1. Relative dispersion & Richardson’s constant Brian Sawford Dept. Mechanical Engineering Monash University PK Yeung and Jason Hackl Georgia Tech

2. Outline Background; turbulent dispersion; absolute & relative dispersion, mixing Similarity theory/scaling; turbulence; relative dispersion; Richardson’s law & constant Standard compensated plots; DNS data Cube-root plots; modifications Estimate for Richardson’s constant PHYSICS OF FLUIDS 20, 065111 2008

3. Instantaneous plume dispersion L=1 km; w * = 2 ms -1 K T = 2000 m 2 s -1  = 10 -5 m 2 s -1 Re=w * L/v ~ 10 8 R ~ 50,000  LRe -3/4 ~1 mm Kolmogorov similarity theory

5. Average concentration & fluctuations

6. Fluctuations are important Odour: human nose responds to peaks pheremones source detection

7. Relative dispersion – simplifications How does the mean-square separation increase with time? z ▬ z 1. Fluid particles 2. 2 particles t x

8. The 1 st Lagrangian Expt?

9. Relative dispersion – calculation Direct numerical simulation z t rz0rz0 rz(t)rz(t) N ~ 10 4 - 10 5

10. Richardson’s t 3 -law Re → ∞ (L >> η); r 0 >> η log t < r 2 ( t )>-  r 0 2 log t  << t << t/t 0 t >> T L r0r0 log

11. Why are we interested in g? Fundamental constant, difficult to measure Connection with turbulent mixing, concentration fluctuations, growth of puffs & instantaneous growth of plumes Model comparison/testing - predictions range from 0.01 to 4 DNS & lab results for R λ ≤ 280 - g ≈ 0.5 – 0.7

12. DNS Data Sets NR λ /idν  10 4 L/ηL/ηTL/tηTL/tη k max ηT/tηT/tη M pair 64 13 432502.67 37 5.41.777424576 128 13 86711.17 105 8.61.4124576 256 13 140281.19 217 13.11.4116724576 512 13 235111.08 473 19.81.4162196608 1024 240111.20 488 20.02.813749152 1024 8 283k2.890.0719 625 23.4216≥32768 1024 12 284b8.80.812 628 282.5141~10 6 1024 13 3904.371.14 1010 31.11.378149152 1024 390n4.371.30 1010 31.81.44178196608 1024 390n24.371.30 1010 31.81.44118196608 2048 13 6501.7321.17 2164 43.81.4428498304 2048 650n1.7321.34 2164 52.41.44107196608

13. Richardson’s t 3 -law “compensated” plot log t/t  < r 2 ( t )>-  r 0 2 log t  << t << t/t 0 t >> T L t3t3 log

14. Dissipation sub-range scaling r 0 /       

15. Ballistic/Batchelor range

16. Richardson’s t 3 -law “compensated” plot log t/t  < r 2 ( t )>-  r 0 2 log t  << t << t/t 0 t >> T L t3t3 log

17. Inertial sub-range scaling r 0 /       

18. Cube-root plots Ott & Mann, JFM, 422, 207, (2000) r 0 /  45.8. 40.1 34.3 28.6 22.9 17.2 11.4 5.7 t ≈ 0.7T L 1/3 ~ A 0 + (g  ) 1/3 t (t 0 << t << T L ) Ishihara & Kaneda, Phys Fluids, 14, L69, (2002) R =280

19. Cube-root plots Ishihara & Kaneda, Phys Fluids, 14, L69, (2002) R =280

20. Cube-root plots TLTL 2TL2TL Our data R λ = 390 r 0 /  256 128 64 32 16 8 4 1 1/4

21. Local slope plots Richardson constant

22. Local slope plots 0 ≤ t ≤ 0.7T L r 0 /  45.8. 40.1 34.3 28.6 22.9 17.2 11.4 5.7 Ishihara & Kaneda, Phys Fluids, 14, L69, (2002)

23. Cube-root slope plots r 0 /  256√2 256 256√2 128 64√2 64 32√2 32 16√2 16 8√2 8 4√2 4 0 ≤ t ≤ 3.9T L Our data R λ = 390

24. Cube-root slope plots r 0 /  16 0 ≤ t ≤ 3.9T L Our data R λ = 390

25. Cube-root slope plots r 0 /  64 32 √2 32 16√2 16 8√2 8 0 ≤ t ≤ 1.4T L Our data R λ = 390

26. Richardson number

27. Conclusions 1.For R λ up to 650 see Richardson range in ~ t 3 plots in inertial sub-range scaling ; imprecise g 2.Cube-root plots in principle give convergence with r 0, but not previously tested with Re. 3.Cube-root local slope plots give clear convergence with r 0 & well-defined Richardson range. g smaller than in simple cub-root plots because of over-shoot. 4.Estimate g = 0.55 – 0.57 at large Re