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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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{\ |
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large$$\Delta x = L_x / n_x$$} |
-Momentum Equations
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x$}{latex} in order to resolve the smallest vorticies. As a rule of thumb. One typically needs about 20 grid points across the shear layers, where the vorticies are going to develop. The boundary conditions are periodic in the x and y directions. The fluid phase satisfies the Navier-Stokes Equations: -Momentum Equations {latex} {\large \begin{eqnarray*} \rho_f (\frac{d \mathbf{u}_f}{dt}+\mathbf{u}_f \cdot \nabla \mathbf{u}_f)=- \nabla p + \mu \nabla ^2 \mathbf{u}_f + \mathbf{f} \end{eqnarray*} } {latex} |
-Continuity
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Equation
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} {\large \begin{align*} \frac{\partial \rho_f}{\partial t} + \nabla \cdot (\rho_f \mathbf{u}_f)=0 \end{align*} } {latex} where {latex}{\large$ |
where
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{\large$$\mathbf{u} |
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$$} |
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{\large$$p$$} |
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{\large$$\rho_f$$} |
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{\large$$\mathbf{f} |
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$$} |
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{\large$$\phi$$} |
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{\large$$M=\phi \rho_p/\rho_ |
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f$$} |
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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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{\ |
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large$$\mathbf{f} |
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$$} |
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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
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both
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conditions
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{\ |
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large$$\phi \ll |
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1$$} |
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{\large$$M \ll |
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1$$} |
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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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{\ |
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large$$\rho_ |
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p$$} |
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{latex}. Newton's second law written for the particle i stipulates: {latex}{\large $m$$m_p \frac{d \mathbf{u}_p^i}{dt}=\mathbf{f}_{ex}^i$}{latex} where {latex}{\large$^i$$} |
where
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{\large$$\mathbf{u}_ |
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p^i$$} |
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{\large |
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$$\mathbf{f}_{ex} |
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^i$$} |
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{\large |
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$$m_ |
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p$$} |
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In
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order
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to
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know
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accurately
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the
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hydrodynamic
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forces
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exerted
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on
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a
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particle
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one
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needs
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to
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resolve
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the
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flow
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to
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a
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scale
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significantly
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smaller
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than
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the
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particle
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diameter.
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This
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is
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computationally
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prohibitive.
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Instead,
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the
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hydrodynamic
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forces
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can
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be
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approximated
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roughly
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to
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be
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proportional
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to
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the
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drift
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velocity ref3:
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}{\large $$$\frac{d \mathbf{u}_p^i}{dt}=\frac{\mathbf{u}_f-\mathbf{u}_p^i}{\tau_p}$}{latex} where {latex}$$} |
where
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{\large |
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$$\tau |
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_p=\rho_p D^2/(18\mu)$$} |
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{\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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{\ |
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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.
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{\large$$St0$}{latex} might adapt quickly or slowly to the fluid speed variations depending on the relative variation of the flow and the particle response time. This rate of adaptation is measured by a non-dimensional number called Stokes number representing the ratio of the particle response time to the flow characteristic time scale. {latex}{\large$St = \frac{\tau_p}{\tau_f}$}{latex} In these $$} |
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
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tutorial,
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the
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growth
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rate
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is
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{\ |
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large$$\gamma = 0.1751 s^{-1} = \frac{1}{\tau_f} |
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$$} |
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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
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{\ |
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large$$St \gg |
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1$$} |
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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
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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
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deviation.
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Intermediate
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cases
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{\ |
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large$$St \approx |
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1$$} |
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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