Aerodyamics of external flows.
[MECA2323]
[30h + 30h] 5 credits
This course is taught in the 2d semester.
Teacher(s): François Dupret, Grégoire Winckelmans
Language: French
Level: Second
Aims
Develop the student's ability to use concepts and tools outils in aerodynamics (hydrodynamics) of external flows, to understand real and complex situations, to model them in a simplified yet sufficient way using an adequate mathematical formalism, and to obtain a physically acceptable solution. Develop the aptitude of the student to also work outside of directed class sessions (exercices and laboratories) and to produce quality and concise written reports.
Main themes
Content and teaching methods
- General reminder of the classical formulation of the Navier-Stokes equations.
- Dimensional analysis : proof of Vaschy-Buckingham theorem; applications.
- Thermodynamics of compressible flows.
2. Vortex dynamics (8 hrs)
- 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.
3. Compressible flow of a perfect fluid (5 hrs)
- 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.
4. Laminar boundary layers (4 hrs)
- Similarity for the case with power law velocity : Falkner-Skan.
- Polhausen method for the general case, and improved method due to Thwaites.
5. Hydrodynamic stability and transition (1 hr)
- 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.
6. Turbulent boundary layers (5 hrs)
- 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.
7. Modelisation of turbulence (2 hrs)
- 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.
Other information (prerequisite, evaluation (assessment methods), course materials recommended readings, ...)
Courses of "fluid mechanics and transfers I and II"
Practical exercices and laboratories :
The practical exercices will be, in part, done in class sessions (roughly 12 hrs, to further develop concepts and applications partially covered during the courses). Students will also have to work on exercices outside of class sessions (roughly 8 hrs), this work leading to a written and graded report. The students must also participate to the laboratories (roughly 8 hrs), this work also leading to a written and graded report.
Some references :
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 (paperback).
L. Rosenhead, "Laminar boundary layers", Oxford University Press 1963, Dover Publications (paperback).
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.