## 澳洲代写｜CIVL3612｜Fluid Mechanics流体力学 悉尼大学

This unit of study aims to provide an understanding of the conservation of mass and momentum in differential forms for viscous fluid flows. It provides the foundation for advanced study of turbulence, flow around immersed bodies, open channel flow, pipe flow and pump design.

A common observation in big rivers or other fast-flowing bodies of water (e.g. during floods) is shown in the figures and sketch below. A fast moving stream of water that is steadily flowing along suddenly decelerates and the position of the free surface ‘jumps’ upwards. After a lot of local turbulent motion, the flow settles down again but is now steadily moving at a significantly slower speed.

We will represent the free surface height as $h(x)$ and the velocity by the function $u(x)$. The fluid has constant density $\rho$ and we will treat the problem as one-dimensional. You can assume that viscous stresses along the control surfaces of the volume shown above are negligibly small, and neglect the density of air.
PART I:
a) consider a streamline drawn (line $\mathrm{AB}$ in the figure) just above the smooth flat lower surface of the channel. How is the static pressure in the fluid along this line related to the height of the river? How does the static pressure vary along the line DEA?

(a):
The pressure distribution on line $\mathrm{AB}$ follows the hydrostatic rule. It is true that the flow is not static but by picking an arbitrary control volume at any point on line AB (green dashed control volume in Figure 1) one can see that the balance of forces in the $y$-direction will tell us that the difference between the pressure at the bottom and the ambient pressure should balance the weight of the liquid inside the control volume. This simply implies that the static pressure on line $\mathrm{AB}$ should be equal
The pressure distribution on line DEA also follows the hydrostatic change merely due to the fact that there is no curvature in the streamlines as one integrates the Euler equation normal to them and thus the only change in pressure when one moves from $\mathrm{E}$ to A will be the hydrostatic part. Ignoring the density of air one can see that the pressure is constant from D to E and then start to grow linearly with height as we move from $\mathrm{E}$ to $\mathrm{A}$. The result is shown in Figure

b) Using the control volume shown in the sketch develop two expressions that relate the velocity and height of the stream at station 1 and the velocity and height of the stream at station 2. Developing a table of relevant quantities along each face of the control volume ABCDEA is highly recommended!
c) [2 points] Combine your expressions from (a) and (b) together to show that the speed of the river can be simply evaluated from simple measurements of the river height (e.g. using marked yardsticks attached to the channel floor):
$$u_1=\sqrt{\frac{g h_2}{2 h_1}\left(h_1+h_2\right)}$$

}(b) and (c): The selected control volume is shown in Figure 3 (dashed green line). One can subtract the ambient pressure from the entire problem and knowing that the net effect of uniform $P_a$ acting on the control volume is zero then there will be no change in the problem analysis if we only deal with gauge pressures $\left(P(x, y)-P_a\right)$Table 1 summarizes all the important parameters acting on different control surfaces for the selected control volume:

Now we can start by writing the conservation rules using the RTT. It is important to notice that due to the turbulent mixing happening in the region of the hydraulic jump, energy will not be conserved and thus either applying the conservation of energy or the Bernoulli equation will not be the right approach. If we write the conservation of mass for the selected control volume then we will have:

$$\text { C.O.Mass: } 0=\frac{d}{d t} \int_{\text {c.v. }} \rho d V+\int_{\text {c.s. }} \rho\left(v-v_c\right) \cdot n d A$$

Knowing that the problem is steady state and using the tabulated quantities, conservation of mass can be simplified to:
$\rho u_1 h_1=\rho u_2 h_2 \Rightarrow u_1 h_1=u_2 h_2$
The conservation of linear momentum in the $x$ direction can also be written in the RTT form:

$$\text { C.O.Momentum: } \frac{1}{W} \sum F_x=\frac{d}{d t} \int_{c . v .} \rho v_x d V+\int_{\text {c.s. }} \rho v_x\left(v-v_c\right) \cdot n d A$$

where $W$ is the width into the page.
The net of external forces acting in the $x$-direction on the control volume neglecting the wall shear effect is a result of pressure forces acting on the (AD) and (BC) control surfaces:

$$\frac{1}{W} \sum F_x=\int_{A D}\left(P-P_a\right) d y-\int_{B C}\left(P-P_a\right) d y=\int_0^{h_1} \rho g y d y-\int_0^{h_2} \rho g y d y=\rho g\left(\frac{h_1^2}{2}-\frac{h_2^2}{2}\right)$$

The right hand side of the RTT for the conservation of linear momentum can also be simplified to (knowing that the problem is steady and using the tabulated identities):

$$\text { R.H.S. of RTT for C.O. Momentum }=\rho u_2^2 h_2-\rho u_1^2 h_1$$

thus the conservation of linear momentum implies that:

$$\rho g\left(\frac{h_1^2}{2}-\frac{h_2^2}{2}\right)=\rho u_2^2 h_2-\rho u_1^2 h_1 \Rightarrow \frac{g}{2}\left(h_1^2-h_2^2\right)=u_2^2 h_2-u_1^2 h_1$$

using the result from conservation of mass (equation (1)) one can eliminate $u_2$ from equation (2) to give:

$$\frac{g}{2}\left(h_1^2-h_2^2\right)=h_1 u_1^2\left(h_1 / h_2-1\right) \Rightarrow u_1=\sqrt{\frac{g h_2}{2 h_1}\left(h_1+h_2\right)}$$

where we have used the identity $h_1^2-h_2^2=\left(h_1-h_2\right)\left(h_1+h_2\right)$.

A deeper question to answer is why is the water moving so fast locally to begin with. To answer this we must consider the topography of the river bed that is upstream of station 1 , as shown in the drawing below. We denote the height of the fluid stream above the river bed as $h(x)$ and the height of the riverbed by $b(x)$ :
d) [1 point] Consider a slice of river $d x$ and show that conservation of mass can be written in the form:
$$u(x) \frac{d h(x)}{d x}+h(x) \frac{d u(x)}{d x}=0$$

(d):
For the selected control volume (Figure 4 ) one can easily write the conservation of mass using Taylor series to obtain expressions for $u(x+\Delta x)$ and $h(x+\Delta x)$ :
$$u(x) h(x)=u(x+\Delta x) h(x+\Delta x) \rightarrow u(x) h(x)=\left(u(x)+\frac{d u}{d x} \Delta x\right)\left(h(x)+\frac{d h}{d x} \Delta x\right)$$
which after ignoring the second order terms such $\left(\Delta x^2\right)$ it can be rewritten as:
$$\Delta x\left(u(x) \frac{d h}{d x}+h(x) \frac{d u}{d x}\right)=0 \Rightarrow u \frac{d h}{d x}+h \frac{d u}{d x}=0$$
Another way to reach the same result is to say that since the flow is incompressible then the volumetric flow rate should remain unchanged thus $d(u h) / d x=0$ which will lead to the same result we just derived in equation (4).

