Periodic Reporting for period 1 - NoisyFluid (Noise in Fluids)
Reporting period: 2023-01-01 to 2025-06-30
Turbulence is a major problem of Physics and Mathematics. Among various fields it is linked with, there is the area of Stochastic Partial Differential Equations, central to this project. The project title, Noise in Fluids, summarizes that we consider various fluid dynamic models (e.g. Navier-Stokes and Euler equations, heat conduction in fluids, particles and polymers in fluids, kinetic equations) and incorporate in them stochastic elements representing a modelisation of turbulent features.
The main purpose of this approach is understanding the reaction of different systems and components to the presence of turbulence. Let us list some of them.
A classical problem is the dissipative effect (the increase of viscosity) that small-scale turbulence has on large scales, identified by Bousinnesq and implemented by Large Eddy Simulation models. We have developed a mathematical scheme where this problem can be rigorously investigated. The extension to 3D models is particularly challenging.
A turbulent fluid then acts on other fields and objects. The modified diffusion properties of temperature is the most natural effect, analogous to the increase of viscosity mentioned above. But the effect on vector fields, like a magnetic field in a conducting fluid, is more complex, involving a deep understanding of the stochastic stretching mechanism.
And a fluid may act on particles embedded into it, like air on water droplets, raising the question of enhanced collision rate due to turbulence. We consider this problem, as well as the problem of stretching of polymers in fluids, problems which increase our understanding of transport and stretching mechanisms.
Ultimately, turbulence in confined plasma, dispersion barriers and zonal flows are even more difficult problems that we hope to approach with suitable stochastic modeling. Here, inverse cascade mechanisms are one of the main concern.
On the theoretical side, referring to the millenium prize problem of singularities for the Navier-Stokes equations, a relevant question is whether turbulence may prevent the development of singularities, thanks to its chaotic rearrangements. A simplified portrait of this question is whether noisy modeled turbulent small scales may prevent the emergence of singularities, which is then one of our main objectives.
The main purpose of this approach is understanding the reaction of different systems and components to the presence of turbulence. Let us list some of them.
A classical problem is the dissipative effect (the increase of viscosity) that small-scale turbulence has on large scales, identified by Bousinnesq and implemented by Large Eddy Simulation models. We have developed a mathematical scheme where this problem can be rigorously investigated. The extension to 3D models is particularly challenging.
A turbulent fluid then acts on other fields and objects. The modified diffusion properties of temperature is the most natural effect, analogous to the increase of viscosity mentioned above. But the effect on vector fields, like a magnetic field in a conducting fluid, is more complex, involving a deep understanding of the stochastic stretching mechanism.
And a fluid may act on particles embedded into it, like air on water droplets, raising the question of enhanced collision rate due to turbulence. We consider this problem, as well as the problem of stretching of polymers in fluids, problems which increase our understanding of transport and stretching mechanisms.
Ultimately, turbulence in confined plasma, dispersion barriers and zonal flows are even more difficult problems that we hope to approach with suitable stochastic modeling. Here, inverse cascade mechanisms are one of the main concern.
On the theoretical side, referring to the millenium prize problem of singularities for the Navier-Stokes equations, a relevant question is whether turbulence may prevent the development of singularities, thanks to its chaotic rearrangements. A simplified portrait of this question is whether noisy modeled turbulent small scales may prevent the emergence of singularities, which is then one of our main objectives.
We have constructed a variety of stochastic processes to be used as a stochastic description of small-scale turbulence; some based on Fourier analysis, others on vortex structures, smoothed versions of point vortices in 2D and vortex filaments in 3D. We have also introduced models based on fractional processes and models incorporating boundaries and transport barriers like zonal flows.
