Lagrangian Vortex Method Research Articles

Account of ambient turbulence for turbine wakes using a Synthetic-Eddy-Method

The present paper aims at describing the use of a Synthetic-Eddy-Method (SEM), initially proposed by Jarrin et al. [12], in the 3D Lagrangian Vortex method framework. The SEM method is used here in order to generate a far-field incoming flow with a prescribed ambient turbulence intensity. However, for the account of the diffusive term in the Navier-Stokes equations, a classical Particle Strength Exchange model with a LES eddy viscosity is used.Firstly, the general characteristics of the Synthetic-Eddy-Method will be presented together with its integration in the framework of the developed 3D unsteady Lagrangian Vortex software [27]. The capability of the ambient turbulence model to reproduce a perturbed flow that verifies any turbulence intensity I∞ and any anisotropic ratio (σu:σv:σw) will be discussed and validated. Then, the capability of the presented ambient turbulence model to compute turbine wakes will also be presented together with first results. Finally, comparisons will be made between the obtained numerical results against experimental data [22, 23] for two levels of ambient turbulence, namely I∞ = 3% and I∞ = 15%. Although the present study was initially performed in the framework of tidal energy, its application to wind energy is straightforward.

STOCHASTIC VORTEX BLOBS METHOD TO STUDY AXISYMMETRIC FLOWS OF INCOMPRESSIBLE VISCOUS FLUID

This paper presents the mathematical and computational formulations of the stochastic Vortex Blobs Method (VBM). It isshow that how the method can be used to cover axisymmetric flows of incompressible viscous fluid. Also, the initialboundary problem is solved by using the Lagrangian vortex method. This method seems to be an extension of the well known two dimensional vortex blob method. When applying and extending this method two steps are required. First, wehave to design an axisymmetric vorticity carrier by using the standard functions as complete elliptic integrals and Legendre polynomials. Second, it is necessary to formulate the appropriate Neumann problem and boundary integral equation to find the potential velocity fields. Both steps are used to describe and compute the total velocity field and formulate the Ito stochastic equations which describing the motion of vorticity carriers.

Vortex kinematics around a submerged plate under water waves. Part II: Numerical computations

This paper presents numerical computations of the flow generated by a horizontal plate immersed in a regular wave field, and associated loads acting on the plate. This numerical work is the continuation of the Poupardin et al. (2012) experimental study. This numerical study is original in the way that the vortical aspects of the flow are not neglected. Therefore, a 2D Lagrangian Vortex method is used as a numerical scheme. These methods are particularly well suited for the computation of unsteady and highly rotational flows in an open domain. The velocity field is decomposed into rotational and potential components, using the Helmholtz decomposition. The rotational part of the velocity is calculated by the Biot–Savart equation using vortex particles. The plate is modelled by a distribution of normal dipoles and the wave field is taken into account by means of a Stokes formulation, which completes the potential part of the velocity.Firstly, the numerical code is validated by means of comparison with the experimental results of Poupardin et al. (2012). In particular, the complex vortical activity and the mean flow velocity field are well reproduced and physically analysed. Secondly, forces acting on the plate are analysed on a wide range of parameters by varying the plate immersions and lengths. In the end, a new scaling is found for the lift forces acting on the plate based on the modified Stokes velocity (i.e. the bottom Stokes velocity for a water depth equals to the plate immersion) and the square of the plate length.

Iterative solver approach for turbine interactions: application to wind or marine current turbine farms

This paper presents a numerical investigation for the computation of wind or marine current turbines in a farm. A 3D unsteady Lagrangian vortex method is used together with a panel method in order to take into account for the turbines. In order to enforce the boundary condition onto the panel elements, a linear matrix system is defined. Solving general linear matrix systems is a topic with important scientific literature. But the main concern here is the application to a dedicated matrix which is non-sparse, non-symmetric, neither diagonally dominant nor positive-definite. Several iterative approaches were tested and compared. But after some numerical tests, a Bi-CGSTAB method was finally chosen. The main advantage of the presented method is the use of a specific preconditioner well suited for the desired application. The chosen implementation proved to be very efficient with only 3 iterations of our preconditioned Bi-CGSTAB algorithm whatever the turbine geometrical configuration. Although developed for wind or marine turbines, the proposed algorithm is absolutely not restricted to these cases, and can be applied to many others. At the end of the paper, some applications (specifically, wake computations) in a farm are presented, along with a quantitative assessment of the computational time savings brought by the iterative approach.

