Two dimensional incompressible ideal flow around a thin obstacle tending to a curve

In this work we study the asymptotic behavior of solutions of the incompressible two dimensional Euler equations in the exterior of a single smooth obstacle when the obstacle becomes very thin tending to a curve. We extend results by Iftimie, Lopes Filho and Nussenzveig Lopes, obtained in the context of an obstacle tending to a point, see [D. Iftimie, M.C. Lopes Filho, H.J. Nussenzveig Lopes, Two dimensional incompressible ideal flow around a small obstacle, Comm. Partial Differential Equations 28 (1–2) (2003) 349–379].     

Travelling Waves with a Singularity in Magnetic Nanowires

We model the evolution of the magnetization in an infinite cylinder by harmonic map heat flow with an additional external field. Using variational methods, we prove the existence of corotationally symmetric travelling wave solutions with a moving vortex. We moreover show that for weak and strong fields the travelling waves connect the original state anti-parallel to the external magnetic field with the totally reversed state in direction of the external field. Our results match numeric simulations. For thicker wires several groups have found a reversal mode where a domain wall with a corotational symmetry and a vortex is propagating through the wire.     

CONVEXITY AND SYMMETRY OF TRANSLATING SOLITONS IN MEAN CURVATURE FLOWS

This paper proves that any rotationally symmetric translating soliton of mean curvature flow in R3 is strictly convex if it is not a plane and it intersects its symmetric axis at one point. The authors also study the symmetry of any translating soliton of mean curvature flow in Rn.     

Asymptotic behavior of vortices in axisymmetric flow without swirl

We investigate the confinement properties of bounded, nonnegative, compactly supported vortices of axisymmetric incompressible Euler flows without swirl. We show that along one direction of the symmetry axis, its support can grow no faster than O[(t log t)^1/2]. The rate at which it approaches the symmetry axis is also estimated. Together with the result of Maffei–Marchioro on the radial growth rate of the support, it is contained in a slowly expanding tubular region. The techniques of the above‐mentioned authors, Iftimie–Lopes–Nussenzveig and Iftimie–Sideris–Gamblin, are used. Copyright © 2008 John Wiley & Sons, Ltd.     

Spherical rarefaction flow in the neighbourhood of a reflection point of a “boundary” characteristic

The structure of the one-dimensional steady spherically symmetric rarefaction flow of an ideal (inviscid and non-heat-conducting) gas in the neighbourhood of a reflection point of a “boundary” C⁻-characteristic is investigated in principal order. The “boundary” C⁻-characteristic separates the gas at rest from the flow due to the outward motion of a piston which confines the gas. In the rt plane, where r is the distance from the centre of symmetry and t is the time, the reflection point, which coincides with the point of arrival on the t axis of the boundary characteristic, coincides with the origin of coordinates. The initial velocity of the piston may be zero (for positive acceleration) or finite. When two symmetrical plane pistons advance, the “derived” derivatives of all the flow parameters on the C⁻-characteristic at the origin of coordinates, which in this case lies on the plane of symmetry, are finite. When a cylindrical and spherical piston advance, the derived derivative of the pressure (velocity) of the gas on the C⁻-characteristic at the origin of coordinates becomes minus (plus)-infinity although without intersecting characteristics of the same family [1–4].     

Stability and Structure of Stretched Vortices

The radially symmetric Burgers vortex is an example of a solution to the Navier-Stokes equations in which the intensification of vorticity due to vortex stretching is balanced by the diffusion of vorticity through viscosity. The linear stability of the symmetric Burgers vortex to a class of two-dimensional perturbations is demonstrated. Analytical solutions obtained from a perturbation analysis and numerical computations are presented of nonsymmetric Burgers vortices in which the radial flow field in a plane perpendicular to the vorticity is nonsymmetric.     

Numerical study of vortex eccentricity in a gas cyclone

This paper presents a numerical study of the vortex eccentricity in a gas cyclone and its effect on the performance of the cyclone. The gas flow in the cyclone was modeled as an unsteady flow by the Navier–Stokes equations with the Reynolds Stress Model (RSM) as the turbulence model. The particles were modelled by the Lagrangian particle tracking (LPT) approach in an unsteady gas flow. Gas cyclones with the same dimensions and total flow rates but different numbers of inlets were simulated with the inlet velocity varying from 12 to 20 m/s. The vortex eccentricities in different cases were analyzed in terms of radial deviation and angular deviation. In addition, the frequency of the precessing vortex core (PVC) was analyzed by the fast Fourier transform (FFT). The results show that the vortex center in the single inlet cyclone has a great eccentricity and its precession center is also different from the geometric center, which reduces the particle collection efficiency. The increase in the symmetry of the inlet causes only a very small increase in the pressure drop in the simulated cases, but it can significantly reduce the vortex eccentricity, particularly by eliminating the eccentricity of the PVC center. The improvement of the vortex eccentricity can generally increase the collection efficiency for particles greater than 2.0 µm. The increase of the collection efficiency is mainly because the symmetrical vortex can restrain the short-circuiting flow of particles. The results can improve the understanding of the vortex flow in gas cyclones which may guide the optimization of gas cyclones.     

