Theoretical predictions for the elliptical instability in a two-vortex flow

2002 ◽  
Vol 471 ◽  
pp. 169-201 ◽  
Author(s):  
STÉPHANE LE DIZÈS ◽  
FLORENT LAPORTE
Keyword(s):  
Growth Rate ◽  
Inclination Angle ◽  
Asymptotic Expression ◽  
Point Vortex ◽  
Normal Modes ◽  
Vortex Pair ◽  
Local Perturbation ◽  
Global Mode ◽  
Elliptical Instability ◽  
Elliptical Flow

Two parallel Gaussian vortices of circulations Γ1 and Γ2 radii a1 and a2, separated by a distance b may become unstable by the elliptical instability due the elliptic deformation of their cores. The goal of the paper is to analyse this occurrence theoretically in a general framework. An explicit formula for the temporal growth rate of the elliptical instability in each vortex is obtained as a function of the above global parameters of the system, the Reynolds number Γ1/v and the non-dimensionalized axial wavenumber kzb of the perturbation. This formula is based on a known asymptotic expression for the local instability growth rate at an elliptical stagnation point which depends on the local characteristics of the elliptical flow and the inclination angle of the local perturbation wavevector at this point. The elliptical flow characteristics are estimated by considering each Gaussian vortex alone in a weak uniform external strain field whose properties are provided by a point vortex modelling of the vortex pair. The inclination angle is obtained from the dispersion relation for the Gaussian vortex normal modes and the local expression near each vortex centre for the two helical modes of azimuthal wavenumber m = 1 and m = −1 which constitute the elliptical instability global mode. Both the final formula and the hypotheses made for its derivation are tested and validated by direct numerical simulations and large-eddy simulations.

Nonlinear evolution of the elliptical instability: an example of inertial wave breakdown

1999 ◽  
Vol 396 ◽  
pp. 73-108 ◽  
Author(s):  
D. M. MASON ◽  
R. R. KERSWELL
Keyword(s):  
Finite Amplitude ◽  
Complex Conjugate ◽  
Small Scale ◽  
Conjugate Pair ◽  
Weakly Nonlinear ◽  
Elliptical Instability ◽  
Elliptical Flow ◽  
Complex Conjugate Pair ◽  
Generic Instability ◽  
Secondary Instabilities

A direct numerical simulation is presented of an elliptical instability observed in the laboratory within an elliptically distorted, rapidly rotating, fluid-filled cylinder (Malkus 1989). Generically, the instability manifests itself as the pairwise resonance of two different inertial modes with the underlying elliptical flow. We study in detail the simplest ‘subharmonic’ form of the instability where the waves are a complex conjugate pair and which at weakly supercritical elliptical distortion should ultimately saturate at some finite amplitude (Waleffe 1989; Kerswell 1992). Such states have yet to be experimentally identified since the flow invariably breaks down to small-scale disorder. Evidence is presented here to support the argument that such weakly nonlinear states are never seen because they are either unstable to secondary instabilities at observable amplitudes or neighbouring competitor elliptical instabilities grow to ultimately disrupt them. The former scenario confirms earlier work (Kerswell 1999) which highlights the generic instability of inertial waves even at very small amplitudes. The latter represents a first numerical demonstration of two competing elliptical instabilities co-existing in a bounded system.

Tidal excitation of hydromagnetic waves and their damping in the Earth

1994 ◽  
Vol 274 ◽  
pp. 219-241 ◽  
Author(s):  
R. R. Kerswell
Keyword(s):  
Growth Rate ◽  
Growth Rates ◽  
Outer Core ◽  
Decay Rates ◽  
Wave Disturbance ◽  
Instability Mechanism ◽  
The Earth ◽  
Elliptical Instability ◽  
Hydromagnetic Waves

We examine the possibility that the Earth's outer core, as a tidally distorted fluid-filled rotating spheroid, may be the seat of an elliptical instability. The instability mechanism is described within the framework of a simple Earth-like model. The preferred forms of wave disturbance are explored and a likely growth rate supremum deduced. Estimates are made of the Ohmic and viscous decay rates of such hydromagnetic waves in the outer core. Rather than a conclusive disparity of scales, we find that typical elliptical growth rates, Ohmic decay rates and viscous decay rates all have the same order for plausible core fields and core-to-mantle conductivities. This study is all the more timely considering the recent realization that the Earth's precession may also drive similar instabilities at comparable strengths in the outer core.

