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Measurement of water particle velocities in waves
TECHNICAL NOTE Measurement of water particle velocities in waves B R E N D A N P. M C N A M E E
Research Fellow, Department of Civil Engineering, University of Melbourne BRUCE B. SHARP
Reader, Department of Civil Engineering, University of Melbourne LEONARD K. STEVENS
Professor, Department of Ovil Engineering, University of Melbourne Water surface profiles and horizontal and vertical water particle velocity components have been measured to investigate the properties of intermediate depth waves generated in the laboratory. The data has been compared with linear wave theory. It was found that linear theory predicted the attenuation of velocity field with depth successfully and that it overestimates both components of velocity slightly.
The water depth was kept at about 3 ft 6in. (1.06 m) for these tests. The wave periods ranged between 1.1 and 2.5 s. The wavelengths were not measured but determined using Airy theory, from the measured wave period, for the purpose of comparing the water particle velocity components. For the range of waves studied, previous work 2 has found that the linear theory estimation of wavelength has agreed quite well with the measured wavelengths. In these present tests, the wave steepness: H/L, varied fron~ 0.008 to 0.049, and the depth to wavelength ratio: d/L, varied between 0.15 and 0.50; hence, indicating the intermediate depth range of wave conditions.
INTRODUCTION The design of offshore structures relies upon an understanding of the water particle kinematics of ocean waves. The Morison approach has been used extensively for the calculation of hydrodynamic loading on cylindrical structures in the Gulf of Mexico. s Generally, the water particle kinematics are determined from fifth order Stokes theory. However, the reported values of the empirical coefficients of inertia and drag: CM and Co, show large scatter, which Dean 3 attributes in part to the fact that water particle velocities and accelerations have not been measured, but determined from the measured surface profile, using an assumed wave theory. Linear diffraction theory has been used for the design of the large, monolithic structures favoured in the North Sea. The gross loads and local pressure distribution predicted using linear theory, have been validated for large vertical cylinders by Chakrabarti and Tam. 1 The measurement of water particle velocities is very difficult because of the small magnitude of the velocities encountered. 6 Consequently, large scatter in the data may occur. The consistency of the data can to some degree be determined by the observed variation of the velocity field with depth. A continuous profile would at least indicate that large fluctuations in measured values will probably not occur. However, there still remains the task of obtaining a successful calibration of the instrument used to measure the velocity components. Previous tests 2'7 have compared wave tank d~ta with the predictions of available wave theories. The scatter of the test data is large in some instances and further experiments are required to correlate the data with the analytical theories. Measurements of water surface profiles and water particle kinematics were therefore made to further investigate the properties of the intermediate, regular waves generated in the laboratory. The comparison with the analytical theories is restricted to Airy theory. 0309-1708/83/010049-05 $2.00 © 1983 CML Publications
EQUIPMENT A Salter duck-type wave generator was used for the experiments. The experimental layout is shown in Fig. 1. The tank is about 200 ft (63 m) long, 6 ft (2.03 m) wide and an operating depth of about 3 ft 6 in. (1.06 m) was used. A capacitance depth probe was used to measure the water surface elevation. This was placed in line with the velocity probe across the channel. The velocity probe was a two-component suspension wire device developed by the second writer, s It was used to measure the vertical and
wave probe
.J
¢q
velc'i o'-typrobe 63m. Figure 1. Experimental layout Applied Ocean Research, 1983, Vol. 5, No. 1 49
Technical No te horizontal water particle velocities. The probe, shown in Fig. 2, basically consists of a strain gauged cantilever with a length of thread suspended between the tip of the cantilever and a second support. The hydrodynamic force exerted by the flow of water past the thread, exerts a force on the cantilever, which is related to the velocity of flow. The
Sk strain gauged A /f~-----~upport frame cantli ~ _t h~rre/ a~ d ~ ~
~ _ _ , tensioner
Figure 2.
Velocity probe
Output500
/
(mV)
/
.
/
/-
/.
,/
/
/
// ./.
oo~
2E
Velocity (cm/s)
i
Figure 3.
