Course Unit


Mathematical Methods in Fluid Dynamics

  • Unit Coordinator: Libor Čermák
  • Programme: Double Degrees
  • ECTS Credits: 4
  • Semester: 1
  • Year: 2
  • Campus: Brno University of Technology
  • Language: English
  • Aims:

    The course is intended as an introduction to the computational fluid dynamics. Considerable emphasis will be placed on the inviscid compressible low: namely, the derivation of Euler equations, properties of hyperbolic systems and an introduction of several methods based on the finite volumes. Methods for computations of viscous lows will be also studied, namely the pressure-correction method and the spectral element method. Students ought to realize that only the knowledge of substantial physical and mathematical aspects of particular types of lows enables them to choose an effective numerical method and an appropriate software product. The development of individual semester assignement constitutes an important experience enabling to verify how the subject matter was managed.

  • Content:

    1. Material derivative, transport theorem, mass, momentum and energy conservation laws.

    2. Constitutive relations, thermodynamic state equations, Navier-Stokes and Euler equations, initial and boundary conditions.

    3. Trauc low equation, acoustic equations, shallow water equations.

    4. Hyperbolic system, classical and week solution, discontinuities.

    5. The Riemann problem in linear and nonlinear case, wave types.

    6. Finite volume method in one and two dimensions, numerical lux.

    7. Local error, stability, convergence.

    8. The Godunov's method, lux vector splitting methods: the Vijayasundaram, the Steger- Warming, the Van Leer.

    9. Viscous incompressible low: finite volume method for orthogonal staggered grids, pressure correction method SIMPLE.

    10. Pressure correction method for colocated variable arrangements, non-orthogonal and unstructured meshes.

    11. Stokes problem, spectral element method.

    12. Steady Navier-Stokes problem, spectral element method.

    13. Unsteady Navier-Stokes problem.

  • Pre-requisites:

    Evolution partial differential equations, functional analysis, numerical methods for partial differential equations.

  • Reading list:

    R.J. LeVeque: Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002.

    E.F. Toro: Riemann Solvers and Numerical Methods for Fluid Dynamics, A Practical Introduction, Springer, Berlin, 1999.

    S.V. Patankar: Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York, 1980.

    J.H. Ferziger, M. Peric: Computational Methods for Fluid Dynamics, Springer-Verlag, New York, 2002.

    M.O. Deville, P.F. Fischer, E.H. Mund: High-Order Methods for Incompressible Fluid Flow. Cambridge University Press, Cambdrige, 2002.

    A. Quarteroni, A. Valli: Numerical Approximatipon of Partial Differential Equations. Springer- Verlag, Berlin, 1994

  • Additional info:

    Basic physical laws of continuum mechanics: laws of conservation of mass, momentum and energy. Theoretical study of hyperbolic conservation laws, particularly of Euler equations that describe the motion of inviscid compressible luids. Numerical modelling based on the finite volume method. Numerical modelling of incompressible lows: Navier-Stokes equations, pressure-correction method, spectral element method.


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