Curvature instability of a curved Batchelor vortex

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).

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Elliptic instability of a curved Batchelor vortex

Journal of Fluid Mechanics ◽

10.1017/jfm.2016.533 ◽

2016 ◽

Vol 804 ◽

pp. 224-247 ◽

Cited By ~ 8

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.

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Elliptic instability in a strained Batchelor vortex

Journal of Fluid Mechanics ◽

10.1017/s0022112007004879 ◽

2007 ◽

Vol 577 ◽

pp. 341-361 ◽

Cited By ~ 38

Author(s):

LAURENT LACAZE ◽

KRIS RYAN ◽

STÉPHANE LE DIZÈS

Keyword(s):

Reynolds Number ◽

Strain Field ◽

Flow Parameter ◽

Numerical Study ◽

Normal Modes ◽

Axial Flow ◽

Base Flow ◽

Large Reynolds Number ◽

Resonant Coupling ◽

Flow Parameters

The elliptic instability of a Batchelor vortex subject to a stationary strain field is considered by theoretical and numerical means in the regime of large Reynolds number and small axial flow. In the theory, the elliptic instability is described as a resonant coupling of two quasi-neutral normal modes (Kelvin modes) of the Batchelor vortex of azimuthal wavenumbers m and m + 2 with the underlying strain field. The growth rate associated with these resonances is computed for different values of the azimuthal wavenumbers as the axial flow parameter is varied. We demonstrate that the resonant Kelvin modes m = 1 and m = −1 which are the most unstable in the absence of axial flow become damped as the axial flow is increased. This is shown to be due to the appearance of a critical layer which damps one of the resonant Kelvin modes. However, the elliptic instability does not disappear. Other combinations of Kelvin modes m = −2 and m = 0, then m = −3 and m = −1 are shown to become progressively unstable for increasing axial flow. A complete instability diagram is obtained as a function of the axial flow parameter for several values of the strain rate and Reynolds number.The numerical study considers a system of two counter-rotating Batchelor vortices in which the strain field felt by each vortex is due to the other vortex. The characteristics of the most unstable linear modes developing on the frozen base flow are computed by direct numerical simulations for two axial flow parameters and compared to the theory. In both cases, a very good agreement is obtained for the most unstable modes. Less unstable modes are also identified in the numerics and shown to correspond to peculiar resonances involving Kelvin modes from branches of different labels.

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Multiple Solutions for Buoyancy-Induced Flow in Saturated Porous Media for Large Peclet Numbers

Journal of Heat Transfer ◽

10.1115/1.3247025 ◽

1986 ◽

Vol 108 (4) ◽

pp. 866-871 ◽

Cited By ~ 24

Author(s):

Rafiqul M. Islam ◽

K. Nandakumar

Keyword(s):

Porous Media ◽

Flow Parameter ◽

Axial Flow ◽

Internal Heat ◽

Saturated Porous Media ◽

Vortex Pattern ◽

Induced Flow ◽

Model Equations ◽

Buoyancy Induced Flow ◽

Two Parameter

The problem of buoyancy-induced secondary flow in fluid-saturated porous media is examined using a numerical model. The natural convection is coupled either with a forced axial flow or uniform internal heat generation. In both cases the model equations are shown to exhibit dual solutions over certain ranges of flow parameter. In the two-parameter space of aspect ratio and Grashof number, the flow transition between the two-vortex and four-vortex pattern can be explained in terms of a tilted cusp.

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Dynamics and Instability of a Vortex Ring Impinging on a Wall

Communications in Computational Physics ◽

10.4208/cicp.150115.210715s ◽

2015 ◽

Vol 18 (4) ◽

pp. 1122-1146 ◽

Cited By ~ 3

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.

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Numerical Results for Axial Flow Compressor Instability

Journal of Turbomachinery ◽

10.1115/1.3262291 ◽

1989 ◽

Vol 111 (4) ◽

pp. 434-441 ◽

Cited By ~ 11

Author(s):

F. E. McCaughan

Keyword(s):

Guide Vane ◽

Axial Flow ◽

Pressure Rise ◽

Companion Paper ◽

Extensive Study ◽

Rotating Stall ◽

Numerical Results ◽

Flow Diagram ◽

Inlet Guide Vane ◽

Flow Compressor

Using Cornell’s supercomputing facilities, we have carried out an extensive study of the Moore–Greitzer model, which gives accurate and reliable information about compressor instability. The bifurcation analysis in the companion paper shows the dependence of the mode of compressor response on the shape of the rotating stall characteristic. The numerical results verify and extend this with a more accurate representation of the characteristic. The effect of the parameters on the shape of the rotating stall characteristic is investigated, and it is found that the parameters with the strongest effects are the inlet length, and the shape of the compressor pressure rise versus mass flow diagram (i.e., tall diagrams versus shallow diagrams). We also discuss the effects of inlet guide vane loss on the characteristic. An evaluation is made of the h′ = −g approximation, and a spectral analysis of the rotating stall cell given by the full model suggests why this breaks down.

