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.

scholarly journals Numerical Simulation of the Solar Granulation

1980 ◽  
Vol 58 ◽  
pp. 293-299
Author(s):  
Lawrence D. Cloutman
Keyword(s):  
Numerical Simulation ◽  
Numerical Solution ◽  
Stokes Equations ◽  
Solar Physics ◽  
Quantitative Agreement ◽  
Time Dependent ◽  
Navier Stokes ◽  
Vortex Rings ◽  
Solar Granulation ◽  
Navier Stokes Equations

AbstractThe solar granulation has been simulated by numerical solution of the multidimensional, time-dependent, nonlinear Navier-Stokes equations applied to the solar atmosphere. Granules may be explained as buoyantly rising bubbles created at the level where T = 8000 K, and which have collapsed into vortex rings. The calculation is in quantitative agreement with observations and has a number of implications for solar physics and convection theory.

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.

Viscous interactions of two co-rotating vortices before merging

2002 ◽  
Vol 467 ◽  
pp. 389-410 ◽  
Author(s):  
STÉPHANE LE DIZÈS ◽  
ALBERTO VERGA
Keyword(s):  
Reynolds Number ◽  
Vortex Core ◽  
Strain Field ◽  
Stokes Equations ◽  
Separation Distance ◽  
Relaxation Processes ◽  
Single Vortex ◽  
Hyperbolic Point ◽  
Initial Vorticity ◽  
Non Gaussian

The viscous evolution of two co-rotating vortices is analysed using direct two-dimensional numerical simulations of the Navier–Stokes equations. The article focuses on vortex interaction regimes before merging. Two parameters are varied: a steepness parameter n which measures the steepness of the initial vorticity profiles in a given family of profiles, and the Reynolds number Re (between 500 and 16 000). Two distinct relaxation processes are identified. The first one is non-viscous and corresponds to a rapid adaptation of each vortex to the external (strain) field generated by the other vortex. This adaptation process, which is profile dependent, is described and explained using the damped Kelvin modes of each vortex. The second relaxation process is a slow diffusion phenomenon. It is similar to the relaxation of any non-Gaussian axisymmetrical vortex towards the Gaussian. The quasi-stationary solution evolves on a viscous-time scale toward a single attractive solution which corresponds to the evolution from two initially Gaussian vortices. The attractive solution is analysed in detail up to the merging threshold a/b ≈ 0.22 where a and b are the vortex radius and the separation distance respectively. The vortex core deformations are quantified and compared to those induced by a single vortex in a rotating strain field. A good agreement with the asymptotic predictions is demonstrated for the eccentricity of vortex core streamlines. A weak anomalous Reynolds number dependence of the solution is also identified. This dependence is attributed to the advection–diffusion of vorticity towards the hyperbolic points of the system and across the separatrix connecting these points. A Re1/3 scaling for the vorticity at the central hyperbolic point is obtained. These findings are discussed in the context of a vortex merging criterion.

scholarly journals A steady separated viscous corner flow

1991 ◽  
Vol 231 ◽  
pp. 1-34 ◽  
Author(s):  
Robert McLachlan
Keyword(s):  
Reynolds Number ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Separated Flow ◽  
Leading Edge ◽  
Vortex Sheet ◽  
Quantitative Agreement ◽  
Free Shear Layer ◽  
Single Vortex ◽  
Secondary Separation

An example is presented of a separated flow in an unbounded domain in which, as the Reynolds number becomes large, the separated region remains of size 0(1) and tends to a non-trivial Prandtl-Batchelor flow. The multigrid method is used to obtain rapid convergence to the solution of the discretized Navier-Stokes equations at Reynolds numbers of up to 5000. Extremely fine grids and tests of an integral property of the flow ensure accuracy. The flow exhibits the separation of a boundary layer with ensuing formation of a downstream eddy and reattachment of a free shear layer. The asymptotic (’triple deck’) theory of laminar separation from a leading edge, due to Sychev (1979), is clarified and compared to the numerical solutions. Much better qualitative agreement is obtained than has been reported previously. Together with a plausible choice of two free parameters, the data can be extrapolated to infinite Reynolds number, giving quantitative agreement with triple-deck theory with errors of 20% or less. The development of a region of constant vorticity is observed in the downstream eddy, and the global infinite-Reynolds-number limit is a Prandtl-Batchelor flow; however, when the plate is stationary, the occurrence of secondary separation suggests that the limiting flow contains an infinite sequence of eddies behind the separation point. Secondary separation can be averted by driving the plate, and in this case the limit is a single-vortex Prandtl-Batchelor flow of the type found by Moore, Saffman & Tanveer (1988); detailed, encouraging comparisons are made to the vortex-sheet strength and position. Altering the boundary condition on the plate gives viscous eddies that approximate different members of the family of inviscid solutions.

