CHAPTER 8
NEAR-SURFACE ORBITAL VELOCITIES
IN IRREGULAR WAVES
K.-F. Daemrich* and A. Gotschenberg**
ABSTRACT
The paper deals with measurements of horizontal orbital velocities
near the surface of mechanically generated waves in a wave flume.
Due to the characteristics of most velocity probes, it is difficult
or impossible to measure with a fixed probe in the area above the
lowest trough. As the probe is not submerged continuously, failures
or uncertainties in the measurements may occur.
To overcome these limitations,
a movable frame for the velocity
probe was designed, which can be moved vertically up and down with
the surface elevation by a disc rotor servo motor, controlled by a
wave gauge. By that continuously velocities up to 3 cm below the
surface could be measured.
Theoretical velocities have
different simulation methods
wave theory.
been calculated for
for irregular waves,
comparison with
based on linear
INTRODUCTION
Investigations and measurements of orbital velocities have been
conducted for a long time. Whereas the most earlier publications
were related to investigations in regular waves to check the various
wave theories, the subsequent dealt with the development of simulation methods
for the theoretical treatment of irregular waves.
Especially the velocities in very steep and breaking waves have
always been an important subject of investigations and the research
on the influence of the directionality will extend, as adequate
test facilities for the generation of directional waves are available now.
Due to the characteristics of most velocity probes, however, in
general measurements close to the surface of the waves, especially
in the wave crests, are difficult or impossible when the probe is
fixed at a certain height above the lowest trough. The sensor is
submerged then only a short time (especially in scaled model tests)
what might create problems due to the time constants of the instrumentation or disturbing air enclosures.
1) Dr.-Ing., Senior Research Engineer
2) Dipl.-Ing., Research Engineer
both Franzius-Institut, University of Hannover, FRGermany
97
98
COASTAL ENGINEERING-1986
Earlier own measurements in irregular waves with inductive type
probes had to be restricted to this area, too, and recently in a
publication of SVENDSEN (1986) on surf zone turbulence it is summarized that "all investigations have concentrated on the region
below trough level where measurements are available".
To overcome these limitations and to be able to perform investigations on velocities in the crest area of irregular waves, a vertical
movable frame for the velocity probe was constructed. This frame
can be moved up and down with the variation of the water surface by
a disc rotor servo motor controlled by the output of a wave gauge.
So the velocity probe is always submerged and able to measure
velocities very high in the wave crest.
INSTRUMENTATION
The vertical movable frame was designed for measurements in hydraulic models with wave heights up to about 0.50 m, equivalent to the
range of the often used wave gauges of the DELFT type.
It consists of a U-profiled aluminium bar sliding between guide
rollers and driven via toothed rack and pinion directly connected
to the servo motor (120W rated output power). A flat coil spring is
used to compensate for the weight of the movable parts and the
velocity probe.
Figure 1 gives a principal sketch of the set-up in the wave flume.
Control
System
® (6)
<?>
©
t®
w-m
-Velocity Probe
Fig. 1: Principal sketch of set-up of vertical movable
frame in the wave flumb
The wave gauge (1) measures the actual surface elevation of the
water. Its output signal (2) provides the electronic control system
which generates the set value (3) for the servo motor. The U-profiled aluminium bar (4) with the velocity probe fixed at the lower
IRREGULAR WAVES
99
end (and connected to the motor via toothed rack and pinion) is
moved up and down by the servo motor following the set value (3) .
The potentiometer transducer (5) gives the actual position (6) of
the U-profile (4) to the control system (actual value) and allows
also to compare surface elevation and position of the probe for
analysis purpose.
Figure 2 shows the device and the electronic control system,
Figure 3 the set-up in the wave flume, fixed to a measuring carriage.
Fig. 3: Set-up of vertical movable frame in the wave flume
100
COASTAL ENGINEERING-1986
For a certain test, the movable frame is mechanically placed in a
middle position and the velocity probe is shifted to the pretended
submerged depth in still water conditions. After starting the tests
the probe is then moved up and down with the surface elevation,
measuring in positions approximately parallel to the surface.
Due to the characteristics of the electronic control system, the
measured location is not exact constant below the wave surface.
However, from the potentiometer transducer, the position of the
probe can always be determined and related to the signal from the
wave gauge. So the exact position as a function of time is available.
As an example, which demonstrates at the
the system, in Figure 4 the time series
location, and the difference between both
wave train was measured in the area of a
steep and some breaking waves.
same time the limits of
of water level and probe
signals, are plotted. The
1:30 slope and contained
Wave [ml
"T—I—I—I—I—I—I—|—
30.0
25.0
35.0
40.0
Position Velocity-Probe [m]
20.0
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40.0
45.0
50.0
55.0
60.0
I ' ' ' ' i
25.0
30.0
Wave - Position Probe
[m]
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1
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r,„.j„
25.0
r
T...|
|—i- i
30.0
T i •! 'i- r 'i
35.0
,.., ,....,,•!
40.0
r.
,..| ,...,, T. ,-y
45.0
T
50.0
,,.,,.
T
••!
