Unsteady two-dimensional jet with flexible flaps at the channel exit

Abstract

The present work studies the effect of passive exit flexibility on a two-dimensional starting jet. The exit flexibility is introduced by attaching two flexible (deformable) flaps at the jet exit of a high aspect ratio rectangular duct with the flaps initially being parallel to the channel walls. A controlled piston motion is used to generate the starting jet, which is composed of a rapid acceleration to a constant velocity ($U_{p}$) that is maintained for a given duration of time, after which it is brought to rest impulsively. The parameters which are varied include the flexural rigidity ($EI$) of the flaps, flap length ($L_{f}$) and piston speed ($U_{p}$), with measurements of the flap kinematics and flow field in each case. The flaps initially bulge due to the acceleration of the piston from rest, with this bulge growing in size and moving downstream as the flow develops, culminating in a large opening at the flap exit. Subsequently, the flaps return to their initial parallel position and remain there as long as the piston is in motion. Once the piston stops, the flaps collapse inwards due to fluid deceleration causing additional flow out of the flap region in the form of a jet that adds to the net amount of fluid pushed by the piston. We find that the flap kinematics is affected by the flap $EI$ and $L_{f}$ besides $U_{p}$. We define a non-dimensional flexural rigidity $EI^{\ast }=EI_{eq}/(1/2\unicode[STIX]{x1D70C}U_{p}^{2}L_{f}^{2}d)$, where $EI_{eq}$ is an equivalent flexural rigidity which takes the self-weight of the flaps into account ($d=\text{channel width}$; $\unicode[STIX]{x1D70C}=$ fluid density). We find that across different $EI_{eq}$, $L_{f}$, and piston speeds, the maximum opening of the flap tip and the time taken to reach this maximum opening in terms of $L/L_{f}$ (where $L=\text{fluid slug length}$) fall on a single curve for all the cases studied, when plotted with $EI^{\ast }$. Particle image velocimetry measurements show that the motion of the flaps results in the formation of additional pairs of vortices when compared to the single vortex pair formed in the absence of flaps. The total final circulation coming out of the flap region remains nearly the same as that of the rigid exit case. However, the final fluid impulse is always found to be higher in the flap cases, with the fluid impulse in most flap cases being approximately two times the fluid impulse of the rigid exit case. This increase in impulse is shown to be linked to the fact that the centroids of vorticity get spread out more in the lateral direction due to the opening of the flaps. The increased impulse and the higher time rate of change of impulse, which is linked with force, suggest that introduction of flexible flaps can help improve thrust performance when looked at from a propulsion point of view.