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.

References in corpus (3)

• Experimental investigation of flow behind a cube for moderate Reynolds numbers
• The chaotic dynamics of large-scale structures in a turbulent wake
• Bifurcation dynamics of a particle-encapsulating droplet in shear flow

Cited by in corpus (2)

• Stability and dynamics of the laminar flow past rectangular prisms
• Wake transition and aerodynamics of a dragonfly-inspired airfoil