Computational Methods for Fluid Flow (MA4110/MA5020)
Module 1: Governing Equations of Fluid Flow
- Conservation laws
- Continuity equation
- Euler equations
- Navier–Stokes equations
- Classification of PDEs in fluid dynamics
- Characteristics and wave propagation
- Non-dimensionalization
- Reynolds and Mach numbers
Module 2: Finite Difference Methods
- Taylor series expansion
- Consistency, Stability, and Convergence
- Von Neumann stability analysis
- Upwind schemes
- Lax–Friedrichs scheme
- Lax–Wendroff scheme
- Applications to advection and diffusion equations
Module 3: Finite Volume Methods
- Integral form of conservation laws
- Control volume formulation
- Numerical fluxes
- Discretization of convection and diffusion terms
- Godunov's method
- Boundary conditions
- MUSCL reconstruction
Module 4: High-Resolution Methods for Conservation Laws
- Total Variation Diminishing (TVD) schemes
- Flux limiters
- SSP Runge–Kutta methods
- ENO schemes
- WENO schemes
- Numerical treatment of shocks and discontinuities
Module 5: Systems of Conservation Laws
- Review of scalar conservation laws
- Systems of conservation laws
- Jacobian matrix and characteristic structure
- Eigenvalues and eigenvectors
- Characteristic decomposition
- Linearized Riemann problems
- Hyperbolicity
- Numerical fluxes for systems
Module 6: Numerical Methods for the Euler Equations
- One-dimensional Euler equations
- Conservative variables and flux functions
-
Approximate Riemann solvers
-
Roe solver
- HLL solver
- HLLC solver
- High-order finite volume methods for Euler equations
- Shock tube problems
- Two-dimensional extensions
- Selected applications in fluid dynamics
Textbooks
- Randall J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press.
- E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer.
- John D. Anderson, Computational Fluid Dynamics: The Basics with Applications, McGraw-Hill.
- Charles Hirsch, Numerical Computation of Internal and External Flows, Volumes 1 & 2.
- C. B. Laney, Computational Gasdynamics, Cambridge University Press.