some novel ideas in turbulence studies. practical aspect: mixing and segregation in random flows....
Post on 20-Dec-2015
213 views
TRANSCRIPT
![Page 1: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/1.jpg)
Some novel ideas in turbulence studies.
Practical aspect: Mixing and segregation in random flows.
Fundamental aspect: Strong fluctuations, symmetries and anomalies.
Gregory Falkovich
Imperial College, December 2005
![Page 2: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/2.jpg)
![Page 3: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/3.jpg)
![Page 4: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/4.jpg)
![Page 5: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/5.jpg)
![Page 6: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/6.jpg)
![Page 7: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/7.jpg)
![Page 8: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/8.jpg)
~
![Page 9: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/9.jpg)
Passive scalar decay
Small spherical spot released in a smooth flow with
![Page 10: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/10.jpg)
entropy
→ singular (fractal) SRB Measure
![Page 11: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/11.jpg)
An anomalous scaling corresponds to slower divergence of particles to get more weight.Statistical integrals of motion (zero modes) of the backward-in-time evolution compensate the increase in the distances by the mass decrease inside the volume.
Coarse-grained density
![Page 12: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/12.jpg)
u
v
Inertial particles
![Page 13: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/13.jpg)
![Page 14: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/14.jpg)
Spatially smooth flow
![Page 15: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/15.jpg)
Equivalent in 1d to Anderson localization :localization length = Lyapunov exponent
One-dimensional model
![Page 16: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/16.jpg)
Super-symmetry broken
Lyapunov exponent
![Page 17: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/17.jpg)
![Page 18: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/18.jpg)
![Page 19: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/19.jpg)
![Page 20: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/20.jpg)
![Page 21: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/21.jpg)
![Page 22: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/22.jpg)
![Page 23: Some novel ideas in turbulence studies. Practical aspect: Mixing and segregation in random flows. Fundamental aspect: Strong fluctuations, symmetries and](https://reader035.vdocuments.us/reader035/viewer/2022062714/56649d4e5503460f94a2d9dd/html5/thumbnails/23.jpg)
Conclusion
All known cases of anomalies (symmetry remains
broken when symmetry breaking factor goes to zero)
can be traced to conserved quantities.
Anomalous scaling is due to statistical conservation laws.