Reference no: EM131755054

Final Exam

Q1. As a continuation of the question from Exam 3 on unsteady aerodynamic characteristics of a NACA 2412 airfoil, further experimental results obtained by the engineer at Boeing is provided. The figures illustrate the flow field far downstream of the airfoil undergoing oscillatory, unsteady motion. The experiments were conducted at a velocity, V_∞ = 50m/s on an airfoil of chord length, c = 1m. By applying our discussions on the Kelvin's circulation theorem, Kutta condition, the starting vortex, and Kutta-Joukowski theorem answer the following questions:

(a) What is the angle of attack amplitude and frequency of the oscillatory motion of the airfoil for data set 1 and data set 2. Assume the vortices are convecting or moving at the freestream velocity.

(b) Discuss the possible implications of the unsteady motion on the lift and drag of the airfoils for the two cases.

Q2. Consider a jet exhausting into quiescent air. An ensemble-averaged velocity vector field has been calculated from 2D particle image velocimetry measurements, which are provided on Canvas as the "velocity data.mat". The data is formatted as a structure array that contains the fields x, y, u, v, where x, y are in mm and u, v are in ms^-1. Thus, velocity_data.u would return a field of u velocity components. Using this dataset, complete the following:

(a) Plot z vorticity ω_z, y with a 2 component vector overlay. HINT: type doc quiver in the command window

(b) Using imagesc(), inspect u velocity. HINT: imagesc() may display your image rotated, the jet should be exhasting from the left to the right. It is clear that as x increases, the width of the jet also increases. Calculate the width of the jet as a function of x. Define the width b of the jet as the distance between the two y locations where u(x) = 0.03u_max(x). Overlay the width of the jet on the u velocity. An analytical solution derived in 1933 gives an expression for width as b = Ax^2/3

where A is a constant. Do your calculated results follow this trend? Provided a comparison and discuss.

(c) Define an appropriate volume and, using momentum analysis, calculate the thrust produced by the jet.

Q3. Pressure distribution measurements at the midspan section of 76.2 cm by 12.7 cm model at Re = 3:1 x 10⁶ which had a NACA 4412 airfoil section are provided in the pdf file labeled "Q3-data". Using only the data provided to you answer the following questions:

(a) Graph C_p as a function of x/c for these three angles of attack. Comment on the movement of the stagnation point, the changes in the magnitude of adverse pressure gradient towards the trailing edge of the upper surface. How does this relate to possible boundary layer separation (or stall)?

(b) Calculate the maximum value of the local velocity at the edge of the boundary layer both on the upper and lower surfaces for all three angles of attack. If these velocities are representative of the changes with angle of attack, how does the circulation (or lift) change with the angle of attack? HINT: use equation C_p = 1 - V/V_∞².

Q4. During recitation (10/18/2017) we discussed how to account for the effect of the ground plane on an elementary vortex flow using the "method of images." Using your analysis from homeworks and recitations, extend the same principles to a 3D wing of an airplane operating near the ground (referred to "in ground effect") as shown in the figure. Considering these "ground effects" answer the following questions:

(a) Determine the coefficient of pressure distribution along the ground for a wing that is located at a distance x away from the ground. HINT: assuming inviscid, incompressible flow determine velocity field and then use C_p = 1 - V/V_∞².

(b) Using arguments based on changes in induced downwash and angle of attack due to the presence of the ground (method of images), explain how the lift experienced by the wing will change in the presence of a ground plane.

(c) BONUS: Using mathematical and physics based arguments, show at what non-dimensional distance away from the ground x/b for a given wing tip vortex strength Γ, the effect of the ground on the wing is negligible?

Attachment:- Assignment Files.rar