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Math
behind
Turbulence

Zoran Grujic

Professor of Mathematics, University of Virginia
zg7c@virginia.edu

Developing a rigorous theory of turbulence consistent with the experiments, computational simulations and the phenomenological theories has been a grand challenge in the mathematical fluids community since the 1930s. Richard Feynman described turbulence as “the most important unsolved problem of classical physics.'' While working on visualizations and simulations of turbulent flows, we thought, "Why not ‘gamify’ a key geometric aspect of this complicated problem and ask players around the world to help us?”

We are studying the phenomenon of turbulent dissipation in 3D viscous incompressible flows modeled by the 3D Navier-Stokes (NS) equations, focusing on the vorticity description of the turbulent flow. Recall that the vorticity field is essentially a differential spin of an infinitesimal region of fluid in view. We consider ‘free’ turbulence (the external force is switched off), away from the boundary, following a geometric approach pioneered by G.I. Taylor in 1930s [1]

Math behind Turbulence equation

A closely related mathematical problem is whether a singularity can form in solutions to the 3D NS equations, this is the so called 3D NS regularity problem [2][3]. The physical significance of the problem in conjunction with its mathematical difficulty (the problem is super-critical) were the main reasons that the 3D NS regularity problem was included in the list of seven Millennium Prize Problems put forth by the Clay Mathematics Institute in 2000. The super-criticality is reflected in the fact that there is a conceptual barrier, or ‘scaling gap ’ between what can be proved and what is needed to rule out singularities.

Turbulence game is based on a recent mathematical work by Z. Grujic and collaborators on the interplay between the spatial complexity of the regions of intense vorticity (the RIVs seen in the game) and the turbulent dissipation [4][5][6][7][8]. Essentially, the mechanism of vortex stretching generates thin geometric structures, most notably 'vortex filaments', which are then transported and folded by a turbulent flow, generating a high level of spatial complexity, and in particular, `sparseness' of the RIVs.

A key player in this novel theory is a suitably defined 'scale of sparseness' of the RIVs; let us call it r. It would be very beneficial to have strong computational evidence revealing the scaling dependence of r on the theoretical prediction for the diffusion scale; this would provide a measure of the scaling gap in a typical turbulent flow. In particular, it would be quite interesting (and timely) to compare the data with a very recent rigorous work [8] featuring the first algebraic reduction of the scaling gap in the NS regularity problem (all previous improvements were logarithmic in nature).

Luckily, this investigation can be reduced to a very concrete task: for a statistically significant number of simulation time-slices, determine the size of the largest ball that can fit within the RIV, visualized based on the data imported from the computational simulations. This is precisely the main challenge in the Turbulence game.

To read more about Janet Rafner and Prof. Zoran Grujic’s collaboration see this article in the University of Virginia math department newsletter.

REFERENCES

  1. Taylor, G.I.: Production and dissipation of vorticity in a turbulent fluid. Proc. Roy. Soc. A 164, 15—23 (1938)
  2. Leray, J.: Etude de diverses equations integrales non lineaires et de quelque problèmes que pose l'Hydrodynamique. J.Math. Pures Appl.12, 1—82 (1933)
  3. Leray, J.: Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. 63, 193—248 (1934)
  4. Grujic, Z.: A geometric measure-type regularity criterion for solutions to the 3D Navier-Stokes equations. Nonlinearity 26, 289--296 (2013)
  5. Dascaliuc R., Grujic, Z.: Vortex stretching and criticality for the three-dimensional Navier-Stokes equations. J. Math. Phys. 53, 115613 (2012)
  6. Bradshaw, Z., Grujic, Z.: A spatially localized L log L estimate on the vorticity in the 3D NSE. Indiana Univ. Math. J. 64, 433—440 (2015)
  7. Grujic, Z.: Vortex stretching and anisotropic diffusion in the 3D Navier-Stokes
  8. https://arxiv.org/abs/1704.05546
Math behind Turbulence
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