- 20th Apr. 2006. Review of part I of this course (WS 2005/06). Outline of part II (SS 2006)
- Review of part I of this course (WS 2005/06)
- Scalar linear conservation laws
- Scalar diffusion laws
- Scalar nonlinear conservation laws (Burgers equation)
- Hyperbolic systems of linear conservation laws
- Euler equations (inviscid fluids)
- Compressible Navier-Stokes equations (viscous fluids)
- Theory of characteristics
- Fundamentals of gas dynamics (expansion, compression, shock waves)
- Riemann solver for scalar problems
- Godunov method
- Upwind schemes and central schemes with artificial dissipation

- Outline of part II (SS 2006)
- Godunov for linear hyperbolic systems
- Approximative Riemann solvers for the Euler equations (Methods by Godunov, Roe, Harten)
- Overview of an industrial finite volume solver
- Structured/unstructured FVM
- Grid generation
- Computation of the control volumes
- Calculation of the numerical fluxes
- Data structures of a finite volume code
- Implementation of boundary conditions
- Iterative solution methods (explicit Runge-Kutta, semi-implicit LU-SGS)

- Generation of the primary grid.
- Structured grids (ijk-structure).
- Hybrid grids (prismatic elements near surfaces, tetraeder in outer domain)

- Computation of the control volumes (dual grid approach)

- Review of part I of this course (WS 2005/06)
- 27th Apr. 2006. Godunovs method for linear hyperbolic systems.
- Review: Euler equations. Finite volume method. Godunov method for linear advection type problems in 1D-space domain.
- Godunov method for linear hyberbolic systems. Transformation to characteristic variables.
- Reading list:
- [LeVeque] Sect. 10.7 Upwind methods for linear hyperbolic systems.
- [LeVeque] Sect. 13.3 Godunovs method for linear hyperbolic systems.

- 3rd May 2006. No lecture.

- 10th May 2006. Euler equations. Conservative variable form.
Primitive variable form. Characteristic variable form
- Conservative variable form. Flux Jacobian for Euler equations. Eigenvalues of flux Jacobian.
- Primitive variable form. Computation of Eigenvalues and Eigenvectors.
- Characteristic variable form.
- Reading list:
- [Laney] Sect. 2.2 The Conservation Form of the Euler Equations
- [Laney] Sect. 2.3 The Primitive Variable Form of the Euler Equations
- [Laney] Sect. 3.3 The Characteristic Variable Form of the Euler Equations
- Feynman, Lectures on Physics, Volume I, Chapter 47 Sound. The wave equation

- 19th May 2006. Euler equations. Characteristic variable form. Shock tube problem
- Characteristic variable form (cont.)
- Aerodynamics Module: Shocks in aerodynamic flows
- Mach number regimes (subsonic, transonic, supersonic)
- Shock waves in supersonic and hypersonic flows

- Shock tube problem
- Phenomenological description
- Elementary wave solutions (shock wave, contact discontinuity, rarefaction wave)

- Reading list:
- [Laney] Sect. 3.3: The Characteristic Variable Form of the Euler Equations
- [Anderson] Sect. 1.10.4: Mach number regimes (subsonic, transonic, supersonic).
- [Anderson] Sect. 7.6: Some aspects of supersonic flow: Shock waves. Picture gallery !!
- [Laney] Sect. 5.1: The Riemann problem for the Euler equations
- [Toro] Sect. 2.4.4 and Sect. 3.1.3: Elementary-wave solutions of the Riemann problem

- 24th May 2006. Euler equations. Shock tube problem
- Shock tube problem (cont.). Elementary wave solutions
- Shock waves
- Normal shock-wave relations
- Physical interpretation
- Entropy condition and second law of thermodynamics

- Contact discontinuities
- Rarefaction waves
- Putting it all together: Solution of the shock-tube problem
- Reading list:
- [Toro] Sect. 2.4.4 and Sect. 3.1.3 Elementary-wave solutions of the Riemann problem
- [Anderson] Sect. 8.6 Calculation of normal shock-wave properties
- [Laney] Sect. 2.1.5 Entropy and the Second Law of Thermodynamics
- [Laney] Sect. 3.5 and Example 3.7 Expansion waves, Sect. 3.6 Compression waves and shock waves, Sect. 3.7 contact discontinuities
- [Laney] Sect. 5.1 The Riemann problem for the Euler equations