Symmetric Intersections of Rauzy Fractals

In this article we study symmetric subsets of Rauzy fractals of unimodular irreducible Pisot substitutions. The symmetry considered is reflection through the origin. Given an unimodular irreducible Pisot substitution, we consider the intersection of its Rauzy fractal with the Rauzy fractal of the reverse substitution. This set is symmetric and it is obtained by the balanced pair algorithm associated with both substitutions.     

Nonself-similar flow with a shock wave reflected from the center of symmetry and new self-similar solutions with two reflected shocks

In some problems concerning cylindrically and spherically symmetric unsteady ideal (inviscid and nonheat-conducting) gas flows at the axis and center of symmetry (hereafter, at the center of symmetry), the gas density vanishes and the speed of sound becomes infinite starting at some time. This situation occurs in the problem of a shock wave reflecting from the center of symmetry. For an ideal gas with constant heat capacities and their ratio γ (adiabatic exponent), the solution of this problem near the reflection point is self-similar with a self-similarity exponent determined in the course of the solution construction. Assuming that γ on the reflected shock wave decreases, if this decrease exceeds a threshold value, the flow changes substantially. Assuming that the type of the solution remains unchanged for such γ, self-similarity is preserved if a piston starts expanding from the center of symmetry at the reflection time preceded by a finite-intensity reflected shock wave propagating at the speed of sound. To answer some questions arising in this formulation, specifically, to find the solution in the absence of the piston, the evolution of a close-to-self-similar solution calculated by the method of characteristics is traced. The required modification of the method of characteristics and the results obtained with it are described. The numerical results reveal a number of unexpected features. As a result, new self-similar solutions are constructed in which two (rather than one) shock waves reflect from the center of symmetry in the absence of the piston.     

The number of spanning trees of plane graphs with reflective symmetry

A plane graph is called symmetric if it is invariant under the reflection across some straight line (called symmetry axis). Let G be a symmetric plane graph. We prove that if there is no edge in G intersected by its symmetry axis then the number of spanning trees of G can be expressed in terms of the product of the number of spanning trees of two smaller graphs, each of which has about half the number of vertices of G.     

Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows

We continue the work of Lopes Filho, Mazzucato and Nussenzveig Lopes [10] on the vanishing viscosity limit of circularly symmetric viscous flow in a disk with rotating boundary, shown there to converge to the inviscid limit in L ²-norm as long as the prescribed angular velocity α(t) of the boundary has bounded total variation. Here we establish convergence in stronger L ² and L ^p -Sobolev spaces, allow for more singular angular velocities α, and address the issue of analyzing the behavior of the boundary layer. This includes an analysis of concentration of vorticity in the vanishing viscosity limit. We also consider such flows on an annulus, whose two boundary components rotate independently. Supported in part by NSF grant DMS-0456861.     

The conditions for the symmetric instability of the vortex motions of an ideal stratified liquid

The problem of the symmetric instability of the steady-state motions of an incompressible ideal liquid which is stratified with respect to its density is investigated in the case of two types of motion, axially symmetric and with translational symmetry. It is shown that the sufficient condition for stability obtained in [1] using a variational method (the direct Lyapunov method) for the motions under consideration is closely related to the extremal nature of their energy; stable motions are characterized by a conditional minimum of the energy. A minimum of the energy holds in the class of states for which a potential vortex, expressed in terms of the Lagrangian invariants, angular momentum and density, is represented by the same function of these invariants. Conditions for instability are formulated and estimates of the increase in the kinetic energy of perturbations are given.     

Twisted Yangians for symmetric pairs of types B,C, D

We study a class of quantized enveloping algebras, called twisted Yangians, associated with the symmetric pairs of types B, C, D in Cartan’s classification. These algebras can be regarded as coideal subalgebras of the Yangian for orthogonal or symplectic Lie algebras. They can also be presented as quotients of a reflection algebra by additional symmetry relations. We prove an analogue of the Poincaré–Birkoff–Witt Theorem, determine their centres and study also extended reflection algebras.     

Generation of large-scale vortex dislocations in a three-dimensional wake-type flow

Numerical study of three-dimensional evolution of wake-type flow and vortex dislocations is performed by using a compact finite diffenence-Fourier spectral method to solve 3-D incompressible Navier-Stokes equations. A local spanwise nonuniformity in momentum defect is imposed on the incoming wake-type flow. The present numerical results have shown that the flow instability leads to three-dimensional vortex streets, whose frequency, phase as well as the strength vary with the span caused by the local nonuniformity. The vortex dislocations are generated in the nonuniform region and the large-scale chain-like vortex linkage structures in the dislocations are shown. The generation and the characteristics of the vortex dislocations are described in detail.     

