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This book gives a mathematical insight--including intermediate derivation steps--into engineering physics and turbulence modeling related to an anisotropic modification to the Boussinesq hypothesis (deformation theory) coupled with the similarity theory of velocity fluctuations.

Through mathematical derivations and their explanations, the reader will be able to understand new theoretical concepts quickly, including how to put a new hypothesis on the anisotropic Reynolds stress tensor into engineering practice. The anisotropic modification to the eddy viscosity hypothesis is in the center of research interest, however, the unification of the deformation theory and the anisotropic similarity theory of turbulent velocity fluctuations is still missing from the literature. This book brings a mathematically challenging subject closer to graduate students and researchers who are developing the next generation of anisotropic turbulence models.

Indispensable for graduate students, researchers and scientists in fluid mechanics and mechanical engineering.

### Chapter 1. Introduction

Abstract
In this chapter, a brief literature review has been carried out considering those contributions which are aligned with the objectives of the present book. Since, there are thousands of works dealing with internal and external turbulent flows, therefore, we consider a selection of those contributions which are relevant to the understanding of the new hypothesis on the anisotropic Reynolds stress tensor in Chap. 5. For the sake of completeness, the governing equations of incompressible turbulent flows have been derived in conjunction with the generalised Boussinesq hypothesis on the Reynolds stress tensor. Intermediate mathematical steps are included in the derivations to make graduate and postgraduate students familiar with the heart of the closure problem of anisotropic turbulence. The shortcomings of the generalised Boussinesq hypothesis have also been discussed to emphasise the necessity of a new hypothesis on the Reynolds stress tensor.
László Könözsy

### Chapter 2. Theoretical Principles and Galilean Invariance

Abstract
This chapter focuses on those theoretical principles which are required to formulate physically correct mathematical closure equations for modelling turbulent flows. The importance of the Galilean invariance in the Newtonian physics is to ensure that the conservation laws of turbulent flow motions remain the same in any two reference frames. Therefore, we devote a particular attention to the Galilean transformation and the derivation of the Galilean invariance of the Reynolds momentum equation (1.​43), the Reynolds stress tensor (1.​54), the rate-of-strain tensor (1.​114) and the generalised Boussinesq hypothesis on the Reynolds stresses (1.​113). The principle of Galilean invariance for the Reynolds stress tensor will also be taken into account in the proposal to the new hypothesis on the anisotropic Reynolds stress tensor in Chap. 5. In addition to the Galilean invariance, the consistency of physical dimensions, the coordinate system independence of physical laws and the realisability condition have also been considered as relevant criteria in the mathematical description of the Reynonds stress tensor. The derivations included in the present chapter make an attempt to bring closer a theoretically demanding advanced subject to a wider audience.
László Könözsy

### Chapter 3. The k- Shear-Stress Transport (SST) Turbulence Model

Abstract
This chapter focuses on the mathematical formulations of the turbulent kinetic energy k and specific dissipation rate $$\omega$$ Shear-Stress Transport (SST) turbulence model proposed by Menter [3, 4] to provide a closure model to the Boussinesq-type counterparts of the new hypothesis on the anisotropic Reynolds stress tensor discussed in Chap. 5. The k-$$\omega$$ SST closure model of Menter [3, 4] is relying on the generalised Boussinesq hypothesis on the Reynolds stress tensor (1.​113) with a modification to the definition of the scalar eddy viscosity coefficient. In other words, the k-$$\omega$$ SST turbulence model assumes that the Reynolds stress tensor (1.​54) is related to the mean rate-of-strain (deformation) tensor (1.​114) and the turbulent kinetic energy k defined by Eq. (1.63). The reason for the choice of the k-$$\omega$$ SST model as a baseline closure model is that it is a well-known fact that the k-$$\omega$$ SST formulation of Menter [3, 4] is validated against many industrially relevant turbulent flow problems with great success . It is also assumed that the k-$$\omega$$ SST turbulence model can capture the shear stress distribution correctly in the boundary layer and it is applicable to adverse pressure gradient flows . However, it is important to highlight from theoretical and practical aspects that any other existing eddy viscosity closure model can be employed in conjunction with the Boussinesq-type counterparts of the new hypothesis on the anisotropic Reynolds stress tensor proposed in Chap. 5.
László Könözsy

### Chapter 4. Three-Dimensional Anisotropic Similarity Theory of Turbulent Velocity Fluctuations

Abstract
This chapter focuses on the three-dimensional anisotropic similarity theory of turbulent oscillatory motions or Galilean invariant turbulent velocity fluctuations as a necessary theoretical background to understand the new hypothesis on the anisotropic Reynolds stress tensor in Chap. 5. The anisotropic similarity theory of three-dimensional turbulent velocity fluctuations was developed by Czibere [2, 3] in conjunction with a stochastic turbulence model (STM) to describe the Reynolds-averaged velocity fluctuations in the anisotropic Reynolds stress tensor (1.​54). The three-dimensional theory of Czibere [2, 3] introduces an anisotropic similarity tensor—which is related to the dimensionless vector potential of the mechanically similar local velocity fluctuations—to distribute anisotropically the principal (dominant) turbulent shear stress in the fluid flow field. It is important to note that certain components of the anisotropic similarity theory presented in this chapter—e.g. the definition of the unit base vectors of the fluctuating natural coordinate system—are discussed in a slightly different way compared to the original theory of Czibere [2, 3]. The reason for minor modifications to the original theory is to introduce a fully Galilean invariant formulation of the anisotropic Reynolds stress tensor (1.​54). The objective is to put the anisotropic similarity theory of velocity fluctuations into practice and make it available to those researchers who are intended to develop the next generation of anisotropic turbulence models.
László Könözsy

### Chapter 5. A New Hypothesis on the Anisotropic Reynolds Stress Tensor

Abstract
In this chapter, a new hypothesis on the anisotropic Reynolds stress tensor has been proposed which is relying on the unification of the generalised Boussinesq hypothesis (1.​113) (deformation theory) and the fully Galilean invariant three-dimensional anisotropic similarity hypothesis (4.​121) of turbulent velocity fluctuations (similarity theory). The anisotropic modification to the generalised Boussinesq hypothesis (1.​113) is in the centre of research interest nowadays , however, the hybridisation of the generalised version of the Boussinesq hypothesis  and the recently developed anisotropic similarity theory of turbulent velocity fluctuations [8, 9] is still missing from the literature. In other words, the new hypothesis proposed here is an anisotropic modification to the generalised Boussinesq hypothesis (1.​113) based on the fully Galilean invariant version of the three-dimensional anisotropic similarity theory of turbulent velocity fluctuations—which is discussed in Chap. 4—in conjunction with the mathematical description of the Reynolds stress tensor (1.​54). In addition to this, a possible anisotropic hybrid k-$$\omega$$ SST/Stochastic Turbulence Model (STM) closure approach has also been proposed related to the new hypothesis on the anisotropic Reynolds stress tensor in this chapter. Computational engineering simulations is the subject of the second volume of this book.
László Könözsy