Blob Method Research Articles

THE VORTEX BLOB METHOD AS A SECOND-GRADE NON-NEWTONIAN FLUID

We show that a certain class of vortex blob approximations for ideal hydrodynamics in two dimensions can be rigorously understood as solutions to the equations of second-grade non-Newtonian fluids with zero viscosity and initial data in the space of Radon measures M (R 2). The solutions of this regularized PDE, also known as the isotropic Lagrangian averaged Euler or Euler-α equations, are geodesics on the volume preserving diffeomorphism group with respect to a new weak right invariant metric. We prove global existence of unique weak solutions (geodesics) for initial vorticity in M (R 2) such as point-vortex data, and show that the associated coadjoint orbit is preserved by the flow. Moreover, solutions of this particular vortex blob method converge to solutions of the Euler equations with bounded initial vorticity, provided that the initial data is approximated weakly in measure, and the total variation of the approximation also converges. In particular, this includes grid-based approximation schemes as are common in practical vortex computations.

A Turbulence Model for the Three-Dimensional Vortex Blob Method.

The purpose of this paper is to propose a turbulence model for the three-dimensional vortex blob method. This model uses a modified version of the core-spreading method introduced by Leonard and Chua (1989) in the spirit of the Smagorinsky type of eddy viscosity. The diffusion of vorticity by the molecular viscosity is represented by the core spreading whose rate is approximated by that of a rectilinear viscous vortex tube. The linear combination of the two viscosities yields a Lagrangian turbulence model, which includes the effects of Reynolds number. We apply each model to an impulsively started round jet forced by two helical disturbances rotating in counter directions and compare the vortical structures of the jet among the models to show that the combined model can be a reasonable turbulence model.

Flow in a 2D duct with moving boundaries - vortex blob approach

In the paper we develop Random Vortex Blob method for the case of flows in periodic domains. It is shown how the formulation of the problem must be modied to account for the periodicity constraint. Properties of periodic vorticity particles are discussed. We also derive a rigorous method to satisfy the boundary conditions. Example computations concern the flow in an Xperiodic duct with moving boundaries and varying in time pressure gradient.

Numerical simulations of wake structure generated by rotating blades using a time marching, free vortex blob method

Vorticity fields in the wake generated by rotating blades are calculated using a time-accurate, free vortex blob method without a non-physical model of the far wake. The computed free-wake geometry of a single rotor in hover is represented by three wake regions: a well-defined tip vortex region, an intermediate region, and an initially generated wake bundle. The simulated wake geometries in the radial and axial directions agree well with the experimental data for a two-bladed rotor for the overall regions of the wake. This agreement can be obtained by slowly increasing the rotational speed of the rotor blade, instead of using an impulsively rotating blade. The unsteady generation process of the wake structure is calculated for a single rotor with two blades.

A Turbulence Model for the Three-Dimensional Vortex Blob Method.

A turbulence model is proposed for the three-dimensional vortex blob method. This model employs a modified version of the core spreading method introduced by Leonard and Chua (1989) in the spirit of Smagorinsky type of eddy viscosity. The diffusion of vorticity by the molecular viscosity is represented by the core spreading whose rate is approximated by that of the rectilinear viscous vortex tube. The linear combination of the two viscosities yields a Lagrangean turbulence model which includes effects of Reynolds number. Each model is applied to an impulsively started round jet forced by two helical disturbances rotating in the counter directions. Vortical structures in the jet are compared among the models to show that the combined model can be a reasonable turbulence model.

Simulating vortex pairs interacting with mixing-layer vortices

Numerical simulations of interactions between rolled-up vortices in a plane mixing layer and a vortex pair by a vortex blob method, with the aim of enhancing the growth of the mixing layer and its turbulence level are made. Evolution of vortex patterns and velocity fields during the interaction are presented. Parameter studies are made for the ratio of circulation of the vortex pair to that of the mixing layer vortices γ, the initial position of introduction of the vortex pair, and the attitude of the vortex pair. The momentum thickness of the mixing layer increases with increasing circulation of the vortex pair, while the increase tends to saturate at γ≈2.5, where the vortex pair passes through the mixing layer. Moreover, the momentum thickness attains a maximum at a particular longitudinal position of the introduction because this position determines how the vortex pair encounters the vortices in the mixing layer.

Spices1 — A smart-particle code for kinetic plasma simulation

Spices1 (Smart Particle In Cell ElectroStatic 1D) simulates 1D plasmas governed by the Vlasov equation. A linear Fokker—Planck operator can be used to represent collisions between plasma species and a given background. The electrostatic approximation is used and no magnetic field is present. Spices1 uses computational particles with internal degrees of freedom (smart particles) to represent better the evolution of the many physical particles allocated to one single computational particle, employing a modified version of the Blob method. The Blob approach leads to inexpensive low noise calculations and has the additional advantage to allow an accurate and deterministic treatment of collision operators. The present work describes the steps taken to implement the Blob method to create Spices1. All the instructions required to use Spices1 are given in details. Several test cases and paradigmatic applications are presented to illustrate the capabilities of Spices1 and to provide benchmarks for testing purposes.

