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Pre-Analysis & Start-Up
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This site is currently under construction. Please come back after it is fully built. Thank you! |
In the Pre-Analysis & Start-Up step, we'll review the following:
- Theory for Fluid Phase
- Theory for Particle Phase
- Choosing the Cases
Pre-Analysis:
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:
...
In
...
the
...
simulations
...
considered
...
for
...
this
...
tutorial,
...
the
...
fluid
...
flow
...
is
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a
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2D
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perturbed
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periodic
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double
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shear
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layer
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as
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described
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in
...
the
...
first
...
section.
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The
...
geometry
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is
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Lx
...
=
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59.15m,
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Ly
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=
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59.15m,
...
and
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the
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mesh
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size
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is
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chosen
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as
...
| Latex |
|---|
{\large$$\Delta x = L_x / n_ |
...
x$$} |
...
...
order
...
to
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resolve
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the
...
smallest
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vorticies.
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As
...
a
...
rule
...
of
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thumb. One
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typically
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needs
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about
...
20
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grid
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points
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across
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the
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shear
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layers,
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where
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the
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vorticies
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are
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going
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to develop.
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The
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boundary
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conditions
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are
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periodic
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in
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the
...
x
...
and
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y
...
directions.
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The
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fluid
...
phase
...
satisfies the
...
Navier-Stokes
...
Equations:
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-Momentum
...
Equations
| Latex |
|---|
{latex} \large
\begin{eqnarray*}
\rho_f (\frac{d \textbfmathbf{u}_f}}{dt}+\textbfmathbf{u}_f} \cdot \nabla \textbfmathbf{u}_f})=- \nabla p + \mu \nabla ^2 \textbfmathbf{u}_f} + \textbfmathbf{f}
\end{eqnarray*} {latex
}
|
-Continuity
...
Equation
| Latex |
|---|
{latex} \large
\begin{align*}
\frac{\partial \rho_f}{\partial t} + \nabla \cdot (\rho_f \textbfmathbf{vu}_f)=0 \\ \Rightarrow \nabla \cdot \textbf{v} = 0 \\ \
\end{align*} {latex
}
where {latex}$\textbf{u_f}${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 |
where
| Latex |
|---|
{\large$$\mathbf{u}$$} |
| Latex |
|---|
{\large$$p$$} |
| Latex |
|---|
{\large$$\rho_f$$} |
| Latex |
|---|
{\large$$\mathbf{f}$$} |
| Latex |
|---|
{\large$$\phi$$} |
| Latex |
|---|
{\large$$M=\phi \rho_p/\rho_ |
...
f$$} |
...
...
very
...
small,
...
it
...
is
...
legitimate
...
to
...
neglect
...
the
...
effects
...
of
...
the
...
particles
...
on
...
the
...
fluid:
...
| Latex |
|---|
{ |
...
\ |
...
large$$\ |
...
mathbf{f} |
...
$$} |
...
...
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
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a
...
few
...
dozens
...
of
...
micron.
...
This
...
satisfies both
...
conditions
...
| Latex |
|---|
{\large$$\phi \ll |
...
1$$} |
...
...
| Latex |
|---|
{\large$$M \ll |
...
1$$} |
...
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
| Latex |
|---|
{\large$$\rho_p$$} |
| Latex |
|---|
{\large $$m_p \frac{d \mathbf{u}_p^i}{dt}=\mathbf{f}_{ex}^i$$} |
where
| Latex |
|---|
{\large$$\mathbf{u}_p^i$$} |
...
One way-coupling is legitimate here. See ANSYS documentation (16.2) for further details about the momentum exchange term.
Expected Results
...
the velocity of particle i,
| Latex |
|---|
{\large $$\mathbf{f}_{ex}^i$$} |
| Latex |
|---|
{\large $$m_p$$} |
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 ref3:
| Latex |
|---|
{\large $$\frac{d \mathbf{u}_p^i}{dt}=\frac{\mathbf{u}_f-\mathbf{u}_p^i}{\tau_p}$$} |
where
| Latex |
|---|
{\large $$\tau_p=\rho_p D^2/(18\mu)$$} |
| Latex |
|---|
{\large $$\rho_p$$} |
Choosing the Cases:
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$$\tau_p \neq 0$$} |
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$$St = \frac{\tau_p}{\tau_f}$$} |
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$$\gamma = 0.1751 s^{-1} = \frac{1}{\tau_f}$$} |
When
| Latex |
|---|
{\large$$St \gg 1$$} |
| Latex |
|---|
{\large$$St \approx 1$$} |
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.
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