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• Foundations of Data Science 数据科学基础

## 物理代写|流体力学代写Fluid Mechanics代考|Two-fluid model

The code NEPTUNE_CFD is based on an Eulerian approach with a finite volume discretization. The flow motion is followed by using the two-fluid model by Ishii (1975) extended to n-phase. In this model, the density, the viscosity and the local velocity are defined for each field in each cell. The following governing equations are solved for each field $\mathrm{k}$ :

• The mass balance equation:
$$\partial_{t} \varepsilon_{k}^{} \rho_{k}+\partial_{x_{i}}\left(\varepsilon_{k}^{} \rho_{k} u_{i, k}\right)=\Gamma_{k}$$
where $\varepsilon_{k}^{*}$ is the volume fraction of field $\mathrm{k}, \rho_{k}$ is its density, $u_{i, k}$ is the ith component of the velocity of field $\mathrm{k}$, and $\Gamma$ is the mass transfer term.
• The momentum balance equation in each space direction i:
$$\begin{array}{r} \partial_{t}\left(\varepsilon_{k}^{} \rho_{k} u_{i, k}\right)+\partial_{x_{j}}\left(\varepsilon_{k}^{} \rho_{k} u_{i, k} u_{j, k}\right)=\mu_{k} \partial_{x_{j}}\left(\varepsilon_{k}^{} S_{i j, k}\right)-\varepsilon_{k}^{} \partial_{x_{j}} P \ +\varepsilon_{k}^{*} \rho_{k} g_{i}+F_{i, k} \end{array}$$
where $\mu_{k}$ denotes the viscosity of field $\mathrm{k}, S$ denotes the viscous stress tensor, $P$ denotes the pressure, $g$ denotes the gravitational constant, and $F_{i, k}$ denotes the specific source terms depending on the field (continuous or dispersed).
• The energy balance equation:
$$\begin{array}{r} \partial_{t}\left(\varepsilon_{k}^{} \rho_{k} H_{k}\right)+\partial_{x_{j}}\left(\varepsilon_{k}^{} \rho_{k} H_{k} u_{j, k}\right)=-\partial_{x_{j}}\left(\varepsilon_{k}^{} Q_{j, k}\right)+\mu_{k} \partial_{x_{j}}\left(\varepsilon_{k}^{} \mu_{k} S_{i j, k} u_{j, k}\right) \ +\varepsilon_{k}^{} \partial P+\varepsilon_{k}^{} \rho_{k} g_{i} u_{i, k}+E_{k}^{I n t} \end{array}$$
where $H_{k}$ is the total enthalpy, $\boldsymbol{Q}{k}=-\lambda{k} \nabla T_{k}$, is the conductive thermal flux, $\lambda_{k}$ is the thermal conductivity, $T_{k}$ is the temperature, and $E_{k}^{\text {Int }}$ are some other interfacial energy transfers. The energy balance equation is only solved when non-isothermal flows are considered, which will only be discussed in the final section.

In the code NEPTUNE_CFD, the assumption of a common pressure for all fields is made.

## 物理代写|流体力学代写Fluid Mechanics代考|Filtered two-fluid equations

As previously done in Labourasse et al. (2007) and Vincent et al. (2008) with the single-fluid model and Lakehal (2004) with the two-fluid model including a dispersed field, the LES filter is applied to the two-fluid model equations for two continuous fields within the LBMo. In this section, the flows are considered isothermal. Thus, the filtered energy balance equation is not detailed. No dispersed fields are taken into account.

• The filtered mass balance equation:
$$\rho_{k} \partial_{t} \overline{\varepsilon_{k}^{}}+\rho_{k} \nabla \cdot\left(\overline{\varepsilon_{k}^{}} \overline{u_{i, k}}\right)+\tau_{\text {interf }}=0$$
where $\overline{\varepsilon_{k}^{*}}$ is the filtered volume fraction of field $\mathrm{k}$ and $\tau_{\text {interf }}$ is a subgrid term related to the relationship between the filtered velocity $\overline{u_{i, k}}$ and the interface topology (see Table 2,1),
• The filtered momentum balance equation:
\begin{aligned} \rho_{k} \partial_{t}\left(\overline{\varepsilon_{k}^{}} \overline{u_{i, k}}\right)+& \tau_{\text {time }}+\rho_{k} \nabla \cdot\left(\overline{\varepsilon_{k}^{}} \overline{u_{i, k}} \overline{u_{j, k}}\right)+\tau_{\text {canv }} \ =& \mu_{k} \nabla \cdot\left(\overline{\varepsilon_{k}^{}} \overline{S_{k}}\right)+\tau_{\text {diff }}-\overline{\varepsilon_{k}^{} \nabla \bar{P}-\tau_{\text {pressure }}} \ &+\overline{\varepsilon_{k}^{*}} \rho_{k} g_{i}+\widehat{F_{C S F}}+\tau_{\text {supperf }}+\widehat{F_{\text {drag }}}+\tau_{\text {drag }} \end{aligned}

