Periodic boundaries are only used when the flow field is assumed to have an N -fold periodicity about the z-axis. This is the common case, for example, when the in-cylinder geometry of a diesel engine is represented by a sector mesh. For example, a 45-degree sector mesh would correspond to an eight-fold ( N =8) periodicity.

When periodic boundaries are used, the computational region is composed of points in a pie-shaped sector , where θ satisfies and . The periodic boundaries are those for which θ = 0 and θ = 2 π / N . The conditions imposed on these boundaries can be inferred from the assumed N -fold periodicity. For a scalar quantity q a constraint sets q(r, θ ,z) = q(r, θ + 2 π /N, z) , where r = g . For a vector v the constraint is that v (r, θ +2 π /N, z) = R ⋅ v (r, θ , z) , where R is the rotation matrix corresponding to the angle 2 π /N.