ISSN 0169-6548
Velocity and water-layer thickness of
overtopping flows on sea dikes
Report of measurements and formulas development
Trung Lˆ
e Hai*
February, 2014
*
PhD student, Section of Hydraulic Engineering, Faculty of Civil Engineering and Geosciences, Delft University of Technology, P.O. Box 5048, 3600 GA Delft, The Netherlands.
Tel. + 31 15 27 83348; Fax: +31 15 27 85124 e-mail: H.T.Le@tudelft.nl, trunglh@wru.vn
ISSN 0169-6548
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This publication has been produced in cooperation with Water Resources University, Viet Nam.
©2014
TU Delft, Department of Hydraulic Engineering, Water Resources University, Viet Nam
1 Introduction 1
1.1 Objectives and method . . . 1
1.2 Theory of the overtopping flows . . . 1
1.2.1 Parameter definitions . . . 2
1.2.2 Overtopping discharge and volume . . . 2
1.2.3 Flow parameters . . . 3 1.3 Report contents . . . 4 2 Flow measurements 5 2.1 Vechtdijk measurement . . . 5 2.2 Tholen measurement . . . 5 2.2.1 Measurement set-up . . . 6 2.2.2 Data analysis . . . 7 2.2.3 Flow velocity . . . 10 2.2.4 Overtopping time . . . 11 2.3 Belgium measurement . . . 14
2.4 Measurements in Viet Nam . . . 15
2.4.1 Devices and measurement method . . . 15
2.4.2 Image processing . . . 17
2.4.3 Results . . . 18
3 Formulas development 20 3.1 Overtopping experiments . . . 20
3.2 At the seaward crest edge . . . 21
3.2.1 Overtopping volume . . . 21 3.2.2 Water-layer thickness . . . 21 3.2.3 Flow velocity . . . 22 3.3 On the crest . . . 23 3.3.1 Effect of position . . . 23 3.3.2 Effect of friction . . . 25
3.4 On the landward slope . . . 26
3.4.1 Effect of position . . . 26
3.4.2 Effect of friction . . . 27
3.4.3 Relationship between flow velocity and water-layer thickness . . . 29
3.5 Summary . . . 30
4 Applications of the new formulas 31 4.1 Equation adaptations . . . 31
4.2 Measurements in the Netherlands and Belgium . . . 32
4.2.1 In front of the simulator . . . 32
4.2.2 Vechtdike slope . . . 33
4.2.3 Tholen slope . . . 34
4.3 Measurements in Vietnam . . . 35
Bibliography . . . 40
A Measurements with the simulators 42 A.1 Vechtdijk . . . 42
A.2 Tholen dike . . . 44
A.3 Belgium . . . 48
A.4 Measurements in Viet Nam . . . 49
B Dimensional analysis 52 B.1 Overtopping volume . . . 52
B.2 Flow depth . . . 53
Introduction
1.1
Objectives and method
The study aims to establish a set of formulas estimating velocity and depth of overtopping flows at sea dikes including seaward-side slope, crest and landward-side slope. These formulas are applicable to flows which are generated by natural waves and the simulator giving associated characteristics of any arbitrary value of exceedance probability. Note that the probability is based on the total number of incident waves and not to be confused with the total number of overtopping events.
The difference between the fictitious wave run-up level and the crest level (Ru,2%− Rc) is widely
considered as a measure of the (potential) energy of wave overtopping. This term is therefore used to estimate values of the mean overtopping discharges, flow velocities, water-layer thickness and overtop-ping volumes. In line with many existing research works and guidlines, the potential term is crucial to derive other parameters. Experimental data of Van Gent [2002] is reanalysed with a slightly different perspective to achieve the study objective. The analysis procedure basically consists of three main steps:
• from wave conditions and dike cross-section to calculate wave run-up levels; • from wave run-up levels to estimate flow parameters on the dike crest;
• from flow parameters on the dike crest to further compute the corresponding parameters on the landward slope.
The obtained equations are verified with flow measurements of the simulator on real dike slopes. Consequently, feasible values representing the roughness of different grass coverings are determined.
1.2
Theory of the overtopping flows
Wave overtopping is usually represented by a mean discharge q over time. An overtopping flow is characterised by its velocity u and thickness h. These parameters change when propagating on the crest and then the landward slope depending on position and surface friction. Often the maximum values during an overtopping event are taken. Previous research often focused on the 2%-value of the parameters. The design of the wave overtopping simulator was mainly based on existing relationship between overtopping volumes and flow velocities at the beginning of the crest of a dike [van der Meer et al., 2006].
For each released volume the overtopping simulator simulates an overtopping wave with a specific flow velocity and flow depth. This overtopping wave flows over the remaining crest and then down the landward slope of the dike. The velocity of a small overtopping wave may decrease significantly along a gentle landward-side slope. While the velocity of a larger wave volume is likely to increase along a steep surface.
The present section introduces the theory on overtopping flow velocity and flow thickness developed in previous works of Van Gent [2002], Sch¨uttrumpf [2001] and Pullen et al. [2007]. The section starts with some definitions of relevant parameters and then the formulas of overtopping discharge and volume are introduced.
1.2.1
Parameter definitions
The report incorporates figures obtained in previous works which were conducted at different organi-sations and institutes. Therefore, names and usages of corresponding parameters are not always the same and can slightly vary. Table 1.1 is aimed to list more or less all relevant parameters used in these works for the sake of clarity and unity in further parts of the report.
Table 1.1: Definitions of parameters relating to wave overtopping phenomenon.
Parameter Symbol Unit Description
mean overtopping discharge q l/s per m significant wave height Hsand Hm0 m
peak period Tp s
seaward slope tan α
mean wave period Tm s
number of coming waves Nw
number of overtopping waves Nov
All parameters related to the seaward slope are described with an index A, those related with the dike crest by an index B and parameters on the landward side by an index C.
1.2.2
Overtopping discharge and volume
This part rewrites information given in the Overtopping Manual [Pullen et al., 2007]. The steepness of a wave with height Hm0 and period Tm−1,0 is defined as
s0= 2π g · Hm0 T2 m−1,0 (1.1) The breaker parameter or surf similarity is ξ0= tan α/
√
s0. The mean wave overtopping discharge
q produced by the wave height Hm0on a dike with the free board Rc is
q pg · H3 m0 =√0.067 tan α· ξ0· exp −4.75 · Rc Hm0 · 1 ξ0 (1.2) The 2% wave run-up level is
Ru,2%
Hm0
= 1.65 · γb· γf· γβ· ξ0 (1.3)
Probability of overtopping Pov in case of a crest freeboard Rc is
Pov= exp " − √ −ln0.02 · Rc Ru,2% 2# (1.4) And with different value of x%, run-up level Ru,x% can be calculated as
Ru,x= Ru,2%·
r lnPov
ln0.02 (1.5)
Probability that a wave overtops the dike crest is Pov=
Nov
Nw
(1.6) The exceedance (smaller?) probability PV of an overtopping volume V per wave is
PV = P (V ≤ V ) = 1 − exp " − V a 0.75# (1.7)
in which the scale factor a is related to wave period, mean discharge and overtopping probability a = 0.84 · Tm·
q Pov
(1.8) The wave volume V can be calculated from a and PV as
V = a[−ln(1 − PV)]4/3 (1.9)
In case of V2% we have (1 − PV2%) = 0.02/Pov.
