Lecture Objectives:

1) Discuss HW3 problems

2) Define Relaxation (example)

3) Define boundary conditions for CFD- Model Boundary conditions

Residual calculation for CFD

• Residual for the cell Rijk=k

ijk-k-1ijk

• Total residual for the simulation domain Rtotal=Rijk|

• Scaled (normalized) residual R=Rijk|/F

iteration

cell positionVariable: p,V,T,…

For all cells

Flux of variable used for normalizationVary for different CFD software

RelaxationRelaxation with iterative solvers:

When the equations are nonlinearit can happen that you get divergency in iterative procedure for solving consideredtime step

Under-Relaxation is often required when you have nonlinear equations!

iteration

convergence

variabledivergence

solution

Solution is Under-Relaxation:

Y*=f·Y(n)+(1-f)·Y(n-1) Y – considered parameter , n –iteration , f – relaxation factor

For our example Y*in iteration 101=f·Y(100)+(1-f) ·Y(99)

f = [0-1] – under-relaxation -stabilize the iterationf = [1-2] – over-relaxation - speed-up the convergence

Value which is should be used for the next iteration

Example of relaxation(example from homework 3 assignment)

N1NNNN1-NN fTcTbTa

Example: Advection diffusion equation, 1-D, steady-state, 4 nodes

1NNN1-NNNNNN T/bcT/bafb/1T 1 2 3 4

1) Explicit format:

2) Guess initial values:

..T ..,T ..,T ...,T 04

03

02

01

3) Substitute and calculate:2

011111

1 T/bcfb/1T

30

2211

222221 T/bcT/bafb/1T

40

3321

333331 T/bcT/bafb/1T

31

444441 T/bafb/1T

..T ..,T ..,T ...,T 14

13

12

11

Substitute and calculate:4) ..T ..,T ..,T ...,T 24

23

22

21

………………………….

.... ,f)T-(1fTT ,f)T-(1fTT 02

12

1r2

01

11

1r1

.... ,f)T-(1fTT ,f)T-(1fTT 12

22

2r2

11

21

2r1 Substitute and calculate:

Relaxation example in Excel

Boundary Conditions

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

Define Boundary Conditions at:

– Surfaces (wall functions)• Velocity • Temperature• Concentration

– Inlets and outlets• Diffusers and outlets• Windows and cracks

Diffuser Types

swirl diffusers

wall or ceiling

floor

Valve diffuser

ceiling diffuser

Diffuser Types

Linear diffusersGrill (side wall) diffusers

Horizontal one side

Vertical

Displacement ventilationdiffusers

Diffuser modeling

Complex geometry - Δ~10-4mWe can spend all our computing power for one small

detail

momentum sources

Momentum method

Diffuser Modeling

Fine mesh or box method for diffuser modeling

Diffuser modeling

High Aspiration diffuser

D

L

D

L

Jet through one opening only

Jet parameters

xA

KVVm 0

0

A0 - effective area of the diffuser

V0 – initial jet velocity

X - distance from the diffuser

Vm – maximum jet velocity at distance x from the diffuser

K – property of diffuser

Diffuser properties (ASHRAE)

Examples in Airpak

Diffuser’s Macro

Surface boundary conditions

temperatureand

velocity

Surface boundarieswall functions

Wall surface

Use wall functions to model the micro-flow in the vicinity of surfaceUsing relatively large mesh (cell) size.

Surface boundary conditions and log-wall functions

)log(1*

2/1

EyyV

E is the integration constant and y* is a length scale

dydV

t

The assumption of ‘constant shear stress’ is used here. Constants k = 0.41 and E = 8.43 fit well to a range of boundary layer flows.

Laminar sub-layer

Turbulent profile

Surface cell

y*- thickness of boundary layer