Lagrangian Vortex Method Research Articles

Effect of Surface Roughness on Aerodynamic Loads of Bluff Body in Vicinity of Smoothed Moving Wall

This paper contributes to a new Lagrangian vortex method for the statistical control of turbulence in two-dimensional flow configurations around a rough circular cylinder in ground effect when considering higher subcritical Reynolds numbers, namely 3 × 104 ≤ Re ≤ 2 × 105. A smoothed moving wall (active control technique) is used to include the blockage effect in association with the variation in cylinder surface roughness (passive control technique), characterizing a hybrid approach. In contrast with the previous approaches of our research group, the rough cylinder surface is here geometrically constructed, and a new momentum source term is introduced and calculated for the investigated problem. The methodology is structured by coupling the random Discrete Vortex Method, the Lagrangian Dynamic Roughness Model, and the Large Eddy Simulation with turbulence closure using the truncated Second-Order Velocity Structure Function model. This methodological option has the advantage of dispensing with the use of both a refined near-wall mesh and wall functions. The disadvantage of costly processing is readily solved with Open Multi-Processing. The results reveal that intermediate and high roughness values are most efficient for Reynolds numbers on the orders of 105 and 104, respectively. In employing a moving wall, the transition from the large-gap to the intermediate-gap regime is satisfactorily characterized. For the conditions studied with the hybrid technique, it was concluded that the effect of roughness is preponderant and acts to anticipate the characteristics of a lower gap-to-diameter ratio regime, especially with regard to intermittency.

Evolutionary Airfoil Shape Optimization Coupling Parameterization Methods with Lagrangian Vortex Method

In the present paper, the evolutionary algorithm for single-objective optimization is developed using a genetic algorithm and employing polynomial-based (PARSEC) and radial basis function (RBF) functions for NACA 2412 airfoil parameterization. The determination of the objective functions, being aerodynamic coefficients, are performed using the Lagrangian vortex particle method. The results of the lift coefficient at the wide range of angle of attack using the vortex particle method shows a good agreement with experimental data listed in the literature. For the optimization results, the lift coefficient obtained from the PARSEC method is optimized to be larger for the whole range of angle of attack; while it still keeps the stall region at the upper surface of the airfoil to be the same as that of the original airfoil. In addition, the RBF method illustrates the lift coefficients larger at the range of angle of attack from -50 to 140 but stall occurs earlier than the original airfoil.

Singularity resolving in solution of the boundary integral equation in two-dimensional vortex methods

The problem of 2D incompressible flow simulation around airfoils with sharp edges and corner point is considered. The solution of the boundary integral equation with respect to vortex sheet intensity arising in Lagrangian vortex method has weak singularity that cannot be resolved correctly in the framework of the existing Galerkin-type numerical schemes. It is shown that for piecewise-smooth bounded solutions the known schemes allow for solution reconstruction with high quality and provide the 2-nd order of accuracy, while for singular solution their order of accuracy goes down to the 1-st. A numerical scheme is suggested that allows for solution singularity resolving and provides the 2-nd order of accuracy. As a model problem, the added mass tensor components computation is considered, since its exact value is known for the Joukowsky wing airfoil with sharp edge (cusp point).

Neural vortex method: From finite Lagrangian particles to infinite dimensional Eulerian dynamics

In fluid analysis, there has been a long-standing problem: lacking a rigorous mathematical tool to map from a continuous flow field to finite discrete particles, hurdling the Lagrangian particles from inheriting the high resolution of a large-scale Eulerian solver. To tackle this challenge, we propose a novel learning-based framework, the neural vortex method (NVM). NVM builds a neural-network description of the Lagrangian vortex structures and their interaction dynamics to reconstruct the high-resolution Eulerian flow field in a physically-precise manner. The key components of our infrastructure consist of two networks: a vortex detection network to identify the Lagrangian vortices from a grid-based velocity field and a vortex dynamics network to learn the underlying governing interactions of these finite structures. By embedding these two networks with a vorticity-to-velocity Poisson solver and training its parameters using the fluid data obtained from grid-based numerical simulation, we can predict the accurate fluid dynamics on a precision level that was infeasible for all the previous conventional vortex methods. We demonstrate the efficacy of our method in generating highly accurate prediction results with low computational cost by predicting the evolution of the leapfrogging vortex rings system, the turbulence system, and the systems governed by Navier–Stokes (NS) equations with different external forces. We compare the prediction results made by NVM and the Lagrangian vortex method (LVM) for solving the NS equation in the periodic box and find that the relative error of the predicted velocity using NVM is more than 10 times lower than that of the LVM. Moreover, our method only requires data collected from a very short training window, more than 100 times smaller than the prediction period, which potentially facilitates data acquisition in real systems.

