Mathematics of Fluid Flow

Course Description

The course aims at introducing students to computational fluid dynamics (CFD). Delivery will include background reading of the pre-requisite material, a series of lectures, conduction of supervised practical exercises and undertaking mini projects. As a motivation, expository lectures will be given on Mathematics and other Sciences as well as Mathematics in Africa will be presented.

Students will do some preparatory reading on the following concepts: Eigenvalues, Eigenvectors, Norms and spectral radius. Algorithms for solving linear systems of equations: Gaussian Elimination; Factorisation Methods for tri-diagonal systems (Crout’s and Choleski’s); iterative methods (Jacobi, Gauss-Seidel, SOR); Derivation of finite difference formulae basing on Taylor series; classification of partial differential equations (PDEs) (elliptic, parabolic, hyperbolic) and some physical problems involving PDEs (Poisson equation, wave equation, diffusion equation). Book: Richard L. Burden and J. Douglas Faires, Numerical Analysis, PWS-KENT Publishing Company, Boston

Lectures will be on the description of some fluid flow problems and an overview of CFD and discussion of some specific aspects. Numerical solutions of some simple fluid flow problems Students will work in groups on a flow problem. They will discretise, choose a numerical scheme, write an algorithm and solve manually on a few grid points. Each group will be given a mini project to solve independently and write a report, which will be assessed.

A tentative course description is given below. Some adjustments of the contebnts are possible. Students are assumed to have some prior knowledge: elementary physics, basics of differential equations, matrices and numerical analysis. Some experience with programming and software tools (Fortran or C/C+/C++, Maple or Mathematica) is assumed.

Students Preparatory Reading (6h, text to be assigned later)
Lectures
General about flow models (4h)
-Introduction of the CFD equations and basic models
-Description of some Fluid Flow problems and an overview of CFD
-PDEs, classification etc.
-Discussion about specific aspects (boundary conditions, asymptotics, boundary layers, non linearities .) Introduction to the numerical solutions (6h)
-discretisation techniques and numerical methodsb
-Introduction to software usage.

Exercise work in groups (6h)
Independent work, miniproject (12h)
Report writing (8h)

Expository lectures
-Mathematics in Other Sciences (1h)
-Mathematics in Africa (1h)

Prepatory reading

Numerical analysis and numerical linear algebra Eigenvalues, Eigenvectors, Norms and spectral radius. Algorithms for solving linear systems of equations: Gaussian Elimination; Factorisation Methods for tri-diagonal systems (Crout’s and Choleski’s); Iterative methods (Jacobi, Gauss-Seidel, SOR);

Elements of numerical methods for PDE:s Derivation of finite difference formulae based on Taylor series; Classification of partial differential equations (PDEs) (elliptic, parabolic, hyperbolic) Some physical problems involving PDEs (Poisson equation, wave equation, diffusion equation).

Literature:

Alternative books:

4. Juha Haataja, Jussi Heikonen, Yrjö Leino, Jussi Rahola, Juha Ruokolainen, Ville Savolainen : Numeeriset menetelmät käytännössä, Tieteen tietotekniikan keskus CSC http://www.csc.fi/oppaat/num.kayt/num.kayt.pdf
luku 3 lineaariset yhtälöryhmät,
luku 7.5 reuna-arvotehtävä,
luku 9 ominaisarvot