Aerospace Engineering - Aerodinamica

Full exam

Politecnico di Milano Department of Aerospace Science and Technology Aerodynamics 051172Final Examination Date: September 5, 2019 Time: 8:30-11:30 hrs Closed notes, closed books. No calculator, mobile telephone, tablet,laptop, or any other WIFI enabled device. In any event, this document should be returned to the exam invigilator. Note to students, please read carefully: Students have the responsibility to write a clearly written exam. If your handwrite is hard to read or unreadable, please print. In the case an exam is unreadable, it will not be graded and the student will automatically fail the exam. If sections of the exam are unreadable, these sections will not be graded and the corresponding points will be detracted from the nal grade.Students have the responsibility to write a well organized exam. The steps of the solutions should be clearly indicated, the rationale of the derivations should be concizely explained in English. If space is needed, the back of the answer sheets can be used, but it should be clearly indicated. Exams poorly organized, where it is impossible to follow the solution, will not be graded and the student will automatically fail the exam. If sections of the exam are poorly organized, where it is impossible to follow the solution, these sections will not be graded and the corresponding points will be detracted from the nal grade.Students are not allowed to change the notations of the problems. A problem solved using a dierent notation will be penalized, as the mark of the problem will be halved.Last name (print)SignatureStudent ID # (Matricola #) 1 1. (8/31 points) In 1908, Blasius obtained an exact solution to the boundary layer equations for a two- dimensional, incompressible, uniform, steady state ow over a semi-in nite at surface, rep- resented here by the positivex-axis . Blasius made the simplifying assumption that there is no pressure gradient. In this case, therefore, the boundary layer equations reduce to @ u@ x + @ v@ y = 0 ; andu@ u@ x + v@ u@ y = @ 2 u@ y 2; with boundary conditionsu(x; y= 0) =v(x; y= 0) = 0; andu(x; y=) =U 1; whereU 1is a uniform freestream velocity and is the boundary layer thickness. (a) The streamfunction , de ned as u=@ @ y ; v =@ @ x ; was used by Blasius to simplify the problem. Rewrite the boundary layer equations and boundary conditions in terms of ,xandy. Brie y, discuss the simpli cation and complication generated by the transformation and its impact on the boundary conditions. (b) Blasius had the clever idea to look for a similarity solution. In order to nd a similaritysolution, consider a similarity variable de ned as =y n qx U 2 1; and a similarity function asf() = (x; y) px ; wherenis a real number andis a constant. Perform the change of variables and determinenandin order to transform the problem in terms of the streamfunction into a valid similarity equation, i.e. an ordinary dierential equation in terms offand only, with appropriate boundary conditions. (c) Brie y, discuss how one can solve such an equation. 2 Answer Page 3 Answer Page 4 2. (8/31 points) In this problem we want to explore the dynamics of a pair of trailing vortices left behind by an aircraft while taking o. The trailing vortices are assumed to be close to the ground and therefore their dynamics will be aected by the ground topography. For simplicity, the take-o strip and surroundings are assumed to be perfectly at and the ground is therefore represented by thex-axis, the real axis. Above the ground, two point vortices are located symmetrically with respect to they-axis (the imaginary axis), on the right, atz R= a+ib, and on the left, atz L= a+ib, wherea >0,b >0 and, in general, a6 =b. The vortex to the right is a positive vortex of strength , while the vortex to the left is a negative vortex of opposite strength. Recall that the complex potential of a positive (counter-clockwise) vortex located atz=z 0is 2 ilog( zz 0) : (a) Construct the complex potential so that the no-penetration boundary condition on thex-axis is satis ed. (b) Verify that the real axis is a streamline.(c) Compute the velocity distribution along the real axis. Sketch the velocity distribution.Does the velocity distribution allow to detect the location of the vortices? (d) Compute the velocity of the right vortex located atz R. Sketch the tra jectories of the right and left vortices. Brie y, explain why the vortices move along the tra jectories you sketched. How the velocity depends on the size ofaandband what is the impact of them? How the presence of the ground, i.e. whenb