...
Wiki Markup |
---|
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}{\large$large$$\Delta x = L_x / n_x$x$$}{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} {\large \begin{eqnarray*} \rho_f (\frac{d \mathbf{u}_f}{dt}+\mathbf{u}_f \cdot \nabla \mathbf{u}_f)=- \nabla p + \mu \nabla ^2 \mathbf{u}_f + \mathbf{f} \end{eqnarray*} } {latex} -Continuity Equation {latex} {\large \begin{align*} \frac{\partial \rho_f}{\partial t} + \nabla \cdot (\rho \mathbf{u}_f)=0 \end{align*} } {latex} where {latex}{\large$large$$\mathbf{u}$$$}{latex} is the fluid velocity, {latex}{\large$p$large$$p$$}{latex} the pressure, {latex}{\large$large$$\rho_f$f$$}{latex} the fluid density and {latex}{\large$large$$\mathbf{f}$$$}{latex} is a momentum exchange term due to the presence of particles. When the particle volume fraction {latex}{\large$large$$\phi$phi$$}{latex} and the particle mass loading {latex}{\large$Mlarge$$M=\phi \rho_p/\rho_f$f$$}{latex} are very small, it is legitimate to neglect the effects of the particles on the fluid: {latex}{\large$large$$\mathbf{f}$$$}{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 satisfies both conditions {latex}{\large$large$$\phi \ll 1$1$$}{latex} and {latex}{\large$Mlarge$$M \ll 1$1$$}{latex} |
One way-coupling is legitimate here. See ANSYS documentation (16.2) for further details about the momentum exchange term.
...
Wiki Markup |
---|
The suspended particles are considered as rigid spheres of same diameter d, and density {latex}{\large$large$$\rho_p$p$$}{latex}. Newton's second law written for the particle i stipulates: {latex}{\large $$m_p \frac{d \mathbf{u}_p^i}{dt}=\mathbf{f}_{ex}^i$$}{latex} where {latex}{\large$$\mathbf{u}_p^i$$}{latex} is the velocity of particle i, {latex}{\large $$\mathbf{f}_{ex}^i$$}{latex} the forces exerted on it, and {latex}{\large $$m_p$$}{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}{\large $$\frac{d \mathbf{u}_p^i}{dt}=\frac{\mathbf{u}_f-\mathbf{u}_p^i}{\tau_p}$$}{latex} where {latex}{\large $$\tau_p$$}{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). |
...
Wiki Markup |
---|
The particle response time measures the speed at which the particle velocity adapts to the local flow speed. Non-inertial particles, or tracers, have a zero particle response time: they follow the fluid streamlines. Inertial particles with {latex}{\large$large$$\tau_p \neq 0$0$$}{latex} might adapt quickly or slowly to the fluid speed variations depending on the relative variation of the flow and the particle response time. This rate of adaptation is measured by a non-dimensional number called Stokes number representing the ratio of the particle response time to the flow characteristic time scale. {latex}{\large$Stlarge$$St = \frac{\tau_p}{\tau_f}$$$}{latex} In these simulations, the characteristic flow time is the inverse of the growth rate of the vortices in the shear layers. This is also predicted by the Orr-Sommerfeld equation. For the particular geometry and configuration we used in this tutorial, the growth rate is {latex}{\large$large$$\gamma = 0.1751 s^{-1} = \frac{1}{\tau_f}$$$}{latex}. When St = 0 the particles are tracers. They follow the streamlines and, in particular, they will not be able to leave a vortex once caught inside. When {latex}{\large$Stlarge$$St \gg 1$1$$}{latex}, particles have a ballistic motion and are not affected by the local flow conditions. They are able to shoot through the vorticies without a strong trajectory deviation. Intermediate cases {latex}{\large$Stlarge$$St \approx 1$1$$}{latex} have a maximum coupling between the two phases: particles are attracted to the vorticies, but once they reach the highly swirling vortex cores they are ejected due to their non zero inertia. In this tutorial, we will consider a nearly tracer case St = 0.2, an intermediate case St = 1 and a nearly ballistic case St = 5. |
...