A Numerical Study on the Effect of Hole Inclination Angle With Imperfection on Film Cooling Effectiveness

2019 ◽  
Author(s):  
Taha Rezzag ◽  
Bassam A. Jubran
Keyword(s):  
Inclination Angle ◽  
Film Cooling ◽  
Numerical Study ◽  
Vortex Pair ◽  
Dimensionless Temperature ◽  
Film Cooling Effectiveness ◽  
Blowing Ratio ◽  
Laser Percussion Drilling ◽  
Inclination Angles ◽  
Lift Off

Abstract The present study numerically evaluates the influence of hole inclination angle with a hole imperfection on film cooling performance. Here, the hole imperfection due to laser percussion drilling is modelled as a half torus. Three hole inclination angles were investigated: 35°, 45° and 55°. Furthermore, every case was evaluated at three blowing ratios: 0.45, 0.90 and 1.25. Each case is compared to a baseline case where the hole imperfection is absent. The results indicate that the hole inclination angle has a strong influence on the film effectiveness performance when a hole imperfection is present. Centerline effectiveness plots reveal a maximum effectiveness deterioration of 89% for a blowing ratio of 0.90 in the vicinity of the hole exit. Dimensionless temperature contours show that the jet produced in the presence of an imperfection is much more compact causing the counter rotating vortex pair to be closer to each other. This enhances the jet to lift off from the plate.

scholarly journals Some estimates on the space scales of vortex pairs emitted from river mouths

2001 ◽  
Vol 8 (1/2) ◽  
pp. 1-7 ◽  
Author(s):  
V. P. Goncharov ◽  
V. I. Pavlov
Keyword(s):  
Point Vortex ◽  
Vortex Pair ◽  
Vortex Structures ◽  
Shelf Zone ◽  
Vortex Pairs ◽  
Polygonal Boundary ◽  
Dipole Vortex ◽  
Qualitative Classification ◽  
River Mouths

Abstract. Two-dimensional vortex pairs are frequently observed in geophysical conditions, for example, in a shelf zone of the ocean near river mouths. The main aims of the work are to estimate the space scales of such vortex structures, to analyze possible scenarios of vortex pair motion and to give the qualitative classification of their trajectories. We discuss some features of the motion of strong localized vorticity concentrations in a given flow in the presence of boundaries. The analyses are made in the framework of a 2D point vortex mo-del with an open polygonal boundary. Estimations are made for the characteristic parameters of dipole vortex structures emitted from river mouths into the open ocean.

Linear three-dimensional global and asymptotic stability analysis of incompressible open cavity flow

2015 ◽  
Vol 768 ◽  
pp. 113-140 ◽  
Author(s):  
Vincenzo Citro ◽  
Flavio Giannetti ◽  
Luca Brandt ◽  
Paolo Luchini
Keyword(s):  
Growth Rate ◽  
Reynolds Number ◽  
Stability Analysis ◽  
Closed Orbit ◽  
Three Dimensional ◽  
Base Flow ◽  
Open Cavity ◽  
Critical Conditions ◽  
Global Mode ◽  
Structural Sensitivity

