Global linear and nonlinear stability of viscous confined plane wakes with co-flow

The global stability of confined uniform density wakes is studied numerically, using two-dimensional linear global modes and nonlinear direct numerical simulations. The wake inflow velocity is varied between different amounts of co-flow (base bleed). In accordance with previous studies, we find that the frequencies of both the most unstable linear and the saturated nonlinear global mode increase with confinement. For wake Reynolds number Re = 100 we find the confinement to be stabilising, decreasing the growth rate of the linear and the saturation amplitude of the nonlinear modes. The dampening effect is connected to the streamwise development of the base flow, and decreases for more parallel flows at higher Re. The linear analysis reveals that the critical wake velocities are almost identical for unconfined and confined wakes at Re ≈ 400. Further, the results are compared with literature data for an inviscid parallel wake. The confined wake is found to be more stable than its inviscid counterpart, whereas the unconfined wake is more unstable than the inviscid wake. The main reason for both is the base flow development. A detailed comparison of the linear and nonlinear results reveals that the most unstable linear global mode gives in all cases an excellent prediction of the initial nonlinear behaviour and therefore the stability boundary. However, the nonlinear saturated state is different, mainly for higher Re. For Re = 100, the saturated frequency differs less than 5% from the linear frequency, and trends regarding confinement observed in the linear analysis are confirmed.

Impedance control balances stability with metabolically costly muscle activation.

Humans are able to stabilize their movements in environments with unstable dynamics by selectively modifying arm impedance independently of force and torque. We further investigated adaptation to unstable dynamics to determine whether the CNS maintains a constant overall level of stability as the instability of the environmental dynamics is varied. Subjects performed reaching movements in unstable force fields of varying strength, generated by a robotic manipulator. Although the force fields disrupted the initial movements, subjects were able to adapt to the novel dynamics and learned to produce straight trajectories. After adaptation, the endpoint stiffness of the arm was measured at the midpoint of the movement. The stiffness had been selectively modified in the direction of the instability. The stiffness in the stable direction was relatively unchanged from that measured during movements in a null force field prior to exposure to the unstable force field. This impedance modification was achieved without changes in force and torque. The overall stiffness of the arm and environment in the direction of instability was adapted to the force field strength such that it remained equivalent to that of the null force field. This suggests that the CNS attempts both to maintain a minimum level of stability and minimize energy expenditure.