8.36: PROBLEM DEFINITION
Situation:
A large venturi meter is calibrated with a scale model using a prototype liquid.
1
10 scale model, p=400kPa.
Find:
The discharge ratio (Qm/Qp)
Pressure difference (∆pp)expected for the prototype.
SOLUTION
Match Reynolds number
Multiply both sides of Eq. (1) by Am/Ap=L2
m/L2
p:
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8.37: PROBLEM DEFINITION
Situation:
Drag is to be measured with a scale model of a bathysphere.
Find:
The ratio of towing speeds (ratio of speed of the model to the speed of the proto-
type).
PLAN
Dynamic similarity based on matching Reynolds number of the model and prototype.
SOLUTION
Reynolds number matching
Assume νm=νp
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8.38: PROBLEM DEFINITION
Situation:
A spherical balloon is tested by towing a scale model in a lake.
Dm=1.4ft, Dp=16.8ft.
Vm=5ft/sec, Fm=37lbf.
Find:
Drag force on the prototype (lbf,N).
Properties:
Air (60 ◦F), Table A.3: νp=1.58 ×10−4ft2/s, ρp=0.00237 kg/m3.
Water (60 ◦F) ,Table A.5: νm=1.22 ×10−5ft2/s, ρm=1.94 kg/m3.
PLAN
Dynamic similarity based on Reynolds number and same force coefficient.
SOLUTION
Match Reynolds numbers
Match force coefficients
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8.39: PROBLEM DEFINITION
Situation:
Lift force for an airplane.
The π−group for lift coefficient is
CL=2 FL
ρV 2S
Air: ρ=1.1kg/m3,CL=0.4.
V=80m/s,S=15m
2.
Find:
Lift force (N).
PLAN
Use the specified value of CL=0.4along with the definition of this π-group.
SOLUTION
REVIEW
Thisliftforceisabout4750lbf.
44
8.40: PROBLEM DEFINITION
Situation:
A scale model of a plane is tested in a wind tunnel.
1
8scale model.
Find:
Density of the air in tunnel.
Properties:
Air (10 ◦C,100 kPa),TableA.4:R=287J/kg K.
Air (10 ◦C,100 kPa) ,Table A.3: μp=1.76 ×10−5Ns/m2.
Air (25 ◦C,100 kPa) ,Table A.3: μm=1.83 ×10−5Ns/m2.
PLAN
Dynamic similarity based on matching Reynolds number and Mach number.
SOLUTION
Match Reynolds number
MatchMachnumber
Density of air at 100 kPa and 10oC
Combining Eqs. (1) and (2):
46
8.41: PROBLEM DEFINITION
Situation:
A windtunnel is used to do tests for the Airbus.
z= 10000 m.
Lm=1m,Lp=79.8m.
Find:
Wind tunnel pressure to have Reynolds number similitude.
Properties:
patm =101.3kPa.
Air (20 ◦C), Table A.4: R=287J/kg K.
PLAN
Find the velocity ratio for the same Mach number, set the Reynolds numbers the
same and solve for the pressure.
SOLUTION
The properties at 10,000 m. From Chapter 3
The density at 10000 m is
For Mach number similitude
For Reynolds number similitude
The model density must be
From the ideal gas law
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8.42: PROBLEM DEFINITION
Situation:
Wind tunnel test done for a Beoing 787 to simulate forces on model with only Mach
number similitude.
z= 30000 m,Lm=1m.
Lp=52m,V=94km/h.
cm=340m/s,M=0.85.
Find:
Theratioofforceonprototypetoforceonmodel.
Properties:
patm =101.3kPa.
Air, Table A.4: R=287J/kg K.
Air, Table A.3: ρ=0.98 kg/m3.
PLAN
Assume Reynolds number similitude is not important and find velocity ratio using
Mach number similitude. Equate force coefficients to find force ratio.
SOLUTION
From Chapter 3, the temperature at 10,000 m is
The pressure is
The density is
The velocity in the wind tunnel is
49
Speed of the airplane
Equating the force coefficients
50
8.43: PROBLEM DEFINITION
Situation:
Flow in a pipe is being tested with air and water.
Find:
Velocity ratio: Vair/Vwater for dynamic similitude.
Assumptions:
T=60◦F.
Properties:
Air (60 ◦F), Table A.3: vA=1.6×10−4.