Modified Lagrangian vortex method with improved boundary conditions for water waves past a thin bottom‐standing barrier

ABSTRACTHerein, the modified Lagrangian vortex method (LVM), a hybrid analytical‐numerical algorithm per se, is devised to simulate the process of vortex formation and shedding from the sharp edge of a zero‐thickness vertical plate under linear water‐wave attack. Application of the Helmholtz decomposition facilitates a convenient switch between the inviscid‐ and viscous‐flow models, thereby enabling easy incorporation of vorticity effects into the potential‐flow calculations for the viscous‐dominated region. In evaluating the potential‐flow component, making good use of the quickly convergent technique with singular basis functions, correctly capturing the singular behavior in velocity fields near the tip of the plate, leads to a considerable reduction of computational burdens and to 12‐decimal‐place accuracy. The viscous correction is carried out via the meshless LVM with improved boundary conditions. Comparisons with previously published results show good agreement. Simulations of vortex generation and evolution illuminate the ability of the present method, and provide a supplement to pertinent experimental works. The hybrid scheme proposed herein allows flexibility for the former LVM and convenience in the code development. Such a compromise fits particularly well for the high‐resolution modeling of sharp‐edged vortex shedding without heavy numerical developments. Copyright © 2014 John Wiley & Sons, Ltd.

Vortical Inviscid Flows with Two-Way Solid-Fluid Coupling

Vortex methods increasingly receive attention from the computer graphics community for simple and direct modeling of complex flow phenomena such as turbulence. The coupling between free-form solids, represented by arbitrary surface meshes, and fluids simulated with vortex methods, leads to visually rich simulations. In this paper, we introduce a novel approach for simulating the interaction between solids and inviscid fluids for high-quality simulations using Lagrangian vortex particles. The key aspect of our method is simulating the creation of vorticity at a solid's surface. While previous vortex simulators only focus on modeling the solid as a boundary for the fluid, our approach allows the accurate simulation of two processes of visual interest. The first is the introduction of surface vorticity in the main flow as turbulence (vortex shedding). The second is the motion of the solid induced by fluid forces. We also introduce to computer graphics the concept of source panels to model nonturbulent flow around objects. To the best of our knowledge, this is the first work on two-way coupling of 3D solids and fluids using Lagrangian vortex methods in computer graphics.

Suppression of vortex shedding on a bluff body

Previous experimental observations suggest that a moving ground can be numerically simulated avoiding vorticity generation on it. In this research line the current paper presents a two-dimensional Lagrangian Vortex Method, which is able to analyze the vortex shedding suppression on a circular cylinder near a moving ground in a high Reynolds number flow of Re=1.0×105. The main purpose is to utilize numerical results to construct the instantaneous velocity and pressure fields for this kind of unsteady viscous flow. The analyses demonstrate that the physical mechanisms involved are associated with an additional circulation caused by the fluid viscosity and with the Venturi effect. Both of these effects are responsible for the increase in lift force as well as for the decrease in drag force acting on body surface.

