Numerical simulations provide us the capability to test hypothetical cases, which are quite difficult to test experimentally with live fish, e.g., we cannot ask a fish to swim with a desired kinematics. We have developed computational tools to simulate self-propelled swimming under realistic Reynolds numbers using large-eddy simulations with a sharp-interface immersed boundary method in a general non-inertial frame of reference. Using our simulations, we tested the effect of tail geometry on fish propulsion under similar conditions. We discovered that at low Strouhal number (St~0.25) a stable leading-edge vortex (LEV) is generated on all the fish tails, but at high St~0.6 the LEV becomes detached . In addition, our simulations of a stingray showed that at low St ~0.27 the LEV on the pectoral disk of swimming stingrays generates a low-pressure region, whereas at high St~0.34 the low- and high-pressure regions are in the back half of the wing and not close to any vortical structures . The undulatory motion generates thrust by the added mass mechanism (accelerating the adjacent fluid), while the LEV enhances thrust by increasing the circulation around the fin. We discuss the difficulties in disentangling the contribution of the LEV and the added mass to total thrust in aquatic swimming, but we show that at low St LEV contributes more to thrust than high St. The ubiquity of the LEV on fish, insect, bat, bird fins/wings suggest that it might be a convergent evolutionary feature in biological propulsion systems, which should serve as an inspiration to enhance force generation/energy harvesting in engineering applications.