8.65: PROBLEM DEFINITION
Situation:
A model of spillway modeled in laboratory.
1
40 scale model.
Vm=3.2ft/s,Qm=3.53 ft3/s.
Find:
Prototype velocity (ft/s) .
Prototype discharge (ft/s) .
PLAN
UseFroudenumberscaling.
SOLUTION
Match Froude number
Multiply both sides of Eq. (1) by Ap/Am=(Lp/Lm)2
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8.66: PROBLEM DEFINITION
Situation:
Flow around a bridge pier is studied using a scale model.
1
12 scale model.
Vm=0.9m/s,Lm=2.5cm.
Find:
(a) Velocity.
(b)Waveheightinprototype.
PLAN
UseFroudenumbermatching.
SOLUTION
Match Froude numbers
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8.67: PROBLEM DEFINITION
Situation:
A scale model of a spillway is tested.
1
250 scale model.
Qm=0.1m
3/s,tm=1min.
Find:
Time for a particle to move along a corresponding path in the prototype (min).
Prototype discharge (m
3/s).
PLAN
UseFroudenumbermatching.
SOLUTION
Match Froude numbers
Then
Also
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8.68: PROBLEM DEFINITION
Situation:
Vp=3.6m/s,tp=12.5h.
Find:
Velocityandperiodinthemodel.
PLAN
UseFroudenumbermatching.
SOLUTION
Match Froude number
because gm=gp.Then
The time varies as t∼L/V so
The model period is
80
8.69: PROBLEM DEFINITION
Situation:
A scale sea wall tests maximum wave force.
1
36 scale model.
Fm=80N.
Find:
Force on the wall for the prototype (N).
Assumptions:
Fresh water (model) and seawater (prototype).
Properties:
Sea Water (10 ◦C), Table A.4: ρp=1026kg/m3.
Water (10 ◦C),TableA.5:ρm=1000kg/m3.
PLAN
Dynamic similarity based on force coefficient and Froude number.
SOLUTION
Match Froude numbers
Match force coefficients.
Using result from Froude number scaling
Force on wall
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8.70: PROBLEM DEFINITION
Situation:
A scale model of a spillway is built to test flow conditions.
1
80 scale model.
Qp=800m
3/s,F
m=51N.
Find:
Water discharge in model for dynamic similarity (m
3/s).
Force on the prototype (kN).
PLAN
Dynamic similitude based on matching force coefficients and Froude numbers.
SOLUTION
Match Froude number
Using velocity ratio from Froude number equality
Discharge for model
Match force coefficients,
Using Froude number matching
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Force on prototyp e
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8.71: PROBLEM DEFINITION
Situation:
A scale model of a dam will be constructed in a laboratory.
Lp=1200m,Wp= 300 m.
Qp=5000m
3/s,Qm=0.9m
3/s.
Am=50m×20 m = 1000 m2.
Find:
The largest feasible scale ratio.
SOLUTION
Check the scale ratio as dictated by Qm/Qp
Then with this scale ratio
Therefore, model will fit into the available space, so use
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8.72: PROBLEM DEFINITION
Situation:
A scale model of a ship is tested in a towing tank.
Lm=4ft,Lp=100ft.
Vm=5ft/s.
Find:
Speed for the prototype that corresponds to the model test.
PLAN
Dynamic similarity based on Froude number.
SOLUTION
Match Froude number
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8.73: PROBLEM DEFINITION
Situation:
A scale model of a ship is tested in tank.
1
25 scale model.
Vm=5ft/s,Fm=2lbf.
Find:
Velocity of the prototype (ft/s).
Wave resistance of the prototype (lbf).
PLAN
Dynamic similarity based on Froude number.
SOLUTION
Match the Froude number
Equating the force coefficients
With Froude number matching
Force on prototyp e
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8.74: PROBLEM DEFINITION
Situation:
A scale model of a building is being tested.
1
20 scale model.
Fm=200N,Vm=20m/s.
Vp=40m/s.
Find:
Drag on the prototype building.
Assumptions:
Reynolds sufficiently high that dynamic similitude is satisfied.
SOLUTION
Match force coefficients
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8.75: PROBLEM DEFINITION
Situation:
A scale model of a building is being tested in a wind tunnel.
1
550 scale model.
Vm=20m/s,Vp= 200 km/h.
Fm=20N,Cpm=1,−2.7,−0.8.
Find:
Pressure values on the prototype.
•windward wall
•side wall
•leeward wall
Lateral force on the prototype in a 150 km/hr wind.
Assumptions:
Assume Reynolds number sufficiently large that dynamic similitude is satisfied.
PLAN
Match the pressure coefficients and the force coefficient.
SOLUTION
Match pressure coefficients
Velocity for prototype is
or
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Extremes of pressure are therefore:
Match force coefficient
Lateral Force:
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8.76: PROBLEM DEFINITION
Situation:
Drag force is measured in a water tunnel and a wind tunnel on three different
models.
Model sizes: 5cm,8cm,15 cm.
Find:
Find the relevant π-groups.
Write a computer program and reduce the given data.
Plot the data using the relevant π-groups.
SOLUTION
The drag force on an object is quantified by the force coefficient. The only significant
π-group is Reynolds number so
2ρV 2L2=fµVL
The data are reduced and plotted below.
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8.77: PROBLEM DEFINITION
Situation:
Pressure drop is measured in a pipe with water and oil.
Find:
Find the relevant π-groups.
Write a computer program and reduce the given data.
Plot the data using the relevant π-groups.
SOLUTION
Performing a dimensional analysis on the equation for pressure drop shows which is
0.016
0.018
0.020
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