An overtopping volume V can be found in different series generated by different overtopping discharges. For example, there are two flow rates and one is larger than the other q2 > q1. Then
we have (PV)q2 < (PV)q1, i.e. discharge q2 gives a smaller opportunity that a wave volume is less
than V . This is due to the fact that there are more larger waves produced by q2 rather than by
q1. Accordingly, overtopping probability of q2 is higher than that of q1, (Pov)q2 > (Pov)q1 because
q2 creates more overtopping waves than q1. When the two discharges are derived from the same
significant wave height then crest level in case of q2 must be lower, (Rc)q2 < (Rc)q1. For q2 a smaller
wave height is therefore required to cause a certain volume, (HV)q2 < (HV)q1 in comparison with q1.
A smaller wave containing a lower wave energy might probably leads to lower values of flow depth and velocity. These arguments are likely to partly explain the deviation from calculations of flow depth and velocity by van der Meer et al. [2010a]. Assuming a Rayleigh distribution for flow velocity and flow depth, he estimated flow characteristics generated by increasing wave volumes for various values of flow rates. For the same volume, lower values of velocity and depth are derived from higher discharges.
1.2.3
Flow parameters
Formulas developed in previous works of Sch¨uttrumpf [2001] and Van Gent [2002] are given here. The key parameters estimating the flow velocities and depths are the crest freeboard Rc and friction
factors that account for frictional energy loss as the overtopping flow travels across the crest and down the landward slope.
At the seaward edge of the crest
At this position, parameters are denoted by the subscript letter A. At the seaward edge of the dike crest the flow parameters are given by:
hA,2% Hs = cA,h Ru,2%− Rc Hs (1.10) and uA,2% √ gHs = cA,u s Ru,2%− Rc Hs (1.11) The values of hA,2%and uA,2% were determined from the peaks of the overtoping wave time series
and these parameters represent the levels exceeded by only 2 percent of the total waves during the tests. The formulae are for maximum velocity at the leading front of the overtopping wave. Note that velocity and depth of a single wave decrease after the passege of the wave front. Values of the empirical coefficients were determined differently by the two authors. Besides, Van Gent [2002] estimated Ru,2%
using a formula which was developed earlier [van Gent], H1/3and Tm−1,0 are wave parameters. While
Sch¨uttrumpf used the equation of de Waal and van der Meer [1992] with wave height H1/3 and wave
period Tm.
On the dike crest
When traveling across the dike crest, the velocity decreases as a function of the surface friction factor fF. The water-layer thickness at a certain position xB on the dike crest (crest width is Bc) can
be estimated as hB,2%= hA,2%· exp −c3 xB B (1.12) different values of the empirical coefficient c3 can be derived from various publications. Flow
uB,2%= uA,2%· exp − xBfF 2hB,2% (1.13) the equation validity is limited to crest widths larger than the wave height at the dike toe (Bc>
Hs−toe). In which according to Hughes [2008] the Fanning friction factor is one-fourth of the Darcy
friction factor, i.e., fD= 4fF.
fF =
2gn2
h1/3 (1.14)
Equation (1.14) is based on the assumption that friction factors and Manning’s n associated with steady supercritical overflow that has reached equilibrium (e.g., Chezy or Manning equation) will be the same for unsteady, rapidly varying flows due to wave overtopping. And it is worth to note that (1.14) has not been validated yet. The factor n is an empirical constant that relates the resistance of the channel to the roughness of the boundary.
On the landward-side slope
The theory is based on the steady state considerations. Sch¨uttrumpf [2001] describes the devel-opment of the theory of flow over a down slope in full depth. The final equations describing the flow along the landward-side slope is as follows:
uC= u0+k1fhtanh(k21t) 1 + f u0 hk1tanh( k1t 2 ) (1.15) with k1= r 2f g sin β hC (1.16) t ≈ − u0 g sin β + s u2 C g2sin β2 + 2s g sin β (1.17) hC = u0h0 uC (1.18) The terminal velocity can be described by:
uB=
s
2hBg sin β
f (1.19)
In equation 1.19, the termp(2g)/f is equal to the Chezy coefficient.
Equation 1.15 is an implicit expression, but is fully controlled by the initial conditions at the start of the down slope with u0and h0and by the slope conditions such as slope angle β and friction
f and by the position along the slope s.
1.3
Report contents
The first chapter presents the objectives and methods of the present study. Chapter 2 describes the set-up and presents the results of the flow measurements with the simulator in the Netherlands, Belgium and Viet Nam. Based upon an available set of data, Chapter 3 develops new formulas to estimate the velocity and water-layer thickness. Chapter 4 compares the new equations to the measurements with the simulators.
Flow measurements
2.1
Vechtdijk measurement
Basically, the present section reproduces information and data given in the project report evaluating dike stretch Vechtdijk in the Netherlands [van der Meer et al., 2010b]. Minor modifications might be found to accommodate the unity of the report. Measurement set-up of overtopping flow is illustrated in Figure 2.1. Positions of paddle wheels (PW) and surfboards (SB) are given in Figure 2.2 and Table 2.1. The horizontal axis gives the distance from the devices to the crest edge. Surfboard SB5 was located at the crest edge which was about 3 m from the simulator. Paddle wheels PW3a and PW1 were attached to surfboards PW3 and PW1, respectively. Paddle wheel PW3b was fixed on the slope surface with the same distance from the crest edge as PW3a.
Figure 2.1: Set-up of the overtopping measurement on Vechtdijk slope.
The flow parameters of 18 waves in the high series, 5 waves in the middle series and 16 waves in the low series are graphically depicted in Figures 2.3 and 2.4. While flow velocity was measured at two positions, water-layer thickness was measured at five positions from the crest to toe. Note that not all paddle wheels worked for every waves.
All values of flow velocity and water-layer thickness measured at Vechtdijk are given in Table A.1 in Appendice A.
