Elliptic instability of a curved Batchelor vortex

2016 ◽  
Vol 804 ◽  
pp. 224-247 ◽  
Author(s):  
Francisco J. Blanco-Rodríguez ◽  
Stéphane Le Dizès
Keyword(s):  
Growth Rate ◽  
Vortex Ring ◽  
Flow Parameter ◽  
Axial Flow ◽  
Vortex Rings ◽  
Single Vortex ◽  
General Stability ◽  
Helical Vortices ◽  
Helical Vortex ◽  
Instability Growth

The occurrence of the elliptic instability in rings and helical vortices is analysed theoretically. The framework developed by Moore & Saffman (Proc. R. Soc. Lond. A, vol. 346, 1975, pp. 413–425), where the elliptic instability is interpreted as a resonance of two Kelvin modes with a strained induced correction, is used to obtain the general stability properties of a curved and strained Batchelor vortex. Explicit expressions for the characteristics of the three main unstable modes are obtained as a function of the axial flow parameter of the Batchelor vortex. We show that vortex curvature adds a contribution to the elliptic instability growth rate. The results are applied to a single vortex ring, an array of alternate vortex rings and a double helical vortex.

Dynamics and Instability of a Vortex Ring Impinging on a Wall

2015 ◽  
Vol 18 (4) ◽  
pp. 1122-1146 ◽  
Author(s):  
Heng Ren ◽  
Xi-Yun Lu
Keyword(s):  
Vortex Ring ◽  
Axial Flow ◽  
Core Region ◽  
Vortical Flow ◽  
Vortex Rings ◽  
Flow Transition ◽  
Azimuthal Component ◽  
Vortical Structures ◽  
The Core ◽  
Eddy Simulation

AbstractDynamics and instability of a vortex ring impinging on a wall were investigated by means of large eddy simulation for two vortex core thicknesses corresponding to thin and thick vortex rings. Various fundamental mechanisms dictating the flow behaviors, such as evolution of vortical structures, formation of vortices wrapping around vortex rings, instability and breakdown of vortex rings, and transition from laminar to turbulent state, have been studied systematically. The evolution of vortical structures is elucidated and the formation of the loop-like and hair-pin vortices wrapping around the vortex rings (called briefly wrapping vortices) is clarified. Analysis of the enstrophy of wrapping vortices and turbulent kinetic energy (TKE) in flow field indicates that the formation and evolution of wrapping vortices are closely associated with the flow transition to turbulent state. It is found that the temporal development of wrapping vortices and the growth rate of axial flow generated around the circumference of the core region for the thin ring are faster than those for the thick ring. The azimuthal instabilities of primary and secondary vortex rings are analyzed and the development of modal energies is investigated to reveal the flow transition to turbulent state. The modal energy decay follows a characteristic –5/3 power law, indicating that the vortical flow has become turbulence. Moreover, it is identified that the TKE with a major contribution of the azimuthal component is mainly distributed in the core region of vortex rings. The results obtained in this study provide physical insight of the mechanisms relevant to the vortical flow evolution from laminar to turbulent state.

scholarly journals XX. The potential of an anchor ring.-Part II

1893 ◽  
Vol 184 ◽  
pp. 1041-1106 ◽  
Keyword(s):  
Vortex Ring ◽  
Cross Section ◽  
Steady Motion ◽  
Vortex Rings ◽  
Laplace’S Equation ◽  
Single Vortex ◽  
Laplace's Equation ◽  
The Stability ◽  
Work Done

This paper is a continuation of that at pp. 43-95 suprd , on “The Potential of an Anchor Bing.” In that paper the potential of an anchor ring was found at all external points; in this/its value is determined at internal points. The annular form of rotating gravitating fluid was also discussed in that paper; here the stability of such a ring is considered. In addition, the potential of a ring whose cross-section is elliptic, being of interest in connection with Saturn, is obtained. The similarity of the methods employed, as well as of the analysis, has led me to give in this paper also a determination of the steady motion of a single vortex-ring in an infinite fluid, and of several fine vortex rings on the same axis. In Section I. solutions of Laplace’s equation applicable to space inside an anchor ring are obtained. These results are applied to obtain the potential of a solid ring at internal points, and also of a distribution of matter on the surface of the ring. The work done in collecting the ring from infinity is obtained.

