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To: 1.5 Organization of the thesis |
In the previous section we discussed several Lagrangian methods for diffusion, each of which has its own characteristics. Each of those methods has difficulty either in handling diffusion accurately or in handling the complex boundary conditions and vorticity fields in many practical flows: The random walk method requires a very large number of vortices for accurate simulations. The PSE method and Fishelov's method require remeshing and vortex overlap to maintain the accuracy of the numerical computations. However, remeshing procedures for flows over complicated boundaries or with complex vorticity fields and maintaining particle overlap in flows with strong convection are difficult. The core expansion method does not represent the convection process correctly for the Navier-Stokes equations. Finally, in the diffusion velocity method, evaluating the diffusion velocity accurately is difficult.
The above difficulties suggest the need for an accurate mesh-free method to handle diffusion. The vorticity redistribution method developed in this work addresses the above difficulties: Unlike the random walk method, the vorticity redistribution method is deterministic and implicitly maintains the vorticity conservation laws. Therefore, our method does not need as many vortices as the random walk method for the same accuracy. Our method has the advantage over the PSE and Fishelov's methods in that it is mesh-free. This mesh-free property of our method provides a significant advantage to compute flows over complicated geometries accurately. Another difficulty with the PSE and Fishelov's methods is the resolution of sharp gradients in the flow; their resolution is asymptotically limited to a size that is asymptotically much larger than the average spacing of the particles, (see section 9.1). However, the resolution in our method is of the order of the average spacing of the particles. We also do not have difficulty in representing convection correctly, unlike the core expansion method. Finally, our method does not face the difficulties of the diffusion velocity method since we do not use the diffusion velocity in our computations.
Next, we will describe the basic idea of the vorticity redistribution method. The vorticity redistribution method is similar to the deterministic particle methods [72,78] in that it changes the strengths of the vortices to simulate diffusion: fractions of the strength, or circulation, of each vortex are moved to neighboring vortices in order to produce the correct amount of diffusion. However, while the deterministic particle methods use simple approximations to find the amounts of circulation to move, instead in section 4.1 we will formulate a special system of equations for it. Also, unlike the deterministic particle methods, the maximum distance that the circulation of a vortex is allowed to move during a time-step is restricted to a chosen distance of the order of the point spacing, rather than large compared to it. This allows scales up to the point spacing to be resolved. Unlike free-Lagrangian methods, no partitioning of the domain is attempted; instead, all available vortices within the allowed distance are included in the discretization.
The key question is to choose the fraction of the circulation of each vortex that is moved (redistributed) to each neighboring vortex. This choice determines the accuracy of the approximation, its stability, and its conservation properties. We will formulate a system of equations from which the redistribution fractions can be found in section 4.1. As will be seen in section 4.2, the equations of this system takes the form of localized conservation laws. This system can be extended to any order of accuracy. A uniform distribution of vortices is not required; however, for uniformly distributed points our method is equivalent to a finite difference scheme. Positivity of the solution of the system is enforced to ensure stability. We use a solution procedure that is guaranteed to find a positive solution to the system of equations, if one exists. If there is no acceptable solution, we add new vortices until there is one.
Fundamentally, our procedure differs from the usual particle methods by separating the computation of the vorticity into two distinct steps: (a) determination of vortex strengths from the localized conservation laws; (b) reconstruction of the vorticity field by convolution. This separation allows us to achieve any chosen order of accuracy regardless of the geometry of the vortex distribution. However, unlike the particle methods, and other numerical methods, in our scheme an individual vortex strength has no identifiable meaning. It is the combination of nearby vortex strengths and positions that determines the local solution. In chapter 9, we discuss the practical implications of these differences.
We next describe the organization of this thesis.