2.2
Tholen measurement
This section reproduces informations which were presented in the technical report of overtopping and run-up flows at Tholen dike [Bakker et al., 2011]. The data obtained during the measurement are
−5 0 5 10 15 0 1 2 3 4 5
Distance from the crest edge [m]
Height [m] Slope profile Surfboard SB5 Simulator SB4 SB3 SB2 SB1 PW3a & PW3b PW1
Figure 2.2: Device positions on Vechtdijk slope.
Table 2.1: Device positions on Vechtdijk slope, overtopping flow measurement. Column 3 is
the horizontal distance from the crest edge.
Paddle wheel Surfboard Distance [m] Height [m]
(1) (2) (3) (4) SB5 0.00 3.61 SB4 1.94 3.12 PW3a and PW3b SB3 3.88 2.64 SB2 7.81 1.90 PW1 SB1 11.74 1.17 Simulator -3.00 3.61
reanalysed giving values of flow velocity, water-layer thickness and overtopping time in the coming paragraphs.
2.2.1
Measurement set-up
Overtopping flowThe measurement set-up at Tholen is illustrated in Figure 2.5. Positions of paddle wheels (PW) and surfboards (SB) are given in Figure 2.6 and Table 2.2. The horizontal axis gives the distance from the devices to the simulator. The simulator and the surfboard SB1 were located on the dike crest. The first six paddle wheels were attached to five surfboards, PW1 and PW2 to SB1, PW3 to SB2, PW4 to SB3, PW5 to SB4, and PW6 to SB5. The last paddle wheel PW7 was fixed on the slope surface with the same distance from the simulator as PW6.
Table 2.2: Device positions on the Tholen inner dike slope.
Paddle wheel Surfboard Distance from the simulator [m] Height [m]
PW2 SB1 2.04 5.10 PW1 1.48 5.15 PW3 SB2 4.85 4.15 PW4 SB3 7.65 3.00 PW5 SB4 10.44 1.80 PW6 and PW7 SB5 13.25 0.55
0 2 4 6 0 2 4 6 8 10
Overtopping wave volume [m3/m]
Flow velocity [m/s]
PW3b PW1
(a) High series
0 2 4 6 0 2 4 6 8 10
Overtopping wave volume [m3/m]
Flow velocity [m/s] PW1 (b) Middle series 0 2 4 6 0 2 4 6 8 10
Overtopping wave volume [m3/m]
Flow velocity [m/s]
PW3a PW3b PW1
(c) Low series
Figure 2.3: Flow velocities of 18 waves in the high series, 5 waves in the middle series and 16
waves in the low series.
Figure 2.7 shows the set-up of run-up measurement on Tholen outer slope. Positions of paddle wheels (PW) and surfboards (SB) are given in Figure 2.8 and Table 2.3. The horizontal axis gives the distance from the devices to the outer crest edge. Paddle wheels PW1 and PW8 were attached to surfboards SB1 and SB8, respectively. Paddle wheel PW1 and surfboard SB1 were located on the dike crest and about 1.5 m from the outer edge. Paddle wheels PW2, PW3, PW4, PW5, PW6 and PW7 were fixed on the slope measuring flow velocity at 3 cm above the surface.
Table 2.3: Device positions on the Tholen outer dike slope, run-up measurement.
Paddle wheel Surfboard Distance from the outer crest edge [m] Height [m]
PW1 SB1 1.55 6.79 PW2 -0.42 6.63 PW3 -2.51 5.93 PW4, PW5 and PW8 SB8 -4.60 5.24 PW6 and PW7 -6.68 4.54
2.2.2
Data analysis
The signal of all devices measuring 26 overtopping flows (7 paddle wheels and 5 surfboards) and 24 run-up flows (8 paddle wheels and 2 surfboards) are depicted in Figure 2.9. The volume of released waves were increased from 0.4 to 5.5 m3/m. The horizontal axis gives the time sequence in second while the vertical axis give the measured signal in Volt.
0 2 4 6 0 0.1 0.2 0.3 0.4
Overtopping wave volume [m3/m]
Flow depth [m] SB5 SB4 SB3 SB2 SB1
(a) High series
0 2 4 6 0 0.1 0.2 0.3 0.4
Overtopping wave volume [m3/m]
Flow depth [m] SB5 SB4 SB3 SB1 (b) Middle series 0 2 4 6 0 0.1 0.2 0.3 0.4
Overtopping wave volume [m3/m]
Flow depth [m] SB5 SB4 SB3 SB1 (c) Low series
Figure 2.4: Flow thickness of 18 waves in the high series, 5 waves in the middle series and 16
waves in the low series.
Figure 2.5: Set-up of the overtopping measurement on the Tholen inner slope.
Measured signal of different devices are converted into SI units as in the Table 2.4 for overtopping measurements and Table 2.5 for run-up measurements. In which, PW denotes paddle wheel and SB stands for surfboard.
The signal series of each overtopping flow are analysed to determine three parameters flow velocity, flow depth and overtopping time. As can be seen in the Figure 2.10, the value of the velocity and the depth are the maximum values of the recorded signal sequences. The overtopping time is the
0 5 10 15 0
2 4 6
Distance from the simulator [m]
Height [m] Slope profile Paddle wheel PW2 PW3 PW4 PW5 PW6 & 7 PW1 SB1 SB2 SB3 SB4 SB5
Figure 2.6: Device positions on the Tholen inner dike slope.
Figure 2.7: Set-up of the run-up measurement on the Tholen outer slope.
−102 −5 0 5
4 6 8
Distance from the outer crest edge [m]
Height [m] Slope profile Paddle wheel SB8 SB1 PW1 PW2 PW3 PW6 & PW7 PW4, PW5 & PW8
Figure 2.8: Device positions on the Tholen outer dike slope, run-up measurement.
continuous interval during which the velocity is above zero. The seven paddle wheels give seven values of the flow velocity and overtopping time. The five surfboards offer five values of the flow depth.
The signal series of each run-up flow consists of two parts the run-up and the run-down, re-spectively. Similar to the overtopping measurement, values of the flow velocity and depth are the maximum values of the corresponding signal parts. Paddle wheels PW4 and PW5 and surfboard SB8 were located at the same distance from the simulator. Their recorded signals are analysed as follows. Signal records of PW4 and PW5 are clearly split into two parts while the SB8 signals are a continuous line as shown in Figure 2.11. SB8 run-rup depth is determined at the moment when PW4 and PW5 lines go across zero after their peaks. SB8 run-down depth is the maximum of the second part of the
0 50 100 150 200 250 300 350 400 450 500 0 1 2 3 4 5 Time [s] Signal [Volt] (a) Overtopping 0 50 100 150 200 250 300 350 400 0 1 2 3 4 5 Time [s] Signal [Volt] (b) Run-up
Figure 2.9: Overtopping and run-up flow measurement at Tholen, signals of all devices.