Figure 4: An arbitrary control volume selected to derive the conservation of mass in the differential form.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|流体力学代写Fluid Mechanics代考|One-Equation Model by Prandtl

statistics-lab™ 为您的留学生涯保驾护航 在代写流体力学Fluid Mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流体力学Fluid Mechanics代写方面经验极为丰富，各种代写流体力学Fluid Mechanics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|流体力学代写Fluid Mechanics代考|One-Equation Model by Prandtl

A one-equation model is an enhanced version of the algebraic models we discussed in previous sections. This model utilizes one turbulent transport equation originally developed by Prandtl. Based on purely dimensional arguments, Prandtl proposed a relationship between the dissipation and the kinetic energy that reads
$$\varepsilon=C_D k^{3 / 2} / l_t$$
where the turbulence length scale $\ell_{\mathrm{t}}$ is set proportional to the mixing length, $\ell_{\mathrm{m}}$, the boundary layer thickness $\delta$ or a wake or a jet width. The velocity scale in Eq. (9.132) is set proportional to the turbulent kinetic energy $V_t \propto k^{1 / 2}$ as suggested independently by Kolmogorov [95] and Prandtl [96]. Thus, the expression for the turbulent viscosity becomes:
$$\mu_t=C_\mu \ell_m k^{0.5}$$
with the constant $C_\mu$ to be determined from the experiment. The turbulent kinetic energy, $k$, as a transport equation is taken from Sect. 9.2.2 in the form of Eqs. (9.111) or (9.126) where the dissipation is implemented. For simple two-dimensional flows where no separation occurs, with the mean-flow component $\overline{V_1} \equiv \bar{U}$ as the significant velocity in $x_1 \equiv x$-direction, and the distance from the wall $x_2 \equiv y$, the following approximation by Launder and Spalding [97] may be used
$$\rho \frac{\mathrm{D} k}{\mathrm{D} t}=\mu_t\left(\frac{\partial \bar{U}}{\partial y}\right)^2+\frac{\partial}{\partial y}\left(\frac{\mu_t}{\sigma_k} \frac{\partial k}{\partial y}\right)-C_D \frac{\rho k^{3 / 2}}{\ell_m},$$
where $\sigma_k=1$ and $C_D=0.08$ are coefficients determined from experiments utilizing simple flow configurations. The one-equation model provides a better assumption for the velocity scale $V_{\mathrm{t}}$ than $\ell_m|\partial \bar{U} / \partial y|$. Similar to the algebraic model, the oneequation one is not applicable to the general three-dimensional flow cases since a general expression for the mixing length does not exist. Therefore the use of a oneequation model does not offer any improvement compared with the algebraic one. The one-equation models discussed above are based on kinetic energy equations. There are a variety of one-equation models that are based on Prandtl’s concept and discussed in [88].

## 物理代写|流体力学代写Fluid Mechanics代考|Two-Equation k − ε Model

The two equations utilized by this model are the transport equations of kinetic energy $k$ and the transport equation for dissipation $\varepsilon$. These equations are used to determine the turbulent kinematic viscosity $v_t$. For fully developed high Reynolds number turbulence, the exact transport equations for $k(9.126)$ can be used. The transport equation for $\varepsilon(9.129)$ includes triple correlations that are almost impossible to measure. Therefore, relative to $\varepsilon$, we have to replace it with a relationship that approximately resembles the terms in Eq. (9.129). To establish such a purely empirical relationship, dimensional analysis is heavily used. Launder and Spalding [98] used the following equations for kinetic energy
$$\frac{\mathrm{D} k}{\mathrm{D} t}=\frac{1}{\rho} \frac{\partial}{\partial x_j}\left(\frac{\mu_t}{\sigma_k} \frac{\partial k}{\partial x_j}\right)+\frac{\mu_t}{\rho}\left(\frac{\partial \bar{V}i}{\partial x_j}+\frac{\partial \bar{V}_j}{\partial x_i}\right) \frac{\partial \bar{V}_i}{\partial x_j}-\varepsilon$$ and for dissipation $$\frac{\mathrm{D} \varepsilon}{\mathrm{D} t}=\frac{1}{\rho} \frac{\partial}{\partial x_j}\left(\frac{\mu_t}{\sigma{\varepsilon}} \frac{\partial \varepsilon}{\partial x_j}\right)+C_{\varepsilon 1} \frac{\mu_t}{\rho} \frac{\varepsilon}{k}\left(\frac{\partial \bar{V}i}{\partial x_j}+\frac{\partial \bar{V}_j}{\partial x_i}\right) \frac{\partial \bar{V}_i}{\partial x_j}-\frac{C{\varepsilon 2} \varepsilon^2}{k},$$
and the turbulent viscosity, $\mu_t$, can be expressed as
$$\mu_t=v_t \rho=\frac{C_\mu \rho k^2}{\varepsilon}$$
The constants $\sigma_{\mathrm{k}}, \sigma_{\varepsilon}, C_{\varepsilon_1}, C_{\varepsilon_2}$ and $C_\mu$ listed in Table 9.2 are calibration coefficients that are obtained from simple flow configurations such as grid turbulence. The models are applied to such flows and the coefficients are determined to make the model simulate the experimental behavior. The values of the above constants recommended by Launder and Spalding [83] are given in Table 9.2.
As seen, the simplified Eqs. (9.165) and (9.166) do not contain the molecular viscosity. They may be applied to free turbulence cases where the molecular viscosity is negligibly small compared to the turbulence viscosity. However, one cannot expect to obtain reasonable results by simulation of the wall turbulence using these equations.

# 流体力学代写

## 物理代写|流体力学代写Fluid Mechanics代考|One-Equation Model by Prandtl

$$\varepsilon=C_D k^{3 / 2} / l_t$$

$$\mu_t=C_\mu \ell_m k^{0.5}$$

$$\rho \frac{\mathrm{D} k}{\mathrm{D} t}=\mu_t\left(\frac{\partial \bar{U}}{\partial y}\right)^2+\frac{\partial}{\partial y}\left(\frac{\mu_t}{\sigma_k} \frac{\partial k}{\partial y}\right)-C_D \frac{\rho k^{3 / 2}}{\ell_m},$$

## 物理代写|流体力学代写Fluid Mechanics代考|Two-Equation k − ε Model

$$\frac{\mathrm{D} k}{\mathrm{D} t}=\frac{1}{\rho} \frac{\partial}{\partial x_j}\left(\frac{\mu_t}{\sigma_k} \frac{\partial k}{\partial x_j}\right)+\frac{\mu_t}{\rho}\left(\frac{\partial \bar{V} i}{\partial x_j}+\frac{\partial \bar{V}j}{\partial x_i}\right) \frac{\partial \bar{V}_i}{\partial x_j}-\varepsilon$$ 和消散 $$\frac{\mathrm{D} \varepsilon}{\mathrm{D} t}=\frac{1}{\rho} \frac{\partial}{\partial x_j}\left(\frac{\mu_t}{\sigma \varepsilon} \frac{\partial \varepsilon}{\partial x_j}\right)+C{\varepsilon 1} \frac{\mu_t}{\rho} \frac{\varepsilon}{k}\left(\frac{\partial \bar{V} i}{\partial x_j}+\frac{\partial \bar{V}j}{\partial x_i}\right) \frac{\partial \bar{V}_i}{\partial x_j}-\frac{C \varepsilon 2 \varepsilon^2}{k},$$ 和湍流粘度， $\mu_t$ ，可以表示为 $$\mu_t=v_t \rho=\frac{C\mu \rho k^2}{\varepsilon}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|流体力学代写Fluid Mechanics代考|Cebeci–Smith Model

statistics-lab™ 为您的留学生涯保驾护航 在代写流体力学Fluid Mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流体力学Fluid Mechanics代写方面经验极为丰富，各种代写流体力学Fluid Mechanics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|流体力学代写Fluid Mechanics代考|Cebeci–Smith Model

Another algebraic model is the Cebeci-Smith [92] which has been used primarily in external high speed aerodynamics with attached thin boundary layer. It is a two-layer algebraic zero-equation model which gives the eddy viscosity by separate expressions in each layer, as a function of the local boundary layer velocity profile. The model is not suitable for cases with large separated regions and significant curvature/rotation effects. The turbulent kinematic viscosity for the inner layer is calculated from
$$v_{t i}=l_m^2\left[\left(\frac{\partial U}{\partial y}\right)^2+\left(\frac{\partial U}{\partial x}\right)^2\right]^{\frac{1}{2}} .$$
For the outer layer kinematic viscosity is
$$v_{t_0}=\alpha U_e \delta_1 F_{K l}(y ; \delta)$$
with
$$F_{K l}(y ; \delta)=\left[1+5.5\left(\frac{y}{\delta}\right)^6\right]^{-1} \text { and } \delta_1=\int_0^\delta\left(1-U / U_e\right) d y$$
$\alpha=0.0168, U_e$ the velocity at the edge of the boundary layer, $\delta_1$ the boundary layer displacement thickness and $F_{K l}$ as the Klebanoff intermittency function [93]. The mixing length in Eq. (9.151) is determined by combining Eqs. (9.143) and (9.144)
$$l_m=\kappa y\left(1-e^{-y^{+} / A^{+}}\right)$$
with $\kappa=0.4$ and $A^{+}=26\left(1+y \frac{d p / d x}{\rho u_\tau^2}\right)^{-1 / 2}$.

