Lecture Objectives
- Unsteady State Simulation
- Example
- Modeling of PM
Unsteady-state (Transient)
CFD simulations
Computationally expensive
Steps
• Identify the problem
– Many problems do not require unsteady-state simulation!
– Identify equations which should be unsteady-state
• Define the simulation period
• Define the required time steps
• Adjust other simulation parameters
– turbulence model, mesh, convergence criteria, number of required
iterations, etc.
– Require substantial investigation for each problem
Computationally very expensive


ρ
 ρ div V  Γ  ,eff grad   S
τ

(t Dτ  t )
ρ
ρ
τ
Dτ
Change of  in volume dxdydz In Time
Discretize equation
a P Φ P  a E Φ E  a W Φ W  a S ΦS  a N Φ N  a H Φ H  a L Φ L  f
System of equation for each time step
ap and f are function of Dt
f is function of previous value for 
x
=
1) Solve the system using the simple algorithm
2) Change the boundary conditions
3) Update the coefficient
4) Solve the new system of equations
A
Φ
F
Steady-state, unsteady-state
or quasi-steady-state
Examples
•
•
•
•
•
•
Airflow around the airplane
Airflow in the room
Airflow around the building
Injection of pollutant in the chamber experiment
Flow in the automobile engine cylinder
DNS simulation of flow in the boundary layer
Simulation period and time step
• Simulation period
– Depends on the boundary condition of
considered phenomenon
• Time step
– Depends on the time scale
– With too large time step quasi-steady-state simulation

D
ρ
ρ
~0
τ
Dτ
Set of steady state simulations
(there is no link in-between previous and next time step)
Time step Dt
• Uniform
• Variable
– Linear
– Piecewise
• User defined
Transient boundaries
• For unsteady-state
airflow created
by transient
boundaries
Particulate matters (PM)
• Properties
– Size, density, liquid, solid, combination, …
• Sources
– Airborne, infiltration, resuspension, ventilation,…
• Sinks
- Deposition, filtration, ventilation (dilution),…
• Distribution
- Uniform and nonuniform
• Human exposure
Properties
ASHRAE
Transaction 2004
Particle size distribution
ASHRAE Transaction 2004
Ventilation system affect the PM concentration in indoor environment !
Two basic approaches
for modeling of particle dynamics
• Lagrangian Model
– particle tracking
– For each particle
ma=SF
• Eulerian Model
– Multiphase flow (fluid and particles)
– Set of two systems of equations
Lagrangian Model
particle tracking
A trajectory of the particle in the vicinity of the spherical
collector is governed by the Newton’s equation
Forces that affect the particle
m∙a=SF
(rVvolume) particle ∙dvx/dt=SFx
(rVvolume) particle ∙dvy/dt=SFy
(rVvolume) particle ∙dvz/dt=SFz
System of equation for each particle
Solution is velocity and direction of each particle
dt  Dt
Lagrangian Model
particle tracking
Basic equations
- momentum equation based on Newton's second law
V
 3
i  Fdrag  F
d r
6 P P t
e
Drag force due to the friction
between particle and air
Fdrag  f  u  u 
p

- dp is the particle's diameter,
- rp is the particle density,
- up and u are the particle and fluid instantaneous velocities in the i direction,
- Fe represents the external forces (for example gravity force).
This equation is solved at each time step for every particle.
The particle position xi of each particle are obtained using the following equation:
dxi
 Vi
dt
For finite time step
dt  Dt
Algorithm for CFD and
particle tracking
Unsteady state airflow
Steady state airflow
Airflow (u,v,w)
Airflow (u,v,w) for time step t
Steady state
Injection of particles
Injection of particles
Particle distribution for time step t
Particle distribution for time step t
Particle distribution for time step t+Dt
Airflow (u,v,w) for time step t+Dt
Particle distribution for time step t+2Dt
Particle distribution for time step t+Dt
…..
…..
Case 1 when airflow is not affected by particle flow
Case 2 particle dynamics affects the airflow
One way coupling
Two way coupling
Eulerian Model
• Solve several sets of NS equations
• Define the boundary conditions in-between phases
Multiphase/Mixture Model
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•
•
•
•
•
•
•
Mixture model
Secondary phase can be granular
Applicable for solid-fluid simulations
Granular physics
Solve total granular pressure to momentum equation
Use Solids viscosity for dispersed solid phase
Density difference should be small.
Useful mainly for liquid-solids multiphase systems
There are models applicable for particles in the air
Multiphase flow
Multiphase flow can be classified in the following
regimes:
- gas-liquid or liquid-liquid flows
- gas-solid flows
– particle-laden flow: discrete solid particles in a continuous gas
– pneumatic transport: flow pattern depends on factors such as solid
loading, Reynolds numbers, and particle properties. Typical patterns are
dune flow, slug flow, packed beds, and homogeneous flow.
– fluidized beds: consist of a vertical cylinder containing particles where
gas is introduced through a distributor.
- liquid-solid flows
- three-phase flows
Multiphase Flow Regimes
Fluent user manual 2006