THE DELTA MODEL
M 284; 1946-1959 M 600; 1959- .... Hydraulic characteristics of the Delta areaThe south-western part of the Netherlands consists of a group of islands formed by the age-long geomorphologic process in which the fresh waters of the rivers Rhine, Maas and Schelde and the tidal flow of the North Sea play their part. For generations Man has interfered with the natural process by building dikes and by adjusting channels to protect the land against the attack of the water.
Tides are strong in this delta: region. They have been studied this last century and more. Tidal constants are available for many places in the delta. It ispossible to predict the "astronomical" water level at each of these places for any time.
The water levels are also influenced by the wind. This influence is important, because the North Sea is a shallow basin where the tangential shear exerted by a strong wind cannot be neglected in respect to the effect ofgravity. As aconsequence there is a difference between the actual water levels and the astronomical ones; this difference is the meteorological effect.
Figure I shows the location of the Netherlands in relation to the North Sea area. With the passing of a barometric low over the North Sea winds veer to Northwest and North along the Scottish and English coasts and in the Southern part of the sea. Thus the water is driven to the Dutch and Belgian coasts and a great positive meteorological effect is caused there. After the depression has moved away to the East, normal conditions gradually return.
The red overprint in figure 1shows a phase of the storm ofJanuary 31and February l, 1953, which caused the disastrans inundations in the Delta region.
Ananalysis of this storm atHook ofHolland is given infigure 2.The meteorological effectreached an extremely high maximum ofmore than three metres (10 ft). The maximum level reached that day by the combined effect ofastronomical tide and wind occurs very rarely. High storm maxima have a smaller frequency than lower ones: a diagram can be made showing the frequency ofany high level. Figure 3isthe frequency of the high water levelat Hookof Holland. The lower part of sucha curve is well established: it isbased upon agreat number of observations and itis not possible to draw another frequency line than the one shown in the figure. It isdifferent, however, for the higher levels, corresponding with very small frequencies. Especially the section of the line which
31 JANUARY 1953 1FEBRUARY1953 2FEBRUARY 19SJ
Fig. 2. Water levelsHook of Holland
isbased on extrapolation is very uncertain. No great values can therefore be attached to the levels in the last twocolumns of the following table. Ô3
~--
----
---+---+
---r
--
--
--
--f,
~~--~----~~----I 0:: >--W ::;: Fig. 3. Frequency ofwater levels at Hook of Holland 5 ...J W > W ...J cr W ~ 2FREQUENCY OF OCCURRENCE PER YEAR
NAP'~---4---+---~---~---~----~~~~
353 1.0.0 to .0.1 .0..01 .0.0.01 .0..0001
Astronomical tides Storm levels
Location
I
I
(figure 4) neap mean sprmg Feb. I Frequency per year
HW
I
LWI
HWI
LWI
HWI
LW 1953 0.5I
0.01 1°.0001 Hook of Holland +0.76 -0.66 +0.90 -0.66 +1.04-0.66 +3.85 +2.42 +3.55 +5.0 Hellevoetsluis . +0.87 -0.78 + 1.03-0.80 +1.16 -0.82 +4.10 +2.65 +3.8 +5.2 Brouwershaven +0.97 -1.07 +1,25 -1.17 +1.47 -1.22 +4.25 +2.75 +3.85 +5.3 Zierikzee. + 1.13-1.33 + 1.40-1.48 +1.61 -1.58 +4.32 +2.90 +4.05 +5.4 Flushing. + 1.42-1.52 + 1.90-1.84 +2.27 -2.05 +4.55 +3.27 +4.25 +5.65Levels in metres with respect to Amsterdam Zero (NAP, approximately mean sea level)
Fig. I. The North Sea. Depths in metres. Pressures in millibars
In 1939a "Storm-Surge Committee" was appointed to make an analysis of the storm levels and to review the consequences for the low lands bordering the sea and the tidal rivers. It drew the attention to the possibility that higher levelswould occur than the highest that had been observed in the period of about one century for which reliable data were available. Much higher storms were bound to cause inundations.
