<b>The transient vortex structure in the wake of an axial-symmetric projectile launched underwater</b>

This paper provides refined wake simulations for an underwater projectile launch using an improved delayed detached eddy simulation with the energy equation, volume of fluid, and the overlapping grid technique. Additionally, the projectile wake vortex was analyzed for different Froude numbers and dimensionless transverse flow speeds. Verifications of the numerical method, grid independence, vortex identification method, and time step size are presented. Through a systematic comparison of the wake morphologies, the flow fields and vortex structures in the wakes were analyzed in detail, and the wake vortex evolution mechanisms were explored. The results show that the Kelvin-Helmholtz instability was observed, and the wake flow of the projectile launched underwater contains a complex vortical system that directly determines the wake instabilities. The resulting multiple sub-vortex structures are compact and closely arranged near the central axis without the transverse flow effect. However, compared with cases having no transverse flow, the large-scale double spiral vortex structure in the wake with a transverse flow is more difficult to fracture. In addition, the U-shaped vortex in the secondary vortex is also obviously generated in the wake during the double spiral vortex structure evolution. With an increase in Froude number, the vortex legs are gradually apparent and, together with the shedding vortex rings in the wake, form a hairpin vortex structure. With an increase in the dimensionless transverse flow speed, the number of sub-vortex rings derived from the shedding vortex in the wake increases significantly, resulting in a more complex interaction mechanism.