Course detail

Mathematical Methods in Fluid Dynamics

FSI-SMM-AAcad. year: 2012/2013

Basic physical laws of continuum mechanics: laws of conservation of mass, momentum and energy. Theoretical study of hyperbolic conservation laws, particularly of Euler equations that describe the motion of inviscid compressible fluids. Numerical modelling based on the finite volume method. Numerical modelling of incompressible flows: Navier-Stokes equations, pressure-correction method, spectral element method.

Language of instruction

English

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

doc. RNDr. Libor Čermák, CSc.

Department

Institute of Mathematics (ÚM)

Offered to foreign students

Of all faculties

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

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations.

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.

Course curriculum

Not applicable.

Work placements

Not applicable.

Aims

The course is intended as an introduction to the computational fluid dynamics. Considerable emphasis will be placed on the inviscid compressible flow: namely, the derivation of Euler equations, properties of hyperbolic systems and an introduction of several methods based on the finite volumes. Methods for computations of viscous flows will be also studied, namely the pressure-correction method and the spectral element method. 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

E.F. Toro: Riemann Solvers and Numerical Methods for Fluid Dynamics, A Practical Introduction, Springer, Berlin, 1999. (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.

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

doc. RNDr. Libor Čermák, CSc.

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 in one and two dimensions, numerical flux.
7. Local error, stability, convergence.
8. The Godunov's method, flux vector splitting methods: the Vijayasundaram, the Steger-Warming, the Van Leer.
9. Viscous incompressible flow: finite volume method for orthogonal staggered grids, pressure correction method SIMPLE.
10. Pressure correction method for colocated variable arrangements, non-orthogonal and unstructured meshes.
11. Stokes problem, spectral element method.
12. Steady Navier-Stokes problem, spectral element method.
13. Unsteady Navier-Stokes problem, time dicretization methods.

Computer-assisted exercise

13 hod., compulsory

Teacher / Lecturer

doc. RNDr. Libor Čermák, CSc.

Syllabus

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