Velocity probe calibration
threads were about 2 in. (50 mm) long and were horizontal. They ran across the tank to take advantage of the twodimensionality of the flow. A typical calibration of the device, obtained by towing the instrument in the channel, is presented in Fig. 3. The measurements were conducted about 130 ft (39 m) from the generator. The velocity probe was clamped at different elevations during continuous wave generation, in order to measure the velocity field profile with depth. The experiments were recorded on-line to an ID 70 minicomputer. The data was recorded in buffer lots of 2048 points at a sampling rate of 40 per second.
EXPERIMENTAL RESULTS A brief investigation of the surface profiles demonstrates that the waves generated in the laboratory closely approximate a sinusoidal wave form. The surface profiles of several waves have been normalized with respect to both amplitude and period. This data is presented in Fig. 4, along with the sinusoidal variation assumed by linear theory. The experimental data is shown with equal crest and trough amplitude about the still water line. It was observed as generally reported 9 for finite amplitude waves, that the mean water level did not necessarily coincide with the still water depth. The crest height above still water level did not vary by more than about 5% from half of the total wave height. It can be seen also, that the wave 'half' periods are not symmetrical for the different waves. In Fig. 5, sample traces of wave height, H, horizontal (u) and vertical (w) water particle velocity are shown. The phase relationship between the wave height and water particle velocity components is clearly illustrated. The horizontal velocity component is in phase with the wave profile and the vertical component is about 90 ° out of phase. The spectral variation of the traces shown in Fig. 5 is presented in Fig. 6. The monotonic nature of the waves
4-a c-
"F~ -I(~ t~
N
% E £. o z
Figure 4.
50
d=1.067m H=O.083m T=1.602s
d=l.060m H:0.068m T:1.482s
Normalized Comparison of wave profiles with Airy theory
Applied Ocean Research, 1983, Vol. 5, No. 1
Wave Period
Technical Note corresponding to trough level of the waves because of the nature of the instrument. The attenuation of the velocity field is most pronounced near the water surface. Although there is some scatter in the data, which may be attributed in part to the velocity device itself and also to the variability in the wave profile as shown in Fig. 4; two main observations can be made. Firstly, linear theory predicts the attenuation of the velocity field with depth for the intermediate range of waves very well. This is well illustrated by the profiles shown in Fig. 7. Secondly, linear theory tends to overestimate the magnitude of both velocity components at all depths. To compare the present data obtained for regular waves generated in the laboratory with published field data, 4 the data was replotted in terms of the spectral transfer functions between wave height and the water particle velocity components:
50
::
E
v
-50
I0
.~.
:=
0'
=
S.(f) - -
=
-
(1)
s (f) r (f) sw(f) s (f) sw(f) 2
-ic
lfi
(2)
-
Twu(f) - - -
(3)
s.(;)
lO-I
-10 Figure 5. velocities
Traces o f
wave height and water particle
generated is evident. Energy at the higher harmonics is significantly lower than at the dominant wave frequency. The water particle velocity components were recorded at several elevations to determine the variation of the velocity field with depth. In order to determine the maximum amplitude of the velocity components, it was necessary to consider that equal maximum positive and negative velocities occur during the wave cycle. This is a result of the calibration procedure of the velocity probe. To examine the fluctuations in amplitudes would require more accurate knowledge of the zero output condition of the velocity probe with time than was possible in these tests. Some drift occurred; however, this is more pronounced under uniform flow conditions than oscillatory flow conditions. Sample velocity profiles are shown in Fig. 7. The continuity of the measured profiles with depth is very good. Those elevations at which readings were repeated, show the consistency of the recording instrument. Some scatter is expected for small values of y/d, where y is the elevation above the bottom. This is caused by the very low magnitude of the velocities, especially the vertical velocity component. In order to correlate all the data, the non-dimensionalised amplitudes of horizontal and vertical water particle velocity components were plotted against the parameter d/L in Fig. 8. Lines of best fit were assumed for the measured velocity profiles and the velocity amplitudes estimated at several elevations. The estimation of the still water depth velocity components is more likely to be in error as no measurements could be performed above that elevation
v
tm
10 -6
0.05
12.5
I Frequency
f
(Hz)
lO-I
t~ =
g N 'F_ o
i0 -s 0.05
1 Frequency
12. f
(Hz)
10-i
i0-6 I
0.05 Frequency
Figure 6.
