**Lecturer: Prof. Hrvoje Jasak**

This course introduces the student to the Finite Volume Method (FVM) of discretisation for low-speed problems in continuum mechanics. The focus of the course are equations with elliptic operators and smooth continuum solutions, usually handled by implicit discretisation methods.

The course will present the unstructured polyhedral method of second order accuracy and introduce the concepts of unstructured mesh handling and operator-based discretization. The basic formulation of FVM is presented on the model problem of a standard transport equation, enhanced by solution algorithms for complex coupled and non-linear sets of partial differential equations.

The students will be able to:

- Apply the Finite Volume Method for development of computational models.
- Develop new numerical models for real engineering problems.
- Analyse the model equations, choose and apply appropriate numerical algorithms for the system.
- Set-up a complete computer simulation of a “single-physics” continuum mechanics problem (fluid flow, heat transfer, solid mechanics).
- Critically evaluate and assess the results of a numerical simulation.

**Lectures:**

- Introduction

Course intro. Computational Continuum Mechanics. Historical developments: early days, expansion of use. Performing a continuum mechanics simulation

- Governing equations:

Transport equations in continuum mechanics: Overview of the basic equations: scalar transport equation, conservation laws. Basics of the discretisation method and numerical solution. Conservation and boundedness, transport properties, steady-state and transient problems, initial and boundary conditions - Mesh handling:

Concept of space and time discretisation. Handling of complex geometry. Mesh structure and organisation (Cartesian, multi-block, body-fitted, unstructured, tetrahedral, polyhedral). Traversing the mesh and accessing mesh metrics. - The Finite Volume Method, 1: Basics

Spatial and temporal variation, volume and surface integrals, properties of discrete systems. Preparation for the unstructured polyhedral Finite Volume discretisation. Implicit and explicit solution method. Evaluation of gradient - The Finite Volume Method, 2: Operator discretisation

Discretisation of rate-of-change, convection, diffusion source and sink terms. Boundary conditions. Peclet number Courant-Friedrichs-Levy number, stability and accuracy concerns. - Solution procedures, 1: Basics

Compressible and incompressible fluids; speed of sound; variable compressibility vs transonic; buoyancy formulation. Incompressibility limit. Non-linearity and the convection term. Effect of convective and pressure-drived transport. - Solution procedures, 2: Pressure-velocity coupling

Solution procedures based on the pressure-velocity coupling.Geometric multigrid, solution acceleration and implicitness. p-U boundary conditions - Solution procedures, 3: Projection methods, block-implicit solution

Segragated and coupled solution algorithms using examples of Navier-Stokes equations.

Steady-state and transient problems, multigrid acceleration, pseudo-time stepping

**Practical Sessions:**

- Scalar transport equation

Examples of convective and diffusive transport; choice of discretization method, local mesh resolution and time-step size (practical). Practical Finite Volume Method: mesh quality metrics and stability/accuracy concerns - Solution of fluid flow equations using segregated and coupled solution methods.

**Prerequisites:**

- Solid understanding of continuum physics equations. Familiarity with basic numerical methods.

**Essential reading:**

- Jasak, Uroic: Practical Computational Fluid Dynamics with the Finite Volume Method; in De Lorenzis, Duster: Modeling in Engineering Using Innovative Numerical Methods for Solids and Fluids, Springer 2020
- Jasak, H.: Error analysis and estimation in the Finite Volume method with applications to fluid flows, PhD Thesis, Imperial College, University of London, 1996

**Recommended reading:**

- Moukalled, Mangani, Darwish: The Finite Volume Methods in Computational Fluid Dynamics, Springer 2016
- Ferziger, J.H. and Peric, M.: Computational Methods for Fluid Dynamics, Springer Verlag, Berlin-New York, 1995
- Versteeg, H.K. and Malalasekera, W.: An Introduction to Computational Fluid Dynamics: The Finite Volume Method, Prentice Hall, 1996