The mathematician behind Turbulence
Zoran Grujic
Collaboration on Turbulence game has been a very stimulating and rewarding experience. The seed idea for the project emerged several years ago while Janet Rafner was participating in an interdisciplinary seminar Turbulence in Art and Math I was running at the University of Virginia. Since my primary research project at the time was the mathematical study of the phenomenon of turbulent dissipation in 3D viscous, incompressible fluid flows through the lens of geometry of the flow, and Janet was curious to try her hand at visualization of turbulent flows, we started thinking about different ways in which the visualization could inform the theory.
The theory is motivated by one of the basic principles of turbulence phenomenology: once a typical length (scale) of the regions of intense fluid activity reaches a certain micro-scale (usually referred to as the dissipation or Kolmogorov scale), the kinetic energy (in the absence of external forces) starts dissipating away. If one is interested in geometric properties, it is beneficial to study dynamics of the flow in terms of the vorticity which is essentially a differential spin of the velocity. This reveals various coherent vortex structures, e.g., vortex sheets and vortex tubes, omnipresent in turbulent flows. In particular, one can study the dissipation scale within the vorticity description of the flow, leading to dissipation of the enstrophy which is the vorticity-analogue of the kinetic energy.
An additional benefit of studying the evolution of the enstrophy is that the control of the enstrophy would ensure that no singularities can form in the flow which is mathematically described by the Navier-Stokes (NS) equations. The question of whether a singularity can form in solutions to the NS equations is usually referred to as the NS regularity problem, and is one of the seven Millennium Prize Problems put forth by the Clay Mathematics Institute.
A key player in this geometric theory is the suitably defined “scale of sparseness” of the (suitably defined) region of intense vorticity. One of the main reasons that the NS regularity problem is hard is that it is super-critical, meaning that there is a `scaling gap' between any known criterion for preventing the singularity formation and any corresponding verifiable property of the solutions. In a recent paper [BFG, arXiv], we presented a mathematical framework—based on the aforementioned scale of sparseness—that enabled us to obtain the first `algebraic/power reduction' of the scaling gap (all the previous reductions where logarithmic in nature).
This was very exciting, and we were extremely interested in a possibility of performing high-resolution computational simulations of some typical turbulent flows that could also be considered as a test ground for singularity-like events (e.g., bursts of the vorticity maximum), and obtaining the estimate on the scale of sparseness. This would provide an invaluable guidance for our future mathematical efforts (with the ultimate goal of closing the scaling gap). One should note that the precise mathematical definition of the scale of sparseness is a geometrically intricate property, and as such it is not amenable to a direct, `blind', algorithmic extraction from the simulation data. In other words, it was necessary to first visualize the regions of intense vorticity in each time-slice of the simulation, and then—depending on the observed spatial structure—try to formulate a `more concrete/usable' description of the scale of sparseness.
The simulations and the initial visualizations were performed in collaboration with the biocomplexity group at the Niels-Bohr Institute led by Joachim Mathiesen, and in particular with Marek Misztal. The visualizations that Marek and Janet produced revealed that the individual regions of intense vorticity (the RIVs in the game) are spatially well-localized; this was great news as it enabled us to come up with a very concrete and straightforward characterization of the scale of sparseness: essentially, the radius of the largest sphere that can be inscribed in an individual RIV within the time-slice in view.
Some of the RIVs turned out to be nearly-convex shapes, and in this case a purely algorithmic approach to identifying the radius of the largest sphere that can fit would have been quite cheap and feasible. However, some of the RIVs displayed a far-from-convex, complex and somewhat `labyrinthine' geometry, and we thought that a `gamification' approach in which the players would be `armed' with a simple algorithmic tool in their search for the largest sphere would be a reasonable and fun way to go.
From this point on Janet got immersed in an intense collaboration with the SAH team (with an assistance from Scott Leinweber) in developing Turbulence game which launched several weeks ago. The scientific data are now being collected from the gameplay, and I can not wait for the results of the initial data analysis! This has been a very exciting, genuinely interdisciplinary collaboration—unlike anything I have done in the past—and I am looking forward to similar adventures in the future.
Read the paper co-authored by Zoran: An algebraic reduction of the `scaling gap' in the Navier-Stokes regularity problem