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• Longitudinal Data Analysis 纵向数据分析
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## 物理代写|流体力学代写Fluid Mechanics代考|Comparison of turbulence models

After identifying the predominant terms, a comparison of the different turbulence models is proposed. Five models are compared: Smagorinsky’s model (Smagorinsky 1963), the wall-adapting local eddy-viscosity (WALE) model (Nicoud and Ducros 1999), Bardina’s model (Bardina et al. 1980), the Approximate Deconvolution Model (ADM) (Adams and Stolz 2002) and the mixed Smagorinsky-Bardina’s model (Bardina et al. 1983). ADM is taken with an order of 6. This choice will be discussed here after.

For this comparison, the relative error of each model is evaluated using the L2 norm:

Figure $2.3$ presents these relative errors for each model and each subgrid term with the most refined mesh. Since Smagorinsky’s and the WALE models are mainly based on the consideration of kinetic energy and inertial effects in turbulent flows, they can only be applied to- $\tau_{\text {conv }}$ – and $\tau_{\text {interf }}$. This figure highlights that ADM is the most appropriate model for all of the subgrid terms, whatever the phase (Vincent et al. 2016). However, for the water phase, the error level of ADM applied to the pressure subgrid term remains high. This has a limited effect since the pressure term is not predominant, as shown in the previous section.

## 物理代写|流体力学代写Fluid Mechanics代考|Effect of the filter

In this section, an analysis of the filter choice is proposed. The previous box filter (equation [2.19]) is compared to a Gaussian filter (equation $[2.20])$
$$G(x)=\frac{1}{\bar{\Delta}} h\left(\frac{\widehat{\Delta}}{2}-|x|\right)$$
where $h$ is the Heaviside function and $\widehat{\Delta}$ is the filtered width equal to $2 \Delta x$ for the two filters.
$$G(x)=\sqrt{\frac{6}{\pi \tilde{\Lambda}^{2}}} \exp \left(\frac{-6 x^{2}}{\tilde{\Delta}^{2}}\right)$$
Figure $2.7$ presents a comparison of the subgrid term weight according to the filter. Table $2.4$ orders the term according to its contribution with the Gaussian filter, and has to be compared with Table 2.3. The weight of the subgrid terms $\tau_{\text {superf, }}, \tau_{\text {conv }}$ and $\tau_{\text {diff }}$ decreases with the Gaussian filter, but increases for $\tau_{\text {interf }}$ and $\tau_{\text {pressure }}$. Globally, the subgrid term contribution is then smaller. This is an important result because if a subgrid term has a limited contribution in the balance equations, the errors made by modeling it will have a limited impact. Thus, it can be preferable to use a filter that limits the contribution of the subgrid terms.

Then, the turbulence models are compared with the two filters in Figure $2.8$ for the oil phase. The same results are obtained with the water phase. The filter has no effect on Smagorinsky’s and the WALE models. These results were expected as these models are not strongly linked to the filter choice, contrary to the Bardina model and ADM. For these two models, the relative error decreases when the Gaussian filter is applied, except for the pressure subgrid term. Once again, the results are in agreement with the expectations. Indeed, in practice, applying a filter corresponds to evaluating a quantity in terms of its value over a few cells. With a box filter with a filter size of 2 , the filtered quantity is evaluated by averaging all of the values over the cells, with at least one common node with the cell of interest. The surrounding cells are considered with the same weight. However, with the Gaussian filter, the same cells are considered, but they are weighted according to the distance from the cell of interest. This approach seems to be better, as the flow in the cells far from the cell of interest can be quite decorrelated from the flow in the cell of interest. This difference can be large enough to induce deconvolution errors and modeling inaccuracies.

## 物理代写|流体力学代写Fluid Mechanics代考|Effect of the filter

$$G(x)=\frac{1}{\bar{\Delta}} h\left(\frac{\widehat{\Delta}}{2}-|x|\right)$$

$$G(x)=\sqrt{\frac{6}{\pi \tilde{\Lambda}^{2}}} \exp \left(\frac{-6 x^{2}}{\tilde{\Delta}^{2}}\right)$$

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