Imperfect bifurcations in a laminar 3D bluff body wake: effect of pitch and yaw

Abstract

We study the effect of pitch and yaw on the bifurcations of the laminar flow past an Ahmed body, a bluff body of width-to-height ratio $W/H=1.2$ and length-to-height ratio $L/H=3$. When perfectly aligned with incoming flow, its wake first undergoes pitchfork bifurcations leading to a static vertical or horizontal deflection, and becomes oscillatory via a secondary Hopf bifurcation at larger Reynolds number $Re$. As commonly observed for imperfect pitchfork bifurcations, any small misalignment preselects one of the steady states, referred to as the "primary" branch, at low $Re$. At larger $Re$, a disconnected "secondary" branch is born via a saddle-node bifurcation. Here, we investigate the effect of misalignment using three-dimensional computations. We first focus on pure pitch and pure yaw with linear stability analysis. In both cases, the secondary branch becomes unstable as soon as the incidence exceeds a small value. In the pure yaw case, the primary branch becomes unstable via a horizontal Hopf bifurcation, destabilised by yaw. In the pure pitch case, the primary branch sees the crossover of vertical Hopf and horizontal pitchfork bifurcations, destabilised and stabilised by pitch, respectively. This motivates use of DNS to study this nonlinear competition, already observed for aligned Ahmed bodies in ground proximity. We then study the effect of simultaneous yaw and pitch with a weakly nonlinear analysis. The competition between the two stationary modes gives rise to rich bifurcation diagrams, allowing us to predict the number of stable solutions and phase space trajectories between different states