Formulation

The two-dimensional incompressible Navier-Stokes equations are:

$$\frac{\partial {\bf v}}{\partial t} + ({\bf v} \cdot \nabla){\bf v} = - \nabla p + \nu \nabla^2 {\bf v} + F,$$

$$\nabla \cdot {\bf v} = 0,$$

where ${\bf v}$ denotes the velocity of the fluid with components ${\bf v} =$ [ u(x,y,t) , v(x,y,t)]^T in the x and y directions; p(x,y,t) is the pressure; F(x,y,t) is a forcing function and $\nu$is the kinematic viscosity. To discretize these equations in time we use a high-order splitting scheme [39]. A variation of this scheme using the spatial operators we have already described is explained and demonstrated in [30].