AME Intermediate Fluid Mechanics. Homework. Updated November 22, 2013

Transcription

1 Please consult the following TAs Michael Arthur (for odd HW sets) Arman Mirhashemi (for even) Books AME Intermediate Fluid Mechanics Homework Updated November 22, 2013 KCD = Fluid Mechanics, 5th Ed., Kundu, Cohen and Dowling MYOH = Fundamentals of Fluid Mechanics, 6th Ed., Munson, Young, Okiishi and Huebsch HW1, due Mon., Sept. 2, KCD Prob KCD Prob KCD Prob KCD Prob MYOH Prob HW2, due Mon., Sept. 9, KCD Prob KCD Prob By using an order of magnitude analysis, the continuity and Navier-Stokes equations can be simplified to the Prandtl boundary layer equations. For steady, incompressible, and two dimensional flow, neglecting gravity the result is u x + v y = 0, u u x + v u y = 1 ρ p x + ν 2 u y 2. Use L and V 0 as characteristic length and velocity, respectively. Nondimensionalize these equations and identify the similarity parameters that result. 4. In an adiabatic atmosphere, the pressure varies with the specific volume in the following manner, pv k = const where k is a constant equal to the ratio of the specific heats c p and c v. Develop an expression for the pressure as a function of elevation for this atmosphere, using the ground as a reference. When z = 0, take p = p 0 and γ = γ 0. Reach the following result p = 1 k k γz + p γ 0. γ 0

2 Figure 1: Wall with corrugations. 5. Which of the following expressions are allowed in index notation? (a) a = b i c ij d j (b) a = b i c i + d j (c) a i = δ ij b i + c i (d) a k = b i c ki (e) a k = b k c + d i e ik (f) a i = b i + c ij d ji e i (g) a i = ǫ ijk b j c k (h) a ij = b ji (i) a ij = b i c j + e jk (j) a kl = b i c ki d l + e ki 6. KCD Prob KCD Prob KCD Prob Consider a wall 100 ft wide and having corrugations (Fig. 1). What are the resultant horizontal and vertical forces on the wall from the air and water? Give the result per unit width of the wall and for n corrugations. 10. A hollow cone is forced into the water by a force F (Fig. 2). Develop equations from which one may determine e. Neglect the weight of the cone and the thickness of the wall. Be sure to state any assumptions you make. HW3, due Mon., Sept. 16, KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob. 3.2.

3 Figure 2: Hollow cone. 7. KCD Prob KCD Prob (MYOH Prob. 4.5) A two-dimensional velocity field is given by u = 1+y and v = 1. Determine the equation of the streamline that passes through the origin. On a graph, plot this streamline. 10. (MYOH Prob. 4.11) Show that the streamlines for a flow whose velocity components are u = c(x 2 y 2 ) and v = 2cxy, where c is a constant, are given by the equation x 2 y y 3 /3 = constant. At which point (points) is the flow parallel to the y-axis? At which point (points) is the fluid stationary? HW4, due Mon., Sept. 23, KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob Compute the components of the strain rate tensor and vorticity vector for the Burgers vortex. The velocity components in cylindrical coordinates are (a,ν and Γ are constants) V r = ar, V z = 2az, V θ = Γ [ )] 1 exp ( r2. 2πr 2ν/a

4 Figure 3: Water jet. 9. (Computer Problem) A jet of water having a velocity of 10 m/s and a diameter of 50 mm is directed against a trough in a horizontal plane, as shown in Fig. 3. The trough can have an angle shown as θ with line A A which is normal to the jet. The water splits up along the trough such that the volume flow rates Q 1 and Q 2 are related as Q 1 = Q 2 cos 2 θ. We will neglect friction and gravity effects on the speed of the water at all times. Plot the algebraic sum of the force components at the Hinge B for θ going from 0 in steps of 2 degrees. Also, what is the torque at B? The trough is held in equilibrium by B at each setting of θ. HW5, due Mon., Sept. 30, KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob (MYOH Prob. 3.10) An incompressible fluid flows steadily past a circular cylinder as shown in Fig. 4. The fluid velocity along the dividing streamline ( x a) is found to be V = V 0 (1 a 2 /x 2 ), where a is the radius of the cylinder and V 0 is the upstream velocity. (a) Determine the pressure gradient along this streamline. (b) If the upstream pressure is p 0, integrate the pressure gradient to obtain the pressure p(x) for x a. (c) Show from the result of part (b) that the pressure at the stagnation point (x = a) is p 0 + ρv0 2 /2, as expected from the Bernoulli equation. 9. (MYOH Prob. 5.6) Water flows out through a set of thin, closely spaced blades as shown in Fig. 5 with a speed of V = 10 ft/s around the entire circumference of the outlet. Determine the mass flow rate through the inlet pipe. HW6, due Wed., Oct. 16, KCD Prob KCD Prob. 5.8