We have improved our understanding of the dissipative and mixing properties of such noise models of small scale turbulence, when they act in the form of transport on passive scalars and on the large scales of the fluid itself. We now know how to quantify the distance from the limit for finite size turbulence, we have first ideas about the role of fractional gaussian noise and alpha stable processes in the description of the noise, we have made numerical simulation to observe the limits of the theory for very unstable flows, we have developed a Smagorinsky type variant in order to explore the dependence of the turbulence on some features of the large scales. The models include interfaces and transport barriers.
We have developed new approaches to understand the improved rate of collision of particles embedded into turbulent fluids, both in the case of small and large Stokes numbers, recovering important physical laws.
We have explored the consequences of stochastic stretching for 3D fluids with particular symmetries. More advanced results have been obtaind on the stretching of passive objects like embedded polymers and of a passive magnetic field.
We have moved first steps in the direction of regularization by turbulence.
We have moved first steps in several complex directions like the understanding of inverse cascade in 2D fluids and the role of larger scale structures in turbulence, the description of anomalous transport and its dissipative properties, the dependence on the Nusselt number on the Rayleigh number in Rayleigh-Benard convection.
We have moved first steps in the direction of turbulence and zonal flows and turbulence and transport barriers in plasma. We are currently working very intensively on turbulence in plasma under the simplified but rich model of Hasegawa and Mima.
We have improved our understanding of the dissipative and mixing properties of such noise models of small scale turbulence, when they act in the form of transport on passive scalars and on the large scales of the fluid itself. We now know how to quantify the distance from the limit for finite size turbulence, we have first ideas about the role of fractional gaussian noise and alpha stable processes in the description of the noise, we have made numerical simulation to observe the limits of the theory for very unstable flows, we have developed a Smagorinsky type variant in order to explore the dependence of the turbulence on some features of the large scales. The models include interfaces and transport barriers.
We have developed new approaches to understand the improved rate of collision of particles embedded into turbulent fluids, both in the case of small and large Stokes numbers, recovering important physical laws.
We have explored the consequences of stochastic stretching for 3D fluids with particular symmetries. More advanced results have been obtaind on the stretching of passive objects like embedded polymers and of a passive magnetic field.
We have moved first steps in the direction of regularization by turbulence.
We have moved first steps in several complex directions like the understanding of inverse cascade in 2D fluids and the role of larger scale structures in turbulence, the description of anomalous transport and its dissipative properties, the dependence on the Nusselt number on the Rayleigh number in Rayleigh-Benard convection.
We have moved first steps in the direction of turbulence and zonal flows and turbulence and transport barriers in plasma. We are currently working very intensively on turbulence in plasma under the simplified but rich model of Hasegawa and Mima.
From large parts of our research will emerge a developed picture about the validity and limitations of Boussinesq conjecture that small scale turbulent transport may act as an improved viscosity or dissipation. The theory of Stochastic Partial Differential Equations will benefit from this important contribution to fundamental physical questions about turbulence, of great relevance also for simulations and engineering.
Our new methodologies and understanding of the collision properties of particles in turbulent fluids may become a reference for future research and allow us to approach some of the still open questions in the field.
Very few have been understood on stretching mechanisms and our results on stretching of polymers and magnetic fields represent a unique example. The question is of fundamental importance for the analysis of singularities of 3D Navier-Stokes equations and for progress on regularization by noise theory.
Our developments may provide new insights on the inverse cascade in 2D fluids, anomalous transport, emergence of large scale structures, in particular zonal flows and possibly allow us to make progress on more difficult questions like turbulence in confined plasma and transport barriers.
Our new methodologies and understanding of the collision properties of particles in turbulent fluids may become a reference for future research and allow us to approach some of the still open questions in the field.
Very few have been understood on stretching mechanisms and our results on stretching of polymers and magnetic fields represent a unique example. The question is of fundamental importance for the analysis of singularities of 3D Navier-Stokes equations and for progress on regularization by noise theory.
Our developments may provide new insights on the inverse cascade in 2D fluids, anomalous transport, emergence of large scale structures, in particular zonal flows and possibly allow us to make progress on more difficult questions like turbulence in confined plasma and transport barriers.