1.3.1 Random walk method

The random walk method to model the diffusion of vorticity was first proposed by Chorin [54]. To simulate the diffusion of vorticity in vortex methods, the positions of the vortices are given random displacements (a random walk) [48]. These random displacements have zero mean and a variance equal to twice the product of the kinematic viscosity and time step. The basic idea of the random walk method is that the random displacements spread out the vortices like the diffusion process spreads out the vorticity.

Several studies investigate the theoretical and numerical aspects of the random walk method: Marchioro & Pulvirenti [144], Goodman [90], and Long [135] have shown that for flows in free-space, the random walk solution converges to that of the Navier-Stokes equations as the number of vortices is increased. However, there are no convergence studies for flows involving solid walls. Puckett [178] gives a survey of the elements of vortex methods; in particular, he discusses in detail the random walk method and its convergence. Chang [41] discusses how to incorporate the random walk method in Runge-Kutta time-stepping schemes. Ghoniem & Sherman [85] studied ways of handling the boundary conditions. They also develop a `gradient random walk' method [51,205] in which the computational points transport derivatives of vorticity instead of the vorticity itself; they show that this procedure produces smoother vorticity distribution than the random walk method.

The random walk method has been used extensively. Here we will mention only some of the applications: Chorin applied the random walk method for simulating flows around cylinders [49,54] and flat plates [49,50,52]. Shestakov [207] has used the random walk method for flow inside a driven cavity. McCracken & Peskin [149] have studied the blood flow through heart valves using the random walk method. Ghoniem [84,86], Sethian [200] and Majda & Sethian [142] have applied the random walk method to problems in combustion; more recently, the random walk method has been used in combustion problems by Melvin [154], Pindera [173], Caldaza [34] and Song [211]. Sod [210] has used random walk method to study the interactions of shock waves with boundary layers. Cheer [42,43] has implemented the random walk method for flows over a cylinder and an airfoil. Van Dommelen [213,237,238] studied flows over impulsively started cylinders, pitching airfoils, jets and cavities. Ghoniem & Cagnon [83] studied the entry flow in a channel and the flow over a backward-facing step using the random walk method. Sethian & Ghoniem [199] studied convergence for a backward-facing step numerically. Martins and Ghoniem [146] have applied the random walk method to simulate the intake flow in a planar piston-chamber device. Smith & Stansby [214,215] have studied flows over impulsively started cylinders. Wang [245] studied various flow control techniques to avoid the dynamic stall of airfoils. Seo [198] has applied the random walk method to flows over translating, oscillating and pitching two-dimensional bodies of arbitrary shape. Tiemroth [221] and Vaidhyanathan [228] have applied the random walk method to study flows over submerged and floating bodies including free-surface interactions. Summers [220] has applied the random walk method to Falkner-Skan boundary layer flows. Chui [58] used the random walk method to study thermal boundary layers. Baden & Puckett [12], and Choi, Humphrey & Sherman have applied the random walk method to compute the flow inside square cavities. Lewis [128] has used the random walk method for flow over airfoil cascades.

The random walk method has some advantages: it is simple to use; and it can easily handle flows around complicated boundaries. The method conserves the total circulation.

However, the random walk method also has some major disadvantages. First, it does not exactly conserve the mean position of the vorticity in free space. Next, the computed solutions are noisy due to the statistical errors. In flow control studies, the statistical errors could mask the effects of varying the control parameters. The statistical errors can also cause symmetric flows to turn asymmetric erroneously. To reduce the statistical errors requires a very large number of vortices.

Many investigations have studied how the errors in the random walk computations vary with the number of vortices. Milinazzo & Saffman [155] tested the random walk method for the case of an initially finite region of vorticity in an unbounded domain. They corrected for the error in the mean position, but found that the error in computed mean size of the vortex system is proportional to the inverse square root of the number of vortices $N$. If the initial vorticity inside the region is constant, the number of vortices must be increased with Reynolds number to keep the relative error in change in size constant at finite times. Roberts [182] showed that if the relative error in size itself is of importance, higher Reynolds numbers do not require additional vortices. In fact, the number of vortices can be reduced if the initial data represent the initial mean size accurately. Fogelson & Dillon [79] have used a simplified one-dimensional version of the problem to study the question how much smoothing should be applied to the random walk results. They found that convergence occurs when the random walk solution is smoothed over a distance that is large compared to the point spacing. Their results show that still a very large number of vortices is needed to improve the accuracy of the random walk method.

From such studies, it follows that the random walk method needs a very large number of vortices for accurate simulations. Correspondingly, the amount of work to convect all the vortices also becomes large. In addition, since the method is not deterministic, the random errors cause difficulties in the physical interpretation of the results. As an alternative to random walk, a number of deterministic methods have been proposed; we will discuss such methods in the following.