A New Hypothesis on the Anisotropic Reynolds Stress Tensor for Turbulent Flows : Volume II: Practical Implementation and Applications of an Anisotropic Hybrid K-omega Shear-Stress Transport/Stochastic Turbulence Model

This self-contained, interdisciplinary book encompasses mathematics, physics, computer programming, analytical solutions and numerical modelling, industrial computational fluid dynamics (CFD), academic benchmark problems and engineering applications in conjunction with the research field of anisotropic turbulence. It focuses on theoretical approaches, computational examples and numerical simulations to demonstrate the strength of a new hypothesis and anisotropic turbulence modelling approach for academic benchmark problems and industrially relevant engineering applications. This book contains MATLAB codes, and C programming language based User-Defined Function (UDF) codes which can be compiled in the ANSYS-FLUENT environment. The computer codes help to understand and use efficiently a new concept which can also be implemented in any other software packages. The simulation results are compared to classical analytical solutions and experimental data taken from the literature. A particular attention is paid to how to obtain accurate results within a reasonable computational time for wide range of benchmark problems. The provided examples and programming techniques help graduate and postgraduate students, engineers and researchers to further develop their technical skills and knowledge. Preface Acknowledgements Contents Acronyms 1 Introduction to Classical Analytical Solutions for Wall-Bounded Turbulence 1.1 Introduction 1.2 Governing Equations of Incompressible Turbulence 1.3 Turbulent Flow in Straight Smooth Circular Pipes 1.3.1 Mathematical Formulation of the Fully-Developed Turbulent Flow Problem in Straight Circular Pipes 1.3.2 The Simplified Reynolds Momentum Equation 1.3.3 The Constant Pressure Gradient Assumption 1.3.4 Integrated Mathematical Form of the Simplified Reynolds Momentum Equation 1.3.5 The Viscous Sublayer and the Definition of the Friction (Shear) Velocity at the Wall 1.3.6 Hydraulic Loss and the Dimensionless Resistance Coefficient (Coefficient of Friction) 1.3.7 The Reynolds Number and Its Related Physical Quantities for Turbulent Flows in Pipes 1.3.8 Hydrodynamic Entrance (Development) Length for Fully-Developed Turbulent Flows in Horizontal Pipes 1.3.9 Relationships Amongst the Pressure Drop, the Reynolds Number and the Volume Flow Rate 1.3.10 Mean Velocity Distribution in the Viscous Sublayer 1.3.11 Fluid Flow Regions of the Dimensionless Mean Velocity Distribution in the Turbulent Boundary Layer 1.3.12 Turbulent Velocity Profiles Using a First-Order (Linear) Mixing-Length Function (Prandtl's Solution) 1.3.13 The Resistance Law for Smooth Circular Pipes 1.3.14 Computation of the Resistance Coefficient Using the Newton–Raphson Iterative Method 1.3.15 An Algorithm for Computing Turbulent Velocity Profiles in Straight Smooth Circular Pipes 1.3.16 Turbulent Velocity Profiles Using a Second-Order Mixing-Length Function (Von Kármán's Solution) 1.3.17 Turbulent Velocity Profiles Using a Third-Order Mixing-Length Function (Czibere's Solution) 1.3.18 Comparison of Analytically Computed Turbulent Mean Velocity Profiles with Experimental Data 1.4 Analogy Between Pipe and Channel Flows 1.4.1 Mathematical Formulation of the Fully-Developed Turbulent Flow Problem in Straight Plane Channels 1.4.2 The Simplified Reynolds Momentum Equation for Two-Dimensional Flows in Channels 1.4.3 Integrated Mathematical Form of the Simplified Reynolds Momentum Equation for Plane Flows 1.4.4 The Definition of the Friction (Shear) Velocity 1.4.5 The Definition of the Reynolds Number for Turbulent Flows in Straight Plane Channels 1.4.6 The Resistance Law for Smooth Channels 1.4.7 Remarks on Channel Flow Computations 1.5 Summary References 2 The Anisotropic Hybrid k-ω SST/Stochastic Turbulence Model 2.1 Introduction 2.1.1 Three-Dimensional Map Space and the Similarity Tensor of the Mechanically Similar Local Velocity Fluctuations 2.1.2 A New Hypothesis on the Anisotropic Reynolds Stress Tensor for Incompressible Turbulent Flows 2.1.3 The Matrix Form of the New Hypothesis on the Anisotropic Reynolds Stress Tensor 2.2 Mathematical Formulation and