FAF3702 Applications of Partial Differential Equations in Fluid Mechanics 7.5 credits

Introduction: Vector and scalar fields in Cartesian, cylindrical and spherical coordinates. Ordinary differential equations. Flow visualization: Streamlines, streamtubes, pathlines, streaklines, timelines. Strain rate tensor. Flux integrals and applications. Norms. Iterative methods for systems of linear and nonlinear equations. Banach fixed-point theorem. Newton method and Gradient method.

Partial differential equations (PDE). Classification of Partial differential equations. Boundary-value problems. Solving PDE using Fourier series and some analytical methods. Finite volume and finite difference numerical methods for PDE. Consistence, convergence and stability of solution methods. Fourier-von Neumann stability analysis. Lax equivalence theorem. Heat equation and diffusion equation in 3D. Laplace equation. Wave equation. Vibrating strings and membranes. Conservation of mass - The continuity equation.The Reynolds Transport equation. Stress tensor and Cauchy’s equation.The Navier-Stokes Equations. Turbulence and its modeling. Reynolds-averaged Navier-Stokes equations. The finite volume method for convection-diffusion problems.The finite volume method for some unsteady flows.

SIMPLE, SIMPLER, SIMPLEC and PISO algorithm.

After finishing the course, the student should be able to

• derive some basic partial differential equations (PDE), described in the course content, in Cartesian, spherical and cylindrical coordinates.
• to use numerical methods (finite volume method), Fourier method and (in some special cases) analytical methods as part of solving PDE.
• to write own computer programs for solving PDE, and programs for the visualization and animation of the solutions.
• analyze consistency, convergence and stability of numerical methods for solving PDE.

1. Entry requirements for PhD studies in architectural engineering or similar knowledge
2. A basic course in fluid mechanics
3. Basic courses in multivariable calculus and differential equations

1. Versteeg, H.K. and Malalasekera, W., “An Introduction to Computational Fluid Dynamics: The Finite Volume Method”.
2. Richard Haberman, Applied Partial Differential Equations.
3. Randall J. Leveque, Finite Volume Methods for Hyperbolic Problems.

If the course is discontinued, students may request to be examined during the following two academic years.

• LAB1 - Laboratory work, 2.0 credits, grading scale: P, F
• TEN1 - Written exam, 5.5 credits, grading scale: P, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

Stefan Larsson

• All members of a group are responsible for the group's work.
• In any assessment, every student shall honestly disclose any help received and sources used.
• In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.