Course detail

Mathematical Methods in Fluid Dynamics

FSI-SMMAcad. 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 that describe the motion of inviscid compressible fluids. Numerical modeling based on the finite volume method and numerical modeling of viscous incompressible flows: pressure-correction method SIMPLE.

Language of instruction

Czech

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

CONDITIONS FOR OBTAINING THE COURSE-UNIT CREDIT: Active participation in seminars, and taking part in a semester project (a protocol with conclusions has to be delivered to the teacher). 

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

FINAL ASSESSMENT: The final classification is based on the sum of the points obtained from the exam.

CLASSIFICATION SCALE: A (excellent): 100-90, B (very good): 89-80, C (good): 79-70, D (satisfactory): 69-60, E (sufficient): 59-50, F (failed): 49-0.


Attendance at lectures is recommended, attendance at seminars is required. Lessons are planned according to the week schedules. 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, and in the case of incompressible flows the pressure-correction method is described. 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.


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  semester assignment.

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.H. Ferziger, M. Peric: Computational Methods for Fluid Dynamics, Springer-Verlag, New York, 2002. (EN)
K. H. Versteeg, W. Malalasekera: An Introduction to Computational Fluid Dynamics, Pearson Prentice Hall, Harlow, 2007. (EN)
M. Feistauer, J. Felcman, I. Straškraba: Mathematical and Computational Methods for Compressible Flow, Oxford University Press, Oxford, 2003 (EN)
V. Dolejší, M. Feistauer: Discontinuous Galerkin Method, Springer, Heidelberg, 2016. (EN)

Recommended reading

L. Čermák: Výpočtové metody dynamiky tekutin, dostupné na http://mathonline.fme.vutbr.cz/

Classification of course in study plans

  • Programme N-MAI-P 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 of selected model tasks on computers. Elaboration of the semester assignment.