FMM-based vortex method for simulation of isotropic turbulence on GPUs, compared with a spectral method

The Lagrangian vortex method offers an alternative numerical approach for direct numerical simulation of turbulence. The fact that it uses the fast multipole method (FMM)—a hierarchical algorithm for N-body problems with highly scalable parallel implementations—as numerical engine makes it a potentially good candidate for exascale systems. However, there have been few validation studies of Lagrangian vortex simulations and the insufficient comparisons against standard DNS codes has left ample room for skepticism. This paper presents a comparison between a Lagrangian vortex method and a pseudo-spectral method for the simulation of decaying homogeneous isotropic turbulence. This flow field is chosen despite the fact that it is not the most favorable flow problem for particle methods (which shine in wake flows or where vorticity is compact), due to the fact that it is ideal for the quantitative validation of DNS codes. We use a 2563 grid with Reλ=50 and 100 and look at the turbulence statistics, including high-order moments. The focus is on the effect of the various parameters in the vortex method, e.g., order of FMM series expansion, frequency of reinitialization, overlap ratio and time step. The vortex method uses an FMM code (exaFMM) that runs on GPU hardware using CUDA, while the spectral code (hit3d) runs on CPU only. Results indicate that, for this application (and with the current code implementations), the spectral method is an order of magnitude faster than the vortex method when using a single GPU for the FMM and six CPU cores for the FFT.

Three-dimensional vortex simulation of unsteady flow in hydraulic turbines

SUMMARY On the basis of the Helmholtz decomposition, a grid-free numerical scheme is provided for the solution of unsteady flow in hydraulic turbines. The Lagrangian vortex method is utilized to evaluate the convection and stretch of the vorticity, and the BEM is used to solve the Neumann problem to define the potential flow. The no-slip boundary condition is satisfied by generating vortex sticks at the solid surface. A semi-analytical regularization technique is applied to evaluate the singular boundary surface integrals of the potential velocity and its gradients accurately. The fast multipole method was extended to evaluate the velocity and velocity gradients induced by the discretized vortex blobs in the Lagrangian vortex method. The successful simulation for the unsteady flow through a hydraulic turbine's runner has manifested the effectiveness of the proposed method. Copyright © 2011 John Wiley & Sons, Ltd.

Subgrid-Scale Modeling of Turbulent Flow Around Circular Cylinder by Mesh-Free Vortex Method

:This paper proposes a numerical model for the two-dimensional, incompressible, unsteady, turbulent flow applied to an impulsively started flow around a circular cylinder in the Reynolds number range from 1×104 to 6×105. A Lagrangian mesh-free vortex method blended with the Large Eddy Simulation theory is employed to simulate the large-scale motion, whereas the turbulent subgrid-scale motion is modeled with an eddy viscosity coefficient, expressed in terms of the Second-Order Velocity Structure Function. The filtered vorticity field is calculated by a superposition of Lamb vortices that are generated near the body surface such that circulation is conserved and the no-slip boundary condition is explicitly imposed at a finite number of points on the cylinder. The no-penetration boundary condition is satisfied exactly on the entire cylinder surface through the application of the circle theorem. The vorticity transport equation is solved using the convective-diffusive operator-splitting algorithm, where vorticity diffusion is simulated with the random walk method and the convective motion of the vortices is integrated in time using the second-order Adams-Bashforth scheme. Numerous simulations for high Reynolds numbers are carried out to determine the numerical parameters of the model. Results for the drag coefficient and the Strouhal number as a function of the Reynolds number present satisfactory agreement with other results from the literature.

Study of the vortex shedding flow around a body near a moving ground

The two-dimensional viscous incompressible fluid flow around a circular cylinder near a moving ground is numerically simulated. In a moving ground one eliminates the influence of the ground boundary layer, which is a crucial factor in the numerical simulation of the flow around a body moving in a close vicinity to a flat ground. A Lagrangian mesh-free vortex method is used to calculate global and local quantities of high Reynolds number flow of 1.0×105. This method is modified to take into account the sub-grid scale phenomena through a second-order velocity structure function model adapted to the Lagrangian scheme. In the present algorithm vortices with a Lamb core are generated only on the circular cylinder surface to ensure that the no-slip condition is satisfied and that the circulation is conserved. On the ground it is only sufficient to ensure the impermeability condition. Based on the experimental results available in the literature, the critical drag behaviour was found to be directly related to a global change in the near wake structure of the cylinder.

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

AbstractThe 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.