Table 2.4: Overtopping measurement, convert functions of 12 channels.
Device Channel Convert function Unit
(1) (2) (3) (4) PW1 1 V olt ∗ 8.57 m/s PW2 3 V olt ∗ 8.57 m/s PW3 5 V olt ∗ 8.57 m/s PW4 7 V olt ∗ 8.57 m/s PW5 9 V olt ∗ 8.57 m/s PW6 11 V olt ∗ 8.57 m/s PW7 12 V olt ∗ 8.57 m/s SB1 2 0.385(V olt − 1.90)1.00 m SB2 4 0.350(V olt − 1.30)1.15 m SB3 6 0.300(V olt − 1.95)1.07 m SB4 8 0.340(V olt − 1.87)1.15 m SB5 10 0.410(V olt − 2.16)1.00 m record.
2.2.3
Flow velocity
The relationships between overtopping flow velocity and depth and wave volume are given in Fig-ure 2.12, while the estimated values are given in Table A.2, Appendice A.
The relationships between velocity and depth of 24 run-up flows and wave volumes are shown in Figure 2.13 and the estimated values are given in Table A.3, Appendice A.
Table 2.5: Run-up measurement, convert functions of 10 channels.
Device Channel Convert function Unit
(1) (2) (3) (4) PW1 1 V olt ∗ 8.57 m/s PW2 3 V olt ∗ 8.57 m/s PW3 5 V olt ∗ 8.57 m/s PW4 7 V olt ∗ 8.57 m/s PW5 9 V olt ∗ 8.57 m/s PW6 11 V olt ∗ 8.57 m/s PW7 12 V olt ∗ 8.57 m/s PW8 13 V olt ∗ 8.57 m/s SB1 2 0.410(V olt − 2.10)1.00 m SB8 10 0.200(V olt − 1.30)1.00 m 200 205 210 215 0 2 4 6 8 10 Time [s] Flow velocity [m/s] PW4 overtopping velocity overtopping time
(a) Flow velocity and overtopping time
200 205 210 215 0 0.05 0.1 0.15 0.2 Time [s] Flow depth [m] SB4 flow depth (b) Flow depth
Figure 2.10: Estimation of overtopping flow velocity, depth and time for a wave volume of
2.5 [m
3/m] using signal recorded by PW4 and SB4, respectively.
in Figure 2.14, whilst the estimated values are given in Table A.4, Appendice A.
2.2.4
Overtopping time
Beside flow velocity and flow depth, overtopping time at five positions on Tholen dike slope were estimated using signal recorded with five surfboards SB1, SB2, SB3, SB4 and SB5. Principle of analysis is described as follows. As shown in Figure2.15, duration of a wave (j) Test can be roughly
estimated by determining its starting start(j) and ending point end(j), respectively. And maximum value of a wave signal (in Volt) aest, which looks like the wave crest, is also estimated and used
105 105.5 106 106.5 107 107.5 108 108.5 109 109.5 110 0 1 2 3 4 Time [s] Flow velocity [m/s] PW8 run−up flow velocity
run−down flow velocity
(a) Flow velocity
105 105.5 106 106.5 107 107.5 108 108.5 109 109.5 110 −0.1 0 0.1 0.2 0.3 Time [s] Flow depth [m] SB8
run−up flow depth run−down flow depth
(b) Flow depth
Figure 2.11: Estimation of run-up and run-down flow velocity and depth for a wave volume
of 2.0 [m
3/m] using signal recorded by PW8 and SB8, respectively.
0 2 4 6 0 2 4 6 8 10 12
Overtopping wave volume [m3/m]
Flow velocity [m/s] PW1 PW2 PW3 PW4 PW5 PW6 PW7
(a) Flow velocity
0 2 4 6 0 0.2 0.4 0.6 0.8
Overtopping wave volume [m3/m]
Flow depth [m] SB1 SB2 SB3 SB4 SB5 (b) Flow depth
Figure 2.12: Overtopping flow velocities and depths of 26 wave volumes.
number of consecutive intervals. Each interval may contents of a certain number of samples ns; for
example about 50 samples are chosen with regard to 142 samples being recorded within 1 second (signal frequency is 142 Hz). Within Test, value of sample (i) is compared to value of sample (i + ns)
and if
sample(i + ns) − sample(i) > a (2.1)
where a < aest, then sample(i) is considered as the starting point of the studied wave (j). Value
of a is usually chosen to equal 30 ∼ 40 % of aest. The ending point of the wave (j) is the starting
point of the next wave (j + 1) in the series.
After the signal peak aest is the wave back with an elongating tail toward the front of the coming
wave. The wave back is constructed of several lines with decreasing slopes. Each separate line usually consists of ns samples, as long as an interval. The steepest line is then extended to intersect the
0 2 4 6 0 2 4 6 8
Overtopping wave volume [m3/m]
Flow velocity [m/s] PW1 PW2 PW3 PW4 PW5 PW6 PW7 PW8
(a) Flow velocity
0 2 4 6 0 0.2 0.4 0.6 0.8
Overtopping wave volume [m3/m]
Flow depth [m]
SB1 SB8
(b) Flow depth
Figure 2.13: Run-up flow velocities and depths of 24 wave volumes.
0 2 4 6 0 2 4 6 8
Overtopping wave volume [m3/m]
Flow velocity [m/s] PW1 PW2 PW3 PW4 PW5 PW6 PW7 PW8
(a) Flow velocity
0 2 4 6 0 0.2 0.4 0.6 0.8
Overtopping wave volume [m3/m]
Flow depth [m]
SB1 SB8
(b) Flow depth
Figure 2.14: Run-down flow velocities and depths of 24 wave volumes.
horizontal axis, the time sequence, at tcut(j). The overtopping time of the studied wave (j) is
Tovt(j) = tcut(j) − start(j) (2.2)
3.45 3.5 3.55 3.6 3.65 3.7 3.75 3.8 3.85 x 105 1.5 2 2.5 3 3.5 Time [s] Signal [Volt] SB1 start (j) wave peak (j + 1) a (j) start (j + 1) T cut (j) T ovt (j) T est (j) a est (j) wave peak (j) wave front wave back end (j) steepest line
Figure 2.15: Underlying principle of estimating the overtopping time of a wave.
from 1 to 5.5 m3/m are plotted in Figure 2.16. All estimated values of overtopping time using both
surfboards and paddle wheels for different wave volumes are given in Table A.5, Appendice A.
