A stable tripole vortex model in two-dimensional Euler flows

2019 ◽  
Vol 878 ◽  
Author(s):  
A. Viúdez
Keyword(s):  
Constant Angular Velocity ◽  
Curvature Radius ◽  
Two Dimensional ◽  
Mathieu Functions ◽  
Vorticity Field ◽  
Vortex Model ◽  
External Branch ◽  
Euler Flows ◽  
Modal Amplitude ◽  
Vortex Centre

An exact solution of a stable vortex tripole in two-dimensional (2-D) Euler flows is provided. The stable tripole is composed of an inner elliptical vortex and two small-amplitude lateral vortices. The non-vanishing vorticity field of this tripole, referred to as here as an embedded tripole because of the closeness of its vortices, is given in elliptical coordinates $(\unicode[STIX]{x1D707},\unicode[STIX]{x1D708})$ by the even radial and angular order-0 Mathieu functions $\text{Je}_{0}(\unicode[STIX]{x1D707})\text{ce}_{0}(\unicode[STIX]{x1D708})$ truncated at the external branch of the vorticity isoline passing through the two critical points closest to the vortex centre. This tripole mode has a rigid vorticity field which rotates with constant angular velocity equal to $\unicode[STIX]{x1D701}_{0}\text{Je}_{0}(\unicode[STIX]{x1D707}_{1})\text{ce}_{0}(0)/2$, where $\unicode[STIX]{x1D707}_{1}$ is the first zero of $\text{Je}_{0}^{\prime }(\unicode[STIX]{x1D707})$ and $\unicode[STIX]{x1D701}_{0}$ is a constant modal amplitude. It is argued that embedded 2-D tripoles may be conceptually regarded as the superposition of two asymmetric Chaplygin–Lamb dipoles, separated a distance equal to $2R$, as long as their individual trajectory curvature radius $R$ is much shorter than their dipole extent radius.

On the application of the Kelvin–Arnol'd energy principle to the stability of forced two-dimensional inviscid flows

1998 ◽  
Vol 356 ◽  
pp. 221-257 ◽  
Author(s):  
P. A. DAVIDSON
Keyword(s):  
Magnetic Field ◽  
Nonlinear Stability ◽  
Broad Class ◽  
Mhd Flow ◽  
Two Dimensional ◽  
Energy Principle ◽  
Inviscid Flows ◽  
Virtual Displacement ◽  
Euler Flows ◽  
Universal Procedure

Arnol'd developed two distinct yet closely related approaches to the linear stability of Euler flows. One is widely used for two-dimensional flows and involves constructing a conserved functional whose first variation vanishes and whose second variation determines the linear (and nonlinear) stability of the motion. The second method is a refinement of Kelvin's energy principle which states that stable steady Euler flows represent extremums in energy under a virtual displacement of the vorticity field. The conserved-functional (or energy-Casimir) method has been extended by several authors to more complex flows, such as planar MHD flow. In this paper we generalize the Kelvin–Arnol'd energy method to two-dimensional inviscid flows subject to a body force of the form −ϕ∇f. Here ϕ is a materially conserved quantity and f an arbitrary function of position and of ϕ. This encompasses a broad class of conservative flows, such as natural-convection planar and poloidal MHD flow with the magnetic field trapped in the plane of the motion, flows driven by electrostatic forces, swirling recirculating flow, self-gravitating flows and poloidal MHD flow subject to an azimuthal magnetic field. We show that stable steady motions represent extremums in energy under a virtual displacement of ϕ and of the vorticity field. That is, d1E=0 at equilibrium and whenever d2E is positive or negative definite the flow is (linearly) stable. We also show that unstable normal modes must have a spatial structure which satisfies d2E=0. This provides a single stability test for a broad class of flows, and we describe a simple universal procedure for implementing this test. In passing, a new test for linear stability is developed. That is, we demonstrate that stability is ensured (for flows of the type considered here) whenever the Lagrangian of the flow is a maximum under a virtual displacement of the particle trajectories, the displacement being of the type normally associated with Hamilton's principle. A simple universal procedure for applying this test is also given. We apply our general stability criteria to a range of flows and recover some familiar results. We also extend these ideas to flows which are subject to more than one type of body force. For example, a new stability criterion is obtained (without the use of Casimirs) for natural convection in the presence of a magnetic field. Nonlinear stability is also considered. Specifically, we develop a nonlinear stability criterion for planar MHD flows which are subject to isomagnetic perturbations. This differs from previous criteria in that we are able to extend the linear criterion into the nonlinear regime. We also show how to extend the Kelvin–Arnol'd method to finite-amplitude perturbations.

