Pier scour has caused bridge failures with catastrophic consequences. The aim of this study was to develop and verify a mathematical model for pier scour predictions. A new approach is taken that combines shallow-water equations with non-hydrostatic pressure corrections and a non-uniform mesh in the horizontal with terrain-following layers in the vertical. This approach significantly improves computational efficiency from the conventional computational fluid dynamics approach. It is concluded that, emerging from the lateral sides of a pier (cylinder), scour deepens while the patterns migrate upstream towards the pier's upstream nose. On the upstream side, scour continues to grow until the bed slope reaches the angle of repose of sediments. On the downstream side, scour grows until equilibrium is reached. The scour hole is shallower downstream than upstream of the pier. The presence of the pier causes a strong downflow near its upstream nose, a strong vortex at its foot on the upstream side and a weak vortex on the downstream side. The predicted flow velocity and scour depth agree well with measurements. The terrain-following layer feature is particularly useful for scour computations; the high efficiency makes the model practical for field-scale applications, which are highly relevant to improved design of pier foundations and pier scour control.