Onset of vortex shedding in flow past Rankine ovals

Preprint, 2025

The Rankine oval is a classical geometry in potential flow, formed by superimposing a uniform stream with velocity U and a source-sink pair separated by distance 2a with strength m, resulting in a closed stagnation streamline whose shape is governed by the dimensionless parameter Ua/m. Although the Rankine body serves as a cornerstone for the classical theory of potential flow, its behavior in viscous flow remains unexplored. The Rankine oval is streamlined in inviscid flow but behaves as a bluff body in viscous flow. The onset of vortex shedding is a critical phenomenon in flows past a bluff body, mapping the transition from steady to periodic wakes. This study systematically investigates the onset of vortex shedding in Rankine oval flows and its associated fluid dynamics by performing direct numerical simulations of incompressible flow past Rankine ovals over Reynolds numbers from 10 to 200 and Ua/m from 0 to 1. The investigation reveals a linear relationship between Ua/m and the critical Reynolds number. This study further characterizes the lift and drag coefficients and Strouhal number, analyzes the vortex formation, and performs a data-driven dimensional analysis. This analysis identifies the dimensionless quantities and empirical formula that determine St and the friction drag coefficient as a function of Re, independent of Ua/m. For sufficiently large Ua/m, the pressure drag can be estimated using potential flow solutions, enabling reliable predictions of the total drag without numerical simulations. These conclusions collectively provide insights into the fluid dynamics of Rankine ovals across diverse flow conditions.