THE INTERACTION OF PERTURBED VORTEX RINGS AND ITS SOUND GENERATION

Interactions of two coaxial vortex rings of thin cores with and without azimuthal instability and the far field sound radiated have been studied numerically using the vortex blob method and acoustic analogy. The azimuthal instability waves of wavenumbers 1≤m≤5 are represented by weak axial and radial sinusoidal perturbations. It was found that the behaviour and the far field sound radiated of two perturbed vortices during the three slip throughs of 3≤m≤4 and during the first slip through of 1≤m≤5 are the same as those of unperturbed casem=0. Because of their mutual induction, an unstable mode of the trailing vortex is found the third slip through atm=1 and 5, significantly lower wavenumbers compared with those of a single vortex ring. The far field sound radiated and their directivity, though still quadruple, are different from those of unperturbed cases, showing the effect of perturbation on the interaction of the two vortices. A criterion for the onset of unstable mode is also established.

A comparison of two vortex methods for large scale geophysical flows

Abstract Two modulated vortex methods for the study of large scale atmospheric or oceanic flow are presented. In particular the methods discussed are based on the equations for the conservation of potential vorticity. The point vortex method is shown to be first order accurate in frequency and second order accurate in group velocity. The blob method presented is shown to be second order accurate in frequency and group velocity. Although the blob method is more accurate it is less efficient than the point vortex method. Currently fast methods exist for the point vortex method but not for the blob vortex method. A methodology for choosing the blob radius is given. These methods are used to calculate the behavior of a modon.

Numerical simulation of the flow through in vitro models of the vocal folds

In order to obtain a reasonable prediction of the flow through the vocal folds, for applications such as a two-mass model for voiced sound production, the use of a quasistationary flow model is considered. This model assumes a quasisteady jet and the flow separation is predicted by means of a boundary layer model. Limitations on the validity of such a model are investigated by means of in vitro experiments which provide global information in the form of flow visualizations and quantitative information in the form of wall pressure and hot wire measurements. A two-dimensional numerical simulation based on a viscous blob method provides further quantitative information on the details of the flow. However, such a simulation cannot predict the transition from laminar to turbulent jet flow which is an essentially three-dimensional effect. Furthermore, both the simplified model and the numerical simulation do not predict the occurrence of asymmetry in the flow which is sometimes observed in the experiments. Imbedding the numerical calculation of the locally incompressible flow in an acoustical model will be discussed. [Work supported by Technology Foundation Grant No. ETN33.2952.]

Blob Method for Kinetic Plasma Simulation with Variable-Size Particles

A new approach to particle in cell simulation is presented for collisionless plasmas. The new method is based on computational particles of variable size (blobs). Each blob represents an element of phase space and its shape and size are evolved in time to represent better the correct evolution. The blob particles can be split to increase the accuracy in selected regions of stronger gradients. Blobs can also be coalesced to keep the total number of blobs constant. The performance of the blob method is analyzed in the case of the Landau damping and in the formation of a sheath in a bounded plasma. The results show the correctness and effectiveness of the new method. When compared to standard PIC methods, the blob technique reduces the noise for a given number of computational particles.

A Method for Determining the Velocity Induced by Highly Anisotropic Vorticity Blobs

Resolution of boundary layer flows at moderate or high Reynolds numbers with the vortex blob method requires a great many isotropic elements. In this paper, an approximate method for determination of the induced velocity from highly anisotropic vorticity blobs is presented, and issues related to use of anisotropic elements in calculations with vortex blob algorithms for high Reynolds number near-wall flows are examined. The method presented here can be used to determine the induced velocity from smooth blob functions of arbitrary form, provided that the vorticity length scale associated with the blob is much less in one direction than in orthogonal directions. The ratio of these length scales is called the blob aspect ratio, ϵ, and is used as a small parameter to construct an asymptotic approximation to the induced velocity field. This method is applied in the present paper to derive induced velocity expressions for anisotropic Gaussian blob functions in both two and three dimensions. It is argued, using test calculations for a Blasius boundary layer, that although direct calculation of the induced velocity requires about an order of magnitude more CPU time for anisotropic Gaussian elements than for isotropic elements, this difference is more than made up for by a reduction of several orders of magnitude in the number of elements needed to resolve boundary layer flows at moderate to high Reynolds numbers. It is also found that the standard vortex blob representation leads to errors in the calculation of wall slip velocity and wall shear stress due to smoothing of the discontinuity between the real and image vorticity fields at the wall, but that these errors can be avoided by placing doublet-type elements along the wall.

