Existence and Stability of Compressible Current-Vortex Sheets in Three-Dimensional Magnetohydrodynamics

Compressible vortex sheets are fundamental waves, along with shocks and rarefaction waves, in entropy solutions to multidimensional hyperbolic systems of conservation laws. Understanding the behavior of compressible vortex sheets is an important step towards our full understanding of fluid motions and the behavior of entropy solutions. For the Euler equations in two-dimensional gas dynamics, the classical linearized stability analysis on compressible vortex sheets predicts stability when the Mach number \(M > \sqrt{2}\) and instability when \(M < \sqrt{2}\) ; and Artola and Majda’s analysis reveals that the nonlinear instability may occur if planar vortex sheets are perturbed by highly oscillatory waves even when \(M > \sqrt{2}\) . For the Euler equations in three dimensions, every compressible vortex sheet is violently unstable and this instability is the analogue of the Kelvin–Helmholtz instability for incompressible fluids. The purpose of this paper is to understand whether compressible vortex sheets in three dimensions, which are unstable in the regime of pure gas dynamics, become stable under the magnetic effect in three-dimensional magnetohydrodynamics (MHD). One of the main features is that the stability problem is equivalent to a free-boundary problem whose free boundary is a characteristic surface, which is more delicate than noncharacteristic free-boundary problems. Another feature is that the linearized problem for current-vortex sheets in MHD does not meet the uniform Kreiss–Lopatinskii condition. These features cause additional analytical difficulties and especially prevent a direct use of the standard Picard iteration to the nonlinear problem. In this paper, we develop a nonlinear approach to deal with these difficulties in three-dimensional MHD. We first carefully formulate the linearized problem for the current-vortex sheets to show rigorously that the magnetic effect makes the problem weakly stable and establish energy estimates, especially high-order energy estimates, in terms of the nonhomogeneous terms and variable coefficients. Then we exploit these results to develop a suitable iteration scheme of the Nash–Moser–Hörmander type to deal with the loss of the order of derivative in the nonlinear level and establish its convergence, which leads to the existence and stability of compressible current-vortex sheets, locally in time, in three-dimensional MHD.     

Collapse and concentration of vortex sheets in two-dimensional flow

Numerical evidence is presented for the existence of collapse configurations of vortex sheets (one-dimensional singular distributions of vorticity) in a two-dimensional ideal fluid. Point vortices are used to approximate the vortex sheets. These and related motions cause a significant concentration of vorticity, with possible relevance to the concentration seen in the evolution of turbulent flows.     

A double vortex sheet model of the viscous flow near the trailing edge of a lifting aerofoil

Classical definitions of boundary layer mass and momentum flux deficiency thicknesses can lead to gross errors when applied to measurements near a trailing edge where the flow curvature in the free stream is appreciable. This paper presents a double vortex sheet model as a development from the single vortex sheet model of Helmholtz and others. Two bound vortex sheets define a potential function which can describe a flow with the same mass and momentum flux deficiencies as the viscous regions. The bound nature of these sheets allows the modelling of the integral properties of these regions while retaining the advantages of a potential flow. The application to the flow near the trailing edge of a lifting aerofoil is given     

Stationary vortex sheets in a stirring flow

Stationary vortex sheets in a two-dimensional stirring flow may be approximated by arrays of stationary point vortices arranged along the support of the sheets. These vortices lie at the roots of a polynomial that satisfies a generalized Lamé differential equation; the polynomial itself (not the roots) determines the complex potential and stream function. In this paper, sufficient conditions for the stirring flow are found so that the differential equation has two independent polynomial solutions with simple closed-form expressions, analogous to hypergeometric polynomials. The corresponding point vortex array then depends on a complex parameter that controls the location of the sheet, so that it may pass through any selected point. Stationary sheets in a periodic flow are approximated by the same method.     

Stability and capillary dynamics of circular vortex sheets

We investigate the motion of circular vortex sheets with surface tension. A linear stability analysis shows that high modes of the circular vortex sheet are stabilized by surface tension, and the sheet is stable if surface tension is larger than a critical value. The modes of perturbations, n = 1 and 2, are always stable, regardless of surface tension, and the mode n = 3 is also stable for large surface tension. The numerical results show that a stable vortex sheet rotates and oscillates weakly. The oscillations of each mode of the interface mainly consist of two travelling waves of different frequencies in time. The amplitude and the period of the oscillation depend on the mode of the perturbation and surface tension. We also perform long-time computations for the unstable evolution of circular sheets. For a high Weber number, ripples are produced on the sheets, as well as pinching and self-intersection. It is found that the appearance of ripples is associated with the growth of noise. For an intermediate Weber number, the sheet evolves to an exotic structure with small spikes on the fingers, while for a low Weber number, it is nonlinearly stable.     

