Aims

To present advanced topics of fluid dynamics building on the introductory concepts developed in MAS2803.

To introduce the implementation of mathematical algorithms within practical computer programs. Students will learn the fundamental programming structures, and their implementation in a widely-used programming language: Fortran. Additionally, to introduce numerical and computational modelling techniques in a number of different mathematical and physical topics. Students will be able to interpret such methods algorithmically, and implement them within practical, problem-solving programs.

Module Summary
To introduce the mathematical tools to model two-dimensional inviscid flows more advanced than seen in MAS2803 and to predict the motion of realistic viscous flows using the Navier-Stokes equation. To illustrate the variety of solutions of the Navier-Stokes equation in the physical world, paying attention to topics such as transitions of flow patterns, and including applications to flying, weather and climate.

Numerical and computational methods are an important part of modern scientific practice, used in almost all branches of mathematics, statistics and physics, and in other fields. These methods are implemented in practice via computer programs, so this module introduces programming as a tool for the implementation of such methods. The emphasis is on the fundamental algorithms and structures, although the material is developed using one specific modern programming languages, to which the students will be introduced: Fortran 95.

The emphasis is on learning in a practical context; we will be writing simple programs from the outset, rather than abstractly studying the language syntax or implementation. The programming will be illustrated and developed through their use in applications from a range of mathematical and physical topics. The algorithmic nature of all the methods covered will be emphasised, highlighting the basic logical structures required.

Outline Of Syllabus

Review of elementary fluid dynamics presented in MAS2803 (e.g. continuity equation, Euler equation, vorticity, stream function, complex potential for two-dimensional flows). More advanced
complex potential methods to include singularities (sources, vortices), boundaries (method of images, Milne-Thomson theorem), Magnus force (lift) and Hamiltonians. Microscopic and macroscopic description of viscosity. The Navier-Stokes equation. The vorticity transport equation. No-slip boundary conditions. Analytic solutions of the Navier-Stokes equation (e.g. channel flows in Cartesian and cylindrical geometries, Couette flows, oscillating flows). Transitions to complex vortex flows (flows past
cylinder, Taylor-Couette, Rayleigh-Benard and Reynolds pipe problems). Dimensionless variables, Reynolds number, introduction to turbulence, drag. Flows in rotating frames: Coriolis, centrifugal and Poincare forces. Simple applications: flying (lift/drag), weather, climate.

Introduction to programming:
Basic program structure and implementation (editing, compiling, executing); simple data types (integer, real, complex, logical, and character); arrays of data; input and output; branching structures ('if' constructs); repeating structures ('do' loops); program blocks (subroutines, functions, modules, interfaces).
Mathematical applications:
Three mathematical topics will be explored in more detail. For each of these, computational methods will be motivated, and practical algorithms derived and converted into programs. The topics may include: numerical integration; numerical solutions of ODEs and PDEs; linear algebra.

Teaching Rationale And Relationship

Non-synchronous online materials are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on assessed work. Present-in-person and synchronous online sessions are used to help develop the students’ abilities at applying the theory to solving problems and to identify and resolve specific queries raised by students, and to allow students to receive individual feedback on marked work. Students who cannot attend a present-in-person session will be provided with an alternative activity allowing them to access the learning outcomes of that session. In addition, office hours/discussion board activity will provide an opportunity for more direct contact between individual students and the lecturer: a typical student might spend a total of one or two hours over the course of the module, either individually or as part of a group.
Alternatives will be offered to students unable to be present-in-person due to the prevailing C-19 circumstances.
Student’s should consult their individual timetable for up-to-date delivery information.

Assessment Rationale And Relationship

A substantial formal examination is appropriate for the assessment of the material in this module. The course assessments will allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.