The purpose of this project is to conduct international collaborative research, between two teams of scientists from the USA and Brazil, in the mathematical analysis and modeling of turbulent incompressible fluids. The topics to be investigated are: small viscosity regime of second grade fluids; uniqueness of weak solutions for certain linear perturbations of the two-dimensional Euler equations; the vanishing viscosity limit of the three-dimensional Navier-Stokes equations with helical symmetry; the search for hypotheses on the structure of invariant measures or stationary statistical solutions of the two and three-dimensional Navier-Stokes equations with anomalous dissipation; the search for energy cascade for flows in domains with physical boundaries; two-dimensional cascades with large gap bimodal forcing. Turbulence is a common phenomenon in fluid motion, in which macroscopic quantities (velocity, pressure, temperatures, etc.) no longer have a deterministic relation with global parameters of the flow. Direct numerical simulations of turbulent flows at large Reynolds numbers, that occur in practical applications, such as in geophysical modeling and mixing in industrial processes, are out of reach even for the state-of-the-art computer power. Therefore, there is an urgent need to pursue this challenging problem analytically, by developing rigorous mathematical and statistical tools to investigate it, and to test these tools computationally.
While turbulence is an everyday occurrence, our understanding is still lacking in many aspects. Quantifying the effect of small scales on the dynamics of large scales is fundamental in modern multiscale science. The goal of the project is to enable a predictive analytical study of turbulent flows. This study will impact wide-ranging applications, from geophysical modeling, such as dispersion of pollutants in the ocean, to biological and industrial modeling, such as design of polymeric materials. The project will consolidate the well-established collaborative efforts of the principal investigators with their Brazilian counterparts, and may lead to new collaboration, especially among the junior research personnel. International collaboration among scientists is a key to economic competitiveness in global markets. Four US and three Brazilian academic institutions are involved in the project. The international dimension of the project is further emphasized through two planned workshops. Training and supervision of at least six Ph.D. students and postdoctoral fellows is also achieved through planned summer schools and scientific workshops. Students and postdoctoral fellows from the US will travel to Brazil to participate in the workshops and the summer schools and interact with the US and Brazilian researchers.
This project is co-funded with the Americas Program of the Office of International Science and Engineering.
The main goal of this project is to utilize the combined expertize of several researchers in the US and Brazil to improve our understanding of fluid flow at high Reynolds numbers, using rigorous mathematics. The Reynolds number measures how viscous the fluid flow is: the higher the number the less viscous the flow. Fluids with high Reynolds numbers display a very complex behavior and often are turbulent, that is, (informally) eddies of many sizes develop in the flow. Turbulent flows arise in many practical situations, from flows in the atmosphere and the oceans, to flows around obstacles, for example the airflow in airplane engines and around the wings and the fuselage. Yet, a thorough understanding of this complex phenomenon, involving many length and time scales, is still lacking. Mathematics can be used in a predictive qualitative fashion (for instance, by validating physically-based theories using the equations that physical quantities of interest must satisfy), as well as in a quantitative way (for example, by providing maximum rates of dissipation of energy in the flow). Direct computer simulations are often prohibitive in terms of cost and computing power, and mathematics can help select reduced models that can be implemented numerically and can still capture the most important features of the phenomenon under investigation. The main outcome of the project is a strengthened collaboration between the Principal Investigator (PI), the other PIs on this collaborative award, and their collaborators in Brazil, in particular at Campinas University (Unicamp) and the Federal University of Rio de Janeiro (UFRJ). The main activities stemming from this collaboration were the organization of a workshop at Unicamp in 2011 and a school at the National Institute of Pure and Applied Mathematics (IMPA) in Rio in 2014. Both the workshop and the school brought together a mix of senior and junior researchers, including several students, from Brazil, China, Europe, the US, and other countries in the Americas, thus enhancing the training component and broader impacts of the project. A major thrust of this project and of the PI's research program in general has been the rigorous analysis of fluid flow at high Reynolds numbers in the presence of walls. A main difficulty is studying this problems is the fact that at any viscosity, no matter how small, walls create vorticity (swirls) in the flow that, in turn, can lead to instabilities in the bulk. As the Reynolds number increases, objects moving in the flow, such as an airplane, experience more drag up to a certain point. The PI and her collaborators have obtained a detailed picture of the behavior of fluids near walls at very large Reynolds numbers, for certain special classes of flows in pipes and channels. In the spirit of considering more complicated and realistic flows, the PI and her collaborators have begun to study helically-symmetric (helical for short) flows. These are flows that respect helical structures. For example, in the absence of viscosity, helical vortices in the flow are transported by the flow itself. These flows arise in nature. The vortices shed by the trailing edge of the blades in a rotating wind turbine are approximately helical. There is evidence that helical flows play an important role in blood circulation under some conditions. From a mathematical point of view, symmetry makes a rigorous analysis more amenable, as it constrains the possible behavior of the flow. It is important and interesting to study what happens to the flow when the geometry of the helices changes, e.g. they become more elongated or wind up more tightly, as this can have a big effect on the flow as a whole. The main result that the PI and her collaborators achieved is to prove rigorously the following. As the pitch of the helices grows (the helices are then stretched along their axis), the flow becomes close (in a sense that can be made mathematically precise) to an almost two-dimensional flow, a so-called "2 and 1/2" dimensional flow, where only two directions of motion in the flow are dynamically active, while the fluid in the third direction is passively transported by the flow in the first two directions and may diffuse at the same time. When the viscosity is absent, (under an additional condition) the flow is well approximated by a truly two-dimensional flow. Flows in two dimensions have special properties and two-dimensional turbulence exhibits characteristics that are very different than three-dimensional turbulence. In the opposite situation in which the pitch of the helices is shrinking (then the helices are compressed along their axis), one expects that the flow is close to a simpler flow, namely a so-called "circularly symmetric" flow. In a circularly-symmetric flow, the fluid moves in circles centered at a given axis and the flow is the same at any height along this axis.
