Masters in Mechanical Engineering Aerodynamics 1 st Semester 2015/16

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1 Masters in Mechanical Engineering Aerodynamics st Semester 05/6 Exam st season, 8 January 06 Name : Time : 8:30 Number: Duration : 3 hours st Part : No textbooks/notes allowed nd Part : Textbooks allowed st Part Indicate if the sentences are true (T) or false (F) in the empty squares. For each theme, any combination of true and false is possible. The classification of each answer is the following: Correct answer 0.5 marks. Empty square 0 marks. Incorrect answer -0.5 marks. In the (numerical) solution of the time-averaged Navier-Stokes equations the mean velocity field is steady and so the effect of the velocity fluctuations is neglected. the application of the no-slip condition depends on the selected turbulence model. the determination of the shear-stress at the wall always requires the determination of the U y. derivative of the mean velocity at the wall ( ) y= 0 eddy-viscosity models depend on the mean velocity field and so they cannot be calculated a priori.. For a two-dimensional boundary-layer over a flat plate, The shear-stress at the wall is proportional to the normal derivative of the velocity at the wall only for laminar flow. the line y=δ (δ stands for the thickness of the boundary-layer) is a streamline if the flow is turbulent. the drag coefficient of the plate always diminishes with the increase of the Reynolds number. flow separation never occurs for any flow regime (laminar or turbulent).

2 3. A turbulence model for the Reynolds-averaged Navier-Stokes equations determines the Reynolds stresses. may not use the concept of eddy-viscosity. is not necessary if the flow is statistically steady. never affects the application of the no-slip condition at the wall. 4. The figure below presents the distributions of the (symmetric) of the pressure coefficient C = τ U ) along the chord (x/c) of a (-C p ) and of the skin friction coefficient ( f w ( ρ ) NACA 00 airfoil at an angle of attack of ( α = ) and a Reynolds number of The results were obtained with the time-averaged Navier-Stokes equations supplemented by the eddy-viscosity k-ω SST turbulence model with and without a model to simulate transition from laminar to turbulent flow. 0 o -C p A B C D C f x/c 0 Line B corresponds to C f distribution obtained with the transition model. The drag coefficient C d determined with the transition model is larger than that obtained without it. The lift coefficient C l obtained in the two simulations is identical. Line D corresponds to the pressure distribution of the calculation without turbulence model.

3 5. The figure below presents the lift coefficient Cl as a function of the drag coefficient Cd determined ed experimentally for an airfoil at three different Reynolds numbers 6, 05,,5 06 e 06. The airfoil is a laminar foil foil. Re C corresponds to the largest Reynolds number, 06. The airfoil as the aerodynamic centre coincident with the pressure centre centre. For Cl close to 0,6, the increase of Cd obtain for Re B is a consequence of the pressure resistance. coefficient of an airfoil with and without three types of 6. The figure below presents the lift coeff flaps (plain,, slotted and Fowler) as a function of the angle of attack α. The airfoil has positive camber camber. The three flaps have different deflections deflections. Lines A and B correspond to flaps with boundary boundary-layer control. Line C corresponds to the plain flap.

4 7. The figure below presents the circulation (Γ) and the lift coefficient (Cl) along the span (y) of two symmetric finite wings at an angle of attack of degrees obtained with the linearized lifting line theory theory. The two wings have the same root chord and no twist, sweep and dihedral angles. One of the wings is rectangular and the other is tapered. tapered Line D corresponds to the lift coefficient (Cl) of the tapered wing. Line B corresponds to the circulation distribution (Γ) of the rectangular wing. wing The aspect ratio (Λ) of the rectangular wing is larger than that of the tapered wing. wing The lift force of the rectangular wing is larger than the lift force of the tapered wing. wing 8. The figure below illustrates three different bodies bodies. Body C exhibits the strongest dependence of the drag coefficient on the Reynolds number. For case C the friction drag coefficient is larger than the pressure drag coefficient. coefficient The average lift coefficient of the three flows is zero zero. The case that leads to the smallest base pressure coefficient is body B.

