Pulsar wind nebulae (PWNe) are clouds of ultra-relativistic highly magnetised particles surrounding rapidly rotating neutron stars. Their emission can range from the radio band up to the very-high-energy (VHE) gamma-ray regime. We focus on the VHE PWN population seen with H.E.S.S. (HESS Collaboration et al., 2018) where many older systems have an observed significant offset of several parsecs between the position of the pulsar and the centroid of VHE emission of the nebula. If due solely to pulsar proper motion such offsets would imply implausible large velocities, above 500 km/s.
We explore, using relativistic (magneto-)hydrodynamic open-source simulations, two possible drivers of such offsets: pulsar proper motion and the inhomogeneity of the medium. We discuss the implications of their physical parameters for the asymmetrical evolution in space of the PWN so as to compare our future results to the observed population of offset VHE middle-aged PWNe.