12.5 f
(Hz)
Spectral variation o f traces in Figure 5
Applied Ocean Research, 1983, 11ol. 5. No. 1
51
Technical Note where Su, Sw and Sn are the spectral density of the horizontal and vertical velocity components and the wave height, respectively. These values are plotted in Fig. 9, with the linear theory prediction• The data is plotted for middepth. The overestimation of the velocity field by Airy theory occurs at all depths and hence only one depth need be considered as representative. The data above and below the mid-depth positions allows a more accurate estimation of the velocity amplitudes by way of interpolation• From these plots, it is again clear that Airy theory tends to overestimate the magnitude of the velocity components• The attenuation of energy with frequency for the horizontal particle velocity is well predicted. The peak in energy of the vertical water particle velocity component is also well correlated, although the scatter of the data is more significant. The transfer function between the vertical and horizontal velocities is most interesting in that it tends to show that Airy theory in fact correctly predicts the relative magnitude of the two velocity components• This is despite the fact that each individual component is overestimated. Haritos 4 presented transfer functions for real sea conditions in deep water• The data was obtained using a similar velocity probe. His data showed that the energy was greater than predicted by Airy theory for both velocity components. The frequency at which the peak in energy for the
i
20
,
•
20
i0
Horizontal Velocity
30
u
(cm/s)
10
20
Vertical Velocity
Y/d
';d:l.O53m. ~T=2.169s. ~H=O.136m.
0 I'0 2'0 30 llorizontal Velocity u (cm/s)
0 10 20 30 Vertical Velocity w {cm/s)
// Ariy
/
/i i
i.
i
i
Y.~
I
g
z
0
0-1
0-6
d/k
I i
I
I
_
i
:
"g o % > _
÷
i + + 4_____i~-~__ 1÷ ~ ~ . 7Y~ . : o
_L
.
~ I'Oo -
•
- , ~
yl. J0.~o.
_
1
0
j o I d/k
06
Correlation of water particle velocities with Airy
theory
w (cm/s)
Airy
0
i
% C 0
Figure 8. 30
~
1.0 D
0'I
d=l.lOOm• H=O.I08m. T=1.440s.
i
"7. 0
c L~ g z /
[
5N
g
ry
i
%
> %
Y/d
'
,
~g >
i
1
i
'
/
vertical components occurred, was in close agreement with Airy theory. However, for both components, no attenuation of energy at higher frequencies was evident. The theoretical transfer function between the horizontal and vertical velocity components is equal to unity for deep water conditions• Haritos found that linear theory underestimated the transfer of energy at the dominant wave frequencies and overestimated it at higher frequencies. The author indicated that some of the above effects observed in the real sea state, may be caused by a wave grouping effect. Chakrabarti2 conducted velocity measurements at two elevations using an electromagnetic flow measuring device. For the range of conditions covered in the present tests, he also observed that the linear theory slightly overestimated the velocity field. CONCLUSIONS
i
/ / ..
0
10 Horizontal Velocity
Figure 7.
2'0 u
3o
(cm/s)
d=l.O67m. H=O.O93m. T=I.950s. io
3o
Sample profiles of water particle velocities versus
depth
52
2o
Vertical Velocity w (cm/s)
Applied Ocean Research, 1983. Vol. 5, No. 1
The water surface elevation and horizontal and vertical water particle velocity components have been measured for laboratory generated regular waves in the intermediate depth range. The waves have been shown to be monotonic and to approximate a sinusoidal wave profile. The correlation of the velocity data with linear theory, has shown in the region: 0.15 > d / L > 0 . 5 , that both the
Technical Note
Transfer
Function
T2un (f) 0
"rl ,-$ e~ e-
horizontal and vertical water particle velocity components are slightly overestimated by theory. Secondly, the attenuation of both velocity components with depth has been found to be satisfactorily described by linear theory. Finally, it was found that Airy theory tended to predict the correct relative magnitude of the two velocity components although overestimating the individual components.
