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