0 2 4 6 0 2 4 6 8 10
Overtopping wave volume [m3/m]
Overtopping time [s] SB1 SB2 SB3 SB4 SB5 (a) Surfboards 0 2 4 6 0 5 10 15
Overtopping wave volume [m3/m]
Overtopping time [s] PW1 PW2 PW3 PW4 PW5 PW6 PW7 (b) Paddle wheels
Figure 2.16: Estimation of overtopping times using surfboards and paddle wheels for different
wave volumes.
2.3
Belgium measurement
This section introduces information given in another technical report [Steendam et al., 2012]. Fig-ure 2.17 shows the measFig-urement set-up including two paddle wheels and one surfboard in front of the simulator. Figure 2.18 illustrates the signals of 22 waves in series K2.
Figure 2.17: Set-up of the measurement in Belgium.
Measured signal of two paddle wheels PW1 and PW2 are converted into SI units as in the Table 2.6. Signals measured by surfboard SB1 are converted in to SI units according to the following rules: • first a value of flow depth is converted as h = 0.38 · (V olt − 2.192)0.88;
• if h > 0.235 m, h is corrected as h = 0.235 + (h − 0.235) · c;
• c = (htoe− hlow)/htoe, with htoe = 0.39 · (V olt − 2.83 − 2.23)0.85 and hlow = 0.335 · (V olt −
2.83 − 2.26)0.50.
The relationships between values of flow parameters and wave volumes are given in Figure 2.19. The estimated values are given in Table A.6, Appendice A.
0 50 100 150 200 250 300 350 400 0 1 2 3 4 5 Time [s] Signal [Volt] PW1 SB1 PW2
Figure 2.18: Flow measurement in Belgium, signals of three devices.
Table 2.6: Convert functions of two paddle wheels PW1 and PW2.
Device Channel Convert function Unit
(1) (2) (3) (4) PW1 1 V olt ∗ 8.564 m/s PW2 3 V olt ∗ 8.564 m/s 0 2 4 6 0 2 4 6 8 10
Overtopping wave volume [m3/m]
Flow velocity [m/s]
PW1 PW2
(a) Flow velocity
0 2 4 6 0 0.1 0.2 0.3 0.4 0.5
Overtopping wave volume [m3/m]
Flow depth [m]
SB1
(b) Flow depth
Figure 2.19: Flow velocities of 22 wave volumes.
2.4
Measurements in Viet Nam
Materials presented in several technical reports such as Trung [2011, 2012] are incorporated into the current section to fully describe the flow measurement with the simulator in Viet Nam. Equipment might be different from what were applied in the Netherlands and Belgium, therefore analysis methods are not the same.
2.4.1
Devices and measurement method
Flows released from the Simulator are very turbulent with high density of air bubbles which normally make flows very white. Presently, there is no reliable device for this kind of flow in Viet Nam. During simulator tests, digital camcorders were applied in determining front velocities on the dike slopes. Based on the time duration for the front part of a flow to run over a given distance, the value of front velocity can be derived. Set-up of a camcorder at Yen Binh dike model is shown in Figure 2.20. The camcorder is mounted on a steel frame to capture the flow over a distance of 10 m. Besides, another
camcorder can be positioned aside to capture the flow as shown in Figure 2.21.
(a) Camcorder position (b) Support frame
Figure 2.20: Set-up of the camcorder to capture overtopping flow on the dike slope
A limited number of representative volumes were selected to measure the front velocity such as 1, 2, 3, 4 and 5 m3/m. Depending on the mean overtopping discharge 40, 60, 80 . . . l/s per m, waves
with small or large volumes can be generated during the corresponding storm. For a flow rate of 40 l/s per m, waves with volume of 2 m3/m can be hardly found but small volumes as 0.5 or 1 m3/m.
In contrast, a discharge of 80 l/s per m may contents of waves with large volumes of up to 4 or even 5 m3/m.
Figure 2.21: The flow captured by a camcorder positioned aside.
A floating device which is basically a light wooden board and can rotate around an axis was designed to measure the flow depth. Overtopping flow can push the free side of the board up a distance from the slope corresponding to the flow depth underneath. This upward movement results in an angle around the axis which is measured by a rotary encoder, left panel in Figure 2.22. The values of rotating angles are recorded by a computer. To determine flow depth, a convert factor from recorded signal to length needed to be validated beforehand. Flow depth measurements were performed properly in Thai Thuy in December 2010. Recorded signal was used in analysing flow depth and flow duration.
To perform hydraulic measurement a team of three staffs was required. One controlled the bottom valves and watched the water level in the Simulator. When the desired level was about reached, about 2 ∼ 3 seconds in advance, he signaled the other two to start recording with camcorders. A name card indicating volume and order of replication was hold in front of these two camcorders for a couple of seconds in order to document properly. The bottom valves were totally opened right when the desired volume was reached. Recording was stopped after the released volume had passed the measurement areas. The pumps discharge qphad to be measured beforehand, then time for filling the representative
(a) a rotary encoder (b) a wave board
Figure 2.22: A wave board connected to a rotary encoder were used in measuring flow depth
on Thinh Long dike slope.
out after the previous one that had been done while the two pumps were kept working stably giving a constant discharge. Measurement of each volume was repeated three times.
2.4.2
Image processing
Data processing is basically done by analysing movies of the overtopping flows on dike slope corre-sponding to certain volumes. Using some programs which are capable of reading files in .avi or .mts format to watch the flow flow movies in a slow speed or in sequence of frames, for instance 25 frames/ second. It can be seen in Figure 2.23, the front part of the flow traveled from xs= 8 m to 11 m within
a time of (t2− t1. The front velocity is then determined as the ratio of the distance and the time as:
u = ∆s ∆t =
s2− s1
t2− t1
(2.3)
(a) flow at t1 (b) flow at t2
Figure 2.23: A series of two images in which the front part of overtopping flow traveled over
a distance of 3 m within ∆t seconds.
2.4.3
Results
Thinh Long and Thai Tho sea dike
Figure 2.24 illustrates the relationship between overtopping volume and front velocity obtained at Thinh Long and Thai Tho sea dikes. Despite the large scattering of data in these figures, it is shown that increasing volume gives increasing front velocity on the dike slope. The data scattering can be explained by different positions along the dike slopes where front velocities were analysed.
0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 V/Hm02 us C /(gH m0 ) 0.5
Velocity on Thinh Long slope
TL1, sC = 3.5 m
TL1, sC = 6.5 m
TL3, sC = 6.5 m
(a) Thinh Long
0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 V/Hm02 us C /(gH m0 ) 0.5
Flow velocity on Thai Tho slope
TT2, sC = 3.5 m
TT3, sC = 3.5 m
TT2, sC = 6.5 m
TT3, sC = 6.5 m
(b) Thai Tho
Figure 2.24: Front velocities on the dike slope of different overtopping volumes, measured at
Thinh Long and Thai Tho sea dikes.
