Content

The aim of the course is to introduce advanced mathematical techniques for analyzing fluid mechanics and for obtaining practical solutions for fluid flow problems.  The course covers a selection from each of the three topics given below.
1. Mathematical theory of fluid mechanics:
- Functions, vectors and tensors;
- Gradient, divergence and rotation operators; theorems of Green and Stokes; properties of irrotational, conservative and potential flow fields;
- Time derivatives; velocity and acceleration, material derivatives; particle paths, equipotential and streamlines; and
- Coordinate systems and transformation rules; Jacobian and Hessian matrices.
2. Partial differential equations for describing fluid dynamics:
- Characteristics and classification of differential equations;
- Properties of first order differential equations; solutions of kinematic wave equations and advection equations;
- Properties of 2nd order elliptic partial differential equations; Laplace and Poisson equations related to stationary flow problems; and
- Properties of 2nd order parabolic partial differential equations; diffusion problems, advection dispersion equations.
3. Numerical techniques:
- Numerical solution of systems of linear equations; relaxation techniques and conjugate gradient methods;
- Numerical solution of nonlinear equations, and systems of nonlinear equations;
- Numerical techniques for interpolation, differentiation and integration; and
- Least squares fitting and optimization techniques.
 
Practical:
- Exercises on functions and vector fields; calculation of potential functions and velocity fields, verification of conservation and rotation properties;
- Calculation of path lines and streamlines for simple fluid flow problems;
- Transformation of coordinate systems;
- Explicit and implicit numerical solutions for the advection-diffusion equation;
- Explicit and implicit numerical solutions for the momentum and continuity equations in 1 dimension;
- Solution of a kinematic wave equation problem, determination of wave velocities and mass transport velocities;
- Computer exercises on solutions of nonlinear problems;
- Computer exercises on interpolation, differentiation and integration of discrete data sets; andComputer exercises on curve fitting techniques.

Course material

Course notes

Format: more information

The background of the mathematics to be used in the practical sessions is explained first in lectures. The students actively participate by making relatively simple exercises with pen, paper and simple scientific calculator.

Content

The aim of the course is to introduce advanced mathematical techniques for analysing fluid mechanics and for obtaining practical solutions for fluid flow problems.  The practical sessions cover a non-exhaustive selection from the following main topics:

• Exercises on functions and vector fields; calculation of potential functions and velocity fields, verification of conservation and rotation properties;
• Calculation of path lines and streamlines for simple fluid flow problems;
• Transformation of coordinate systems;
• Explicit and implicit numerical solutions for the advection-diffusion equation;
• Explicit and implicit numerical solutions for the momentum and continuity equations in 1 dimension;
• Solution of a kinematic wave equation problem, determination of wave velocities and mass transport velocities;
• Computer exercises on solutions of nonlinear problems;
• Computer exercises on interpolation, differentiation and integration of discrete data sets; and
• Computer exercises on curve fitting techniques.

Course material

Application notes on Toledo

Format: more information

In the practical sessions students work out larger size problems. Most of the problems are solved in a computer class with the support of a spreadsheet. The students are also introduced to using a scientific software environment (Matlab) for solving some of the problems.