## 物理代写|流体力学代写Fluid Mechanics代考|Baldwin–Lomax Algebraic Model

The third algebraic model is the Baldwin-Lomax model [94]. The basic structure of this model is essentially the same as the Cebeci-Smith model with the exception of a few minor changes. Similar to Cebeci-Smith, this model is a two-layer algebraic zero-equation model which gives the eddy kinematic viscosity $v_t$ as a function of the local boundary layer velocity profile. The model is suitable for high-speed flows with thin attached boundary-layers, typically present in aerospace and turbomachinery applications. While this model is quite robust and provides quick results, it is not capable of capturing details of the flow field. Since this model is not suitable for calculating flow situations with separation, its applicability is limited. We briefly summarize the structure of this model as follows. The kinematic viscosity for the inner layer is
$$v_{t_i}=l_m^2|\Omega|$$
with
$$l_m=\kappa y\left(1-e^{-y^{+} / A_0}\right)$$
and $\Omega=e_i e_j \omega_{i j}$ as the rotation tensor. The outer layer is described by
$$v_{t 0}=\alpha C_{\mathrm{cp}} F_{\mathrm{wake}} F_{\mathrm{kl}}\left(y, y_{\max } / C_{\mathrm{Kleb}}\right)$$
with the wake function $F_{\text {wake }}$
$$F_{\text {wake }}=\min \left(y_{\max } F_{\max } ; C_{\mathrm{wk}} y_{\max } U_{\mathrm{diff}} / F_{\max }\right)$$
and $F_{\max }$ and $y_{\max }$ as the maximum of the function
$$F(y)=y|\Omega|\left(1-e^{-y^{+} / A_0}\right)$$
The velocity difference $U_{\text {diff }}$ is defined as the difference of the velocity at $y_{\max }$ and $y_{\min }$ :
$$U_{\mathrm{diff}}=\operatorname{Max}\left(\sqrt{U_i U_i}\right)-\operatorname{Min}\left(\sqrt{U_i U_i}\right)$$
with the closure coefficients listed in Table 9.1.
The above zero-equation models are applied to cases of free turbulent flow such as wake flow, jet flow, and jet boundaries.

# 流体力学代写

## 物理代写|流体力学代写Fluid Mechanics代考|Cebeci–Smith Model

$$v_{t i}=l_m^2\left[\left(\frac{\partial U}{\partial y}\right)^2+\left(\frac{\partial U}{\partial x}\right)^2\right]^{\frac{1}{2}}$$

$$v_{t_0}=\alpha U_e \delta_1 F_{K l}(y ; \delta)$$

$$F_{K l}(y ; \delta)=\left[1+5.5\left(\frac{y}{\delta}\right)^6\right]^{-1} \text { and } \delta_1=\int_0^\delta\left(1-U / U_e\right) d y$$
$\alpha=0.0168, U_e$ 边界层边缘的速度， $\delta_1$ 边界层位移厚度和 $F_{K l}$ 作为 Klebanoff 间歇函数 [93]。方程式中的 混合长度。(9.151) 是通过结合等式来确定的。(9.143) 和 (9.144)
$$l_m=\kappa y\left(1-e^{-y^{+} / A^{+}}\right)$$
$$\text { 和 } \kappa=0.4 \text { 和 } A^{+}=26\left(1+y \frac{d p / d x}{\rho u_\tau^2}\right)^{-1 / 2} \text {. }$$

## 物理代写|流体力学代写Fluid Mechanics代考|Baldwin–Lomax Algebraic Model

$$v_{t_i}=l_m^2|\Omega|$$

$$l_m=\kappa y\left(1-e^{-y^{+} / A_0}\right)$$

$$v_{t 0}=\alpha C_{\mathrm{cp}} F_{\text {wake }} F_{\mathrm{kl}}\left(y, y_{\max } / C_{\mathrm{Kleb}}\right)$$

$$F_{\text {wake }}=\min \left(y_{\max } F_{\max } ; C_{\mathrm{wk}} y_{\max } U_{\mathrm{diff}} / F_{\max }\right)$$

$$F(y)=y|\Omega|\left(1-e^{-y^{+} / A_0}\right)$$

$$U_{\mathrm{diff}}=\operatorname{Max}\left(\sqrt{U_i U_i}\right)-\operatorname{Min}\left(\sqrt{U_i U_i}\right)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## ME3103 Fluid Mechanics课程简介

Physical properties of fluids:

Fluids are substances that can flow and take the shape of the container they occupy. The physical properties of fluids include density, viscosity, surface tension, and compressibility.

Density is defined as the mass per unit volume of a fluid. It is denoted by the symbol ρ and is measured in kg/m³.

Viscosity is a measure of a fluid’s resistance to deformation or flow. It is denoted by the symbol μ and is measured in Pa·s or N·s/m².

Surface tension is the force per unit length acting perpendicular to any line drawn on the surface of a liquid. It is denoted by the symbol γ and is measured in N/m.

Compressibility is a measure of how much a fluid’s volume changes when subjected to a pressure change. It is denoted by the symbol β and is measured in 1/Pa.

## PREREQUISITES

Outcome 1: Students will be able to use the fundamental principles and mathematical basis underlying the conservation equations.
Outcome 2: Be able to identify the guiding principles in a given fluid problem, to formulate the governing equations, and so to solve basic engineering problems.
Outcome 3: Recognize the difference between an ideal fluid and a viscous fluid, and to understand the limitations of the solutions for real practical fluid flows. Understand the difference between a simple solution and a real practical problem.

Outcome 4: Understand where their analysis might involve approximations and empirical approaches; for example, pipe flows and boundary layer flows.
Outcome 5: Have improved their ability to formulate an ordered approach to problem solving, using words of explanation in derivations, and algebra before substituting numerical values that allows neat analytical solutions and dimensional analysis.

## ME3103 Fluid Mechanics HELP（EXAM HELP， ONLINE TUTOR）

Problem 3.1 The material description of a fluid motion is given by the pathline equations
\begin{aligned} & x_1=\xi_1, \ & x_2=k \xi_1^2 t^2+\xi_2, \ & x_3=\xi_3 \end{aligned}
with $k$ as a constant having a dimension, such that the dimensional integrity of both sides of the above equation systems is preserved. Show that the Jacobian determinant $J=\operatorname{det}\left(\partial x_i / \partial \xi_j\right)$ does not vanish and obtain the transformation $\boldsymbol{\xi}=\boldsymbol{\xi}(x, t)$.

