Low-Order Prediction of Unsteady Flow Separation Validated Against Lagrangian Material Spike Formation

We investigate unsteady flow separation using a low-order model that couples potential flow with unsteady integral boundary layer equations. The boundary layer equations form a hyperbolic system, and the emergence of a finite-time singularity corresponds to a spike in displacement thickness, indicating the onset of flow detachment. This formulation enables dynamic prediction of the time and location of boundary-layer separation with minimal computational cost. Onset of flow separation on an impulsively started cylinder and a pitching NACA0012 airfoil is studied for a range of Reynolds numbers (Re = 10k–50k) using this approach. The airfoil is pitched using a ramp type Eldrege function with varying pitch rate. DNS for the same cases are performed and Haller's Lagrangian theory of flow separation is used to validate the predictions of the low-order model. In this theory, a material spike emerges when fluid sheets initially aligned with the wall fold sharply away, marking true Lagrangian separation. We find strong agreement between the formation of a singularity in the low-order model and the emergence of a material spike in the DNS data, both in their timing and location on the airfoil surface. The differences in capturing separation point with the zero-shear-stress principle and that by material-spike-formation are also discussed.