1. 2-1

2. You are a fluid dynamicist visiting the Louvre in Paris and are asked by the curator to comment on the above paintings. What do you say? 2-2

3. High-Re flow! Low-Re flow, Van Gogh’s clouds have no small scales! 2-3

4. Leonardo da Vinci Pioneered the flow visualization genre close to 500 years ago. The sketch --- a free water jet issuing from a square hole into a pool---represents perhaps the world's first use of visualization as a scientific tool to study a turbulent flow. Leonardo wrote (translated by Ugo Piomelli, University of Maryland), "Observe the motion of the surface of the water, which resembles that of hair, which has two motions, of which one is caused by the weight of the hair, the other by the direction of the curls; thus the water has eddying motions, one part of which is due to the principal current, the other to the random and reverse motion." According to John L. Lumley, Cornell University, Leonardo may have prefigured the now famous Reynolds turbulence decomposition nearly 400 years prior to Osborne Reynolds' own flow visualization and analysis! In describing the swirling water motion behind a bluff body, da Vinci provided the earliest reference to the importance of vortices in fluid motion: "So moving water strives to maintain the course pursuant to the power which occasions it and, if it finds an obstacle in its path, completes the span of the course it has commenced by a circular and revolving movement." Leonardo accurately sketched the pair of quasi-stationary, counterrotating vortices in the midst of the random wake. Finally, da Vinci's words "... The small eddies are almost numberless, and large things are rotated only by large eddies and not by small ones, and small things are turned by both small eddies and large.." presage Richardson's cascade, coherent structures, and large- eddy simulations, at least. 2-4

5. “ … We know what the equations are … let’s just use DNS?” 2-5

6. 2-6

7. large scale small scale 2-7

8. 2-8

9. “ … DNS is mighty useful, however we don’t have enough IBM SP’s to get to a PBL Reynolds number … We need something else …” 2-9

10. LARGE EDDY SIMULATION (LES) 2-10

11. 2-11

12. TURBULENCE ENERGY CASCADE CONCEPTS LARGESMALL 2-12

13. TURBULENCE ENERGY CASCADE CONCEPTS Re As Re increases the extent of the (universal) inertial subrange expands (Pope, 2000) 2-13

14. 2-14

15. SPATIAL FILTERING LOWPASS FILTER CUTOFF WAVENUMBER  C FULLY RESOLVED 2-15

16. SPATIAL FILTERING LOWPASS FILTER CUTOFF WAVENUMBER  C PARTIALLY RESOLVED FULLY RESOLVED 2-16

17. SPATIAL FILTERING LOWPASS FILTER CUTOFF WAVENUMBER  C MEAN PARTIALLY RESOLVED FULLY RESOLVED 2-17

18. TIME/SPACE FILTERING Instantaneous plume (short time exposure) Ensemble average plume (long time exposure) Smoke visualization downstream of a point source in a wind tunnel 2-18

19. BoxGaussian Sharp spectral cutoff 2-19

20. 2-20

21. 2-21

22. Data in the atmospheric surface layer 2-22

23. 2-23

24. 2-24

25. RANS limit 2-25

26. Taylor series expansion 2-26

27. 2-27

28. FILTERING IN THE INERTIAL RANGE sharp filter cutoff spectrum subgrid-scale energy resolved-scale energy 2-28

29. FILTERING IN THE INERTIAL RANGE Gaussian filter subgrid-scale energy has contributions from above and below the nominal filter cutoff k c resolved-scale energy The effect of finite differencing in (most) simulation codes is to mimic Gaussian filtering 2-29

30. 2-30

31. 2-31

32. ProductionDissipation Buoyancy 2-32