A Lagrangian vortex method for smoke simulation with two-way fluid–solid coupling

We propose a pure Lagrangian vortex particle dynamics framework to simulate the various phenomena of turbulent smoke and two-way coupling between fluid and solid. In the framework, we model the fluid using vorticity and velocity fields that are discretized on the vortex particles. We extend a segment-based vortex stretching strategy with the subdivision of long segments to capture the fine details of smoke while maintaining good numerical stability. In addition, a vorticity field correction mechanism based on the merging and splitting of particles is proposed to avoid excessive particle counts and reduce the associated unnecessary computational overhead. We also present a fast and approximate two-way coupling method for simulating the interaction between solid and fluid. Our method can be used to solve various fluid phenomena, including Kármán vortex streets, jets, and some examples of two-way fluid–solid coupling.

Research on smoke simulation with vortex shedding.

The Lagrangian vortex method has the advantage of producing highly detailed simulations of fluids such as turbulent smoke. However, this method has two problems: the construction of the velocity field from the vorticity field is inefficient, and handling the boundary condition is difficult. We present a pure Lagrangian vortex method, including a nested grid to accelerate the construction of the velocity field, and a novel boundary treatment method for the vorticity field. Based on a tree structure, the nested grid algorithm considerably improves the efficiency of the velocity computation while producing visual results that are comparable with the original flow. Based on the vortex-generating method, the least square method is used to compute the vorticity strength of the new vortex elements. Further, we consider the mutual influence between the generated vortex particles. We demonstrate our method’s benefits by using a vortex ring and various examples of interaction between the smoke and obstacles.

Vortex particle method with iterative Brinkman penalization for simulation of flow past sharp-shape bodies

This paper presents a Lagrangian vortex method combined with iterative Brinkman penalization for the simulation of incompressible flow past a complex geometry. In the proposed algorithm, particle and penalization domains are separately introduced. The particle domain is for the computation of particle convection and diffusion, while the penalization domain is the enforcement of the wall boundary conditions. In iterative Brinkman penalization, the no-slip boundary condition is enforced by applying penalization force in multiple times within each time step. This enables large time step size reducing computational cost and maintains the capability in handling complex geometries. The method is validated for benchmark problems such as an impulsively started flow past a circular cylinder, normal to a flat plate, and a symmetric airfoil at Reynolds numbers ranging from 550 to 1000. The vorticity and streamline contours, drag, and lift coefficients show a good agreement with those reported in literature.

Surface Roughness Effects on Flows Past Two Circular Cylinders in Tandem Arrangement at Co-Shedding Regime

This paper contributes by investigating surface roughness effects on temporal history of aerodynamic loads and vortex shedding frequency of two circular cylinders in tandem arrangement. The pair of cylinders is immovable; of equal outer diameter, D; and its geometry is defined by the dimensionless center-to-center pitch ratio, L/D. Thus, a distance of L/D = 4.5 is chosen to characterize the co-shedding regime, where the two shear layers of opposite signals, originated from each cylinder surface, interact generating counter-rotating vortical structures. A subcritical Reynolds number of Re = 6.5 × 104 is chosen for the test cases, which allows some comparisons with experimental results without roughness effects available in the literature. Two relative roughness heights are adopted, nominally ε/D = 0.001 and 0.007, aiming to capture the sensitivity of the applied numerical approach. Recent numerical results published in the literature have reported that the present two-dimensional model of surface roughness effects is able to capture both drag reduction and full cessation of vortex shedding for an immovable cylinder near a moving ground. That roughness model was successfully blended with a Lagrangian vortex method using sub-grid turbulence modeling. Overall, the effects of relative roughness heights on flows past two cylinders reveal changing of behavior of the vorticity dynamics, in which drag reduction, intermittence of vortex shedding, and wake destruction are identified under certain roughness effects. This kind of study is very useful for engineering conservative designs. The work is also motivated by scarcity of results previous discussing flows past cylinders in cross flow with surface roughness effects.

