Numerical flow simulation

Lecturer(s): Fabien GODEFERD, Christophe CORRE
Course ⋅ 16 hStudy ⋅ 12 h


The goal of the course is to provide the students with an "advanced user / beginner developer" level in computational fluid dynamics, with a focus on compressible flows of interest in aerospace and energy applications. Following the course, the student should be able to properly select and apply a solution method for an engineering problem of practical interest and should understand the observed numerical behaviour (accuracy, robustness). The student will also be able to perform basic developments in existing CFD codes: change of boundary conditions or implementation of a new numerical flux.


Classification of PDEs. Method of characteristics. Finite difference. Finite volumes. Centered and upwind schemes. Riemann solvers. TVD schemes. Structured and unstructured grids. Spectral methods.


Lecture #1: Introduction to CFD. From pioneering works to 21st century challenges. Lectures #2 and #3: Analysis of scalar problems : classification of PDEs, method of characteristics, finite difference schemes for model problems : 1D advection, 1D diffusion, 1D advection-diffusion. Lectures #4 and #5: Extension of 1D finite-difference schemes to non-linear systems of conservation laws (Euler equations): from the 1st-order upwind scheme to high-resolution schemes. Lectures #6 and #7: Finite-Volume Schemes in structured and unstructured grids. From Euler equations in Cartesian grids to the Navier-Stokes equations in triangular grids.
Lecture #8 : Introduction to spectral methods.

Learning Outcomes

  • Understanding the current challenges of CFD. Applying the method of characteristics to analyze exact solutions of scalar conservation laws. Computing truncation erros and amplification factors for finite difference schemes applied to model advection, diffusion and advection-diffusion problems in one and several space dimensions. Implementing a numerical flux in a CFD code solving the traffic flow equation.
  • Analyzing centered and upwind schemes for the solution of 1D Euler equations (smooth flows and flows including discontinuities). Selecting a relevant numerical scheme for the flow under study and using the proper tuning parameters for this scheme (artificial viscosity calibration, choice of a MUSCL-type reconstruction, selection of a slope limiter). Implementing a numerical flux in a CFD code solving the Euler equations.
  • Extending numerical schemes to multi-D flows. Understanding of the generic structure of an unstructured finite-volume CFD solver : from the data structure of the computational grid to the space reconstruction options ensuring a second-order accuracy. Simulating an oblique shockwave / laminar boundary layer interaction using the unstructured FV SU2 CFD code.
  • Understanding of the key principles of spectral methods and ability to apply these methods to a flow simulation.


Grade = 40% knowledge (final exam) + 60% know-how (reports on computer labs) Knowledge grade = 100% final exam grade Know-how grade = 100% average of the 3 computer labs reports Note : when the MOD is (also) followed as a master course (Fluid Mechanics & Energetics, Aerospace Propulsion, Acoustics,...) the 2h MOD final exam is completed with an extra hour devoted to a master-specific problem. In that case, there is on one hand the MOD final exam grade (computed from the problems treated during the 2h of the MOD exam) and a master final exam grade which includes the grade of the master-specific problem. The final master grade for the course is computed with the same balance betwen knowledge and know-how, with the knowledge grade given by the master final exam grade.