The velocity field determines the motion of the vorticity field. On the other hand, it turns out that we can find the velocity field from the vorticity field; in the following, we describe this.
The mass conservation equation (2.1)
We substitute (2.10) and (2.11) in the definition of
vorticity (2.3) and obtain the following Poisson equation
for the stream function,
We can solve (2.12) to find the stream function
and then the velocity field using (2.9).
A standard approach to solve the Poisson equation (2.12) is the
Green's function method [13,95]. Using this method,
for flows in free space (no boundaries) we can obtain from
(2.12) as,
The velocity given by (2.16) is for free space flows since we used the free space Green's function in (2.13). For flows over solid boundaries, this velocity can be corrected to satisfy the boundary condition (2.5); we will describe that in section 6.3.
The vorticity equation (2.7), the velocity equation (2.16), the initial condition and the boundary conditions together describe the evolution of the vorticity. Next we review the conservation laws derived from the vorticity equation.