Computational Fluid Dynamics of Compressible Flows
WEEK 1-4:
1. Introduction: Motivation, Problems in predicting compressible flows, Unique phenomena in compressible flows, Course Objectives, Course Plan, Evaluation of coursework, Required Background, Term Project
[Term Project]
2. Review of Computational Fluid Dynamics (CFD)
- 2.1. Governing equations in CFD: Navier-Stokes equations (N-S), Flux vector formulation of the NS equations, Conservative vs. primitive forms, Euler equations, Model equations
- 2.2. Classification of differential equations: ODE’s vs. PDE’s, Linear vs. non-linear equations, First-order vs. higher-order equations, Conservative vs. non-conservative forms
- 2.3. Classification of Partial Differential Equations (PDE’s): Determining the nature of PDE’s (elliptic, parabolic, hyperbolic), Physical meaning for fluid flows, Computational meaning for fluid flows, Boundary and initial conditions for PDE’s
- 2.5. Turbulence: Free turbulent flows, Boundary layers near solid walls, Turbulence modeling in CFD, Wall functions and implications for grid generation
- 2.6. Numerical solution of PDE’s: Selection of mathematical model, Selection of discretization method (Finite Difference, Finite Volume, Finite Element, Spectral Method)
- 2.7. Grid generation: Structured vs. unstructured grids, Grid transformation, Cartesian grids, Zonal or block-structured grids, Hybrid grids, Moving mesh techniques (Sliding mesh, CHIMERA grids) Deforming mesh techniques, Adaptive grids, Multigrid methods and their relation to grid generation, Basic guidelines for grid generation
- 2.8. Boundary treatment: Boundary conditions, Boundary treatment (Changing the numerical method at edges, Changing the computational domain at edges), Solid Wall boundary treatment, Far-field boundary treatment, Non-reflecting boundaries
- 2.9. Solution techniques for the discretized equations: Explicit vs. implicit formulations, Solutions techniques for explicit method (Lax-Wendroff, MacCormack, Runge-Kutta), Solution techniques for implicit methods (Direct methods /Gaussian elimination, Cramer’s rule/, Indirect methods /Thomas algorithms, point-iterative methods, approximate factorization/)
- 2.10. Errors and uncertainty in CFD: Sources of error, Sources of uncertainty, Stability analysis of numerical errors (Discrete Perturbation analysis, Von Neumann Stability Analysis, Multidimensional considerations), The Courant-Friedrich-Loewy number (CFL), Stability vs. accuracy, Local vs. Global time stepping, Evaluation of convergence (Iterations convergence: residuals, Grid convergence, Time step convergence), Characteristic features related to stability
(Consistency, Boundedness, Transportiveness)
- 2.11. Conclusions: methods best suited for compressible flows
WEEK 5-8:
3. The Finite Volume Method (FVM)
- 3.1. Governing equations for FV formulations: Conservative integral form of the N-S equations, General transport equation, FV discretization of the governing equations,
- 3.2. FVM for diffusion problems: FVM for 1D steady-state diffusion, Worked examples on 1D steady state diffusion, FVM for 2D diffusion problems, FVM for 3D diffusion problems [A1]
- 3.3. FVM for convection-diffusion problems: FVM for 1D steady convection-diffusion, Central differencing scheme (CD), Upwind differencing scheme (UD), Hybrid CD-UD schemes, Powerlaw scheme, Higher order schemes (QUICK), Total Variation Diminishing schemes (TVD), Flux limiters [A2]
- 3.4. FVM for unsteady flows: FVM for unsteady diffusion, Explicit scheme, Crank-Nicolson scheme, Fully implicit scheme, Extension to multidimensions, FVM for unsteady convection-diffusion [A3]
WEEK 9-12:
4. Solution of the compressible Euler equations
- 4.1. Gasdynamics Basics: Expansion waves, Compression waves, Shock waves, Contact discontinuities
- 4.2. Euler equations: Integral form, Conservation form, Vector-Matrix notation, Rankine-Hugoniot relations, Entropy condition
- 4.3. The Riemann problem: Riemann problem for Euler equations, Riemann problem for linear system of equations, Godunov type schemes, Approximate Riemann problems (Motivation for approximating, Roe’s approximate Riemann solver, Secant line and secant plane approximations, Roe averages, Performance, Other approximate Riemann solvers)
- 4.4. Upwind and adaptive stencils: Flux averaging, Flux splitting, Wave speed splitting, Reconstruction-Evolution methods
- 4.5. Selected solution techniques: Roe’s 1st order upwind method, Beam-Warming 2nd order upwind method, Higher order methods: MUSCL interpolation [A4]