Numerical computation of vortex sheet roll-up in the background shear flow

The motion of a planar vortex sheet in the presence of a uniform shear flow is considered. We use the same numerical method as Krasny in his study of vortex sheets, namely, Chorin's vortex blob method. A linear analysis shows that the background shear influences the motion of the vortex sheet in such a way that with respect to low frequency modes it is stabilized in some cases. We examine this phenomenon not only by the linear analysis but also by a numerical simulation of the nonlinear integral equation. In particular, we note carefully how the number and the location of the roll-ups of a vortex sheet change as the strength of shear changes.

Kinetic simulation of bounded plasmas with the BLOB method

In the industrial applications of plasmas, the interaction of charged particles with the surfaces of the reactor chamber is extremely important. Considerable effort is being devoted to study the formation and the properties of the sheath near confining structures or around solid objects immersed in a plasma. In this paper, results obtained by using the BLOB method to simulate collisionless sheaths are presented. The BLOB method is based on the particle-in-cell (PIC) approach, but uses particles with additional degrees of freedom to describe their size and shape.

The pressure problem in the stochastic vortex blob method

The paper deals with the problem of recovering pressure from the given vorticity and velocity fields. Both fields result from the stochastic vortex-blob simulation of the viscous incompressible flow. The vorticity field is a discontinuous function. The method is presented here to find pressure as a solution to the variational problem which forms the weak version of the momentum equation. No explicit boundary condition for pressure are required and no derivatives of vorticity have to be calculated in this approach.

High amplitude vortex-induced pulsations in a gas transport system

High Reynolds number, low Mach number gas flows in pipe systems with closed side branches exhibit spectacular low frequency self-sustained pulsations driven by periodic vortex shedding at specific values of the Strouhal number. A detailed study is presented of the behaviour of the flow in a system with two opposite closed side branches of equal length in a cross configuration. For junctions with both sharp and rounded edges the acoustic flow velocity amplitude is comparable to the main flow velocity. A two-dimensional potential flow model based on the vortex blob method, used to simulate the flow in the junction, describes accurately the flow visualization and laser Doppler data obtained in pipes with square cross-sections and with sharp edged junctions. The numerical simulation is used to calculate the acoustical power generated by the vortical flow at a given amplitude of the acoustic velocity field and Strouhal number. In reality, for a pulsation with constant amplitude, this power is balanced by the viscothermal losses and acoustic radiation, which is the basis for the indirect measurement of the source power. It is shown that, for the acoustic amplitude observed, radiation losses due to the generation of non-resonating harmonics by wavesteepening has to be taken into account in the energy balance. This finding is confirmed by the appearance of shock waves in the geometry with rounded edges.

A Fast Algorithm for Vortex Blob Interactions

We introduce a new fast summation a algorithm which, unlike previous methods, is not restricted to potential functions. Its asymptotic complexity is O(nHd+1), where n is the number of blobs, H is the level of refinement, and d is the domain dimension. For a nonsingular kernel the complexity reduces to O(n). We present a complete error and complexity analysis of the algorithm and describe its implementation. As an example we use the vortex blob method to simulate vortex sheet motion in incompressible flow.

Numerical Simulation of Round Jet by Three-Dimensional Discrete Vortex Method. On the Limitation of Maximum Value of Vorticity.

An incompressible round compound jet issuing from a circular nozzle was simulated by a three-dimensional vortex blob method. The circular nozzle was composed of a source disc and vortex panels. The nascent vortex blobs were released from the edge of the nozzle in such a way that their circulation satisfies Kelvin's conservation law of circulation. The vortex rings released from the edge experienced vortex merging and developed three-dimensional structures with small-scale random structures. Effects of the limitation of maximum vorticity of the vortex blobs on the vortex structures were discussed. The large-scale vortical structure is fairly indendent of the limitation of maximum value of vorticity.

Investigation of the use of Prandtl/Navier–Stokes equation procedures for two-dimensional incompressible flows

The technique of combining solutions of the Prandtl equations with solutions of the Navier–Stokes equations to compute incompressible flow around two-dimensional bodies is investigated herein. Computational evidence is presented which shows that if the ‘‘obvious’’ coupling is used to combine the solutions, then the resulting solution is not accurate. An alternate coupling procedure is described which greatly improves the accuracy of the solutions obtained with the combined equation approach. An alternate coupling that can be used to create a more accurate vortex sheet/vortex blob method is then shown.

Boundary Conditions for Viscous Vortex Methods

This paper presents a Neumann-type vorticity boundary condition for the vorticity formulation of the Navier-Stokes equations. The vorticity creation process at the boundary, due to the no-slip condition, is expressed in terms of a vorticity flux. The scheme is incorporated then into a Lagrangian vortex blob method that uses a particle strength exchange algorithm for viscous diffusion. The no-slip condition is not enforced by the generation of new vortices at the boundary but instead by modifying the strength of the vortices in the vicinity of the boundary.