Module Number: MW2134
Duration: 1 Semester
Recurrence: Winter Semester
Number of ECTS: 4
Professor in charge: Wolfgang Polifke
It is recommended the students attend Wärme- und Stoffübertragung as well as Numerische Methoden für Ingenieure (Lehrstuhl für Numerische Mechanik) before registering for this course
The course offers an overview of fundamental numerical methods used in thermo-fluid dynamics as well as an introduction into good programming practices in a high-level programming language (Matlab). The course approach is to associate both the structure of the partial differential equations describing heat transport and the numerical algorithms that are best suited to solve these equations. This ansatz will be used to solve typical problems in thermo-fluid dynamics, in particular the heat equation (Fourier equation). It is expected that at the end of the course, students increase their understanding in heat transfer phenomena by performing mathematical, numerical and physical analysis of the partial differential equations describing heat transport. During the 8 sessions of the course, the students will write Matlab algorithms that perform specific tasks in order to solve a given thermo-fluid dynamic problems. The students will work in groups of two in the computer lab (Studentenrechnerraum) of the Lehrstuhl für Thermodynamik. At the beginning of each session, a short lecture will introduce both the physical phenomena under study and the numerical algorithms to be used. During the last third of the semester, students will carry out individual projects. It will be assessed how capable the students are in handling thermo-fluid dynamics problems as a sum of two contributions: physical understanding of the equations describing a given phenomenon and implementation of the associated best-suited numerical algorithm. The following methods will be considered (the associated physical problems are stated in brackets): Finite Difference method (2D heat equation with variable thermal diffusivity); Finite Volume method (2D heat equation in a structured irregular mesh, heat transfer by a fin); Finite Element method (1D heat equation); Gauss-Seidel, SOR approaches as iterative methods to solve large linear systems derived from finite differences, finite volumes or finite element methods (1D/2D heat equation); unsteady problems: Fourier analysis of the error, explicit and implicit schemes, Runge-Kutta methods, characteristics and CFL conditions (unsteady 2D heat equation, 1D convection equation); Green functions and numerical integration (2D heat equation with a distributed source); Solving non-linear systems: Newton-Raphson (1D steady Euler equations), Optimization: search methods, gradient methods, constraint optimization (optimization of a fin shape to maximize heat transfer)
Team and individual work, presentations carried out by students, lectures.