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## Über dieses Buch

This volume presents a mathematical development of a recent approach to the modeling and simulation of turbulent flows based on methods for the approximate solution of inverse problems. The resulting Approximate Deconvolution Models or ADMs have some advantages over more commonly used turbulence models – as well as some disadvantages. Our goal in this book is to provide a clear and complete mathematical development of ADMs, while pointing out the difficulties that remain. In order to do so, we present the analytical theory of ADMs, along with its connections, motivations and complements in the phenomenology of and algorithms for ADMs.

## Inhaltsverzeichnis

### Chapter 1. Introduction

This book presents a mathematical development of a recent approach to the modeling and simulation of turbulent flows based on methods for the approximate solution of inverse problems.
William J. Layton, Leo G. Rebholz

### Chapter 2. Large Eddy Simulation

At high Reynolds number the fluid velocity is exponentially sensitive to perturbations of the problem data. This sensitivity, however, is not uniform. The large structures (large eddies) evolve deterministically and are thus not sensitive [BFG02]. The small eddies are sensitive because they have a random character.
William J. Layton, Leo G. Rebholz

### Chapter 3. Approximate Deconvolution Operators and Models

The great challenge in simulation of turbulent flows from applications ranging from geophysics to biomedical device design is that equations for the pointwise flow quantities are well-known but intractable to computational solution and sensitive to uncertainties and perturbation in problem data. On the other hand, closed equations for the averages of flow quantities cannot be obtained directly from the physics of fluid motion.
William J. Layton, Leo G. Rebholz

### Chapter 4. Phenomenology of ADMs

An approximate deconvolution operator denoted by D is an approximate filter inverse that is accurate on the smooth velocity components and does not magnify the rough components.
William J. Layton, Leo G. Rebholz

### Chapter 5. Time Relaxation Truncates Scales

The fundamental requirement for a successful turbulent flow simulation is to truncate scales to those representable on a computationally feasible mesh without substantially changing the large flow structures.
William J. Layton, Leo G. Rebholz

### Chapter 6. The Leray-Deconvolution Regularization

In LES one solves a system whose solution is an approximation to local spacial averages of the NSE. In regularization modeling one solves a system similar to the NSE which has better qualitative properties for numerical simulation than the underlying NSE.
William J. Layton, Leo G. Rebholz

### Chapter 7. NS-Alpha- and NS-Omega-Deconvolution Regularizations

Consider the NSE in rotational form: $$\begin{array}{rrr}\rm{u}_t+(\nabla\times \rm{u})\times\rm{u}-\rm{v}\triangle\rm{u}+\nabla P=f(\rm{x},t), \\ \nabla \cdot = 0, \\ \text{where} P = p + 1/2|u|_2\end{array}$$
William J. Layton, Leo G. Rebholz

### Backmatter

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