5 Figure 4: Flow past cylinder Figure 5: Water through blades

6 3. KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob HW7, due Wed., Oct. 30, KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob HW8, due Wed., Nov. 6, KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob. 8.23

7 HW9, due Wed., Nov. 13, KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob HW10, due Wed., Dec. 4, (MYOH Prob. 8.52) Blood (assume µ = lb s/ft 2, SG = 1.0) flows through an artery in the neck of a giraffe from its heart to its head at a rate of ft 3 /s. Assume the length is 10 ft and the diameter is 0.20 on. If the pressure at the beginning of the artery (outlet of the heart) is equivalent to 0.70 ft Hg, determine ythe pressure at the end of the artery when the head is (a) 8 ft above the heart, or (b) 6 ft below the heart. Assume steady flow. How much of this pressure difference is due to elevation effects, and how much is due to frictional effects? 2. (MYOH Prob. 8.54) Gasoline flows in a smooth pipe of 40-mm diameter at a rate of m 3 /s. If it were possible to prevent turbulence from occuring, what would be the ratio of the head loss for the actual turbulent flow compared to that if it were laminar flow? 3. (MYOH Prob. 8.58) Given 90 threaded elbows used in conjunction with copper pipe (drawn tubing) of 0.75-in. diameter, convert the loss for a single elbow to equivalent length of copper pipe for wholly turbulent flow. 4. (MYOH Prob. 8.74) Air flows through a rectangular galvanized iron duct of size 0.30 m by 0.15 m at a rate of m 3 /s. determine the head loss in 12 m of this duct. 5. (MYOH Prob. 8.80) According to fire regulations in a town, the pressure drop in a commercial steel horizontal pipe must not exceed 1.0 psi per 150 ft of pipe for flowrates up to 500 gal/min. If the water temperature is above 50 F, can a 6-in-diameter pipe be used? 6. (MYOH Prob ) Water is pumped between two large open reservoirs through 1.5 km of smooth pipe. The water surfaces in the two reservoirs are at the same elevation. When the pump adds 20 kw to the water the flowrate is 1 m 3 /s. If minor losses are negligible, determine the pipe diameter. 7. (Currie, Prob. 9.8) Use the momentum integral with the velocity profile u U e = a + bη + cη 2 + dη 3 where η = y/δ, to evaluate the boundary layer thicknesses δ, δ, and θ, and the surface shear stress τ 0 for flow over a flat surface without pressure gradient.

8 8. KCD Prob KCD Prob KCD Prob HW11, due Wed., Dec. 4, KCD Prob KCD Prob KCD Prob KCD Prob (MYOH Prob ) An expression for the value of c p for carbon dioxide as a function of temperature is c p = T T 2 where c p is in ft lb/lbm R and T is in R. Compare the change in enthalpy of carbon dioxide using the constant value of c p = 152 ft lb/lbm R with the change in enthalpy using the expression above, for T 2 T 1 equal to (a) 10 R, (b) 1000 R, (c) 3000 R. Ser T 1 = 540 R. 6. (MYOH Prob ) If the observed speed of sound in steel is 5300 m/s, determine the bulk modulus of elasticity of steel in N/m 2. The density of steel is nominally 7790 kg/m 3. How does does your value of E v for steel compare with E v for water at 15.6 C? Compare the speeds of sound in steel, water, and air at standard atmospheric pressure and 15 C, and comment on what you observe. 7. (MYOH Prob ) An ideal gas flows isentropically through a converging-diverging nozzle. At a section in the converging portion of the nozzle, A 1 = 0.1 m 2, p 1 = 600 kpa(abs), T 1 = 20 C, and Ma 1 = 0.6. For section (2) in the diverging part of the nozzle, determine A 2, p 2, and T 2 if Ma 2 = 3.0 and the gas is air. 8. (MYOH Prob ) For Fanno flow, prove that dv V = fk(ma2 /2)(dx/D) 1 Ma 2 and in doing so show that when the flow is subsonic, friction accelerates the fluid, and when the flow is supersonic, friction decelerates the fluid. 9. (MYOH Prob ) Air enters a 0.5-ft inside diameter duct with p 1 = 20 psia, T 1 = 80 F, and V 1 = 400 ft/s. What frictionless heat addition rate in Btu/s is necessary for an exit gas temperature T 2 = 1500 F? Determine p 2, V 2, and Ma 2 also. 10. (MYOH Prob ) An aircraft cruises at a Mach number of 2.0 at an altitude of 15 km. Inlet air is decelerated to a Mach number of 0.4 at the engine compressor inlet. A normal shock occurs in the inlet diffuser upstream of the compressor inlet at a section where the Mach number is 1.2. For isentropic diffusion, except across the shock, and for standard atmosphere, determine the stagnation temperature and pressure of the air entering the engine compressor.

9 HW12, due Wed., Dec. 11, KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob KCD Prob