pp. 094102 ◽  
Author(s):  
Claire Donnadieu ◽  
Sabine Ortiz ◽  
Jean-Marc Chomaz ◽  
Paul Billant
Keyword(s):  
Three Dimensional ◽  
Vortex Pair ◽  
Transient Growth ◽  
Counter Rotating Vortex Pair

Stability characteristics of a counter-rotating unequal-strength Batchelor vortex pair

2012 ◽  
Vol 696 ◽  
pp. 374-401 ◽  
Author(s):  
Kris Ryan ◽  
Christopher J. Butler ◽  
Gregory J. Sheard
Keyword(s):  
Stokes Equations ◽  
Vortex Pair ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Mode Amplitude ◽  
Nonlinear Growth ◽  
Circulation Ratio ◽  
Secondary Stage ◽  
Aircraft Wake ◽  
Crow Instability

AbstractA Batchelor vortex represents the asymptotic solution of a trailing vortex in an aircraft wake. In this study, an unequal-strength, counter-rotating Batchelor vortex pair is employed as a model of the wake emanating from one side of an aircraft wing; this model is a direct extension of several prior investigations that have considered unequal-strength Lamb–Oseen vortices as representations of the aircraft wake problem. Both solution of the linearized Navier–Stokes equations and direct numerical simulations are employed to study the linear and nonlinear development of a vortex pair with a circulation ratio of$\Lambda = \ensuremath{-} 0. 5$. In contrast to prior investigations considering a Lamb–Oseen vortex pair, we note strong growth of the Kelvin mode$[\ensuremath{-} 2, 0] $coupled with an almost equal growth rate of the Crow instability. Three stages of nonlinear instability development are defined. In the initial stage, the Kelvin mode amplitude becomes sufficiently large that oscillations within the core of the weaker vortex are easily observable and significantly affect the profile of the weaker vortex. In the secondary stage, filaments of secondary vorticity emanate from the weaker vortex and are convected around the stronger vortex. In the tertiary stage, a transition in the dominant instability wavelength is observed from the short-wavelength Kelvin mode to the longer-wavelength Crow instability. Much of the instability growth is observed on the weaker vortex of the pair, although small perturbations in the stronger vortex are observed in the tertiary nonlinear growth phase.

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