A structural model for the vortex tubes of isotropic turbulence

A class of exact solutions of the Navier–Stokes equations is introduced to model the fine-scale, tubular structures of isotropic turbulence. The model vortices exhibit slow algebraic fall-off of the induced velocity, and accurately reproduce the velocity signatures observed in DNS and experiments. The proposed model has interesting implications for the theoretical analysis of turbulence, supporting the view that the inertial range energy scaling may have a link with the near-singular velocity field induced by vortex tubes produced by the roll-up of vortex sheets.      

A multi-vortex model of leading-edge vortex flows

A multi-vortex model of the vortex sheets shed from the sharp leading edges of slender wings is considered. The method, which is developed within the framework of slender-body theory, is designed to deal with those situations in which more than one centre of rotation is formed on the wing, for example on a slender wing with lengthwise camber or with a strake. Numerical results are presented, firstly for situations where comparison can be made with a vortex sheet model and secondly for cases, such as those described above, where a vortex sheet model is unable to describe the flow. Where comparison is available, agreement is good and in the cases where more than one vortex system is present interesting interactions are obtained.     

Abundant singularities

We show that a generic Euler flow or a generic ideal magnetostatic equilibrium contains an intricate structure of tangential discontinuities which are vortex sheets or current sheets, respectively. The discontinuities are present at all spatial scales and are dense at least in some parts of the volume occupied by the fluid. Possible implications for the energy dissipation in turbulent flows and for the structure of equilibria in tokamaks are discussed.     

Study of water waves with submerged obstacles using a vortex method with Helmholtz decomposition

The present study develops a 2‐D numerical scheme that combines the vortex method and the boundary integral method by a Helmholtz decomposition to investigate the interaction of water waves with submerged obstacles. Viscous effects and generation of vorticity on the free surface are neglected. The second kind of Fredholm integral equations that govern the strengths of vortex sheets along boundaries are solved iteratively. Vorticity is convected and diffused in the fluid via a Lagrangian vortex (blob) method with varying cores, using the particle strength exchange method for diffusion, with particle redistribution. A grid‐convergence study of the numerical method is reported. The inviscid part of the method and the simulation of the free‐surface motion are tested using two calculations: solitary wave propagation in a uniform channel and a moving line vortex in the fluid. Finally, the full model is verified by simulating periodic waves travelling over a submerged rectangular obstacle using nonuniform vortex blobs with a mapping of the redistribution lattice. Overall, the numerical model predicts the vortices' evolution and the free‐surface motion reasonably well. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical study on evolution of subharmonic varicose low-speed streaks in turbulent channel flow

The evolution of two spanwise-aligned low-speed streaks in a wall turbulent flow, triggered by the instability of the subharmonic varicose (SV) mode, is studied by a direct numerical simulation (DNS) method in a small spatial-periodic channel. The results show that the SV low-speed streaks are self-sustained at the early stage, and then transform into subharmonic sinuous (SS) low-speed streaks. Initially, the streamwise vortex sheets are formed by shearing, and then evolve into zigzag vortex sheets due to the mutual induction. As the intensification of the SV low-speed streaks becomes prominent, the tilted streamwise vortex tubes and the V-like streamwise vortex tubes can be formed simultaneously by increasing \( + \frac{{\partial u}}{{\partial x}}\). When the SV low-speed streaks break down, new zigzag streamwise vortices will be generated, thus giving birth to the next sustaining cycle of the SV low-speed streaks. When the second breakdown happens, new secondary V-like streamwise vortices instead of zigzag streamwise vortices will be generated. Because of the sweep motion of the fluid induced by the secondary V-like streamwise vortices, each decayed low-speed streak can be divided into two parts, and each part combines with the part of another streak, finally leading to the formation of SS low-speed streaks.     

Formation and interaction of intense vortex sheets in 3-dimensional,incompressible, hydrodynamics

Various questions related to the physics of inviscid flows are reviewed. The emergence of strong vortex sheets has repeatedly been observed in the simulation of the 3-dimensional equations, with a variety of initial conditions. In the case of axisymmetric Euler flows, the origin of these sheets can be easily understood with the help of an analogy with thermally driven flows. A more general mechanism to explain these sheets is proposed. Questions of singularities are briefly reviewed. Lastly, preliminary results on the connection between the sheets forming in inviscid flows, and the vortex tubes observed in high Reynolds number flows are presented.

