Dynamic interactions between a self-propelled flexible plate and multiple wall-mounted compliant blades

A numerical study is conducted on a self-propelled flexible plate moving above an array of wall-mounted compliant blades at Reynolds numbers of order 100, designed to mimic flatfish-like locomotion near the vegetation-covered seabed. Compared with a flat rigid wall located at the same vertical distance from the plate, the blade-covered wall enhances the propulsive performance of the plate, yielding a higher propulsive velocity and lower energy consumption per transport distance. This enhancement intensifies as the blade spacing and vertical distance between the plate and the bottom wall decrease, but diminishes when the flapping frequency approaches the natural frequency of the blades due to resonance. By accounting for the asymmetric motion of the plate during the upper and lower half-periods, a universal scaling law italic Re Subscript c Baseline tilde left parenthesis italic Re Subscript f comma delta Baseline right parenthesis Superscript 3 divided by 2 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Re</mml:mtext> </mml:mrow> </mml:mrow> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>∼</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Re</mml:mtext> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:mi>δ</mml:mi> </mml:mrow> </mml:msub> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mn>3</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> ${\textit{Re}}_c\sim ({\textit{Re}}_{f,\delta })^{3/2}$ is identified between the propulsive Reynolds number italic Re Subscript c <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Re</mml:mtext> </mml:mrow> </mml:mrow> <mml:mi>c</mml:mi> </mml:msub> </mml:math> ${\textit{Re}}_c$ and the corrected flapping Reynolds number italic Re Subscript f comma delta <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Re</mml:mtext> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:mi>δ</mml:mi> </mml:mrow> </mml:msub> </mml:math> ${\textit{Re}}_{f,\delta }$ . Under the excitation of the flapping plate, two distinct vibration patterns are identified for the blade array. The first is the flapping-driven mode, which propagates synchronously with the plate’s motion. The second exhibits forward-propagating travelling waves, which are generated by the cooperative underdamped vibrations of downstream blades, and have a phase velocity much faster than the propulsive velocity of the plate. This study may shed light on the design of bio-inspired underwater vehicles and flow sensing techniques.