Internal structure of vortex rings and helical vortices

2015 ◽  
Vol 785 ◽  
pp. 219-247 ◽  
Author(s):  
Francisco J. Blanco-Rodríguez ◽  
Stéphane Le Dizès ◽  
Can Selçuk ◽  
Ivan Delbende ◽  
Maurice Rossi
Keyword(s):  
Internal Structure ◽  
Strain Field ◽  
Stokes Equations ◽  
External Solution ◽  
Quantitative Agreement ◽  
Vortex Rings ◽  
Radius Of Curvature ◽  
Asymptotic Results ◽  
Single Vortex ◽  
Helical Vortices

The internal structure of vortex rings and helical vortices is studied using asymptotic analysis and numerical simulations in cases where the core size of the vortex is small compared to its radius of curvature, or to the distance to other vortices. Several configurations are considered: a single vortex ring, an array of equally-spaced rings, a single helix and a regular array of helices. For such cases, the internal structure is assumed to be at leading order an axisymmetric concentrated vortex with an internal jet. A dipolar correction arises at first order and is shown to be the same for all cases, depending only on the local vortex curvature. A quadrupolar correction arises at second order. It is composed of two contributions, one associated with local curvature and another one arising from a non-local external 2-D strain field. This strain field itself is obtained by performing an asymptotic matching of the local internal solution with the external solution obtained from the Biot–Savart law. Only the amplitude of this strain field varies from one case to another. These asymptotic results are thereafter confronted with flow solutions obtained by direct numerical simulation (DNS) of the Navier–Stokes equations. Two different codes are used: for vortex rings, the simulations are performed in the axisymmetric framework; for helices, simulations are run using a dedicated code with built-in helical symmetry. Quantitative agreement is obtained. How these results can be used to theoretically predict the occurrence of both the elliptic instability and the curvature instability is finally addressed.

A universal time scale for vortex ring formation

1998 ◽  
Vol 360 ◽  
pp. 121-140 ◽  
Author(s):  
MORTEZA GHARIB ◽  
EDMOND RAMBOD ◽  
KARIM SHARIFF
Keyword(s):  
Vortex Ring ◽  
Vortex Rings ◽  
Water Tank ◽  
Single Vortex ◽  
Piston Cylinder ◽  
Vortex Ring Formation ◽  
Image Velocimetry ◽  
Wide Range ◽  
Formation Number ◽  
Piston Stroke

The formation of vortex rings generated through impulsively started jets is studied experimentally. Utilizing a piston/cylinder arrangement in a water tank, the velocity and vorticity fields of vortex rings are obtained using digital particle image velocimetry (DPIV) for a wide range of piston stroke to diameter (L/D) ratios. The results indicate that the flow field generated by large L/D consists of a leading vortex ring followed by a trailing jet. The vorticity field of the leading vortex ring formed is disconnected from that of the trailing jet. On the other hand, flow fields generated by small stroke ratios show only a single vortex ring. The transition between these two distinct states is observed to occur at a stroke ratio of approximately 4, which, in this paper, is referred to as the ‘formation number’. In all cases, the maximum circulation that a vortex ring can attain during its formation is reached at this non-dimensional time or formation number. The universality of this number was tested by generating vortex rings with different jet exit diameters and boundaries, as well as with various non-impulsive piston velocities. It is shown that the ‘formation number’ lies in the range of 3.6–4.5 for a broad range of flow conditions. An explanation is provided for the existence of the formation number based on the Kelvin–Benjamin variational principle for steady axis-touching vortex rings. It is shown that based on the measured impulse, circulation and energy of the observed vortex rings, the Kelvin–Benjamin principle correctly predicts the range of observed formation numbers.

scholarly journals Curvature instability of a curved Batchelor vortex

2017 ◽  
Vol 814 ◽  
pp. 397-415 ◽  
Author(s):  
Francisco J. Blanco-Rodríguez ◽  
Stéphane Le Dizès
Keyword(s):  
Vortex Ring ◽  
Short Wavelength ◽  
Flow Parameter ◽  
Axial Flow ◽  
Companion Paper ◽  
Relative Strength ◽  
Radius Of Curvature ◽  
Resonant Coupling ◽  
Resonant Phenomenon ◽  
Large Rings

