This paper presents a time-optimal path planning algorithm based on edge-search scheme for Autonomous Underwater Vehicles (AUVs) with turbulent and cluttered current fields. The proposed scheme produces paths with a range of continuous headings to overcome the shortcomings of grid-based path planning: discrete motion model. In this method, the AUV maximally utilizes ocean current to save energy based on the velocity analysis. We verified our approach on a set of eddy current fields with randomly size and center position of eddy current, and the performance of resultant paths is compared with 8-connectivity model and continuous motion model, respectively. Simulation results show that the cost of edge-search path is slightly more than that of continuous motion model path, but it takes the same computation time as the 8-connectivity method.