Accuracy of the random vortex method for a problem with non-smooth initial conditions

The accuracy of the random vortex method, in approximating the solution of the Navier-Stokes equation, is investigated using the model problem of a circular vortex as suggested by Milinazzo and Saffman [13]. The method consists of partitioning the vorticity into "vortex blobs." These blobs are moved via two actions. First, a blob is deterministically moved under the action of the velocity field associated with the other blobs. Then to simulate viscosity a random component is added to the position of the blob. For this model problem the nonlinear terms of the Navier-Stokes equation vanish. Thus the major error inherent in the deterministic component of the method vanishes. Consequently, for this model problem concentration is on the interaction of the deterministic and random components of the method. Results show that the accuracy of the method depends heavily on the initial distribution and strength of the computational elements, i.e., the vortex blobs. With the right choice of initial conditions it is found that e₁(t)=|L(t)-A(t)|A(t) is O(R^{- 1 2}N^{- 1 2}), where L(t) and A(t) are, respectively, the exact and computed angular moment of vorticity distribution at time t for N vortex blobs at a Reynolds number R.