<p>Direct numerical simulations of turbulent boundary layers over two types of discontinuous converging and diverging riblets are performed to validate their drag reduction performances and investigate their impacts on turbulence statistics and coherent structures. To suppress the total drag increases in the diverging region of traditional converging and diverging riblets (T-riblets), we design new converging and diverging riblets (N-riblets) with the heights gradually decreasing from the convergence line to the diverging line. Results show that both riblet configurations reduce the skin-friction drag, but the pressure drag is increased. The N-riblets are able to relieve the net drag increase from 7.42% to 0.93%, suggesting their potential in reducing the total drag. Both the discontinuous converging and diverging riblets cause the generation of large-scale secondary flows that originate from the ribbed walls and persist in the downstream wakes over the smooth walls. Their impacts on the time-averaged flow fields and Reynolds stresses are shown to be different between the T- and N-riblets. Moreover, two-point correlations of streamwise velocity fluctuations are calculated to show their modifications on coherent structures, providing more insights to explain the different impacts of T- and N-riblets on the fluid dynamics over the ribbed walls and in the downstream wakes.</p>