The committee considered this state ofaffairsto be unacceptable. It recommended that the defense against the sea should be strengthtened, even if the chance that a disastar would occur was very,·very small.
In the North and the Southwest part of the country the strengthtening can be
done in one of two ways: raising the sea walls- the dikes - or closing the estuaries. A combination of the two is also possible.
NORTH SEA
Secondary dom
#
Main closing dom with ~ischorgesluice(Honngvliet )
I
II
Flood gote (Hollandse YsseL)I
Salt barrier {Dude Moos}<$> Regulating point tidoL motion
~ Reçutotinç point
If L___j river discharge
o 5 10 15 20 25 30 35 ,0 km.
Fig. 4. The Delta region
In the Delta region two inlets at the least will have to stay open, viz the Scheldt
and the Nieuwe Waterweg, being the entrances forshipping tothe ports of Antwerp
and of Rotterdam. This means that the tides and the storm surges willstill penetrate
into the Delta region after the other inlets have been blocked. It is necessary to know what will be the strength of the tidal currents then in various channels and
also which water levelswill be reached during exceptional strong gales.
It is evident that observations in the past can only serve to predict water levels and currents aslong as the course of the channels remains unchanged. As soon as changes are initiated, for instance by building a dam, prediction can only be done
by means of theory or by experiment.
It is possible to calculate the flowthrough evena complicate network ofchannels,
but the amount of work to bedone before aresultis attained is very great. Variations
in the conditions are many: there are neap tides or spring tides, a moderate gale or a very strong one, the discharge of the rivers may be small or great, there may
be dams on certain locations or at other places, or no dams at all, channels may
be widened by dredging or by natural scouring and shoals may be reclaimed. It is out of the question to calculate all this and therefore the Committee rec -ommended that a small-scale model should be made ofthe Delta region. This rec
-ommendation was followed up by the government and so a start was made at
constructing the model.
The first steps were taken cautiously. In the autumn of 1946 a concrete section of oneof the channels was the first part of the model to be made. Operation started in September 1948.At that time the model represented the Northern part of the
region only.
The model has not eliminated theoretical calculations. The results obtained in it are regularly compared with calculation, which, in its turn, is greatly simplified
by the presence of the model.
Besides the final conditions, after the completion of the Delta works, much
attention has to be given to currents and water levels which will occur temporarily
during the construction of the dams. The operations for blocking the last gaps in
the dams depend on those temporary currents and levels, which have to bemeasured in the model. In the period in which one of the inlets has already been closed and its neighbour has not, very strong currents may scour deep gullies in some channels and make navigation impossible. The model has to indicate the program for the execution of the Delta works and also for the auxiliary dams which have to bemade
in order to prevent unacceptable conditions.
Another task for the model is to provide data for outlining the future fresh water management in the Delta region.
An overall picture of the scheme of the Delta Plan is given in figure 4, which also shows the limits of the model.
The work in the field started in 1954 with the building of a movable storm -surge weir at Capelle, East of Rotterdam. The entire Plan is expected to be ready in 1978.
The model
Basic equations
In principle the model is used as a means of solving the differential equations which define two-dimensional flow of the water in the sea-arms and the rivers. The boundary conditions are realized by controlling the discharge of the rivers
and the tidal motion at the boundaries of the model and by moulding the channels,
all according to the appropriate scale rules. The scales can be deduced from the
two following equations, the acceleration of gravity (g) and the density of water (e) being the same in model and in nature.
The equation of continuity:
B. az = _aQ_,
a
t
l
a
s
where B = width of channel (as a function of the water level),
z
=water level above datum (NAP), Q_,=discharge in channel, s = distance in the direction of the channel, t =time. The equation of motion:a
v
a
v
-+at
v-+2w·a
s
v=
n az g - a---·'"
as
C2 vIvIJ
_
_
'_::__eL=-W---,-Iw__,I (-Iz+-z)+
2 e(Iz+z) where v =velocity in the direction of the channel,Vn =velocity normal to the channel,
w
=
vertical component of angular velocity of the earth (0,57 X10-4 rad/secin latitude 52°), Ii =channel depth below datum, C = Chezy friction coefficient for uniform flow,
J
=wind-water surface friction coefficient (R:> 0.02),eL
= density of air,e
=density of water, W =wind velocity in the direction ofthe channel.The lefthand sideofthe second equation represents the components ofacceleration
ofaunit mass of water, the third term being due to the rotation of the earth (Coriolis). On the right side are the forces causing acceleration, firstly the force due to gravity, secondly bottom friction and finally surface wind friction.
