1. Turbulent Fluid Flow
daVinci [1510]

2. Examples
Turbulent votices separating from a cylinder
wake
"False color image of the far field of a submerged turbulent jet" by C. Fukushima and J. Westerweel, Technical University of Delft, The Netherlands - Own work. Licensed under CC BY 3.0 via Wikimedia Commons -
Pyroclastic flow in Indonesia
Vortices visualized with laser fluoresence
Vortices in a rising jet
Mixing layer

3. Reynold’s Number

4. Karmen Vortices

5. Laminar – Turbulent Flow Regimes
Free stream
plume
Blue dye injected into a clear pipe at different flow regimes
Boundary layer
obstruction
Laminar – Turbulent transition with distance
An album of fluid motion, Milton Van Dyke

6. Flow velocity in a tidal channel
Velocity in all directions = mean + variation
Milne et al. [2013]

7. Velocity variations over 60s at a point in a channel
Milne et al. [2013]

8. Some characteristics Flows become unstable at high Re
Laminar flow becomes perturbed
Perturbation damped (low Re)
Perturbation grows (high Re)
Vortices/eddies form, wide range of scales
Rapid mixing, momentum, mass, heat
Large vortices break up into smaller vortices
Energy dissipation
Largesmall vortex  molecular motion heat

9. How to characterize turbulent flows
Empirical Laws
Manning (channels)
Darcy-Weisbach (conduits)
Izbash (porous)
Forchheimer (porous)
dh/dx q

10. Darcy-Weisbach Eqn. Pressure drop in pipes
is fluid density,
v is average velocity,
d is pipe diameter, and
f is the friction factor.
Low Re

12. Analysis Methods

13. DNS Simulation
LES Simulation

14. RANS Reynolds Average Navier Stokes

15. Strain
Change in length/original length
Change in angle

16. Momentum Eqn. Constitutive law for fluid Einstein Notation
Navier-Stokes for Incompressible Fluid

17. Reynolds’ averaging, Mass
Mean + fluctuation
Substitute
Take average
Averaging rules
Result

18. Reynolds’ averaging, Momentum
Starting eq.
Substitute
Focus on one term
Other terms
Result
Note the fluctuating terms

19. Closure Problem
6 unknowns

20. Turbulent Viscosity Boussinesq (1892)
Turbulence dissipates energy in a way that is analogous to viscous dissipation
In Turbulent flow

21. Classical models based on RANS
Zero equation model: mixing length model.
One equation model: Spalart-Almaras.
3. Two equation models:
k- ε style models (standard, RNG, realizable),
k- ω model
4. Seven equation model: Reynolds stress model.

22. k-e Method k: Turbulent kinetic energy e: Turbulence dissipation rate
Cm: constant
Need equations for k, e
Assume k, e are conserved, use standard approach
A= vc =vkr

23. k-e Method

24. Implementation

25. Boundary Conditions Inlet, Outlet, Wall, Open, other
No slip, wall = default
Specify inlet and outlet
Need to specify pressure somewhere
Dirichlet (specified pressure)
Neumann (specify velocity)
n unit vector normal to boundary
u flux vector
C1 known function

26. Options for boundary conditions
Velocity (uniform) =0.001m/s
Laminar Inflow = m/s
1m entrance length
Pressure = 1
No viscous stress
Pressure=0
No viscous stress
Need to calculate Pressure in top two cases, calculate velocity in bottom case

27. Wall Conditions Need b.c. for k and e
Represent steep gradients at wall in turbulence