From thin plates to Ahmed bodies: linear and weakly non-linear stability of rectangular prisms

Abstract

We study the stability of laminar wakes past three-dimensional rectangular prisms. The width-to-height ratio is set to W/H = 1.2, while the length-to-height ratio 1/6 < L/H < 3 covers a wide range of geometries from thin plates to elongated Ahmed bodies. First, global linear stability analysis yields a series of pitchfork and Hopf bifurcations: (i) at lower Reynolds numbers Re, two stationary modes, A and B, become unstable, breaking the top/bottom and left/right planar symmetries, respectively; (ii) at larger Re, two oscillatory modes become unstable and, again, each mode breaks one of the two symmetries. The critical Re of these four modes increase with L/H, qualitatively reproducing the trend of stationary and oscillatory bifurcations in axisymmetric wakes (e.g. thin disk, sphere and bullet-shaped bodies). Next, a weakly non-linear analysis based on the two stationary modes A and B yields coupled amplitude equations. For Ahmed bodies, as Re increases state (A, 0) appears first, followed by state (0, B). While there is a range of bistability of those two states, only (0, B) remains stable at larger Re, similar to the static wake deflection (across the larger base dimension) observed in the turbulent regime. The bifurcation sequence, including bistability and hysteresis, is validated with fully non-linear direct numerical simulations, and is shown to be robust to variations in W and L in the range of common Ahmed bodies.