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• 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|Presentation of the test case

In this test case, an oil drop with a cubic shape (size of $\mathrm{L} / 2$ ) is initially placed in a cubic box (size $\mathrm{L}=0.1 \mathrm{~m}$ ) containing liquid water (see Figure 2.1). The evolution of the system is driven by the gravity forces in the $\mathrm{z}$ direction $\left(\mathrm{g}=9.81 \mathrm{~m} . \mathrm{s}^{-2}\right)$. At the end of the simulation, the oil phase is supposed to be located in the top part of the box with the liquid water beneath. The fluid properties are given in Table $2.2$. The interfacial tension coefficient between oil and water is equal to $0.045 \mathrm{~N} \cdot \mathrm{m}^{-1}$.

The test case has been simulated with three different mesh refinements: $128^{3}$ cells, $256^{3}$ cells and $512^{3}$ cells. The time steps are kept constant and are, respectively, equal to $0.8 \mathrm{~ms}, 0.2 \mathrm{~ms}$ and $0.05 \mathrm{~ms}$, such that the CFL number is constant and always smaller than 0.9. The simulations have been performed with 144 cores for the first grid and 1152 for the two others during, respectively, 7 hours, 47 hours and 2 months and a half to reach 13 physical seconds.

## 物理代写|流体力学代写Fluid Mechanics代考|Order of magnitude of the subgrid terms

The subgrid terms appearing in the filtered two-fluid equations are compared in terms of order of magnitude to find the predominant and negligible ones.

This first step is important, since the larger the contribution of this subgrid term is, the more the modeling errors will impact the simulation. For this purpose, a uniform box filter is applied to the simulation results extracted at the peak of enstrophy (maximum value of the velocity curl) for the three grids. Only the first neighborhood of each cell (filter size of 2) is considered. The subgrid term $\tau_{\text {time }}$ is not represented in this study, since the analysis is only proposed for one time. Each subgrid term in the momentum balance equation is normalized by the convection resolved term. The interfacial subgrid term $\tau_{\text {interf }}$ appearing in the mass balance equation is normalized by its corresponding resolved part.

The summation of the subgrid term and the convection (respectively interfacial) resolved term is made in the whole domain, only where the phase is present, as illustrated here with the convection subgrid term:The results are presented in Figure 2.2. Table $2.3$ proposes a classification of the subgrid terms according to the phase for the most refined grid. As documented in Labourasse et al. (2007), Vincent et al. (2008) and Larocque et al. (2010), the subgrid term $\tau_{\text {superf }}$ is largely predominant for the two phases. Furthermore, a small contribution of the diffusion subgrid term $\tau_{\text {diff }}$ is observed for the two phases. Nevertheless, for the oil phase, contrary to the single-fluid formulation, this contribution is not negligible and very close to those of $\tau_{\text {cony }}$ and $\tau_{\text {interf }}$. The results obtained with the single-fluid formulation seem to be closer to those obtained here for the water phase.

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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。

assignmentutor™您的专属作业导师
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