Effects of tunable stiffness on the hydrodynamics and flow features of a passive pitching panel

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Abstract

Three-dimensional numerical simulations are carried out to study the hydrodynamic performance and flow features of a bio-inspired underwater propulsor. The propulsor is constituted by a passive pitching panel. The leading edge of the panel is prescribed under a periodic heaving motion while the panel pitches passively due to the employing of a stiffness-lumped torsional spring at the leading edge. Effects of the torsional spring stiffness have been put emphases on. A real-time tunable stiffness strategy is presented and employed in the study. We first study the passive pitching effects on the hydrodynamics and flow features of the panel using a series of constant stiffness and then we study the tunable stiffness effects using cosinusoidal curve based waveforms, in which the effects of phase difference (φ) between the stiffness profile and the oscillation motion and as well as the effects of stiffness fluctuation amplitude (Gk) are investigated, respectively. The stiffness profile beneficial for propulsion efficiency is further applied to cases with different oscillation amplitudes. A high-fidelity immersed boundary method based direct numerical simulation (DNS) solver is employed to acquire the fluid dynamics and to simulate the flow. The panel passive pitching motion is solved by coupling the DNS flow solver and the Euler rigid body dynamic equation. Results show that for the constant stiffness cases, the panel generates sinusoidal-like pitching motion, and in certain stiffness range, flexibility could benefit efficiency while holding some extent of stiffness could enhance the thrust. For the tunable stiffness cases, it is found that both the mean thrust and propulsive efficiency improved when the stiffness change is in-phase with the heaving motion (φ=0°). The largest mean thrust is found at φ=120°. The wake profile shows that in the constant stiffness cases and φ=120° case, the panel in each cycle generates a pair of elongated and twisted vortex tubes, the vortex tubes in each pair interconnected with each other and induces unprofitable interactions. While in the φ=0° case the panel generates a pair of round and closed vortex loops in each cycle and the vortex loops separated directly after they have been detached from the panel and thus avoided the unprofitable vortex interactions. The stiffness fluctuation amplitude (Gk) effects study (all employing the in-phase stiffness profile) shows that all the three tested cases (Gk=0.25G0,0.5G0,0.75G0) acquired thrust and efficiency enhancement while the Gk=0.5G0 case acquired the largest efficiency benefit and the Gk=0.75G0 case had the largest thrust. Wake profiles show entire vortex loops may not be formed when Gk is small while larger Gk may affect the arrangement of the vortex loops thus may generate unprofitable vortex interactions. The results of cases with motions with variable oscillating amplitudes (A) show employing the real-time altering stiffness profile (φ=0°,Gk=0.5G0) improves the propulsion performance in a certain range of A (0.4A0.8). Results from this paper demonstrated the potential of using real-time tunable stiffness in the design of passive pitching propulsors of underwater vehicles that pursue higher performance.

Introduction

Fishes possess remarkable swimming performance such as high swimming speed, high efficiency and enduring swimming ability. Most of these fishes are found to swim by bending their aft-body and pitching their caudal fin, which is thought to be the major way of providing thrust. Despite the active bending or pitching which is actuated by muscle, passive morphing or pitching mechanism is widely found in fish swimming and insect flight in nature (Bergou et al., 2007, Daniel and Combes, 2002, Fish et al., 2006). Previous research has shown that these passive morphing or rotation may be beneficial for the sake of improving efficiency, however, in the case of a robot that uses rigid panel as a propulsor, in which the flexibility is lumped at the joint or leading edge of the panel, employing completely passive pitching motion will decrease mechanical complexity and reduce system mass (Wood, 2008). On the other hand, evidence in nature shows that stiffness of propulsors can be actively controlled for pursuing better performance (Adams and Fish, 2019). Whereas, in recent researches, the passive rotating propulsors are all employing constant stiffness. The research about passive pitching propulsor which employs real-time tunable stiffness is scarce. The main goal of this paper is to study the tunable stiffness effects on a bio-inspired passive pitching propulsor.