In this paper, we analyse the curvature instability of a curved Batchelor vortex. We consider this short-wavelength instability when the radius of curvature of the vortex centreline is large compared with the vortex core size. In this limit, the curvature instability can be interpreted as a resonant phenomenon. It results from the resonant coupling of two Kelvin modes of the underlying Batchelor vortex with the dipolar correction induced by curvature. The condition of resonance of the two modes is analysed in detail as a function of the axial jet strength of the Batchelor vortex. In contrast to the Rankine vortex, only a few configurations involving $m=0$ and $m=1$ modes are found to become the most unstable. The growth rate of the resonant configurations is systematically computed and used to determine the characteristics of the most unstable mode as a function of the curvature ratio, the Reynolds number and the axial flow parameter. The competition of the curvature instability with another short-wavelength instability, which was considered in a companion paper (Blanco-Rodríguez & Le Dizès, J. Fluid Mech., vol. 804, 2016, pp. 224–247), is analysed for a vortex ring. A numerical error found in this paper, which affects the relative strength of the elliptic instability, is also corrected. We show that the curvature instability becomes the dominant instability in large rings as soon as axial flow is present (vortex ring with swirl).

Numerical Investigation of Vortex Ring Ground Plane Interactions

2017 ◽  
Vol 139 (7) ◽  
Author(s):  
K. Bourne ◽  
S. Wahono ◽  
A. Ooi
Keyword(s):  
Vortex Ring ◽  
Ground Plane ◽  
Vortex Structures ◽  
Vortex Rings ◽  
Angle Of Incidence ◽  
Flow Conditions ◽  
Single Vortex ◽  
Wall Flow ◽  
The Impact ◽  
Near Wall

The interaction between multiple laminar thin vortex rings and solid surfaces was studied numerically so as to investigate flow patterns associated with near-wall flow structures. In this study, the vortex–wall interaction was used to investigate the tendency of the flow toward recirculatory behavior and to assess the near-wall flow conditions. The numerical model shows very good agreement with previous studies of single vortex rings for the case of orthogonal impact (angle of incidence, θ = 0 deg) and oblique impact (θ = 20 deg). The study was conducted at Reynolds numbers 585 and 1170, based on the vortex ring radius and convection velocity. The case of two vortex rings was also investigated, with particular focus on the interaction of vortex structures postimpact. Compared to the impact of a single ring with the wall, the interaction between two vortex rings and a solid surface resulted in a more highly energized boundary layer at the wall and merging of vortex structures. The azimuthal variation in the vortical structures yielded flow conditions at the wall likely to promote agitation of ground based particles.

Modal stability analysis of a helical vortex tube with axial flow

2013 ◽  
Vol 738 ◽  
pp. 222-249 ◽  
Author(s):  
Yuji Hattori ◽  
Yasuhide Fukumoto
Keyword(s):  
Boundary Condition ◽  
Growth Rate ◽  
Stability Analysis ◽  
Vortex Tube ◽  
Axial Flow ◽  
Short Wave ◽  
Combined Effects ◽  
Leading Order ◽  
Helical Vortex ◽  
Wave Limit

AbstractThe linear stability of a helical vortex tube with axial flow, which is a model of helical vortices emanating from rotating wings, is studied by modal stability analysis. At the leading order the base flow is set to the Rankine vortex with uniform velocity along the helical tube whose centreline is a helix of constant curvature and torsion. The helical vortex tube in an infinite domain, in which the free boundary condition is imposed at the surface of the tube, is our major target although the case of the rigid boundary condition is also considered in order to elucidate the effects of torsion and the combined effects of torsion and axial flow. The analysis is based on the linearized incompressible Euler equations expanded in $\epsilon $ which is the ratio of the core to curvature radius of the tube. The unstable growth rate can be evaluated using the leading-order neutral modes called the Kelvin waves with the expanded equations. At $O(\epsilon )$ the instability is a linear combination of the curvature instability due to the curvature of the tube and the precessional instability due to the axial flow, both parametric instabilities appearing at the same resonance condition. At the next order $O({\epsilon }^{2} )$ not only the effects of torsion but also the combined effects of torsion and axial flow appear, a fact which has been shown only for the short-wave limit. The maximum growth rate increases for the right-handed/left-handed helix with positive/negative helicity, in which the torsion makes the period of particle motion increase. All results converge to the previous local stability results in the short-wave limit. The differences between the two cases of different boundary conditions are due to the isolated mode of the free boundary case, whose dispersion curve depends strongly on the axial flow.

