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## 物理代写|空气动力学代写Aerodynamics代考|Hypersonic Aerodynamics

According to Newtonian impact theory, which fails to explain the classical lift generation, the pressure exerted by the air particles impinging on a surface is equal to the time rate of change of momentum vertical to the wall. Using this principle we can find the pressure exerted by the air particles on the wall which is inclined with free stream with angle $\theta_w$. Since the velocity, as shown in Fig. 1.8, normal to the wall is $U_n$ the time rate of change of momentum becomes $p=\rho U_n^2$.
If we write $\mathrm{U}n=\mathrm{U} \sin \theta_w$, the surface pressure coefficient reads as $$c_p=\frac{p-p{\infty}}{\frac{1}{2} \rho_{\infty} U^2}=2 \sin ^2 \theta_w-\frac{2}{\gamma M^2}$$
The free stream Mach number $\mathrm{M}$ is always high for hypersonic flows. Therefore, its square $\mathrm{M}^2 \gg 1$ is always true. If the wall inclination under consideration is sufficiently large i.e. $\theta_w$ is greater than $35^{\circ}-40^{\circ}$, the second term in Eq. $1.26$ becomes negligible compared to the first term. This allows us to obtain a simple expression for the surface pressure at hypersonic speeds as follows
$$c_F \simeq 2 \sin ^2 \theta_{\mathrm{W}}$$
Now, we can find the lift and the drag force coefficients for hypersonic aerodynamics based on the impact theory. According to Fig. $1.8$ the sectional lift coefficient reads as
$$c_L=2 \sin ^2 \theta_w \cos \theta_w$$
and the sectional drag coefficient becomes
$$c_D=2 \sin ^3 \theta_w$$

## 物理代写|空气动力学代写Aerodynamics代考|The Piston Theory

The piston theory is an approximate theory which works for thin wings at high speeds and at small angles of attack. If the characteristic thickness ratio of a body is $\tau$ and $\mathrm{M} \tau$ is the hypersonic similarity parameter then for $\mathrm{M} \tau \gg 1$ the Newtonian impact theory works well. For the values of $M \tau<1$ the piston theory becomes applicable. Since $\tau$ is small for thin bodies, at high Mach numbers the shock generated at the leading edge is a highly inclined weak shock. This makes the flow region between the surface and the inclined shock a thin boundary layer attached to the surface. If the surface pressure at the boundary layer is $p$ and the vertical velocity on the surface is $w_a$, then the flow can he modeled as the wedge flow as shown in Fig. 1.9.

The piston theory is based on an analogy with a piston moving at a velocity w in a tube to create compression wave. The ratio of compression pressure created in the tube to the pressure before passing of the piston reads as Lieppmann and Roshko (1963); Hayes and Probstien (1966),
$$\frac{p}{p_{\infty}}=\left[1+\frac{\gamma-1}{2} \frac{w}{a_{\infty}}\right]^{\frac{2 \gamma}{\gamma-1}}$$
Here, $a_{\infty}$ is the speed of sound for the gas at rest. If we linearize Eq. $1.30$ by expanding into the series and retain the first two terms, the pressure ratio reads as
$$\frac{p}{p_{\infty}} \cong 1+\gamma \frac{w_a}{a_{\infty}}$$
wherein, $w_a$ is the time dependent vertical velocity which satisfies the following condition: $w_a \ll a_{\infty}$. The expression for the vertical velocity in terms of the body motion and the free stream velocity is given by
$$w_a=\frac{\partial z_a}{\partial t}+U \frac{\partial z_a}{\partial x}$$
Equation $1.31$ is valid only for the hypersonic similarity values in, $0<\mathrm{M} \tau<0.15$, and as long as the body remains at small angles of attack during the motion while the vertical velocity changes according to Eq. 1.32. For higher values of the hypersonic similarity parameter, the higher order approximations will be provided in the relevant chapter.

# 空气动力学代考

## 物理代写|空气动力学代写空气动力学代考|高超声速空气动力学

$$c_F \simeq 2 \sin ^2 \theta_{\mathrm{W}}$$

$$c_L=2 \sin ^2 \theta_w \cos \theta_w$$
，截面阻力系数为
$$c_D=2 \sin ^3 \theta_w$$

## 物理代写|空气动力学代写空气动力学代考|活塞理论

$$\frac{p}{p_{\infty}}=\left[1+\frac{\gamma-1}{2} \frac{w}{a_{\infty}}\right]^{\frac{2 \gamma}{\gamma-1}}$$

$$\frac{p}{p_{\infty}} \cong 1+\gamma \frac{w_a}{a_{\infty}}$$
，其中$w_a$为随时间变化的垂直速度，满足以下条件:$w_a \ll a_{\infty}$。用身体运动和自由流速度表示的垂直速度的表达式为
$$w_a=\frac{\partial z_a}{\partial t}+U \frac{\partial z_a}{\partial x}$$

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MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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