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Pre-Analysis & Start-Up
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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 |
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{\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 |
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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
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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
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Latex |
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{\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 |
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{ |
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\large$$p$$} |
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pressure,
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Latex |
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{\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 |
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{\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 |
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{\large$$\phi$$} |
Latex |
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{\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 |
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{\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 |
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{\large$$\phi \ll |
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1$$} |
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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 |
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{\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 |
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{\large $$m {latex}$m_p \frac{d \textbfmathbf{u}_p^i}{dt}=\textbfmathbf{f}_ex^i${latexex}^i$$} |
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 |
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{\large $$\mathbf{f}_ |
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{ |
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ex}^i$$} |
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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 |
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{\large $$ {latex}$m_p${latex} its mass. In order to know accurately the hydrodynamic forces exerted on a particle one needs to resolve the flow to a scale significantly smaller than the particle diameter. This is computationally prohibitive. Instead, the hydrodynamic forces can be approximated roughly to be proportional to the drift velocity: {latex}$\frac{d \textbfmathbf{u}_p^i}{dt}=\frac{\textbfmathbf{vu}_f-textbf\mathbf{u}_p^i}{\tau_p}${latex$$} 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 Model(DPM). |
Expected Results
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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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