# 物理代写|流体力学代写Fluid Mechanics代考|CHNG2801

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## 物理代写|流体力学代写Fluid Mechanics代考|Conformal Transformation, Basic Principles

Consider two planes, one is the z-plane, in which the point $z=x+i y$ is located and the other is the $\zeta$-plane in which the point $\zeta=\xi+i \eta$ is to be plotted. Let there be a function $\zeta=f(z)$ that facilitates the transformation of point $z$ in $z$-plane into the $\zeta$-plane. The function $\zeta=f(z)$, thus, defines a mapping or transformation of $z$ plane onto $\zeta$-plane. Figure $6.18$ is a simple example of transformation of points that constitute the straight lines in z-plane onto corresponding points in $\zeta$-plane. Consider the transformation function
$$\zeta=\xi+i \eta=z^2=(x+i y)^2=x^2-y^2+2 i x y .$$
Comparing the real and imaginary parts, it follows that
$$\xi=x^2-y^2 \eta=2 x y .$$
As seen in Fig. 6.18, constant lines $x=C_x$ in the $z$-plane are mapped onto parabolas open to the left. Furthermore, Fig. 6.18 suggests that the magnitudes of angles between the $x=$ const. and $y=$ const. in z-plane are preserved, when transforming into $\zeta$-plane. Eliminating $y$ from Eq. (6.109) leads to
$$\xi=C_x^2-\frac{\eta^2}{4 C_x^2} .$$
For $C_x=0$ ( $y$ axis) the parabolae coincide with the negative $\xi$ axis. Lines $y=C_y$ are mapped onto parabolae open to the right:
$$\xi=\frac{\eta^2}{4 C_y^2}-C_y^2$$
where for $C_y=0$ ( $x$ axis) the parabolas lie along the positive $\xi$ axis. Before getting into transformation details, it is important to know when the transformation equation can be solved for $x$ and $y$ as single-valued functions of $\xi$ and $\eta$, that is, when the transformation has a single-valued inversee. As we saw in Chap. 3, Eq.

## 物理代写|流体力学代写Fluid Mechanics代考|Kutta-Joukowsky Transformation

Before treating the Kutta-Joukowsky transformation, a brief description of the transformation process is given below. We consider the mapping of a circular cylinder from the $z$-plane onto the $\zeta$-plane. Using a mapping function, the region outside the cylinder in the $z$-plane is mapped onto the region outside another cylinder in the $\zeta$-plane. Let $P$ and $Q$ be the corresponding points in the $z$ – and $\zeta$-planes respectively. The potential at the point $P$ is
$$F(z)=\Phi+i \Psi .$$
The point $Q$ has the same potential, and we obtain it by insertion of the mapping function
$$F(z)=F(z(\zeta))=F(\zeta) .$$

Taking the first derivative of Eq. $(6.115)$ with respect to $\zeta$, we obtain the complex conjugate velocity $\overline{\boldsymbol{V}}\zeta$ in the $\zeta$ plane from $$\overline{\boldsymbol{V}}\zeta(\zeta)=\frac{d F}{d \zeta} .$$
Considering $z$ to be a parameter, we calculate the value of the potential at the point z. Using the transformation function $\zeta=f(z)$ we determine the value of $\zeta$ which corresponds to $z$. At this point $\zeta$, the potential then has the same value as at the point z. To determine the velocity in the $\zeta$ plane, we form
$$\frac{d F}{d \zeta}=\frac{d F}{d z} \frac{d z}{d \zeta}=\frac{d F}{d z}\left(\frac{d \zeta}{d z}\right)^{-1}$$
after introducing Eq. (6.116) into Eq. (6.117) and considering $\bar{V}z(z)=d F / d z$, Eq. (6.117) is rearranged as $$\overline{\boldsymbol{V}}\zeta(\zeta)=\overline{\boldsymbol{V}}z(z)\left(\frac{d \zeta}{d z}\right)^{-1}$$ Equation (6.118) expresses the relationship between the velocity in $\zeta$-plane and the one in $z$-plane. Thus, to compute the velocity at a point in the $\zeta$ plane we divide the velocity at the corresponding point in the z plane by $d \zeta / d z$. The derivative $d F d \zeta$ exists at all points where $d \zeta / d z \neq 0$. At singular points with $d \zeta / d z=0$, the complex conjugate velocity in the $\zeta$ plane $\overline{\boldsymbol{V}}\zeta(\zeta)=d F / d \zeta$ becomes infinite, if it is not equal to zero at the corresponding point in the $z$ plane.

## 物理代写|流体力学代写流体力学代考|保形变换，基本原理

.

$$\zeta=\xi+i \eta=z^2=(x+i y)^2=x^2-y^2+2 i x y .$$

$$\xi=x^2-y^2 \eta=2 x y .$$

$$\xi=C_x^2-\frac{\eta^2}{4 C_x^2} .$$

$$\xi=\frac{\eta^2}{4 C_y^2}-C_y^2$$

## 物理代写|流体力学代写流体力学代考|库塔-朱科夫斯基变换

$$F(z)=\Phi+i \Psi .$$

$$F(z)=F(z(\zeta))=F(\zeta) .$$

$$\frac{d F}{d \zeta}=\frac{d F}{d z} \frac{d z}{d \zeta}=\frac{d F}{d z}\left(\frac{d \zeta}{d z}\right)^{-1}$$
，将Eq.(6.116)引入Eq. (6.117 $\bar{V}z(z)=d F / d z$，式(6.117)重新排列为 $$\overline{\boldsymbol{V}}\zeta(\zeta)=\overline{\boldsymbol{V}}z(z)\left(\frac{d \zeta}{d z}\right)^{-1}$$ 式(6.118)表示在 $\zeta$-平面和in平面 $z$-plane。因此，要计算某一点的速度 $\zeta$ 我们用z平面上对应点的速度除以 $d \zeta / d z$。导数 $d F d \zeta$ 存在于所有点 $d \zeta / d z \neq 0$。在奇异点上 $d \zeta / d z=0$的复共轭速度 $\zeta$ 平面 $\overline{\boldsymbol{V}}\zeta(\zeta)=d F / d \zeta$ 变成无穷大，如果它在对应的点上不等于零 $z$ 平面。

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## MATLAB代写

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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