Course detail

Mathematical Methods in Fluid Dynamics

FSI-SMM-AAcad. year: 2025/2026

Basic physical laws of continuum mechanics: laws of conservation of mass, momentum, and energy. Theoretical study of hyperbolic equations, particularly of Euler equations describing the motion of inviscid compressible fluids. Finite volume method. Numerical modelling of the Euler equations based on the finite volume method. Numerical modelling of viscous incompressible flows: pressure-correction method SIMPLE.

Language of instruction

English

Number of ECTS credits

4

Mode of study

Not applicable.

Guarantor

doc. Ing. Luděk Nechvátal, Ph.D.

Department

Institute of Mathematics (ÚM)

Entry knowledge

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

Rules for evaluation and completion of the course

COURSE-UNIT CREDIT IS AWARDED BASED ON THE FOLLOWING CONDITIONS: A semestral project consisting of assigned problems. Active participation in seminars.

EXAMINATION: The exam is oral. The students can obtain up to 100 points from the exam.

The final grade is based on the evaluation of the oral part.

The grading scheme is as follows: excellent (90-100 points), very good (80-89 points), good (70-79 points), satisfactory (60-69 points), sufficient (50-
59 points), failed (0-49 points).

Attendance at lectures is recommended, and attendance at seminars is obligatory and checked. Absence from lessons may be compensated by the agreement with the teacher supervising the seminars.

Aims

The course is intended as an introduction to computational fluid dynamics. In the case of compressible flow, the finite volume method is introduced. 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 assignments constitutes an important part to verify how the subject matter was managed.

Students will be made familiar with the basic principles of fluid flow modeling: physical laws, the mathematical analysis of equations describing flows (Euler and Navier-Stokes equations), the choice of an appropriate method (which issues from the physical as well as from the mathematical essence of equations) and the computer implementation of the proposed method (preprocessing = mesh generation, numerical solver, postprocessing = visualization of desired physical quantities). Students will demonstrate the acquainted knowledge by elaborating on semester assignments.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

E.F. Toro: Riemann Solvers and Numerical Methods for Fluid Dynamics, A Practical Introduction, Springer, Berlin, 1999. (EN)
J.S. Hesthaven, T. Warburton: Nodal Discontinuous Galerkin Methods, Springer-Verlag New York, 2008. (EN)
M. Feistauer, J. Felcman, I. Straškraba: Mathematical and Computational Methods for Compressible Flow, Oxford University Press, 2003. (EN)
R.J. LeVeque: Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. (EN)

Recommended reading

J. Y. Murthy, S. R. Mathur: Numerical Methods in Heat, Mass and Momentum Transfer, Draft Notes ME 608, Purdue University, 2002, http://widget.ecn.purdue.edu/~jmurthy/me608/main.pdf.
L. Čermák: Řešení Navierových-Stokesových rovnic metodou tlakových korekcí. http://mathonline.fme.vutbr.cz/Matematicke-metody-v-teorii-proudeni/sc-1230-sr-1-a-239
L. Čermák: Řešení nestlačitelného proudění tekutin metodou spektrálních prvků. http://mathonline.fme.vutbr.cz/Matematicke-metody-v-teorii-proudeni/sc-1230-sr-1-a-239
M. Lukacova-Medviďová: Mathematical methods in fluid dynamics, CERM, Brno, 2003.

Classification of course in study plans

  • Programme N-AIM-A Master's 2 year of study, winter semester, compulsory
  • Programme N-MAI-A Master's 2 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

doc. Ing. Luděk Nechvátal, Ph.D.

Syllabus

1. Material derivative, transport theorem, laws of conservation of mass and momentum.
2. Law of conservation of energy, constitutive relations, thermodynamic state equations.
3. Navier-Stokes and Euler equations, initial and boundary conditions.
4. Hyperbolic system, examples of hyperbolic systems.
5. Classical solution of the hyperbolic system.
6. Week solution of the hyperbolic system, discontinuities.
7. The Riemann problem in linear and nonlinear case, wave types.
8. Finite volume method, numerical flux,
9. Local error, stability and convergence of the numerical method.
10. Godunov's method, Riemann numerical flux.
11. Numerical fluxes of Godunov's type.
12. Boundary conditions, second order methods.
13. Finite volume method for viscous incompressible flows: the SIMPLE algorithm on a rectangular mesh.

Computer-assisted exercise

13 hod., compulsory

Teacher / Lecturer

doc. Ing. Luděk Nechvátal, Ph.D.

Syllabus

Demonstration of solutions for selected model tasks on the computer. Separately work out the specified tasks on a computer.