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  • Particles in a Periodic Double Shear Flow - Pre-Analysis & Start-Up

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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:

...

In

...

the

...

simulations

...

considered

...

for

...

this

...

tutorial,

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the

...

fluid

...

flow

...

is

...

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

...

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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59.15m,

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Ly

...

=

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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

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Latex

...

{\

...

large$$\Delta x = L_x / n_x$$}
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
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

...

Equation

{
Latex
}
{\large 
\begin{align*} 
\frac{\partial \rho_f}{\partial t} + \nabla \cdot (\rho_f \mathbf{u}_f)=0
\end{align*}
} 
{latex}

where {latex}{\large$

where

Latex
{\large$$\mathbf{u}

...

$$}
is the fluid velocity,
Latex
{\large$$p$$}
the pressure,
Latex
{\large$$\rho_f$$}
the fluid density and
Latex
{\large$$\mathbf{f}

...

$$}
is a momentum exchange term due to the presence of particles. When the particle volume fraction
Latex
{\large$$\phi$$}
and the particle mass loading
Latex
{\large$$M=\phi \rho_p/\rho_

...

f$$}
are very small,

...

it

...

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

...

particles

...

on

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the

...

fluid:

...

Latex

...

{\

...

large$$\mathbf{f}

...

$$}
can be set to zero. This type of coupling is called one-way.

...

In

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these

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simulations

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the

...

fluid

...

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

...

is

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constituted

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of

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about

...

400

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glass

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beads

...

of

...

diameter

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a

...

few

...

dozens

...

of

...

micron.

...

This

...

satisfies

...

both

...

conditions

...

Latex

...

{\

...

large$$\phi \ll

...

1$$}
and
Latex
{\large$$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 
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 as 
...
 rigid 
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 spheres 
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 of 
...
 same 
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 diameter 
...
 d, 
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 and 
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 density 
...
 Latex
...
{\
...
large$$\rho_
...
p$$}
. Newton’s second law written for the particle i stipulates:
 Latex
{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 
 Latex
{\large$$\mathbf{u}_
...
p^i$$}
 is the velocity of particle i, 
 Latex
{\large 
...
$$\mathbf{f}_{ex}
...
^i$$}
 the forces exerted on it, and 
 Latex
{\large 
...
$$m_
...
p$$}
 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 
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 to 
...
 the 
...
 drift 
...
 velocity ref3:

{
 Latex
}{\large $$$\frac{d \mathbf{u}_p^i}{dt}=\frac{\mathbf{u}_f-\mathbf{u}_p^i}{\tau_p}$}{latex}
where {latex}$$}
where 
 Latex
{\large 
...
$$\tau
...
_p=\rho_p D^2/(18\mu)$$}
 is known as the particle response time, 
 Latex
{\large $$\rho_p$$}
 the particle density and D the particle diameter. 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).
 
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 
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 the 
...
 fluid 
...
 streamlines. 
...
 Inertial 
...
 particles 
...
 with 
...
 Latex
...
{\
...
large$$\tau_p \neq 0$$}
 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$$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 $$}
In these simulations, 
...
 the 
...
 characteristic 
...
 flow 
...
 time 
...
 is 
...
 the 
...
 inverse 
...
 of 
...
 the 
...
 growth 
...
 rate 
...
 of 
...
 the 
...
 vortices 
...
 in 
...
 the 
...
 shear 
...
 layers. 
...
 This 
...
 is 
...
 also 
...
 predicted 
...
 by 
...
 the 
...
 Orr-Sommerfeld 
...
 equation. 
...
 For 
...
 the 
...
 particular 
...
 geometry 
...
 and 
...
 configuration 
...
 we 
...
 used 
...
 in 
...
 this 
...
 tutorial, 
...
 the 
...
 growth 
...
 rate 
...
 is 
...
 Latex
...
{\
...
large$$\gamma = 0.1751 s^{-1} = \frac{1}{\tau_f}
...
$$}
...
. 
...
 When 
...
 St 
...
 = 
...
 0 
...
 the 
...
 particles 
...
 are 
...
 tracers. 
...
 They 
...
 follow 
...
 the 
...
 streamlines 
...
 and, 
...
 in 
...
 particular, 
...
 they 
...
 will 
...
 not 
...
 be 
...
 able 
...
 to 
...
 leave 
...
 a 
...
 vortex 
...
 once 
...
 caught 
...
 inside.
...
When 
...
 Latex
...
{\
...
large$$St \gg 
...
1$$}
...
, 
...
 particles 
...
 have 
...
 a 
...
 ballistic 
...
 motion 
...
 and 
...
 are 
...
 not 
...
 affected 
...
 by 
...
 the 
...
 local 
...
 flow 
...
 conditions. 
...
 They 
...
 are 
...
 able 
...
 to 
...
 shoot 
...
 through 
...
 the 
...
 vorticies 
...
 without 
...
 a 
...
 strong 
...
 trajectory 
...
 deviation. 
...
 Intermediate 
...
 cases 
...
 Latex
...
{\
...
large$$St \approx 
...
1$$}
 have a maximum coupling between the two phases: particles are attracted to the vorticies, but once they reach the highly swirling vortex cores they are ejected due to their non zero inertia.
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. 
Go to Step 2: Geometry
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