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Lecture Objectives:
• Review discretization methods for advection diffusion equation– Accuracy– Numerical Stability
• Unsteady-state CFD – Explicit vs. Implicit method
• HW2– Turbulence – http://network.bepress.com/physical-sciences-and-mathematics/physics/fluid-dynamics/– http://www.transportation.anl.gov/engines/multi_dim_model_les.html
0dxS}x
ΦΓ
x
ΦΓ{ΦρVΦρV
e
w
Φw
eff Φ,e
eff Φ,wxex
Steady–state 1D example
ew
P EW
x
xwxe
Ww ΦΦ If Vx > 0,
If Vx < 0, Ee ΦΦ
X direction
Pe ΦΦ
Pw ΦΦ
and
and
Diffusion term:
Source term: xSdxS Φ
e
w
Φ Assumption:Source is constant over the control volume
a)
b)
c)
I)
Convection term - Upwind-scheme:
WxPxwxex ΦρVΦρVΦρVΦρV
PxExwxex ΦρVΦρVΦρVΦρV
x
ΦΦ2ΦΓ
x
ΦΦ
x
ΦΦΓ
x
ΦΓ -
x
ΦΓ
:then
x
ΦΦΓ
x
ΦΓ and
x
ΦΦΓ
x
ΦΓ
WPEeff Φ,
w
WP
e
PEeff Φ,
weff Φ,
eeff Φ,
w
WPeff Φ,
weff Φ,
e
PEeff Φ,
eeff Φ,
When mesh is uniform:X = xe = xw
sourceWEPPEx qTTT2
xx
k/cp TT
x
ρV
source1N1NN1-NNx qTTT2
xx
k/cpTT
x
ρV
N N+1N-1
x
x x
N1NNNN1-NN fTcTbTa
General equation
N
N
i
N
N
i
NN
NNN
iii
f
f
f
f
f
ba
cba
cba
cba
cb
1
2
1
1
2
1
111
222
11
...
...
T
T
...
T
...
T
T
...
...
Advection diffusion equation 1-D, steady-state
Different notation:
1D example multiple (N) volumes
N
N
i
N
N
i
NN
NNN
iii
f
f
f
f
f
ba
cba
cba
cba
cb
1
2
1
1
2
1
111
222
11
...
...
Φ
Φ
...
Φ
...
Φ
Φ
...
...
1 2 i N-1 N
Equation matrix:
For 1D problem3-diagonal matrix
12111 fΦcΦb
2322212 fΦcΦbΦa
3433323 fΦcΦbΦa
3
NNN1-NN fΦbΦa ……………………………
Equation for volume 1
Equation for volume 2 Nequations
N unknowns
3D problem
W E
N
S
H
L
P
Equation in the general format:
fΦaΦaΦaΦaΦaΦaΦa LLHHNNSSWWEEPP
Wright this equation for each discretization volumeof your discretization domain
x =
FΦA
7-diagonal matrix
60,000 cells (nodes)N=60,000
60,000 elements
60
,00
0 e
lem
ent
s
This is the system for only one variable ( )Φ When we need to solve p, u, v, w, T, k, , C
system of equation is very larger
Boundary conditionsfor CFD application - indoor airflow
Real geometry
Model geometry
Where are the boundary Conditions?
CFD ACCURACY
Depends on airflow in the vicinity ofBoundary conditions
1) At air supply device
2) In the vicinity of occupant
3) At room surfaces
Detailed modeling- limited by computer power
Surface boundaries
Wall surface
W use wall functions to model the flow in the vicinity of surfaceUsing relatively large mesh (cell) size.
0.01-20 mmfor forced convection
thickness
Airflow at air supply devices
Complex geometry - Δ~10-4m
We can spend all our
computing power for one small detail
momentum sources
Diffuser jet properties
High Aspiration diffuser
D
L
D
L
How small cells do you need? We need simplified models for diffusers
Peter V. Nielsen
Simulation of airflow in In the vicinity of occupantsHow detailed should we make the geometry?
General Transport Equation Unsteady-state
W E
N
S
H
L
PEquation in the algebraic format:
fΦaΦaΦaΦaΦaΦaΦa LLHHNNSSWWEEPP
S)gradΓ(div )Vdiv( ρτ
ρ eff,
We have to solve the system matrix for each time step !
Unsteady-state 1-D
τρ
τρ
PP
Transient term:
ΦWEPeff Φ,
P(or W)P)E(or x SΦΦΦ2
xx
Γ ΦΦ
x
ρV
τρ
PP
Are these values for step or + ?
If: - - explicit method - + - implicit method
General Transport Equation unsteady-state 1-D
Fully explicit method:
Implicit method:
ΦΔττ
WΔττ
EΔττ
Peff Φ,Δττ
P(or W)Δττ
P)or E(x
τP
ΔττP SΦΦ2Φ
ΔxΔx
Γ ΦΦ
Δx
ρV
Δτ
ΦΦρ
ΦτW
τE
τP
eff Φ,τP(or W)
τP)or E(
xτP
ΔττP SΦΦ2Φ
ΔxΔx
Γ ΦΦ
Δx
ρV
Δτ
ΦΦρ
Value form previous time step (known value)
Make the difference between - Calculation for different time step- Calculation in iteration step