# Mark J. Stock

I am an artist, scientist, and programmer interested in high-performance computing, computational fluid dynamics (CFD), and algorithmic artwork. I expect to donate specifically to pages presenting information on vortexes, vortex methods, and computational tools for fluid dynamics. Here are the pages that I've touched so far:

My credentials for commenting come from a PhD in Aerospace Engineering (University of Michigan, 2006) and a series of papers on vortex sheet and particle methods over the last 5 years available from my web site.

# Vortex Methods

(also Lagrangian Vortex Particle Method (LVPM), Free Vortex Method (FVM), Discrete Vortex Method (DVM))

sssVortex methods are novel computational fluids dynamics (CFD) methods that use Lagrangian discretizations of vorticity to allow computers to perform predictive simulations of fluid phenomena for science and engineering. More traditional CFD methods use Eulerian (fixed) spatial discretizations and velocity-pressure variables, which come with specific computational disadvantages including numerical instability and complex volumetric meshing. Vortex methods address these disadvantages at the expense of increased computational cost per time step and decreased generality.

## History

Lord Kelvin refers to vortex atoms in 1867[1], but not in the sense of fluid dynamics. A.R. Low first expressed the idea of tracking vorticity in 1928[2], but the first calculations using vortex methods were accomplished by Rosenhead[3][4] in 1930-1. A long delay between these early efforts ended in 1959 with the repetition of Rosenhead's experiment by Birkhoff and Fisher[5], and then numerical experiments by Tung and Ting in 1967[6]. Significant research began to appear in the 1970s, three-dimensional vortex methods and parallelization efforts dominated research in the 1980s, and the breadth of computational techniques and applications for vortex methods has been growing rapidly since the 1990s.[7]

## Governing equations

Incompressible assumption eliminates

### Vorticity equation

Starts with taking the curl of the Navier-Stokes equations.

### Velocity inversion

Two major method types exist for determining the velocity field from a distribution of vorticity. One begins with the Biot-Savart law, and the other involves solving the Laplace equation within a Particle-in-cell method.

## Computational Methods

Vortex methods encompass a wide range of computational, numerical, and algorithmic techniques. This section will present the various classifications of historical and modern vortex methods.

### Discretization

Both the earliest and the most capable current vortex methods solvers discretize vorticity onto discrete Lagrangian particles. Still, many vortex methods use alternative discretizations such as filaments and sheets. Each has advantages and disadvantages, though researchers have been focusing most effort recently on particle vortex methods.

Particles have become the most common vortex method discretization technique because they are more capable of accounting for the variety of terms in the vorticity equation. Adapting a particle distribution...

Vortex filament methods benefit from Kelvin's observations that the circulation of a vortex line does not change along its length, and it does not end in free space---only on a solid body. This eliminates the need to calculate the stretching component of the vorticity equation and explains why vortex filament methods were the first three-dimensional vortex simulations. Adaptation of the filaments is accomplished by splitting long segments in two parts and placing the new node either at the geometric center of the old segment or along a spline connecting the original nodes. Adaptation cannot easily occur along any non-tangential direction without extra connectivity information.

Vorticity very commonly enters a flow at interfaces between two fluids, or as the result of shedding from a solid body, lending support to vortex methods based on sheet discretizations. In two dimensions these sheets are continuous linear or higher-order connected segments. In three dimensions sheets are described by connected meshes of triangles or quadrilaterals. Adaptation in the two sheet-tangent directions can take a variety of forms, depending on the element shape. Adaptation in the sheet-normal direction is generally not attempted.

### Velocity inversion method

• Vortex-in-cell - Based on Particle-in-cell methods, VIC first appeared in a 1970 report by J.P. Christiansen[8] and was extended to three dimensions by Who in When.
• Treecode (Barnes-Hut, Anderson?, also multipole treecode)
• Fast Multipole Method
• Domain decomposition method

### Vorticity equation terms

• Stretch
• In a particle vortex method, the stretching term is accounted for through the dot product of the velocity gradient tensor and the particle circulation vector.
• Vortex filament methods, due to their construction, can ignore this term.
• Vortex sheet methods must deal with this term.
• Viscous diffusion (PSE, VRM, grid)
• Inviscid - no diffusion of vorticity, Krasny showed that inviscid vortex sheet motion always leads to discontinuities. Are liquid helium flows inviscid?
• Random vortex method (RVM) - Chorin, vortex particles move in a random walk, with movement distances related to the diffusion length scale.
• Particle Strength Exchange (PSE) - Mas-Gallic, based on concepts from Smoothed Particle Hydrodynamics (SPH), PSE exchanges circulations between particles based on local gradient calculations.
• Vorticity Redistribution Method (VRM) - Shankar, van Dommelin, particles exchange circulations based on a local solution to the diffusion equation. New particles are created when no local solution can be found.
• Grid-based diffusion - Vortex methods in which particles are periodically redistributed onto a regular grid can take advantage of the relative simplicity of standard, Eulerian-like grid-based diffusion techniques.
• Baroclinicity - To date have only been exercised in vortex sheet methods (both 2D and 3D)
• Subfilter-scale dissipation (LES) - Some argue that the vortex discretization itself allows subgrid-scale dissipation. More explicit subfilter-scale dissipation has been shown for vortex methods by Cottet, and a dynamics scheme by Gharakhani.
• Compressibility -- A weakly-compressible vortex method was created by Eldredge, et al.[9]

### Boundary conditions

• Free-space - The Biot-Savart equation...summation of potential flow primitives...
• Periodic (easy with VIC, some summation with non-grid-based)
• Inviscid boundary (BEM) - these are the traditional panel methods.
• Viscous boundary (BEM plus shedding) - These are like panel methods...how different?

## Comparison to Eulerian methods

• Lagrangian helps with convective stability
• vorticity variables
• reduced dimensional meshing (meshes are only D-1 dimensions)
• No difficulty with cons. mass equation
• concentration of computational effort

• Single time step requires more effort (reference Cottet's paper on speed differences)
• algorithmically more complex

## Major practitioners

A list of major researchers who have contributed to vortex methods in the past three decades includes, but is not limited to: A. Chorin, G.H. Cottet, A. Gharakhani, A. Ghoniem, P. Koumoutsakos, R. Krasny, A. Leonard, J. Strickland, G. Winckelmans.

## Uses

• related to vortex lattice methods
• fixed-wake and free-wake models used for helicopter and wing turbine design
• computer graphics fluids