This paper presents a design method for target enclosing of an underactuated autonomous surface vehicle to surround a maneuvering target with time-varying velocity, and both the target and the follower suffer from unknown ocean currents. Specifically, a target-enclosing controller is developed based on a line-of-sight (LOS) guidance and an ocean current estimator, where only the yaw rate is used to stabilize the circular motion around the target. Then, the proposed strategy is extended to the case where the target velocity is unavailable and the ocean currents time-varying. By using the relative range and angle information, an extended state observer is developed to estimate unknown relative velocities together with the ocean currents in real-time. Based on the estimated relative dynamics, a LOS, target-enclosing controller is developed without any velocity information of the target. The stability of the closed-loop system is analyzed via the Lyapunov theory. Both simulations and experiments are conducted to show the performance of the proposed LOS, target-enclosing controller for surrounding a maneuvering target.