Scales oj the model
The model is required to reproduce the motion of the water accurately. As neither waves of short period, nor the movement of sand along the bottom of the channels are taken into consideration, a great distortion of the model is possible.
The required accuracy of measurement of the water levels in the model, corres -ponding to an accuracy of one or two centimeters in prototype, necessitated a vertical scale of about 60. The value chosen is 64, being the square of 8, which is the scale for the velocity of the current. The Reynolds number is high enough to ensure turbulent flow in the model.
The horizontal scale of 2400 was considered to be acceptable in view of the distortion of the model. The available site wasjust large enough for the model to be built on this scal~.
From the foregoing equations the other scales can be deduced. Summarizing: height (z) length (s) velocity (v) time (i) discharge (Q.) friction coëff. (C2) wind velocity (W) angular veloeity (w) 64 2400 8 300(one day
=
4.8 minutes) 1 227800 37.5 l.35 1/300The time scale is a very convenient one. Times are reduced in the model to such an extent that the phenomena that fill a few days (like a gale) in nature, are realized in a quarter of an hour in the model. On the other hand the tidal period is long enough, that sufficient observations can be made.
Boundaries oj the model
On the upstream side the rivers Rhine and Maas are extended as far as the influence of the sea is noticeable.
At the sea end the boundary is at such a distance from the coastline, that altera
-tions made in the channel system, have no perceptible effect at the boundary itself In the early stages only the part of the Delta north of Haringvliet and Hollands Diep was considered. Hence it was decided that in the'model the tides should be generated in each inlet separately. This had real advantages with respect to the operation of the model, because it made possible the independent choice of tidal amplitude, mean level and phase for each sea-arm. Later, when the whole Delta Area was incorporated in the plans, an extension was added to the model, which was provided with a continuous sea area, to examine Oostersehelde and Brouwers -havense Gat closing operations. Haringvliet and Nieuwe Waterweg channels retained their independent tide generators.
The area of the Delta as it is in the model is shown in figure 4.
Construction of t/ze model
The model is constructed in cement mortar, mostly in situ but also partly as precast units. The channels are moulded using templets as shown in figure 5. In order to avoid errors due to the distortion special attention has to be given to the
channel form to prevent eddy formation in the exaggarated deep sections in the model.
The roughness of the channels and shoals is obtained by the use ofvertical bars
(figure 6), the height and spacing of which 'are first calculated, then checked by reproducing a known condition of the prototype and if necessary adjusted until
an accurate reproduction is obtained.
Fig. 5. Construction of a tidal channel
Fig. 6. One of the Deltadams in the model in the (copper)wood of roughness bars
Control of t/ze model
During the operation of the model certain conditions have to be satisfied; namely
discharge in the rivers and waterlevels at the seaboundaries.
During a test the river discharge is generally kept constant. The sea levelsfollow
a preset program of tides, which may be an avarage tide continuously repeated,
areproduction ofa seriesofactual tide cycles,or an artificially" composed" sequence, which may include a storm surge.
For each tidal program the influence ofthe river discharge on the regime in the
Delta can be analysed by repeating the tests for various discharges of the river.
The river discharges are controlled by means of V-notch weirs,fed from a constant
level basin.
The water level at the sea boundary is controlled by an electrical system such
that any tidal sequence of eight days duration may be reproduced, including the possibility of repeating any part of the cycle continuously. The principle of the generation of tides is shown is figure 7. Nieuwe Waterweg and Haringvliet are
controlled by one gate each. The sea area in front of Brouwershavense Gat and
Oostersehelde is controlled at five points, the positions between the control points
of the operating gates being linearly interpolated. The programming of the seven
Ow= a
Slack water
-
°w>O"Ebcurrent
DM 8
Fig. 7. Generation of tides
Fig. 8. Control cabin. Tidal curves for points along the sea boundary are realized by the series of plugs ineachpanel
Fig. 9. Coriolis cylinders in
the North Sea part of
the model
control points is made centrally in a cabin where on plugboards the tidal curves are traced out (figure 8).
