### 物理代写|空气动力学代写Aerodynamics代考|ME471

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## 物理代写|空气动力学代写Aerodynamics代考|Generation of Lift

The very basic theory of aerodynamics lies in the Kutta-Joukowski theorem. This theorem states that for an airfoil with round leading and sharp trailing edge immersed in a uniform stream with an effective angle of attack, there exists a lifting force proportional to the density of air $\rho$, free stream velocity $U$ and the circulation $\Gamma$ generated by the bound vortex. Hence, the sectional lifting force $l$ is equal to
$$l=\rho U \Gamma$$
Figure $1.1$ depicts the pertinent quantities involved in generation of lift.

The strength of the bound vortex is given by the circulation around the airfoil, $\Gamma=\oint \mathbf{V} . \mathbf{d s} .$

If the effective angle of attack is $\alpha$, and the chord length of the airfoil is $c=2 b$, with the Ionkowski transformation the magnitude of the circulation is found as $\Gamma=2 \pi \alpha \mathrm{b} U$. Substituting the value of $\Gamma$ into Eq. $1.1$ gives the sectional lift force as
$$l=2 \rho \pi \alpha b U^2$$
Using the definition of sectional lift coefficient for the steady flow we obtain,
$$c_l=\frac{l}{\rho U^2 b}=2 \pi \alpha$$
The very same result can be obtained by integrating the relation between the vortex sheet strength $\gamma_{\mathrm{a}}$ and the lifting surface pressure coefficient $\mathrm{c}{\mathrm{pa}}$ along the chord as follows. $$\mathrm{c}{\mathrm{pa}}(x)=\mathrm{c}{\mathrm{pl}}-\mathrm{c}{\mathrm{pu}}=2 \gamma_{\mathrm{a}}(x) / \mathrm{U}$$
The lifting presssure coéfficient for an aairfoil with angle of attack reads as
$$c_{p a}(x)=2 \alpha \sqrt{\frac{b-x}{b+x}}, \quad-b \leq x \leq b$$

During rapidly changing unsteady motion of an airfoil the aerodynamic response is no longer the timewise slightly changing steady phenomenon.

For example, let us consider a thin airfoil with a chord length of $2 \mathrm{~b}$ undergoing a vertical simple harmonic motion in a free stream of $U$ with zero angle of attack. If the amplitude of the vertical motion is $\bar{h}$ and the angular frequency is $\omega$ then the profile location at any lime $t$ reads as
$$z_a(t)=\bar{h} e^{i \omega t}$$
If we implement the pure steady aerodynamics approach, because of Eq. $1.3$ the sectional lift coefficient will read as zero. Now, we write the time dependent sectional lift coefficient in terms of the reduced frequency $k=\omega \mathrm{b} / \mathrm{U}$ and the non-dimensional amplitude $\bar{h}^=\bar{h} / b$. $$c_l(t)=\left[-2 i k C(k) \bar{h}^+k^2 \bar{h}^\right] \pi e^{i \omega t}$$ Let us now analyze each term in Eq. $1.6$ in terms of the relevant aerodynamics. (i) Unsteady Aerodynamics: If we consider all the terms in Eq. $1.6$ then the analysis is based on unsteady aerodynamics. $C(k)$ in the first term of the expression is a complex function and called the Theodorsen function which is the measure of the phase lag between the motion and aerodynamic response. The second term, on the other hand, is the acceleration term based on the inertia of the air parcel displaced during the motion. It is called the apparent mass term and is significant for the reduced frequency values larger than unity. (ii) Quasi Unsteady Aerodynamics: If we neglect the apparent mass term in Eq. 1.6 the aerodynamic analysis is then called quasi unsteady aerodynamics. Accordingly, the sectional lift coefficient reads as $$c_l(t)=\left[-2 \pi i k C(k) \bar{h}^\right] e^{i \omega t}$$

## 物理代写|空气动力学代写空气动力学代考|产生升力

$$l=\rho U \Gamma$$

$$l=2 \rho \pi \alpha b U^2$$

$$c_l=\frac{l}{\rho U^2 b}=2 \pi \alpha$$

$$c_{p a}(x)=2 \alpha \sqrt{\frac{b-x}{b+x}}, \quad-b \leq x \leq b$$

## 物理代写|空气动力学代写空气动力学代考|非定常升力系数

$$z_a(t)=\bar{h} e^{i \omega t}$$

## 有限元方法代写

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## MATLAB代写

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