AEAS 103 11.02.20

 

♦ Factors that affect lift generation:

    • Air Velocity

    • Density of air

    • Wing area

    • Shape of the airfoil

    • Angle of attack

since we know that,  $$\begin{align*} & L\propto \rho, ~~ L \propto s ~~ and,~~ L \propto v^2 \\ & ~~~~~~~\therefore L = \frac{1}{2} \rho v^2 s C_L \end{align*}$$  

♦ Effect of AOA(angle of attack) on lift:

    Up to a certain AOA of an airfoil, the airflow follows the profile or shape of the airfoil. Up to this AOA, lift increases almost linearly as the AOA increases. At a certain AOA (usually less than $15 \circ$) the airflow begins to separate from the aerofoil and can not follow the shape of the aerofoil. This certain angle is called the stall angle. After stall angel lift decreases gradually with the increase of Angle of Attack (AOA).

break

   ♦ Proof of Bernoulli's Theorem:  

we know, 

    Total Energy = Work done 

          $$\begin{align*} \textrm{Total Energy} & = \textrm{Work done} \\ & = \textrm{Force} \times \textrm{Distance} \\ & = F \times d \\ & = PAd ~~~~~~~~~\bigg| [\frac{F}{A}= P] \\ & = PV \end{align*}$$  

Work done from $A_1$ to $A_2$ per unit volume =$(P_1 - P_2) \tag{1}\label{eq:1}$

$$\begin{align*} \textrm{Total Energy} & = \textrm{Kinetic energy + Potential energy} \\ & = \frac{1}{2} mv^2 + mgh \\ & = \frac{1}{2} \rho v^3 + \rho vgh \\ \therefore \textrm{ Total energy per unit volume} & = \frac{1}{2} \rho v^2 + \rho gh \tag{2}\label{eq:2} \end{align*}$$

from equation (1) & (2) we get,

  $$\begin{align*} P_1 - P_2 & = \frac{1}{2} \rho v^2 + \rho gh \\ \implies ~P_1 - P_2 & = \frac{1}{2} \rho ( v_2 ^2 - v_1^2) + \rho g(h_2 - h_1) \\ \implies P_1 + \frac{1}{2} \rho v_1 ^2 + \rho g h_1 & = P_2 + \frac{1}{2}\rho v_2 ^2 + \rho g h_2 \\ \therefore P_1 + \frac{1}{2} \rho v^2 + \rho gh & = \textrm{Constant} \end{align*}$$  

 

$$\begin{align*} h_1 = h_2 \\ h = h_1 - h_2 \\ \textrm{and,} ~\rho gh = 0 \end{align*}$$ $$P + \frac{1}{2}\rho v^2 = Const$$ $$KE = \frac{1}{2} \rho v^2 = \frac{1}{2}m v^2$$