Flow depth achieved at TT02 and TT03 in Thai Thuy are plotted in Figure 2.25. At each slope section, measurements were conducted within 2 positions, x(2 ∼ 5) m and x(5 ∼ 8) m, results are symbolised with solid triangles and empty circles alternatively. In general, larger overtopping volume increases the corresponding flow depth on the dike slope.
0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 V/Hm02 hs C /H m0
Water−layer thickness on Thai Tho slope
TT2, sC = 3.5 m
TT3, sC = 3.5 m
TT2, sC = 6.5 m
TT3, sC = 6.5 m
Figure 2.25: Flow depth on dike slope as a function of overtopping volume, sections TT02
and TT03 of Thai Thuy sea dike.
Yen Binh dike model
Yen Binh slope was about 40 m long, front velocities were observed at three areas, from sC= 5 ∼ 8
m, 8 ∼ 11 m (the first camcorder), and 11 ∼ 15 m (the second camcorder). In which sC is the
distance from crest edge. Measurement was performed at the YB3 section which was covered with Carpet grass and had no obstacle. Figure 2.26 illustrates the variation of flow velocity along the slope with increasing wave volumes. Despite the small scattering of data in these figures, it is revealed that increasing volume gives increasing front velocity on the dike slope. Comparing these three figures indicates that along the slope, front velocities tend to increase slightly.
0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 V/Hm02 us C /(gH m0 ) 0.5
Flow velocity on Yen Binh slope
sC = 6.5 m
sC = 9.5 m
sC = 13.0 m
Formulas development
3.1
Overtopping experiments
A series of experiments was performed by Van Gent [2002] in order to establish a set of formulas quantifying the overtopping flow on sea dikes. The combinations of a 1/100 foreshore, a 1/4 seaward slope, two crest widths of 0.2 and 1.1 m, and two landward slopes of 1/2.5 and 1/4 resulted in four dike cross-sections A, B, C and D. Two kinds of material were applied to make dike surface smooth and rough, respectively. Specifications of five cross-sections are described in Table 3.1, where α is the seawards-side slope angle and β is the landward-side slope angle.
Table 3.1: Dike cross-sections tested in Van Gent [2002].
Series tan α tan β Crest level Crest width Surface
A 1:4 1:2.5 1.0 m 0.2 m smooth
B 1:4 1:4.0 1.0 m 0.2 m smooth
C 1:4 1:2.5 1.0 m 1.1 m smooth
D 1:4 1:4.0 1.0 m 1.1 m smooth
D’ 1:4 1:4.0 1.0 m 1.1 m rough
Velocity and water-layer thickness were measured at both crest edges (denoted as P1 and P2, respectively) and at three locations along the landward slope with increasing vertical distance to the crest P3 0.1 m, P4 0.25 m and P5 0.4 m. Flow velocity was measured with micro-impellers. Dike cross-sections and measurement positions are depicted in Figure 3.1.
Figure 3.1: Different dike cross-sections tested in Van Gent [2002].
at least 1000 waves. For each dike configuration from A to D there were 18 combinations of water levels (5 values), wave period (3 values) and wave spectral shapes (single-peak and double-peak). The geometry of the model dike and wave conditions are constrained by the following relationship:
• high crest level 0 < (Ru,2%− Rc)/Hs< 1.0;
• at the dike toe 0.28 < (Hs/htoe) < 0.46.
The obtained data set is reanalysed to relate the flow characteristics including depth and velocity with overtopping wave volumes in the coming sections.
3.2
At the seaward crest edge
3.2.1
Overtopping volume
The simulator was designed with regard to a distribution of overtopping waves [Pullen et al., 2007] on a dike crest. The velocity and water-layer thickness associated with each wave volume are considered as input for the simulator to produce an identical flow overtopping the crest and then running down the landward slope. Accordingly the flow characteristics including velocity, water-layer thickness and overtopping duration are assumed to be related to the corresponding wave volume.
With the assumption that flows generated by the simulator are similar to those produced by overtopping waves, experimental results of the later might be used in examining the former. Among other parameters, values of overtopping volume can be a comparative criterion of these two kinds of flow. First of all, measured values of overtopping volumes should be compared with the distribution proposed in Pullen et al. [2007] that the simulator is based on.
Data given in Van Gent [2002] are plotted together with calculations according to Pullen et al. [2007] in Figure 3.2. Experimental results seem to match relatively well the computed values. Condi-tions of similarity between flow by wave overtopping and the simulator are partly satisfied. Therefore, it is feasible to analyse this data set further to quantify the overtopping and simulator flows.
0 0.5 1 0 0.2 0.4 0.6 0.8 1 V2% / Hm02 computation V2% / H m0 2 measurement series A series B series C series D series Dr
Figure 3.2: Overtopping volumes measured by Van Gent [2002] versus computations with
Pullen et al. [2007], range of measurement 0 ≤ V
2%/H
2m0
≤ 0.89.
3.2.2
Water-layer thickness
Assume that the seaward slope is infinitively long, the water-layer thickness along the slope gets maximum value at the moment when the run-up water reaches the Ru,2% level as shown in Figure 3.3.
At this steady momment, the part of the run-up water above the dike crest level is considered equal to the water volume that might overtop the crest. Besides, the water-layer thickness is assumed to linearly decrease upward the slope. Therefore, the unit overtopping volume can be calculated as the area of a triangle above the crest level as follow:
VA=
1 2hA
Ru,2%− Rc
with α is the angle between the seaward slope and the horizontal. Then the water-layer thickness at the seaward edge, the shortest side of the triangle, can be written as
hA=
2 · VA· sin α
Ru,2%− Rc
(3.2)
(a) Water-layer thickness (b) Flow velocity
Figure 3.3: Water-layer thickness and flow velocity on the seaward slope, definition sketch.
The relationship between two sides of Equation 3.2, the left measured by Van Gent [2002] and the right calculated with Pullen et al. [2007] is determined by performing a linear regression analysis, thus resulting in hA,2%= chA V2%· sin α Ru,2%− Rc (3.3) in which ch
A is 1.1 with 95% confidence bounds of 1.06 and 1.14, respectively. Application range
of Equation 3.2 falls within 0 ≤ hA,2%/Hm0≤ 0.18.
