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In the Pre-Analysis & Start-Up step, we'll review the following:
A particle laden flow is a multiphase flow where one phase is the fluid and the other is dispersed particles. Governing equations for both phases are implemented in Fluent. To run a meaningful simulation, a review of the theory is necessary.
In the simulations considered for this tutorial, the fluid flow is a 2D perturbed periodic double shear layer as described in the first section. The geometry is Lx = 59.15m, Ly = 59.15m, and the mesh size is chosen as {latex}$\Delta x = L_x / n_x${latex} in order to resolve the smallest vorticies. As a rule of thumb. One typically needs about 20 grid points across the shear layers, where the vorticies are going to develop. The boundary conditions are periodic in the x and y directions. The fluid phase satisfies the Navier-Stokes Equations: -Momentum Equations {latex} \begin{eqnarray*} \rho_f (\frac{d \textbf{u}_f}{dt}+\textbf{u}_f \cdot \nabla \textbf{u}_f)=- \nabla p + \mu \nabla ^2 \textbf{u}_f + \textbf{f} \end{eqnarray*} {latex} -Continuity Equation {latex} \begin{align*} \frac{\partial \rho_f}{\partial t} + \nabla \cdot (\rho \textbf{u}_f)=0 \end{align*} {latex} where {latex}$\textbf{u}${latex} is the fluid velocity, {latex}$p${latex} the pressure, {latex}$\rho_f${latex} the fluid density and {latex}$\textbf{f}${latex} is a momentum exchange term due to the presence of particles. When the particle volume fraction and the particle mass loading {latex}$M=\phi \rho_p/\rho_f${latex} are very small, it is legitimate to neglect the effects of the particles on the fluid: {latex}$\textbf{f}${latex} can be set to zero. This type of coupling is called one-way. In these simulations the fluid phase is air, while the dispersed phase is constituted of about 400 glass beads of diameter a few dozens of micron. This satises both conditions {latex}$\phi \ll 1${latex} and {latex}$M \ll 1${latex} |
One way-coupling is legitimate here. See ANSYS documentation (16.2) for further details about the momentum exchange term.
The suspended particles are considered as rigid spheres of same diameter d, and density {latex}$\rho_p${latex}. Newton's second law written for the particle i stipulates: {latex}$m_p \frac{d \textbf{u}_p^i}{dt}=\textbf{f}_ex^i${latex} where {latex}$\texfbf{u}_p^i${latex} is the velocity of particle i, {latex}$\textbf{f}_ex^i${latex} the forces exerted on it, and {latex}$m_p${latex} its mass. In order to know accurately the hydrodynamic forces exerted on a particle one needs to resolve the flow to a scale significantly smaller than the particle diameter. This is computationally prohibitive. Instead, the hydrodynamic forces can be approximated roughly to be proportional to the drift velocity: {latex}$\frac{d textbf{u}_p^i}{dt}=\frac{\textbf{v}-textbf{u}_p^i}{\tau_p}${latex} where {latex}$\tau_p${latex} is known as the particle response time. This equation needs to be solved for all particles present in the domain. This is done in Fluent via the module: Discrete Phase Model(DPM). |
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