Turbulent Couette–Poiseuille and Couette flows at different mean strain rates (velocity ratio of Couette wall to bulk flow, –), in a square duct at a bulk Reynolds number 10 000 are investigated by large eddy simulation. The numerical framework consists of a finite-volume method with a staggered-grid arrangement of dependent variables. Spatial derivatives are approximated using second-order centred differencing, and a fractional-step method is used for temporal integration. Simulations are conducted with grids. Secondary flow near the Couette wall shows a gradual change of vortex size and position as the moving wall velocity increased, where the two clockwise rotating vortices gradually merge in tandem with speed of the moving wall and form a large clockwise vortex. A linear relation is observed to exist between the angle of the two vortices and the parameter , and the angle saturates beyond . Also, at , together with a small counter-clockwise corner vortex, this vortex pattern is similar to that observed in the corner region of the duct flow with a free surface. The change of the vortex patterns also influences the dominant transport terms in the streamwise vorticity transport equation. Near the moving wall due to the reduction of the streamwise velocity fluctuation at the moving wall, turbulence structure gradually moves towards a rod-like axisymmetric turbulence, and as increases beyond , turbulence reverts to the disc-like structure.