Lagrangian Vortex Method Research Articles

A Triangulated Vortex Method for the Axisymmetric Euler Equations

A new method is presented for the prediction of unsteady axisymmetric inviscid flows. By combining a triangulated vortex approach with a novel evaluation technique for the Biot–Savart integrals, a Lagrangian vortex method is developed which eliminates the singularities usually present in axisymmetric methods, without recourse to normalizations or other approximations. Furthermore, the computational effort scales as the number of control points N and, in the large N limit, depends only on the order of quadrature employed. The accuracy and computational effort are assessed by comparison with the velocity field of a Gaussian core vortex ring and the use of the technique is illustrated by computation of the motion of Norbury rings and of vortex ring pairing.

Vibration Analysis of Thick Laminated Composite Cylindrical Shells

Fig. 2 Variation of CD and CL with time for case 9. Ogami and Ayano 2 computed CD = 1.07, using a Lagrangian viscous vortex method (they did not calculate the Strouhal number ). The general agreement between the two numerical methods is acceptable for the mean drag coefe cient, and both results are close to the experimental values. However, the three-dimensional and turbulence effects present in the experiments are nonnegligible for the Reynolds number used in the simulations, and a two-dimensional computation must produce higher values for the drag coefe cient, as obtained in our simulation. Conclusions The vortex method presented here to study the two-dimensional, incompressible, unsteady e ow around a e xed circular cylinder is abletopredictthemainglobalquantitiesofahigh-Reynolds-number e ow. The calculatedaerodynamicforces areclosetothe experimental and numerical data used for comparison. Also, the simulations are able to capturecomplexe owmechanisms,such as separation,as part of the computation. The analysis of the ine uence of the numerical parameters on the simulation has pointed out the importance of choosing suitable values for N, D t, and e , and the trends when these parametersarevariedhavealsobeenidentie ed.Thenumerical results also brought up that e strongly affects the simulations and, therefore, must be modeled correctly. This is an important output of this work. Acknowledgments

Vortex Simulation of Rotor/Stator Interaction in Turbomachinery

A new numerical method is presented in this paper to simulate rotor/stator interaction in turbomachinery by means of a vortex method based on a Lagrangian frame. The algorithm takes the result from steady solution as input, which gives an initial description of the unsteady disturbance flow field. To calculate the unsteady response to these disturbances, the Lagrangian vortex method is used to capture the convective process, and the deterministic vortex scheme to approximate the viscous diffusion process. The application of Baldwin–Lomax turbulence model in wakes is developed, to overcome difficulties such as the much higher calculated viscosity in the outer region than that in the boundary regions, and the difficulty in continuously tracing moving wake centerlines encountered when other numerical methods are used. Generally the agreement between the computational and experimental results is good. The sweeping characteristic of wakes, the influence of unsteadiness on incidence, and the decaying features of unsteady velocities and pressure are studied in the paper.

A coupled lagrangian-eulerian scheme for reacting flow modeling

We describe a coupled Lagrangian-Eulerian numerical scheme for modeling reacting flow in two dimensions. The scheme uses the Lagrangian vortex method for solving the Navier-Stokes equations, and an Eulerian finite difference implementation on a multi-layered adaptive mesh for the scalar conservation equations. Coupling terms involving baroclinic and diffusional circulation source terms as well as expansion sources for dilatation resulting from heat release are implemented as necessary. A boundary element method is used for the potential flow solution. A fast multipole method is used for all velocity field evaluations. The numerical construction is presented along with representative buoyant reacting jet flow results.

