Homogeneous shear flows with an imposed mean velocityU=Syxˆare studied in a period box of sizeLx×Ly×Lz, in the statistically stationary turbulent state. In contrast with unbounded shear flows, the finite size of the system constrains the large‐scale dynamics. The Reynolds number, defined by Re≡SL2y/&ngr; varies in the range 2600⩽Re⩽11300. The total kinetic energy and enstrophy in the volume of numerical integration have large peaks, resulting in fluctuations of kinetic energy of order 30%–50%. The mechanism leading to these fluctuations is very reminiscent of the ‘‘streaks’’ responsible for the violent bursts observed in turbulent boundary layers. The large scale anisotropy of the flow, characterized by the two‐point correlation tensor ⟨uiuj⟩ depends on the aspect ratio of the system. The probability distribution functions (PDF) of the components of the velocity are found to be close to Gaussian. The physics of the Reynolds stress tensor,uv, is very similar to what is found experimentally in wall bounded shear flows. The study of the two‐point correlation tensor of the vorticity ⟨&ohgr;i&ohgr;j⟩ suggests that the small scales become isotropic when the Reynolds number increases, as observed in high Reynolds number turbulent boundary layers. However, the skewness of thezcomponent of vorticity is independent of the Reynolds number in this range, suggesting that some small scale anisotropy remains even at very high Reynolds numbers. An analogy is drawn with the problem of turbulent mixing, where a similar anisotropy is observed. ©1996 American Institute of Physics.