where $\tau_{\text {time }}, \tau_{\text {conv }}, \tau_{\text {diff }}$ are, respectively, the time, convective and diffusive subgrid terms, and $\tau_{\text {pressure }}, \tau_{\text {superf }}$ and $\tau_{\text {drag }}$ are the three specific subgrid terms of the two-fluid model applied to two continuous fields (see Table $2.1$ for the expressions). $\tau_{\text {drag }}$ has the following expression:
$$\tau_{\text {drag }}=\overline{F_{\text {drag }}}-\bar{F}{\text {drag }}$$ with $\overline{F{\text {drag }}}$ being defined as:
\begin{aligned} &\overline{\varepsilon_{2}^{}}<0.3: \overline{\mathbf{F}{\text {bubble }}}=\overline{\varepsilon{1}^{} \varepsilon_{2}^{} \frac{18 \mu_{1}}{\varepsilon_{1}^{} d_{p}^{2}}\left(u_{1}-u_{2}\right)} \ &\overline{\varepsilon_{2}^{}}>0.7: \overline{\mathbf{F}{\text {droplet }}}=\overline{\varepsilon{1}^{} \varepsilon_{2}^{} \frac{18 \mu_{2}}{\varepsilon_{2}^{} d_{p}^{2}}\left(u_{1}-u_{2}\right)} \ &0.3 \leq \varepsilon_{2}^{} \leq 0.7: \overline{\mathbf{F}{\text {mix }}}=\overline{\frac{0.7-\varepsilon{2}^{}}{0.7-0.3} \mathbf{F}{\text {bubble }}+\frac{\varepsilon{2}^{}-0.3}{0.7-0.3} \mathbf{F}{\text {droplet }}} \end{aligned} and $\widetilde{F{d r a g}}$ :
$\overline{\varepsilon_{2}^{}}<0.3: \overline{\mathbf{F}{\text {bubble }}}=\overline{\varepsilon{1}^{}} \overline{\varepsilon_{2}^{}} \frac{18 \mu_{1}}{\bar{\varepsilon}{1} \bar{d}{p}^{2}}\left(\overline{u_{1}}-\overline{u_{2}}\right)$ $0.3 \leq \overline{\varepsilon_{2}^{}} \leq 0.7: \overline{\mathbf{F}{\text {mix }}}=\frac{0.7-\varepsilon{2}^{}}{0.7-0.3} \mathbf{F}{\text {bubble }}+\frac{\overline{\varepsilon{2}^{*}}-0.3}{0.7-0.3} \mathbf{F}_{\text {droplet }}$

## 物理代写|流体力学代写Fluid Mechanics代考|Two-fluid model

• 质量平衡方程：
$$\partial_{t} \varepsilon_{k}^{} \rho_{k}+\partial_{x_{i}}\left(\varepsilon_{k}^{} \rho_{k} u_{i, k}\right)=\Gamma_{k}$$
其中 $\varepsilon_{k}^{*}$ 是场 $\mathrm{k} 的体积分数，\rho_{k}$ 是它的密度，$u_{i, k}$ 是 $\mathrm{k}$ 场速度的第 i 个分量，$\Gamma$ 是传质项。
• 每个空间方向 i 的动量平衡方程：
$$\begin{array}{r} \partial_{t}\left(\varepsilon_{k}^{} \rho_{k} u_{i, k}\right) +\partial_{x_{j}}\left(\varepsilon_{k}^{} \rho_{k} u_{i, k} u_{j, k}\right)=\mu_{k} \partial_{x_ {j}}\left(\varepsilon_{k}^{} S_{ij, k}\right)-\varepsilon_{k}^{} \partial_{x_{j}} P \ +\varepsilon_{k}^ {*} \rho_{k} g_{i}+F_{i, k} \end{array}$$
其中 $\mu_{k}$ 表示场 $\mathrm{k} 的粘度，S$ 表示粘性应力张量，$P$ 表示压力，$g$ 表示引力常数，$F_{i, k}$ 表示取决于场（连续或分散）的具体源项。
• 能量平衡方程：
$$\begin{array}{r} \partial_{t}\left(\varepsilon_{k}^{} \rho_{k} H_{k}\right)+\partial_{x_{j }}\left(\varepsilon_{k}^{} \rho_{k} H_{k} u_{j, k}\right)=-\partial_{x_{j}}\left(\varepsilon_{k}^ {} Q_{j, k}\right)+\mu_{k} \partial_{x_{j}}\left(\varepsilon_{k}^{} \mu_{k} S_{ij, k} u_{j , k}\right) \ +\varepsilon_{k}^{} \partial P+\varepsilon_{k}^{} \rho_{k} g_{i} u_{i, k}+E_{k}^{I nt} \end{数组}$$
其中 $H_{k}$ 是总焓，$\boldsymbol{Q}{k}=-\lambda{k} \nabla T_{k}$ 是传导热通量，$\lambda_{k}$ 是热导率，$T_{k}$ 是温度，$E_{k}^{\text {Int }}$ 是其他一些界面能量转移。只有在考虑非等温流动时才能求解能量平衡方程，这将在最后一节中讨论。