Problem 3.2 The fluid motion is described by:
\begin{aligned} x_1 & =\xi_1, \ x_2 & =\frac{1}{2}\left(\xi_2+\xi_3\right) e^{a t}+\frac{1}{2}\left(\xi_2-\xi_3\right) e^{-a t}, \ x_3 & =\frac{1}{2}\left(\xi_2+\xi_3\right) e^{a t}-\frac{1}{2}\left(\xi_2-\xi_3\right) e^{-a t} \end{aligned}
(a) Show that the Jacobian determinant does not vanish.
(b) Determine the velocity and acceleration components
(1) in material coordinates $V_i\left(\xi_j, t\right), A_i\left(\xi_j, t\right)$,
(2) in spatial coordinates $V_i\left(x_j, t\right), A_i\left(x_j, t\right)$.

Problem 3.4 The motion of a fluid is described in the material coordinate by:
\begin{aligned} & x_1=\xi_1 e^{a t}, \ & x_2=\xi_2 e^{a t}, \ & x_3=\xi_3 e^{-2 a t} \end{aligned}
with $a=$ const. and $\boldsymbol{\xi}=\boldsymbol{\xi}(x, t=0)$.
a) Calculate the velocity and acceleration components $V_i\left(\xi_j, t\right.$ and $A_i\left(\xi_j, t\right)$ in material coordinates.
b) Determine the spatial description of the velocity and acceleration components $V_i\left(x_k, t\right)$ and $A_i\left(x_k, t\right)$ by eliminating the material coordinate $\xi_j=\xi_j\left(x_k, t\right)$ in the results obtained in (a).
c) Find the acceleration components using the substantial derivatives of $V_i\left(x_k, t\right)$.
d) Is this a potential flow? If yes, find the potential function.

Problem 3.7 The components of a velocity flow field $V_i\left(\xi_j\right)$ are given by
\begin{aligned} & V_1=a\left(x_1+x_2\right) \ & V_2=a\left(x_1-x_2\right) \ & V_3=W \end{aligned}
with the constants $a$ and $W$. Determine
(a) the divergence $\nabla \cdot V$ of the flow field,
(b) the rotation $\nabla \times \mathbf{V}$,
(c) the parametric representation of the pathlines $x_i=x_i\left(\xi_j, t\right)$ with $\xi_j=x_j(t=0)$,
(d) nonparametric representation of the projection of the pathlines in $x_1, x_2$-plane by eliminating the curve parameter $t$,
(e) the projection of the streamlines in $x_1, x_2$-plane by integrating the differential equations for the streamlines.

Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

Statistics-lab™可以为您提供ac.uk ME3103 Fluid Mechanics流体力学课程的代写代考辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。

## 物理代写|流体力学代写Fluid Mechanics代考|Intermittency Modeling for Periodic Unsteady Flow

statistics-lab™ 为您的留学生涯保驾护航 在代写流体力学Fluid Mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流体力学Fluid Mechanics代写方面经验极为丰富，各种代写流体力学Fluid Mechanics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|流体力学代写Fluid Mechanics代考|Intermittency Modeling for Periodic Unsteady Flow

The effect of periodic unsteady wake flow on boundary layer transition is discussed more in detail in Chap. 11, Sect. 8.2. The specific problematic of the transition, however, are discussed in this section. To establish an intermittency based transition model that accounts for the periodic unsteady inlet flow impinging on a flat plate, a curved plate, a compressor or turbine blade, we first introduce a dimensionless parameter that characterizes periodic nature of the incoming flow:
$$\zeta=\frac{U_w t}{b} \equiv=\frac{y}{b} \text { with } b=\frac{1}{\sqrt{\pi}} \int_{-\infty}^{+\infty} \Gamma d \xi_2 .$$
Equation (8.45) relates the passing time $t$ of a periodic flow that impinges on the surface with the passing velocity in lateral direction $U_w$ and the intermittency width $b$. The latter is directly related to the wake width introduced by Schobeiri et al. [70]. We define the relative intermittency function $\Gamma$ in Eq. (8.45) as: $$\Gamma=\frac{<\gamma_i\left(t_i\right)>-<\gamma_i\left(t_i\right)>{\min }}{<\gamma_i\left(t_i\right)>{\max }-<\gamma_i\left(t_i\right)>{\min }}$$ In Eq. (8.46), $<\gamma_i\left(t_i\right)>$ is the time dependent ensemble-averaged intermittency function which determines the transitional nature of an unsteady boundary layer. The maximum intermittency $<\gamma_i\left(t_i\right)>{\max }$, shown in Fig. 8.20a, exhibits the time dependent ensemble averaged intermittency value inside the wake vortical core. Finally, the minimum intermittency $<\gamma_i\left(t_i\right)>{\min }$, represents the ensemble averaged intermittency values outside the wake vortical core. Experimental results presented in Fig. $8.20 \mathrm{~b}$ show that the relative intermittency function $\Gamma$ closely follows a Gaussian distribution, which is given by: $$\Gamma=e^{-\zeta^2}$$ Here, $\zeta$ is the non-dimensionalized lateral length scale. The validity of Eq. (8.47) has been verified for different cases $[65,71,72]$, suggesting it is a universal unsteady intermittency function. Using this function as a universally valid intermittency relationship for zero and non-zero pressure gradient cases [71], the intermittency function $<\gamma_i\left(t_i\right)>$ is completely determined if additional information about the minimum and maximum intermittency functions $<\gamma_i\left(t_i\right)>{\min }$ and $\gamma_i\left(t_i\right)>{\max }$ are available. The distribution of $\gamma_i\left(t_i\right)>{\min }$ and $\gamma_i\left(t_i\right)>{\max }$ in the streamwise direction are plotted in Fig. 8.21 (a). The steady case shown in Fig. 8.21 (b) serves as the basis of comparison for these maximum and minimum values. In the steady case, the intermittency starts to rise from zero at a streamwise Reynolds number $\mathrm{Re}{x, s}=2 \times 10^5$, and gradually approaches the unity corresponding to the fully turbulent state. This is typical of natural transition and follows the intermittency function (8.42). The distributions of maximum and minimum turbulence intermittencies $\gamma_i\left(t_i\right)>{\min }$ and $<\gamma_i\left(t_i\right)>{\max }$ in the streamwise direction are shown in Fig. 8.21a.

## 物理代写|流体力学代写Fluid Mechanics代考|Reynolds-Averaged Equations for Fully Turbulent Flow

In most engineering applications, the flow quantities such as velocity, pressure, temperature, and density are generally associated with certain time dependent fluctuations. These fluctuations may be of deterministic or stochastic nature. Turbulent flow is characterized by random fluctuations in velocity, pressure, temperature, and density. Figure 8.22 schematically shows the time dependent turbulent velocity vector as a function of time for statistically steady, statistically unsteady, and periodic unsteady flows. It exhibits three representative cases encountered in engineering application. Case (a) represents a statistically steady flow through a duct (pipe, nozzle, diffuser etc.). Case (b) reveals the statistically unsteady velocity at the exit of a storage facility during a depressurizing process. Case (c) depicts a periodic unsteady turbulent flow (almost sinusoidal) with a time dependent mean that is encountered in combustion engines. Periodic unsteady flows are also found in all sorts of turbines and compressors.