The viscous and inviscid linear stability of the incompressible flow past a square open cavity is studied numerically. The analysis shows that the flow first undergoes a steady three-dimensional bifurcation at a critical Reynolds number of 1370. The critical mode is localized inside the cavity and has a flat roll structure with a spanwise wavelength of about 0.47 cavity depths. The adjoint global mode reveals that the instability is most efficiently triggered in the thin region close to the upstream tip of the cavity. The structural sensitivity analysis identifies the wavemaker as the region located inside the cavity and spatially concentrated around a closed orbit. As the flow outside the cavity plays no role in the generation mechanisms leading to the bifurcation, we confirm that an appropriate parameter to describe the critical conditions in open cavity flows is the Reynolds number based on the average velocity between the two upper edges. Stabilization is achieved by a decrease of the total momentum inside the shear layer that drives the core vortex within the cavity. The mechanism of instability is then studied by means of a short-wavelength approximation considering pressureless inviscid modes. The closed streamline related to the maximum inviscid growth rate is found to be the same as that around which the global wavemaker is concentrated. The structural sensitivity field based on direct and adjoint eigenmodes, computed at a Reynolds number far higher than that of the base flow, can predict the critical orbit on which the main instabilities inside the cavity arise. Further, we show that the sub-leading unstable time-dependent modes emerging at supercritical conditions are characterized by a period that is a multiple of the revolution time of Lagrangian particles along the orbit of maximum growth rate. The eigenfrequencies of these modes, computed by global stability analysis, are in very good agreement with the asymptotic results.

Elliptical instability of the Moore–Saffman model for a trailing wingtip vortex

2016 ◽  
Vol 803 ◽  
pp. 556-590 ◽  
Author(s):  
J. Feys ◽  
S. A. Maslowe
Keyword(s):  
Growth Rates ◽  
Stokes Equations ◽  
Normal Modes ◽  
Axial Flow ◽  
Velocity Profiles ◽  
Wing Loading ◽  
Axial Velocity Component ◽  
Important Conclusion ◽  
Elliptical Instability ◽  
Loading Parameter

In this paper, we investigate the elliptical instability exhibited by two counter-rotating trailing vortices. This type of instability can be viewed as a resonance between two normal modes of a vortex and an external strain field. Recent numerical investigations have extended earlier results that ignored axial flow to include models with a simple wake-like axial flow such as the similarity solution found by Batchelor (J. Fluid Mech., vol. 20, 1964, pp. 645–658). We present herein growth rates of elliptical instability for a family of velocity profiles found by Moore & Saffman (Proc. R. Soc. Lond. A, vol. 333, 1973, pp. 491–508). These profiles have a parameter $n$ that depends on the wing loading. As a result, unlike the Batchelor vortex, they are capable of modelling both the jet-like and the wake-like axial flow present in a trailing vortex at short and intermediate distances behind a wingtip. Direct numerical simulations of the linearized Navier–Stokes equations are performed using an efficient spectral method in cylindrical coordinates developed by Matsushima & Marcus (J. Comput. Phys., vol. 53, 1997, pp. 321–345). We compare our results with those for the Batchelor vortex, whose velocity profiles are closely approximated as the wing loading parameter $n$ approaches 1. An important conclusion of our investigation is that the stability characteristics vary considerably with $n$ and $W_{0}$, a parameter measuring the strength of the mean axial velocity component. In the case of an elliptically loaded wing ($n=0.50$), we find that the instability growth rates are up to 50 % greater than those for the Batchelor vortex. Our results demonstrate the significant effect of the distribution and intensity of the axial flow on the elliptical instability of a trailing vortex.

scholarly journals An experimental investigation on elliptical instability of a strongly asymmetric vortex pair in a stable density stratification

2006 ◽  
Vol 13 (6) ◽  
pp. 641-649 ◽  
Author(s):  
B. Cariteau ◽  
J.-B. Flór
Keyword(s):  
Froude Number ◽  
Vortex Pair ◽  
Azimuthal Velocity ◽  
Buoyancy Frequency ◽  
Nonlinear Interactions ◽  
Secondary Vortex ◽  
Asymmetric Vortex ◽  
Elliptical Instability ◽  
Acceleration And Deceleration ◽  
Instability Growth

