Abstract The question of open boundary conditions of inflow and outflow type for the Navier-Stokes equations and its influence on the convergence to steady state is addressed. Both the continuous and semi-discrete problem are analysed using the energy method and the Laplace transform technique. The energy method is used to derive well-posed boundary conditions for the continuous problem. For the semi-discrete problem we use the energy method to prove that by using the well-posed boundary conditions for the continuous problem and adding a suitable numerical boundary condition well-posedness is preserved. By employing the Laplace transform technique the spectra for different types of boundary conditions are obtained. The spectra are analysed and it is shown how the choice of boundary conditions strongly affects the convergence to steady state. One-dimensional Navier-Stokes calculations are performed and the resulting convergence rates agree well with the theoretical analysis. Finally, the spectra obtained using inflow and outflow types of boundary conditions are compared with spectra obtained using periodic boundary conditions and the choice of a time-integration method for the Navier-Stokes equations is discussed.