A simple point vortex model for two‐dimensional decaying turbulence

1992 ◽  
Vol 4 (5) ◽  
pp. 1036-1039 ◽  
Author(s):  
R. Benzi ◽  
M. Colella ◽  
M. Briscolini ◽  
P. Santangelo
Keyword(s):  
Point Vortex ◽  
Two Dimensional ◽  
Simple Point ◽  
Vortex Model ◽  
Decaying Turbulence ◽  
Point Vortex Model

scholarly journals Vibration behavior of diamondene nano-ribbon passivated by hydrogen

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Lei Wang ◽  
Ranran Zhang ◽  
Jiao Shi ◽  
Kun Cai
Keyword(s):  
Molecular Dynamics ◽  
Slenderness Ratio ◽  
Curvature Radius ◽  
Two Dimensional ◽  
Vibration Frequencies ◽  
Carbon Allotrope ◽  
First Order ◽  
Vibration Behavior ◽  
Temperature Controlled ◽  
Dynamics Simulations

Abstract Diamondene is a new kind of two dimensional carbon allotrope with excellent properties and passivation approaches are often used to reduce the extremely high pressure required during its fabrication. When a one-end-clamped diamondene ribbon is hydrogenated on one surface, the ribbon tends to bend and vibrate due to asymmetric layout of C-H bonds on two surfaces. In the present work, the vibration behavior, including natural curvatures and vibration frequencies of diamondene ribbons, were investigated by molecular dynamics simulations. Results indicate that the natural curvature radius of a narrow diamondene ribbon is close to 12.17 nm at a temperature below 150 K, which is essential for fabricating an arc nanodevice. The first order frequency (f1) of a cantilever beam made from the ribbon follows traditional beam vibration theory if the slenderness ratio is low. In particular, f1 increases logarithmically at temperature below 50 K, but changes slightly between 50 K and 150 K. It suggests a design scheme for a nanoresonator with temperature-controlled frequency.

Casimir cascades in two-dimensional turbulence

2013 ◽  
Vol 729 ◽  
pp. 364-376 ◽  
Author(s):  
John C. Bowman
Keyword(s):  
Stokes Equation ◽  
Navier Stokes ◽  
Fourth Power ◽  
Two Dimensional ◽  
Vorticity Field ◽  
Navier Stokes Equation ◽  
Small Scales ◽  
Open Question ◽  
Incompressible Navier Stokes Equation

AbstractIn addition to conserving energy and enstrophy, the nonlinear terms of the two-dimensional incompressible Navier–Stokes equation are well known to conserve the global integral of any continuously differentiable function of the scalar vorticity field. However, the phenomenological role of these additional inviscid invariants, including the issue as to whether they cascade to large or small scales, is an open question. In this work, well-resolved implicitly dealiased pseudospectral simulations suggest that the fourth power of the vorticity cascades to small scales.

Generating controllable velocity fluctuations using twin oscillating hydrofoils

2012 ◽  
Vol 713 ◽  
pp. 150-158 ◽  
Author(s):  
S. F. Harding ◽  
I. G. Bryden
Keyword(s):  
Inverse Problem ◽  
Time Series ◽  
Flow Velocity ◽  
Two Dimensional ◽  
Velocity Fluctuations ◽  
Instantaneous Flow ◽  
Vortex Model ◽  
Flow Fluctuations ◽  
Wide Range ◽  
Oscillatory Velocity

AbstractAn experiment apparatus has been previously developed with the ability to independently control the instantaneous flow velocity in a water flume. This configuration, which uses two pitching hydrofoils to generate the flow fluctuations, allows the unsteady response of submerged structures to be studied over a wide range of driving frequencies and conditions. Linear unsteady lift theory has been used to calculate the instantaneous circulation about two pitching hydrofoils in uniform flow. A vortex model is then used to describe the circulation in the wakes that determine the velocity perturbations at the centreline between the foils. This paper introduces how the vortex model can be discretized to allow the inverse problem to be solved, such that the foil motions required to recreate a desired velocity time series can be determined. The results of this model are presented for the simplified cases of oscillatory velocity fluctuations in the vertical and stream-wise directions separately, and also simultaneously. The more general case of two-dimensional aperiodic velocity fluctuations is also presented, which demonstrates the capability of configuration between the suggested frequency limits of $0. 06\leq k\leq 1. 9$.

scholarly journals The spiral wind-up of vorticity in an inviscid planar vortex

1998 ◽  
Vol 371 ◽  
pp. 109-140 ◽  
Author(s):  
ANDREW P. BASSOM ◽  
ANDREW D. GILBERT
Keyword(s):  
Reynolds Number ◽  
Differential Rotation ◽  
Coarse Grained ◽  
Far Field ◽  
Two Dimensional ◽  
Vorticity Field ◽  
Test Function ◽  
Smooth Test ◽  
Linear Evolution ◽  
Vorticity Distribution

The relaxation of a smooth two-dimensional vortex to axisymmetry, also known as ‘axisymmetrization’, is studied asymptotically and numerically. The vortex is perturbed at t=0 and differential rotation leads to the wind-up of vorticity fluctuations to form a spiral. It is shown that for infinite Reynolds number and in the linear approximation, the vorticity distribution tends to axisymmetry in a weak or coarse-grained sense: when the vorticity field is integrated against a smooth test function the result decays asymptotically as t−λ with λ=1+(n2+8)1/2, where n is the azimuthal wavenumber of the perturbation and n[ges ]1. The far-field stream function of the perturbation decays with the same exponent. To obtain these results the paper develops a complete asymptotic picture of the linear evolution of vorticity fluctuations for large times t, which is based on that of Lundgren (1982).

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