scholarly journals Generalized Navier–Stokes Equations and Dynamics of Plane Molecular Media

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 288
Author(s):  
Alexei Kushner ◽  
Valentin Lychagin
Keyword(s):  
Carbon Dioxide ◽  
Internal Structure ◽  
Stokes Equation ◽  
Hydrogen Cyanide ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations

The first analysis of media with internal structure were done by the Cosserat brothers. Birkhoff noted that the classical Navier–Stokes equation does not fully describe the motion of water. In this article, we propose an approach to the dynamics of media formed by chiral, planar and rigid molecules and propose some kind of Navier–Stokes equations for their description. Examples of such media are water, ozone, carbon dioxide and hydrogen cyanide.

Velocity Field Response of a Forced Swirl Stabilized Premixed Flame

2017 ◽  
Author(s):  
Kiran Manoharan ◽  
Travis Smith ◽  
Benjamin Emerson ◽  
Christopher M. Douglas ◽  
Tim Lieuwen ◽  
...  
Keyword(s):  
Heat Release ◽  
Shear Layer ◽  
Recirculation Zone ◽  
Stokes Equations ◽  
Quantitative Agreement ◽  
Forced Response ◽  
Response Analysis ◽  
Annular Jet ◽  
Experimental Response ◽  
Prediction Approach

This study is motivated by the necessity to develop a low order prediction approach for unsteady heat release response characteristics in lean premixed gas turbine combustors. This in turn requires an accurate description of the coherent hydrodynamic oscillations induced in the combustor flow by acoustic forcing. Time resolved velocity and flame position fields are obtained using sPIV and OH-PLIF measurements on a single nozzle, swirl-stabilized, premixed, methane-air flame in a model “unwrapped” annular combustor rig. A natural acoustic oscillation in the rig at 115 Hz results in a coherent flow oscillation that is concentrated primarily within the shear layer between the annular jet flow and the central recirculation zone. A linear stability analysis performed about time averaged base flow fields shows that the flow does not have any self-excited hydrodynamic modes. We then compare predictions from a forced response analysis at a forcing frequency of 115 Hz, based on the linearized Navier-Stokes equations for this coherent response. Good qualitative agreement between linear forced response analysis predictions and experimental response results, is seen for the spatial variation of velocity oscillation amplitude fields, away from the burner centerline. Further, good quantitative agreement between predictions and the experimental response is seen for the phase speed of velocity oscillations along the shear layer between the annular jet and the central recirculation zone. This phase velocity is an important flow field characteristic that has a significant impact on the heat release response that results from these coherent velocity oscillations. Present methods for forced response analysis assume uniform forcing amplitude along the radial direction at the forcing location, as well as, open flows along the streamwise direction. Both these assumptions are not strictly true for the present burner which has a center body on its axis. This maybe the reason for somewhat poor qualitative and quantitative agreement between experiments and predictions at the centerline.

Structural and superconducting properties of melt-grown Y–Ba–Cu–O superconductors

1996 ◽  
Vol 11 (10) ◽  
pp. 2406-2415 ◽  
Author(s):  
R. Gopalan ◽  
T. Rajasekharan ◽  
T. Roy ◽  
G. Rangarajan ◽  
V. Ganesan ◽  
...  
Keyword(s):  
Electron Microscopy ◽  
Strain Field ◽  
Crack Width ◽  
Growth Process ◽  
Critical Radius ◽  
Radius Of Curvature ◽  
Superconducting Properties ◽  
X Ray Diffraction ◽  
X Ray ◽  
Transmission Electron

YBa2Cu3O7 (123) samples with varying Y2BaCuO5 (211) concentrations (0 mol%, 20 mol%, 28 mol%, and 50 mol%) were synthesized by the melt-growth process. Microstructural characterizations were done using x-ray diffraction (XRD), optical microscopy, scanning electron microscopy, and transmission electron microscopy (TEM). It was found that 123 platelet width, crack width between the platelets, and 211 particle size decreased systematically with increasing 211 concentration. TEM study showed that there is a critical radius of curvature (rc ≤ 0.2 μm-0.3 μm) of the 123/211 interface where defects/contrasts of strain field start to appear, and these defects are believed to be responsible for pinning the magnetic flux. Microhardness measurements showed that Vickers hardness (VHN) increases with increasing 211 content. Critical current density (Jc) values obtained from magnetization measurements using a SQUID magnetometer were found to increase in melt-grown samples by the addition of 211 content.

scholarly journals Local and global pairing instabilities of two interlaced helical vortices