## 物理代写|流体力学代写Fluid Mechanics代考|Filtered two-fluid equations

• 过滤后的质量平衡方程：
$$\rho_{k} \partial_{t} \overline{\varepsilon_{k}^{}}+\rho_{k} \nabla \cdot\left(\overline{\varepsilon_{k }^{}} \overline{u_{i, k}}\right)+\tau_{\text {interf }}=0$$
其中 $\overline{\varepsilon_{k}^{*}}$ 是场 $\mathrm{k}$ 和 $\tau_{\text {interf }}$ 的过滤体积分数是与过滤速度 $\overline{u_{i, k}}$ 和接口拓扑（见表 2,1），
• 滤波后的动量平衡方程：
\begin{aligned} \rho_{k} \partial_{t}\left(\overline{\varepsilon_{k}^{}} \overline{u_{i, k}}\right )+& \tau_{\text {时间 }}+\rho_{k} \nabla \cdot\left(\overline{\varepsilon_{k}^{}} \overline{u_{i, k}} \overline{ u_{j, k}}\right)+\tau_{\text {canv }} \ =& \mu_{k} \nabla \cdot\left(\overline{\varepsilon_{k}^{}} \overline{ S_{k}}\right)+\tau_{\text {diff }}-\overline{\varepsilon_{k}^{} \nabla \bar{P}-\tau_{\text {压力}}} \ & +\overline{\varepsilon_{k}^{*}} \rho_{k} g_{i}+\widehat{F_{CSF}}+\tau_{\text {superf }}+\widehat{F_{\text {拖动 }}}+\tau_{\text {拖动}} \end{对齐}

$$\tau_{\text {drag }}=\overline{F_{\text {drag }}}-\bar{F}{\text {拖动 }}$$ 其中 $\overline{F{\text {drag }}}$ 被定义为：
\begin{aligned} &\overline{\varepsilon_{2}^{}}<0.3: \overline{ \mathbf{F}{\text {bubble }}}=\overline{\varepsilon{1}^{} \varepsilon_{2}^{} \frac{18 \mu_{1}}{\varepsilon_{1}^ {} d_{p}^{2}}\left(u_{1}-u_{2}\right)} \ &\overline{\itempsilon_{2}^{}}>0.7: \overline{\mathbf{F}{\text {droplet}}}=\overline{\itempsilon{1}^{}\itempsilon_{2}^ {} \frac{18\mu_{2}}{\value_psilon_{2}^{}d_{p}^{2}}\left(u_{1}-u_{2}\right)} \ &0 \leq \varepsilon_{2}^{} \leq 0.7: \overline{\mathbf{F}{\text {mix }}}=\overline{\frac{0.7-\varepsilon{2}^{}}{0.7 -0.3 } \mathbf{F}{\text {bubble}}+\frac{\varepsilon{2}^{}-0.3}{0.7-0.3} \mathbf{F}{\text {droplet}}} \end {对齐} 和 $\width{drag}}$ ：
$\overline{\varepsilon_{2}^{}}<0.3: \overline{\mathbf{F}{\text {bubble }}}=\overline{\varepsilon{1}^{}} \overline{\varepsilon_ {2}^{}} \frac{18 \mu_{1}}{\bar{\varepsilon}{1} \bar{d}{p}^{2}}\left(\overline{u_{1} }-\overline{u_{2}}\right)$ $0.3 \leq \overline{\varepsilon_{2}^{}} \leq 0.7: \overline{\mathbf{F}{\text {mix }}}= \frac{0.7-\varepsilon{2}^{}}{0.7-0.3} \mathbf{F}{\text {气泡}}+\frac{\overline{\varepsilon{2}^{*}}-0.3 }{0.7-0.3} \mathbf{F}_{\text {液滴}}$

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