On bifurcations and traction forces on an obstacle in incompressible flow

A systematic numerical investigation of flow-regime transitions in the two-dimensional incompressible Navier-Stokes flow past a confined circular cylinder is presented. For a fixed benchmark geometry, we observe a clear empirical correspondence between qualitative changes in steady traction profiles, understood here as the pointwise force density given by the Cauchy stress tensor on the obstacle boundary, and bifurcations in the long-time behavior of the unsteady Navier-Stokes equations. The observed transitions include onset of time-periodic oscillations, the appearance of multiple steady solutions and loss of effective symmetry. The well-known planar Sch\"afer-Turek benchmark is considered for Reynolds numbers up to 500. Several numerical techniques are employed to compute steady solutions, boundary traction profiles, and linear stability spectra such as duality-based approach for traction evaluation, deflation methods for detecting multiple steady states, and both two- and three- dimensional linear stability analyzes. The results suggest that steady boundary traction profiles can serve as a sensitive diagnostic indicator of critical Reynolds numbers at which qualitative changes in flow dynamics occur. This suggests a computationally inexpensive, complementary approach for detecting flow-regime transitions within this benchmark configuration.