Simulation of the Piston Driven Flow Inside a Cylinder With an Eccentric Port

A grid-free Lagrangian approach is applied to simulate the high Reynolds number unsteady flow inside a three-dimensional domain with moving boundaries. For this purpose, the Navier-Stokes equations are expressed in terms of the vorticity transport formulation. The convection and stretch of vorticity are obtained using the Lagrangian vortex method, while diffusion is approximated by the random walk method. The boundary-element method is used to solve a potential flow problem formulated to impose the normal flux condition on the boundary of the domain. The no-slip condition is satisfied by a vortex tile generation mechanism at the solid boundary, which takes into account the time-varying boundary surfaces due to, e.g., a moving piston. The approach is entirely grid-free within the fluid domain, requiring only meshing of the surface boundary, and virtually free of numerical diffusion. The method is applied to study the evolution of the complex vortical structure forming inside the time-varying semi-confined geometry of a cylinder equipped with an eccentric inlet port and a harmonically driven piston. Results show that vortical structures resembling those observed experimentally in similar configurations dominate this unsteady flow. The roll-up of the incoming jet is responsible for the formation of eddies whose axes are nearly parallel to the cylinder axis. These eddies retain their coherence for most of the stroke length. Instabilities resembling conventional vortex ring azimuthal modes are found to be responsible for the breakup of these toroidal eddies near the end of the piston motion. The nondiffusive nature of the numerical approach allows the prediction of these essentially inviscid phenomena without resorting to a turbulence model or the need for extremely fine, adaptive volumetric meshes.

Inviscid Axisymmetrization of an Elliptical Vortex

The inviscid evolution of elliptical, nonuniform vorticity distributions is studied computationally using a high resolution Lagrangian (vortex) method with minimal numerical dissipation. The simulations reveal that the vortices evolve, through a process of filamentation, to a configuration consisting of a vortex surrounded by weak filamentary structures. The shape of the final configuration depends on the profile of the initial vorticity distribution. For the same ellipticity, relatively smooth profiles evolve to axisymmetric vortical structures, whereas sharper initial vorticity distributions result in robust non-axisymmetric configurations. A systematic convergence study is conducted to establish the accuracy of the method for long time inviscid simulations. To further assess the issue of axisymmetrization we compare our results with related numerical and experimental studies.

Three-Dimensional Vortex Simulation of Time Dependent Incompressible Internal Viscous Flows

A hybrid random vortex-boundary element method is developed for the solution of time-dependent incompressible three-dimensional internal flow problems. The numerical scheme is grid-free within the flow domain and is based on a combination of the Lagrangian vortex method to capture the convection and stretch of the vortical field, the random walk method to describe the diffusion process, and the boundary element method to superimpose a potential flow on the vortical field such that the normal flux boundary condition is satisfied. The no-slip boundary condition is satisfied by generating vorticity tiles on solid boundaries, which are subsequently diffused and convected into the flow interior. Additionally, a boundary condition is devised for the application of fully developed flow properties at the exit plane. In this paper, the formulation and the numerical scheme are presented, followed by a parametric study of the accuracy of the method using the model problem of the flow in a duct with square cross section at Re= 100. We show that the method converges to the analytical solution of the problem as the resolution of the time integration and the discretization are improved, and we discuss the impact of each resolution parameter on the accuracy. In addition, selected results from the simulation of an impulsively started flow over a cube at Re= 100 are presented. We use the results of this test case to demonstrate that the method captures the effect of sharp edges, parallel and normal to the streamwise flow direction, on the flow dynamics.

A coupling finite difference/particle method for the resolution of 2D Navier-Stokes equations in velocity-vorticity form

A domain decomposition method which combines a vortex method and a finite difference method is presented to solve the Navier-Stokes equations in velocity-vorticity form. The lagrangian vortex method is used in the flow region where convective effects are dominant whereas the finite difference method using velocity-vorticity in this paper is used in the flow region where the convective effects may be dominant. Unsteady flow around an isolated NACA 0012 is studied. The comparison between numerical and experimental data show that the method is consistent for simulating two dimensional flows. Flow interaction between an airfoil and a circular cylinder has been analyzed showing the abilities of the method in multibodies flow simulation. The velocity-vorticity formulation can be extended easily in three dimensional flow simulation.

Numerical vorticity creation based on impulse conservation.

The problem of creating solenoidal vortex elements to satisfy no-slip boundary conditions in Lagrangian numerical vortex methods is solved through the use of impulse elements at walls and their subsequent conversion to vortex loops. The algorithm is not uniquely defined, due to the gauge freedom in the definition of impulse; the numerically optimal choice of gauge remains to be determined. Two different choices are discussed, and an application to flow past a sphere is sketched.

