## 物理代写|流体力学代写Fluid Mechanics代考|MEC2404

2023年3月22日

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## 物理代写|流体力学代写Fluid Mechanics代考|Stability of Small Disturbances

We consider a statistically steady flow motion, on which a small disturbance is superimposed. This particular flow is characterized by a constant mean velocity vector field $\overline{\mathbf{V}}(\mathbf{x})$ and its corresponding pressure $\bar{p}(\mathbf{x})$. We assume that the small disturbances we superimpose on the main flow is inherently unsteady, three-dimensional and is described by its vector filed $\tilde{\mathbf{V}}(\mathbf{x}, t)$ and its pressure disturbance $\tilde{p}(\mathbf{x}, \mathbf{t})$. In contrast to the random fluctuations which characterize turbulent flows, the disturbance field is of deterministic nature that is why we denote the disturbances with a tilde $(\sim)$ as opposed to a prime ( $)$ ), which we use for random fluctuations. Thus, the resulting motion has the velocity vector field:
$$\mathbf{V}(\mathbf{X}, t)=\overline{\mathbf{V}}(\mathbf{X})+\tilde{\mathbf{V}}(\mathbf{X}, t)$$
and the pressure field:
$$\mathbf{p}(\mathbf{X}, t)=\overline{\mathbf{p}}(\mathbf{X})+\tilde{\mathbf{p}}(\mathbf{X}, t) .$$
Assuming that $|\tilde{\mathbf{V}}(\mathbf{X}, t)| L L|\overline{\mathbf{V}}(\mathbf{X})|$ and $\tilde{p}(\mathbf{X}, t) L L \overline{\mathbf{p}}(\mathbf{X})$, we introduce Eqs. (8.2) and (8.3) into the Navier Stokes equation (4.43):
$$\frac{\partial(\overline{\mathbf{V}}+\tilde{\bar{V}})}{\partial t}+(\overline{\mathbf{V}}+\tilde{\mathbf{V}}) \cdot \nabla(\overline{\mathbf{V}}+\tilde{\mathbf{V}})=-\frac{1}{\rho} \nabla(\bar{p}+\tilde{p})+v \Delta(\overline{\mathbf{V}}+\tilde{\mathbf{V}}) .$$
Performing the differentiation and multiplication, we arrive at:
$$\frac{\partial \tilde{\mathbf{V}}}{\partial t}+\overline{\mathbf{V}} \cdot \nabla \overline{\mathbf{V}}+\overline{\mathbf{V}} \cdot \nabla \tilde{\mathbf{V}}+\tilde{\mathbf{V}} \cdot \nabla \overline{\mathbf{V}}+\tilde{\mathbf{V}} \cdot \nabla \tilde{\mathbf{V}}=-\frac{1}{\rho}(\nabla \bar{p}+\nabla \tilde{p})+v(\Delta \overline{\mathbf{V}}+\Delta \tilde{\mathbf{V}})$$
The small disturbance leading to linear stability theory requires that the nonlinear disturbance terms be neglected. This results in
$$\frac{\partial \tilde{\mathbf{V}}}{\partial t}+\overline{\mathbf{V}} \cdot \nabla \overline{\mathbf{V}}+\overline{\mathbf{V}} \cdot \nabla \tilde{\mathbf{V}}+\tilde{\mathbf{V}} \cdot \nabla \overline{\mathbf{V}}=-\frac{1}{\rho} \nabla \bar{p}+v \Delta \overline{\mathbf{V}}-\frac{1}{\rho} \nabla \tilde{p}+v \Delta \tilde{\mathbf{V}}$$

