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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.
Fluid Phase:
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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. .
Particle Phase:
The suspended particles are considered as rigid spheres of same diameter d, and density
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{latex}$\rho_p${latex} |
. Newton’s second law written for the particle i stipulates:
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{latex}$m_p \frac{d \textbf{u}_p^i}{dt}${latex} |
where uip is the velocity of particle i, Fiex the forces exerted on it, and mp 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 computationnaly pro- hibitive. Instead, the hydrodynamic forces can be approximated roughly to be proportional to the drift velocity:
d u ip = v − u ip dt τp
where τp = XX 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 ?.
Expected Results
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Under Construction |
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