55.0
1—f—T-
60.0
y = -0.05 m (below surface)
Test 22108613. d = 0.39 m, 1:30
Fig. 4: Comparison of surface elevation and probe location
101
IRREGULAR WAVES
In the not extreme steep waves the system works with deviations of
less than 1 cm. However, in the very steep waves, with the present
layout of the motor and the control electronics, the system works
not yet satisfying. For further measurements in steep and breaking
waves, the system has to be improved by taking a stronger motor and
a controller of better quality.
Finally in Figure 5 examplarily maxima (crest) and minima (trough)
of the wave time-series and of the time-series of the position of
the probe are compared in scatter plots.
3
osition velocity-Probe [ml
•m
20
'oaition Velocity -Probe Em]
.13-
.13-
.10-
.10-
.05-
.0B-
0.00-
/
0.000.00
Wave [ml
Crest, y - -0.03 m tbelow surface)
Teat 31108606. d - 1.00 m
, Position Velocity-Probe [m]
••/
Have (ml
Crest, y - -0.05 m (below surface]
Test 22108613, d - 0.39 ra. 1:30
,0
'oaition Velocity-Probe [m]
/
.13-
/
.10'
•/
.03-
/
Nave Em)
Trough, y - -0.03 tn (below surface)
Test 21108606. d - 1.00 m
Have [m]
Trough, y - -0.05 in [below surface)
Test 22108613. d - 0.39 m. 1:30
Fig. 5: Comparison of extremal values of wave surface in crest and
trough and pertinent position of the probe
a) moderate wave conditions, design condition for equipment
b) very steep and breaking waves
102
COASTAL ENGINEERING -1986
As the motor of the system can be controlled by any external
signal, the equipment can also be used for a dynamic calibration of
the velocity probes. For the tests discussed in the following, a
Delft Propeller Probe was used as a velocity probe. However, also a
Minilab SD-12 system was available and included in the test series.
HYDRAULIC BOUNDARY CONDITIONS
The measurements discussed in the following were performed in the
wave flume of the FRANZIUS-INSTITUT. The total length is about
120 m, the width 2.20 m. At the end of the flume a 1:30 slope is
installed. The water depth during the tests was 1.0 m in the
horizontal part.
The results presented here are from a JONSWAP spectrum with
Peak enhancement factor
% =3.3
Spectral peak period
T = 1.8 s
Significant wave height
H ^ = 0.18 n
The control signals for the wave machine were calculated according
to the random phase spectrum method with a sequence length of
204.8 s for pusher movement of the wave paddle.
Measurements were taken in two sections:
- water depth d = 1.00 m horizontal slope (no breaking waves)
- water depth d = 0.39 m 1:30 slope (some waves breaking)
The submerged depth of the velocity probe was 3, 5, 10, and 15 cm
below actual water surface with the moving frame and for comparison,
25 cm and 50 cm below mean water with a fixed height of the probe.
RESULTS OF THE TESTS
During the tests, time-series of waves, horizontal velocities, and
probe locations were measured simultaneously. An HP1000/A600+ computer system was used for data acquisition and analysis.
Theoretical results, based on linear wave theory, have been calculated for comparison, with simulation methods described in detail in
earlier publications (DAEMRICH, EGGERT, KOHLHASE (1980), DAEMRICH,
EGGERT, CORDES (1982)).
In general for the analysis of such tests the transfer function
method (or linear filtering method) would be prefered. With this
method the theoretical velocity is calculated from the FOURIER
coefficients of the wave train, by applying the linear transfer
function and getting the time-series by an inverse FOURIER transformation.
For the measurements with the probe location varying with the
surface elevation, however, this method leads to a considerable
extension of the computer work, and it was decided to use for the
present the complementary method, a simulation method in the time
domain, which gives results of similar quality, at least in the
more pronounced waves. In this method, the maxima of the orbital
velocities under crest and trough of a wave were calculated from
half-wave parameters. For the horizontal velocities, this half-wave
IRREGULAR WAVES
103
parameters are the amplitude of a wave crest or a wave trough (zero
crossing wave crest height a and zero crossing wave trough excursion a ) and the pertinent half-period (crest period T and trough
period T ) . The parameter designation is according to the List of
Sea State Parameters, IAHR (1986) . For further explanations and
definition sketches see DAEMRICH et al.(1980), (1982).
As an example, in
velocity maxima for
y = 0.25 m below the
scatter diagrams. It
ce in the tendency.
Figure 6 measured and calculated horizontal
a constant submerged depth of the probe of
mean water level are computed and plotted in
is obvious, that there is not to much differen-
, u Meaaurement [m/sl
u TF Theory [m/al
Crest, y - -0.25 m (canst.)
Test 21108602. d - 1.00 m
. u Measurement [m/a)
u Theory [m/s]
Crest., y - -0.2S tt (const.)
Test 2110S602. d - 1.00 m
u Measurement [m/a]
u TF Theory [m/al
Trough, y - -0.25 m (const.)
Test 2110B602. 0 - 1.00 m
u Theory [m/a)
Trough, y - -0.25 n (const.)
