以力元理論分析在中低雷諾數流場中撓性平板擺動之推力機制

Abstract

本研究基於力元理論觀點以數值模擬來探討在低雷諾數流場中，以有限長度撓性平板進行上下擺動(heave)模擬魚類尾鰭在BCF泳動模式中的受力機制。藉由調整不同雷諾數Re=200, 500 & 800、史卓荷數St=0.1~0.6與變形幅度a0=0.0, 0.1, 0.15, & 0.2，來觀察平板受力分布。透過力元理論來分析在整個擺動週期中撓性平板的受力機制，由CD - t曲線圖發現推力貢獻分別來自於平板加速變形運動所產生(C_Da)、流場中的渦流(C_Dv)以及物體表面渦度(C_Ds^')。當固定擺動頻率只有上下擺動(a0=0.0)會增加推力並且減少摩擦力，而固定擺動振幅進行擺動會使得兩種作用力會一起增加。在兩種不同的運動條件下隨著變形幅度增加，推力會有明顯成長，然而摩擦力也會有顯著提升。由文獻指出影響尾鰭受力主要來源是尾流，然而從區域體渦度分析可以清楚了解環繞於平板週圍的渦流皆會對結構造成受力變化。在固定擺動頻率的運動情況下，其分析結果觀察到尾流只佔整體推力20~30%，其次推力貢獻為平板上下區域與兩側區域。而固定擺動振幅的運動情況下，其分析結果觀察到尾流仍是佔整體推力約25%，其次推力貢獻為兩側區域與平板前緣區域。此外，受非定常擺動影響的勢流力與平板的表面渦度是無法被忽略力元貢獻。另一方面，在較高的史卓荷數與變形幅度條件下推力會遠大於阻力，但是仍會有所極限。同時也以力元理論仔細分析在不同擺動模式下的推進效率，結果顯示當高於臨界St=0.2，平板會產生正推進效率。

A numerical study is conducted to investigate the force mechanisms for a 3D heaving flexible plate from the perspective of a diagnostic force element analysis. The problem is relevant to a simplified flapping fish-tail with the leading edge held heaving in the space. The flow is assumed to be laminar with the Reynolds numbers fixed at Re=200, 500 & 800, and the Strouhal number St ranging from 0.1 to 0.6, and the flexure amplitude of the plate a0 from 0.1 to 0.2 (dimensionless). As the finite plate is set into unsteady motion, complicated vortex patterns around the plate are generated. It is shown that in the case of a rigid plate (a0=0.0) under a constant heaving frequency (f), the generation of thrust is increased, while friction drag is reduced. As in the case of a rigid plate heaving upon a fixed amplitude (h0), both the thrust and friction drag are enhanced. When the flexibility (a0) of the plate increases from zero, the thrust generation and frictional drag on the plate both increase significantly under the two distinguished cases of having constant heaving frequency (f) and heaving amplitude (h0). In the literature, the force (thrust) exerted on the tail-mimicking plate is largely credited to the vortices (vorticity) in the wake. However, this study performs a regional force analysis to show that the vorticity in the wake region supplies approximately 20-30% of the total thrust, especially in the cases of strong thrust generation under the constant heaving frequency. Comparable contributions come also from the regions direct above and below the heaving plate (mainly including the attached vortices) as well as from the two side regions (mainly including the tip vortices) next to the flexible plate. In the case of fixed h0, 25% of the thrust is contributed from the wake region, and the secondary thrust contribution is obtained from the two side regions of the plate. In addition, the potential motion associated with the unsteady flapping and the contribution from the surface vorticity are non-negligible constituent force components. The net thrust generated is larger at higher St and a0, but this is of course limited by the maximum flexibility and heaving amplitude allowed. The relative importance of each of the various force contributions was analyzed in detail, and the results may shed light on how a flapping tail generates propulsive efficiency above the critical Strouhal number St=0.2.