Any turbulent quantity can be decomposed in a mean and a fluctuation part, where the mean may be time dependent itself as we saw in the ensemble averaging process. For a statistically steady flow, the velocity vector is decomposed in a mean and fluctuation term: $$\mathbf{V}(\mathbf{x}, t)=\overline{\mathbf{V}}(\mathbf{x})+\mathbf{V}^{\prime}(\mathbf{x}, t)$$
The velocity components are obtained from Eq. (8.51) as:
$$V_i\left(x_j, t\right)=\bar{V}i\left(x_j\right)+V_i^{\prime}\left(x_j, t\right) .$$ For a statistically unsteady flow, the flow velocity $$\mathbf{V}(\mathbf{x}, t)=<\overline{\mathbf{V}}(\mathbf{x}, t)>+\mathbf{V}^{\prime}(\mathbf{x}, t)$$ with $<\mathbf{V}(\mathbf{x}, \mathrm{t})>$ as the ensemble averaged velocity according to Eq. (8.54). $$<\mathbf{V}(\mathbf{x}, \mathrm{t})>=\frac{1}{M} \sum{j=1}^M \mathbf{V}\left(\mathrm{t}_{\mathrm{i}}, \mathrm{j}\right)$$
where the flow is realized $M$ times and each time the velocity $\mathbf{V}(\mathbf{x}, \mathrm{t})$ is determined at the same position $\mathbf{x}$ and the same instant of time $t$. The velocity components are obtained from Eq. (8.53):
$$V_i\left(x_j, t\right)=+V_i^{\prime}\left(x_j, t\right) .$$

# 流体力学代写

## 物理代写|流体力学代写Fluid Mechanics代考|Intermittency Modeling for Periodic Unsteady Flow

$$\zeta=\frac{U_w t}{b} \equiv=\frac{y}{b} \text { with } b=\frac{1}{\sqrt{\pi}} \int_{-\infty}^{+\infty} \Gamma d \xi_2 .$$

$$\Gamma=\frac{<\gamma_i\left(t_i\right)>-<\gamma_i\left(t_i\right)>\min }{\left\langle\gamma_i\left(t_i\right)>\max -<\gamma_i\left(t_i\right)>\min \right.}$$

$$\Gamma=e^{-\zeta^2}$$

## 物理代写|流体力学代写Fluid Mechanics代考|Reynolds-Averaged Equations for Fully Turbulent Flow

$$\mathbf{V}(\mathbf{x}, t)=\overline{\mathbf{V}}(\mathbf{x})+\mathbf{V}^{\prime}(\mathbf{x}, t)$$

$$V_i\left(x_j, t\right)=\bar{V} i\left(x_j\right)+V_i^{\prime}\left(x_j, t\right)$$

$$\mathbf{V}(\mathbf{x}, t)=<\overline{\mathbf{V}}(\mathbf{x}, t)>+\mathbf{V}^{\prime}(\mathbf{x}, t)$$

$$<\mathbf{V}(\mathbf{x}, \mathrm{t})>=\frac{1}{M} \sum j=1^M \mathbf{V}\left(\mathrm{t}_{\mathrm{i}}, \mathrm{j}\right)$$

$$V_i\left(x_j, t\right)=+V_i^{\prime}\left(x_j, t\right)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|流体力学代写Fluid Mechanics代考|Intermittency Modeling for Steady Flow at Zero

statistics-lab™ 为您的留学生涯保驾护航 在代写流体力学Fluid Mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流体力学Fluid Mechanics代写方面经验极为丰富，各种代写流体力学Fluid Mechanics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|流体力学代写Fluid Mechanics代考|Intermittency Modeling for Steady Flow at Zero

The transition process was first explained by Emmons [61] through the turbulent spot production hypothesis. Adopting a sequence of assumptions, Emmon arrived at the following intermittency relation:
$$\gamma(\mathbf{x})=\mathbf{1}-\mathrm{e}^{\frac{-\mathbf{x g x ^ { 3 }}}{30}}$$
with $\sigma$ as the turbulent spot propagation parameter, $g$ the spot production parameter, $x$ the streamwise distance, and $U$ the mean stream velocity. While the Emmon’s spot production hypothesis is found to be correct, Eq. (8.40) does not provide a solution compatible with the experimental results. As an alternative, Schubauer and Klebanoff [62] used the Gaussian integral curve to fit the $\gamma$-distribution measured along a flat plate. Synthesizing the Emmon’s hypothesis with the Gaussian integral, Dhawan and Narasimha [63] proposed the following empirical intermittency factor for natural transition:
$$\gamma(\mathbf{x})=\mathbf{1}-\mathrm{e}^{-\mathrm{A} \xi^2}$$
with $\xi=\left(x-x_s\right) / \lambda, \lambda=(x){\gamma=0.75}-(x){\gamma=0.25}$ and $x_S$ as the streamwise location of the transition start and $A$ as constant. The solution of Eq. (8.41) requires the knowledge of $\lambda$ which contains two unknowns and the location of transition start $x_S$. In [64] the constant $A$ was set equal to 0.412 . Thus, we are dealing with three unknowns, namely $x_S$, and the two streamwise positions at which the intermittency factor assumes values of 0.75 and 0.25 . While the transition start $x_S$ can be estimated, the two streamwise positions $(x){\gamma=0.75}$ and $(x){\gamma=0.25}$ are still unknown. Further more, the quantity $A$ which was set equal to 0.412 , may be itself a function of several parameters such as the pressure gradient and the free-stream turbulence intensity. As we discuss in the following section, a time dependent universal unsteady transition model was presented in [64] for curved plate channel under periodic unsteady flow condition and generalized in [65] for turbomachinery aerodynamics application. The intermittency model for steady state turned out to be a special case of the unsteady model presented in $[65,66]$, it reads:
with $C_1=0.95, C_2=1.81$. With the known intermittency factor, the averaged velocity distribution in a transitional region is determined from:
$$\overline{\mathbf{V}}=(1-\bar{\gamma}) \mathbf{V}_L+\gamma \mathbf{V}_T$$

## 物理代写|流体力学代写Fluid Mechanics代考|Identification of Intermittent Behavior

The flow through a significant number of engineering devices is of periodic unsteady nature. Steam and gas turbine power plants, jet engines, turbines, compressors and pumps are a few examples. Within these devices unsteady interaction between individual components takes place. Figure 8.15 schematically represents the unsteady flow interaction between the stationary and rotating frame of a turbine stage.

A stationary probe traversing downstream of the stator at station (2) records a spatially periodic velocity distribution. Another probe placed on the rotor blade leading edge that rotates with the same frequency as the rotor shaft, registers the incoming velocity signals as a temporally periodic. The effect of this periodic unsteady inlet flow on the blade boundary layer is qualitatively and quantitatively different from those we discussed in the preceding section. The difference is shown in a simplified sketch presented in Fig. 8.16.

While the boundary layer thickness $\delta$ in case (a) is temporally independent, the one in case (b) experiences a temporal change. To predict the transition process under unsteady inlet flow condition using the intermittency approach, we first consider Fig. 8.17.

Figure 8.17 includes three sets of unsteady velocity data taken at three different times but during the same time interval $\Delta t$ (corresponding to the sequence $i=0$ to $i=N$ ). Each of these sets is termed an ensemble. Considering the velocity distribution at an arbitrary position vector $\mathbf{x}$ and at an ensemble $j$ such as $\mathbf{V}(\mathrm{t}, \mathrm{j})$, we use the same procedure we applied to the statistically steady flow discussed above to identify the nature of the periodic unsteady boundary layer flow. The corresponding intermittency function $\mathbf{I}(t, j)$ at a given position vector $\mathbf{x}$ is shown in Fig. 8.17. For a particular instant of time identified by the subscript $i$ for all ensembles, the ensemble average of $\mathbf{I}(\mathrm{t}, \mathrm{j})$ over $N$ number of ensembles results in an ensemble averaged intermittency function $\langle\gamma(\mathbf{x}, \mathrm{t})\rangle$. This is defined as:
$$<\gamma(\mathbf{x}, \mathrm{t})>=\frac{1}{\mathrm{M}} \sum_{\mathrm{j}=1}^{\mathrm{M}} \mathrm{I}\left(\mathrm{t}_{\mathrm{i}}, \mathrm{j}\right)$$

