Due to the COVID19 crisis, the information below is subject to change,
in particular that concerning the teaching mode (presential, distance or in a comodal or hybrid format).
5 credits
30.0 h + 30.0 h
Q2
Teacher(s)
Chatelain Philippe; Winckelmans Grégoire;
Language
English
Main themes
Reminder of the conservation equations for incompressible and compressible flows, dimensional analysis (VaschyBuckingham theorem) and applications. Vorticityvelocity formulation of the equations and general results: entropy, vortex tubes (Kelvin and Helmholtz theorems), velocity induced by vorticity (BiotSavart) in 3D and in 2D, vorticity production (at walls, baroclinic term) and diffusion, reformulation of Bernoulli's equation. Incompressible irrotational flows : vortex sheets at wall and in wake, impulsive start of an airfoil, wing of finite span in steady state (Prandtl model, optimal wing). Compressible flows : 2D steady supersonic flows : small perturbations and acoustic waves, method of characteristics, expansion waves and compression (shock) waves, applications; 1D unsteady flow : method of characteristics. Laminar boundary layer for the case with variable external velocity (FalknerSkan, Polhausen, Thwaites). Flow stability (OrrSommerfeld) and transition to turbulence. Turbulent boundary layer : law of the wall (Prandtl, von Karman), law of the wake, unification (Millikan, Coles), case with variable external velocity and concept of equilibrium boundary layer (Clauser, Coles). Modelisation of turbulence : Statistical approach (Reynolds) and equations for the averaged fields, closure models (algebraic, with one or two conservation equations), exemples of application.
Aims
At the end of this learning unit, the student is able to :  
1 
In consideration of the reference table AA of the program "Masters degree in Mechanical Engineering", this course contributes to the development, to the acquisition and to the evaluation of the following experiences of learning:

Content
1. General theory (5 hrs)
 General reminder of the classical formulation of the NavierStokes equations.
 Dimensional analysis : proof of VaschyBuckingham theorem; applications.
 Thermodynamics of compressible flows.
 Conservation equations in vorticityvelocity formulation, for incompressible and compressible flows.
 Resultats on the conservation equations and on control volume budgets
 Vortex tube in 3D : theorems of Kelvin and of Helmholtz, applications.
 Velocity induced by vorticity : BiotSavart; application to 3D vortex tubes and to 2D vortices (gaussian, etc.).
 Vorticity production : at walls, baroclinic term; vorticity diffusion; reformulation of Bernoulli's equation (incompressible and compressible).
 2D irrotational flows : starting airfoil and vortex sheets; KuttaJoukowski; Blasius theorem for lift and moment.
 Prandtl model for wing of finite span: lift and induced drag, applications (optimal elliptical wing, rectangular wing), Oswald efficiency.
 2D steady supersonic flows : concept of characteristics; small perturbations and acoustic waves; method of characteristics; isentropic expansion waves (PrandtlMeyer); non isentropic compression waves (shock waves: normal and oblique shocks); applications (e.g., "diamond" profile); wave drag.
 1D unsteady flows (subsonic or supersonic) : method of characteristics and Riemann invariants; application to propagation to traveling shock and expansion system.
 Similarity for the case with power law velocity : FalknerSkan.
 Polhausen method for the general case, and improved method due to Thwaites.
 Linearisation in small perturbations of the NavierStokes equation, and stability of viscous flows; simplification for parallel flows (OrrSommerfeld): application to boundary layer and comparison with experimental results. Case of inviscid flows (Rayleigh): application to the shear layer.
 "Route" to turbulence : phenomenological description of transition in a boundary layer.
 Reminders, classical approach and global results for the case with constant external velocity.
 Von Karman and Prandtl approach for the effective turbulence viscosity: law of the wall (with logarithmic law), Millikan's argument
 Case with general external velocity: experimental results (Clauser, etc.), unification by Coles : law of the wall and law of the wake, composite velocity profiles; computational method for the boundary layer development up to separation.
 Concept of "equilibrium turbulent boundary layer" : similarity parameters by Clauser and by Coles.
 Statistical approach by Reynolds and averaged equations.
 Closure models : algebraic, with one transport equation, with two transport equations (e.g., ke, kw) ; calibration and boundary conditions; applications and comparisons with experimental resultats.
Teaching methods
Due to the COVID19 crisis, the information in this section is particularly likely to change.
Lectures : there are typically 13 lectures in class, each of 2 hours.Sessions of practical exercices are also organised in class, each of 2 hours, to further develop concepts covered during the lectures and to do some applications
The students must also perform a number (typically 3) of homeworks which require to be able to use programing tools such as Python or Matlab. These homeworks are mandatory and they must be done during the quadrimester, each with a start date and a deadline date for the report, which is graded. Depending on the amplitude of the work/effort expected, these homeworks are done alone or in team of two.
The students must also participate to the laboratories (typically 2) that are organised in small groups; each group must produce one laboratory report, which is also graded.
Evaluation methods
Due to the COVID19 crisis, the information in this section is particularly likely to change.
The notes for the homeworks and for the laboratories correspond to work to be performed during the quadrimester, each within a given time slot.The final note of each student takes into account the notes obtained for the homeworks and for the laboratories, and the note obtained at the final written exam.
The homeworks and the laboratories are essential for this course. It is not possible to obtain a credit for this course by soley presenting the final exam.
Online resources
Bibliography
 G. K. Batchelor, "An introduction to fluid dynamics", Cambridge University Press 1967 (reprinted paperback 1994).
 F. M. White, "Viscous fluid flow" second edition, Series in Mechanical Engineering, McGrawHill, Inc., 1991.
 P. A. Thompson, "Compressiblefluid dynamics", advanced engineering series, Maple Press, 1984.
 H. Lamb, "Hydrodynamics", sixth edition, Cambridge University Press 1932, Dover Publications.
 L. Rosenhead, "Laminar boundary layers", Oxford University Press 1963, Dover Publications.
 P. G. Drazin and W. H. Reid, "Hydrodynamic stability", Cambridge University Press 1985.
 M. Van Dyke, "An album of fluid motion", The Parabolic Press, 1982.
 H. Schlichting, "Boundarylayer theory", Mc GrawHill, NY, 1968.
 H.W. Liepmann and A. Roshko, « Elements of gasdynamics », Dover Publications, 2001.
 D. J. Tritton, « Physical Fluid Dynamics », Clarendon Press, 1988.
Teaching materials
 Notes et/ou transparents des cotitulaires
Faculty or entity
MECA
Force majeure
Teaching methods
The lectures are given either live using videoconferencing or using prerecorded podcasts accompanied by live Q&A sessions. The exercise session are given live using videoconferencing and/or with prerecorded videos. The labs are replaced by virtual labs, given as homeworks.
Evaluation methods
The written indivual exam is then organized remotely.