Test 21108502. d - 1.00 m
Fig. 6: Comparison of measured and calculated maxima of horizontal
orbital velocities in constant submerged depth (y = 0.25 m)
a) Transfer function method b) Complementary method
upper Part: Velocities under the crests
lower Part: Velocities under the troughs
104
COASTAL ENGINEERING-1986
Well known and typically is the tendency of overestimating the
velocities under the crests and underestimating the velocities under
the troughs, which is somewhat more pronounced in the complementary
method. This general deviation in the theoretical results is due to
using linear wave theory without any modifications as e.g. stretching of coordinates.
In Figure 7 a part of a measured wave time-series is plotted
together with theoretical velocity profiles for crests and troughs
of individual waves and measured velocities 3, 5, 10, and 15 cm
below the actual surface elevation. For comparison results from
measurements in constant submerged depths of 25 and 50 cm below
mean water level are inserted.
0.0
.9 1.0
.0
.5 .0
.5 1.0.0
.S
.0
.5 1.0.0
.5 .0
.9 1.0.0
.9
.0
.9 1.0.0
.9
Fig. 7: Velocity profiles in crests and troughs of
irregular waves (Theory: Complementary method)
The overestimation of velocities under the crests and the underestimation under the troughs are clearly visibel, however, under these
wave conditions there is no change in the general trend closer to
the surface.
In Figure 8 and 9 results from the complete time-series are shown
as scatter plots. Compared are maxima of measured and theoretically
calculated velocities in various submerged depth (upper part: wave
crest data, lower part: wave trough data).
IRREGULAR WAVES
105
Figure 8 gives results measured in the horizontal section of the
flume (water depth of 1.0 m), 10, 5, and 3 cm below the surface.
The scatter of the data is not too high, also compared to earlier
measurements. Overestimation and underestimation are more pronounced
closer to the surface.
In Figure 9 similar results are plotted from measurements with the
same wave train, but measured in 0.39 m waterdepth in the area of
the 1:30 slope. Due to the steeper waves the minimum submerged
depth of the velocity probe was 5 cm to avoid surfacing, the
maximum submerged depth was 10 cm to avoid bottom contact. To show
the trend, results from measurement in a constant submerged depth
of 25 cm were added. The scatter of the data measured near the
surface has increased due to some very steep or just breaking
waves. The highest velocities are measured in a few just breaking
waves.
MODIFICATION OF LINEAR WAVE THEORY
To consider the problem of overestimation and underestimation for
design purpose the theoretical calculations were conducted exemplarily with a modification of the linear wave theory, similar to the
stretching of coordinates (WHEELER (1969)).
Instead of using the actual location of the velocity probe to
calculate theoretical values, e.g. 3 cm below surface elevation, we
have assumed a constant location of the probe of 3 cm below mean
water level for the calculation of the maxima of the near-surface
orbital velocities. By this modification, the trend of over- and
underestimation is clearly diminished in the complementary method
and almost compensated when using the transfer function method as a
simulation method.
In Figure 10 scatter diagrams are plotted exemplarily with results
of those calculations.
Methods of modification of wave theories, however, have to be
tested further for general application. But with the results of
several contributions presented at this conference, there seems to
be a good chance for simulation methods on the basis of linear wave
theories.
ACKNOWLEDGEMENT
The investigation described in this paper was carried out as part
of the research programme of the Sonderforschungsbereich 205 at the
FRANZIUS-INSTITUT for Hydraulic Research and Coastal Engineering,
University of Hannover. The authors gratefully acknowledge the Deutsche Forschungsgemeinschaft for their financial support throughout
the investigations.
106
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, u Measurement («/s)
u TF Theory [m/sj
Crest, y - -0.03 m (const.)
Test 21108606. d - 1.00 in
Crest, y - -0.03
TEST 21108606
u Measurement Im/s]
. u Measurement fffi/s]
u Theory tm/s]
Trough, y - -0.03
TEST 21108606
u TF Theory [m/s]
Trough, y - -0.03 m (const.)
Test 21108606. 0 - 1.00 in
a
b
Fig. 10: Effect of modification of linear wave theory
a) Complementary method b) Transfer function method
REFERENCES
DAEMRICH,K.-F., EGGERT,W.D. and KOHLHASE, S. (1980). Investigations
on Irregular Waves in Hydraulic Models. Proc. 17th Int. Conf.
Coastal Engineering, Sydney, pp. 186-203
DAEMRICH.K.-F., EGGERT.W.D. and CORDES.H. (1982). Investigations on
Orbital Velocities and Pressures in Irregular Waves. Proc. 18th
Int. Conf. Coastal Engineering, Cape Town, pp. 297-311.
IAHR (1986) . List of Sea State Parameters. Supplement to bulletin 52
(1986) PIANC. Brussels, Belgium.
SVENDSEN,I.A. (1986). Analysis of Surf Zone Turbulence. Danish Center for Appl. Math, and Mechanics, Techn. Univ. of Denmark, Report
No. 330.
WHEELER,J.D. (1969). Method for Calculating Forces Produced by Irregular Waves. Preprints 1969 offshore Technology Conference, Vol.1,
Paper No. OTC 1006, pp. 1-71 - 1-82.