# Propulsion Performance of the Full-Active Flapping Foil in Time-Varying Freestream
## 2 Problem Description and Methodology  
### Problem Description  
Referring to [PERSON] [26] and [PERSON] [27], we note that the shear flow is mainly affected by the seafloor topography, while the marine vehicles propelled by flapping foil are mostly used in the vast ocean, the influence of the terrain on the flow field can be ignored. Therefore, we only discuss the effect of time-varying flow velocity on the propulsion performance of active flapping wings in this paper. The sketch of full-active flapping foil under time-varying freestream velocity is illustrated in Figure 1. The foil has two degrees of freedom, which can only translate in the vertical direction and rotate around the pivot, other degrees of freedom are strictly prohibited. The heaving and pitching motions of the flapping foil are fully prescribed and defined as a sinusoidal motion. A two-dimensional NACA0012 foil is used in this paper, \(c\) and \(d\) are defined as the chord length and the thickness of the foil, the foil's pivot is located at one third of chord.  
The heaving and pitching motion of the full-active flapping foil are defined as:  
\[y(t)=y_{0}\sin(2\pi ft) \tag{1}\]  
\[\theta(t)=\theta_{0}\sin(2\pi ft+\Phi) \tag{2}\]  
Figure 1: Sketch of full-active flapping foil under time-varying freestream velocity.  
where \(y_{0}\) and \(\theta_{0}\) are defined as the heaving and pitching amplitudes, respectively. \(f\) is the motion frequency, \(f^{*}\) is the non-dimensional coefficient of motion frequency, named reduced frequency, \(f^{*}=fc/U_{\infty}\). \(\Phi\) is the phase different between the heaving and pitching motions, \(\Phi=\pi/2\). Referring to the assumptions of time-varying flow velocity made by [PERSON] [34] and [PERSON] [35], the time-varying freestream in this paper is considered as a superposition of steady flow and sinusoidal pulsating flow. Therefore, the time-varying flow \(U(t)\) is defined as:  
\[U(t)=U_{\infty}+\gamma\sin(2\pi f_{\circ}t)U_{\infty} \tag{3}\]  
where \(U_{\infty}\) is the constant term of flow, \(\gamma\sin(2\pi f_{\circ}t)U_{\infty}\) is the pulsating term of flow. \(\gamma\) and \(f_{\circ}\) are defined as the oscillating amplitude and oscillating frequency of the time-varying flow. \(\tau\) is defined as the oscillating frequency coefficient of the time-varying flow, \(f_{\circ}=2\tau f\). The instantaneous thrust coefficient, instantaneous lift coefficient and instantaneous moment coefficient of full-active flapping foil are expressed as:  
\[C_{X}(t)=F_{X}(t)/\frac{1}{2}\rho U^{2}c \tag{4}\]  
\[C_{Y}(t)=F_{Y}(t)/\frac{1}{2}\rho U^{2}c \tag{5}\]  
\[C_{M}(t)=M(t)/\frac{1}{2}\rho U^{2}c^{2} \tag{6}\]  
where \(F_{X}\), \(F_{Y}\) and \(M\) are defined as the thrust, lift and moment. \(\rho\) is the fluid density. The instantaneous input power \(P_{I}(t)\) and propulsion power \(P_{I}(t)\) are defined as:  
\[P_{I}(t)=P_{Y}(t)+P_{\theta}(t)=F_{Y}(t)V_{y}(t)+M(t)\Omega(t) \tag{7}\]  
\[P_{O}(t)=F_{X}(t)U(t) \tag{8}\]  
where \(V_{y}\) is the heaving velocity, \(\Omega\) is the pitching velocity. The mean input power coefficient and mean propulsion power coefficient are expressed as:  
\[\overline{C_{PI}}=\frac{1}{\mathrm{T}}\int_{0}^{\mathrm{T}}{(P_{I}(t)/\frac{ 1}{2}\rho U^{3}c)dt} \tag{9}\]  
\[\overline{C_{PO}}=\frac{1}{\mathrm{T}}\int_{0}^{\mathrm{T}}{(P_{O}(t)/\frac{ 1}{2}\rho U^{3}c)dt} \tag{10}\]  
where \(T\) is defined as the period of the heaving and pitching motion of the flapping foil, \(T=1/f\). The propulsion efficiency of the full-active flapping foil is defined as:  
\[\eta=\overline{C_{PO}}/\overline{C_{PI}} \tag{11}\]