Course detail

Mathematical Methods in Fluid Dynamics

FSI-SMM-AAcad. year: 2021/2022

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

Mode of study

Not applicable.

Guarantor

doc. Ing. Luděk Nechvátal, Ph.D.

Department

Institute of Mathematics (ÚM)

Learning outcomes of the course unit

Students will be made familiar with basic principles of the fluid flow modelling: 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 proposed method (preprocessing = mesh generation, numerical solver, postprocessing = visualization of desired physical quantities). Students will demonstrate the acquinted knowledge by elaborating semester assignement.

Prerequisites

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

Co-requisites

Not applicable.

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.

Assesment methods and criteria linked to learning outcomes

CONDITIONS FOR OBTAINING THE COURSE-UNIT CREDIT: Active participation in seminars, taking part in a semester project (a protocol with conclusions has to be delivered to the teacher). Students can obtain up to 30 points from the seminars - they will be included in the final course classification.

EXAM: The exam is oral. The students can obtain up to 70 points from the exam.

FINAL ASSESSMENT: The final classicifation is based on the sum of the points obtained from both the seminars and 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.

Course curriculum

Not applicable.

Work placements

Not applicable.

Aims

The course is intended as an introduction to the computational fluid dynamics. In case of compressible flow, the finite volume method and the discontinuous Galerkin method are introduced, and in case of incompressible flows the pressure-correction method and the finite element method are 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.

Specification of controlled education, way of implementation and compensation for absences

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.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

R.J. LeVeque: Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. (EN)
E.F. Toro: Riemann Solvers and Numerical Methods for Fluid Dynamics, A Practical Introduction, Springer, Berlin, 1999. (EN)

Recommended reading

M. Lukacova-Medviďová: Mathematical methods in fluid dynamics, CERM, Brno, 2003.
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

eLearning

eLearning: currently opened course

Classification of course in study plans

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

  • Programme M2A-A Master's

    branch M-MAI , 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 hours, optionally

Teacher / Lecturer

Mgr. Jitka Zatočilová, 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. Traffic flow equation, acoustic equations, shallow water equations.
5. Hyperbolic system, classical and week solution, discontinuities.
6. The Riemann problem in linear and nonlinear case, wave types.
7. Finite volume method, numerical flux, local error, stability, convergence.
8. The Godunov's method.
9. Flux vector splitting methods: Vijayasundaram, Steger-Warming, Van Leer, Roe.
10. Boundary conditions, secon order methods.
11. Discontinuous Galerkin method for compressible inviscid flow: introduction to DGM, discretization of 2D Euler equations.
12. Finite volume method for viscous incompressible flows: the SIMPLE algorithm on a rectangular mesh.
13. Finite volume method for viscous incompressible flows: the SIMPLE algorithm for an unstructured mesh.

Computer-assisted exercise

13 hours, compulsory

Teacher / Lecturer

Mgr. Jitka Zatočilová, Ph.D.

Syllabus

Demonstration of solutions of selected model tasks on computers. Elaboration of the semester assignment.

eLearning

eLearning: currently opened course