List of (prospective) Projects
FRG fixed points of the 1D Burgers equation for randomly stirred fluids. (LC) The project aims to derive the FRG flow equations for the 1D stochastic Burgers equation in a simple approximation, integrate numerically these equations to find the fixed point structure of the 1D Burgers equation and analyse the different scaling regimes.
Lagrangian intermittency and irreversibility (MC) In this project, we use a small DB of Lagrangian trajectories to do some analysis of Lagrangian structure functions and or Lagrangian power to discuss irreversibility.
Spontaneous stochasticity of the Kuramoto-Sivashinsky equation (ES)
Two-Point Statistics from Fiber Tracking Velocimetry in a turbulent channel flow (CB) This project aims at evaluating two-point statistics of a turbulent channel flow using rigid fibers transported by the flow. This technique, recently validated experimentally and numerically in homogeneous isotropic turbulence (Brizzolara et al., PRX, 2021), allows to measure flow transverse velocity increments and even the local turbulent dissipation rate for sufficiently short fibers. Using numerical data, we will apply this technique to a turbulent channel flow, which is inhomogenous and anisotropic. The length and inertia of the fibers will be varied to quantify the effect of these parameters on the measurements.
Dynamical models for the turbulent cascade (JB) This project explores simplified dynamical models for the turbulent energy cascade that go beyond phenomenological approaches. The aim is to understand how intermittency, as described by Kolmogorov’s 1962 refined self-similarity hypothesis, emerges dynamically from the coupling between scales. Students will construct and analyze stochastic evolution equations for coarse-grained energy transfers, investigating how fluctuations at small scales lead to intermittent statistics. A central aspect is to illustrate Eulerian spontaneous stochasticity, viewed through the lens of renormalization-group ideas, where multiplicative randomness arises from unresolved interactions. Model predictions will be compared with direct numerical simulations, focusing on scaling exponents, intermittency corrections, and the validity of refined similarity across scales.
Zero-modes in a multifractal random shell model for scalar transport (ST) The project aims to analyse numerically and theoretically the emergence of non-Gaussian flatness in multifractal extension of random dyadic model, which can be seen as a shell model version of the Kraichnan model of scalar transport.
Predictability of singular initial vorticity problems (AM & ST) The goal of this project is to numerically investigate examples of simple singular initial data for which the 2D Euler equations or their variants, have non-unique solutions. One example is the 2D Kelvin-Helmoltz instability. When perturbed by a small noise it is known to give rise to spontaneous stochasticity of vortex sheets. The goal is to numerically investigate the universality of this mechanism when varying the range of interactions between the vortices. A second and perhaps simpler case is the initial data proposed by Bressan and Shen, 2019 (arxiv 2002.01962) where vorticity has a point singularity.
Stochastic modeling of fluid tracers and time reversibility (RZ) A stochastic model describing the interactions of a fluid particle with all other fluid particles of a turbulent flow has been proposed based on a local relationship between the force on the fluid particle, kinetic energy and dissipation. This model exhibits some of the remarkable characteristics of fluid particle dynamics, in particular the occurrence of extreme events and the emergence of an anomalous scaling law. This model can also produce an asymmetry in certain statistics (such as the rate of kinetic energy variation) through time reversal, a sign of dissipative dynamics. This project aims to characterize and analyze the conditions under which such temporal asymmetry is possible and to identify the minimum ingredients to be incorporated into a stochastic model in order to obtain this behavior. In particular, initial empirical tests seem to show that it is necessary to have non-Markovian dynamics and a non-diagonal diffusion tensor. Python or C++ programs for simulating this stochastic process are available.
Make a DNS wish (PM) If anyone has never done a DNS, we can do one. Any set of pdes! Any BC! Any cluster! Any manifold!