Numerical investigation of the turbulent ISM In this work we introduced basic turbulence theory into the framework of the interstellar medium. In many cases turbulence simulations are applied to the interstellar medium (ISM) merely because it is a medium where extremely high Reynolds numbers are actually realised and the parameters of the ISM are only taken into account as far as they are needed for the turbulence research. Here however we investigated the basic turbulence properties while at the same time we modelled the properties of the ISM as thoroughly as possible. The important point is that there are many physical processes going on in the ISM which should be incorporated in the corresponding simulations. These processes reachfrom external influences of the radiation field originating from hot stars to the internal interaction of the particles culminating in the intricate chemistry of the molecular cloud medium. Each of the different phases of the ISM has its own dominant processes to be taken into account for a realistic modelling ...