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

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## 物理代写|流体力学代写Fluid Mechanics代考|Viscous penalty method

The viscous penalty method, used in this chapter to perform particle-resolved direct numerical simulations, is a fictitious domain method where fixed staggered Cartesian grids are used to discretize both fluid and solid media. As explained by Kataoka (1986) for fluid/fluid two-phase flows and Vincent et al. (2014) for particle flows, the resulting model implicitly takes into account the coupling between different phases separated by resolved interfaces, i.e. larger than the mesh cell size. Given that all the configurations simulated in this chapter involve fixed particles, the motion equations then read:
\begin{aligned} \nabla \cdot \mathbf{u} &=0 & \text { [3.1] } \ \rho\left(\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u} \cdot \nabla) \mathbf{u}\right) &=-\nabla p+\rho \mathbf{g}+\nabla \cdot\left[\mu\left(\nabla \mathbf{u}+\nabla^{t} \mathbf{u}\right)\right]+\mathbf{F}{m} & \text { [3.2] } \end{aligned} where $\mathbf{u}$ is the velocity, $p$ is the pressure, $t$ is the time, $\mathbf{g}$ is the gravity vector, and $\rho$ and $\mu$ are, respectively, the density and viscosity of the equivalent fluid. The source term $\mathbf{F}{m}$ is used to impose a flow rate to the fluid if required.
The one-fluid model is almost identical to the classical incompressible Navier-Stokes equations, except that the local properties of the equivalent fluid $(\rho$ and $\mu$ ) depend on $C$. In this chapter, an arithmetic average is used for density $\left(\rho=C \rho_{s}+(1-C) \rho_{f}\right)$ and a harmonic average is considered for viscosity $\left(\mu=\frac{\mu_{s} \mu_{f}}{C \mu_{f}+(1-C) \mu_{s}}\right)$ (Vincent et al. 2014).

Satisfying the incompressible and solid constraints in fluid and particles requires developing a specific model. Two penalty approaches are proposed and detailed in the next section to tackle with these constraints:

• ensuring the solid behavior in the solid zones where $C=1$ requires a specific rheological law to be defined for the rigid fluid part without imposing the velocity, as the particle velocities are not always known a priori in particulate motions (particle sedimentation, fluidized beds, turbulence particle interaction). A specific model is implemented for handling the solid particle behavior in the one-fluid Navier-Stokes equations. It is based on a decomposition of the viscous stress tensor and on a penalty method that acts on the viscosity, which tends to large values in the particles (Caltagirone and Vincent 2001) to implicitly impose the solid behavior and also the coupling between fluid and solid motion. For fixed particles, the velocity of the cell containing the centroid of the particle is imposed equal to zero. The viscous penalty method propagates the zero velocity in the whole solid medium;

## 物理代写|流体力学代写Fluid Mechanics代考|Validation of Aslam extension

To validate these extensions, we used the same example as Aslam (2004). i.e. a $[-\pi, \pi]^{2} 2 \mathrm{D}$ domain with a particle located at the center of the domain and a function $g$ to be extrapolated given by:
$$g(\boldsymbol{x})=\left{\begin{array}{ll} \cos \left(\boldsymbol{x}{1}\right) \sin \left(\boldsymbol{x}{2}\right) & \text { if } \psi(\boldsymbol{x})>0 \ 0 & \text { otherwise } \end{array}\right. \text { (i.e. inside the particle) }$$
In this example, the function $g$ is defined inside the particle, i.e. in $\Omega_{1}$. It has to be extrapolated in the band $\mathcal{B}$, as illustrated in Figure $3.5$.

This example was extended to non-spherical particles, with an ellipse and a square. Figure $3.6$ shows the contours of $g$ inside (a) a circle, (b) an ellipse and (c) a square, which will be extrapolated in $\mathcal{B}$, i.e. the delimited white zone outside the particle, using equations [3.9].

