### 物理代写|流体力学代写Fluid Mechanics代考|MECH3261

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• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• 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|Flow Classification

Laminar flow is characterized by the smooth motion of fluid particles with no random fluctuations present. This characteristic is illustrated in Fig. 1.3a by measuring the velocity distribution $\mathbf{V}=V(\mathbf{x})$ of a statistically steady flow at an arbitrary position vector $\mathbf{x}$. As Fig. $1.3$ reveals, the velocity distribution for laminar flow does not have any time-dependent random fluctuations. In contrast, random fluctuations are inherent characteristics of a turbulent flow. Figure $1.3 \mathrm{~b}$ shows the velocity distribution for a turbulent flow with random fluctuations. For a statistically steady flow, the velocity distribution is time dependent, given by $\mathbf{V}=V(\mathbf{x}, t)$.

It can be decomposed as a constant mean velocity $\bar{V}(\mathbf{x})$ and random fluctuations $\mathbf{V}^{\prime}(\mathbf{x}, \mathbf{t})$ :
$$V(\mathbf{x}, t)=\overline{\mathbf{V}}(x)+\mathbf{V}^{\prime}(\mathbf{x}, t) .$$
At this point, the question may arise under which condition the flow pattern may change from laminar to turbulent. To answer this question, consider the experiment by Reynolds [5] late nineteenth century, who injected dye streak into a pipe flow as shown in Fig. 1.4.

At a lower velocity, Fig. 1.4a, no fluctuation was observed and the dye filament followed the flow direction. At certain distances, the diffusion process that was gradually taking place caused a complete mixing of the dye with the main fluid. Increasing the velocity, Fig. 1.4b however, changed the flow picture completely. The orderly motion of the dye with a short laminar length, shown in Fig. 1.4b, changed into a transitional mode that started with a sinus-like wave, which we discuss in detail in Chap. 8. The transitional mode was followed by a strong fluctuating turbulent motion. This resulted in a rapid mixing of the dye with the main fluid. To explain this phenomenon, Reynolds introduced a dimensionless parameter, named after him later as the Reynolds number.

## 物理代写|流体力学代写Fluid Mechanics代考|Change of Density, Incompressible, Compressible Flow

Fluid density generally changes with pressure and temperature. As the Mollier diagram for steam shows, the density of water in the liquid state changes insignificantly with pressure. In contrast, significant changes are observed when water changes the state from liquid to vapor. A similar situation is observed for other gases.

Considering a statistically steady liquid flow with negligibly small changes in density, the flow is termed incompressible. For gas flows, however, the density change is a function of the flow Mach number.

Figure $1.8$ depicts relative changes of different flow properties as functions of the flow Mach number. Up to $M=0.3$, the relative changes of density may be considered negligibly small meaning that the flow may be considered incompressible. For Mach numbers $M>0.3$, density changes cannot be neglected. In case the flow velocity approaches the speed of sound, $M=1.0$, the flow pattern undergoes a drastic change associated with shock waves.

The density classification based on flow Mach number gives a practical idea about the density change. A more adequate definition whether the flow can be considered compressible or incompressible is given by the condition $D \rho / D t=0$, which in conjunction with the continuity equation results in $\nabla \cdot \mathbf{V}=0$. This is the condition for a flow to be considered incompressible. This issue is discussed in more detail in Chap. $4 .$

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

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

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