1. Objective Introduce Reynolds Navier Stokes Equations (RANS)
Discuss HW 1

2. Energy Cascade Concept in Turbulence
Kinetic energy is continually being transferred from the mean flow to the turbulent motion by large-scale eddies
The process of vortex stretching leads to a successive reduction in eddy size and to a steepening of velocity gradients between adjacent eddies.
Eventually the eddies become so small that viscous dissipation leads to the conversion of kinetic energy into heat.

3. Indoor airflow jet jet The question is: What we are interested in:
exhaust
supply
jet
The question is:
What we are interested in:
main flow or
turbulence?
turbulent

4. Method for solving of Navier Stokes (conservation) equations
Analytical
Define boundary and initial conditions. Solve the partial deferential equations.
Solution exist for very limited number of simple cases.
Numerical
- Split the considered domain into finite number of volumes (nodes). Solve the conservation equation for each volume (node).
Infinitely small difference
finite “small” difference

5. Numerical method
Simulation domain for indoor air and pollutants flow in buildings
3D space
Solve p, u, v, w, T, C
Split or “Discretize”
into smaller volumes

6. Mesh size for direct Numerical Simulations (DNS)
~1000
~2000 cells
For 2D wee need ~ 2 million cells
Also, Turbulence is 3-D phenomenon !

7. Capturing the flow properties2”
nozzle
Eddy ~ 1/100 in
Mesh (volume) should be smaller than eddies !
(approximately order of value)

8. Mesh size For 3D simulation domain 2.5 m Mesh size 0.1m → 50,000 nodes
3D space (room)

9. We need to model turbulence!
Reynolds Averaged Navier Stokes equations

10. First Methods on Analyzing Turbulent Flow
- Reynolds (1895) decomposed the velocity field into a time average
motion and a turbulent fluctuation
vx’ Vx
- Likewise
f stands for any scalar: vx, vy, , vz, T, p, where:
From this class
We are going to make a difference
between large and small letters
Time averaged component

11. Time Averaging Operations

12. Averaging Navier Stokes equations
Substitute into Navier Stokes equations
Instantaneous velocity
fluctuation
around
average
velocity
Average
velocity
Continuity equation:
time
Average whole equation:
Average
Average of average = average
Average of fluctuation = 0

13. Example: of Time Averaging
Write continuity equations in a short format:
=0 continuity
Short format of continuity equation in x direction:

14. Averaging of Momentum Equation

15. Time Averaged Momentum Equation
Instantaneous velocity
Average velocities
Reynolds stresses
For y and z direction:
Total nine

16. Time Averaged Continuity Equation
Instantaneous velocities
Averaged velocities
Time Averaged Energy Equation
Instantaneous temperatures and velocities
Averaged temperatures and velocities

17. Reynolds Averaged Navier Stokes equations
Reynolds stresses
total are unknown
same
Total 4 equations and = 10 unknowns
We need to model the Reynolds stresses !

18. Modeling of Reynolds stresses Eddy viscosity models
Average velocity
Boussinesq eddy-viscosity approximation
Is proportional to deformation
Coefficient of proportionality
k = kinetic energy
of turbulence
Substitute into Reynolds Averaged equations

19. Reynolds Averaged Navier Stokes equations
Continuity: 1)
Momentum: 2) 3) 4)
Similar is for STy and STx
4 equations 5 unknowns → We need to model

20. 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

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

22. 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

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

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

25. 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

26. 1-D example of discretization of general transport equation
Steady state 1dimension (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.

27. HW Example
The figure below shows a turbulent boundary layer due to forced convection above the flat plate.
The airflow above the plate is steady-state.
Consider the points A and B above the plate and line l parallel to the plate.
Point A y
Flow direction
Point A
Point B
line l
For the given time step presented on the figure above plot the velocity
Vx and Vy along the line l.
b) Is the stress component txy lager at point A or point B? Why?
c) For point B plot the velocity Vy as function of time.

28. 3-D