scholarly journals Numerical stability analysis of a vortex ring with swirl

2019 ◽  
Vol 878 ◽  
pp. 5-36 ◽  
Author(s):  
Yuji Hattori ◽  
Francisco J. Blanco-Rodríguez ◽  
Stéphane Le Dizès
Keyword(s):  
Numerical Simulation ◽  
Growth Rate ◽  
Direct Numerical Simulation ◽  
Vortex Ring ◽  
Stokes Equations ◽  
Base Flow ◽  
Critical Layer ◽  
Navier Stokes ◽  
Instability Mode ◽  
Navier Stokes Equations

The linear instability of a vortex ring with swirl with Gaussian distributions of azimuthal vorticity and velocity in its core is studied by direct numerical simulation. The numerical study is carried out in two steps: first, an axisymmetric simulation of the Navier–Stokes equations is performed to obtain the quasi-steady state that forms a base flow; then, the equations are linearized around this base flow and integrated for a sufficiently long time to obtain the characteristics of the most unstable mode. It is shown that the vortex rings are subjected to curvature instability as predicted analytically by Blanco-Rodríguez & Le Dizès (J. Fluid Mech., vol. 814, 2017, pp. 397–415). Both the structure and the growth rate of the unstable modes obtained numerically are in good agreement with the analytical results. However, a small overestimation (e.g. 22 % for a curvature instability mode) by the theory of the numerical growth rate is found for some instability modes. This is most likely due to evaluation of the critical layer damping which is performed for the waves on axisymmetric line vortices in the analysis. The actual position of the critical layer is affected by deformation of the core due to the curvature effect; as a result, the damping rate changes since it is sensitive to the position of the critical layer. Competition between the curvature and elliptic instabilities is also investigated. Without swirl, only the elliptic instability is observed in agreement with previous numerical and experimental results. In the presence of swirl, sharp bands of both curvature and elliptic instabilities are obtained for $\unicode[STIX]{x1D700}=a/R=0.1$, where $a$ is the vortex core radius and $R$ the ring radius, while the elliptic instability dominates for $\unicode[STIX]{x1D700}=0.18$. New types of instability mode are also obtained: a special curvature mode composed of three waves is observed and spiral modes that do not seem to be related to any wave resonance. The curvature instability is also confirmed by direct numerical simulation of the full Navier–Stokes equations. Weakly nonlinear saturation and subsequent decay of the curvature instability are also observed.

Suppression of Baroclinic Instabilities in Buoyancy-Driven Flow over Sloping Bathymetry

2017 ◽  
Vol 47 (1) ◽  
pp. 49-68 ◽  
Author(s):  
Robert D. Hetland
Keyword(s):  
Growth Rate ◽  
Linear Instability ◽  
Linear Stability Theory ◽  
Buoyancy Frequency ◽  
Bottom Slope ◽  
Coriolis Parameter ◽  
Plume Front ◽  
Instability Theory ◽  
Instability Growth ◽  
Flow Configurations

AbstractBaroclinic instabilities are ubiquitous in many types of geostrophic flow; however, they are seldom observed in river plumes despite strong lateral density gradients within the plume front. Supported by results from a realistic numerical simulation of the Mississippi–Atchafalaya River plume, idealized numerical simulations of buoyancy-driven flow are used to investigate baroclinic instabilities in buoyancy-driven flow over a sloping bottom. The parameter space is defined by the slope Burger number S = Nf−1α, where N is the buoyancy frequency, f is the Coriolis parameter, and α is the bottom slope, and the Richardson number Ri = N2f2M−4, where M2 = |∇Hb| is the magnitude of the lateral buoyancy gradients. Instabilities only form in a subset of the simulations, with the criterion that SH ≡ SRi−1/2 = Uf−1W−1 = M2f−2α 0.2, where U is a horizontal velocity scale and SH is a new parameter named the horizontal slope Burger number. Suppression of instability formation for certain flow conditions contrasts linear stability theory, which predicts that all flow configurations will be subject to instabilities. The instability growth rate estimated in the nonlinear 3D model is proportional to ωImaxS−1/2, where ωImax is the dimensional growth rate predicted by linear instability theory, indicating that bottom slope inhibits instability growth beyond that predicted by linear theory. The constraint SH 0.2 implies a relationship between the inertial radius Li = Uf−1 and the plume width W. Instabilities may not form when 5Li > W; that is, the plume is too narrow for the eddies to fit.

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