Course detail
Mathematical Methods in Fluid Dynamics
FSI-SMM-A Acad. year: 2025/2026 Winter semester
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 modelling based on the finite volume method and discontinuous Galerkin method. Discontinuous Galerkin method for viscous compressible flows. Numerical modelling of viscous incompressible flows: pressure-correction method SIMPLE.
Language of instruction
English
Number of ECTS credits
4
Supervisor
Department
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.
The study programmes with the given course
Programme N-AIM-A: Applied and Interdisciplinary Mathematics, Master's, compulsory
Programme N-MAI-A: Mathematical Engineering, Master's, compulsory
Type of course unit
Lecture
26 hours, optionally
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 hours, compulsory
Syllabus
Demonstration of solutions for selected model tasks on the computer. Separately work out the specified tasks on a computer.