Particle Simulation of Swirling Flows

The vortex particle method has been applied to the axisymmetric swirling flow of a viscous fluid. The formulation used yields two transport equations which have been solved within the lagrangian framework of particle method. The diffusion operator for both equations has been approximated by means of a Particle Strength Exchange scheme. Applications to the cases of one isolated vortex ring and two co-rotating vortex rings illustrate the interest of this new method. Special attention has been devoted to the vorticity production resulting from the interaction between the azimuthal components of vorticity and velocity. The generation of small eddies at the boundary of the vortex ring cross-section has been particularly investigated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Diffusion velocity for a three-dimensional vorticity field

A discussion is presented on the existence of a diffusion velocity for the vorticity vector that satisfies extensions of the Helmholtz vortex laws in a three-dimensional, incompressible, viscous fluid flow. A general form for the diffusion velocity is derived for a complex-lamellar vorticity field that satisfies the property that circulation is invariant about a region that is advected with the sum of the fluid velocity and the diffusion velocity. A consequence of this property is that vortex lines will be material lines with respect to this combined velocity field. The question of existence of diffusion velocity for a general three-dimensional vorticity field is shown to be equivalent to the question of existence of solutions of a certain Fredholm equation of the first kind. An example is given for which it is shown that a diffusion velocity satisfying this property does not, in general, exist. Properties of the simple expression for diffusion velocity for a complex-lamellar vorticity field are examined when applied to the more general case of an arbitrary three-dimensional flow. It is found that this form of diffusion velocity, while not satisfying the condition of circulation invariance, nevertheless has certain desirable properties for computation of viscous flows using Lagrangian vortex methods. The significance and structure of the noncomplex-lamellar part of the viscous diffusion term is examined for the special case of decaying homogeneous turbulence.     

Large eddy simulation (2D) using diffusion–velocity method and vortex‐in‐cell

A large eddy Simulation based on the diffusion‐velocity method and the discrete vortex method is presented. The vorticity‐based and eddy viscosity type subgrid scale model simulating the enstrophy transfer between the large and small scale appears as a convective term in the diffusion‐velocity formulation. The methodology has been tested on a spatially growing mixing layer using the two‐dimensional vortex‐in‐cell method and the Smagorinsky subgrid scale model. The effects on the vorticity contours, momemtum thickness, mean streamwise velocity profiles, root‐mean‐square velocity and vorticity fluctuations and negative cross‐stream correlation are discussed. Comparison is made with experiment and numerical work where diffusion is simulated using random walk. Copyright © 2004 John Wiley & Sons, Ltd.     

A three‐dimensional vortex particle‐in‐cell method for vortex motions in the vicinity of a wall

A new vortex particle‐in‐cell method for the simulation of three‐dimensional unsteady incompressible viscous flow is presented. The projection of the vortex strengths onto the mesh is based on volume interpolation. The convection of vorticity is treated as a Lagrangian move operation but one where the velocity of each particle is interpolated from an Eulerian mesh solution of velocity–Poisson equations. The change in vorticity due to diffusion is also computed on the Eulerian mesh and projected back to the particles. Where diffusive fluxes cause vorticity to enter a cell not already containing any particles new particles are created. The surface vorticity and the cancellation of tangential velocity at the plate are related by the Neumann conditions. The basic framework for implementation of the procedure is also introduced where the solution update comprises a sequence of two fractional steps. The method is applied to a problem where an unsteady boundary layer develops under the impact of a vortex ring and comparison is made with the experimental and numerical literature. Copyright © 2001 John Wiley & Sons, Ltd.     

湍流一般机理及其应用

本文综述了湍流机理发展中最重要的一些文章,包括:Brown-Roshko的混合剪切层中大尺度涡的发展;Perry和Chong的湍流边界层中A形涡结构的湍流机理;以及笔者提出的关于3维湍流场中涡结构的涡量脉动对流和扩散并存的一般机理。在一般机理基础上建立了判别流场的准则,列举了在固壁边界附近的猝发现象并展望它的未来应用。      

Large eddy simulation (2D) of spatially developing mixing layer using vortex‐in‐cell for flow field and filtered probability density function for scalar field

A large eddy simulation based on filtered vorticity transport equation has been coupled with filtered probability density function transport equation for scalar field, to predict the velocity and passive scalar fields. The filtered vorticity transport has been formulated using diffusion‐velocity method and then solved using the vortex method. The methodology has been tested on a spatially growing mixing layer using the two‐dimensional vortex‐in‐cell method in conjunction with both Smagorinsky and dynamic eddy viscosity subgrid scale models for an anisotropic flow. The transport equation for filtered probability density function is solved using the Lagrangian Monte‐Carlo method. The unresolved subgrid scale convective term in filtered density function transport is modelled using the gradient diffusion model. The unresolved subgrid scale mixing term is modelled using the modified Curl model. The effects of subgrid scale models on the vorticity contours, mean streamwise velocity profiles, root‐mean‐square velocity and vorticity fluctuations profiles and negative cross‐stream correlations are discussed. Also the characteristics of the passive scalar, i.e. mean concentration profiles, root‐mean‐square concentration fluctuations profiles and filtered probability density function are presented and compared with previous experimental and numerical works. The sensitivity of the results to the Schmidt number, constant in mixing frequency and inflow boundary conditions are discussed. Copyright © 2005 John Wiley & Sons, Ltd.     