5 Masters in Mechanical Engineering Aerodynamics st Semester 05/6 Exam st season, 8 January 06 Time : 8:30 Duration : 3 hours st Part : No textbooks/notes allowed nd Part : Textbooks allowed nd Part. The time-averaged Navier-Stokes equations supplemented by the k-ω SST eddy-viscosity model were solved to determine the skin friction coefficient C = τ ρu (illustrated in figure ) along a flat plate. ReL f w e 7 = = 0. Two calculation were performed with U L ν the standard k-ω SST model applying the no-slip condition with and without wall functions. A third simulation was made with an additional model to simulate transition from laminar to turbulent flow. 0 C f A B C Blasius Re x Figure a) Identify the letter of the legend that corresponds to each simulation justifying your choice. b) Estimate the largest and smallest values obtained for the drag coefficient C F in the three simulations. c) For case A and Re = 5 0 y / of 6 x, estimate the dimensionless distance to the wall L the location where the mean horizontal velocity component is U = 0, 6U.

6 η c c b ξ Figure. Consider the steady, bi-dimensional, potential and incompressible flow around a circular cylinder. The radius of the cylinder is m and its centre is located at ( c + ic ) of the coordinate system ζ=ξ+iη. The uniform incoming flow makes an angle α, ( α <π/4), with the real axe ξ and the magnitude of the velocity is U. At the centre of the cylinder, there is a line vortex with the required intensity to guarantee that there is a stagnation point at the intersection of the cylinder with the positive real axe, ξ=b. a) Write the complex potential that represents the flow as a function of the angle of attack α, c and c. Indicate clearly what is the coordinate system adopted. b) Determine the range of values of c, c and angle of attack α for which the locations with maximum and minimum pressure coefficient are in the intersections of the cylinder with the ξ and η axis and the minimum pressure coefficient is equal to -3.5 C = 3.5 ( ) ) p min Consider the Joukowski transformation given by transforms the cylinder into an airfoil. b z = ζ + with z = x + i y that ζ c) Sketch the flow in the transformed plane, identifying the shape of foil, for the values of c and c of the previous question and for the angle of attack α where the pressure centre is located at x = 0. 4c ( x = 0 at the leading edge).

7 3. The flow around the Eppler 374 was determined with the time-averaged Navier-Stokes equations supplemented by an eddy-viscosity model at angle of attack of zero degrees ( α = 0 o ) and a Reynolds of Table and figure 3 present the values of the pressure ( C drag coefficients obtained in 5 geometrically similar d C and friction ( d grids. h i / h is the grid refinement ratio. Table and figure also include the observed order of grid convergence p and the estimate of the solutions for h / h = i 0 obtained with a least-square fit to a power series expansion truncated to a single term. The drag coefficient C d measured experimentally is (C d (C d (C d p=,9 (C d p= h i /h Figure 3 h i / h p ( C d ,9 ( C d Table a) Estimate the numerical uncertainty of the pressure ( C d and friction ( d coefficients obtained in the finest grid. b) Estimate the numerical uncertainty of the ratio ( d ( d solutions of the finest grid. C drag C / C obtained from the c) Estimate the modeling error of the drag coefficient C d of the selected mathematical model.

8 4. A small aircraft weights 3.6kN and has a cruise speed of 80 km/h. The aircraft has a trapezoidal wing without sweep and dihedral and the root chord is,5m and the taper ratio /3. The wing area is S=8m and its section does not change along the span. The drag coefficient of the wing section at small angles of attack is approximately constant and equal to C d = 0, 005. The lift coefficient of the wing C L at small angles of attack is C = 4,77 α + 0,035, α in radians. L ( ) Assume that the drag coefficient of the aircraft is equal to the drag coefficient of the wing. 5 3 ν =.5 0 m /s, ρ =.kg/m. ar ar a) Determine the lift coefficient of the wing when the aircraft flies at cruise speed and constant height. b) Determine the minimum propulsion power in the conditions of the previous question. c) If the aircraft looses 0m of height for each km that it flies without changing the wing configuration, estimate the time required for the aircraft loose 00m of height.

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