Resistance + Propulsion: Scaling laws.

...

V

F

P = F⋅V

10 km/h

2.78 m/s

2.85 N

8 W

20 km/h

5.56 m/s

11.39 N

63 W

30 km/h

8.3 m/s

25.63 N

212 W

40 km/h

11.11 m/s

45.60 N

506 W

100 km/h

27.78 m/s

284.72 N

7910 W

200 km/h

55.60 m/s

1138.9 N

63272 W

- Geometric similarity gives:

[pic 18]

Then the model length is:

[pic 19] m

- The ship speed is:

[pic 20] m/s

The Froude number of the ship is:

[pic 21]

Model and full-scale ship have the same Froude number, thus:

[pic 22] m/s

- First, we compute the Reynolds numbers of ship and model:

[pic 23]

[pic 24]

The frictional resistance coefficients are:

[pic 25]

[pic 26]

The total resistance coefficient for the ship is:

[pic 27]

We assume the same wave resistance coefficient for full scale and model scale. Thus:

[pic 28]

This yields a resistance (with [pic 29]):

[pic 30] kN

- The necessary power to tow the model:

[pic 31]W

For the car we have:

[pic 32]10.8 m/s

= 38.9 km/h

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Resistance + Propulsion: Sailing yacht test

A sailing yacht has been tested. The full-scale dimensions are Lpp = 9.00 m, S = 24.0 m2, ∇ = 5.150 m3.

The yacht will operate in sea water of ρ = 1025 kg/m3, ν = 1.19⋅10−6 m2/s.

The model was tested with scale λ = 7.5 in fresh water with ρ = 1000 kg/m3, ν = 1.145⋅10−6 m2/s.

The experiments yield for the model:

Vm

[m/s]

0.5

0.6

0.75

0.85

1.0

1.1

1.2

RT,m

[N]

0.402

0.564

0.867

1.114

1.584

2.054

2.751

- Determine the form factor following Hughes-Prohaska.

- Determine the form factor following ITTC'78. For simplification assume the exponent n for Fn to be 4 and determine just α and k in regression analysis.

Solution

The model data are: Lm = Ls/λ = 9/7.5 = 1.2 m

Sm = Ss/λ2 = 24/7.52 = 0.427 m2

We consider only the lowest 4 speeds as for the others a considerable wave influence is to be expected.

- We compute the total resistance coefficient of the model, using [pic 33], [pic 34],

[pic 35]

Vm

RT,m

Fn

Rn⋅10−5

cF0 ⋅103

cT ⋅103

cT/cF0

Fn4/cF0

0.50 m/s

0.402

0.146

5.24

4.843

7.532

1.555

0.091

0.60 m/s

0.564

0.175

6.29

4.643

7.338

1.580

0.197

0.75 m/s

0.867