"h
REFERENCES N
Transfer
Function
T2wn (f)
3
~
4
~
5
~
6
~
7
g 8 9
Transfer
Function
T~wu (f)
Chakrabarti, S. K. and Tam, W. A. Gross and local wave loads on a large vertical cylinder - theory and experiment, Proc. OTC, Texas, 1973, OTC 1818, pp. 813-824 Chakrabarti, S, K. Laboratory generated wavesand wavetheories, Proc. ASCE, August 1980, WW3, pp. 349-368 Dean, R. G. Methodology for evaluating suitability of wave and wave force data for determining drag and inertia coefficients, Proc. BOSS 76, Norway, Vol. II, 1976, pp. 40-64 Haritos, N. Wave spectra and energy transfer to dynamic structural response. PhD thesis, University of Melbourne, 1979 Hogben, N. Fluid loading on offshore structures, a state of art appraisal: wave loads, Maritime Technology Monograph No. 1, The Royal Institution of Naval Architects, 1974 McNamee, B. P. A study of wave forces on cylinders in the inertial regime, PhD thesis, University of Melbourne, 1980 Le Mehaute, B., Divoky, D. and Lin, A. Shallow water waves: a comparison of theories and experiments, Proc. l lth Conf. Coastal Engineering, London, 1968, Vol. I, pp. 86-107 Sharp, B. B. Flow measurement with a suspension wire, Proc. ASCE HY92 1964, 90, 37-53 Silvester, R. Development in Geotechnieal Engineering 4 A Coastal EngineeringI, Elsevier Scientific Publishing Co., 1974
NOMENCLATURE d C
f H L T
N
Figure 9. Comparison o f transfer funct~ns for water l~rticN re.cities with Airy theory
T2 Y
still water depth wave frequency wave height, peak to peak wavelength wave period transfer function of wave height to horizontal velocity transfer function of wave height to vertical velocity transfer function of horizontal to vertical velocity elevation above bottom wave height
Applied Ocean Research, 1983, Vol. 5, No. 1
53
Research Fellow, Department of Civil Engineering, University of Melbourne BRUCE B. SHARP
Reader, Department of Civil Engineering, University of Melbourne LEONARD K. STEVENS
Professor, Department of Ovil Engineering, University of Melbourne Water surface profiles and horizontal and vertical water particle velocity components have been measured to investigate the properties of intermediate depth waves generated in the laboratory. The data has been compared with linear wave theory. It was found that linear theory predicted the attenuation of velocity field with depth successfully and that it overestimates both components of velocity slightly.
The water depth was kept at about 3 ft 6in. (1.06 m) for these tests. The wave periods ranged between 1.1 and 2.5 s. The wavelengths were not measured but determined using Airy theory, from the measured wave period, for the purpose of comparing the water particle velocity components. For the range of waves studied, previous work 2 has found that the linear theory estimation of wavelength has agreed quite well with the measured wavelengths. In these present tests, the wave steepness: H/L, varied fron~ 0.008 to 0.049, and the depth to wavelength ratio: d/L, varied between 0.15 and 0.50; hence, indicating the intermediate depth range of wave conditions.
