Course detail

# Mathematical Methods in Fluid Dynamics

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 and finite element method.

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.

Recommended optional programme components

Not applicable.

M. Feistauer, J. Felcman, I. Straškraba: Mathematical and Computational Methods for Compressible Flow, Oxford University Press, Oxford, 2003 (EN)
L. Čermák: Výpočtové metody dynamiky tekutin, dostupné na http://mathonline.fme.vutbr.cz/
V. Dolejší, M. Feistauer: Discontinuous Galerkin Method, Springer, Heidelberg, 2016. (EN)
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)

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

COURSE-UNIT CREDIT IS AWARDED ON THE FOLLOWING CONDITIONS: Active participation in practicals. Elaboration of a semester assignment, where the students prove their knowledge acquired. Students, who gain course-unit credits, will also obtain 0--30 points, which will be included in the final course classification.
FORM OF EXAMINATIONS: The exam is oral. As a result of the exam students will obtain 0--70 points.
FINAL ASSESSMENT: The final point course classicifation is the sum of points obtained from both the practisals (0--30) and the exam (0--70).
FINAL COURSE CLASSIFICATION: 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.
If we measure the exam success in percentage points, then the classification grades are: 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.

Language of instruction

Czech

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.

Classification of course in study plans

• Programme M2A-P Master's

branch M-MAI , 2. year of study, winter semester, 4 credits, compulsory

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Syllabus

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, numerical flux, local error, stability, convergence.
7. The Godunov's method, flux vector splitting methods: Vijayasundaram, Steger-Warming, Van Leer, Roe.
8. Boundary conditions, secon order methods.
9. Discontinuous Galerkin method for compressible inviscid flow: introduction to DGM, discretization of 2D Euler equations.
10. Discontinuous Galerkin method for compressible inviscid flow: elementary matrices and vectors, assembly process, time discretization.
11. Discontinuous Galerkin method for 2D compressible viscous flow.
12. Finite volume method for viscous incompressible flows: the SIMPLE algorithm on rectangular mesh.
13. Finite volume method for viscous incompressible flows: the SIMPLE algorithm for unstructured mesh.

Computer-assisted exercise

13 hours, compulsory

Teacher / Lecturer

Syllabus

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