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# 美国代写|流体力学代写Fluid Dynamics代写|MER332

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## 美国代写|流体力学代写Fluid Dynamics代写|Where does the Vorticity come from?

To summarize, vorticity may be redistributed throughout space by advection and diffusion. Moreover, in three-dimensional motion, it may be intensified or diminished through the stretching or compression of the vortex tubes. However, none of these processes can create vorticity in a region of fluid which had none to start with. One might ask, therefore, where the vorticity in the Kármán vortex street shown in Figure $2.15$ has come from.

Here, once again, the analogy between heat transport and vorticity in a two-dimensional flow is useful. Heat cannot be created or destroyed within the interior of a fluid by advection or diffusion. However, if the cylinder in Figure $2.15$ were heated, then heat would get into the fluid by diffusing in from the surface of the cylinder. The same is true of vorticity. In the absence of rotational body forces, vorticity can get into a fluid only by diffusing in from the boundaries. So, just as there is a thermal boundary layer on a heated cylinder, through which heat diffuses into the fluid, so there is a viscous boundary layer, though which vorticity diffuses into the flow. Indeed, all of the vorticity downstream of the cylinder shown in Figure $2.15$ started off in a thin viscous boundary layer on the upstream surface of the cylinder.

## 美国代写|流体力学代写Fluid Dynamics代写|Enstrophy and its Governing Equation

The intensification of vorticity by vortex-line stretching is often quantified using enstrophy, $\omega^{2} / 2$. A budget equation for enstrophy comes from the dot product of $\omega$ with ( $\left.2.45\right)$,
$$\frac{D}{D t}\left(\frac{\omega^{2}}{2}\right)=\omega_{i} \omega_{j} \frac{\partial u_{i}}{\partial x_{j}}+\nu \omega \cdot\left(\nabla^{2} \omega\right)$$
which is usually rewritten as
$$\frac{D}{D t}\left(\frac{\omega^{2}}{2}\right)=\omega_{i} \omega_{j} S_{i j}-\nu(\nabla \times \boldsymbol{\omega})^{2}+\nu \nabla \cdot[\boldsymbol{\omega} \times(\nabla \times \boldsymbol{\omega})]$$
The first term on the right of $(2.52)$ corresponds to the generation of enstrophy through vortex-line stretching, or else a reduction in enstrophy via compression of the vortex lines. The second term, by contrast, represents the destruction of enstrophy by viscous forces, while the third term is often unimportant as the divergence integrates to zero for a localized distribution of $\omega$. We conclude that enstrophy, just like mechanical energy, is destroyed by friction. However, unlike mechanical energy, there is a natural generation mechanism in the absence of body forces. Enstrophy plays an important role in turbulence where, as noted in $₫ 2.7 .2$, there is a delicate balance between vorticity intensification by chaotic vortex-line stretching and the viscous destruction of enstrophy.

## 美国代写|流体力学代写FLUID DYNAMICS代写|ENSTROPHY AND ITS GOVERNING EQUATION

$$\frac{D}{D t}\left(\frac{\omega^{2}}{2}\right)=\omega_{i} \omega_{j} \frac{\partial u_{i}}{\partial x_{j}}+\nu \omega \cdot\left(\nabla^{2} \omega\right)$$
which is usually rewritten as
$$\frac{D}{D t}\left(\frac{\omega^{2}}{2}\right)=\omega_{i} \omega_{j} S_{i j}-\nu(\nabla \times \boldsymbol{\omega})^{2}+\nu \nabla \cdot[\boldsymbol{\omega} \times(\nabla \times \boldsymbol{\omega})]$$
The first term on the right of $(2.52)$ corresponds to the generation of enstrophy through vortex-line stretching, or else a reduction in enstrophy via compression of the vortex lines. The second term, by contrast, represents the destruction of enstrophy by viscous forces, while the third term is often unimportant as the divergence integrates to zero for a localized distribution of $\omega$. We conclude that enstrophy, just like mechanical energy, is destroyed by friction. However, unlike mechanical energy, there is a natural generation mechanism in the absence of body forces. Enstrophy plays an important role in turbulence where, as noted in $₫ 2.7 .2$,在混沌涡线拉伸引起的涡度增强和熵的粘性破坏之间存在微妙的平衡。

## 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。