The function $g$ is extrapolated using Aslam extension, i.e. resolving equations [3.9], for four different orders:

• Constant Aslam extension $m=1$ : this consists of resolving equation $\frac{\partial g}{\partial \sigma}+H \nabla g \cdot \mathbf{n}-0$ in $\mathcal{B}$ until $g$ reaches a steady state, i.e. $\frac{\partial g}{\partial \sigma}-0$, i.e. $\frac{\partial g}{\partial \mathbf{n}}=\nabla g \cdot \mathbf{n}=0$, which means that extrapolated $g$ is constant in the Linear Aslam extension $m=2$ : this consists of extrapolating the first normal derivative $g_{2}=\frac{\partial g}{\partial \mathbf{n}}$ from $\Omega_{1}$, where it is computed from $g$ using [3.10], to the band $\mathcal{B}$ using constant Aslam extension. Then, $g$ is extrapolated from $\Omega_{1}$ to $\mathcal{B}$ by resolving equation $\frac{\partial g}{\partial \sigma}+H \nabla g \cdot \mathbf{n}=g_{2}$ until $g$ reaches a steady state. An illustration of such a function is given in Figure 3.8, where we can observe that $g$ is no more constant in the normal direction to the particle surface in $\mathcal{B}$ $\left(\nabla g \cdot \mathbf{n}=g_{2}\right)$

## 物理代写|流体力学代写Fluid Mechanics代考|Viscous penalty method

$$\nabla \cdot \text { 在 }=0 \quad[3.1] r\left(\frac{\partial \text { 在 }}{\partial R^{4}}+(\text { 在 } \cdot \nabla) \text { 在 }\right)=-\nabla p+r \mathbf{G}+\nabla \cdot\left[\nVdash\left(\nabla \text { 在 }+\nabla^{\text {在 }}\right)\right]+\mathbf{F} \text { 羊 }$$

• 确保固体区域中的固体行为 $C=1$ 需要在不施加速庻的情况下为刚性㳘体部分定义特 定的流变学定律，因为粒子速度在粒子运动（粒子沉降、流化床、湍流粒子相互作 用) 中并不总是先验已知的。实现了一个特定模型来处理单流体 Navier-Stokes 方 程中的固体粒子行为。它基于粘性应力张量的分解和作用于粘度的惩恩方法，该方 法倾向于在颗粒中使用较大的值 (Caltagirone 和 Vincent 2001) 以隐含地施加固 体行为以及流体和固体之间的耦合运动。对于固定粒子，包含粒子质心的单元的速 席被强制为零。粘性惩罚法在整个固体介质中传播零速度;

## 物理代写|流体力学代写Fluid Mechanics代考|Validation of Aslam extension

$\$ \$$g(\backslash b o l d s y m b o l{x})=\backslash left { \cos (x 1) \sin (x 2) \quad if \psi(x)>00 \quad otherwise 、正确的。Itext { (即在粒子内部) } \ \$$

(b) 椭圆和 (c) 正方形内，将外推 $\mathcal{B}$ ，即使用方程 $[3.9]$ ，在粒子外部划定的白色区域。

• 持续 Aslam 扩展 $m=1$ ：这包括解决方程 $\frac{\partial g}{\partial \sigma}+H \nabla g \cdot \mathbf{n}-0$ 在B直到 $g$ 达到稳 定状态，即 $\frac{\partial g}{\partial \sigma}-0 ， \mid \mathrm{E} \frac{\partial g}{\partial \mathbf{n}}=\nabla g \cdot \mathbf{n}=0$ ，这意味着外推 $g$ 在线性 Aslam 扩展中 是常数 $m=2$ : 这包括外推一阶正态导数 $g_{2}=\frac{\partial g}{\partial \mathbf{n}} 从 \Omega_{1}$ ，它是从哪里计算出来的 $g$ 使用 [3.10]，到乐队 $\mathcal{B}$ 使用恒定的 Aslam 扩展。然后， $g$ 推断自 $\Omega_{1}$ 至B通过求解方 程 $\frac{\partial g}{\partial \sigma}+H \nabla g \cdot \mathbf{n}=g_{2}$ 直到 $g$ 达到稳定状态。图 $3.8$ 给出了这种函数的说明，涐 们可以观察到 $g$ 在粒子表面的法线方向上不再是常数 $B\left(\nabla g \cdot \mathbf{n}=g_{2}\right)$

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