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†A new finite volume method for the incompre ssible Navier -Stokes equations, expressed in arbitrary Lagrangian -Eulerian (ALE) form, is used to simulate the forced -motion hydrodynamics of a single isolated flapping foil propulsor undergoing simple harmonic motion with two rotational degrees -of -freedom . The flapping foil is used for both primary propulsion and maneuvering control on a biomimetic flapping foil autonomous underwater vehicle (BFFAUV) employing four identical flapping foils in a sea turtle configuration. Simulation results are first present ed for two rudimentary problems with exact solutions to validate the ALE algorithm for both viscous and invsicid flows with moving boundaries. A mesh movement algorithm based on linear elastostatics is then described and shown to perform well for a benchma rk mesh movement problem. The Euler equations are then used to compute the inviscid hydrodynamic forces acting on a single isolated flapping foil with a uniform forward velocity. The computed mean and instantaneous thrust are in good agreement with availab le experimental data for the single set of kinematics investigated. Finally, a definition of the hydrodynamic mean propulsive efficiency of the flapping foil is presented and evaluated for the BFFAUV propulsor. ver the last decade or so, one class of pro blems in computational fluid dynamics that has undergone substantial development is the class where the fluid domain boundary is either explicitly time -dependent or is unknown a priori and determined, in a coupled fashion, as part of an unsteady flow solut ion. Free -surface, fluid -structure interaction, and forced -motion flows are typical of problems in this class. A natural way to formulate moving boundary problems is the so -called arbitrary Lagrangian -Eulerian (ALE) form of the fundamental conservation law s where the domain boundary and interior control surfaces are allowed to move arbitrarily in time and which recovers the Eulerian and Lagrangian forms as special limiting cases of the general ALE form. Here, a new finite volume method for the incompressibl e Euler and Navier -Stokes equations, expressed in ALE form, is used to simulate the forced -motion hydrodynamics of a single isolated flapping foil propulsor undergoing simple harmonic motion with two rotational degrees -of -freedom. The flapping foil is used for both primary propulsion and maneuvering control on a biomimetic flapping foil autonomous underwater vehicle (BFFAUV) employing four identical flapping foils in a sea turtle configuration 1 . Recently the method was used to simulate the inviscid forced -motion hydrodynamics of the robotic fish RoboTuna 2 . Results are first presented for two rudimentary flow problems with exact solutions to establish the accuracy of the method. A mesh movement algorithm based on a modified form of the equilibrium equation go verning classical linear elasticity is then described and shown to perform well for a benchmark grid movement problem as well as for the BFFAUV propulsor. The computed force and power time histories for a single flapping foil are then presented and compare d with available experimental data 3 . Finally, a definition of the hydrodynamic mean propulsive efficiency of the flapping foil is presented and evaluated for single set of kinematic parameters that are typical of the BFFAUV in uniform forward motion.