1. Lecture Objectives: Define 1) Reynolds stresses and
2) K-ε turbulence models
Analyze General CFD (Transport) Equation
Discretization of Transport Equation
Introduce Boundary Conditions

2. Modeling of Turbulent Viscosity
Fluid property – often called laminar viscosity
Flow property – turbulent viscosity
MVM: Mean velocity models
TKEM: Turbulent kinetic energy equation models
Additional models:
LES: Large Eddy simulation models
RSM: Reynolds stress models

3. Prandtl Mixing-Length Model (1926)
One equation models:
Prandtl Mixing-Length Model (1926) Vx y x l
Characteristic length (in practical applications:
distance to the closest surface)
-Two dimensional model
-Mathematically simple
-Computationally stable
-Do not work for many flow types
There are many modifications of Mixing-Length Model:
- Indoor zero equation model: t =  V l
Distance to the closest surface
Air velocity

4. Kinetic energy and dissipation of energy
Kolmogorov scale
Eddy breakup and decay to smaller length scales where dissipation appear

5. Two equation turbulent model
Kinetic energy
Energy dissipation
From dimensional analysis
constant
We need to model
Two additional equations:
kinetic energy
dissipation

6. Reynolds Averaged Navier Stokes equations
Continuity: 1)
Momentum: 2) 3) 4)
General format:

7. General CFD Equation Values of , ,eff and S Equation  ,eff S
Continuity 1
x-momentum V1
+ t
-P/x+Sx
y-momentum V2
-P/y-g(T∞-Twall)+Sy
z-momentum V3
-P/z+Sz
T-equation T
/l + t/t ST
k-equation k
(+ t)/k
G- +GB
-equation 
(+ t)/
[ (C1G-C2)/k] +C3GB(/k)
Species C
(+ t)/c SC
Age of air t 
 t =Ck2/ , G= t (Ui/xj +Uj/xi) Ui/xj , GB=-g(/CP)( t/T,t) T/ xi
C1=1.44, C2=1.92, C3=1.44, C=0.09 , t=0.9, k =1.0,  =1.3, C=1.0

8. Discretization - Available methods:
Computers do not solve a partial differential equation
(computers do only 1 + 1, 1 + 0, and, )
- Convert partial differential equation into algebraic equations
obtain finite number of numerical values instead of continuous solution
- Available methods:
- finite volume method
- finite difference method
- finite element method

9. Finite Volume Method - Conservation of f for the finite volume
Divide the whole computation domain into sub-domains
One dimension: n h W P dx E dx w e s Dx l e w
- Finite volume is a fixed space in the flow domain with imaginary boundaries
that allow the fluid to flow in and out.
- Integral conservation of the quantities such as mass, momentum and energy. f

10. General Transport Equation -3D problem steady-stateH N W P E S L
Equation for node P in the algebraic format:

11. A 1D example of discretization of general transport equation
Steady state 1dimetion (x): W
dxw P
dxe E Dx w e
Point W and E represent the cell center of the west and east neighbors of cell P
and w, e the neighboring surfaces. Integrating with Gaussian theorem on this control
volume gives:
To obtain the equations for the value at point P,
assumptions are used to convert the surface values to the center values.

12. 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:

13. 1D example After substitution a), b) and c) into I):
We started with partial differential equation:
same
and developed algebraic equation:
We can write this equation in general format:
Unknowns
Equation coefficients

14. General Transport Equation -3D problem steady-stateH N
Equation in the algebraic format: 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 larger

15. Boundary conditions in CFD application in indoor airflow
Real geometry
Model geometry
Where are the boundary
Conditions?

16. 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

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

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

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

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

21. AIRPAK Software