1. Lecture Objectives:
Review discretization methods for advection diffusion equation
Accuracy
Numerical Stability
Unsteady-state CFD
Explicit vs. Implicit method
HW2
Turbulence

2. Steady–state 1D exampleI)
X direction
If Vx > 0,
If Vx < 0,
Convection term - Upwind-scheme: W P
dxw
dxe E
and a)
and Dx
Diffusion term: w e b)
When mesh is uniform:
DX = dxe = dxw
Assumption:
Source is constant over the control volume c)
Source term:

3. Advection diffusion equation 1-D, steady-stateDx N Dx
N+1
Different notation: Dx
General equation

4. 1D example multiple (N) volumes
N unknowns 1 2 3 i
N-1 N
Equation for volume 1 N
equations
Equation for volume 2
……………………………
Equation matrix:
For 1D problem
3-diagonal matrix

5. 3D problem Equation in the general format: H N W P E S L
Wright this equation for each discretization volume
of your discretization domain A F
60,000 elements
60,000 cells (nodes)
N=60,000 x =
60,000 elements
7-diagonal matrix
This is the system for only
one variable ( )
When we need to solve p, u, v, w, T, k, e, C
system of equation is very larger

6. Boundary conditions for CFD application - indoor airflow
Real geometry
Model geometry
Where are the boundary
Conditions?

7. CFD ACCURACY Depends on airflow in the vicinity of Boundary conditions
1) At air supply device
2) In the vicinity of occupant
3) At room surfaces
Detailed modeling
limited by
computer power

8. Surface boundaries
thickness mm
for forced
convection
Wall surface
W use wall functions to model the flow in the vicinity of surface
Using relatively large mesh (cell) size.

9. Airflow at air supply devices
momentum sources
Complex geometry - Δ~10-4m
We can spend all our
computing power for one small detail

10. Diffuser jet properties
High Aspiration diffuser D D L L
How small cells do you need?
We need simplified models for diffusers

11. Simulation of airflow in In the vicinity of occupants
How detailed should we make the geometry?
Peter V. Nielsen

12. General Transport Equation Unsteady-stateH N
Equation in the algebraic format: W P E S L
We have to solve the system matrix for each time step !
Transient term:
Are these values for step  or + ?
Unsteady-state 1-D
If: -  explicit method
- + - implicit method

13. General Transport Equation unsteady-state 1-D
Fully explicit method:
Implicit method:
Value form previous time step (known value)
Make the difference between
- Calculation for different time step
- Calculation in iteration step