Water (60 ◦F),TableA.5:vA=1.2×10−5.
SOLUTION
Match Reynolds number
The correct choice is c)
51
8.44: PROBLEM DEFINITION
Situation:
Crude oil flows in a smooth pipe.
dp=47in,dm=4in.
Vp=2ft/s.
Find:
Mean velocity of water in model to insure dynamic similarity.
Properties:
Crude Oil: ρp=1.75 slug/ft3,μp=4×10−4lbf s/ft2.
Water (60 ◦F),TableA.5:ρm=1.94 slug/ft3,μm=2.36 ×10−5lbf s/ft2.
SOLUTION
Match Reynolds number
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8.45: PROBLEM DEFINITION
Situation:
A student team is designing a radio-controlled blimp.
Dragforceischaracterizedwithacoefficient of drag:.
CD=2 FD
ρV 2Ap
=0.3
V=800mm/s,D=0.475 m.
Ap=πD2/4.
Find:
a.) Reynolds number.
b.) Force of drag (N).
c.) Power in watts (W).
Sketch:
V
Assumptions:
Assume the blimp cross section is round.
Properties:
Air (20 ◦C), Table A.3: ρ=1.2kg/m3,μ=18.1×10−6N·s/m2.
PLAN
Find the Reynolds number by direct calculation. Find the drag force using the
definition of CD.Find power (P)by using the product of force and speed: P=FDragV.
SOLUTION
Reynolds number
Projected area
Drag force
Power
REVIEW
1. The drag force is about 1/50th of a Newton, which is about 1/200th of a lbf.
2. The power is about 16 milliwatts. The supplied power would need to be higher
to account for factors such as propeller efficiency and motor efficiency.
54
8.46: PROBLEM DEFINITION
Situation:
Flow in a conduit (on earth) to be used to characterize a prototype that will be
build on the moon.
Find:
Kinematic viscosity of fluid for model on earth.
Properties:
Fluid: v=0.5×10−5m2/s.
PLAN
Dynamic similarity based on Reynolds number and Froude number.
SOLUTION
Match Froude number
Match Reynolds number
55
8.47: PROBLEM DEFINITION
Situation:
A scale model of a drying tower is to be tested with water.
Vp=12m/s.
1
15 scale model.
Find:
Entry velocity of the model for similitude.
Properties:
Air: vp=4×10−5m2/s.
Water: vm=10
−6m2/s.
PLAN
Similitude based on matching Reynolds numbers.
SOLUTION
Match Reynolds number
56
8.48: PROBLEM DEFINITION
Situation:
A scale model is of a discharge meter for oil is tested using water.
1
9scale model.
Vm=1.6m/s,∆pm=3kPa.
Find:
Velocity for the prototype for dynamic similitude (m/s).
Pressure difference for the prototype (kPa).
Properties:
Oil (20 ◦C):vp=10
−5m2/s,ρp=860kg/m.
Water: vm=10
−6m2/s,ρm= 998 kg/m.
PLAN
Dynamic similarity based on Reynolds number and pressure coefficients.
SOLUTION
Match Reynolds number
Match pressure coefficients
57
8.49: PROBLEM DEFINITION
Situation:
Water flowing through a rough pipe is modeled with air.
D=10cm,V=1.5m/s.
ppipe =150kPa,∆pa=780Pa.
Find:
Air velocity to achieve dynamic similarity (m/s).
Pressure difference for the water flow (kPa).
Properties:
Water (10 ◦C),TableA.5:μw=1.31 ×10−3Ns/m2,ρw=1000kg/m.
Air (20 ◦C), Table A.5: μw=1.81 ×10−5Ns/m2,ρa=1.2kg/m.
PLAN
Dynamic similitude based on Reynolds number and pressure coefficients.
SOLUTION
Match Reynolds number
Then
Match pressure coefficients
58
59
8.50: PROBLEM DEFINITION
Situation:
An “acoustic” minesweeper (a noisemaker) will be studied by using a scale model
in a water tunnel.
1
5scale model.
Vprot. =5m/s.
Find:
Velocity to use in the water tunnel (m/s).
Drag force on the prototype (N).
Assumptions:
Sea water under the same conditions is used in both tests.
PLAN
Dynamic similarity based on matching Reynolds number and drag force with force
coefficient.
SOLUTION
Match Reynolds number
Match force coefficients
60