M.Sc. in Space and Astronautical Engineering · 6 CFU · 2nd year, 1st semester · course code 10606304

Course information

• Instructor: Matteo Bernardini

• Credits / hours: 6 CFU -  42 h of lectures + 18 h of computer lab

• Language: English

• Track: Space Transportation (also valid for the dual-degree programmes with Georgia Institute of Technology)

• Office hours: Monday 16:00–18:00, building RM031, room 28 (please email in advance: matteo.bernardini@uniroma1.it)

• Course material: lecture notes and slides are distributed through Microsoft Teams (the access code is given during the first lecture)

• Official course sheet: Sapienza course catalogue

Learning objectives

By the end of the semester you should be able to:

• describe the steps involved in a successful CFD project;

• understand the fundamentals of what CFD is and how it works;

• set up and run steady and unsteady CFD analyses for typical aerospace applications and critically assess the results.

Prerequisites

Fundamentals of aerodynamics and gas dynamics.

Programme

Weeks 1–4 · Discretisation fundamentals

• Week 1 — Course presentation, introduction to CFD. Conservation principles. General transport equation for a scalar variable. Introduction to discretisation methods: finite-difference and finite-volume methods.

• Week 2 — The finite-volume method. Discretisation strategy, approximation of surface and volume integrals. Mid-point rule. Derivation of the discrete system of algebraic equations.

• Week 3 — Iterative methods for the solution of linear systems: the Jacobi and Gauss–Seidel methods. Discretisation of the diffusion term. Derivation of the algebraic equation for orthogonal grids. Boundary conditions for the heat equation.

• Week 4 — Methods for unsteady problems, discretisation of the transient term. Two-level and explicit/implicit methods: forward and backward Euler schemes.

Weeks 5–8 · Convection–diffusion on general grids

• Week 5 — Discretisation of the diffusion term on general unstructured non-orthogonal grids. Minimum-correction, orthogonal and over-relaxed approaches for cross-diffusion. Deferred-correction approach. Scarborough criterion. Explicit and implicit under-relaxation. Gradient computation: cell-based and node-based Green–Gauss approaches.

• Week 6 — Discretisation of the convection term. Analytical solution of the 1D convection–diffusion equation. Péclet number. Numerical solutions with central-difference and upwind schemes.

• Week 7 — Modified equation and truncation error, numerical (artificial) viscosity. Second-order linear schemes, κ-scheme formulation. Popular methods: CDS, QUICK, CUI, FROMM, SOU. Godunov theorem and onset of oscillations. Non-linear schemes, total variation, TVD schemes.

• Week 8 — Flux-limiter functions and Sweby diagram. TVD schemes and popular limiters (minmod, superbee, harmonic). Basic concepts on the solution of the Navier–Stokes equations: pressure-based and density-based solvers. Projection methods, Helmholtz decomposition. Incompressible flows, pressure–velocity coupling.

Weeks 9–10 · Navier–Stokes solvers and turbulence

• Week 9 — SIMPLE method for incompressible flows. Outer and inner iterations. Derivation of the pressure-correction equation. Introduction to turbulence: transition, sensitivity to initial and boundary conditions. Energy cascade and dissipation rate. Kolmogorov scales and viscous dissipation. Numerical strategies for turbulent flows: direct numerical simulation (DNS).

• Week 10 — DNS and large-eddy simulation (LES), computational cost estimates. Turbulence modelling, averaging and ergodicity. Reynolds decomposition and Reynolds-averaged Navier–Stokes (RANS) equations. Reynolds stress tensor and Boussinesq approximation. Eddy viscosity.

Weeks 11–12 · Turbulence models and near-wall treatment

• Week 11 — Analysis of the Boussinesq approximation. Turbulent-viscosity models: zero-, one- and two-equation models. Mixing-length hypothesis.

• Week 12 — Two-equation models, k–ε and k–ω families: pros and cons. Wall-bounded flows and the law of the wall. Near-wall modelling strategies. Wall functions and their application; standard and scalable wall functions.

Textbooks

Lecture notes and slides are provided by the instructor. Suggested textbooks:

• F. Moukalled, L. Mangani, M. Darwish, The Finite Volume Method in Computational Fluid Dynamics, Springer.

• J.H. Ferziger, M. Perić, R.L. Street, Computational Methods for Fluid Dynamics, Springer.

Examination

Near the end of the course a CFD project is assigned, to be carried out in groups of up to four students. The project requires a technical report (max 10 pages), to be delivered at least one week before the exam date, and a 15–20 minute presentation by all members of the group on the day of the oral exam. The discussion may cover theoretical topics from the course programme. The final mark accounts for both the group result and the individual contribution.