Lagrangian Vortex Computations of a Four Tidal Turbine Array: An Example Based on the NEPTHYD Layout in the Alderney Race

This study investigates the wake interaction of four full-scale three-bladed tidal turbines with different ambient turbulence conditions, in straight and yawed flows. A three-dimensional unsteady Lagrangian Vortex Blob software is used for the numerical simulations of the turbines’ wakes. In order to model the ambient turbulence in the Lagrangian Vortex Method formalism, a Synthetic Eddy Method is used. With this method, turbulent structures are added in the computational domain to generate a velocity field which statistically reproduces any ambient turbulence intensity and integral length scale. The influence of the size of the structures and their density (within the study volume) on the wake of a single turbine is studied. Good agreement is obtained between numerical and experimental results for a high turbulence intensity but too many structures can increase the numerical dissipation and reduce the wake extension. Numerical simulations of the four turbine array with the layout initially proposed for the NEPTHYD pilot farm are then presented. Two ambient turbulence intensities encountered in the Alderney Race and two integral length scales are tested with a straight flow. Finally, the wakes obtained for yawed flows with different angles are presented, highlighting turbine interactions.

Flow-induced cross-flow vibrations of long flexible cylinder with an upstream wake interference

Flow-induced vibration (FIV) of a flexible cylinder with an upstream wake interference at a subcritical Reynolds number is numerically investigated in this study. Two cylinders are installed in a tandem arrangement with the tandem separation between the cylinder centers set at 5.0 diameters. The downstream cylinder is flexible and placed in the wake of the stationary rigid upstream cylinder. A quasi-three-dimensional fluid-structure interaction (FSI) numerical methodology that couples the strip theory-based Lagrangian discrete vortex method with the finite-element method (FEM) for structural dynamics is developed to simulate the FIV response of the flexible cylinder with the upstream wake interference. The vortex-induced vibration (VIV) of an identical isolated cylinder is also numerically simulated as a contrast. This numerical study characterizes the dynamic response of the cylinder FIV with the upstream wake interference and sheds light on the FSI mechanisms responsible for the structural dynamic response. With the upstream wake interference, the cylinder FIV response shows two features distinct from the isolated VIV response: the vibration of large amplitude during the modal resonance branch transition and the extension of the modal resonance branch. The hydrodynamic coefficients database is constructed by the rigid cylinder forced vibration experiment to help explain the FSI properties of the FIV dynamic response. The lower added mass coefficient for the FIV with the upstream wake interference than the VIV of the isolated cylinder guarantees the synchronization between the vortex shedding frequency and the “true” natural frequency of the structure persisting to higher reduced velocity in a certain modal resonance response branch. The excitation coefficient distribution indicates that the cylinder FIV with the upstream wake interference reaches higher amplitude at high reduced velocity, instead of ceasing resonance as the isolated cylinder. The numerical wake visualization is shown and used to explain the correlation between the distribution of hydrodynamic coefficients along the cylinder span and the wake vortex mode. It is found that the upstream wake interference effect is strongly correlated with the vortex–structure interaction pattern between the upstream wake vortices and the downstream motion. When the upstream vortex impinges on the downstream cylinder and splits into subvortices, the effect of the upstream wake interference acting on the downstream cylinder reduces. When the downstream cylinder enters the gap between the upstream vortices over the entire vibration process, the upstream wake has a stronger interference effect on the downstream FIV response.

Fluid–structure coupling in time domain for dynamic stall using purely Lagrangian vortex method

This paper presents the purely fast Lagrangian vortex method (FLVM) for the simulation of the external incompressible flows past heaving and pitching bodies with high-frequency oscillation. The Nascent vortex element is introduced to the flow field to retain the Lagrangian characteristics of the solver. The viscous effect is modeled using a core spreading method coupled with the splitting and merging spatial adaptation scheme. The particle’s velocity is calculated using Biot–Savart formulation. To accelerate computation, a fast multipole method (FMM) is employed. The validity of FLVM solver is verified by temporal and spatial convergence studies for the case of flows past an impulsively started cylinder at the Reynolds numbers ranging from 50 to 9500. The accuracy of FLVM is then confirmed for the simulation of flows around the pitching flat plate and oscillating airfoil. The time history of drag and lift coefficients and the vorticity contours show a good agreement with those reported in literature. Furthermore, the FLVM is employed to determine the flutter derivatives and flutter speed of an oscillating flat plate. Results are compared with theoretical solutions based on Theodorsen’s function. In general, the results agree well with those obtained by the inviscid theory.

On the Scope of Lagrangian Vortex Methods for Two-Dimensional Flow Simulations and the POD Technique Application for Data Storing and Analyzing.