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Sommario Sono considerate varie questioni correlate con la fisica dei flussi non-viscosi. La nascita di strati di forti vortici è stata ripetutamente osservata nella simulazione delle equazioni tridimensionali, per diverse condizioni iniziali. Nel caso di flussi di Eulero assialsimmetrici, l'origine di questi piani può essere facilmente compresa con l'aiuto di una analogia con i flussi guidati termicamente. Viene inoltre proposto un più generale meccanismo per giustificare questi strati e si passano in rassegna brevemente questioni riguardanti le singolarità. Infine, vengono presentati alcuni risultati preliminari sulla connessione tra i piani formantisi in flussi non viscosi ed i vortici tubolari osservati nei flussi ad alti numeri di Reynolds.

     

涡识别技术及其在船用螺旋桨伴流场中的应用

涡识别是很重要的流体问题, 为了在船用螺旋桨伴流场中寻找一种合理的涡识别方法, 本文结合实践, 研究了六种涡识别技术理论, 其中使用Burgers涡流和Lamb-Oseen涡流作了必要的解释, 讨论了各种识别方法的优缺点. 局部低压标准比较直观, 但深究其黏性和非定常影响后, 明显不足; 迹线或流线显然不能满足伽利略不变性, 会使辨别变得混乱; 涡度大小需要规定其阈值, 具有一定不确定性, 且也会识别不是涡的涡片; 速度梯度张量的复特征值也会有识别不出的区域; 速度梯度张量的第二不变量标准和对称张量的第二特征值标准能更好地识别涡核, 这两种标准有时等效. 螺旋桨伴流场的数值模拟是在开源软件OpenFOAM平台上实现的, 湍流大涡模型由一种局部动态方程建模, 此模型优于动态Smagorinsky模型. 最终的结果显示: 对于船用螺旋桨伴流场中的涡, 采用速度梯度张量的第二不变量和对称张量的第二特征值的结果基本一致, 而最小压力标准、流线或迹线标准、涡度值标准和速度张量的复特征值标准都存在一定的缺陷, 不适用于船用螺旋桨伴流场中的涡识别.      

VISCOUS FLOWFIELD BASED ON DISCONTINUOUS BOUNDARY ELEMENT METHOD AND VORTEX METHOD

基于非协调边界元方法和涡方法的联合应用, 模拟了二维和三维黏性不可压缩流场. 计算中利用离散涡元对漩涡的产生、凝聚和输送过程进行模拟, 并将整体计算域分解为采用涡泡模拟的内部区域和用涡列模拟的数字边界层区域. 计算域中涡量场的拉伸和对流由Lagrangian涡方法模拟, 用随机走步模拟涡量场的扩散. 内部区域涡元涡量场速度由广义Biot-Savart公式计算, 势流场速度则采用非协调边界元方法计算. 非协调边界元将所有节点均取在光滑边界处, 从而避免了法向速度的不连续现象; 而对于系数矩阵不对称的大型边界元方程组,引入了非常高效的预处理循环型广义极小残余(the generalized minimum residual, GMRES)迭代算法, 使得边界元法的优势得到了充分发挥, 同时, 在内部涡元势流场计算中对近边界点采用了正则化算法, 该算法将奇异积分转化为沿单元围道上一系列线积分, 消除了势流计算中速度及速度梯度的奇异性. 二维、三维流场算例证明了所用方法的正确性, 也验证了该算法可以大幅度提高模拟精度和效率.     

带尾涡面的三维简化航天机亚音速有限元计算

本文采用Ｇａｌｅｒｋｉｎ有限元方法以全位势方程为控制方程计算了航天机三维简化模型的亚声速流动。为模拟实际流动,在后部加一尖劈形后体,并拖出一尾涡面。为局部超场 速区解的稳定,采用人工密度修正。对密度和环量进行双重迭代,得到了亚声速下三维航天机的压分分布和气动力系数。     

An inviscid model for vortex shedding from a deforming body

An inviscid vortex sheet model is developed in order to study the unsteady separated flow past a two-dimensional deforming body which moves with a prescribed motion in an otherwise quiescent fluid. Following Jones (J Fluid Mech 496, 405–441, 2003) the flow is assumed to comprise of a bound vortex sheet attached to the body and two separate vortex sheets originating at the edges. The complex conjugate velocity potential is expressed explicitly in terms of the bound vortex sheet strength and the edge circulations through a boundary integral representation. It is shown that Kelvin’s circulation theorem, along with the conditions of continuity of the normal velocity across the body and the boundedness of the velocity field, yields a coupled system of equations for the unknown bound vortex sheet strength and the edge circulations. A general numerical treatment is developed for the singular principal value integrals arising in the solution procedure. The model is validated against the results of Jones (J Fluid Mech 496, 405–441, 2003) for computations involving a rigid flat plate and is subsequently applied to the flapping foil experiments of Heathcote et al. (AIAA J, 42, 2196–2204, 2004) in order to predict the thrust coefficient. The utility of the model in simulating aquatic locomotion is also demonstrated, with vortex shedding suppressed at the leading edge of the swimming body.      

