This integrated thesis documents a series of complementary numerical investigations aimed at an improved understanding of turbulent flows and heat transfer in a square duct with ribs of different shapes mounted on one wall. Direct numerical simulation (DNS) is used to accurately resolve the spatial and temporal scales of the simulated flows. The first DNS investigates the turbulent flow in a ribbed square duct of different blockage ratios. The results are compared with those of a smooth duct flow. It is observed that an augmentation of the blockage ratio concurrently generates stronger turbulent secondary flow motions, which drastically alter the turbulent transport processes between the sidewall and duct center, giving rise to high-degrees of non-equilibrium states. The dynamics of coherent structures are studied by examining characteristics of the instantaneous velocity field, swirling strength, spatial two-point auto-correlations, and velocity spectra. The impact of the blockage ratio on the turbulent heat transfer is investigated in the second numerical study. The results show that owing to the existence of the ribs and confinement of the duct, organized secondary flows appear as large streamwise-elongated vortices, which have profound influences on the transport of momentum and thermal energy. This study also shows that the spatial distribution and magnitude of the drag and heat transfer coefficients are highly sensitive to the rib height. The final study focuses on a comparison of highly-disturbed turbulent flows in a square duct with inclined and V-shaped ribs mounted on one wall. The turbulence field is highly sensitive to not only the rib geometry but also the boundary layers developed over the side and top walls. Owing to the difference in the pattern of the cross-stream secondary flow motions of these two ribbed duct cases, the flow physics in the inclined rib case is significantly different from the V-shaped rib case. It is found that near the leeward and windward faces of the ribs, the mean inclination angle of turbulence structures in the V-shaped rib case is greater than that of the inclined rib case, which subsequently enhances momentum transport between the ribbed bottom wall and the smooth top wall.