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To: 1.2 Hybrid methods for diffusion |
One of the major difficulty in mesh-based computations of high Reynolds number separated flows around complex geometries like multi-element airfoils or even more complex geometries like fighter aircrafts, is to generate an effective mesh to solve the governing flow equations. The common strategy of generating a mesh with fine resolution near solid walls may not be adequate in high Reynolds number separated flows. The reason is that the steep flow gradients, such as in boundary layers, do not always occur only normal to a solid wall. For example, to compute the flow around an impulsively started cylinder, the mesh does not only need to be refined normal to the cylinder wall. When separation occurs, sharp gradients also develop in the direction along the wall. Van Dommelen & Shen [241] showed that in fact very fine resolution is required in the direction along the wall to resolve the rapid evolution of the vorticity layers in that direction. More often the steep flow gradients also occur due to separated vorticity fields, like the forebody vortices from an aircraft, for example. It is very difficult to predict a priori the evolution of the separated vorticity fields.
An accurate representation of the separated vorticity fields is crucial to determine the aerodynamic forces on bodies accurately, yet it is very difficult to accurately resolve the separated vorticity fields using a mesh. At high Reynolds numbers, the numerical diffusion due to insufficient resolution could overwhelm the actual diffusion, and hence the separated vorticity fields will be diffused erroneously; this will result in large errors in the computed aerodynamic forces on vehicles such as fighter planes and cars. It can cause difficulties for computing airplane control forces when the inaccurate separated vorticity from the main wing interacts with the tail surfaces. On the other hand, since the vortex methods are based on following the vorticity, they provide excellent adaptive resolution of the separated vorticity fields.
In vortex methods, the convection of vortices can be achieved using mesh-free algorithms, as mentioned earlier. However, the diffusion of vortices must also be handled in a mesh-free manner to avoid the difficulties of mesh-based computations, and this is a more difficult problem. One of the main difficulties is the chaotic distribution of the vortices. High Reynolds number flows are characterized by a strong mixing of the vortices, and a regular vortex distribution is almost impossible to maintain.
Some previous attempts to deal with this difficulty have been based on interpolating to a mesh for at least some of the computation (section 1.2). However, this loses a significant part of the advantage that a Lagrangian computation attempts to achieve over conventional computation: it is very hard to produce efficient meshes for complex geometries, especially for separated flows where they need to resolve sharp gradients that are not aligned with the boundaries. Further, using a mesh introduces interpolation errors between the mesh and the vortices, and the resulting wide variety of errors tends to make the final accuracy uncertain.
Alternate approaches to deal with the irregular vortex locations restore order periodically, by periodically selecting a new set of vortices with strengths found by interpolation from the previous set (subsections 1.3.3 and 1.3.4). One difficulty is that the generation of an effective new vortices distribution is not really different from generating a mesh; another difficulty is that at high Reynolds numbers, to be truly effective, the regeneration has to be done frequently. This requires again effective solution adaptive meshes, which would be a very difficult problem for ``real life" high Reynolds number separated flows about complex configurations. The interpolation errors and trade-offs in choosing the times at which to redefine the vortex distribution again introduce considerable uncertainty about the optimal procedure and the final errors.
In order to actually achieve the advantages that a Lagrangian computation promises, such as the elimination of the mesh generation problem and the accurate representation of separation processes and separated vorticity without excessive mesh points, better approaches are needed. Those approaches must directly handle the irregular vortex distribution produced by high Reynolds number flows. A number of methods that can do this have been proposed (subsections 1.3.1, 1.3.2 and 1.3.5), of which the ``random walk" method of subsection 1.3.1 has without doubt turned out to be the most effective. However, although this method works (e.g. see figure 8.23), in practice it is quite inaccurate. This may in part be due to the fact that the random walk method does not satisfy the various physical conservation laws exactly. Furthermore, the method is of a statistical nature, which means that the results are not easily reproducible, and the errors may be even larger if you happen to be unlucky. In parameter studies, it is very difficult to separate the effect of the parameters from the random errors. There is further no obvious way to improve the order of accuracy of the method.
In this thesis we will propose another method to deal with chaotic vortex distributions. It could be called a ``computed finite difference" method, since we compute the equivalent of a finite difference formula for the diffusion of each vortex at each time. However, since the effect of the finite difference formula is to redistribute part of the strength of each vortex among its neighbors, we call it the ``Redistribution Method". We will show that this method can be implemented efficiently and is significantly more accurate than the random walk method. It also does not have the inherent limitations in order of accuracy of the random walk method. In the following, we briefly describe the various existing methods for handling diffusion. We will show that these existing methods cannot handle our requirements for a mesh-free accurate procedure. Finally, we will introduce our new method that can.