# Design of a Human Muscle-Powered Flying Machine
## 5 Selected Design Aspects of the Wing
### Wing Aerodynamics  
Two of the fundamental aerodynamic forces acting on the wing are lift and drag. They depend on the velocity, air density, shape, and surface area of the body. The lift force, denoted as \(P_{z}\), and the drag force, \(P_{x}\), can be determined using the formulas [48]:  
\[P_{z}=\frac{\rho v^{2}}{2}SC_{l}\, \tag{2}\]  
Figure 3: Stages of designing a flying machine (adopted from [45]). \[P_{x}=\frac{\rho v^{2}}{2}SC_{d}\, \tag{3}\]  
where \(P_{z}\)--lift force N, \(P_{x}\)--drag force N, \(\rho\)--air density kg/m\({}^{3}\), \(v\)--velocity of the body m/s, \(C_{l}\)--lift coefficient, and \(C_{d}\)--drag coefficient. As a rule, the aforementioned forces can be applied at different locations, generating a pitching moment \(M\), expressed using the formula [48]:  
\[M=\frac{\rho v^{2}}{2}SC_{a}C_{m}\ \, \tag{4}\]  
where \(M\)--pitching moment N-m, \(c_{a}\)--chord length m, and \(C_{m}\)--pitching moment coefficient. Lift on a wing is generated due to pressure distribution, with higher pressure on the lower surface and lower pressure on the upper surface. According to [PERSON]'s principle, when the velocity of the airflow increases, the pressure decreases, and vice versa. The airflow over the upper surface of the wing moves at a higher velocity than the airflow along the lower surface, creating a pressure difference that results in lift. The shape of the wing, or airfoil, and the angle of attack (AoA)--the angle between the wing's chord line and the oncoming airflow--are both critical in generating this pressure difference. As the angle of attack increases, the airflow over the upper surface speeds up further, lowering the pressure on the top surface and increasing the pressure difference between the two sides of the wing, leading to greater lift. However, if the angle of attack becomes too large, the airflow can separate from the wing surface, causing a sudden loss of lift, known as stall. In addition to lift, the second fundamental aerodynamic force is wing drag. Five components of wing drag can be distinguished. The first one is _skin friction drag_, resulting from the viscosity of air molecules adhering to the wing's surface. As the distance from the surface increases until reaching the point of undisturbed flow velocity, the speed of moving particles increases. This phenomenon occurs in a layer known as the boundary layer. The movement of particles in the boundary layer can be laminar (layers move without mixing) or turbulent (mixing of particle layers occurs). The critical velocity is understood as the speed at which laminar flow transforms into turbulent flow. It is expressed using the Reynolds number \(Re\), which determines the ratio of inertial forces to viscous forces [48]:  
\[Re=\frac{v\cdot c_{a}}{v}\ \, \tag{5}\]  
where \(Re\)--the Reynolds number, \(v\)--flow velocity m/s, and \(v\)--kinematic viscosity of air m\({}^{2}\)/s. Therefore, it is important for the laminar flow of particles to dominate on the wing surface, as turbulent motion increases skin friction drag. Another component of wing drag is _form drag_, depending on the size of the vortices formed behind the body. The size of the vortices is influenced by the object's characteristics such as size and shape. Reduced pressure, created in the wake of vortices, acts on the body, attempting to pull it backward, generating drag. The next component is _induced drag_, generated by vortices occurring behind the trailing edge, resulting from the interaction of air streams. Particles at the wingtips move from areas of high pressure to areas of lower pressure, causing movement towards the fuselage on the upper surface and away from the fuselage on the lower surface. The aspect ratio of the wing significantly affects induced drag--longer wings with smaller chord at the tips lead to reduced airflow and thus, induced drag. Another important factor is flight altitude. When the wing operates close to the ground, ground effect is observed.