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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:
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In
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the
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simulations
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considered
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for
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this
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tutorial,
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the
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fluid
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flow
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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
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the
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first
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section.
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The
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geometry
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is
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Lx
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=
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59.15m,
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Ly
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=
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59.15m,
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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
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| Latex |
|---|
{\large$$\Delta x = L_x / n_ |
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x$$} |
...
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order
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to
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resolve
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the
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smallest
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vorticies.
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As
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a
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rule
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of
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thumb. One
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typically
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needs
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about
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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
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x
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and
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y
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directions.
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The
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fluid
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phase
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satisfies the
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Navier-Stokes
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Equations:
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-Momentum
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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}
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-Continuity
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Equation
| Latex |
|---|
{latex} \large
\begin{align*}
\frac{\partial \rho_f}{\partial t} + \nabla \cdot (\rho_f \textbfmathbf{u}_f)=0
\end{align*}
{latex}
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where
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| Latex |
|---|
{\large$$\mathbf{u} |
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$$} |
...
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the
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fluid
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velocity,
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| Latex |
|---|
...
{ |
...
\large$$p$$} |
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pressure,
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| Latex |
|---|
{\large$$\rho_ |
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f$$} |
...
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fluid
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density
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and
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| Latex |
|---|
{\large$$\mathbf{f} |
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$$} |
...
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a
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momentum
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exchange
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term
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due
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to
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the
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presence
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of
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particles.
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When
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the
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particle
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volume
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fraction
| Latex |
|---|
{\large$$\phi$$} |
| Latex |
|---|
{\large$$M=\phi \rho_p/\rho_ |
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f$$} |
...
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very
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small,
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it
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is
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legitimate
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to
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neglect
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the
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effects
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of
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the
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particles
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on
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the
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fluid:
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| Latex |
|---|
{\large$$\mathbf{f} |
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$$} |
...
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be
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set
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to
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zero.
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This
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type
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of
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coupling
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is
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called
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one-way.
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In
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these
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simulations
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the
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fluid
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phase
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is
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air,
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while
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the
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dispersed
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phase
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is
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constituted
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of
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about
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400
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glass
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beads
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of
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diameter
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a
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few
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dozens
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of
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micron.
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This
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satisfies both
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conditions
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| Latex |
|---|
{\large$$\phi \ll |
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1$$} |
...
...
| Latex |
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{\large$$M \ll |
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1$$} |
One way-coupling is legitimate here. See ANSYS documentation (16.2) for further details about the momentum exchange term.
Particle Phase:
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The
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suspended
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particles
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are
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considered
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as
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rigid
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spheres
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of
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same
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diameter
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d,
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and
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density
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| Latex |
|---|
{\large$$\rho_ |
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p$$} |
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Newton’s second
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law
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written
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for
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the
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particle
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i
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stipulates:
| Latex |
|---|
{\large $$m {latex}$m_p \frac{d \textbfmathbf{u}_p^i}{dt}=\textbfmathbf{f}_{ex}^i${latex^i$$} |
where
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| Latex |
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{\large$$\mathbf{u}_ |
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p^i$$} |
...
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the
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velocity
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of
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particle
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i,
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| Latex |
|---|
{\large $$\mathbf{f}_{ex} |
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^i$$} |
...
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forces
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exerted
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on
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it,
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and
| Latex |
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{\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 $$ {latex}$m_p${latex} its mass. In order to know accurately the hydrodynamic forces exerted on a particle one needs to resolve the flow to a scale significantly smaller than the particle diameter. This is computationally prohibitive. Instead, the hydrodynamic forces can be approximated roughly to be proportional to the drift velocity: {latex}$\frac{d \textbfmathbf{u}_p^i}{dt}=\frac{\textbfmathbf{vu}_f-\textbfmathbf{u}_p^i}{\tau_p}${latex$$} 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 |
where
| Latex |
|---|
{\large $$\tau_p=\rho_p D^2/(18\mu)$$} |
| Latex |
|---|
{\large $$\rho_p$$} |
Choosing the Cases:
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The
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particle
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response
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time
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measures
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the
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speed
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at
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which
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the
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particle
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velocity
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adapts
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to
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the
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local
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flow
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speed.
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Non-inertial
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particles,
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or
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tracers,
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have
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a
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zero
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particle
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response
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time:
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they
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follow
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the
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fluid
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streamlines.
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Inertial
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particles
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with
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| Latex |
|---|
{\large$$\tau_p \neq |
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0$$} |
...
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adapt
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quickly
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or
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slowly
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to
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the
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fluid
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speed
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variations
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depending
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on
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the
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relative
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variation
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of
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the
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flow
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and
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the
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particle
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response
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time.
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This
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rate
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of
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adaptation
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is
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measured
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by
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a
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non-dimensional
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number
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called
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Stokes
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number
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representing
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the
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ratio
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of
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the
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particle
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response
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time
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to
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the
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flow
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characteristic
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time
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scale.
| Latex |
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{\large$$St {latex}$\St = \frac{\tau_p}{\tau_f}${latex} In these $$} |
In these simulations,
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the
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characteristic
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flow
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time
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is
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the
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inverse
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of
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the
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growth
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rate
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of
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the
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vortices
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in
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the
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shear
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layers.
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This
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is
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also
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predicted
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by
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the
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Orr-Sommerfeld
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equation.
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For
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the
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particular
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geometry
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and
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configuration
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we
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used
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in
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this
...
tutorial,
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the
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growth
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rate
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is
...
| Latex |
|---|
{\large$$\gamma = 0.1751 s^{-1} = \frac{1}{\tau_f}$$} |
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When
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St
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=
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0
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the
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particles
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are
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tracers.
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They
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follow
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the
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streamlines
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and,
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in
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particular,
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they
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will
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not
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be
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able
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to
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leave
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a
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vortex
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once
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caught
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inside.
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When
| Latex |
|---|
{\large$$St \gg |
...
1$$} |
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particles
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have
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a
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ballistic
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motion
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and
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are
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not
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affected
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by
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the
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local
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flow
...
conditions.
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They
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are
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able
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to
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shoot
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through
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the
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vorticies
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without
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a
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strong
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trajectory
...
deviation.
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Intermediate
...
cases
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| Latex |
|---|
{\large$$St \approx |
...
1$$} |
...
...
a
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maximum
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coupling
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between
...
the
...
two
...
phases:
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particles
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are
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attracted
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to
...
the
...
vorticies,
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but
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once
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they
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reach
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the
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highly
...
swirling
...
vortex
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cores
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they
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are
...
ejected
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due
...
to
...
their
...
non
...
zero
...
inertia.
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In
...
this
...
tutorial,
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we
...
will
...
consider
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a
...
nearly
...
tracer
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case
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St
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=
...
0.2,
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an
...
intermediate
...
case
...
St
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=
...
1
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and
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a
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nearly
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ballistic
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case
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St
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=
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5.
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Expected Results
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Go to all FLUENT Learning Modules
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