As mentioned above, the acceleration due to the earth's rotation affects the waterlevels in the Delta Area. In this area, in lat. 52ON., this rotational acceleration
causes a slope of more than one centimeter per kilometer perpendicular to the
direction of motion of a mass of water flowing with a velocity of one meter pcr
second. For the sea area this is ofthe same order of magnitude as the friction slope at the same velocity sothat the introduction of Coriolis effect is indispensable in order to obtain the correct pattern of flow in the model. The situation isdifferent for the channels which are of limited width, and where in consequence the Coriolis
force has no noticable influence on the flow conditions.
The Coriolis effect is produced in the model by means of rotating cylinders
(figure 9), which introduce a force on the water perpendicular to the direction of flow (Magnus effect). The spacing of the cylinders and the characteristics as a
function of the diameter and angular velocity were deduced by experiments. The sea area of the model contains 150"Coriolis rotors" running at 12 revolutions per
second and together providing a force on the water equivalent to the Coriolis acceleration.
Fig. 10. Waterlevel recording apparatus Wavo
Observations in the model
In the model waterlevels and currents are measured.
For the calibration of datum level, point gauges are used. During operation the waterlevels at any predetermined points can be obtained continuously using "Wavo" electric recorders (see figure 10).
The main pen records the waterlevel under theinstrument. This is equipped with a second stylus with which it is possible to record simultaneously the waterlevel at the measuring point together with the difference in level between this point and a distant measuring point. This provides an accuracy far better than ispossible by the deduction of slopes from two individual records, because time errors are avoided.
The currents may be expressedeither as velocity, or asdischarge. The propagation of tides and storm surges isbetter expressed in terms ofdischarge, while the hinder to navigation and the scour of channels are more closely related to the velocity. Discharges are measured by apendulum type current meter (figure 11), calibrated in place, taking account of the waterlevel variations. The electrical signal is fed to the waterlevel recorder and recorded through the second stylus.
Velocities are measured either with the pendulum current meter or with "micro mills" (very small rotating current meters), the latter being alsoused for measuring the velocities in the openings of the dams during the various stages of closure.
A method of measuring discharges by feeding the impulses of a micro mill into an electronic computing system isin a stage of trial.
Accuracy oj the results
In general theprobable errorsin the model measurements arcdirectly comparable with those in prototype.
In both model and prototype the error in the waterlevel measurements is less than five centimeters, generally one or two (prototype), except during gales, when greater errors occur.
Zero errors in timing may be as much as five minutes, which during periods of rapid rise or fall of the tide may cause further errors in the waterlevel of fiveto ten centimeters.
Velocities can be measured to within five percent, but the discharges are less
accurate and in some complex channels errors may occasionally exceed ten or
even twenty percent.
In the model repetition of tests, facilitated by automation, enables the accuracy of the results to be increased.
Besides the measurement errors discussed above there are also systematic errors in the model due to the fact that some prototype phenomena, such as local wind
effects and density currents, have not been reproduced. However, asfar as possible
corrections are made to compensate for the neglected factors, for example, by
modifying the tidal programmes to allow for local wind effect..There may also be local errors due to incorrect detailing of the model.
Finally itshould be emphasized that the purpose of the model is not to reproduce the prototype conditions, but to predict the consequences of alterations in the existing situation of the prototype. That is to say that the absolute accuracy of reproduction of one arrangement of the model is not soimportant asthe accuracy
of reproduction of the difference between various arrangements. In the difference
the systematic errors are compensated to a great extent.
Generally the change of waterlevel can be predicted to within 0.05 m at normal tides anel 0.1 m at storm surges. Changes in the velocity are expected to be within
0.1 m/sec. .
Fig.II.
Pendulum
currentmeter
