**Lecturer:** Dr Hrvoje Jasak

This course consists of 12 hours or lectures and two practical sessions.

**Lectures:**

1. 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

2. 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.

3. 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

4. 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.

5. Linear solver methodology:

Matrix structure and properties. Relationship between the computational mesh and discretisation matrix; order of accuracy, boundedness and stability concerns. Definition and properties of large sparse matrices: matrix type, diagonal dominance, matrix band and consequences, eigen-values and eigen-vectors, spectral properties of the matrix. Basic fixed point iterative methods, Krylov space solvers and algebraic multigrid

6. 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.

7. Solution procedures, 2: Pressure-velocity coupling

Solution procedures based on the pressure-velocity coupling.Geometric multigrid, solution acceleration and implicitness. p-U boundary conditions

8. Solution procedures, 3: Projection methods, block-implicit solution

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

9. Solution procedures, 4: Solution acceleration

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

10. Other equation sets:

Solid mechanics, free surface flows. Inter-equation coupling probles.

11. Dynamic mesh simulations:

Moving deforming mesh, topological changes, adaptive mesh refinement. Aspects of dynamic mesh simulations, problem setup and data analysis. Overset mesh method. Immersed Boundary Method

12. High performance computing and large-scale simulations:

Computer architecture and message-passing. Operation of discretisation and linear solver algorithms in distributed mode. Parallel decomposition and reconstruction; dynamic load balancing

**Practical sessions:**

Practical 1: Scalar transport equation

Examples of convective and diffusive transport; choice of discretisation method, local mesh resolution and time-step size (practical). Practical Finite Volume Method: mesh quality metrics and stability/accuracy concerns

Practical 2: Solution of fluid flow equations using segregated and coupled solution methods.