1. Introduction: Motivation, Problems in predicting compressible flows, Unique phenomena in compressible flows, Course Objectives, Course Plan, Evaluation of coursework, Required Background, Term Project
[Term Project]
2. Review of Computational Fluid Dynamics (CFD)
- 2.1. Governing equations in CFD: Navier-Stokes equations (N-S), Flux vector formulation of the NS equations, Conservative vs. primitive forms, Euler equations, Model equations
- 2.2. Classification of differential equations: ODE’s vs. PDE’s, Linear vs. non-linear equations, First-order vs. higher-order equations, Conservative vs. non-conservative forms
- 2.3. Classification of Partial Differential Equations (PDE’s): Determining the nature of PDE’s (elliptic, parabolic, hyperbolic), Physical meaning for fluid flows, Computational meaning for fluid flows, Boundary and initial conditions for PDE’s
- 2.5. Turbulence: Free turbulent flows, Boundary layers near solid walls, Turbulence modeling in CFD, Wall functions and implications for grid generation
- 2.6. Numerical solution of PDE’s: Selection of mathematical model, Selection of discretization method (Finite Difference, Finite Volume, Finite Element, Spectral Method)
- 2.7. Grid generation: Structured vs. unstructured grids, Grid transformation, Cartesian grids, Zonal or block-structured grids, Hybrid grids, Moving mesh techniques (Sliding mesh, CHIMERA grids) Deforming mesh techniques, Adaptive grids, Multigrid methods and their relation to grid generation, Basic guidelines for grid generation
- 2.8. Boundary treatment: Boundary conditions, Boundary treatment (Changing the numerical method at edges, Changing the computational domain at edges), Solid Wall boundary treatment, Far-field boundary treatment, Non-reflecting boundaries
- 2.9. Solution techniques for the discretized equations: Explicit vs. implicit formulations, Solutions techniques for explicit method (Lax-Wendroff, MacCormack, Runge-Kutta), Solution techniques for implicit methods (Direct methods /Gaussian elimination, Cramer’s rule/, Indirect methods /Thomas algorithms, point-iterative methods, approximate factorization/)
- 2.10. Errors and uncertainty in CFD: Sources of error, Sources of uncertainty, Stability analysis of numerical errors (Discrete Perturbation analysis, Von Neumann Stability Analysis, Multidimensional considerations), The Courant-Friedrich-Loewy number (CFL), Stability vs. accuracy, Local vs. Global time stepping, Evaluation of convergence (Iterations convergence: residuals, Grid convergence, Time step convergence), Characteristic features related to stability
(Consistency, Boundedness, Transportiveness)
- 2.11. Conclusions: methods best suited for compressible flows
WEEK 5-8:
3. The Finite Volume Method (FVM)
- 3.1. Governing equations for FV formulations: Conservative integral form of the N-S equations, General transport equation, FV discretization of the governing equations,
- 3.2. FVM for diffusion problems: FVM for 1D steady-state diffusion, Worked examples on 1D steady state diffusion, FVM for 2D diffusion problems, FVM for 3D diffusion problems [A1]
- 3.3. FVM for convection-diffusion problems: FVM for 1D steady convection-diffusion, Central differencing scheme (CD), Upwind differencing scheme (UD), Hybrid CD-UD schemes, Powerlaw scheme, Higher order schemes (QUICK), Total Variation Diminishing schemes (TVD), Flux limiters [A2]
- 3.4. FVM for unsteady flows: FVM for unsteady diffusion, Explicit scheme, Crank-Nicolson scheme, Fully implicit scheme, Extension to multidimensions, FVM for unsteady convection-diffusion [A3]
WEEK 9-12:
4. Solution of the compressible Euler equations
- 4.1. Gasdynamics Basics: Expansion waves, Compression waves, Shock waves, Contact discontinuities
- 4.2. Euler equations: Integral form, Conservation form, Vector-Matrix notation, Rankine-Hugoniot relations, Entropy condition
- 4.3. The Riemann problem: Riemann problem for Euler equations, Riemann problem for linear system of equations, Godunov type schemes, Approximate Riemann problems (Motivation for approximating, Roe’s approximate Riemann solver, Secant line and secant plane approximations, Roe averages, Performance, Other approximate Riemann solvers)
- 4.4. Upwind and adaptive stencils: Flux averaging, Flux splitting, Wave speed splitting, Reconstruction-Evolution methods
- 4.5. Selected solution techniques: Roe’s 1st order upwind method, Beam-Warming 2nd order upwind method, Higher order methods: MUSCL interpolation [A4]
