Viscous dissipation of monochromatic waves in still water of finite constant depth in a channel of finite width

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Viscous dissipation of monochromatic waves in still water of finite

constant depth in a channel of finite width

by

I. K. Suastika

January 1999

Delft University ofTechnology

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1. Introduction

Energy dissipation ofwater waves propagating in a smooth-walled channel offinite water depth and finite width is mainly due to viscous effects. Providing that the wave length is sufficiently large compared to the water depth and a finite channel width, wave energy is principaIly dissipated in the boundary layers on the bed and on the walIs. Hunt [1952] studied the attenuation of periodic water waves travelling in shallow water in a channel of finite width. He derived theoretically a formulation for a damping modulus

K

,

representing the viscous attenuation as

A

=

Aoexp(-

Xx), where

A

is local wave amplitude,

x

is position along the wave travel and the subscript

0

refers to a reference position. The damping modulus K is expressed as

where

h

is water depth,

b

is channel width,

k

is wave number, o is angular frequency and v is kinematic viscosity of water.

In the present study we investigate experimentally the wave attenuation of monochromatic waves travelling in still water of finite constant depth in a channel of finite width. In the

experiments

,

we measured the water surface elevation at a number of locations along the

flume. First, Fourier analysis is performed to the surface elevation data, in which the surface elevation is represented as a sum of many harmonie components.

This

analysis is performed to the surface elevation data obtained at each station, resulting in discrete values ofwave

amplitudes and phase shifts as function of wave travel distance. Next, the values ofwave amplitudes and phase shifts so obtained are modelIed by using a wave attenuation model in order to estimate the damping modulus

K

.

Finally,the damping modulus

Kso

determined is compared with that calculated using the formulation ofHunt.

2. Experimental arrangement and procedures

A series of experiments has been conducted at the Laboratory ofFluid Mechanics, Delft University of Technology, The Netherlands. The flume is 40.0 m long, 0.80 m wide and 1.0 m deep. Figure 1 shows the longitudinal section ofthe flume. In all the experiments, the water depth was kept constant at 0.80 m. Waves were generated at the right end ofthe flume, as viewed in Figure 1,by oscillating translations ofthe wave paddie. The motion ofthe wave paddie was controIled electronically by signals sent from a personal computer. At the left end ofthe flume an arti:ficial wooden beach was mounted to absorb the incoming wave energy.

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flume, as shown in Figure 1.

In this series of experiments, six monochromatic waves were tested: three monochromatic waves ofperiods 1.20, 1.45 and 1.70 s,each with initial amplitude ofabout 0.06 m, and three monochromatic waves of the same periods as above, but each with initial amplitude of about 0.09m.

Two measurement methods have been used in the experiments. The fust one is asynchronous measurement of surface elevation with two gauges separated longitudinally by about a quarter wave length, allowing to determine the incoming and reflected waves at each station. This measurement procedure was repeated at a nwnber of stations. In the following, we

will

refer this method to as 'Auke' method. The second method is asynchronous measurement of surface elevation at six stations, using one probe at each station. This method does not allow to determine the incoming and reflected waves separately from the measurement at each station.

The sampling interval ofthe Auke data acquisition is 0.05 s and that ofthe six synchronous measurement method is 0.02 s. The data were stored in a disk for further analysis.

3. Data analysis and results

First, we analyse the data of Auke - measurement, using the software package Auke - pc which bas been developed by Delft Hydraulics, The Netherlands. First, Fourier analysis was performed to the surface elevation data, including the 0tb, 1st, 2ndand 3rdharmonie

components. Next, the harmonie components ofthe surface elevation measured synchronously at the two positions were used in the determination ofthe incoming and reflected waves. An example of result of Auke analysis performed to the surface elevation data of station 1 for the case ofwaves ofperiod T=1.20 s and mean amplitude along the flwne ofabout 0.06 m gives an incoming wave amplitude of

0 .

0565

mand a reflected wave amplitude of

0 .

0010

m. In this

A

case, the reflection coefficient, defined as

Cr

=(~)(lOO%), whereä,

andx,

are incoming and reflected wave amplitude, respectively, is about~.73 per cent.

The same analysis bas been performed to surface elevation data of stations 1 to 6. Having the values of incoming wave amplitude at stations 1 to 6, a damping modulus K is estimated by least-square fitting an exponential attenuation model expressed as

At=At

;

oexp(

-Kx). A typical

semi-log plot ofnormalised incoming wave amplitude as function ofwave travel distance for the case ofmonochromatic waves ofperiod T

=

1.20 s and mean amplitude along the flwne of about 0.06 mis shown in Figure 2. The damping modulus

K

is equal to the absolute value of the slope ofthe line given by Ln(A(x)/AO>, which is, in this case, equal to 0.0026 mol.

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0.005 0 -0.005 3 6 9 12 15

•

-0.01

•

Ö -0.015

•

~ -0.02

•

I

·

_Ln(A/Ao)

J

$ _-0_.₀₂₅

•

_• _Predicted _Ln(A/Ao) c

•

....I -0.03

_•

-0.035 -0.04

•

-0.045

•

-0.05 x [m]

Figure 2. A typical plot of incoming wave amplitude as function of wave travel distance,for the

case T = 1.20 s and mean wave amplitude over the tlume length of about 0.06 m. The wave

amplitude is normalized by that measured at station 1. Tbe positions x=0,3,...15 correspond to stations 1,2, ...6,respectively.

The results ofthe determination ofthe damping modulus as described above for the six monochromatic waves having been tested are presented in Table I. We see that the damping modulus determined from Auke method agrees fairly weIl with the theoretical value of Hunt. In almost all the cases, the experimental value is slightly larger than the theoretical one.

Next, we analyse the data of simultaneous measurement of surface elevation at six stations,

using one probe at each station. As in Auke -

analysis,

Fourier analysis was fust performed to the surface elevation data, resulting in harmonie components of the surface elevation at stations 1 to 6. In modelling the wave attenuation, only the fundamental component is included. Since this measurement method does not alIow to separate the incoming and reflected waves from a single measurement at each station, the wave attenuation is modelIed as the sum of damped incoming and reflected waves.

This

is represented as a superposition of exponentially damped incoming and reflected waves:

TJ(x,t)=Aj;oexp(-Kx)cos(kx-ot)+Ar;oexp(Kx)cos(kx+ot+cI»

where "

is

water surface elevation, t is time and

cl> is

phase shift between the incoming and reflected waves. The damping modulus

K

is estimated by least square fitting the model to the data. Some problems arising from this method have still not been solved yet, and the results presented here are not reliable.

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T [sJ Mean Hunt [m-I] Auke- Simultaneous Ar(O)/Ai(O) amplitude [m] method [mol] measurement at calculated from

six positions simultaneous [mol] _measuremen_t _[~] L20 0_06 0.0021 0.0026 0.0015 2.2 1.45 0.06 0.0016 0.0014

_-

-1.70 0.06 0.0013 0.0015 0.0010 1.1 1.20 0.09 0.0021 0.0021 0.0020 0.75 1.45 0.09 0.0016 0.0022

-

-1.70 0.09 0.0013 0.0017 0.0011 1.1

Table 1:Estimation of damping modulus by Auke - method and simultaneous measurement at six

location along the flume.

Note: Both time series ofT

=

Viscous dissipation of monochromatic waves in still water of finite constant depth in a channel of finite width