Simulating star formation in molecular cores II. The effects of different levels of turbulence

(Abridged) We explore, by means of a large ensemble of SPH simulations, how the level of turbulence affects the collapse and fragmentation of a star-forming core. All our simulated cores have the same, except that we vary (a) the initial level of turbulence (as measured by the ratio of turbulent to gravitational energy, $\alpha_{\rm turb} \equiv U_{\rm turb}/|\Omega| = 0, 0.01, 0.025, 0.05, 0.10 {\rm and} 0.25$)and(b),forfixed$\alpha_{\rm turb}$, the details of the initial turbulent velocity field (so as to obtain good statistics). A low level of turbulence (${\alpha }_{\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{b}}\sim 0.05$) suffices to produce multiple systems. As ${\alpha }_{\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{b}}$ is increased, the number of objects formed and the companion frequency both increase. The mass function is bimodal, with a flat low-mass segment representing single objects ejected from the core before they can accrete much, and a Gaussian high-mass segment representing objects which because they remain in the core grow by accretion and tend to pair up in multiple systems.