In this chapter we present boundary conditions for the vorticity–velocity formulation of the Navier–Stokes equations and we describe their implementation in the context of vortex methods. We restrict our discussion to flows bounded by impermeable, solid walls, although several of the ideas can be extended to other cases such as free-surface flows.

The direct numerical simulation of wall-bounded flows requires accurately resolving the unsteady physical processes of vorticity creation and evolution in small regions near the boundary. Vortex methods directly resolve the vorticity field, and they automatically adapt to resolve strong vorticity gradients in regions near the wall, but they are faced with the algorithmic complication of dealing with the no-slip boundary condition. The no-slip boundary condition is expressed in terms of the velocity field at the wall and does not involve explicitly the vorticity.

Mathematically we may understand this difficulty by considering the kinematic and dynamic description of the flow motion and observing that there is an inconsistency between the number of equations and the number of boundary conditions. The kinematic description of the flow, relating the velocity to the vorticity, is an overdetermined set of equations if we prescribe all the components of the velocity at the boundary. On the other hand, no vorticity boundary condition is readily available for the Navier–Stokes equations that govern the dynamic description of the flow.

Physically, the no-slip boundary condition expresses the requirement that the flow field must adhere to the boundary.