Collaborative research: Turbulent cascades and dissipation in the 3D Navier-Stokes model

University Of Virginia Main Campus, Charlottesville VA

Investigators

Abstract

Grujic DMS-1515805 Dascaliuc DMS-1516487 Turbulence is everywhere, both disruptive and helpful in technological applications, such as safety and efficiency of transportation, biomedical research, climate studies, or infrastructure design. On one hand, suppressing turbulent drag is key in designing and engineering energy efficient vehicles. Understanding the genesis and dynamics of a turbulent wake behind a large plane is key in maintaining safety of the airspace in the proximity of an airport. On the other hand, turbulent mixing may be desirable -- an example being designing more efficient drug delivery systems. The main theme of the project is a rigorous study of various manifestations of turbulence in three-dimensional fluid flows modeled by the Navier-Stokes equations. This is considered both from the perspective of the mathematical theory of turbulence, and as a physical mechanism underlying possible blow-ups (singularities) of the solutions of the system. Ruling out the possibility of singularity formation in any physical model is a fundamental question, even more so when the model should be applicable to such an omnipresent class of physical phenomena as are 3D fluid flows. Graduate students are included in the work of the project. The project branches into three directions: (1) vortex stretching and local anisotropic diffusion, (2) isotropic diffusion of the velocity field and induced scaling laws, and (3) turbulent transport in non-homogeneous Navier-Stokes model. The first two directions follow from the recent work of the investigators and their collaborators in presenting a numerically and analytically motivated geometric scenario of vortex filaments formation that exhibits logarithmic sub-criticality in the context of the 3D Navier-Stokes regularity problem. In this scenario, the transversal scale of the filaments -- a natural, anisotropic micro-scale of the flow -- triggers the mechanism of local, anisotropic diffusion, preventing the possible formation of singularities. The motivation for the third direction of the study comes from the realm of building and reinforcing mathematical support for Kolmogorov phenomenology in physical scales of the flow. The investigators have recently managed to adopt their physical scales methodology to the study of forced turbulence. This opens up an avenue for studying the influence of the spatial distribution of the force on formation of turbulent cascades, as well as for the study of boundary effects via the masking feedback/volume penalization approach. Graduate students are included in the work of the project.