Computational fluid dynamics Volume 3

The primary goal in the third volume is to review the fundamentals of turbulence and turbulent flows and to extend the governing equations and numerical schemes developed in Volume II to include turbulence. This volume begins with the basic definitions and concepts in turbulence and turbulent flows. Subsequently, the modification of the governing equations and numerical schemes is introduced. There are three approaches by which turbulent flow fields may be computed. The first approach is based on the averaged Navier-Stokes equations either in the form of Reynolds-Averaged Navier-Stokes (RANS) equations or Favre-Averaged Navier-Stokes (FANS) equations. These formulations, along with several turbulence models and numerical considerations for the solutions of equations, are presented in Chapter 21.The second and third approaches are the Large Eddy Simulations (LES) and the Direct Numerical Simulations (DNS), which are presented in Chapter 23.Since typical computations involved in turbulence and, in particular, in DNS, require higher-order schemes such as compact finite difference formulations, these formulations are introduced in Chapter 22. Finally, a computer code based on the RANS equations and several turbulence models have been developed and included in the text Student Guide for CFD-Volume III. Title page ......Page 1 Date-line ......Page 2 CONTENTS ......Page 3 Preface ......Page 7 20.1 Introductory Remarks ......Page 9 20.2 Fundamental Concepts and Definitions ......Page 12 20.3 Introduction to Transition ......Page 13 20.3.1 Stability Theory ......Page 15 20.4 Conceptual Model for Turbulent Flows ......Page 17 20.5 Production, Diffusion, and Dissipation of Turbulence ......Page 20 20.6 Length and Time Scales of Turbulence ......Page 21 20.7 Free Shear Layer Flows ......Page 23 20.8 Numerical Techniques for Turbulent Flows ......Page 24 20.9 Concluding Remarks ......Page 26 21.2 Fundamental Concepts ......Page 28 21.2.1 Universal Velocity Distribution ......Page 31 21.3 Modification of the Equations of Fluid Motion ......Page 35 21.3.1 Reynolds Averaged Navier-Stokes Equations ......Page 36 21.3.1.1 Turbulent Shear Stress and Heat Flux ......Page 42 21.3.1.2 Flux Vector RANS Formulation ......Page 45 21.4 Turbulence Models ......Page 46 21.4.1 Zero-Equation Turbulence Models ......Page 47 21.4.2 One-Equation Turbulence Models ......Page 50 21.4.2.1 Baldwin-Barth One-Equation Turbulence Model ......Page 51 21.4.2.1.1 Nondimensional form ......Page 52 21.4.2.1.2 Baldwin-Barth turbulence model in computational space ......Page 53 21.4.2.2 Spalart-Allmaras One-Equation Turbulence Model ......Page 56 21.4.2.2.1 Nondimensional form ......Page 58 21.4.2.2.2 Spalart-Allmaras turbulence model in computational space ......Page 59 21.4.3 Two-Equation Turbulence Models ......Page 61 21.4.3.1 $k-\varepsilon$ Two-Equation Turbulence Model ......Page 62 21.4.3.1.1 Low Reynolds number $k-\varepsilon$ model ......Page 64 21.4.3.1.3 Initial and boundary conditions ......Page 66 21.4.3.2 $k-\omega$ Two-Equation Turbulence Model ......Page 67 21.4.3.3 Combined $k-\varepsilon$/$k-\omega$ Two-Equation Turbulence Model ......Page 68 21.4.3.3.1 Baseline model ......Page 71 21.4.3.3.2 Initial and boundary conditions ......Page 72 21.4.3.3.5 Nondimensional form ......Page 73 21.4.3.3.6 $k-\varepsilon$/$k-\omega$ turbulence model in computational space ......Page 74 21.5 Numerical Considerations ......Page 76 21.6 Finite Difference Formulations ......Page 77 21.6.1.1 Implicit Formulation ......Page 78 21.6.1.2 Explicit Formulation ......Page 83 21.6.2.1 Implicit Formulation ......Page 85 21.6.2.2 Explicit Formulation ......Page 92 21.6.3 Two-Equation Turbulence Models ......Page 94 21.7.1 Shock/Boundary Layer Interaction ......Page 100 21.7.2 Two-Dimensional Base Flow ......Page 103 21.7.3 Axisymmetric, Supersonic, Turbulent Exhaust Flow ......Page 108 21.8 Favre-Averaged Navier-Stokes Equations ......Page 113 21.8.1 Turbulent Viscosity ......Page 121 21.9 Concluding Remarks ......Page 123 22.1 Introductory Remarks ......Page 125 22.2 Compact Finite Difference Formulations for the First-Order Derivatives ......Page 126 22.2.1 One-Sided Approximations ......Page 133 22.3 Compact Finite Difference Formulations for the Second-Order Derivatives ......Page 134 22.4 Error Analysis ......Page 135 22.5 Application to a Hyperbolic Equation ......Page 138 22.6 Problems ......Page 145 23.1 Introductory Remarks ......Page 147 23.2.1 Filtered Navier-Stokes Equations ......Page 148 23.2.2 Subgridscale Models ......Page 154 23.2.2.1 Eddy Viscosity ......Page 155 23.2.2.1.1 Smagorinsky model ......Page 156 23.2.2.1.2 Dynamic model ......Page 157 23.3 Direct Numerical Simulation ......Page 160 Appendix J: Transformation of Turbulence Models from Physical Space to Computational Space ......Page 163 Appendix K: The Transport Equation for the Turbulence Kinetic Energy ......Page 167 References ......Page 174 Index ......Page 178