lab b: kinematics of surface waves, and added resistance...

Lab B:Kinematics of Surface Waves, and

Added Resistance of Ship in Waves

13.021 Pre-Lab HandoutFall 2004

Kinematics of Surface Waves-

Background

Definitions•Coordinate system: x, z• Gravity: -g•Water depth: H•Free surface elevation: η

Potential Flow

g

k̂

• A is wave amplitude• λ is the wavelength, is the wave number (λ=2π/k)• ω is the wave frequency, T is the wave period (ω=2π/T)• Linear wave theory valid for h/λ > 1/7

Linear free-surface gravity waves are defined by their amplitude A, wavelength λ = 2π/k, and frequency ω.

x

z

A

kπλ 2=

)cos(),( txkAtx ωη −=

k

2 tanh( )gk kHω =Approximations to dispersion relation

As (shallow water/ long waves)

As (deep water/ short waves)

0kH→ tanh( )kH kH→ 2 2gk Hω∴ ≅

kH→∞ tanh( ) 1kH → 2 gkω∴ ≅

For linear free-surface gravity waves ω and k, are uniquely related, given H by theDISPERSION RELATION

Hk

kg

2ω

1

linea

r

π

Phase and Group Velocities

pV kω=

gdVdkω=

Phase Velocity = Velocity of wave crests (what the eye sees)

In general: Vp ≥ Vg. In particular: shallow water Vp = Vg, deep water Vp/2 = Vg

Group Velocity = Energy propagation (property of each wave)

∆l = (t2 – t1) * Vp [For (t2 – t1) =T λ = T * Vp]

t1 t2x

t1 t2 x

gV

pV)tt( 12 −⋅pV

)tt( 12 −⋅gVWave packet: waves appear & disappear

Particle OrbitsUnder the waves particles follow distinct orbits depending on whether the water is shallow, intermediate or deep. Water is considered deep when water depth is greater than one-half the wavelength of the wave.

Kinematics of Surface Waves-

Lab

1. Calibrate wave probes #1 and #2 ONLY to obtain voltage – wave amplitude relationship.

2. Measure wave depth and distance between probes.

3. Input wave frequency and amplitude to wave maker.

4. Observe waves from tow tank window. Take direct measurement of wave amplitude.

5. Create wave packets by inputting wave frequency and amplitude to wave maker for short time.

6. Use data acquisition to record wave probes measurements of free surface elevation as a function of time. Repeat steps 5-6 for multiple frequencies and amplitudes.

7. Repeat step 3 and observe surface contraction and dilation with provided glitter.

8. Observe particle orbits below surface with glitter, if possible.

Wave paddle

Wave train

BeachVp,Vg

WaveProbe

#2

WaveProbe

#1

WaveProbe

#3

H

Lab Setup

Outline of Procedures

Measurements in Lab• H = water depth (~ 1m)

• Lij = Distance between probes i and j

• f = input signal

• Rate at which data is recorded

• A = wave amplitude from wave probes & direct measurement from tow tank window

• η(x,t) = free surface elevation from wave probes

Post Lab Analysis• ω = 2π f, verify from data

• λ = Wave length (from 2 adjacent probes simultaneously and ω or from Vp and ω)

• Vp = Phase velocity (from 2 adjacent probes and ω)

• Vg= Group velocity (from 2 distant probes and ω)

Report• Introduction: Scope of lab

• Describe and provide calibration of the wave probes

• Detailed but concise description of the procedure followed to obtain measurements

• Detailed but concise description of the post-lab analysis

• Relationship between wave length and frequency (dimensionless plot)

• Relationship between phase velocity and frequency (dimensionless plot)

• Relationship between group velocity and frequency (dimensionless plot)

• Discussion on dependence of above results on wave amplitude

• Discussion of observed particle orbits

• Conclusions

Added Resistance of Ship in Waves -

Backgroundfrom Faltinsen, 1990

Added Resistance of Ship in Waves -

Lab

1. Tow the vessel at constant velocity Uo. Use data acquisition to record TOTAL drag.

2. Input wave frequency fo and amplitude to wave maker.

3. Tow the vessel at constant velocity Uo. Use data acquisition to record drag and wave measurements from wave probe #1

4. Repeat steps 1-2 for multiple amplitudes, but keeping Uo, fo constant.

Outline of Procedures

Post Lab Analysis• D = Total drag from step 1

• A = wave amplitude, from measurements from wave probe #1 and calibration chart.

• Raw = Added resistance (2 zero measurements) from D and drag from 4.

Report• Introduction: Scope of lab

• Detailed but concise description of the procedure followed to obtain measurements

• Detailed but concise description of the post-lab analysis

• Relationship between added resistance and wave amplitude (dimensionless plot)

• Discussion on dependence of added resistance on wave length

•Conclusions