INTRODUCTION The design of offshore structures relies upon an understanding of the water particle kinematics of ocean waves. The Morison approach has been used extensively for the calculation of hydrodynamic loading on cylindrical structures in the Gulf of Mexico. s Generally, the water particle kinematics are determined from fifth order Stokes theory. However, the reported values of the empirical coefficients of inertia and drag: CM and Co, show large scatter, which Dean 3 attributes in part to the fact that water particle velocities and accelerations have not been measured, but determined from the measured surface profile, using an assumed wave theory. Linear diffraction theory has been used for the design of the large, monolithic structures favoured in the North Sea. The gross loads and local pressure distribution predicted using linear theory, have been validated for large vertical cylinders by Chakrabarti and Tam. 1 The measurement of water particle velocities is very difficult because of the small magnitude of the velocities encountered. 6 Consequently, large scatter in the data may occur. The consistency of the data can to some degree be determined by the observed variation of the velocity field with depth. A continuous profile would at least indicate that large fluctuations in measured values will probably not occur. However, there still remains the task of obtaining a successful calibration of the instrument used to measure the velocity components. Previous tests 2'7 have compared wave tank d~ta with the predictions of available wave theories. The scatter of the test data is large in some instances and further experiments are required to correlate the data with the analytical theories. Measurements of water surface profiles and water particle kinematics were therefore made to further investigate the properties of the intermediate, regular waves generated in the laboratory. The comparison with the analytical theories is restricted to Airy theory. 0309-1708/83/010049-05 $2.00 © 1983 CML Publications
EQUIPMENT A Salter duck-type wave generator was used for the experiments. The experimental layout is shown in Fig. 1. The tank is about 200 ft (63 m) long, 6 ft (2.03 m) wide and an operating depth of about 3 ft 6 in. (1.06 m) was used. A capacitance depth probe was used to measure the water surface elevation. This was placed in line with the velocity probe across the channel. The velocity probe was a two-component suspension wire device developed by the second writer, s It was used to measure the vertical and
wave probe
.J
¢q
velc'i o'-typrobe 63m. Figure 1. Experimental layout Applied Ocean Research, 1983, Vol. 5, No. 1 49
Technical No te horizontal water particle velocities. The probe, shown in Fig. 2, basically consists of a strain gauged cantilever with a length of thread suspended between the tip of the cantilever and a second support. The hydrodynamic force exerted by the flow of water past the thread, exerts a force on the cantilever, which is related to the velocity of flow. The
Sk strain gauged A /f~-----~upport frame cantli ~ _t h~rre/ a~ d ~ ~
~ _ _ , tensioner
Figure 2.
Velocity probe
Output500
/
(mV)
/
.
/
/-
/.
,/
/
/
// ./.
oo~
2E
Velocity (cm/s)
i
Figure 3.
Velocity probe calibration
threads were about 2 in. (50 mm) long and were horizontal. They ran across the tank to take advantage of the twodimensionality of the flow. A typical calibration of the device, obtained by towing the instrument in the channel, is presented in Fig. 3. The measurements were conducted about 130 ft (39 m) from the generator. The velocity probe was clamped at different elevations during continuous wave generation, in order to measure the velocity field profile with depth. The experiments were recorded on-line to an ID 70 minicomputer. The data was recorded in buffer lots of 2048 points at a sampling rate of 40 per second.
EXPERIMENTAL RESULTS A brief investigation of the surface profiles demonstrates that the waves generated in the laboratory closely approximate a sinusoidal wave form. The surface profiles of several waves have been normalized with respect to both amplitude and period. This data is presented in Fig. 4, along with the sinusoidal variation assumed by linear theory. The experimental data is shown with equal crest and trough amplitude about the still water line. It was observed as generally reported 9 for finite amplitude waves, that the mean water level did not necessarily coincide with the still water depth. The crest height above still water level did not vary by more than about 5% from half of the total wave height. It can be seen also, that the wave 'half' periods are not symmetrical for the different waves. In Fig. 5, sample traces of wave height, H, horizontal (u) and vertical (w) water particle velocity are shown. The phase relationship between the wave height and water particle velocity components is clearly illustrated. The horizontal velocity component is in phase with the wave profile and the vertical component is about 90 ° out of phase. The spectral variation of the traces shown in Fig. 5 is presented in Fig. 6. The monotonic nature of the waves
4-a c-
"F~ -I(~ t~
N
% E £. o z
Figure 4.
50
d=1.067m H=O.083m T=1.602s
d=l.060m H:0.068m T:1.482s
Normalized Comparison of wave profiles with Airy theory
Applied Ocean Research, 1983, Vol. 5, No. 1
Wave Period
Technical Note corresponding to trough level of the waves because of the nature of the instrument. The attenuation of the velocity field is most pronounced near the water surface. Although there is some scatter in the data, which may be attributed in part to the velocity device itself and also to the variability in the wave profile as shown in Fig. 4; two main observations can be made. Firstly, linear theory predicts the attenuation of the velocity field with depth for the intermediate range of waves very well. This is well illustrated by the profiles shown in Fig. 7. Secondly, linear theory tends to overestimate the magnitude of both velocity components at all depths. To compare the present data obtained for regular waves generated in the laboratory with published field data, 4 the data was replotted in terms of the spectral transfer functions between wave height and the water particle velocity components:
50
::
E
v
-50
I0
.~.
