Dissipation, cascade and singularities in turbulence

Turbulent flows are governed by the Navier-Stokes equations which describe the motion of fluids from macroscopic to microscopic scales, where dissipation phenomena transform kinetic energy into heat. Two characteristics make it difficult, if not impossible, to use these equations directly to numerically simulate the behaviour of the fluids that surround us: 1) the flows extend over gigantic ranges of scales, implying a number of degrees of freedom too great to be managed by our current super-computers; 2) they lead to the development of localised structures characterised by very strong gradients, which we do not know how to manage numerically, and which could be the signature of singularities or quasi-singularities developing in the limit of large Reynolds numbers. In this thesis, we propose to use this second phenomenon to counter the problems related to the first one, i.e. to improve the modelling of turbulent flows by taking into account their irregularity. Thus, we propose models for the correction of the Navier-Stokes equations at macroscopic or mesoscopic scales. These models are guided by observations and confronted with experimental data or numerical simulations. We first present the Navier-Stokes equations and the functioning of numerical simulations for real turbulent flows, from the algorithm to the high-performance computing. Then, we focus on two macroscopic quantities of turbulent flows: the topology of large recirculations and the energy flow. For these quantities, we propose two models: the first one for the behaviour far from the fluid walls, using a large eddy simulation method, and the second one taking into account the walls and studying the impact of the roughness on the fluid entrainment. After this macroscopic study, we take a statistical look at the macroscopic, mesoscopic and microscopic properties of the turbulence using structure functions and local inter-scale energy flows. Our statistical analysis highlights the low regularity of the observed flows which do not seem to respect the classical Kolmogorov modelling which predicts a fixed regularisation scale. To take this phenomenon into account, we propose a new universal normalisation of the flows using the properties of the multifractal model. This normalisation is interpreted in a thermodynamic framework, and linked to the extensivity of a free energy. We then turn to the understanding of local dissipation and the observed variability of the dissipation scale. We observe that the latter is correlated with the existence of extreme values of energy transfer at the smallest scales, which we interpret as a signature of the existence of singularities or quasi-singularities. Based on this vision, we propose a model to describe fluids including singularities called pincons that interact with each other and with the regular flow that surrounds them. All these models, from the largest scales of the flow to the point singularities, are ultimately a reflection of the impact of the singularities (point defects) at different scales. If the pincons do not necessarily exist, taking them into account would make it possible to explain or model several phenomena: the determination of the topologies of the large recirculations of the flow, the appearance of energy wells independent of the viscosity and maintained by the non-equilibrium character and the universality of the local filtered statistics. (author)