## 5 Thrust and Efficiency of a Wing Performing Two Harmonic Motions
### Pitching Oscillations of a Rectangular Wing in Extreme Ground Effect  
Consider the case of rotational oscillations of the rectangular wing near a wall \(y_{w}=h+\theta_{0}(x-a_{0})\cos kt\), where \(a_{0}\)- the abscissa of the rotation center, measured from the trailing edge. Thus, in this case we have \(q_{i}(t)=\theta(t),\hat{q}_{i}(t)=\dot{\theta}(t),f_{q_{i}}(x)=x-a_{0}\). To avoid cumbersome expressions we restrict ourselves to presenting the plots of the derivatives of the lift coefficient with respect to kinematic parameters in Figures 12 and 13.  
Figure 11: Ideal efficiency of a rectangular wing heaving in extreme ground effect (\(h\to 0\)) and out of ground effect [33] for Strouhal numbers tending to zero.  
Figure 10: Ideal efficiency of a flat rectangular wing of different aspect ratio \(\lambda\) in heaving oscillations in extreme ground effect versus Strouhal number. Dashed is the efficiency curve for a heaving foil out of ground effect (Note that the efficiency of heaving foil out of ground effect for a flat plate shown in Figure 10 by dashed line was evaluated with use of classical formula \(\eta_{i}=[F^{2}(k/2]+G^{2}(k/2)/F(k/2)\), where \(F\) and \(G\) are real and imaginary parts of Theodorsen function [1]).  
The period-averaged thrust coefficient for rotational oscillations is given by the formula  
\[\langle C_{t}\rangle=\frac{1}{2}\theta_{0}^{2}\bigg{(}C_{s}^{\theta\theta}+k^{2} C_{s}^{\dot{\theta}\dot{\theta}}-C_{y}^{\theta}\bigg{)}; \tag{109}\]  
where  
\[C_{s}^{\theta\theta}=\frac{1}{2h}\sum_{n=0}^{\infty}\bigg{(}\frac{\zeta_{n}a_{1 n}}{\cosh\xi_{n}}+b_{1n}\zeta_{n}\text{tanh}\zeta_{n}\bigg{)}^{2}, \tag{110}\]  
\[C_{s}^{\dot{\theta}\dot{\theta}}=\frac{1}{2 hk^{2}}\sum_{n=0}^{\infty}\bigg{(} \frac{\zeta_{n}a_{2n}}{\cosh\xi_{n}}+b_{2n}\zeta_{n}\text{tanh}\zeta_{n}-b_{1n }k\bigg{)}^{2}, \tag{111}\]  
The coefficients \(a_{1n},a_{2n},b_{1n}\) and \(b_{2n}\) can be determined from the formulas below.  
\[a_{1n}=\frac{w_{1n}-w_{2n}\text{tanh}\zeta_{n}/\zeta_{n}}{1+k^{2}\text{tanh}^ {2}\zeta_{n}/\zeta_{n}^{2}},a_{2n}=\frac{w_{2n}+w_{1n}\text{tanh}\zeta_{n}/ \zeta_{n}}{1+k^{2}\text{tanh}^{2}\zeta_{n}/\zeta_{n}^{2}},b_{1n}=\frac{4(-1)^ {n}}{\zeta_{n}^{3}}, \tag{112}\]  
\[b_{2n}=\frac{b_{1n}k(1+a_{0})}{2};w_{1n}=\frac{k^{2}b_{1n}}{2\zeta_{n}}\bigg{(} \frac{1+a_{0}}{\cosh\zeta_{n}}-1+a_{0}\bigg{)},w_{2n}=\frac{kb_{1n}}{\zeta_{n} }\bigg{(}1+\text{tanh}\zeta_{n}\text{tanh}\frac{\zeta_{n}}{2}\bigg{)}. \tag{113}\]  
Figure 12: Derivative \(hc_{y}^{\theta}\) for a rectangular wing in extreme ground effect versus aspect ratio for \(a_{0}=0.5\) and different Strouhal numbers.  
Figure 13: Derivative \(hc_{y}^{\theta}\) for a rectangular wing in extreme ground effect versus aspect ratio for \(a_{0}=0.5\) and different Strouhal numbers.  
Plots of averaged thrust coefficient versus Strouhal number for different aspect ratios are presented below.  
In the limiting case of pitch oscillations of the infinite aspect-ratio wing in extreme ground effect, we obtain  
\[C_{s}^{\theta\theta}=\frac{\left[2+k^{2}(a_{0}+2/3)\right]^{2}}{4h(1+k^{2})^{2}} \mathcal{C}_{s}^{\hat{\theta}\hat{\theta}}=\frac{\left[2(a_{0}-1)+k^{2}(a_{0}- 2/3)\right]^{2}}{4h(1+k^{2})^{2}}, \tag{114}\]  
\[C_{y}^{\theta}=\frac{12+5k^{2}+k^{4}}{12h(1+k^{2})}+k^{2}a_{0}\frac{2-k^{2}}{6h (1+k^{2})},\mathcal{C}_{y}^{\hat{\theta}}=\frac{(1-a_{0})(2+k^{2})}{4h(1+k^{2} )}-\frac{2}{3h(1+k^{2})}. \tag{115}\]