Mathematics of large eddy simulation of turbulent flows

Turbulence is ubiquitous in nature and central to many applications important to our life. (It is also a ridiculously fascinating phenomenon.) Obtaining an accurate prediction of turbulent flow is a central difficulty in such diverse problems as global change estimation, improving the energy efficiency of engines, controlling dispersal of contaminants and designing biomedical devices. It is absolutely fundamental to understanding physical processes of geophysics, combustion, forces of fluids upon elastic bodies, drag, lift and mixing. Decisions that affect our life must be made daily based on predictions of turbulent flows. Direct numerical simulation of turbulent flows is not feasible for the foreseeable future in many of these applications. Even for those flows for which it is currently feasible, it is filled with uncertainties due to the sensitivity of the flow to factors such as incomplete initial conditions, body forces, and surface roughness. It is also expensive and time consuming-far too time consuming to use as a design tool. Storing, manipulating and post processing the mountain of uncertain data that results from a DNS to extract that which is needed from the flow is also expensive, time-consuming, and uncertain. The most promising and successful methodology for doing these simulations of that which matters in turbulent flows is large eddy simulation or LES. LES seeks to calculate the large, energetic structures (the large eddies) in a turbulent flow. The aim of LES is to do this with complexity independent of the Reynolds number and dependent only on the resolution sought. The approach of LES, developed over the last 35 years, is to filter the Navier-Stokes equations, insert a closure approximation (yielding an LES model), supply boundary conditions (called a Near Wall Model in LES), discretize appropriately and perform a simulation. The first three key challenges of LES are thus: Do the solutions of the chosen model accurately reflect true flow averages? Do the numerical solutions generated by the chosen discretization, reflect solutions of the model? And, With the chosen model and method, how is simulation to be performed in a time and cost effective manner? Although all three questions are considered herein, we have focused mostly on the first, i.e. the mathematical development of the LES models themselves. The second and third questions concerning numerical analysis and computational simulation of LES models are essential. However, the numerical analysis of LES should not begin by assuming a model is a correct mathematical realization of the intended physical phenomenon (in other words, that the model is well posed). To do so would be to build on a foundation of optimism. Numerical analysis of LES models with sound mathematical foundations is an exciting challenge for the next stage of the LES adventure. One important approach to unlocking the mysteries of turbulence is by computational studies of key, building block turbulent flows (as proposed by von Neumann). The great success of LES in economical and accurate descriptions of many building block turbulent flows has sparked its explosive growth. Its development into a predictive tool, useful for control and design in complex geometries, is clearly the next step, and possibly within reach in the near future. This development will require much more experience with practical LES methods. It will also require fundamental mathematical contributions to understanding ``How", ``Why", and ``When" an approach to LES can work and ``What" is the expected accuracy of the combination of filter, model, discretization and solver. The extension of LES from application to fully developed turbulence to include transition and wall effects and then to the delicate problems of control and design is clearly the next step in the development of large eddy simulation. Progress is already being made by careful experimentation. Even as ``[The universe] is written in mathematical language" (Galileo), the Navier-Stokes equations are the language of fluid dynamics. Enhancing the universality of LES requires making a direct connection between LES models and the (often mathematical formidable) Navier-Stokes equations. One theme of this book is the connection between LES models and the Navier-Stokes equations rather than the phenomenology of turbulence. Mathematical development will complement numerical experimentation and make LES more general, universal, robust and predictive. We have written this book in the hope it will be useful for LES practitioners interested in understanding how mathematical development of LES models can illuminate models and increase their usefulness, for applied mathematicians interested in the area and especially for Ph.D. students in computational mathematics trying to make their first contribution. One of the themes we emphasize is that mathematical understanding, physical insight and computational experience are the three foundations of LES! Throughout, we try to present first steps at a theory as simple as possible, consistent with correctness and relevance, and no simpler. We have tried, in this balancing act, to find the right level of detail, accuracy and mathematical rigor. This book collects some of the fundamental ideas and results scattered throughout the literature of LES and embeds them in a homogeneous and rigorous mathematical framework. We also try to isolate and focus on the mathematical principles shared by apparently distinct methodologies in LES and show their essential role in robust and universal modeling. In part I we review basic facets of on the Navier-Stokes equations; in parts II and III we highlight some promising models for LES, giving details on the mathematical foundation, derivation and analysis. In part IV we present some of the difficult challenges introduced by solid boundaries; part V presents a syllabus for numerical validation and testing in LES. We are all too aware of the tremendous breadth, depth and scope of the area of LES and of the great limitations of our own experience and understanding. Some of these gaps are filled in other excellent books on LES. In particular, we have learned a lot ourselves from the books of Geurts, John, Pope, and Sagaut. We have tried to complement the treatment of LES in these excellent books by developing mathematical tools, methods, and results for LES . Thus, many of the same topics are often treated herein but with the magnifying glass of mathematical analysis. This treatment yields new perspectives, ideas, language and illuminates many open research problems. We offer this book in the hope that it will be \textit{useful} to those who will help develop the field of LES and fill in many of the gaps we have left behind herein. It is a pleasure to acknowledge the help of many people in writing this book. We thank Pierre Sagaut for giving us the initial impulse in the project and for many detailed and helpful comments along the way. We owe our friend and colleague Paolo Galdi a lot as well for many exciting and illuminating conversations on fluid flow phenomena. Our first meeting came through one such interaction with Paolo. We also thank Volker John, who throughout our LES adventure has been part of our day to day ``battles''. Our understanding of LES has advanced through working with friends and collaborators Mihai Anitescu, Jeff Borggaard, Adrian Dunca, Songul Kaya, Roger Lewandowski, and Niyazi Sahin. The preparation of this manuscript has benefited from the financial support of the National Science Foundation and Ministero dell'Istruzione, dell'Universit\`a e della Ricerca.