• Language of instruction: English. The course is taught entirely in English, all teaching materials are in English, and all assessments take place in English.
• Prerequisites: Mathematical Tools for Engineers 1 (I3003ME), Basic Numerical Methods (I3010ME), Continuum Mechanics (I3011ME), Fluid Mechanics 1 (I3012ME), Heat Transfer and Environmental Challenges (I3015ME), Fluid Mechanics 2 (I3018ME), Mathematical Tools for Engineers 2 (I3020ME), Numerical Methods for Mechanics 1 (I3021ME), Fluid Mechanics 3 (I4002ME), Numerical Methods for Mechanics 2 (I4007ME).
  
• Co-requisites: Numerical Methods for Mechanics 3 (I5005ME).
• Objectives: this course aims to provide an understanding of the physical mechanisms underlying the transition to turbulence, as well as the physical and theoretical foundations of the models used to simulate turbulent flows, so that students can use them with confidence in the simulation of an industrial flow. By the end of the course, students will be able to:
  –      describe and characterise a turbulent flow (Reynolds number, instabilities and transition, length and time scales, effects on mixing and energy dissipation);
  –      decompose a flow into mean and fluctuating fields, derive the Reynolds equations and the transport equations for the Reynolds stress tensor, the turbulent kinetic energy K and its dissipation rate ε, and interpret each of their terms physically (production, transport, pressure redistribution, dissipation);
  –      explain the origin of the closure problem and why the averaged equations must be modelled;
  –      describe homogeneous turbulence statistically (one- and two-point correlations, integral and Taylor scales) and use isotropy properties to simplify the equations;
  –      analyse the effect of a mean velocity gradient (strain, shear, rotation) on the energy and anisotropy of turbulence, and explain the return to isotropy;
  –      characterise free (jet) and wall-bounded (boundary layer, channel) turbulent flows through similarity properties, wall scales, velocity laws and friction laws;
  –      interpret the energy cascade and turbulence spectra, and estimate the Kolmogorov scales;
  –      formulate and critically assess eddy-viscosity RANS models (K–ε and its variants) and compare them with Reynolds stress models, identifying their assumptions, ranges of validity and limitations;
  –      set up and run a turbulent flow simulation with an industrial code (Ansys Fluent): geometry, mesh, boundary conditions, choice and setting of the turbulence model, wall treatment;
  –     critically analyse simulation results (convergence, sensitivity to mesh and model, comparison with reference data) and communicate the results and their limitations.  

The competences developed in this course are:

• Characterisation and definition of turbulent flows: Reynolds' experiment and the Reynolds number, instabilities and transition to turbulence (small perturbations of a fluid and linearised Navier–Stokes equations; Couette flows, jets), grid turbulence; consequences of turbulence for transport (mixing) and energy dissipation.
• Statistical description and averaged equations: ensemble averaging, Reynolds decomposition, Reynolds equations, physical meaning of the Reynolds stress tensor, transport equations for the Reynolds stress tensor, the turbulent kinetic energy K and the dissipation rate ε; the closure problem; turbulent energy budgets in channel flow.
• Homogeneous isotropic turbulence: one-point and two-point correlations, integral and Taylor scales, consequences of isotropy, basic equations, experimental results on grid turbulence (energy decay, evolution of scales) and modelling.
• Homogeneous anisotropic turbulence and mean velocity gradients: return to isotropy, modelling of the pressure–strain correlation (Rotta model), plane pure strain and rapid distortion, homogeneous shear, effect of rotation (Coriolis force) on turbulence.
• Free turbulent flows: turbulent plane jet, similarity properties of mean velocity profiles, closure by the eddy-viscosity hypothesis (Boussinesq) and its limits of validity.
• Wall-bounded turbulent flows: wall scales, mean velocity profile, logarithmic law and velocity-defect law, matching, friction laws, boundary layer equations, von Kármán integral equation, turbulent kinetic energy budget; study of turbulent plane Couette flow and comparison with a direct numerical simulation (DNS).
• Prediction of turbulent flows: the K–ε model: eddy-viscosity hypothesis, modelling of the K and ε equations, improvements (RNG, realizable), alternatives (K–ω, SST, Reynolds stress model), wall treatment, ranges of validity and limitations.
• Spectral analysis and the energy cascade: three-dimensional and one-dimensional spectra, dissipation spectrum, energy transfer between scales, Kolmogorov theory (dissipation scales, −5/3 law), relations between Reynolds numbers.
• Application to industrial configurations with CFD software (Ansys Fluent): practical work on concrete cases: channel flow with and without rotation (K–ε and Reynolds stress model compared with DNS), dynamic meshing (four-stroke cycle in a cylinder), turbulent diffusion flame and spray, two-phase flow with the VOF method (air pocket in water); good practice in meshing, turbulence model selection and analysis of results.
• Mathematical concepts covered: tensor calculus (index notation), statistical averages and correlation functions, Fourier analysis (spectra), scaling analysis and scaling laws, systems of partial differential equations.

• Take-home assignment (individual): 25% of the final mark.
• Ansys Fluent design project (individual), assessed by:
  –      1 written report (approximately 15 pages, excluding front and back matter) presenting the approach (model set-up, meshing, choice of turbulence models) and a critical analysis of the results and their limitations, by comparison with direct numerical simulation data;
  –      criteria: relevance of modelling choices, quality of the mesh and of convergence, rigour of the physical interpretation, clarity of communication;
  –      weighting: 25% of the final mark.

• Final examination (2 h, one personal handwritten A4 sheet allowed): 50% of the final mark.

• Take-home assignment, design project and final examination – Level 1: no AI
  The assessment is carried out entirely without the use of AI, in a controlled environment, which ensures that you rely solely on your existing knowledge and skills, and on those developed in this course. You must not use AI at any point during your assessment. You must demonstrate that you have acquired the fundamental skills and knowledge at the level expected for this course.