Two-way lock-in and scaling relations for a heaving compliant membrane in a cylinder wake

We numerically investigate the propulsion of a two-dimensional compliant membrane executing prescribed harmonic heave in the near wake of a stationary circular cylinder at Reynolds number italic Re equals 3000 Re = 3000 $\textit{Re}=3000$ . Using a partitioned high-fidelity fluid–structure interaction solver with nonlinear iterative force correction, we map the coupled response over upper A Superscript asterisk Baseline element of left bracket 0.05 comma 0.5 right bracket A ∗ ∈ [ 0.05 , 0.5 ] $A^* \in [0.05,0.5]$ and f Superscript asterisk Baseline element of left bracket 0.1 comma 0.6 right bracket f ∗ ∈ [ 0.1 , 0.6 ] $f^* \in [0.1,0.6]$ , where upper A Superscript asterisk A ∗ $A^*$ is dimensionless flapping amplitude and f Superscript asterisk f ∗ $f^*$ is dimensionless flapping frequency. The parameter sweep reveals a sharp transition in the force maps from weakly forced, wake-following behaviour to a high-performance regime in which cycle-averaged lift and lift-to-drag increase abruptly. This transition coincides with intermittent exposure of the membrane to higher-momentum fluid and a pronounced amplification of deformation. A frequency-resolved analysis, combined with a body-frame Fourier mode decomposition of the flow, identifies four distinct flapping states, namely a wake-dominated state, two flapping-dominated states associated with lift reduction and lift gain and a two-way lock-in state. In the latter, vortex shedding locks onto the imposed actuation, while the first fluid-loaded structural mode approaches the actuation frequency, producing selective amplification of low-order deformation. The resulting curvature-induced camber intensifies leading-edge suction and increases the cross-membrane pressure difference, establishing feedback that reorganises upstream shedding. Motivated by these mechanisms, we derive scaling relations for the cycle-averaged lift, drag and power that separate quasi-steady motion, added-mass effects, curvature-induced contributions, wake-momentum deficit and transverse shear. For the present reference compliance, the scaling clarifies how flexibility is detrimental when the membrane remains fully immersed within the wake core, yet beneficial when intermittent wake exposure permits passive camber amplification to offset momentum deficit with limited drag penalty. These results provide a mechanistic framework for the present reference-compliance configuration and identify how wake exposure, imposed heaving and membrane response combine to produce two-way lock-in.