A Remark on the Global Existence of a Third Order Dispersive Flow into Locally Hermitian Symmetric Spaces

We prove time-global existence of solutions to the initial value problem for a third order dispersive flow into compact locally Hermitian symmetric spaces. The equation under consideration generalizes two-sphere-valued completely integrable systems modeling the motion of vortex filament. Unlike one-dimensional Schrödinger maps, our third order equation is not completely integrable under the curvature condition on the target manifold in general. The idea of our proof is to exploit two conservation laws and an “almost conserved quantity” which prevents the formation of a singularity in finite time.     

激波在异种气体中传播及诱导的剪切混合研究

利用二阶迎风TVD格式求解多组分,层流全N－S方程,针对直通道和突扩直通道,研究了马赫数为2和4的激波在H2和空气界面上的传播及诱导的燃料剪切混合,计算结果表明：（1）直通道中,剪切层中的激波阵面要发生畸变,存在对混合起主要作用的卷吸涡,激波马赫数不同,卷吸涡结构和横向混合的尺寸也不同,激波马赫数低,剪切混合效果好,（2）在突扩直通道中,马赫数为2和4的激波在H2中产生不同强度激波,在剪切层中都能产生顺时针,尺度较大的卷吸涡,后台阶增强了剪切层的混合。     

On the development of a wake vortex in inviscid flow

The evolution of an initial perturbation in an axisymmetric subsonic normal inviscid gas flow through a pipe is directly simulated. The basic (unperturbed) flow has a zero radial velocity component, while its axial velocity component (along the axis of symmetry) increases or decreases linearly with the radius. The perturbation is specified as a swirl (rotation about the axis) with a positive or negative velocity vanishing on the central axis and the lateral surface. Irrespective of its direction, the swirl gives rise to a steady-state vortex carried by the flow. It shape is spherical (contiguous to the rotation axis) or circular (sliding along the impermeable lateral surface).     

Relative equilibria of symmetric n-body problems on a sphere: Inverse and direct results

In this paper, the author proves that the existence of a class of relative equilibria for SO(3)-symmetric n-body problems on a sphere implies that these problems are in fact O(3)-symmetric in a time-reversing sense. Besides this “inverse” result, the author also proves a set of “direct” results, in which the existence of certain symmetric relative equilibria are deduced purely from symmetry considerations of the n-body problems on a sphere. The author then formulates an explicit method for the reduction of this class of symmetric Hamiltonians, which leads to symplectic shape or relative variables for the further study of the relative equilibria and equilibria. This class of problems includes the point vortex problem on a sphere, and the author applies the main results in this paper to that problem. © 1998 John Wiley & Sons, Inc.     

Computational study of decaying annular vortex flow using the Rε/k − ε turbulence model

In this article, we present three dimensional CFD study of turbulent vortex flow in an annular passage using OpenFOAM 1.6. The vortex flow is generated by introducing the flow through a tangential entry to the passage. For the analysis presented in this article, turbulence was modeled using the R_ε/k − ε model, in addition, a comparison between such model with the standard k − ε model was conducted and discussed. The main characteristics of the flow such as vortex structure and recirculation zone were investigated. It was found that flow is subjected to Rankine vortex structure with three forced vortex regimes and a free vortex region near to the outer wall. The phenomenon of vortex decay was investigated by depicting the swirl number trend along the axial direction of the flow domain. It was found that the vortex decay is subjected to an exponential decay behavior. New coefficients for the exponential decay correlation were derived based on local values of velocity components in different radial planes.     

风力机叶尖涡尾迹结构PIV测量研究

空气动力研究与发展中心低速空气动力研究所依托工程型大风洞（实验段直径3.2 m）,采用高分辨率CCD相机（4 008像素×2 672像素）,针对旋转状态下的风力机叶片尾流开展大视场（单个观测区域达到570 mm×380 mm）PIV(particle image velocimetry)测量,以NREL UAE Phase Ⅵ风力机叶片1/8缩比模型为实验对象,获取了叶尖涡产生、发展的流动数据,为研究风力机叶尖涡结构和流动机理研究提供重要的基础数据．观测结果表明,叶尖涡从后缘脱落后首先有一个短时间的向内运动,然后随着尾流的膨胀向外运动,其涡强度则先是短时间内降低,然后随着涡的卷起而增强,从而形成一个强大的叶尖涡．在实验观察范围内叶尖涡在来流方向的迁移规律近似线性．