Vortex‐in‐cell method combined with a boundary element method for incompressible viscous flow analysis

In this study, an immersed boundary vortex‐in‐cell (VIC) method for simulating the incompressible flow external to two‐dimensional and three‐dimensional bodies is presented. The vorticity transport equation, which is the governing equation of the VIC method, is represented in a Lagrangian form and solved by the vortex blob representation of the flow field. In the present scheme, the treatment of convection and diffusion is based on the classical fractional step algorithm. The rotational component of the velocity is obtained by solving Poisson's equation using an FFT method on a regular Cartesian grid, and the solenoidal component is determined from solving an integral equation using the panel method for the convection term, and the diffusion term is implemented by a particle strength exchange scheme. Both the no‐slip and no‐through flow conditions associated with the surface boundary condition are satisfied by diffusing vortex sheet and distributing singularities on the body, respectively. The present method is distinguished from other methods by the use of the panel method for the enforcement of the no‐through flow condition. The panel method completes making use of the immersed boundary nature inherent in the VIC method and can be also adopted for the calculation of the pressure field. The overall process is parallelized using message passing interface to manage the extensive computational load in the three‐dimensional flow simulations. Copyright © 2011 John Wiley & Sons, Ltd.     

Diffusion of a ring vortex

The diffusion of a ring vortex is investigated in the present paper with allowance for the influence of the initial radius of the toroidal vorticity distribution on the flow structure. The statement of the problem in such a formulation makes it possible to classify and reinterpret results obtained previously. A vortex pair is studied together with a vortex ring. The toroidal vorticity and stream function distributions are obtained analytically. The self-induced lift velocity of the ring vortex is found. The influence of inertial terms is investigated numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 10–15, November–December, 1987.     

平行涡管三维合并的直接数值模拟

采用拟谱方法求解Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程，对两个同向平行涡管的三维合并过程进行直接数值模拟，分析了它的不同于二维合并的“缠绕式”特性，特别地，对在合并过程中产生的垂直于初始涡管方向的非主体涡量的意义和物理机制了探讨。     

Adaptive Euler simulations of airfoil–vortex interaction

A grid redistribution method is used together with an improved spatially third‐order accurate Euler solver to improve the accuracy of direct Euler simulations of airfoil–vortex interaction. The presented numerical results of two airfoil–vortex interaction cases indicate that with combination of the two methods, the numerical diffusion of vorticity inherent in the direct Euler simulations is drastically reduced without increasing the number of grid points. With some extra works due to grid redistribution, the predicted vortex structure is well preserved after a long convection and much sharper acoustic wave front resulting from airfoil–vortex interaction is captured. Copyright © 2006 John Wiley & Sons, Ltd.     

Vortex particle‐in‐cell method for three‐dimensional viscous unbounded flow computations

A new vortex particle‐in‐cell (PIC) method is developed for the computation of three‐dimensional unsteady, incompressible viscous flow in an unbounded domain. The method combines the advantages of the Lagrangian particle methods for convection and the use of an Eulerian grid to compute the diffusion and vortex stretching. The velocity boundary conditions used in the method are of Dirichlet‐type, and can be calculated using the vorticity field on the grid by the Biot–Savart equation. The present results for the propagation speed of the single vortex ring are in good agreement with the Saffman's model. The applications of the method to the head‐on and head‐off collisions of the two vortex rings show good agreement with the experimental and numerical literature. Copyright © 2000 John Wiley & Sons, Ltd.     

CALCULATION OF STEADY AND OSCILLATING FLOWS IN TUBES USING A VORTICITY TRANSPORT ALGORITHM

Steady and oscillating axisymmetric tube flows are modelled using a vorticity transport algorithm. The axisymmetric convective –diffusive Navier–Stokes equations are solved using a splitting technique. Axisymmetric ring vortex filaments are introduced on the walls and subsequently convected and diffused throughout the flow field. An axisymmetric equation similar to the Oseen diffusion equation is used to diffuse the ring vortex filaments. Vorticity is reflected from the tube walls using two techniques. Results are presented for the developing Poiseuille flow and for the developed flow in the form of the entrance length and the axial velocity and vorticity profiles. Good agreement is achieved with a finite difference method in the developing region of Poiseuille flow. The developed flow results are compared with the analytical solutions. The developed profiles of velocity and vorticity have errors of less than 0ċ3 per cent for both methods of dealing with reflection of diffusion at the bounding surfaces and similar accuracy is obtained for the velocity profiles in oscillating flow except at the wall. Oscillating flow is produced with a discretized sinusoidal piston motion. Velocity profiles, boundary layer thickness and entrance length are presented for oscillating flow. Good agreement is achieved for low-Womersley-number non-dimensional frequency. At higher values of this parameter, flows are inaccurately simulated, because the number of piston positions used to discretize the piston motion is inversely proportional to the non-dimensional frequency.     

