SimCafe

Space shortcuts

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

Versions Compared

Old Version 36

changes.mady.by.user user-e0eda

Saved on

compared with

New Version Current

changes.mady.by.user user-b349d

Saved on

Key

  • This line was added.
  • This line was removed.
  • Formatting was changed.
Comment: Migration of unmigrated content due to installation of a new plugin

...

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,

...

the

...

fluid

...

flow

...

is

...

a

...

2D

...

perturbed

...

periodic

...

double

...

shear

...

layer

...

as

...

described

...

in

...

the

...

first

...

section.

...

The

...

geometry

...

is

...

Lx

...

=

...

59.15m,

...

Ly

...

=

...

59.15m,

...

and

...

the

...

mesh

...

size

...

is

...

chosen

...

as

...

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
}
{\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}
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 
...
 legitimate 
...
 to 
...
 neglect 
...
 the 
...
 effects 
...
 of 
...
 the 
...
 particles 
...
 on 
...
 the 
...
 fluid: 
...
 Latex
...
{\large$$\mathbf{f}$$}
...
 can 
...
 be 
...
 set 
...
 to 
...
 zero. 
...
 This 
...
 type 
...
 of 
...
 coupling 
...
 is 
...
 called 
...
 one-way. 
...
 In 
...
 these 
...
 simulations 
...
 the 
...
 fluid 
...
 phase 
...
 is 
...
 air, 
...
 while 
...
 the 
...
 dispersed 
...
 phase 
...
 is 
...
 constituted 
...
 of 
...
 about 
...
 400 
...
 glass 
...
 beads 
...
 of 
...
 diameter 
...
 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 
...
 as 
...
 rigid 
...
 spheres 
...
 of 
...
 same 
...
 diameter 
...
 d, 
...
 and 
...
 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 $$m_p \frac{d \mathbf{u}_p^i}{dt}=\mathbf{f}_{ex}^i$$}{latex}
where {latex}
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 
...
 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 
...
 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$$St = \frac{\tau_p}{\tau_f}$$}{latex}

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
Go to all FLUENT Learning Modules
...