# 流体力学代写

## 物理代写|流体力学代写Fluid Mechanics代考|Intermittency Modeling for Steady Flow at Zero

Emmons [61] 首先通过湍流点产生假说解释了过渡过程。采用一系列假设，Emmon 得出以下间歇关系：
$$\gamma(\mathbf{x})=\mathbf{1}-\mathrm{e}^{\frac{-\mathrm{xgx}^3}{30}}$$

$$\gamma(\mathbf{x})=\mathbf{1}-\mathrm{e}^{-\mathrm{A} \xi^2}$$

$$\overline{\mathbf{V}}=(1-\bar{\gamma}) \mathbf{V}_L+\gamma \mathbf{V}_T$$

## 物理代写|流体力学代写Fluid Mechanics代考|Identification of Intermittent Behavior

$$<\gamma(\mathbf{x}, \mathrm{t})>=\frac{1}{\mathrm{M}} \sum_{\mathrm{j}=1}^{\mathrm{M}} \mathrm{I}\left(\mathrm{t}_{\mathrm{i}}, \mathrm{j}\right)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## MST326 Fluid Mechanics课程简介

Physical properties of fluids:

Fluids are substances that can flow and take the shape of the container they occupy. The physical properties of fluids include density, viscosity, surface tension, and compressibility.

Density is defined as the mass per unit volume of a fluid. It is denoted by the symbol ρ and is measured in kg/m³.

Viscosity is a measure of a fluid’s resistance to deformation or flow. It is denoted by the symbol μ and is measured in Pa·s or N·s/m².

Surface tension is the force per unit length acting perpendicular to any line drawn on the surface of a liquid. It is denoted by the symbol γ and is measured in N/m.

Compressibility is a measure of how much a fluid’s volume changes when subjected to a pressure change. It is denoted by the symbol β and is measured in 1/Pa.

## PREREQUISITES

Outcome 1: Students will be able to use the fundamental principles and mathematical basis underlying the conservation equations.
Outcome 2: Be able to identify the guiding principles in a given fluid problem, to formulate the governing equations, and so to solve basic engineering problems.
Outcome 3: Recognize the difference between an ideal fluid and a viscous fluid, and to understand the limitations of the solutions for real practical fluid flows. Understand the difference between a simple solution and a real practical problem.

Outcome 4: Understand where their analysis might involve approximations and empirical approaches; for example, pipe flows and boundary layer flows.
Outcome 5: Have improved their ability to formulate an ordered approach to problem solving, using words of explanation in derivations, and algebra before substituting numerical values that allows neat analytical solutions and dimensional analysis.

## MST326 Fluid Mechanics HELP（EXAM HELP， ONLINE TUTOR）

Problem 4.1 Incompressible Newtonian fluid with constant density and viscosity flows between two parallel plates with infinite width. Body forces are neglected. Given are the plate height $h$, the components of the pressure gradient, $\frac{\partial p}{\partial x_1}=-K \frac{\partial p}{\partial x_2} \equiv 0, \frac{\partial p}{\partial x_3} \equiv 0$, the velocity field between the plates $u_1\left(x_2\right)=\frac{K}{2 \mu}\left(\frac{h^2}{4}-x_2^2\right), u_2 \equiv 0, u_3 \equiv 0$, the density $\varrho$ and the absolute viscosity $\mu$.
(a) Show that the given velocity field satisfies the continuity and the Navier-Stokes equation.
(b) Determine the components of the stress tensor.
(c) Calculate the dissipation function $\Phi$.
(d) Find the energy per unit depth, length, and time dissipated in heat within the gap.
(e) Calculate the principal stresses and their directions.

Problem 4.2 Newtonian fluid flows through the sketched channel, Fig. 4.4, with infinite extensions in $x_1$ – and $x_3$-direction and the height $h$. The plane flow is steady, the density $\varrho$ and the viscosity $\mu$ are assumed to be constant, and body forces are neglected. The top and bottom wall are porous such that a constant normal velocity component $V_W$ can be established at the walls. The pressure gradient in $x_1$-direction is constant $\left(\partial p / \partial x_1=-K\right)$. Because of the infinite extension of the channel, the velocity distribution does not depend upon $x_1$. The variables $\varrho, \mu, K, h, V_W$ are given.
(a) Using the continuity equation, calculate the distribution of the velocity component in $x_2$-direction $u_2\left(x_2\right)$.
(b) Simplify the $x_1$-component of the Navier-Stokes equation for this problem.
(c) Give the boundary condition for the velocity component $u_1$.
(d) Calculate the velocity distribution $u_1\left(x_2\right)$. (Hint: After solving the homogeneous differential equation, the particular solution of the inhomogeneous differential equation can be found setting $u_{1_p}=$ const. $x_2$ ).

Problem 4.3 A Newtonian fluid with constant density and viscosity flows steadily through a two dimensional vertically positioned channel with the width $h$ shown in Fig. 4.5. The motion of the fluid is described by the Navier Stokes equations. The flow is subjected to the gravitational acceleration $\mathbf{g}=e_1 g$ and a constant pressure gradient in flow direction $x_1$. Assume that $V_2=V_3=0$

(a) Determine the solution of the Navier-Stokes equations.
(b) Write a computer program; show the velocity distributions for the following cases: (a) For $K=0$, (b) $K>0$, and (c) $K<0$.
(c) For which $K$ there is no flow?

Problem 4.4 A Newtonian fluid with constant density and viscosity flows steadily through a two dimensional channel positioned at an angle $\alpha$ shown in Fig. 4.6 with the width $2 \mathrm{~h}$. The motion of the fluid is described by the Navier Stokes equations. The flow is subjected to the gravitational acceleration $\mathbf{g}=\mathbf{e}_{\mathbf{1}} \mathbf{g}_1+\mathbf{e}_2 \mathbf{g}_2$ and a constant pressure gradient in flow direction $x_1$. Assume that $V_2=V_3=0$.
(a) Determine the solution of the Navier-Stokes Equations. Write a computer program and plot the velocity distributions for: (a) For $K=0$, (b) $K>0$, and (c) $K<0$.
(b) For which $K$ there is no flow?

Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

Statistics-lab™可以为您提供ac.uk MST326 Fluid Mechanics流体力学课程的代写代考辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。

## 物理代写|流体力学代写Fluid Mechanics代考|CIVL3612

statistics-lab™ 为您的留学生涯保驾护航 在代写流体力学Fluid Mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流体力学Fluid Mechanics代写方面经验极为丰富，各种代写流体力学Fluid Mechanics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|流体力学代写Fluid Mechanics代考|Curved Channel, Negative Pressure Gradient

Once the solution of Eq. (7.20) is found, the dimensionless velocity distribution is obtained from Eq. (7.17):
$$\frac{U}{U_m} \equiv \frac{V^}{V_m^}=\Phi e^{\left.\frac{1}{2} b \xi_{2 \max }-\xi_2\right)} .$$
As seen earlier, the solution $\Phi=\Phi\left(\xi_2\right)$ is a function of the coordinate $\xi_2$ only and incorporates the Reynolds number as a parameter. Thus, the velocity distributions represented by Eq. (7.31) exhibit similar solutions. An asymmetrically curved channel with convex and concave walls is generated by choosing $a=-1$ and $b=1$. As shown in Fig. 7.1, the negative pressure gradient is established by an asymmetrically convergent channel with convex and concave walls. For Reynolds number $\mathrm{Re}=500$ the velocity distributions at the coordinate $\xi_1=3.8$ exhibit an almost parabolic shape with the maximum close to $\xi_2=0.3$. For the similarity reasons explained above, similar velocity distribution is found and plotted at $u=0.38$ for the same Reynolds number. Increasing the Reynolds number to $\mathrm{Re}=750,1000$ respectively results in steeper velocity slopes at both walls (Fig. 7.1). As a consequence, the velocity profile tends to become fuller, particularly for higher Reynolds numbers. As shown, the viscosity effect is restricted predominantly to the wall regions and continuously reduces by increasing the Reynolds number. This behavior again justifies the Prandtl assumption for higher Reynolds number to divide the flow field into a viscous and an inviscid flow zone. For Reynolds numbers up to $\mathrm{Re}=5000$, velocity distributions can be calculated without convergence problems. Thus for an accelerated flow, the stability of the laminar flow and the transition from laminar into turbulent flow are apparently extended to higher Reynolds numbers as expected.