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.     

Unsteady flow calculation in a radial flow centrifugal pump with spiral casing

The prediction of the two-dimensional unsteady flow established in a radial flow centrifugal pump is considered. Assuming the fluid incompressible and inviscid, the velocity field is represented by means of source and vorticity surface distributions as well as a set of point vortices. Using this representation, a grid-free (Lagrangian) numerical method is derived based on the coupling of the boundary element and vortex particle methods. In this context the source and vorticity surface distributions are determined through the non-entry boundary condition together with the unsteady Kutta condition. In order to satisfy Kelvin's theorem, vorticity is shed at the trailing edges of the impeller blades. Then the vortex particle method is used to approximate the convection of the free vorticity distribution. Results are given for a pump configuration experimentally tested by Centre Technique des Industries Mécaniques (CETIM). Comparisons between predictions and experimental data show the capability of the proposed method to reproduce the main features of the flow considered.     

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.     

Reconnections of vortex filaments

The process of break-down and reconnection of vortex filaments is considered by the method of three-dimensional vortex singularities (vortons) in various situations, including oblique interaction of a vortex ring with a boundary in shear flow, shedding of a vortex ring from a horseshoe vortex, instability of elliptic vortex ring, Crow instability of two perturbed antiparallel vortex filaments, merging and subsequent splitting of vortex rings. Special attention is paid to the global integrals (vorticity, momentum, angular momentum) and to the inviscid dissipation of energy. The visualization of the effective vortex core, created by the interference of the vorticity fields of vortons, is presented. The comparisons with other methods of simulation of three-dimensional vortex interactions and with the observations have been made.     

Bifurcation of an elliptic vortex ring

Time-dependent vorticity fields of elliptic vortex rings of aspect ratios 2, 3 and 4 were measured by means of hot-wire anemometry. The time evolution of their vorticity fields was analyzed and the processes of vortex ring formation, advection, interaction and decay, and the mechanism of vortex bifurcation are studied. The following crosslinking model is proposed: A thick vortical region composed of many equivalent vortex filaments with distributed cores is initially formed at the orifice and they behave as inviscid filaments. The elliptic ring deforms and the end parts of its major axis get closer. Then, the vortex filaments interact at the touching point and the ring partially bifurcates. Almost simultaneously, turbulent spot appears at this point, and propagates around the ring cross section, thus preventing further bifurcation. And it becomes a turbulent blob. This model is also supported by numerical simulation by a high-order vortex method and the Navier-Stokes solution.     

A combined vortex and panel method for numerical simulations of viscous flows: a comparative study of a vortex particle method and a finite volume method

This paper describes and compares two vorticity‐based integral approaches for the solution of the incompressible Navier–Stokes equations. Either a Lagrangian vortex particle method or an Eulerian finite volume scheme is implemented to solve the vorticity transport equation with a vorticity boundary condition. The Biot–Savart integral is used to compute the velocity field from a vorticity distribution over a fluid domain. The vorticity boundary condition is improved by the use of an iteration scheme connected with the well‐established panel method. In the early stages of development of flows around an impulsively started circular cylinder, and past an impulsively started foil with varying angles of attack, the computational results obtained by the Lagrangian vortex method are compared with those obtained by the Eulerian finite volume method. The comparison is performed separately for the pressure fields as well. The results obtained by the two methods are in good agreement, and give a better understanding of the vorticity‐based methods. Copyright © 2005 John Wiley & Sons, Ltd.     

Steady Vortex Ring with Isochronous Flow in the Vortex Core

Steady-state solutions to the problem of a thin vortex ring in an inviscid incompressible fluid in infinite space are investigated. The Fraenkel procedure is used to construct the steady-state solutions. In this procedure a given vorticity distribution in plane flow with circular streamlines is transformed into a steady vortex ring using an expansion in the ring thinness parameter. For example, a two-dimensional vortex of constant vorticity is transformed into a steady vortex ring with the uniform distribution in which the absolute value of vorticity is proportional to the distance from the axis of symmetry. The principal aim of our study is to construct the algorithm of finding the flow for an isochronous vortex ring in which the periods of revolution are the same for all the liquid particles in the vortex core. The problem is that the two-dimensional distribution which goes over in the isochronous ring in accordance with the Fraenkel procedure is unknown in advance. In particular, the ring with the uniform distribution is not isochronous despite the isochronism of the initial two-dimensional flow. In this connection the Fraenkel procedure is significantly modified so that the initial two-dimensional vorticity distribution is determined in each of the steps of the iteration procedure. The solution for the vortex ring with the uniform distribution obtained in the present study is significantly used to construct the isochronous solution. The necessary corrections to the former solution are calculated in each step. Obtaining of the isochronous flow is the key step for the investigation of stability of three-dimensional oscillations of the vortex ring since the oscillation spectrum of this flow is discrete.