Least-Squares and Triangulated Vortex Methods

There are at least two more deterministic approaches for inclusion of diffusion in vortex methods. As we remark below, they have been later combined together. The first approach is the so-called “free Lagrangian” method [176], which is based on a discrete approximation of the Laplacian on an irregular grid. A Voronoi triangulation is built using as nodes the particle positions, and discrete differential operators are approximated on this mesh. The construction of a Voronoi diagram can be computationally expensive, and it has to be updated at every time step as the particle positions are advected. In subsequent work [177] a fast method is used to construct a Delaunay triangulation that is of O(NlogN), incorporating also a fast summation technique for the velocity evaluation, but restricting this time to inviscid flows. (A Delaunay triangulation is not the same as the Voronoi

diagram, but one can be obtained from the other). This last work includes commentaries on how to generalize the fast triangulated vortex method to viscous flows, flows with boundaries and three- dimensional flows, but neither of these generalizations was carried out. Actual application of the Delaunay triangulated vortex method to two-dimensional viscous flows with solid boundaries is presented in [89, 90], where the concept of diffusion velocity is incorporated as well as a coupled panel method. This work also integrates a least-squares approach to calculate derivatives, described below, hence the combined use of the two methods that we remarked upon.

The other concept used in vortex methods for inclusion of viscous effects is the “moving least- squares method” of [126]. The authors were motivated by the loss of accuracy of both the Fishelov method and PSE when the particles become irregularly spaced, and wanted to provide an alterna- tive means of obtaining the derivatives of vorticity. The least-squares method consists of fitting a polynomial of order two to the vorticity field in a neighbourhood of the point where the derivative is desired. When the points are regularly spaced and only close neighbours are considered, the approach is equivalent to a centered difference. The drawback is having to solve a two-by-two linear system per particle, at each time step. Consider this one-dimensional example, defineζm(x) as the

approximation to the vorticityω(x) in the neighbourhood of a control pointxm, denoteωm≡ω(xm),

and write

ζm(x) =ωm+Bm(x−xm) +Cm(x−xm)2, (2.67)

where the constants Bm and Cm correspond to approximations of the first and second derivatives

of the vorticity at the control point. Define the error at the control point by

Jm≡

N

X

n=1

Lnm[ωn−ζm(xn)]2. (2.68)

The coefficientsLnmare used to establish the width of the neighbourhood around the control point.

Defining a length scaleδmto set up the locality ofLnm, the authors of [126] use

Lnm= exp · −(xn−xm)2 δ2 m ¸ . (2.69)

Now, minimizing the error, the following equations are obtained for the coefficientsBmandCm N

X

n=1

Lnm(xn−xm)i[ωn−ζm(xn)] = 0, i= 1,2. (2.70)

The least-square method is basically a way for calculating derivatives. Once a polynomial function has been fit onto the control points in the neighbourhood of the point of interest, then the derivatives of vorticity at this point are approximated by differentiating the polynomial fit. The method was tested in [126] by measuring theRMS error in the second derivative of a one-dimensional vorticity

field (a simple Gaussian) and plottingversus the overlap ratio. This error was compared with the Fishelov method and PSE for different irregular grids. The results show how the error increases severely with both Fishelov and PSE methods when the grid becomes irregular. In contrast, the least-squares calculation remains accurate with very irregular grids. For tests using an error-function initial vorticity field, it was necessary to add image vortices near the discontinuity (which is anal- ogous to a “wall”) at x= 0. The results showed that the least-squares calculation exhibits errors comparable to a central difference formula when sufficient images are included.

The application of the least-squares approach to two- or three-dimensional vortex methods was implemented by combining it with a diffusion velocity concept which is used to spread the vortic- ity support. The details of the full-fledged vortex method are not given in [126], which basically reports the preliminary one-dimensional numerical experiments to obtain error measurements. The implementation of the vortex method is only sketched and described to be underway. But in [127] it is shown how the diffusion velocity method is combined with the moving least-squares scheme to calculate derivatives, although this work is restricted by the simplifications of axial symmetry.

Recently, the authors of the moving least-squares method have combined it with the triangulation method of [177]. In [128] a three-dimensional viscous vortex method is described which relies on the moving least-squares approach to approximate the derivatives for calculation of the stretching and diffusion terms, but instead of using vortex blobs the vorticity field is interpolated on a tetrahedral mesh that is fitted to the Lagrangian points. We stress that the triangulated methods arenotvortex blob methods, and therefore do not require cutoff functions. Hence, the advantage is attained that the vorticity field does not penetrate the surface of the body. Additionally, it is possible to have node points in a distribution with high anisotropy in one direction, which is advantageous when boundary layer flows are computed. The method —given the name “tetrahedral vortex element” or TVE method— was benchmarked on the flow past a sphere at Reynolds number of 100, and has recently been applied to a more complicated three-dimensional flow in [78] with notable results. But perhaps one could consider this a method closer to vortex-in-cell than to vortex blob methods, where the mesh is now unstructured. It is clearly not a grid-less method, as a tetrahedral mesh is constructed and fit to the Lagrangian points on each time step.

In conclusion, with so many different approaches to construct a viscous vortex method, it is clear that the field is still maturing and is yet far from a consensus in regards to the “best” viscous scheme. Each method reviewed above has some desirable characteristics as well as some disadvantages. Particle strength exchange is very sensitive to having an ordered distribution of particles, and so there has been a great amount of work on remeshing schemes. The method of superposing derivatives of the cutoff function, as used by Anderson and Greengard for vortex stretching and Fishelov for diffusion, is also dependent on a regular particle arrangement to maintain accuracy; diffusion velocity,

as well, requires constant overlap of blobs. It would seem that the problem for an accurate viscous vortex method is not the viscous scheme, but spatial adaption. Indeed, to liberate themselves from the problem generated by irregular particle fields, the authors of the vortex redistribution method resort to elaborate and computationally expensive algorithms, while the tetrahedral vortex element method does away with the particle representation altogether (at the cost of constructing a mesh at every time step). Even the core spreading method suffers from the problem of how to limit the size that vortex blobs can grow, so needs a form of spatial adaption. But in contrast to the other methods, core spreading is utterly simple in its approach to satisfying diffusion effects. If the spatial accuracy can be maintained with some form of adaptive refinement, the core spreading method seems to offer the opportunity of a truly grid-less viscous vortex method.