Simulation of three-dimensional internal flows by the random vortex and boundary element methods

A vortex boundary element method is developed for the grid-free simulation of time-dependent, incompressible, viscous flow in three dimensional configurations. The numerical scheme is based on a combination of the Lagrangian vortex method to capture the convection and stretch of the vortical field, the random walk method to describe the diffusion process, and the boundary element method to impose the normal flux boundary condition on the boundary surfaces. The no-slip boundary condition is satisfied by an extended vortex tile generation mechanism. A new boundary condition is devised to impose the fully developed flow properties at the exit plane. The formulation of the numerical scheme is presented, followed by a parametric study of the accuracy of the method using the model problem of flow in a duct with square cross-section at Re=100. Additionally, results from an example of piston driven flow in a cylinder with square cross-section and an offset port at Re=350 (based on the piston side and maximum speed) are presented.

Nonlinear unstable viscous fingers in Hele–Shaw flows. II. Numerical simulation

The nonlinear stages of two-dimensional immiscible displacement processes in Hele–Shaw flows are investigated by means of large scale numerical simulations based on a purely Lagrangian vortex method. The vortex sheet at the interface between the two fluid phases is discretized into circular arcs with a continuous distribution of circulation, which renders our numerical technique highly accurate. A complicated unsteady growth mechanism is observed for the emerging viscous fingers, involving a combination of spreading, shielding, and tip splitting. As the surface tension is further reduced, smaller length scales arise and the fingertip exhibits a new splitting pattern in which three new lobes emerge instead of two. Monitoring the velocity as well as the radius of curvature at the fingertip demonstrates that the instability of the finger evolves in an oscillatory fashion. The two-lobe and the three-lobe splitting can thus be explained as different manifestations of the same instability mode. Comparison with experiment shows good qualitative but only fair quantitative agreement. By imposing a constraint on the curvature at the fingertip, experimental results, which show fingers of width considerably smaller than half the cell width and exhibit ‘‘dendritic’’ instability modes, are reproduced.

Generalized discrete vortex method for cylinders without sharp edges

VISCOUS flow around a circular cylinder may be simulated by the fractional step method of Chorin, where the inviscid part of the calculation is solved by convecting discrete vortices in a Lagrangian vortex-in-cell scheme with a radially expanding polar mesh and by superimposing random walks to simulate diffusion. This approach is extended to allow nearly arbitrary cylinder shapes, without sharp edges, by a conformal transformation of the shape to a circle. The attractions of the Lagrangian vortex method concerning numerical diffusion, stability, and efficiency are thus transferred to more general shapes, while maintaining high accuracy where it is required. Forces, pressures, and vortex shedding frequencies for eight different shapes were found to be in reasonable agreement with experiments at subcritical Reynolds numbers. Contents Flows around several cylinders without sharp edges have been computed by the Lagrangian discrete-vortex scheme first proposed by Chorin for the solution of the vorticity equation.1 At each time step, a vortex sheet is created along the cylinder surface in order to satisfy the boundary condition of zero tangential velocity. The sheet is then discretized into new vortices, which are added to those representing the vorticity field of the flow. The process of vorticity diffusion is simulated by adding a random perturbation to the position of each vortex. Finally the vortices are convected in the velocity field. Good flow simulations require a very large number of vortices, and their convection may be handled efficiently by the vortex-in-cell method.2 The computation is performed in a transformed complex plane f. The Theodorsen and Garrick transformation [z =/(£)] is used to map the region outside the cylinder in the physical complex plane z onto the region outside the unit circle If I =1 in the transformed plane. The function / contains a set of mapping coefficients, which are determined for each cylinder by the fast-Fourier-transform method of Ives.3'4 The vorticity field is modeled, in the physical plane, by a distribution of vortices outside the cylinder and, in the transformed plane, by a distribution of vortices of equal circulations at corresponding points outside the unit circle. 5fD, the random perturbation simulating diffusion, is given by