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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
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simulations
...
considered
...
for
...
this
...
tutorial,
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the
...
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 |
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{\large$$\Delta x = L_x / n_ |
...
x$$} |
...
...
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 |
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{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 |
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{latex} \large \begin{align*} \frac{\partial \rho_f}{\partial t} + \nabla \cdot (\rho_f \textbfmathbf{u}_f})=0 \end{align*} {latex} |
where
...
Latex |
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{\large$$\mathbf{u}$$} |
Latex |
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{\large$$p$$} |
Latex |
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{\large$$\rho_f$$} |
Latex |
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{\large$$\mathbf{f}$$} |
Latex |
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{\large$$\phi$$} |
Latex |
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{\large$$M=\phi \rho_p/\rho_ |
...
f$$} |
...
...
very
...
small,
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it
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is
...
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
...
the
...
fluid:
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Latex |
---|
{ |
...
\ |
...
large$$\ |
...
mathbf{f} |
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$$} |
...
...
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
...
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
...
satisfies both
...
conditions
...
Latex |
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{\large$$\phi \ll |
...
1$$} |
...
...
Latex |
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{\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 |
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{\large$$\rho_p$$} |
Latex |
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{\large $$m_p \frac{d \mathbf{u}_p^i}{dt}=\mathbf{f}_{ex}^i$$} |
where
Latex |
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{\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
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the velocity of particle i,
Latex |
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{\large $$\mathbf{f}_{ex}^i$$} |
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 |
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{\large $$\frac{d \mathbf{u}_p^i}{dt}=\frac{\mathbf{u}_f-\mathbf{u}_p^i}{\tau_p}$$} |
where
Latex |
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{\large $$\tau_p=\rho_p D^2/(18\mu)$$} |
Latex |
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{\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 |
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{\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 |
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{\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 |
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{\large$$\gamma = 0.1751 s^{-1} = \frac{1}{\tau_f}$$} |
When
Latex |
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{\large$$St \gg 1$$} |
Latex |
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{\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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