## Mathematical Methods in Fluid Dynamics

• ECTS credits: 4
• Semester: 1
• University: Brno University of Technology
• Prerequisites:

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

• Objectives:

The course is intended as an introduction to the computational fluid dynamics. Considerable emphasis will be placed on the inviscid compressible flow: 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 flows 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 flows 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.

• Topics:

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. Traffic flow 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 flux.

7. Local error, stability, convergence.

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

9. Viscous incompressible flow: 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.

• Books:

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.