- 31st May 2006. Euler equations. Shock tube problem
- (slides of this lecture , in pdf format)
- (C-program for analytical solution of the shock-tube problem)
- Characterisation of elementary waves.
- Genuinely nonlinear and linearly degenerate waves.
- Rarefaction waves: self similar smooth solutions of the form v(x/t)
- (Generalized) k-Riemann invariants. k-Riemann invariants for contact discontinuity.
- k-Riemann invariant for rarefaction fan. Entropy. Isentropic relations for ideal gases. Anlytical solution of the Euler equations for a rarefaction fan.

- Solution of the shock tube problem (for the special situation: rarefaction - contact - shock)
- Analytical solution across rarefaction, contact, shock
- Newton method for solution of the implicit equation for pressure
- Implementation to a computer code
- Illustration for two test cases

- Reading list:
- [Laney] Sect. 5.1 The Riemann problem for the Euler equations

- Suggestion for advanced study on the Riemann problem for the Euler equations:
- [LeVeque] Chapter 7, 8, 9
- [Godlewski, Raviart] Chapter I Nonlinear hyperbolic systems in one space dimension (very advanced ...)

- No lecture on Wednesday, 7th June !!

- 14th June 2006. Exact and approximative Riemann solvers. Scheme by Harten, Lax and van Leer.

- (slides of this lecture , in pdf format)
- Review of Godunov method for nonlinear systems. Godunov method in conservation form. Intercell fluxes
- Exact intercell fluxes for Godunov method
- Approximative Riemann solver by Harten, Lax and van Leer (HLL).
- Reading list:
- [Toro] Chapter 6., Section 6.1-6.3: The Method of Godunov for Non-linear Systems.
- [Toro] Chapter 10. The HLL and HLLC (C stands for contact) Riemann solvers.

- 21st June 2006. Approximative Riemann solvers (cont). Roe's scheme.

- 28th June 2006. Discretization of the viscous fluxes. Gradient reconstruction. Finite volume method in 2D/3D.

- 28th June 2006. Informal introduction to aerodynamics.

- 5th July 2006. Industrial tools for CFD. Commercial grid-generation tool CENTAUR. Finite volume solver DLR TAU Code.
- A simple geometry-description format (the MegaCADs dat-format)
- Commercial grid-generation tool CENTAUR
- Grids for solving the Euler equations
- Grids for solving the Navier-Stokes equations

- Numerical solution of the Euler equations
- TAU parameter file
- Role of the flux-discretisation scheme for the inviscid fluxes
- Engineering study: Mach number dependence of the results (subsonic, transonic, supersonic flow)
- Engineering study: Influence of the angle of attack

- 12th July 2006. Industrial tools for CFD (cont.) Finite volume solver DLR TAU Code.
- Numerical solution of the Euler equations (cont.)
- Numerical solution of the Navier-Stokes equations (cont.)
- TAU parameter file
- Role of the solver parameters
- Discretization of the inviscid fluxes
- Gradient reconstruction

- Engineering study: Mach number dependence of the results (subsonic, transonic, supersonic flow)
- Engineering study: Influence of the angle of attack
- Engineering study: Comparison of inviscid and viscous results

- [Godlewski, Raviart] Edwige Godlewski, Pierre-Arnaud Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, 1996.
- [Laney] Culbert B. Laney, Computational Gasdynamics
- [LeVeque] Randall J. LeVeque, Numerical Methods for Conservation Laws
- [Alonso,Finn] Alonso, Finn, Physics, Addison-Wesley, 1992.
- [Pope] Pope, Turbulent Flows, Cambridge University Press, 2000.
- [Toro] Eleuterio F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, 1997.
- [Anderson] John D. Anderson, Jr. Fundamentals of Aerodynamics, Mc Graw Hill 2001.

Last update: June 1st 2006, 21:50pm