In various natural phenomena or industrial processes, quantities such as mass, energy and quantity of movement, are transferred by two joint mechanisms: diffusion and convection (or advection). There are numerous examples: thermal transfer, particle transfer, propagation of pollutants, flows …
The convective aspect (transfer by a velocity field) is represented by hyperbolic type terms in balance equations. While diffusion poses no particular problem, convection leads to instabilities and discontinuities in solutions; this leads in turn to difficulties in both theoretical and numerical terms and to specific solution techniques.
With this Unit, students will learn how to identify these phenomena, to mathematically model them in the form of partial differential equations and to solve the equations, using principally the finite element method. They will know how to apply these methods to simple cases by treating specific problems of instability and discontinuity (fluid-fluid interfaces, for example).
| On completion of the unit, the student will be capable of: | Classification level | Priority |
|---|---|---|
| Identifying and modelling transfer phenomena, writing balance equations (convection-diffusion) | 2. Understand | null |
| Writing variational formulations of balance equations and finite element method based formulations | 2. Understand | null |
| Analysing and predicting solution behaviour (collision, stability) | 4. Analyse | null |
| Discretising by using basics of adequate functions, notably to avoid instabilities | 3. Apply | null |
| Applying numerical computing solutions (writing algorithms) to an example | 3. Apply | null |
| Testing and validating results | 6. Assess | null |
| Organising and communicating results in the form of a practical course report | 5. Summarise | null |
| Percentage ratio of individual assessment | Percentage ratio of group assessment | ||||
|---|---|---|---|---|---|
| Written exam: | 50 | % | Project submission: | 0 | % |
| Individual oral exam: | 0 | % | Group presentation: | 0 | % |
| Individual presentation: | 0 | % | Group practical exercise: | 50 | % |
| Individual practical exercise: | 0 | % | Group report: | 0 | % |
| Individual report: | 0 | % | |||
| Other(s): 0 % | |||||
| Type of teaching activity | Content, sequencing and organisation |
|---|---|
| Course | Identification of transfer phenomena, writing balance equations, weak formulations. Analysis of stability, study of collisions. Discretisation by finite elements, streamline-upwind methods (SUPG) Interface tracking – Level set methods Practical course: Matlab application to a flow problem in a porous medium - tests and report |