Derivations 2.2.1 Tensor Divergence of the New Anisotropic Reynolds Stress Tensor for Incompressible Turbulent Flows 2.2.2 The New Turbulent Kinetic Energy Production Term 2.2.3 Dissipation and Diffusion Terms of the Turbulent Kinetic Energy Transport Equation 2.2.4 The New Specific Dissipation Rate Production Term 2.2.5 Dissipation and Diffusion Terms of the Transport Equation of the Specific Dissipation Rate 2.3 Governing Equations of the Anisotropic Hybrid k-ω SST/STM Closure Model 2.4 Summary References 3 Implementation of the Anisotropic Hybrid k-ω SST/STM Closure Model 3.1 Introduction 3.2 Implementation of the Three-Dimensional Stochastic Turbulence Model (STM) 3.2.1 An Algorithm for Computing the Elements of the Anisotropic Similarity Tensor and Its Deviatoric Part 3.2.2 A MATLAB Code Implementation of the Stochastic Turbulence Model (STM) and Its Explanation 3.2.3 An Example How to Compute the Elements of the Anisotropic Similarity Tensor and Its Deviatoric Part 3.3 A Computer Code for the Anisotropic Hybrid k-ω SST/STM Closure Model 3.3.1 A C Code Based User-Defined Function (UDF) Implementation of the Anisotropic k-ω SST/STM Closure Model in the ANSYS-FLUENT Environment 3.4 Summary References 4 Two-Dimensional Simulations with an Anisotropic Hybrid k-ω SST/STM Approach 4.1 Introduction 4.2 Two-Dimensional Classical Benchmark Problems 4.2.1 Turbulent Flow over a Flat Plate with Zero Pressure Gradient (The Klebanoff Problem) 4.2.2 Turbulent Flow over a NACA 0012 Airfoil 4.2.3 Axisymmetric Turbulent Shear Flows in Straight Circular Pipes at Low and High Reynolds Numbers 4.2.4 Turbulent Shear Flow in a Rotationally Symmetric Coaxial Curved Duct with Varying Cross Sections 4.2.5 Turbulent Flow over a Plane Backward-Facing Step 4.3 Summary References 5 Three-Dimensional Simulations with an Anisotropic Hybrid k-ω SST/STM Approach 5.1 Introduction 5.2 Turbulent Flow in a Horizontal Cylindrical Pipe 5.2.1 Mean Pressure and Velocity Distributions 5.2.2 Anisotropic Reynolds Stress Distributions 5.2.3 Turbulent Energy Production and Dissipation 5.2.4 Concluding Remarks 5.3 Turbulent Flow over a NACA 0013 Wing 5.3.1 The Experimental Background 5.3.2 Simulation Results 5.3.3 Concluding Remarks 5.4 Turbulent Flow over the Jetstream 31 Aircraft 5.5 Summary and Future Work References Appendix A Supplementary Mathematical Derivations A.1 Derivations in the Cylindrical Coordinate System A.1.1 Unit Vectors and Their Derivatives in the Cylindrical Coordinate System A.1.2 Definition of the Hamilton (nabla/del) Vector-Type Differential Operator in Cylindrical Coordinates A.1.3 Divergence of a Vector Field A.1.4 The Convective Term of the Momentum Equation A.1.5 The Pressure Gradient Term in Cylindrical Coordinates A.1.6 Definition of the Second-Order Scalar-Type Laplace Differential Operator in Cylindrical Coordinates A.1.7 The Viscous Term of the Momentum Equation A.1.8 Tensor Divergence of the Reynolds Stress Tensor A.2 Integration of the Turbulent Velocity Profile u(r)/uτ A.2.1 Prandtl's Solution A.2.2 Von Kármán's Solution A.3 Absolute and Relative Errors A.3.1 Computation of the Absolute Error A.3.2 Computation of the Relative Error Appendix B Supplementary Computer Codes B.1 A MATLAB Code of Analytical Solutions in Pipes B.2 User-Defined Function (UDF) C Codes B.2.1 A UDF C Code for the Two-Dimensional Version of the Anisotropic Hybrid k-ω SST/STM Closure Model B.2.2 A UDF C Code for Computing the Scalar Elements of the Reynolds Stress Tensor relying on the Boussinesq Hypothesis for Two-Dimensional Flows B.2.3 A UDF C Code for Computing the Scalar Elements of the Reynolds Stress Tensor relying on the Boussinesq Hypothesis for Three-Dimensional Flows Appendix C Digitalised Experimental Data C.1 Digitalised Experimental Data for Two- and Three-Dimensional Simulations C.1.1 Digitalised Experimental Data of Klebanoff (1955) for a Turbulent Flow over a Flat Plate with Zero Pressure Gradient C.1.2 Digitalised Experimental Data of Nikuradse (1932) for Turbulent Mean Velocity Distributions in a Straight Smooth Circular Pipe C.1.3 Digitalised Experimental Data of Laufer (1954) for Turbulent Flows in a Straight Smooth Circular Pipe C.1.4 Experimental Data of Szabó and Kecke (2001) for Turbulent Flows in a Rotationally Symmetric Coaxial Curved Duct with Varying Cross Sections