3.2 Vortex methods for viscous flows

For Newtonian viscous flows, the vorticity equation (2.4) is

$$\frac{D \omega}{Dt} \; = \; \nu\,\nabla^2 \omega \quad \ .$$ (3.17)

To solve (3.17) numerically, we use the viscous splitting algorithm mentioned in the introduction of this chapter. Mathematically, this algorithm is expressed by splitting each time-step into a convection step and a diffusion step as follows (Chorin et al. [53]):

Convection step:

$$\frac{d\vec{x}_i}{dt} = \sum_{j}\,\Gamma_j \, \vec{K}_\delta(\vec{x}_i(t);\vec{x}_j(t))$$ (3.18)

$$\frac{d\Gamma_i}{dt} = 0 \quad \ ,$$ (3.19)

Diffusion step:

$$\frac{d\vec{x}_i}{dt} = 0$$ (3.20)

$$\frac{ \partial \omega}{\partial t} = \nu\,\nabla^2 \omega \quad \ .$$ (3.21)

In the above equations, $\vec{x}_i$ and $\Gamma_i$ are the position and circulation of vortex $i$ respectively; $\vec{K}_\delta$ is the velocity kernel $\vec{K}\ast\phi_\delta$; and $\omega$ is the smooth vorticity distribution represented by the vortices $\Gamma_i$.

A number of theoretical studies have shown that the velocity field from the viscous splitting algorithm converges to the velocity field of the Navier-Stokes equations. Beale & Majda [19] have shown the convergence for flows in free space. Ying [251] and Beale & Greengard [15] showed convergence for flows over solid boundaries.