ANALYSIS OF CARANGIFORM UNDULATION BASED ON VIRTUAL POWER PRINCIPLE
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摘要: 自然界中的大多数鱼类通过波状摆动的方式实现推进, 这是动态变形的鱼体和周围流体相互作用的结果, 研究推进中流体对鱼体变形的响应不仅可以增强对波状推进的认识, 还可以为流动控制提供依据. 以鲹科模式推进的仿生二维模型为研究对象, 通过数值计算获得鱼体波动产生的流场以及鱼体受到的流体力数据. 基于虚功率原理, 将鱼体受力分解为4部分, 分别是鱼体边界变速运动的瞬时贡献, 流场中流体旋转和应变速率相对大小的贡献, 壁面剪切应力的类摩阻分量和壁面摩阻分量. 结果表明, 当鱼体波状摆动产生推力时, 鱼体边界变速运动是主要的正推力来源, 并且该项80%的推力贡献来源于20%的鱼尾部分的边界变速运动. 鱼尾两侧边界层中的流体旋转和应变速率的相对大小和壁面摩阻对推力都是负贡献. 对于低雷诺数的情况, 流体旋转和应变速率的相对大小的负贡献低于壁面摩阻的负贡献, 而在高雷诺数的情况下, 流体旋转和应变速率的相对大小的负贡献强于壁面摩阻的负贡献. 壁面类摩阻分量相对于其他3项总是较小的. 结合标度律分析, 在摆动推进的标度关系中, 与雷诺数无关的推力部分是由边界的变速运动、流场中流体旋转和应变速率共同提供, 且流体旋转和应变速率也贡献了摆动推力中与雷诺数有关的部分, 而这一部分接近类摩阻和摩阻的一半, 同时, 类摩阻和摩阻还提供了常阻力分量.Abstract: Most fish in nature achieve propulsion through undulatory movements, which are the result of the interaction between the deforming fish body and the surrounding fluid. To study the response of the fluid can enhance our understanding of undulatory propulsion and flow control. A two-dimensional deforming airfoil is used to model the carangiform fish. The flow field generated by fish body and the fluid forces acting on the fish body were obtained by using computational fluid dynamics. Using the principle of virtual power, the thrust on the fish body was decomposed into four parts, which are the instantaneous contribution of the boundary acceleration, the contribution of the relative magnitude of fluid rotation and strain rate in the flow field, the wall friction-like component and the wall friction component. The results show that the instantaneous contribution of the boundary acceleration is the main source of positive thrust. The rear 80% of the thrust contribution of this term comes from the instantaneous boundary acceleration movement of the rear 20% of the fish body. The fluid rotation and strain rate in the boundary layer on both sides of the fish tail and the friction contribute to resistance. For low Reynolds number, the negative contribution of the relative magnitude of fluid rotation and strain rate is lower than that of wall friction, while for high Reynolds number, the negative contribution of the relative magnitude of fluid rotation and strain rate is stronger than that of wall friction. However, the wall friction-like component is always smaller compared to the other three terms. In the analysis of the scaling law of undulatory propulsion, it was found that there is a component independent of the Reynolds number which is provided by the first two parts, while the component that is dependent on the Reynolds number is provided by the last three parts. Furthermore, the frictional force and the friction-like force provide constant resistance.
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Key words:
- virtual power principle /
- carangiform /
- undulatory propulsion /
- thrust decomposition
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图 2 一个波动周期不同时刻的鱼体中心线及其包络线
Figure 2. The centerline of the fish body at various time instants for one undulation cycle and their envelops
图 3 推力系数在一个周期内的变化
Figure 3. The time-dependent thrust coefficient in an undulation cycle
图 5
$t = 0T,t = 1/8T,t = 1/4T,t = 3/8T $ 时刻的涡量场(第1列)、Q分布图(第2列)、Q对推力的贡献分布图(第3列)和虚速度势场(第4列)Figure 5. Vorticity field (first column), Q distribution diagram (second column), Q distribution diagram of contribution to thrust (third column) and virtual velocity potential field
$\phi $ (fourth column) at$t = 0T,t = 1/8T,t = 1/4T,t = 3/8T $ 
图 8 不同尾迹区的
$Q$ 分布对推力的贡献Figure 8. The time-dependent contribution of
$Q$ distribution to thrust in different wake regions
图 9 不同雷诺数
$Re$ 条件下$\left\langle {{C_{T,a}}} \right\rangle ,\left\langle {{C_{T,Q}}} \right\rangle ,\left\langle {{C_{T,v}}} \right\rangle 和\left\langle {{C_{T,f}}} \right\rangle$ 随运动参数$St$ 的变化规律Figure 9. The time-averaged thrust coefficient
$\left\langle {{C_{T,a}}} \right\rangle ,\left\langle {{C_{T,Q}}} \right\rangle ,\left\langle {{C_{T,v}}} \right\rangle ,\left\langle {{C_{T,f}}} \right\rangle $ as a function of$St$ at different Reynolds number
图 10 不同雷诺数
$Re$ 条件下$\left\langle {{C_{T,a}}} \right\rangle ,\left\langle {{C_{T,Q}}} \right\rangle ,\left\langle {{C_{T,v}}} \right\rangle 和\left\langle {{C_{T,f}}} \right\rangle$ 的贡献堆叠柱状图Figure 10. Stacked histogram of the time-averaged thrust coefficient
$\left\langle {{C_{T,a}}} \right\rangle ,\left\langle {{C_{T,Q}}} \right\rangle ,\left\langle {{C_{T,v}}} \right\rangle ,\left\langle {{C_{T,f}}} \right\rangle $ at different Reynolds number
图 11 在
$t = 0 T,t = 1/8 T,t = 1/4 T和t = 3/8 T$ 时刻以及不同摆幅条件下($A = 0.06,0.08,0.10,0.12和0.14$ ),$\Delta \phi (x)$ 以及$\Delta \phi (x)/A$ 沿着鱼体中线的分布Figure 11. Distribution of
$\Delta \phi (x)$ and$\Delta \phi (x)/A$ along the center line of fish at different time($ t = 0 T,t = 1/8 T,t = 1/4 T,t = 3/8 T $ ) and under different amplitude condition($A = 0.06,0.08,0.10,0.12,0.14$ )
图 12 惯性力对推力贡献的标度关系及其拟合参数与雷诺数的关系
Figure 12. Scaling law of the contribution of the inertial force to thrust and the relationship between fitting parameters and Reynolds number
图 13 内应力对推力贡献的标度关系极其拟合参数与雷诺数的关系
Figure 13. Scaling law of the contribution of the internal stress to thrust and the relationship between fitting parameters and Reynolds number
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