A Lagrangian vortex method for unbounded flows

AbstractA technique is presented for velocity calculations on the highly distorted node distributions typical of those found in Lagrangian vortex methods. The method solves the partial differential equation for streamfunction directly on the nodes, via a sparse, symmetric system of equations that can be solved using standard iterative solvers. When implemented in a triangulated vortex method, the technique gives computation times which scale as N1.23, where N is the number of nodes. The computation scheme is derived for two‐dimensional problems and applied to the prediction of the evolution of perturbed multipolar vortices. Due to the numerical performance of the method, it has been possible to examine such evolution at higher and lower Reynolds numbers than have been considered in published numerical studies. Copyright © 2008 John Wiley & Sons, Ltd.

Modeling the flow past an oscillating airfoil by the method of viscous vortex domains

The Lagrangian vortex method for solving the Navier-Stokes equations is applied for numerically modeling the unsteady flow past a wing airfoil executing angular oscillations in a viscous incompressible flow. Formulas relating the unsteady forces on the airfoil and the vorticity field are derived. The calculated results are compared with the experimental data for the NACA-0012 airfoil executing harmonic oscillations in an air flow at the Reynolds number Re = 4.4 × 104.

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.

Modified interpolation kernels for treating diffusion and remeshing in vortex methods

A scheme treating diffusion and remeshing, simultaneously, in Lagrangian vortex methods is proposed. The vorticity redistribution method is adopted to derive appropriate interpolation kernels similar to those used for remeshing in inviscid methods. These new interpolation kernels incorporate diffusion as well as remeshing. During implementation, viscous splitting is employed. The flow field is updated in two fractional steps, where the vortex elements are first convected according to the local velocity, and then their vorticity is diffused and redistributed over a predefined mesh using the extended interpolation kernels. The error characteristics and stability properties of the interpolation kernels are investigated using Fourier analysis. Numerical examples are provided to demonstrate that the scheme can be successfully applied in complex problems, including cases of nonlinear diffusion.

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

AbstractThis 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.

Numerical simulation of unsteady interaction of centrifugal impeller with its diffuser using Lagrangian discrete vortex method

In this study, an advanced Lagrangian vortex- boundary element method is applied to simulate the unsteady impeller-diffuser interactions in a diffuser pump not only for design but also for off-design considerations. In velocity calculations based on the Biot-Savart law we do not have to grid large portions of the flow field and the calculation points are concentrated in the regions where vorticity is present. Lagrangian representation of the evolving vorticity field is well suited to moving boundaries. An integral pressure equation shows that the pressure distribution can be estimated directly from the instantaneous velocity and vorticity field. The numerical results are compared with the experimental data and the comparisons show that the method used in this study can provide us insight into the complicated unsteady impeller-diffuser interaction phenomena in a diffuser pump.

Lagrangian 보오텍스 방법에서의 압력장 계산

In the Lagrangian vortex particle method based on the vorticity-velocity formulation for solving the incompressible Navier-Stokes equations, a numerical scheme for calculating pressure fields is presented. Implementation of the numerical method is directly connected with the well-established surface panel methods, just by dealing with the dynamic coupling among vorticity field. Assuming the vorticity and the velocity fields are to be calculated in time domain analysis, the pressure calculation for a complete set of solution at present time step is performed in a similar way to the one used in the Eulerian description. For a validation of the present method, we illustrate the early development of the viscous flow about an impulsive started circular cylinder for Reynolds number 550. The comparative study with the Eulerian finite Volume method provides an extensive understanding and application of the mesh-free Lagrangian vortex methods for numerical simulation of viscous flows around arbitrary bodies of general shape.

Stability of a vortex sheet roll-up

The stability of vortex sheet roll-up is studied using a Lagrangian vortex method. We consider an initially unstable (Kelvin–Helmholtz) vortex sheet. During its nonlinear evolution, a perturbation is added to test it for a secondary instability. The growth of the perturbation depends on its phase and on the local strain rate. In the linear stage of this secondary instability, the dispersion relation is calculated. It is found that the growth rate and the cutoff wave number are fixed by the regularization parameter of the Birkhoff–Rott equation.