## 物理代写|流体力学代写Fluid Mechanics代考|Curved Channel, Positive Pressure Gradient

The positive pressure gradient within the asymmetrically curved channel discussed above is created by reversing the flow direction. Figure 7.2 shows the flow at different Reynolds numbers. As shown in Fig. 7.2, for $\mathrm{Re}=500$, the velocity distribution on the concave wall is fully attached. The fluid particles moving in streamwise direction are exposed to three different type of forces: (1) the wall shear stress force acting in opposite direction decelerates the fluid particle; (2) the decelerating effect of the wall shear stress is intensified by the pressure forces which also act in opposite direction causing the flow to further decelerate; and (3) the centrifugal force caused by the channel curvature pushes the fluid particle away from the convex wall towards the concave one increasing the susceptibility of flow to separation. The interaction of these three forces increase the tendency for separating along the convex wall. Increasing the Reynolds number to $\mathrm{Re}=1500$ causes the flow separation on the convex channel wall. In this case the laminar low along the convex surface is, while the non-separated portion appears as a laminar jet attaching to the concave wall.

As shown in Fig. 7.2, the combination of the channel curvature and the positive pressure gradient has caused a flow separation on the convex wall, whereas no separation occurred on the concave wall. From fluid mechanical point of view, we are interested in determining the effect of pressure gradient on the velocity distribution in the absence of curvature. To investigate this, we generate a channel with straight wall geometry by setting $a=-2$, and $b=0$. With these new constants, Eq. (7.20) reduces to:
$$\Phi^{\prime \prime}+4 \Phi+\operatorname{Re} \Phi^2+C_1=0 .$$
This special case constitutes a purely radial laminar flow through a channel with straight walls and is known as the Hamel-flow [41]. The results are shown in Fig. 7.3, where the velocity distributions are plotted for three different Reynolds numbers. Close to the wall at $\mathrm{Re}=500$, the flow exhibits a tendency for separation on both walls. Increasing the Reynolds number to $\mathrm{Re}=750$ and 1500 respectively causes the flow separation on both walls. A comparison with the results in Fig. 7.2 clearly indicates that the difference in velocity distributions is attributed to the nature of wall curvature.

# 流体力学代写

## 物理代写|流体力学代写Fluid Mechanics代考|Curved Channel, Negative Pressure Gradient

$\$ \$$\ \$$

## 物理代写|流体力学代写Fluid Mechanics代考|Curved Channel, Positive Pressure Gradient

$$\Phi^{\prime \prime}+4 \Phi+\operatorname{Re} \Phi^2+C_1=0$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|流体力学代写Fluid Mechanics代考|AMME2261

statistics-lab™ 为您的留学生涯保驾护航 在代写流体力学Fluid Mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流体力学Fluid Mechanics代写方面经验极为丰富，各种代写流体力学Fluid Mechanics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

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

To determine the influence of curvature and pressure gradient on temperature distribution, the velocity distribution must be known. This requires the solution of continuity and the Navier-Stokes equations. As the first conservation law, the continuity equation in coordinate invariant form is: $$\nabla \cdot \mathbf{V}=\mathbf{0}$$
For a curvilinear coordinate system, Equation (7.1) can be written as [see Eqs. (4.7) and (A.36)]:
$$V_i^i+V^k \Gamma_{k i}^i=0$$
with $\mathbf{V}$ as the velocity vector that is decomposed in its contravariant components $V^i$ in normal to the flow direction must vanish. As a result, the integration of Eq. (7.2) must fulfill both the continuity and the Navier-Stokes equations. This is possible only if the Christoffel symbols $\Gamma_{k i}^i$ are not functions of the coordinates themselves. The corresponding channel with the curvilinear coordinate is then obtained from the transformation:
$$w=-\frac{2}{a+i b} \ln z \text { with } z=x+i y \text { and } w=\xi_1+i \xi_2$$
with $\xi_i$ as the orthogonal curvilinear coordinate system.
\begin{aligned} & x=e^{-\frac{1}{2}\left(a \xi_1-b \xi_2\right)} \cos \left(\frac{a \xi_1+b \xi_2}{2}\right) \ & y=-e^{-\frac{1}{2}\left(a \xi_1-b \xi_2\right)} \sin \left(\frac{a \xi_2+b \xi_1}{2}\right) \end{aligned}
with $a$ and $b$ as real constants that define the configuration of the channel and $\xi_1$ and $\xi_2$ as the orthogonal curvilinear coordinates. The corresponding metric coefficients and Christoffel symbols are:
\begin{aligned} & g^{11}=g^{22}=\frac{4}{a^2+b^2} e^{a \xi_1-b \xi_2}, g^{12}=g^{21}=0 \ & \Gamma_{k l}^1=\frac{1}{2}\left(\begin{array}{cc} -a & b \ b & +a \end{array}\right) \quad \Gamma_{k l}^2=-\frac{1}{2}\left(\begin{array}{cc} b & a \ a & -b \end{array}\right) . \end{aligned}

## 物理代写|流体力学代写Fluid Mechanics代考|Solution of the Navier-Stokes Equation

Equation (7.20) describes the motion of viscous flows through curved channels pertaining to the coordinate transformation discussed in Sect. 7.1.1. It includes both the Navier-Stokes and continuity equations that are reduced to a single, ordinary, nonlinear, second-order differential equation. The solutions of Eq. (7.20), $\Phi=\Phi\left(\xi_2\right)$ are functions of the coordinate $\xi_2$ and incorporate the Reynolds number as parameter. Special cases of Eq. (7.20) are the purely radial flow, where $a=-2$ and $b=0$, and the flow through concentric cylinders with $a=0$ and $b=1$. For those cases analytical and numerical solutions were found in [40, 41]. Based on Jeffery-Hammel’s solutions, Milsaps and Pohlhausen [45] calculated the temperature distribution within the straight wall diffuser and nozzle. Extensive discussions by Schlichting [39] underscore the importance of those flows from a general theoretical point of view. To show the effect of the curvature and pressure gradient on the temperature and velocity distribution, an asymmetrically curved channel with convex and concave walls is generated by choosing $a=-1$ and $b=1$, Schobeiri $[42,43]$.

