Computational Fluid Dynamics for Industrial Aerodynamics II

• 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)
  
  
• 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

Lecture notes:

(lecture notes , in ps format) (lecture notes , in pdf format)

Reading list:

• [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