1. Objective Discus Turbulence
Introduce Reynolds Navier Stokes Equations (RANS)

2. Conservation Equationsy z x

3. Turbulence Forced convection on flat plate

4. Turbulence

5. Size of eddies
hurricane
nozzle
2 in
Eddy ~ 1/100 in
~200 miles

6. Transition from laminar to turbulent flow

7. Turbulence in the vicinity of human body
PT-Teknik.dk

8. 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.

9. 3-D

10. 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

11. 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.

12. 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

13. 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

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

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

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

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

18. 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

19. Time Averaging Operations

20. 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