Student learning objectives

You should be able to achieve the following goals by the end of the semester:

• Get knowledge of the various steps involved in a successful CFD project

• Understand the basics of what CFD is and how it works

• Conduct steady state and transient CFD analyses for typical aerospace applications

Prerequisites

Fundamentals of aerodynamics and gasdynamics.

Program

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

Week 2 - The finite volume method. Discretization 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 a linear system of equations. The Jacobi and Gauss-Seidel method. Discretization of the diffusion term. Derivation of the algebraic equation for orthogonal grids. Boundary conditions for the heat equation.

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

Week 5 - Discretization of the diffusion term on general unstructured non-orthogonal grids. Minimum correction, orthogonal, and over-relaxed approaches for the treatment of cross-diffusion. Deferred correction approach. Scarborough criterion. Explicit and implicit under-relaxation. Computation of the gradient: cell-based and node-based Green-Gauss approaches.

Week 6 - Discretization of the convection term. Analytical solution for the 1D convection-diffusion equation. Peclet number. Numerical solution for the central difference and upwind scheme.

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

Week 8 - Non-linear schemes, flux limiter function, and Sweby diagram. TVD schemes and popular flux 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.

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

Week 10 - Direct numerical simulation (DNS) and Large Eddy Simulation (LES), computational cost estimates. Turbulence modeling, averaging procedure, and ergodicity. Reynolds decomposition and derivation of Reynolds averaged Navier Stokes (RANS) equations. Reynolds stress tensor and Boussineq approximation. Eddy viscosity.

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

Week 12 - Two-equation models, k-eps and k-om families. Pros and cons of the various models. Wall-bounded flows, the law of the wall. Strategies for near-wall modeling. Wall functions and their application. Standard and scalable wall functions.

Textbooks

Lecture notes and slides are provided by the instructor.

Suggested textbooks

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

- Computational methods for Fluid Dynamics  - J.H. Ferziger, M. Peric & R.L. Street

Examination method

Near the end of the course, a CFD project will be assigned, to be carried out in groups of max 4 people. The project involves the writing of a technical/scientific report (max 10 pages) which must be delivered at least one week before the date of the exam and the results of which will be illustrated on the day of the oral test with a presentation of approximately 15-20 minutes by all members of the working group. During the discussion of the project, the interview may touch on theoretical topics that are part of the course program. The evaluation will take into account both the result obtained by the working group and the individual contribution.