:=
0'
=
S.(f) - -
=
-
(1)
s (f) r (f) sw(f) s (f) sw(f) 2
-ic
lfi
(2)
-
Twu(f) - - -
(3)
s.(;)
lO-I
-10 Figure 5. velocities
Traces o f
wave height and water particle
generated is evident. Energy at the higher harmonics is significantly lower than at the dominant wave frequency. The water particle velocity components were recorded at several elevations to determine the variation of the velocity field with depth. In order to determine the maximum amplitude of the velocity components, it was necessary to consider that equal maximum positive and negative velocities occur during the wave cycle. This is a result of the calibration procedure of the velocity probe. To examine the fluctuations in amplitudes would require more accurate knowledge of the zero output condition of the velocity probe with time than was possible in these tests. Some drift occurred; however, this is more pronounced under uniform flow conditions than oscillatory flow conditions. Sample velocity profiles are shown in Fig. 7. The continuity of the measured profiles with depth is very good. Those elevations at which readings were repeated, show the consistency of the recording instrument. Some scatter is expected for small values of y/d, where y is the elevation above the bottom. This is caused by the very low magnitude of the velocities, especially the vertical velocity component. In order to correlate all the data, the non-dimensionalised amplitudes of horizontal and vertical water particle velocity components were plotted against the parameter d/L in Fig. 8. Lines of best fit were assumed for the measured velocity profiles and the velocity amplitudes estimated at several elevations. The estimation of the still water depth velocity components is more likely to be in error as no measurements could be performed above that elevation
v
tm
10 -6
0.05
12.5
I Frequency
f
(Hz)
lO-I
t~ =
g N 'F_ o
i0 -s 0.05
1 Frequency
12. f
(Hz)
10-i
i0-6 I
0.05 Frequency
Figure 6.
12.5 f
(Hz)
Spectral variation o f traces in Figure 5
Applied Ocean Research, 1983, 11ol. 5. No. 1
51
Technical Note where Su, Sw and Sn are the spectral density of the horizontal and vertical velocity components and the wave height, respectively. These values are plotted in Fig. 9, with the linear theory prediction• The data is plotted for middepth. The overestimation of the velocity field by Airy theory occurs at all depths and hence only one depth need be considered as representative. The data above and below the mid-depth positions allows a more accurate estimation of the velocity amplitudes by way of interpolation• From these plots, it is again clear that Airy theory tends to overestimate the magnitude of the velocity components• The attenuation of energy with frequency for the horizontal particle velocity is well predicted. The peak in energy of the vertical water particle velocity component is also well correlated, although the scatter of the data is more significant. The transfer function between the vertical and horizontal velocities is most interesting in that it tends to show that Airy theory in fact correctly predicts the relative magnitude of the two velocity components• This is despite the fact that each individual component is overestimated. Haritos 4 presented transfer functions for real sea conditions in deep water• The data was obtained using a similar velocity probe. His data showed that the energy was greater than predicted by Airy theory for both velocity components. The frequency at which the peak in energy for the
i
20
,
•
20
i0
Horizontal Velocity
30
u
(cm/s)
10
20
Vertical Velocity
Y/d
';d:l.O53m. ~T=2.169s. ~H=O.136m.
0 I'0 2'0 30 llorizontal Velocity u (cm/s)
0 10 20 30 Vertical Velocity w {cm/s)
// Ariy
/
/i i
i.
i
i
Y.~
I
g
z
0
0-1
0-6
d/k
I i
I
I
_
i
:
"g o % > _
÷
i + + 4_____i~-~__ 1÷ ~ ~ . 7Y~ . : o
_L
.
~ I'Oo -
•
- , ~
yl. J0.~o.
