1 Introduction -- 1.1 Historical Background and Literature Review -- 1.2 Governing Equations of Incompressible Turbulent Flows -- 1.3 Summary -- References -- 2 Theoretical Principles and Galilean Invariance -- 2.1 Introduction -- 2.2 Basic Principles of Advanced Turbulence Modelling -- 2.3 Summary -- References -- 3 The k-w Shear-Stress Transport (SST) Turbulence Model -- 3.1 Introduction -- 3.2 Mathematical Derivations -- 3.3 Governing Equations of the k-w SST Turbulence Model -- 3.4 Summary -- References -- 4 Three-Dimensional Anisotropic Similarity Theory of Turbulent Velocity Fluctuations -- 4.1 Introduction -- 4.2 Similarity Theory of Turbulent Oscillatory Motions -- 4.3 Summary -- References -- 5 A New Hypothesis on the Anisotropic Reynolds Stress Tensor -- 5.1 Introduction -- 5.2 The Anisotropic Reynolds Stress Tensor -- 5.3 An Anisotropic Hybrid k-w SST/STM Closure Model for Incompressible Flows -- 5.4 Governing Equations of the Anisotropic Hybrid k-w SST/STM Closure Model -- 5.5 On the Implementation of the Anisotropic Hybrid k-w SST/STM Turbulence Model -- 5.6 Summary -- References -- Appendices: Additional Mathematical Derivations -- A.1 The Unit Base Vectors of the Fluctuating OrthogonalCoordinate System -- A.2 Galilean Invariance of the Unsteady Fluctuating VorticityTransport Equation -- A.3 The Deviatoric Part of the Similarity Tensor.