Turbulence characteristics in the spatially developing fractal-generated turbulence are investigated by means of the direct numerical simulation (DNS). Three kinds of fractal grids, namely, cross, I and square types, are numerically constructed using the immersed boundary method. The results show that fractal grids generate high turbulence intensities compared with the classical grid turbulence using a biplane square mesh grid at the same mesh Reynolds number. Among the three types, the fractal square grid is the best to generate nearly homogeneous and isotropic turbulence in the far downstream decaying region. The integral length scale in the fractal generated turbulence remains approximately constant in the decaying region, whereas it grows downstream in the classical grid turbulence. These results are in good agreement with the previous experiments.