The possibilities of applying the pure Lagrangian vortex methods of computational fluid dynamics to viscous incompressible flow simulations are considered in relation to various problem formulations. The modification of vortex methods—the Viscous Vortex Domain method—is used which is implemented in the VM2D code developed by the authors. Problems of flow simulation around airfoils with different shapes at various Reynolds numbers are considered: the Blasius problem, the flow around circular cylinders at different Reynolds numbers, the flow around a wing airfoil at the Reynolds numbers and , the flow around two closely spaced circular cylinders and the flow around rectangular airfoils with a different chord to the thickness ratio. In addition, the problem of the internal flow modeling in the channel with a backward-facing step is considered. To store the results of the calculations, the POD technique is used, which, in addition, allows one to investigate the structure of the flow and obtain some additional information about the properties of flow regimes.

Study of Surface Roughness Effect on a Bluff Body—The Formation of Asymmetric Separation Bubbles

Turbulent flows around bluff bodies are present in a large number of aeronautical, civil, mechanical, naval and oceanic engineering problems and still need comprehension. This paper provides a detailed investigation of turbulent boundary layer flows past a bluff body. The flows are disturbed by superficial roughness effect, one of the most influencing parameters present in engineering applications. A roughness model, recently developed by the authors, is here employed in order to capture the main features of these complex flows. Starting from subcritical Reynolds number simulations (Re = 1.0 × 105), typical phenomena found on critical and supercritical flow regimes are successfully captured, like non-zero lift force and its direction change, drag crisis followed by a gradual increase on this force, and separation and stagnation points displacement. The main contribution of this paper is to present a wide discussion related with the temporal history of aerodynamic loads of a single rough circular cylinder capturing the occurrence of asymmetric separation bubbles generation. The formation of asymmetric separation bubbles is an intrinsic phenomenon of the physical problem, which is successfully reported by our work. Unfortunately, there is a lack of numerical results available in the literature discussing the problem, which has also motivated the present paper. Previous study of our research group has only discussed the drag crisis, without to investigate its gradual increase and the change on lift force direction. Our results again confirm that the Lagrangian vortex method in association with Large-Eddy Simulation (LES) theory enables the development of two-dimensional roughness models.

Lagrangian Vortex computations of turbine wakes: recent improvements using Poletto’s Synthetic Eddy Method (SEM) to account for ambient turbulence

This paper presents possible techniques for modelling ambient turbulence in the Lagrangian Vortex Method formalism. Due to the fact that regular Synthetic Eddy Method (SEM) already presented in previous studies is not divergence free by definition; improvements were necessary to develop a similar SEM method with such a divergence free property. The recent improvements formulated by R. Poletto give the way to such a possibility. This new Divergence Free Synthetic Eddy Method (DFSEM) is presented here in comparison with the regular SEM. Obtained numerical velocity fields are compared in terms of convergence properties, Power Spectral Density and also Taylor macro-scale. Finally, turbine wakes are computed with both the recent Poletto’s DFSEM and the regular Jarrin’s SEM to highlight differences. At this stage of development, the DFSEM seems very promising even though some improvements are still necessary.

Numerical experiments of the flow around a bluff body with and without roughness model near a moving wall

Numerical tests at high Reynolds number flows were taken on circular cylinder placed near and parallel to a moving ground. A moving ground running at the same speed as the free stream eliminates the confusing effects of the boundary layer formed on the ground being, therefore, more effective to establish a better understanding of the relationship between complete vortex shedding suppression and surface roughness. A detailed quantitative measurement of the flow field around the cylinder using Lagrangian vortex method with roughness model was carried out. The effect of higher surface roughness heights is studied because it introduces greater instabilities in the boundary layer of bluff bodies. The purpose is to investigate supercritical flow patterns from subcritical Reynolds number flow simulations. The present results are compared against those measured for smooth cylinder under the same flow conditions. It is found that certain critical gap ratio between the rougher cylinder bottom and the moving wall significantly reduces the drag force. The lift force points away from the wall plane. The full vortex shedding suppression is successfully anticipated. In addition, the Strouhal number vanishes. The contribution of this research is to report that von Kaman-type vortex shedding totally ceases and instead two nearly parallel shear layers are formed behind the cylinder in moving ground when employing two-dimensional modeling of roughness. Previous numerical results for flow around smooth cylinder placed closer to the moving ground did not capture the behavior of Strouhal number equal to zero. Unfortunately, there is a lack of experimental results for rough cylinder near a moving wall, which motives the present study.

Cost-Efficient Numerical Schemes for the Boundary Integral Equation Solution in 2D Vortex Methods

The problem of the flow simulation around airfoils using Lagrangian vortex methods is considered. Numerical schemes of the second order of accuracy for free vortex sheet intensity distribution along the airfoil are developed for smooth and non-smooth airfoils. The schemes are based on the Galerkin approach with piecewise-constant and piecewise-linear basis functions. The finite element method ideas are used and the resulting piecewise-linear scheme has the same numerical complexity as the scheme with piecewise-constant numerical solution. The modification of a FEM-type scheme is developed for non-smooth airfoils which permits to take into account the discontinuity of the solution at the specified points; its computational cost increases insignificantly.

