Computational Fluid Dynamics S2

Module aims

This module provides a solid foundation of the theory and implementation of primarily 1D numerical methods for computational fluid dynamics and allows the student to gain a basic understanding of the advantages and limitations of the use of CFD in an industrial environment.

Learning outcomes

On successfully completing this module, you should be able to: 1. Select and state appropriate model equations for different types of flows such as incompressible/compressible, viscous/inviscid, potential;  2. Distinguish between the mathematical classifications of partial differential equation and explain their physical significance;  3. Design and implement stable numerical schemes for the 1-D advection and diffusion equations;  4. Analyse the stability of 1-D linear equations using the Von Neumann stability analysis and explain how this technique can be used to determine diffusion and dispersion errors;  5. Explain how the 1-D theory can be applied to multi-dimensional problems involving system of equations, complicated geometries and discontinuities;  6. Recognise the theoretical/practical limitations of current numerical algorithms in CFD  7. Identify the computational methods used in industry and be aware of appropriate tests to validate and assess numerical results.

Module syllabus

Introduction:  Governing equations: conservative/integral form.  Reduced models and range of applicability and limitations.  Classification of Model Equations:  (Elliptic, parabolic and hyperbolic) and their relation to fluid problems.  Construction of model 1-D problems (linear advection-diffusion equations).  Construction of Basic Numerical Schemes:  Finite Differences (FD), Finite Volume (FV) and Finite Elements (FE).  Analysis and Solution of Finite Difference Schemes:  Order, truncation error and consistency of a scheme using Taylor expansions.  Solution of algebraic systems (direct and basic iterative methods).  Explicit and implicit time integration.  Courant-Friedrichs-Lewy condition and diffusive time step restrictions.  Lax theorem: consistency, stability and convergence.  Von Neumann linear analysis for stability and dispersion/diffusion properties.  Non-Linear Conservation Laws:  1-D theory.  Examples of 1-D hyperbolic conservation laws.  Characteristics.  Discontinuities and jump conditions.  Weak solutions and entropy condition.  Linear versus non-linear advection.  Systems of Conservation Laws:  Jacobian matrices, linearized equations, conservative and characteristic variables.  Rankine-Hugoniot jump conditions.  Boundary conditions.  Numerical Representation of Discontinuities:  Requirements on numerical schemes.  Conservative discretisation: Lax-Wendroff theorem.  First versus second order schemes.  Representation of discontinuities: physical aspects, shock fitting/capturing.  Numerical Schemes for Non-Linear Conservation Laws:  Centred schemes: one-step and two-step Lax Wendroff, MacCormack predictor-corrector.  Artificial dissipation.  Upwind schemes: flux vector and flux difference splitting. Monotone schemes:  Godunov and Harten theorems. Exact and approximate Riemann solvers. High-order upwind schemes: the TVD property.  The construction of TVD schemes using slope and flux limiters.  Numerical Schemes for Multi-Dimensional Problems:  Finite differences and finite volume.  Computational domain and boundary conditions.  Discretization of viscous terms.  

Pre-requisites

Teaching methods

The module will be delivered primarily through large-class lectures introducing the key concepts and methods, supported by a variety of delivery methods combining the traditional and the technological.  The content is presented via a combination of slides, whiteboard and visualizer.Learning will be reinforced through tutorial question sheets and coursework, featuring analytical and computational tasks representative of those carried out by practising engineers.

Assessments

You will receive feedback following the coursework submissions.

You will receive feedback on examinations in the form of an examination feedback report on the performance of the entire cohort.

You will receive feedback on your performance whilst undertaking tutorial exercises, during which you will also receive instruction on the correct solution to tutorial problems.

Further individual feedback will be available to you on request via this module’s online feedback forum, through staff office hours and discussions with tutors.

Module leaders