The stiffness-lumped passive pitching mechanism has been studied by some researchers in robot design or fluid mechanism studies. These studies are related to both insect flying and fish swimming. Ishihara studied the lift generation in the crane fly’s flapping wing motions using a dynamically scaled model that employs a plate spring on the wing joint. High angle of attack was maintained passively during the flapping motion and sufficient lift was generated to support the insect weight (Ishihara et al., 2009). Whitney and Wood used the theoretical method to study the aeromechanics of passive rotation in flapping flight in which the aerodynamics is predicted by a blade-element model and the passive rotation is acquired by coupling the blade-element model and rigid body dynamics equation (Whitney and Wood, 2010). Zhu et al. (2019) used two joints to mimic the lateral bending in the tuna fish swimming in his design of a tuna-like robot which achieved a maximum swimming speed of 4.0 body lengths (BL) per second. Note that most of the documented fish-like robots can only achieve body velocities of 0.251.5 BL/s. Zhu uses one joint to connect the actuation system and posterior support structure and employs another joint to connect the posterior support structure and the caudal fin. Elastic bands were attached to link the posterior support structure and the caudal fin, so the posterior joint could rotate passively due to the interaction between the elastic bands, fluid pressure, and structural inertia. The research mainly talked about the capability of high-frequency swimming and swimming performance such as midline kinematics, speed and cost of transport (COT), etc. Fluid dynamics and flow features about this kind of stiffness-lumped flapping panels have been studied by some researchers. Zhong et al. (2019) studied the interaction between dorsal and caudal fin using a tuna-inspired platform which also employs this kind of passive pitching mechanism in the peduncle part of the fish just like Zhu et al. (2019) did in the tuna robot. Some other researchers using this kind of passive pitching model to study the aero/hydrodynamics and flow features about flapping wing or panels can be found in (Moore, 2014, Wang et al., 2018, Zeyghami et al., 2018, Zhong et al., 2017).

However, as mentioned above, the previous studies all employ constant stiffness. We here hypothesize that particularly controlling the stiffness in this kind of stiffness-lumped passive pitching panel may improve the thrust or efficiency. We first employ a series of constant stiffness to study the constant stiffness effects on the propulsion performance of the panel, then we study the real-time tunable stiffness effects using sinusoidal curve based waveforms. Effects of the phase difference between the stiffness profile and the oscillation motion and as well as the effects of stiffness fluctuation amplitude (Gk) are investigated. A high-fidelity immersed boundary method based direct numerical simulation (DNS) solver is employed to simulate the flow and hydrodynamics and the pitching motion is solved by coupling the flow solver and the Euler rigid body dynamic equation.

Section snippets

Problem definition

The current research employs a bio-inspired trapezoidal panel. The aspect ratio (AR) of the panel is defined as the ratio of the square of the longer base of the panel to the area of the panel (L2S, where L is the longer base, and S is the area) and is set to be 2.025 in the current work. A virtual torsional spring is employed at the leading edge to connect the actuation motion and the panel. The leading edge undergoes a prescribed harmonic oscillating motion which is described by Eq. (1) xt=Ax

Kinematics and hydrodynamics

In this section, a series of constant stiffness tests are carried out to study the stiffness (constant) effects on the propulsion performance. In order to study the effects of the stiffness specifically, the reduced frequency, Reynolds number, Mass number and oscillation amplitude are kept the same (k=1, Re = 500, M = 0.1, A=0.4), while the Cauchy numbers are set as the following values: Ch = 0.089, 0.133, 0.178, 0.222, 0.267, 0.311, 0.356, 0.4, 0.445, 0.051.

Table 1 gives the cycle-averaged

Conclusion

In the present study, three-dimensional numerical simulations are carried out to study the propulsion performance and flow features of a torsional-spring-derived passive pitching panel. The torsional spring stiffness effects have been put emphases on and a real-time altering stiffness idea is pointed out and employed in the study.

The research found that employing constant stiffness may let the panel generate sinusoidal-form-like pitch motions in spite of that the pitching amplitudes and phase

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported in part by the Office of Naval Research MURI Grant N000141410533 and NSF CNS-1931929 to HD, and China scholarship council and National Natural Science Foundation of China (11972060 & 1721202) to HH and YW.

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