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1 Mestrado Integrado em Engenharia Mecânica Aerodynamics 1 st Semester 212/13 Exam 2ª época, 2 February 213 Name : Time : 8: Number: Duration : 3 hours 1 st Part : No textbooks/notes allowed 2 nd Part : Textbooks allowed 1 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.25 marks. Empty square marks. Incorrect answer -.15 marks 1. In the mathematical models for the simulation of turbulent flows, in direct numerical simulation (DNS), the flow is most of the times steady. the dependent variables (unknowns) of Large-Eddy Simulation (LES) are identical to the dependent variables of the Reynolds-Averaged Navier-Stokes (RANS) equations. the numerical requirements for their resolution depend of the adopted formulation. in the Reynolds-Averaged Navier-Stokes (RANS) equations, mass and momentum conservation/balance are satisfied on average. 2. For a finite wing at small angles of attack, the drag coefficient depends on the lift coefficient. the lifting line theory can only be used for an ideal fluid. The lift force does not depend on the aspect ratio of the wing. The induced angle of attack is equal to the difference between the effective and geometric angles of attack, α = α α. i ef geom

2 3. The figure below illustrates the velocity profile of a flat plate boundary-layer. U is the velocity component parallel to the plate, U e is the velocity of the external flow and y is the distance to the wall. The displacement of the outer flow streamlines is equivalent to area B. The area A defines the momentum thickness θ. The velocity component normal to the wall is zero. For y=c, the shear-stress is exactly equal to zero. 4. Mass conservation and momentum balance for a two-dimensional flow may be written as: u v + = (1) x y u u u + v x y p u u = + ν + (2) 2 2 ρ x x y 2 2 v v 1 p v v u + v = + ν + (3) 2 2 x y ρ y x y where u and v are the velocity components, p is the pressure, ρ is the fluid density and ν is the fluid kinematic viscosity. The equations are applicable to steady, compressible or incompressible flows. If p stands for the pressuree relative to the hydrostatic pressure, the weight of the fluid is included in the components of the pressure gradient. The equations may be applied to statistically steady turbulent flow if u, v and p represent mean values and ν stands for the effective viscosity. The left-hand side of equations (2) and (3) represents the change of momentum per unit mass of an element of fluid.

3 5. The figure below presents the lift coefficient of four different airfoils as a function of the angle of attack for the same Reynolds number. Airfoil D is the thickest. Airfoil A should exhibit the most intense suction peak at 4º angle of attack. Airfoil B presents thin airfoil stall whereas airfoil C exhibits trailing edge stall. At 7º angle of attack, the flow around airfoil A should present flow separation close to the trailing edge. 6. The figure below presents the mean velocity profiles of three turbulentt boundary-layers that have the same value of external velocity U e. In region D, the Reynolds stresses are negligible. The velocity profile C corresponds to adverse pressure gradient. The three velocity profiles have the same wall shear-stress. U C e y f ξ =. ν 2

4 7. The figure below presents the pressure coefficient C p on the surface of an airfoil at two angles of attack for which Cl > and the minimum pressure coefficient as a function of the angle of attack. The plots are typical of a NACA 4 digits airfoil. The pitching moment around the aerodynamic centre of the airfoil is zero zero. The angle of attack B is larger than the angle of attack A. The drag coefficient of this airfoil does not depend significantly on the Reynolds number. 8. The figure below presents the flow around three bluff bodies. Body C presents the drag coefficient with the smallest dependence on the Reynolds number. Vortex shedding appears only in the wake of body B. The mean lift coefficient of the three flows is zero. The smallest base pressure coefficient (near-wake wake pressure coefficient) is obtained for case A.

5 Mestrado Integrado em Engenharia Mecânica Aerodynamics 1 st Semester 212/13 Exam 2ª época, 2 February 213 Time : 8: Duration : 3 hours 1 st Part : No textbooks/notes allowed 2 nd Part : Textbooks allowed 2 nd Part 1. Figure 1 (at the end of the exam) presents several aerodynamic coefficients of a NACA airfoil at four different Reynolds numbers obtained numerically with a panel method that simulates the boundary-layer effects with a blowing velocity. Cl is the lift coefficient. ( C d ) 1, ( C d ) 2 and ( d ) 3 components. C are the airfoil drag and its friction and pressure C Mc is the pitching moment around the centre of the airfoil. The pressure distributions on the airfoil surface were obtained for angles of attack α of -2, and 2 degrees. ν ar = m /s, ρar = 1.2kg/m 3 a) Identify the plots that correspond to the friction and pressure resistance coefficients and explain the change of these coefficients with the lift coefficient. b) For a Reynolds of and zero degrees angle of attack ( α = ), estimate the location of the transition point when we assume zero pressure gradient boundary-layers. Discuss the result based on the surface pressure distribution at zero degrees angle of attack. c) Taking into account the pressure distribution included in figure 1, estimate the friction resistance coefficient for a Reynolds number of o and an angle of attackα = 2. Compare the result with that obtained from the panel method. d) Calculations of the flow around the airfoil at the three angles of attack and four Reynolds numbers given in figure 1 are to be performed with a Reynolds-Averaged Navier-Stokes

6 (RANS) solver. The program available includes two eddy-viscosity models: k-ε standard and Wilcox k-ω model with specific damping functions for transition modeling. Which model would you select to perform each of the twelve calculations? Give a clear justification to your choice. 2. Consider the steady, bi-dimensional, potential and incompressible flow around a circular cylinder. The radius of the cylinder is 1m and its centre is located at (.1;i.2) 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. Indicate clearly what is the coordinate system adopted. b) Determine the range of angles of attack ( α min and α max ) for which the absolute value of the real coordinate of the location with the minimum pressure coefficient ( ξ ( C p )min ) is less or equal than.5 and the minimum pressure coefficient ( C is larger than ξ ( C ).5 ( C p ) p min min > 3.4. ) min p

7 Consider the Joukowski conformal mapping given by z = ζ + 2 b ζ with z = x + i y that transforms the cylinder into an airfoil. c) Make a qualitative sketch of the flow in the transformed plane for zero angle of attack α =. Identify clearly the airfoil shape. ( ) d) Determine the angle(s) of attack for which the pressure coefficient at the leading edge is C =.. equal to.64, ( ) 64 p leading edge 3. A finite wing of a small aircraft has an average chord of 1.4m a plan form area of 8m 2, no twist and its section is a NACA airfoil ( C l and Cd given in figure 1). The weight of the aircraft is 2.4kN and the cruise speed at constant altitude is equal to 18km/h. Assume that the drag force of the aircraft is equal to the drag force of the wing. a) For the wing section, determine the location of the aerodynamic centre and the pitching moment around the aerodynamic centre. b) Determine the lift coefficient of the wing. c) Estimate the minimum propulsion power at cruise speed. d) For the conditions of question c), determine the pitching moment around the centre of the airfoil for the root section of the wing.

8 C l.2 (C d ) 3.6 (C d ) α (graus) (C d ) C l C l C l C Mc C p α A -C p α (graus) α B x/c Figure 1 Aerodynamic characteristics of a NACA airfoil. -C p x/c α C x/c

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