Flow simulation around 3D bodies by using Lagrangian vortex loops method with boundary condition satisfaction with respect to tangential velocity components

Abstract A new approach is developed for incompressible 3D flow simulation around bodies by Lagrangian vortex method. Closed vortex loops are considered as vortex elements, which are generated on all the body surface and provide the satisfaction of the boundary condition on the body surface. The procedure of the double layer potential density reconstruction, called T-scheme is proposed, which consists of two steps. Firstly, the integral equation with respect to vortex sheet intensity is solved, which expresses the equality between the tangential components of flow velocity limit value and the body surface velocity. It is solved by using Galerkin approach. Secondly, the least-squares procedure is implemented, which permits to find nodal values of the double layer potential density. It is shown that the developed algorithm makes it possible to improve significantly the quality of solution for the complex-shaped bodies with low-quality surface meshes. The combination of this approach with vortex wake modeling with vortex loops permits to simulate unsteady flows with acceptable computational complexity. It can be useful for CFD applications and visual effects reproducing in computer graphics.

Numerical simulation of vortex-induced vibration of long flexible risers using a SDVM-FEM coupled method

This paper presents a grid-independent numerical methodology that couples the strip theory based discrete vortex method (SDVM) with the finite element method (FEM) to simulate the vortex-induced vibration (VIV) of a long flexible vertical riser. Based on the strip theory, a three-dimensional flow filed is approximately simulated by a series of computational ‘flow strips’. A Lagrangian discrete vortex method is employed to numerically solve the unsteady vorticity transport equations of each ‘flow strip’. The flexible riser is modelled as a tensioned Bernoulli-Euler beam with the dynamical equation solved by the finite element method in time domain. The two-dimensional DVM code is firstly validated for the VIV simulation of a rigid cylinder that was experimentally studied by Khalak and Williamson (1996). Referring to two typical experimental configurations of Lehn (2003) and Chaplin et al. (2005), the VIV of a long flexible riser immersed in a uniform and stepped incoming flow are numerically simulated, respectively. A good agreement was achieved through detailed comparisons between the present numerical prediction and the experimental data, including the structural in-line and cross-flow VIV response modes, root mean square amplitudes and the dominant frequency. The occurrence of standing and travelling wave responses, dual resonance between in-line and cross-flow motions and figure-eight trajectory are reported. The wake patterns corresponding to two response waves are also presented and investigated. Related to structural local vibration amplitude, two principle vortex patterns resembling the ‘2S’ and ‘2P’ modes are identified in the wake. The standing wave component of structural response determines the vortex shedding pattern. The travelling wave component affects the spanwise vortices shedding at different phases.

Development of a new Lagrangian vortex method for evaluating effects of surfaces roughness

Abstract The main contribution of this paper is to develop a new methodology to include effects of surface roughness on aerodynamic of bodies. The numerical approach simulates the mechanism of vortex shedding of a rough circular cylinder in the subcritical flow regime ( R e = 1 0 5 ) using a mesh-free vortex method. The local turbulence effects are considered during the vorticity diffusion process; as consequence a connection between the larger scales and the smaller ones is made by eddy viscosity. The eddy viscosity computation is necessary for each Lamb vortex element that constructs the Karman vortex street; it is also the way employed to make feasible the purposed surface roughness model. Simultaneous measurements of the spanwise-integrated unsteady aerodynamic forces and time averaged local surface pressures on a two-dimensional rough circular cylinder are investigated aiming to reach supercritical Reynolds number flow patterns from subcritical flows. It is shown that the surface roughness model captures successfully the drag crisis in a qualitative way.

A Vortex-In-Cell Method for the Simulation of Viscous Flows

A numerical technique of solving incompressible, vortex-dominated flows is described. The Navier-Stokes equations in velocity-vorticity formulation are integrated by a fractional step algorithm which treats convective motions and diffusive phenomena separately. Convection of vorticity is treated in a Lagrangian fashion, whereas the vorticity diffusion is computed on a grid by finite differences. The two problems are solved and combined at each time-step. The technique is as simple and robust as classical finite difference methods. Although being a bit more dissipative, it still preserves the spectral accuracy of fully Lagrangian vortex methods. The numerical procedure is validated against theoretical, experimental and numerical data from literature.