A finite element numerical model was established to simulate the local scour of a submarine pipeline under steady flow to obtain the bed morphology at different times during the local scour development process. Based on the unsteady Reynolds-average N-S equations and the k-ε standard turbulence model, the Exner equation was solved to simulate the change of seabed. A dynamic mesh technique was used to show the change process of seabed. The equations were discretized by the characteristics method in time and by finite element method in space. The model was verified by the flow around a two-dimensional square bluff body and Mao’s clear water scour test. The results show that the simulation results are in good agreement with the test data, and the model can be used to simulate the local scour of submarine pipelines.