U steady hydrodynamic propulsion has long been studied as an alternative to conventional propulsion types with the allure that it brings capabilities for higher speed, better economy, and wider versatility [1, 2]. These propulsors generate circulation (lift-based) forces and fluid acceleration (added mass-based) forces. Typically these systems are simplified to periodically heaving and pitching foils in a steady free stream velocity. Using these simplifications, experimentally validated analytical models [3, 4] can accurately predict the propulsive performance of a foil based upon input kinematics and flow conditions. One important observation these models have given us is the influence of the foil quasi-steady drag. This drag acts as an offset to the thrust produced by the foil unsteadiness, and dictates the peak efficiency of the propulsor [5]. Our expectation is that the foil shape will strongly influence this steady drag, which can be used to improve performance.

A wide array of specialized foil shapes exist in nature and application, as shown in figure 1. The foil shape influences lift/drag, weight, radar reflection, shock location, turbulence transition, and acoustic noise. Thus, optimizing foil shape, given an objective, has become absolutely critical to industry [6]. Here, we are particularly interested in how the foil shape influences parameters like lift and drag, something that has been widely documented since the onset of aerodynamics [7]. Often, optimizations are complex and have multiple objectives. For example, when designing a wing that minimizes radar reflection it must also maintain reasonable flight performance [6]. In the simplified framework of unsteady propulsion, we adopt a single objective: to maximize efficiency which we expect to be synonymous with minimizing foil steady drag.