- 530.328 Fluid Mechanics II, 2007 Final Exam
Closed books, closed notes. Each problem has the same weight. Time: 35 minutes or less. Develop your answers on the available place. (33%) NAME: ...................... 1. A rough pipe is connected to a pressurized vessel (gauge pressure p) as shown. The viscous water flow exits to atmospheric pressure at speed V below the water level at height h. About the speed V, which of these statements is TRUE? (a) The speed V is larger than (b) The speed V is smaller than (c) The speed V is equal to (d) Cannot say (e) The velocity is smaller than
2( p ! patm ) / " + 2 gh . 2( p ! patm ) / " + 2 gh . 2( p ! patm ) / " + 2 gh . ( p ! patm ) / h + 2 "h .
p h V
2. A pipe divides into two as shown, each arm discharging to atmospheric pressure. The flow rate at the entrance is Q0. All pipe segments have the same length, diameter and roughness. Which of the following statements is FALSE? (a) Q1=Q2 (b) pM-p N < p N-patm (c) p N<pM (d) pM-p N > p N-patm (e) Q2 =Q0/2
Q1 patm Q0 M N
Q2 patm
3. A Pelton wheel deflects a water jet at 180 degrees as shown. Evaluate the reaction torque Tz applied at the support S.
V, A,
y H z x
V, A, S
H/3
4. The Concorde used to fly at supersonic speeds. From the figure below showing the Mach cone, estimate the Mach cone angle assuming standard atmospheric conditions everywhere. V=650 m/s
5. Why is oil generally considered to be a better lubricant than water? (a) Because as a side-product of gasoline refining process it is cheaper. (b) Because it increases the shear stresses needed to slide surfaces along each other. (c) Because oil heats up less and thus dissociation and ionization effects are prevented. (d) Because oils higher viscosity causes a larger pressure build-up between surfaces (e) Because oils higher viscosity causes a smaller pressure build-up between surgaces 6. Consider real flow over a cylinder. Which one of these statements is FALSE? (a) Bernoulli equation may be applied far from the cylinder in the flow in the front portion. (b) The boundary layer near the front stagnation point is nearly of constant height (c) The minimum pressure point in the flow occurs at infinity. (d) The drag coefficient at high Reynolds number may depend on Reynolds number (e) If the flow were modeled using potential flow, the resulting drag force would be zero.
xB
7. How would compute the expression
xA
) % !y " cy( dx $ '
# !u
&
using discrete approximations. c is a
constant, and u is a function of (x,y): N #1 & ui , j +1 " ui , j "1 " c ui +1, j " ui " 2 , j ( !y, (a) ) % 2 $ ' i = 1 2 !x
(
)(
)
)
where !x = ( x A " x B ) / N and !y = y / c
(b) (c) (d)
) % 2 !y (u $
N i =1 N
#1
i , j +1
& " ui , j "1 " cy j ( !x, '
)
where !x = ( x A " x B ) / N and y j = j !y
) % !y (u $
i =1 2
#1
i + 1, j + 1
& " ui "1, j "1 " vi , j ui +1, j " ui "1, j ( !x, '
(
)
where !x = ( x A " x B ) / N and !y = y / c
" (( u
N i =1
i + 1, j + 1
! ui !1, j !1 ! cui , j #x 2 ,
)
)
where #x = ( x A ! x B ) / N
8. The thermodynamic process associated with propagation of sound is: (a) Isothermal (constant T) (b) Isobaric (constant pressure) (c) Constant density (d) Isentropic (constant s) (e) Constant enthalpy 9. If M1>1, in the diverging nozzle shown (isentropic flow), which of the following is correct? (a) V2<V1 and p2<p1 (b) V2<V1 and p2>p1 (c) V2>V1 and p2<p1 1 2 (d) V2>V1 and p2>p1 (e) There must be a normal shock between 1 and 2
10. Express the of acceleration the rocket in terms of the variables given on the sketch (show control-volume used): External air density air Drag coeff = Cd Rocket cross-sect area = AR Rocket instantaneous velocity = U Total Mass = M g
Gas density j Nozzle area A
Exit velocity of gases relative to rocket = Vj
11. What does the integral shown below stand for? !! 2 "" !(h + 12 V )(V idA)
CS
(a) Total torque applied to control surface (b) Rate of change of kinetic energy in control volume (c) Flux of potential energy across control surface (d) Flux of kinetic, internal energy and flow-work across control surface (e) Has no physical meaning
530.328 - Fluid Mechanics II, 2006 Final Exam Open books, open notes. Each problem has approximately same weight. (67%) Time: 2.4 hrs. NAME................................................................. 1. (30%) An American west windmill is an axial turbine. Suppose it is operating at a constant angular velocity, 60 rpm and that at these conditions the wind velocity (axial velocity) at the rotor is 5 m/s (use air density of =1.2 kg/m3). Assume the flow occurs through an annulus of fairly small radius difference, i.e. you may use the mean radius for your analysis. The mean radius is Rm = 0.7 m, and the frontal cross-flow area is Ac = 1.3 m2. Naturally the wind enters the windmill without swirl. The mean blade angle at the back-side (2) is 300. (a) Determine the torque and power output generated by the windmill. (b) Determine the inlet angle of the blade (1) so that the incoming flow is well aligned with the windmill at the design condition for 60 rpm and wind-speed of 5 m/s. (c) Assume that the shaft is connected to a water pump and assume that the transmission and pump have 100% efficiency. To pump water through a 1-inch rough pipe (e/D = 0.001) of length 200 meters, what flow-rate can I get for my farm? Dont be afraid to iterate 1-2 times
Ac=1.3 m2 V1 (magnitude 5 m/s)
Rm= 70 cm =60 rpm
1=? Rm 2=30 0 V2
D=2.54 cm m L=200 m
2. (18%) During the design of a wind-tunnel experiment, we wish to ascertain how close a support structure T (whose 2-D shape is well approximated by a Rankine half-body) can be brought to the measurement point A located upstream, without affecting the velocity there by more than 5% (i.e. the velocity at A should only be changed by 5% as compared to its value if T was not there at all). The unperturbed velocity is U=20 m/s and the thickness of the structure is h=15cm. Use 2-D potential flow. (a) Find the source strength and placement distance s. (b) Find the minimum distance d for 5% disturbance. (c) Under these conditions, find the percentage (compared to 0.5U2) by which the pressure at A is perturbed by the presence of T. air, =1.2 kg/m3 U=20 m/s A T d=? s=? h=15 cm
3. (19%) A smooth converging nozzle is attached to a large tank of compressed air (ideal gas). The pressure in the tank is 300,000 Pa, and density is 4.0 kg/m3. The flow that is established is isentropic within the nozzle. The back pressure is atmospheric pressure (100,000 Pa) away from the nozzle exit. Point 1 is right at the nozzle throat. (a) Find the Mach number at point 1. Is the flow there subsonic, sonic, or supersonic? (b) Find the temperature 0 in the tank, and T1 at the nozzle throat. (c) Find the density 1 of the air at the nozzle throat. (d) Find the speed of sound c1 in the air at point 1. (e) Find the air speed at the nozzle, V1. (f) The area of the nozzle is A1=2. x 10-4 m2. Evaluate the mass flow rate exiting the tank. M1 = ? 1= ? T1 = ? c1 = ? V1 = ? m=?
Air tank at p0 = 3.0 105 Pa 0= 4.0 kg/m3 T0 = ??
pb = 1.0 105 Pa
A1 = 2.0 x 10-4 m2