For the solution of Eq. (7.20), a numerical integration procedure is applied. Starting from the initial conditions specified below and the determination of constant $C_1$, an iteration method is developed that reduces the boundary-value problem to an initial one. The solution of differential Eq. (7.20) must fulfill the governing initial and boundary conditions. The boundary conditions are given by the non-slip conditions at the channel walls:
\begin{aligned} & \xi_2=\xi_{2_{B 1}} \equiv 0.1, \Phi=\Phi_{B 1} \equiv 0 \ & \xi_2=\xi_{2_{B 2}} \equiv 0.5, \Phi=\Phi_{B 2} \equiv 0 \end{aligned}
where the indices $B 1$ and $B 2$ refer to the convex and concave channel walls. The initial condition is described by the maximum value of the velocity distribution and its position $\xi_2=\xi_{2_{\max }}$, which is unknown for the time being:
$$\xi_2=\xi_{2_{\max }}, \Phi=\Phi_{\max }= \pm 1, \Phi^{\prime}=\Phi_{\max }^{\prime}=0$$
The positive sign of $\Phi$ indicates an increase of the cross-section area in direction of decreasing $\xi_1$, which is associated with the positive pressure gradient. The negative sign characterizes the accelerated flow in direction of increasing $\xi_1$, where negative pressure gradient prevails. The constant $C_1$ in Eq. (7.20) specifies the solution of Eq. (7.20) and significantly affects the convergence speed. It must be determined so that the above boundary and initial conditions are identically fulfilled. The following iteration method enables precise calculation of $C_1$. Starting from Eq. (7.20),
$$\Phi^{\prime \prime}=\Psi^{\prime \prime}+C_1$$
where
$$\Psi^{\prime \prime} \equiv-2 b \Phi^{\prime}+\left(a^2+b^2\right) \Phi+\frac{a^2}{4} \operatorname{Re} \Phi^2$$

# 流体力学代写

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

$$\nabla \cdot \mathbf{V}=\mathbf{0}$$

$$V_i^i+V^k \Gamma_{k i}^i=0$$

$$w=-\frac{2}{a+i b} \ln z \text { with } z=x+i y \text { and } w=\xi_1+i \xi_2$$

$$x=e^{-\frac{1}{2}\left(a \xi_1-b \xi_2\right)} \cos \left(\frac{a \xi_1+b \xi_2}{2}\right) \quad y=-e^{-\frac{1}{2}\left(a \xi_1-b \xi_2\right)} \sin \left(\frac{a \xi_2+b \xi_1}{2}\right)$$

## 物理代写|流体力学代写Fluid Mechanics代考|Solution of the Navier-Stokes Equation

$$\xi_2=\xi_{2_{B 1}} \equiv 0.1, \Phi=\Phi_{B 1} \equiv 0 \quad \xi_2=\xi_{2_{B 2}} \equiv 0.5, \Phi=\Phi_{B 2} \equiv 0$$

$$\xi_2=\xi_{2_{\max }}, \Phi=\Phi_{\max }= \pm 1, \Phi^{\prime}=\Phi_{\max }^{\prime}=0$$

$$\Phi^{\prime \prime}=\Psi^{\prime \prime}+C_1$$

$$\Psi^{\prime \prime} \equiv-2 b \Phi^{\prime}+\left(a^2+b^2\right) \Phi+\frac{a^2}{4} \operatorname{Re} \Phi^2$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|流体力学代写Fluid Mechanics代考|ENGR30002

statistics-lab™ 为您的留学生涯保驾护航 在代写流体力学Fluid Mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流体力学Fluid Mechanics代写方面经验极为丰富，各种代写流体力学Fluid Mechanics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|流体力学代写Fluid Mechanics代考|Effect of Solidity on Blade Profile Losses

Equation (5.181) exhibits a fundamental relationship between the lift coefficient, the solidity, the inlet and exit flow angle, and the loss coefficient $\zeta$. The question is, how the profile loss $\zeta$ will change if the solidity $\sigma$ changes. The solidity has the major influence on the flow behavior within the blading. If the spacing is too small, the number of blades is large and the friction losses dominate. Increasing the spacing, which is identical to reducing the number of blades, at first causes a reduction of friction losses. Further increasing the spacing decreases the friction losses and also reduces the guidance of the fluid that results in flow separation leading to additional losses. With definite spacing, there is an equilibrium between the separation and friction losses. At this point, the profile loss $\zeta=\zeta_{\text {tnctoon }}+\zeta_{\text {separaton }}$ is at a minimum. The corresponding spacing/chord ratio has an optimum, which is shown in Fig. 5.33. To find the optimum solidity for a variety of turbine and compressor cascades, a series of comprehensive experimental studies have been performed by several researchers. A detailed discussion of the results of these studies is presented in [23].

The relationship for the lift-solidity coefficient derived in the preceding sections is restricted to turbine and compressor stages with constant inner and outer diameters. This geometry is encountered in high pressure turbines or compressor components, where the streamlines are almost parallel to the machine axis. In this special case, the stream surfaces are cylindrical with almost constant diameter. In a general case such as the intermediate and low pressure turbine and compressor stages, however, the stream surfaces have different radii. The meridional velocity component may also change from station to station. In order to calculate the blade lift-solidity coefficient correctly, the radius and the meridional velocity changes must he taken into account. Detailed discussions on this and turbomachinery aero-thermodynamic topics are found in [23].

## 物理代写|流体力学代写Fluid Mechanics代考|Inviscid Potential Flows

As discussed in Chap. 4 , generally the motion of fluids encountered in engineering applications is described by the Navier-Stokes equations. Considering today’s computational fluid dynamics capabilities, it is possible to numerically solve the Navier-Stokes equations for laminar flows (no turbulent fluctuations), transitional flows (using appropriate intermittency models), and turbulent flow (utilizing appropriate turbulence models). Given today’s computational capabilities, one may argue at this juncture that there is no need to artificially subdivide the flow regime into different categories such as incompressible, compressible, viscid or inviscid ones. However, based on the degree of complexity of the flow under investigation, a computational simulation may take up to several days, weeks, and even months for direct Navier-stokes simulations (DNS). The difficulties associated with solving the Navier-Stokes equations are caused by the existence of the viscosity terms in the Navier-Stokes equations.

Measuring the velocity distributions encountered in engineering applications such as in a pipe flow, flow around a compressor or turbine blade, or along the wing of an aircraft, we find that the effect of viscosity is confined to a very thin layer called the boundary layer with a local thickness $\delta$. As we discuss in Chap. 11, comprehensive experimental investigations performed earlier by Prandtl $[26,27]$ show that the boundary layer thickness $\delta$ compared to the length $L$ of the subject under investigation is very small. In the vicinity of the wall, because of the no-slip condition, the velocity is $V_{\text {wall }}=0$. Moving away from the wall towards the edge of the boundary layer, the velocity continuously increases until it reaches the velocity at the edge of the boundary layer $V=V_\delta$. Within the boundary layer, the flow is characterized by non-zero vorticity $\nabla \times V \neq 0$. No major changes in velocity magnitude is expected outside the boundary layer, provided that the surface of the subject under investigation does not have a curvature. In case of surfaces with convex or concave curvatures, the velocity outside the boundary layer changes in lateral direction.

Outside the boundary layer, the effect of the viscosity can be neglected as long as the Reynolds number is high enough ( $\operatorname{Re}=100,000$ and above) indicating that the convective flow forces are much larger than the shear stress forces. Theoretically, the boundary layer thickness approaches zero as the Reynolds number tends to infinity. In this case, the flow can be assumed as irrotational, which is then characterized by zero vorticity $\nabla \times V=0$. Thus, as Prandtl suggested, the flow may be decomposed into two distinct regions, the vortical inner region, called the boundary layer, where the viscosity effect is predominant, and the non-vortical region outside the boundary layer.

The flow in the outer region can be calculated using the Euler equation of motion, while the boundary layer method can be applied for calculating the viscous flow within the inner region. Combining these two methods allows calculation of the flow field in a sufficiently accurate manner as long as the boundary layer is not separated. Figure $6.1$ exhibits the velocity distributions along the suction surface of an airfoil. While in case (a) the viscosity is accounted for, in case (b) it is neglected. Thus, the flow is assumed irrotational, which is characterized by $\nabla \times V=0$. As a consequence of this assumption, the velocity on the surface has a non-zero tangential component, which is in contrast to the reality. These type of flows are called potential flows which is the subject of the following sections.

## 有限元方法代写

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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。