_
1
0
j o I d/k
06
Correlation of water particle velocities with Airy
theory
w (cm/s)
Airy
0
i
% C 0
Figure 8. 30
~
1.0 D
0'I
d=l.lOOm• H=O.I08m. T=1.440s.
i
"7. 0
c L~ g z /
[
5N
g
ry
i
%
> %
Y/d
'
,
~g >
i
1
i
'
/
vertical components occurred, was in close agreement with Airy theory. However, for both components, no attenuation of energy at higher frequencies was evident. The theoretical transfer function between the horizontal and vertical velocity components is equal to unity for deep water conditions• Haritos found that linear theory underestimated the transfer of energy at the dominant wave frequencies and overestimated it at higher frequencies. The author indicated that some of the above effects observed in the real sea state, may be caused by a wave grouping effect. Chakrabarti2 conducted velocity measurements at two elevations using an electromagnetic flow measuring device. For the range of conditions covered in the present tests, he also observed that the linear theory slightly overestimated the velocity field. CONCLUSIONS
i
/ / ..
0
10 Horizontal Velocity
Figure 7.
2'0 u
3o
(cm/s)
d=l.O67m. H=O.O93m. T=I.950s. io
3o
Sample profiles of water particle velocities versus
depth
52
2o
Vertical Velocity w (cm/s)
Applied Ocean Research, 1983. Vol. 5, No. 1
The water surface elevation and horizontal and vertical water particle velocity components have been measured for laboratory generated regular waves in the intermediate depth range. The waves have been shown to be monotonic and to approximate a sinusoidal wave profile. The correlation of the velocity data with linear theory, has shown in the region: 0.15 > d / L > 0 . 5 , that both the
Technical Note
Transfer
Function
T2un (f) 0
"rl ,-$ e~ e-
horizontal and vertical water particle velocity components are slightly overestimated by theory. Secondly, the attenuation of both velocity components with depth has been found to be satisfactorily described by linear theory. Finally, it was found that Airy theory tended to predict the correct relative magnitude of the two velocity components although overestimating the individual components.
"h
REFERENCES N
Transfer
Function
T2wn (f)
3
~
4
~
5
~
6
~
7
g 8 9
Transfer
Function
T~wu (f)
Chakrabarti, S. K. and Tam, W. A. Gross and local wave loads on a large vertical cylinder - theory and experiment, Proc. OTC, Texas, 1973, OTC 1818, pp. 813-824 Chakrabarti, S, K. Laboratory generated wavesand wavetheories, Proc. ASCE, August 1980, WW3, pp. 349-368 Dean, R. G. Methodology for evaluating suitability of wave and wave force data for determining drag and inertia coefficients, Proc. BOSS 76, Norway, Vol. II, 1976, pp. 40-64 Haritos, N. Wave spectra and energy transfer to dynamic structural response. PhD thesis, University of Melbourne, 1979 Hogben, N. Fluid loading on offshore structures, a state of art appraisal: wave loads, Maritime Technology Monograph No. 1, The Royal Institution of Naval Architects, 1974 McNamee, B. P. A study of wave forces on cylinders in the inertial regime, PhD thesis, University of Melbourne, 1980 Le Mehaute, B., Divoky, D. and Lin, A. Shallow water waves: a comparison of theories and experiments, Proc. l lth Conf. Coastal Engineering, London, 1968, Vol. I, pp. 86-107 Sharp, B. B. Flow measurement with a suspension wire, Proc. ASCE HY92 1964, 90, 37-53 Silvester, R. Development in Geotechnieal Engineering 4 A Coastal EngineeringI, Elsevier Scientific Publishing Co., 1974
NOMENCLATURE d C
f H L T
N
Figure 9. Comparison o f transfer funct~ns for water l~rticN re.cities with Airy theory
T2 Y
still water depth wave frequency wave height, peak to peak wavelength wave period transfer function of wave height to horizontal velocity transfer function of wave height to vertical velocity transfer function of horizontal to vertical velocity elevation above bottom wave height
Applied Ocean Research, 1983, Vol. 5, No. 1
53