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Mathematical and numerical foundations of turbulence models and applications. (English) Zbl 1328.76002

Modeling and Simulation in Science, Engineering and Technology. New York, NY: Birkhäuser/Springer (ISBN 978-1-4939-0454-9/hbk; 978-1-4939-0455-6/ebook). xvii, 517 p. (2014).
The book under review provides a comprehensive interdisciplinary reference for mathematical modeling, theoretical and numerical analysis of 3D incompressible turbulent flows. The table of contents is as follows:
1. Introduction. 2. Incompressible Navier-Stokes equations. 3. Mathematical basis of turbulence modeling. 4. The \(k\)-\(\varepsilon\) model. 5. Laws of turbulence by similarity principles. 6. Steady Navier-Stokes with wall laws and fixed eddy viscosities. 7. Analysis of the continuous steady NS-TKE model. 8. Evolutionary NS-TKE model. 9. Finite element approximation of the steady Smagorinski model. 10. Finite element approximation of evolution Smagorinski model. 11. A projection-based variational multiscale model. 12. Numerical approximation of NS-TKE model. 13. Numerical experiments. A Tool Box. Index.
The book is divided into three parts.
1.
Chapters 2 to 5. The authors start from well accepted basic notions and heuristic reasonings in fluid dynamics and turbulence modeling and clarify as much as possible the mathematical arguments which lead to the today’s widely used issues in mathematical turbulence models, such as one-equation models, the \(k\)-\(\varepsilon\) model and large eddy simulation models. We particularly mention: a careful discussion of boundary conditions, boundary layers and wall laws; derivation of the \(k\)-\(\varepsilon\) model; similarity principles (isotropy and Kolmogorov law). These chapters give a mathematically rigorous presentation of the physical foundations of the above mentioned turbulence models and are of an independent interest.
2.
Chapters 6 to 8. Chapter 6 is devoted to two methods of proof of the existence of weak solutions to the steady Navier-Stokes equations with a given eddy viscosity and a wall law as boundary condition, namely the standard Galerkin method and the linearization method by Schauder’s fixed-point theorem. Chapter 7 is concerned with the proof of the existence of a weak solution to the steady Navier-Stokes equations coupled with a PDE for the turbulent kinetic energy \(k\). This system of PDEs is completed by a wall law for the velocity \(\mathbf{v}\) and an inhomogeneous boundary condition for \(k\). The method of proof consists in establishing the existence of weak solution to a regularized system and then the passing to the limit. Unbounded eddy viscosities are discussed in subsection 7.5.3 (pp. 241-243). Chapter 8 is concerned with the unsteady Navier-Stokes system for the velocity \(\mathbf{v}\) coupled with the turbulent energy equation (TKE) \[ \partial_t k+\mathbf{v} \cdot \nabla k = \nabla \cdot \Big[ \Big(\mu+\mu_t(k,t,\text\textbf{x})\Big)\nabla k\Big] + \nu_t(k,t,\text\textbf{x}) \; |D\text\textbf{v}|^2 - kE(k,t,\text\textbf{x}) \] where \(\nu_t, \mu_t\) and \(E\) are given non-negative, uniformly bounded continuous functions on \(\mathbb{R} \times \mathbb{R}_+ \times \Omega\) (\(\Omega\) = computational domain in \(\mathbb{R}^3\)) (cf. p. 252). Notice that \(E\) replaces the term \(\sqrt{k}/\ell(t,\mathbf{x})\) where \(\ell(t,\mathbf{x})\) denotes the mixing length (cf. pp. 79, 140) (in one-equation turbulence models, the term \(k^{3/2}/\ell\) represents the rate of dissipation of turbulent kinetic energy; cf., e.g., [D. C. Wilcox, Turbulence modeling for CFD. 3rd ed. DCW Industries, Inc. (2006)]). Theorem 8.2 (p. 269) states the existence of a weak solution to the initial/boundary-value problem under consideration, the equality in (1) being replaced by an inequality (cf. also the remarks on p. 302).
3.
Chapters 9 to 13. These chapters are concerned with the numerical analysis of several turbulence models. In Chapter 12 the authors perform a finite element approximation of the turbulence models studied in Chaps. 7 and 8 where the unbounded terms are regularized by truncation, in particular the production term \(\nu_t |D\mathbf{v}|^2\) on the right hand side of (1).
The appendix “Tool box” contains some results from Sobolev spaces, functional analysis and integration theory which are used in the text. Each chapter ends with a bibliography. Reading the book requires some familiarity with reasonings in functional analytic methods for PDEs.
Summary: The book is a valuable reference for the study of the mathematical foundations of several widely used turbulence models. It can be recommended for advanced graduate students and researchers in mathematical and numerical turbulence theory.\)\)\)\]\)\)

MSC:

76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
76F02 Fundamentals of turbulence
76D05 Navier-Stokes equations for incompressible viscous fluids
76M10 Finite element methods applied to problems in fluid mechanics
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