Abstract. We investigate the elliptical instability of a strongly asymmetric vortex pair in a stratified fluid, generated by the acceleration and deceleration of the rotation of a single flap. The dominant parameter is the Froude number, Fr=U/(NR), based on the maximum azimuthal velocity, U, and corresponding radius, R, of the strongest vortex, i.e. the principal vortex, and buoyancy frequency N. For Fr>1, both vortices are elliptically unstable while the instability is suppressed for Fr<1. In an asymmetric vortex pair, the principal vortex is less – and the secondary vortex more – elliptical than the vortices in an equivalent symmetric dipolar vortex. The far more unstable secondary vortex interacts with the principal vortex and increases the strain on the latter, thus increasing its ellipticity and its instability growth rate. The nonlinear interactions render the elliptical instability more relevant. An asymmetric dipole can be more unstable than an equivalent symmetric dipole. Further, the wavelength of the instability is shown to be a function of the Froude number for strong stratifications corresponding to small Froude numbers, whereas it remains constant in the limit of a homogenous fluid.

Three-dimensional vortex dynamics in oscillatory flow separation

2011 ◽  
Vol 674 ◽  
pp. 408-432 ◽  
Author(s):  
MIGUEL CANALS ◽  
GENO PAWLAK
Keyword(s):  
Time Scale ◽  
Oscillatory Flow ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Outer Region ◽  
Vortex Pair ◽  
Propagation Distance ◽  
Oscillation Period ◽  
Number Range ◽  
Elliptical Instability

The dynamics of coherent columnar vortices and their interactions in an oscillatory flow past an obstacle are examined experimentally. The main focus is on the low Keulegan–Carpenter number range (0.2 < KC < 2), where KC is the ratio between the fluid particle excursion during half an oscillation cycle and the obstacle size, and for moderate Reynolds numbers (700 < Rev < 7500). For this parameter range, a periodic unidirectional vortex pair ejection regime is observed, in which the direction of vortex propagation is set by the initial conditions of the oscillations. These vortex pairs provide a direct mechanism for the transfer of momentum and enstrophy to the outer region of rough oscillating boundary layers. Vortices are observed to be short-lived relative to the oscillation time scale, which limits their propagation distance from the boundary. The instability mechanisms leading to vortex decay are elucidated via flow visualizations and digital particle image velocimetry (DPIV). Dye visualizations reveal complex three-dimensional vortex interactions resulting in rapid vortex destruction. These visualizations suggest that one of the instabilities affecting the spanwise vortices is an elliptical instability of the strained vortex cores. This is supported by DPIV measurements which identify the spatial structure of the perturbations associated with the elliptical instability in the divergence field. We also identify regions in the periphery of the vortex cores which are unstable to the centrifugal instability. Vortex longevity is quantified via a vortex decay time scale, and the results indicate that vortex pair lifetimes are of the order of an oscillation period T.

scholarly journals Entrapping of a vortex pair interacting with a fixed point vortex revisited. I. Point vortices

2018 ◽  
Vol 30 (9) ◽  
pp. 096603 ◽  
Author(s):  
Konstantin V. Koshel ◽  
Jean N. Reinaud ◽  
Giorgio Riccardi ◽  
Eugene A. Ryzhov
Keyword(s):  
Fixed Point ◽  
Point Vortex ◽  
Vortex Pair ◽  
Point Vortices

Notes on the stability problem for inhomogeneous equilibria

1983 ◽  
Vol 29 (2) ◽  
pp. 275-286 ◽  
Author(s):  
K. R. Symon
Keyword(s):  
Growth Rate ◽  
Normal Mode ◽  
Stability Problem ◽  
Normal Modes ◽  
Electromagnetic Mode ◽  
Spectral Matrix ◽  
First Order ◽  
Real Frequency ◽  
Special Cases ◽  
The Stability

Several conclusions regarding the stability of inhomogeneous Vlasov equilibria are drawn from earlier work. A technique is presented for generating first-order formulae for the change δω in the frequency of any normal mode, when a parameter λ characterizing the equilibrium is changed slightly. Several applications are given, including a first-order calculation of the growth rate or damping of an electromagnetic mode due to the presence of plasma. A condition is derived for the existence of a normal mode with real frequency. When there are ignorable co-ordinates, the normal modes can be written in the form of waves propagating in the ignorable directions. The character of the modes depends on certain symmetries in the dynamic spectral matrix. Special cases arise when the orbits can be approximated in certain ways.

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