2019 ◽  
Vol 863 ◽  
pp. 927-955 ◽  
Author(s):  
Hugo Umberto Quaranta ◽  
Mattias Brynjell-Rahkola ◽  
Thomas Leweke ◽  
Dan S. Henningson
Keyword(s):  
Growth Rates ◽  
Double Helix ◽  
Water Channel ◽  
Three Dimensional ◽  
Rotation Frequency ◽  
Quantitative Agreement ◽  
Driving Mechanism ◽  
Helical Vortices ◽  
Local Pairing ◽  
Real Flow

We investigate theoretically and experimentally the stability of two interlaced helical vortices with respect to displacement perturbations having wavelengths that are large compared to the size of the vortex cores. First, existing theoretical results are recalled and applied to the present configuration. Various modes of unstable perturbations, involving different phase relationships between the two vortices, are identified and their growth rates are calculated. They lead to a local pairing of neighbouring helix loops, or to a global pairing with one helix expanding and the other one contracting. A relation is established between this instability and the three-dimensional pairing of arrays of straight parallel vortices, and a striking quantitative agreement concerning the growth rates and frequencies is found. This shows that the local pairing of vortices is the driving mechanism behind the instability of the helix system. Second, an experimental study designed to observe these instabilities in a real flow is presented. Two helical vortices are generated by a two-bladed rotor in a water channel and characterised through dye visualisations and particle image velocimetry measurements. Unstable displacement modes are triggered individually, either by varying the rotation frequency of the rotor, or by imposing a small rotor eccentricity. The observed unstable mode structure, and the corresponding growth rates obtained from advanced processing of visualisation sequences, are in good agreement with theoretical predictions. The nonlinear late stages of the instability are also documented experimentally. Whereas local pairing leads to strong deformations and subsequent breakup of the vortices, global pairing results in a leapfrogging phenomenon, which temporarily restores the initial double-helix geometry, in agreement with recent observations from numerical simulations.

Boundary layer leading-edge receptivity to sound at incidence angles

2001 ◽  
Vol 444 ◽  
pp. 383-407 ◽  
Author(s):  
ERCAN ERTURK ◽  
THOMAS C. CORKE
Keyword(s):  
Acoustic Waves ◽  
Stokes Equations ◽  
Incidence Angle ◽  
Leading Edge ◽  
Quantitative Agreement ◽  
Navier Stokes ◽  
Angle Of Incidence ◽  
Sound Waves ◽  
Nose Radius ◽  
Navier Stokes Equations

The leading-edge receptivity to acoustic waves of two-dimensional parabolic bodies was investigated using a spatial solution of the Navier–Stokes equations in vorticity/streamfunction form in parabolic coordinates. The free stream is composed of a uniform flow with a superposed periodic velocity fluctuation of small amplitude. The method follows that of Haddad & Corke (1998) in which the solution for the basic flow and linearized perturbation flow are solved separately. We primarily investigated the effect of frequency and angle of incidence (−180° [les ] α2 [les ] 180°) of the acoustic waves on the leading-edge receptivity. The results at α2 = 0° were found to be in quantitative agreement with those of Haddad & Corke (1998), and substantiated the Strouhal number scaling based on the nose radius. The results with sound waves at angles of incidence agreed qualitatively with the analysis of Hammerton & Kerschen (1996). These included a maximum receptivity at α2 = 90°, and an asymmetric variation in the receptivity with sound incidence angle, with minima at angles which were slightly less than α2 = 0° and α2 = 180°.

Sound generation by shock–vortex interactions

1999 ◽  
Vol 380 ◽  
pp. 81-116 ◽  
Author(s):  
OSAMU INOUE ◽  
YUJI HATTORI
Keyword(s):  
Shock Wave ◽  
Stokes Equations ◽  
Early Stage ◽  
Near Field ◽  
Time Integration ◽  
Flow Fields ◽  
Navier Stokes ◽  
Single Vortex ◽  
Vortex Interactions ◽  
Reflected Shock

Two-dimensional, unsteady, compressible flow fields produced by the interactions between a single vortex or a pair of vortices and a shock wave are simulated numerically. The Navier–Stokes equations are solved by a finite difference method. The sixth-order-accurate compact Padé scheme is used for spatial derivatives, together with the fourth-order-accurate Runge–Kutta scheme for time integration. The detailed mechanics of the flow fields at an early stage of the interactions and the basic nature of the near-field sound generated by the interactions are studied. The results for both a single vortex and a pair of vortices suggest that the generation and the nature of sounds are closely related to the generation of reflected shock waves. The flow field differs significantly when the pair of vortices moves in the same direction as the shock wave than when opposite to it.

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