This module provides an introduction to basic numerical methods used to simulate the flow of Newtonian fluids. Different formulations of the governing equations will be introduced and their relative merits discussed. The module will cover the discretization of the governing equations using the finite volume and finite element methods. The treatment of the convection term will be described and the effect this has on the overall stability of the schemes in the case of convection-dominated flows. The solution of the resulting systems of equations will be described using coupled and decoupled approaches.

Most fluid flow problems cannot be solved by analytical means. In these cases one needs to resort to numerical techniques. In this module two distinct numerical methods will be described for discretizing the governing equations: the finite volume and finite element methods. The finite volume method is based on the integration of the governing equations over a control volume and the discretization of fluxes across the edges using finite difference approximations. The finite element method is based on the weak formulation of the problem, the partition of the flow domain into finite elements and the development of local polynomial-based approximations to the solution on each element. Important features of both methods will be discussed.

Prerequisite Modules: MA3303 Theoretical and Computational PDE's

Recommended Modules: MA0332 Fluid Dynamics

• Derive and non-dimensionalise the Navier-Stokes equations.
• Compare the advantages and disadvantages of different formulations of the Navier-Stokes equations.
• Analyse the properties of numerical methods for solving convection-diffusion problems on one and two dimensions.
• Analyse the properties of numerical methods for solving the Navier-Stokes equations.
• Compare the stability properties of different finite volume and finite difference schemes especially when applied to convection-dominated problems.
• Categorise upwind schemes according to their properties.
• Construct and analyse temporal schemes for time-dependent problems.

30 - 50 minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 120 hours private study including preparation of worked solutions for problem classes.

Skills:

The ability to discretize the equations governing the flow of a Newtonian fluid.

Transferable Skills:

Appreciation of the properties of numerical methods for solving the governing equations for Newtonian fluids.

Awareness of important issues in the solution of the governing equations using numerical methods such as accuracy, stability and efficiency.

Formative assessment is carried out in the problem classes.  Feedback to students on their solutions and their progress towards learning outcomes is provided during these classes.  

In addition students will be expected to carry out a substantial piece of assessed coursework. This will require students to perform some computational work involving numerical simulations of fluid flow and writing a report on their findings.

The major component of summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has a choice of four from five equally weighted questions.

• Navier-Stokes Equations. Conservation of mass. Conservation of linear momentum. Constitutive equation for a Newtonian fluid.
• Formulations of the Governing Equations. Primitive variable formulation. Stream function/vorticity formulation. Stream function formulation.
• Finite Difference and Finite Volume Methods for Convection/Diffusion Problems. Model problems. Central and upwind differences. Finite volume methods. The SIMPLE family of algorithms.
• Finite Element Methods. Model problems in one and two dimensions. Treatment of convection using SUPG. Compatible approximation spaces and the LBB condition for well-posedness.
• Methods of solution. Direct and iterative methods.
• Time-dependent problems. Semi-implicit methods. Projection methods.

Computational Methods for Fluid Dynamics, Ferziger, J. H., Peric, M., Springer,2002

Computational Fluid Dynamics – The Finite Volume Method,  Versteeg, H. K.& Malalasekera, W., Pearson, 2007

Computational Rheology, Owens, R. G., & Phillips, T. N., Imperial College Press, 2002

Finite Elements and Fast Iterative Solvers, Elman, H. C., Silvester, D. J., & Wathen, A. J., Oxford University Press, 2005

Numerical Heat Transfer and Fluid Flow, Patankar, S. V., Hemisphere Publishing, 1980

Computational Techniques for Fluid Dynamics (Volume I), Fundamental and General Techniques for Different Flow Categories, Fletcher, C. A. J., Springer-Verlag, 1991

Computational Techniques for Fluid Dynamics (Volume II), Specific Techniques for Different Flow Categories, Fletcher, C. A. J., Springer-Verlag, 1991