The formation of bow shocks in freely expanding supersonic jets is simulated using the method of Smoothed Particle Hydrodynamics (SPH), suitably reformulated for application to flows with axial symmetry. It is found that a stable bow shock is formed when a jet encounters an obstruction of diameter d less than about half the jet diameter D. Results are presented for flow simulations with adiabatic indices of 4/3 and 5/3, representing the two cases in which the pressure due to relativistic particles is dominant and negligible respectively, and for pre-shock Mach numbers of 3 and 5. In each of these simulations the jet has an initial diameter of D = 5d. The frozen-in magnetic field configuration is calculated for each flow simulation, and it is found that a small pre-shock toroidal field component may become dominant in a localized region immediately behind the apex of the bow shock. These models form the basis for detailed surface brightness, polarization and spectral calculations to be presented in a subsequent paper.