## 3 Lift, Moment and Thrust Coefficients of a Lifting Surface in Extreme Ground Effect  
Once the problem described by the Equations (1)-(3) or, in the case of harmonic oscillations by (9)-(11) is solved, the lift coefficient \(C_{y}\) and the moment coefficient \(m_{z}\) can be obtained with use of the formulas  
\[C_{y}(t)=\frac{\lambda}{l^{2}}\iint\limits_{S_{w}}[p]dS_{w}=\frac{\lambda}{l^{ 2}}\iint\limits_{S_{w}}([p]^{q_{i}}q_{i}+[p]^{\hat{q}_{i}}\dot{q}_{i})dS_{w}=C ^{q_{i}}_{y}q_{i}+C^{\hat{q}_{i}}_{y}\dot{q}_{i}, \tag{12}\]  
\[m_{z}(t)=\frac{\lambda}{l^{2}}\iint\limits_{S_{w}}[p]xdS_{w}=\frac{\lambda}{l^ {2}}\iint\limits_{S_{w}}([p]^{\hat{q}_{i}}q_{i}+[p]^{\hat{q}_{i}}\dot{q}_{i}) xdS_{w}=m_{z}^{q_{i}}q_{i}+m_{z}^{\hat{q}_{i}}\dot{q}_{i}, \tag{13}\]  
Here the moment coefficient is defined with respect to the trailing edge \(x=0\). It can be easily recalculated for any other abscissa \(a_{0}\) of the center of rotation. The thrust force, acting upon oscillating and deforming wing in the direction of motion is defined as a sum of the _suction force_ and horizontal projection of _normal loads_. Therefore, within assumptions of linear theory the thrust force coefficient can be written as follows  
\[C_{t}=C_{s}+\overline{C}_{x}, \tag{14}\]  
where \(C_{s}\) is a suction force coefficient, \(\overline{C}_{x}\)- coefficient of the resultant of normal loads. For \(C_{t}>0\) the wing generates thrust and for \(C_{t}<0\) the wing experiences drag. Let us determine the main components of the thrust coefficient \(C_{t}\). Notably, the suction force is due to low pressures in the flow around highly curved leading edge. It can be shown that near the leading edge all components of perturbed velocity of the flow are finite excepting the one in the plane normal to its planform contour. If the latter component behaves as  
\[u_{\
u}=\frac{A}{\sqrt{-\
u}},\
u<0, \tag{15}\]where \(\
u\) is an outer normal to the edge planform contour lying in the plane of the wing, then the suction force coefficient can be calculated with use of the formula  
\[C_{s}=\frac{2\pi\lambda}{l^{2}}\int\limits_{l_{1}}A^{2}(s)\cos(\
u,x)ds=\frac{2 \pi\lambda}{l^{2}}\int\limits_{-l/2}^{l/2}A^{2}(z)dz \tag{16}\]  
where \(s\) is arc differential. Investigation of the local flow in vicinity of the leading edge within a theory of a lifting surface in extreme ground effect, leads to the formula  
\[A(s)=\sqrt{\frac{h}{2\pi}}\frac{\partial\Gamma}{\partial\
u}\text{ at }l_{1} \tag{17}\]  
Substituting (17) into (16) one comes to the following expression for the suction force coefficient of a wing, moving in immediate proximity of flat wall:  
\[C_{s}=\frac{h\lambda}{l^{2}}\int\limits_{l_{1}}\left(\frac{\partial\Gamma}{ \partial\
u}\right)_{\
u=0}^{2}\cos(\
u,x)ds=\frac{h\lambda}{l^{2}}\int\limits _{-l/2}^{l/2}\left(\frac{\partial\Gamma}{\partial\
u}\right)_{\
u=0}^{2}dz \text{ at the leading edge }l_{1}, \tag{18}\]  
Using expansion (6) of \(\Gamma\) in terms of kinematic parameters, we can rewrite (18) as  
\[C_{s}=\frac{h\lambda}{l^{2}}\int\limits_{-l/2}^{l/2}\left(\frac{\partial\Gamma ^{q_{i}}}{\partial\
u}q_{i}+\frac{\partial\Gamma^{\hat{q}_{i}}}{\partial\
u} \hat{q}_{i}\right)^{2}dz= C_{s}^{q_{i}q_{i}}q_{i}^{2}+C_{s}^{\hat{q}_{i}\hat{q}_{i}}q_{i} \hat{q}_{i}+C_{s}^{\hat{q}_{i}\hat{q}_{i}}\hat{q}_{i}^{2}, \tag{19}\]  
where  
\[C_{s}^{q_{i}q_{i}}=\frac{h\lambda}{l^{2}}\int\limits_{-l/2}^{l/2}\left(\frac{ \partial\Gamma^{q_{i}}}{\partial\
u}\right)_{\
u=0}^{2}dzC_{s}^{q_{i}\hat{q}_ {i}}=\frac{2h\lambda}{l^{2}}\int\limits_{-l/2}^{l/2}\left(\frac{\partial\Gamma ^{q_{i}}}{\partial\
u}\frac{\partial\Gamma^{\hat{q}_{i}}}{\partial\
u}\right) _{\
u=0}dz; \tag{20}\]  
Notably, in a number of practical cases the suction force coefficient and its derivatives with respect to kinematic parameters can be found in analytical form. As noted above, the thrust of the oscillating wing is due not only to the suction force, but also to projection of the normal hydrodynamic loads onto direction of motion. The contribution of the latter is accounted for by the coefficient  
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