  We present a model of a pulsar wind nebula evolving inside its associated
supernova remnant. The model uses a hydrodynamics code to simulate the
evolution of this system when the pulsar has a high velocity. The simulation
distinguishes four different stages of pulsar wind nebula evolution: the
supersonic expansion stage, the reverse shock interaction stage, the subsonic
expansion stage and ultimately the bow shock stage. The simulation bears out
that, due to the high velocity of the pulsar, the position of the pulsar is
off-centered with respect to its pulsar wind nebula, after the passage of the
reverse shock. Subsequently the pulsar wind nebula expands subsonically untill
the event of the bow shock formation, when the motion of the pulsar becomes
supersonic. The bow shock formation event occurs at roughly half the crossing
time, when the pulsar is positioned at 0.677 times the radius of the supernova
remnant blastwave, in complete agreement with analytical predictions. The
crossing time is defined by the age of the supernova remnant, when the pulsar
overtakes the blastwave bounding the supernova remnant.
  The results of the model are applied to three supernova remnants: N157B,
G327.1-1.1 and W44. We argue that the head of the pulsar wind nebula,
containing the active pulsar, inside the first two systems are not bounded by a
bow shock. However, in the case of W44 we argue for a scenario in which the
pulsar wind nebula is bounded by a bow shock, due to the supersonic motion of
the pulsar.
