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1 MAE101B: Advanced Fluid Mechanics Winter Quarter Name: Final This is a three hour open-book exam. Please put your name on the top sheet of the exam. Answer all six questions. Make any necessary assumptions and state them. Explain your working. If units are not given, they are SI (using m, kg, s and K). Problem 1. (25) 2. (10) 3. (10) 4. (10) 5. (10) 6. (10) TOTAL = Points HW = % MIDTERM = % FINAL = % COURSE GRADE = Exams will be available from staff in EBU II 573 with proper identification from March 27. The final exam will have your course grade on it. 1

2 1 (25 points) Answer the following questions. Be brief. A (5 points) Using dimensional analysis, derive the dependence of the friction factor for flow in a pipe in the form p 1 = l ( 2 ρu2 D φ Re, ε ). D The above result uses one assumption in addition to standard dimensional analysis. What is it? B (5 points) A heat exchanger consists of flow through an annular pipe with outer radius 20 cm containing water at 80 C surrounding a circular pipe of radius 10 cm containing water at 20 C. The pipes are both 1 meter long and constructed of tubing, and the flow rate is 10, 000 cm 3 s 1 in both pipes. What are the hydraulic diameters and Reynolds number of each flow? Are the flows turbulent? 2

3 C (2 points) Why are there dimples on golf balls and not on ping pong balls? D (3 points) Compute the drag coefficient for flow over a rectangular plate at high Reynolds number with no pressure gradient, starting from the expression for the shear stress corresponding to the Blasius boundary layer profile. E (2 points) Explain why the stagnation temperature is constant for isentropic and for Fanno flow and across a normal shock. 3

4 F (3 points) How long does it take to empty a 500 ml can of cola through a 4-mm diameter straw as quickly as possible if the flow is to remain laminar? [Take the properties of cola to be the same as water.] G (5 points) Air enters a length of constant area pipe with p 1 = 200 kpa(abs), T 1 = 500 K and V 1 = 400 m/s. If 500 kj/kg of energy is removed from the air by frictionless heat transfer along the pipe, determine the stagnation temperature at the entrance and exit. 4

5 2 (10 points) Water at 5 C is pumped through a 7.5-cm diameter, 600 m-long steel pipe with an elevation gain of 100 m from a reservoir at a rate of 5,000 liters/s. If it is necessary to maintain a pressure of 10 atm at the exit, determine the power added to the water by the pump. Neglect minor losses. 5

6 3 (10 points) A smooth ball with diameter 38.1 mm and weight N is released underwater. What is its terminal velocity as it rises? 6

7 4 (10 points) The Concorde supersonic transport flew at Mach 2.2 at 20 km altitude. What is the Mach angle? How far behind the aircraft was it heard on the ground? What effect are you neglecting in this calculation? Concorde has four engines, each of which produces a thrust of 170 kn. Its mass on take-off is 185 tons and fuel is consumed at a rate of 22.6 tons/h (metric tons). What is the ratio of lift to drag two hours into flight? What is the lift coefficient if the wing surface is m 2? 7

8 5 (10 points) Standard atmospheric air (T 0 = 59 F, p 0 = 14.7 psia) is drawn steadily through a frictionless and adiabatic converging nozzle into an adiabatic, constant crosssectional area duct. The duct is 10 ft long and has an inside diameter of 0.5 ft. The average friction factor for the duct may be estimated as being equal to What is the maximum mass flowrate in slugs/s through the duct? For this maximum flowrate, determine the values of static temperature, static pressure, stagnation temperature, stagnation pressure, and velocity at the inlet and exit of the duct. 8

9 6 (10 points) Air from a reservoir at standard conditions flows isentropically through a converging-diverging duct and forms a normal shock downstream of the throat. The temperature increases by a factor of 2 across the shock. Compute the flowrate in BG units through the duct if the throat has area 3 in 2. 9

10 Useful values and parameters Units and constants 1 hp = 550 lb ft/s 1 in = 2.54 cm Acceleration of gravity: g = 9.81, g = 32.2 ft/s 2 R = F Material properties (in SI units unless otherwise stated) Water: ρ = and µ = at 5 C. Water: ρ = and µ = at 20 C. Water: ρ = and µ = at 80 C. Air: ρ = and µ = at 20 C. US standard atmosphere: p = at 0 m altitude. Air (ideal gas): c p = 1004, R = (SI), R = 1, ft lb/slug R (BG), R = ft lb/lbm R (EE) Formulas Blasius boundary layer: τ w = 0.332U 3/2 ρµ/x. Moody chart 10

11 Drag coefficients for cylinders and spheres Isentropic, Fanno, normal shock relations for an ideal gas with k =

12 12

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