3.2.3
Flow velocity
On the seaward slope, the flow velocity uAdecreases when running up the seaward slope and becomes
zero at Ru,2% due to gravity force and friction. If the friction effect is neglected, we can have the
following equation system
uv A− gt = 0 uv At − gt2 2 = Ru,2%− Rc (3.4) where uv
Ais the vertical component of uA as shown in Figure 3.3(a). Solve the system, we have
t =p2(Ru,2%− Rc)/g
uvA=p2g(Ru,2%− Rc)
(3.5) At the seaward crest edge, the horizontal component of uA is given as follow, the superscript (h)
denoting horizontal is left out for simplicity
uA= uvA· cot α = cot α
q
2g(Ru,2%− Rc) (3.6)
With the assumption that the overtopping flow on the dike crest is supercritical, uA is modified
to relate topghA,2% as follow
uA,2%= cot α s g(Ru,2%− Rc) hA,2% Hm0 (3.7) The numerator is multiplied by the water-layer thickness hA,2% whilst the denominator is
multi-plied by the wave height Hm0to keep dimensionally equal. The water-layer thickness hA,2%is replaced
with Equation (3.3) and chA is left out, then Equation (3.7) is written as
uA,2% = cos α
r gV2%
sin αHm0
Similar to water-layer thickness, the relationship between two sides of Equation 3.8 is determined by performing a linear regression analysis, thus resulting in
uA,2%= cuA· cos α
r gV2%
sin αHm0
(3.9) in which cuA is 0.88 with 95% confidence bounds of 0.85 and 0.90, respectively. Note that Equa-tion (3.9) is applicable when the velocities satisfy 0 ≤ uA,2%/
√ gHm0≤ 1.49. 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 cAh V2% sinα / (Ru,2% − Rc) / Hm0 hA,2% / H m0 measurement
(a) Water-layer thickness
0 0.5 1 1.5 2 0 0.5 1 1.5 2 c A u cosα/H m0 (V2%/sinα) 0.5 u A,2% /(gH m0 ) 0.5 measurement series A series B series C series D series Dr (b) Flow velocity
Figure 3.4: Water-layer thickness and flow velocity measured by Van Gent [2002] vs. results
computed with Equations (3.3) and (3.9) using c
hA= 1.1 and c
uA= 0.88, respectively.
The values of water-layer thickness and flow velocity from Van Gent [2002] are now compared to hA,2% and uA,2% calculated with Equations (3.3) and (3.9), respectively in Figure 3.4.
3.3
On the crest
The water amount subtracted from a run-up flow would come over the crest level, thus resulting in an overtopping flow. The water left over might run further on the seaward slope but is not going to pass the crest level; and then flows downward as wave run-down. Evaluating the measured data Van Gent [2002] found that roughness affects the velocities but not the thickness and the crest width affects the velocity but not the water-layer thickness on the crest. Here, the surface roughness and the position along the crest are hypothesised to influence both these two flow parameters. Sketch of the overtopping flow on the dike crest is given in Figure 3.5.
(a) Water-layer thickness (b) Flow velocity
Figure 3.5: Water-layer thickness and flow velocity on a dike crest, definition sketch.
3.3.1
Effect of position
When propagating on the crest, the water mass tends to become flatter to spread over the crest if the friction is small and its velocity is usually decreased. Variation of water-layer thickness and flow
velocity on the crest are illustrated in Figures 3.6. 0 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0 x B / Lm−1,0 (h B,2% − h A,2% ) / h A,2%
(a) water-layer thickness
0 0.5 1 −0.4 −0.2 0 0.2 0.4 0.6 x B / Lm−1,0 (u B,2% − u A,2% ) /u A,2% series A series B series C series D (b) flow velocity
Figure 3.6: Variation of the water-layer thickness and flow velocity on the crest measured by
Van Gent [2002].
Some other authors such as Van Gent [2002] and an der MeerVAN DER MEER et al. [2010a] suggested an exponential function to express the steady decrease in flow velocity and water-layer thickness along the dike crest. Distance to the seaward crest edge xB is related to different length
parameters including crest width B and wave length. In line with this concept, the exponential function is deployed here to satisfy the assumptions that
• at the seaward crest edge xB = 0, hxB=0,2% = hA,2% and uxB=0,2% = uA,2%
• at the end of an infinitively long crest xB= +∞, hxB=0,2%= 0 and uxB=0,2% = 0.
Water-layer thickness
As can be seen in Figure 3.6, water-layer thickness not clearly decrease or increase along the crest. In case of a smooth crest, for the sake of simplicity, the water-layer thickness is assumed to lightly decrease along the crest. To do so, a coefficient presenting the effect of position on the dike crest is introduced exp(bh BxB/Lm−1,0), thus resulting in hxB,2%= exp bhB xB Lm−1,0 hA,2% (3.10) or in long form hxB,2%= exp bhB xB Lm−1,0 chA V2%· sin α Ru,2%− Rc (3.11) A multiple linear regression analysis of experimental results was conducted giving bh
B = -2.24.
Flow velocity
As shown in Figure 3.6, the flow velocity is likely to decrease slightly along the dike crest. Similar to water-layer thickness, an exponential function is proposed to present the variation of the flow velocity on the dike crest as follow
uxB,2%= exp buB xB Lm−1,0 uA,2% (3.12) or in long form
uxB,2%= exp buB xB Lm−1,0 cuAcos α r gV2% sin αHm0 (3.13) where, bu
B is an empirical coefficient which are determined by performing a multiple linear
anal-ysis of data measured by Van Gent. In comparison, the variation of the flow velocity on the crest was determined analytically depending on the momentum equation by Sch¨uttrumpf [2001] [see also Schuttrumpf and van Gent, 2003]. The equation was verified by experimental data derived from measurements on scale model tests.
0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 h B,2% / Hm0 computation h B,2% / H m0 measurement
(a) water-layer thickness
0 0.5 1 1.5 2 0 0.5 1 1.5 2 u B,2% /(gHm0) 0.5 computation u B,2% /(gH m0 ) 0.5 measurement series A series B series C series D series Dr (b) flow velocity
Figure 3.7: Water-layer thickness and flow velocity at the landward crest edge, measurement
of Van Gent vs. results computed with Equations (3.11) and (3.13) using b
hB= -2.24 and b
uB= -1.82, respectively. Friction coefficients f
Bh= 0.1 and f
Bu= 0.3.
The experimental data are now compared to values obtained with Equations (3.11) and (3.13) in Figure 3.7 using bh
B = -2.24 and buB = -1.82, respectively.
3.3.2
Effect of friction
Comparing the results of configurations D with smooth surface and Dr with rough surface reveals the friction effect on water-layer thickness and flow velocity as shown in Figure 3.8. Velocities measured in series D are higher than in series Dr at every positions. In contrast, most of the values of water-layer thickness are smaller in series D. In general, the rough surface of Dr reduces the front velocity but facilitates the water-layer thickness.