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Analysis of axisymmetric boundary layers

Journal of Fluid Mechanics ◽

10.1017/jfm.2018.449 ◽

2018 ◽

Vol 849 ◽

pp. 927-941 ◽

Cited By ~ 2

Author(s):

Praveen Kumar ◽

Krishnan Mahesh

Keyword(s):

Boundary Layer ◽

Pressure Gradient ◽

Boundary Layers ◽

Axial Flow ◽

Streamwise Velocity ◽

Skin Friction Coefficient ◽

Radius Of Curvature ◽

Normal Velocity ◽

Displacement Thickness ◽

Analytical Relations

Axisymmetric boundary layers are studied using integral analysis of the governing equations for axial flow over a circular cylinder. The analysis includes the effect of pressure gradient and focuses on the effect of transverse curvature on boundary layer parameters such as shape factor ($H$) and skin-friction coefficient ($C_{f}$), defined as $H=\unicode[STIX]{x1D6FF}^{\ast }/\unicode[STIX]{x1D703}$ and $C_{f}=\unicode[STIX]{x1D70F}_{w}/(0.5\unicode[STIX]{x1D70C}U_{e}^{2})$ respectively, where $\unicode[STIX]{x1D6FF}^{\ast }$ is displacement thickness, $\unicode[STIX]{x1D703}$ is momentum thickness, $\unicode[STIX]{x1D70F}_{w}$ is the shear stress at the wall, $\unicode[STIX]{x1D70C}$ is density and $U_{e}$ is the streamwise velocity at the edge of the boundary layer. Relations are obtained relating the mean wall-normal velocity at the edge of the boundary layer ($V_{e}$) and $C_{f}$ to the boundary layer and pressure gradient parameters. The analytical relations reduce to established results for planar boundary layers in the limit of infinite radius of curvature. The relations are used to obtain $C_{f}$ which shows good agreement with the data reported in the literature. The analytical results are used to discuss different flow regimes of axisymmetric boundary layers in the presence of pressure gradients.

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Sound generation by coaxial collision of two vortex rings

Journal of Fluid Mechanics ◽

10.1017/s0022112000002123 ◽

2000 ◽

Vol 424 ◽

pp. 327-365 ◽

Cited By ~ 19

Author(s):

O. INOUE ◽

Y. HATTORI ◽

T. SASAKI

Keyword(s):

Vortex Ring ◽

Sound Pressure ◽

Stokes Equations ◽

Near Field ◽

Time Integration ◽

Relative Strength ◽

Navier Stokes ◽

Vortex Rings ◽

Sound Waves ◽

Change Of Direction

Sound pressure fields generated by coaxial collisions of two vortex rings with equal/unequal strengths are simulated numerically. The axisymmetric, unsteady, compressible Navier–Stokes equations are solved by a finite difference method, not only for a near field but also for a far field. 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 results show that the generation of sound is closely related to the change of direction of the vortex ring motion induced by the mutual interaction of the two vortex rings. For the case of equal strength (head-on collision), the change of direction is associated with stretching of the vortex rings. Generated sound waves consist of compression parts and rarefaction parts, and have a quadrupolar nature. For the case of unequal strengths, the two vortex rings pass through each other; the weaker vortex ring moves outside the stronger vortex ring which shows a loop motion. The number of generated waves depends on the relative strength of the two vortex rings. The sound pressure includes dipolar and octupolar components, in addition to monopolar and quadrupolar components which are observed for the case of a head-on collision.

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Measurement of permittivity by means of an open resonator. I. Theoretical

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1982.0029 ◽

1982 ◽

Vol 380 (1778) ◽

pp. 49-71 ◽

Cited By ~ 67

Keyword(s):

Dielectric Material ◽

Open Resonator ◽

Variational Formula ◽

Dielectric Materials ◽

Companion Paper ◽

Radius Of Curvature ◽

Complex Source ◽

Spherical Mirrors ◽

Diffraction Effects ◽

Plane Slab

The objective of this paper is to provide a reliable theoretical foundation for the open-resonator method of measuring permittivity and loss-tangent of dielectric materials. From an exact solution of Maxwell’s equations obtained by means of the complex-source-point method, simplified formulae for the six Cartesian components of the electromagnetic field are obtained. These are then used in a variational formula to obtain an accurate formula for the resonant frequency of an open resonator with spherical mirrors (of equal radius of curvature) having a parallel-plane slab of dielectric material located centrally between the two mirrors. It is assumed that the mirrors are sufficiently large for diffraction effects to be negligible. In the following companion paper II, experimental results are presented which verify the accuracy, within the usual limits of practical measurement, of the simple correction formulae deduced here.

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Effects of convex transverse curvature on wall-bounded turbulence. Part 1. The velocity and vorticity

Journal of Fluid Mechanics ◽

10.1017/s0022112094004490 ◽

1994 ◽

Vol 272 ◽

pp. 349-382 ◽

Cited By ~ 43

Author(s):

João C. Neves ◽

Parviz Moin ◽

Robert D. Moser

Keyword(s):

Turbulent Flows ◽

Plane Channel ◽

Axial Flow ◽

Structural Data ◽

Streamwise Velocity ◽

Momentum Equation ◽

Radius Of Curvature ◽

Curvature Effects ◽

Transverse Curvature ◽

Logarithmic Region

Convex transverse curvature effects in wall-bounded turbulent flows are significant if the boundary-layer thickness is large compared to the radius of curvature (large γ = δ/a). The curvature affects the inner part of the flow if a+, the cylinder radius in wall units, is small.Two direct numerical simulations of a model problem approximating axial flow boundary layers on long cylinders were performed for γ = 5 (a+ ≈ 43) and γ = 11 (a+ ≈ 21). Statistical and structural data were extracted from the computed flow fields. The effects of the transverse curvature were identified by comparing the present results with those of the plane channel simulation of Kim, Moin & Moser (1987), performed at a similar Reynolds number. As the curvature increases, the skin friction increases, the slope of the logarithmic region decreases and turbulence intensities are reduced. Several turbulence statistics are found to scale with a curvature dependent velocity scale derived from the mean momentum equation. Near the wall, the flow is more anisotropic than in the plane channel with a larger percentage of the turbulent kinetic energy resulting from the streamwise velocity fluctuations. As the curvature increases, regions of strong normal vorticity develop near the wall.

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