8.51: PROBLEM DEFINITION
Situation:
Forces due to wind on a building are to be modeled by using a scale model in wind
tunnel.
1
500 scale model.
Vp=47ft/s,Vm=300ft/s.
Fm=50lbf.
Find:
Density needed for the air in the wind tunnel ¡slug/ft3¢.
Force on the full-scale building (prototype) (lbf).
Properties:
Air, ρp=0.0024 slug/ft3.
PLAN
Equate the Reynolds number for dynamic similitude and equate force coefficients for
force ratio.
SOLUTION
Reynolds number
Equating force coefficients
Force on prototype
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8.52: PROBLEM DEFINITION
Situation:
Performanceofalargevalveinpetroleumpipeline is characterized by recording
data on a scale model in water.
D=60cm,Q=0.5m
3/s.
1
3scale model,C
p=1.07.
Find:
Flow rate to be used in the model (laboratory) test (m
3/s).
The pressure coefficient for the prototype.
Properties:
Petroleum: S=0.82,μ=3×10−3Ns/m2.
Water: μ=10
−3Ns/m2.
PLAN
Equate flow rates by Reynolds number matching With dynamic similitude, the pres-
sure coefficients will be the same for model and prototype.
SOLUTION
Matching Reynolds number
Multiply both sides of above equation by Am/Ap=(Dm/Dp)2
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8.53: PROBLEM DEFINITION
Situation:
The moment acting on the rudder of submarine in sea water is studied using a 1/50
scale model in a water tunnel.
1
40 scale model.
Vm=6.6m/s,Mm=2Nm.
Find:
Speed of the prototype that corresponds to the speed in the water tunnel (m/s).
Moment that corresponds to the data from the model (Nm).
Properties:
Sea Water (10 ◦C) ,Table A.4: vp=1.4×10−6m2/s,ρp=1026slug/ft3.
Water (10 ◦C),TableA.5:vm=1.31 ×10−6m2/s,ρm= 1000 slug/ft3.
PLAN
Match the Reynolds number for dynamic similitude and extend force coefficient to
model the moment.
SOLUTION
Match Reynolds numbers
Inthesamewayasaforcecoefficient was defined there is a corresponding moment
64
=46.875
Moment on the submarine
Mp=93.75 N ·m
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8.54: PROBLEM DEFINITION
Situation:
A scale model hydrofoil is tested in a water tunnel.
Vm=15m/s,Fm=25kN.
Find:
Lift force on the prototype.
PLAN
Match the Reynolds number to obtain model-prototype velocity ratio and match force
coefficients for force ratio.
SOLUTION
Match Reynolds numbers
Match force coefficients
Using velocity ratio from Reynolds number matching
When the same fluid is used for the model and prototype
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8.55: PROBLEM DEFINITION
Situation:
A scale model of an automobile will be tested in a pressurized wind tunnel.
1
10 scale model.
Vp=100km/h,c= 345 m/s.
Find:
Pressure in tunnel test section for same Mach number and Reynolds numbers.
PLAN
Match the Mach number and Reynolds number and solve for pressure.
SOLUTION
MatchMachnumber
Match Reynolds number
Thedynamicviscosityisthesameformodelandprototypewithsametemperature
so ρm
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8.56: PROBLEM DEFINITION
Situation:
A scale model of an automobile will be tested in a wind tunnel at atmospheric
pressure.
1
8scale model.
Vm=80km/h,cm=345m/s.
Find:
Speed of air in the wind tunnel to match the Reynolds number of the prototype.
Determine if Mach number effects would be important in the wind tunnel.
PLAN
Match the Reynolds number and find velocity for model. Evaluate Mach number for
compressibility effects.
SOLUTION
Match Reynolds number
Mach number
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8.57: PROBLEM DEFINITION
Situation:
Water droplets are in an air stream.
We/√Re = 0.5,Vair =12m/s.
Find:
Droplet diameter for break up (mm).
Properties:
pair =1.01 kPa,σ=0.041 N/m.
Air (20 ◦C), Table A.3: ρ=1.2kg/m3,μ=1.81 ×10−5Ns/m2.
PLAN
Apply the We/√Re = 0.5criteria, combined with the equations for Weber number
and Reynolds number.
SOLUTION
WebernumberandReynoldsnumber
σ=0.5
Solve for diameter
Calculations
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8.58: PROBLEM DEFINITION
Situation:
Breakupofajetofwaterintodropletsinairstream.
V=30m/s,We=6.0.
Find:
Estimated diameter of droplets.
Properties:
Table A.3: ρ=1.20 kg/m3.
Table A.5: σ=0.073 N/m.
PLAN
Set the Weber number equal to 6 and solve for droplet size.
SOLUTION
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8.59: PROBLEM DEFINITION
Situation:
Satisfying both Froude number and Reynolds number criteria for model testing.
Find:
Relationship between kinematic viscosity ratio and scale ratio.
PLAN
Match Froude number and Reynolds numbers and solve for ratios.
SOLUTION
Match Froude numbers
Match Reynolds numbers
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8.60: PROBLEM DEFINITION
Situation:
The spillway of a dam is simulated using a scale model.
1
20 scale model.
tm=2s,Lm=8cm.
Find:
Wave height (prototype).
Wave period (prototype).
PLAN
Dynamic similarity based on Froude number.
SOLUTION
Since it is a scale model, the wave height will scale directly
Match Froude number
The time will scale as
Thus
72
8.61: PROBLEM DEFINITION
Situation:
A prototype of a dam spillway is represented with a scale model.
1
25 scale model.
V=2.5m/s,Q=0.10 m3/s.
Find:
Velocity for prototype (m/s).
Discharge for prototype (m
3/s).
PLAN
Dynamic similarity based on Froude number.
SOLUTION
Match Froude number
The discharge ratio is
Discharge for prototype
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8.62: PROBLEM DEFINITION
Situation:
A seaplane model has a scale to simulate take-off.
1
6scale model.
Vp=117km/h.
Find:
Model speed to simulate a takeoffconditions (m/s).
PLAN
UseFroudenumberscaling.
SOLUTION
Match Froude number
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8.63: PROBLEM DEFINITION
Situation:
A model spillway has a scale.
1
36 scale model.
Q=3000m
3/s.
Find:
Velo city ratio.
Discharge ratio.
Model discharge (m
3/s) .
PLAN
Dynamic similarity based on Froude number.
SOLUTION
Match Froude number
or for this case
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8.64: PROBLEM DEFINITION
Situation:
Flow in a river is to be studied using a scale model.
1
64 scale model.
dp=20ft,Vp=15ft/s.
Find:
Model velocity (ft/s) .
Model depth (ft/s) .
PLAN
UseFroudenumberscaling.
SOLUTION
Match Froude number
Geometric similitude
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