Stabilization Effect of Magnetic Fields on Two-Dimensional Compressible Current-Vortex Sheets

We analyze the linear stability of rectilinear compressible current-vortex sheets in two-dimensional isentropic magnetohydrodynamics, which is a free boundary problem with the boundary being characteristic. In the case when the magnitude of the magnetic field has no jump on the current-vortex sheets, we find a necessary and sufficient condition of linear stability for the rectilinear current-vortex sheets, showing that magnetic fields exert a stabilization effect on compressible vortex sheets. In addition, a loss of regularity with respect to the source terms, both in the interior domain and on the boundary, occurs in a priori estimates of solutions to the linearized problem for a rectilinear current-vortex sheet, as the Kreiss–Lopatinskii determinant associated with this linearized boundary value problem has roots on the boundary of frequency spaces. In this study, the construction of symmetrizers for a reduced differential system, which has poles at which the Kreiss–Lopatinskii condition may fail simultaneously, plays a crucial role in the a priori estimates.     

The shedding of vorticity from a smooth surface

This work is concerned with the development of a numerical model for laminar flow separation from a smooth boundary. The concept of irreversible vorticity generation is used to formulate an algorithm to predict both the location of separation and the shedding rate of vorticity. Results of flow separation from a circular cylinder are presented. The calculated vortex sheets are visualized as streak lines in the wake of the cylinder. Streamline patterns are constructed from these calculations showing location of separation and wake structure.     

An advanced design method for propellers behind bodies of revolutionCalcul de l'hélice fonctionnant en arrière d'un corps à symétrie axiale

The flow around the unshrouded marine propellers operating in the wake of an axisymmetric body is rotational and tridimensional. An inverse method based on the model of inviscid and rotational fluid and coupling two complementary steps (axisymmetric computation + 3D panel method) is proposed for the design of the marine propellers. The meridional flow computation leads to the determination of axisymmetrical stream sheets as well as the approximate camber surface of the blades and gives a good estimation of the surface of the free vortex wake. The new aspects developed in this method are the involvement of the contraction and the stretching of the free vortex wake, the rotational character of the incoming flow in the axisymmetric computation with tridimensional effects due to 3D panel method. To cite this article: N. Settou, B. Viney, C. R. Mecanique 331 (2003).     

Direct and large eddy simulation of stratified turbulence

Simulations of geophysical turbulent flows require a robust and accurate subgrid-scale turbulence modeling. To evaluate turbulence models for stably stratified flows, we performed direct numerical simulations (DNSs) of the transition of the three-dimensional Taylor–Green vortex and of homogeneous stratified turbulence with large-scale horizontal forcing. In these simulations we found that energy dissipation is concentrated within thin layers of horizontal tagliatelle-like vortex sheets between large pancake-like structures. We propose a new implicit subgrid-scale model for stratified fluids, based on the Adaptive Local Deconvolution Method (ALDM). Our analysis proves that the implicit turbulence model ALDM correctly predicts the turbulence energy budget and the energy spectra of stratified turbulence, even though dissipative structures are not resolved on the computational grid.     

Multisensor Hot Wire Vorticity Probe Measurements of the Formation Field of Two Corotating Vortices

Experimental evidence is reported, regarding the formation of a pair of co-rotating tip vortices by a split wing configuration, consisting of two half wings at equal and opposite angles of attack. Simultaneous measurements of the three-dimensional vector fields of velocity and vorticity were conducted on a cross plane at a downstream distance corresponding to 0.3 cord lengths (near wake), using an in-house constructed 12-sensor hot wire anemometry vorticity probe. The probe consists of three closely separated orthogonal 4-wire velocity sensor arrays, measuring simultaneously the three-dimensional velocity vector at three closely spaced locations on a cross plane of the flow filed. This configuration makes possible the estimation of spatial velocity derivatives by means of a forward difference scheme of first order accuracy. Velocity measurements obtained with an X-wire are also presented for comparison. In this near wake location, the flow field is dictated by the pressure distribution established by the flow around the wings, mobilizing large masses of air and leading to the roll up of fluid sheets. Fluid streams penetrating between the wings collide, creating on the cross plane flow a stagnation point and an “impermeable” line joining the two vortex centres. Along this line fluid is directed towards the two vortices, expanding their cores and increasing their separation distance. This feeding process generates a dipole of opposite sign streamwise mean vorticity within each vortex. The rotational flow within the vortices obligates an adverse streamwise pressure gradient leading to a significant streamwise velocity deficit characterizing the vortices. The turbulent flow field is the result of temporal changes in the intensity of the vortex formation and changes in the position of the cores (wandering).