At the end of this learning unit, the student is able to :
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:
The contribution of this Teaching Unit to the development and command of the skills and learning outcomes of the programme(s) can be accessed at the end of this sheet, in the section entitled “Programmes/courses offering this Teaching Unit”.
- General reminder of the classical formulation of the Navier-Stokes equations.
- Dimensional analysis : proof of Vaschy-Buckingham theorem; applications.
- Thermodynamics of compressible flows.
- Conservation equations in vorticity-velocity formulation, for incompressible and compressible flows.
- Resultats on the conservation equations and on control volume budgets
- Vortex tube in 3-D : theorems of Kelvin and of Helmholtz, applications.
- Velocity induced by vorticity : Biot-Savart; application to 3-D vortex tubes and to 2-D vortices (gaussian, etc.).
- Vorticity production : at walls, baroclinic term; vorticity diffusion; reformulation of Bernoulli's equation (incompressible and compressible).
- 2-D irrotational flows : starting airfoil and vortex sheets; Kutta-Joukowski; 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.
- 2-D steady supersonic flows : concept of characteristics; small perturbations and acoustic waves; method of characteristics; isentropic expansion waves (Prandtl-Meyer); non isentropic compression waves (shock waves: normal and oblique shocks); applications (e.g., "diamond" profile); wave drag.
- 1-D 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 : Falkner-Skan.
- Polhausen method for the general case, and improved method due to Thwaites.
- Linearisation in small perturbations of the Navier-Stokes equation, and stability of viscous flows; simplification for parallel flows (Orr-Sommerfeld): 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., k-e, k-w) ; calibration and boundary conditions; applications and comparisons with experimental resultats.
Due to the COVID-19 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.
Due to the COVID-19 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.
- 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, McGraw-Hill, Inc., 1991.
- P. A. Thompson, "Compressible-fluid 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, "Boundary-layer theory", Mc Graw-Hill, NY, 1968.
- H.W. Liepmann and A. Roshko, « Elements of gasdynamics », Dover Publications, 2001.
- D. J. Tritton, « Physical Fluid Dynamics », Clarendon Press, 1988.
- Notes et/ou transparents des cotitulaires