This paper outlines a velocity–vorticity based numerical simulation method for modelling perturbation development in laminar and turbulent boundary layers at large Reynolds numbers. Particular attention is paid to the application of integral conditions for the vorticity. These provide constraints on the evolution of the vorticity that are fully equivalent to the usual no-slip conditions. The vorticity and velocity perturbation variables are divided into two distinct primary and secondary groups, allowing the number of governing equations and variables to be effectively halved. Compact finite differences are used to obtain a high-order spatial discretization of the equations. Some novel features of the discretization are highlighted: (i) the incorporation of the vorticity integral conditions and (ii) the related use of a co-ordinate transformation along the semi-infinite wall-normal direction. The viability of the numerical solution procedure is illustrated by a selection of test simulation results. We also indicate the intended application of the simulation code to parametric investigations of the effectiveness of spanwise-directed wall oscillations in inhibiting the growth of streaks within turbulent boundary layers.