In this paper, the Crank-Nicholson + component-consistent pressure correctionmethod for the numerical solution of the unsteady incompressible Navier-Stokesequation of [1] on the rectangular half-staggered mesh has been extended to thecurvilinear half-staggered mesh. The discrete projection, both for the projectionstep in the solution procedure and for the related differential-algebraic equations,has been carefully studied and verified. It is proved that the proposed method isalso unconditionally (in△t) nonlinearly stable on the curvilinear mesh, providedthe mesh is not too skewed. It is seen that for problems with an outflow bound-ary, the half-staggered mesh is especially advantageous. Results of preliminarynumerical experiments support these claims.