## 物理代写|流体力学代写Fluid Mechanics代考|The Orr-Sommerfeld Stability Equation

Before proceeding with the stability analysis, for the sake of simplicity, we set in Eq. (8.9) $\bar{V}1 \equiv U, \tilde{V}_1 \equiv \tilde{u}, \tilde{V}_2 \equiv \tilde{v}, x_1=x, x_2=y$ and find \begin{aligned} & \frac{\partial \tilde{u}}{\partial t}+U \frac{\partial \tilde{u}}{\partial x}+\tilde{v} \frac{\partial U}{\partial y}=-\frac{1}{\rho} \frac{\partial \tilde{p}}{\partial x}+v\left(\frac{\partial^2 \tilde{u}}{\partial x^2}+\frac{\partial^2 \tilde{u}}{\partial y^2}\right) \ & \frac{\partial \tilde{v}}{\partial t}+U \frac{\partial \tilde{v}}{\partial x}=-\frac{1}{\rho} \frac{\partial \tilde{p}}{\partial y}+v\left(\frac{\partial^2 \tilde{v}}{\partial x^2}+\frac{\partial^2 \tilde{v}_2}{\partial y^2}\right) . \end{aligned} For the disturbance field superimposed on the main laminar flow we introduce the following complex stream function: $$\psi(x, y, t)=\phi(y) e^{i(\alpha x-\beta t)} .$$ In Eq. (8.13) $\phi$ is the complex function of disturbance amplitude which is assumed to be a function of $y$ only. The stream function can be decomposed into a real and an imaginary part: $$\psi(x, y, t)=\psi{\Re}(x, y, t)+i \psi(x, y, t){\Im}$$ from which only the real part $$\Re(\psi)=e^{i \beta t}\left[\phi{\Re} \cos \left(\alpha x-\beta_{\Re} t\right)-\phi_{\mathfrak{Y}} \sin \left(\alpha x-\beta_{\Re} t\right)\right]$$
has a physical meaning. Similarly the complex amplitude is decomposed into a real and an imaginary part:
$$\phi(y, t)=\phi_{\Re}(x, y, t)+i \phi(y, t)_3$$
While $\alpha$ is a real quantity and is related to the wavelength $\lambda=2 \pi / \alpha$, the quantity $\beta$ is complex and consists of a real and an imaginary part
$$\beta=\beta_r+i \beta_i$$

## 物理代写|流体力学代写Fluid Mechanics代考|Stability of Small Disturbances

$$\mathbf{V}(\mathbf{X}, t)=\overline{\mathbf{V}}(\mathbf{X})+\tilde{\mathbf{V}}(\mathbf{X}, t)$$

$$\mathbf{p}(\mathbf{X}, t)=\overline{\mathbf{p}}(\mathbf{X})+\tilde{\mathbf{p}}(\mathbf{X}, t) .$$

$$\frac{\partial(\overline{\mathbf{V}}+\tilde{\bar{V}})}{\partial t}+(\overline{\mathbf{V}}+\tilde{\mathbf{V}}) \cdot \nabla(\overline{\mathbf{V}}+\tilde{\mathbf{V}})=-\frac{1}{\rho} \nabla(\bar{p}+\tilde{p})$$

$$\frac{\partial \tilde{\mathbf{V}}}{\partial t}+\overline{\mathbf{V}} \cdot \nabla \overline{\mathbf{V}}+\overline{\mathbf{V}} \cdot \nabla \tilde{\mathbf{V}}+\tilde{\mathbf{V}} \cdot \nabla \overline{\mathbf{V}}+\tilde{\mathbf{V}} \cdot \nabla \tilde{\mathbf{V}}=$$导致线性稳定性理论的小扰动要求忽略非线性扰动项。这 导致
$$\frac{\partial \tilde{\mathbf{V}}}{\partial t}+\overline{\mathbf{V}} \cdot \nabla \overline{\mathbf{V}}+\overline{\mathbf{V}} \cdot \nabla \tilde{\mathbf{V}}+\tilde{\mathbf{V}} \cdot \nabla \overline{\mathbf{V}}=-\frac{1}{\rho} \nabla \bar{p}$$

## 物理代写|流体力学代写Fluid Mechanics代考|The Orr-Sommerfeld Stability Equation

\begin{aligned} & \bar{V} 1 \equiv U, \tilde{V}1 \equiv \tilde{u}, \tilde{V}_2 \equiv \tilde{v}, x_1=x, x_2=y \text { 并找到 } \ & \frac{\partial \tilde{u}}{\partial t}+U \frac{\partial \tilde{u}}{\partial x}+\tilde{v} \frac{\partial U}{\partial y}=-\frac{1}{\rho} \frac{\partial \tilde{p}}{\partial x}+v\left(\frac{\partial^2 \tilde{u}}{\partial x^2}+\frac{\partial^2 \tilde{u}}{\partial y^2}\right) \end{aligned} 对于喤加在主层流上的扰动场，我们引入如下复流函数: $$\psi(x, y, t)=\phi(y) e^{i(\alpha x-\beta t)} .$$ 在等式中。 $(8.13) \phi$ 是扰动振幅的复函数，假定为以下函 数的函数 $y$ 仅有的。流函数可以分解为实部和虚部: $$\psi(x, y, t)=\psi \Re(x, y, t)+i \psi(x, y, t) \mathfrak{I}$$ 从中只有实部 $$\mathfrak{R}(\psi)=e^{i \beta t}\left[\phi \mathfrak{R} \cos \left(\alpha x-\beta{\mathfrak{R}} t\right)-\phi_{\mathfrak{Y}} \sin \left(\alpha x-\beta_{\mathfrak{R}} t\right)\right]$$

$$\phi(y, t)=\phi_{\mathfrak{R}}(x, y, t)+i \phi(y, t)_3$$

$$\beta=\beta_r+i \beta_i$$

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