−0.8 −0.6 −0.4 −0.2 0 −0.2 −0.1 0 0.1 0.2 0.3 0.4 ∆ u ∆ h P2 P3 P4 P5
Figure 3.8: Comparisons of flow velocity and water-layer thickness along the landward slope
(P2, P3, P4 and P5 positions) between rough and smooth configurations Dr and D,
respec-tively.
The twofold effect implies an assumption of an empirical coefficient fh
B representing the influence
of the roughness on the water-layer thickness and fu
B for the flow velocity. As a result, Equations
(3.10) and (3.12) are adjusted to take into account the friction effect as follows hxB,2%= exp bhB xB Lm−1,0 (1 + fBh)hA,2% (3.14) uxB,2%= exp buB xB Lm−1,0 (1 − fBu)uA,2% (3.15)
The friction coefficients fBh and fBu are estimated by conducting regression analyses of the data measured in series Dr and D. The obtained results are
fh
B= 0.1
fBu= 0.3 (3.16)
It can be seen that roughness exerts more considerable effect on the flow velocity than water-layer thickness. It is worth noting that Van Gent [2002] stressed on the roughness effects on the velocities but not the thickness. Deploy these above values into Equations (3.14) and (3.15), obtained results are plotted together with the measured values of Van Gent [2002] in Figure 3.7.
3.4
On the landward slope
Sketch of the overtopping flow on the landward slope is given in Figure 3.9, where yC and sC are the
distance to the crest edge in vertical direction and on the slope, respectively.
(a) Water-layer thickness (b) Flow velocity
Figure 3.9: Water-layer thickness and flow velocity on the dike crest, definition sketch.
3.4.1
Effect of position
Variation of the flow velocity and water-layer thickness measured by van Gent on the landward slope are plotted in Figure 3.10. In general, the water-layer thickness reduces while the flow velocity becomes larger in the downward direction. The amplitudes of variation are larger on steeper slopes, inclination of 1/2.5 compared to 1/4.
The variation of the flow velocity ∆u and the water-layer thickness ∆h along the landward slope are compared in Figure 3.11. The values of ∆u are positive while the values of ∆h are always negative. In the other words, flow velocity becomes larger whilst water-layer thickness tends to reduce when propagating downward. The amplitudes of the velocity variation are often more significant than those of the water-layer, difference can be on the order of two times.
The effect of position is taken into account by introducing empirical coefficients which include the influence of the landward slope (yC/ sin β) as follows
• expbh C
yC
tan βLm−1,0
for water-layer thickness • 1 + bu
C yC
tan βLm−1,0
0 0.5 1 −0.8 −0.6 −0.4 −0.2 0 ∆ h y C / tanβ / Lm−1,0 (a) water-layer thickness
0 0.5 1 0 0.5 1 1.5 2 ∆ u yC / tanβ / Lm−1,0 series A series B series C series D (b) flow velocity
Figure 3.10: Variation of water-layer thickness and flow velocity on the landward slope,
measurement of Van Gent [2002].
0 0.5 1 1.5 2 −0.8 −0.6 −0.4 −0.2 0 ∆ u ∆ h series A series B series C series D
Figure 3.11: Relationship between variation of the flow velocity ∆u and water-layer thickness
∆h on the landward-side slope.
Thus the water-layer thickness and the flow velocity at a position, which has a vertical distance from the crest level yC, are written as
hyC,2%= exp bhC yC tan βLm−1,0 hB,2% (3.17) uyC,2%= 1 + buC yC tan βLm−1,0 uB,2% (3.18)
In which, hB,2% and uB,2% are water-layer thickness and flow velocity calculated with
Equa-tion (3.11) and (3.13), respectively, at xB= B which is the crest width.
Measured data of water-layer thickness and flow velocity on the landward-side slope and calculation by Equations (3.17) and (3.18) are compared in Figure 3.12. Here bhC= -5.4 and buC= 2.82 are derived from multiple regression analyses.
3.4.2
Effect of friction
Coefficients representing friction effect on water-layer thickness fh
C and on flow velocity fCu are
em-ployed for Equation (3.17) and (3.18) as follows hyC,2%= exp bhC yC tan βLm−1,0 (1 + fCh)hB,2% (3.19)
0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 h yC,2%/ Hm0 computation h yC,2% / H m0 measurement
(a) water-layer thickness
0 1 2 3 4 0 1 2 3 4 uyC,2% / (gHm0)0.5 computation u yC,2% / (gH m0 ) 0.5 measurement series A series B series C series D series Dr (b) flow velocity
Figure 3.12: Water-layer thickness and flow velocity on the landward slope, measurement of
Van Gent [2002] vs. computation with Equations (3.17) using b
hC= -5.4 and Equation (3.18)
using b
uC= 2.82.
uyC,2% = 1 + buC yC tan βLm−1,0 (1 − fCu)uB,2% (3.20)In which, hB,2%and uB,2%are calculated with Equation (3.14) and (3.15) taking into account the
friction effect of the crest. Linear regression analysises of series Dr and D were performed giving the values of fCh and fCu as
fCh= 0.1
fCu= 0.1 (3.21)
With fh
C= fCu surface roughness is assumed to similarly affect the water-layer thickness and flow
velocity along the landward slope. Water-layer thickness and flow velocity computed with Equa-tions (3.19) and (3.20) are compared to measurements in Figure 3.13. Circle symbols represent series Dr with a rough surface while square dots are for configuration D constructed of smooth material.
0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 h yC, 2% / Hm0 computation h yC, 2% / H m0 measurement
(a) water-layer thickness
0 1 2 3 4 0 1 2 3 4 uyC, 2% / (gHm0)0.5 computation u YC, 2% / (gH m0 ) 0.5 measurement series Dr series D (b) flow velocity
Figure 3.13: Water-layer thickness and flow velocity on the landward slope, measurement of
Van Gent [2002] vs. computation with Equation (3.19) using f
Ch= 0.1 and Equation (3.20)
utilising f
Cu= 0.1.
All values of the friction coefficients due to surface roughness obtained from regression analyses are given in Table 3.2. Value of fh
the slope. These relationships can be used in estimating the friction coefficients of other dike surface materials in reality, e.g. a grass cover.
Table 3.2: Coefficients due to friction affecting flow depth and water-layer thickness.
Flow parameter Crest Landward slopeWater-depth layer h 0.1 0.1 Flow velocity u 0.3 0.1
Performing calculation with Equations of Sch¨uttrumpf [2001] shows that for a small friction co-efficient velocity increases gradually while water-layer thickness becomes smaller. On the contrary, a large friction causes velocity to decrease along a slope while to facilitate the thickness [e.g., Steendam et al., 2012]. Note that both Sch¨uttrumpf [2001] and Van Gent [2002] base their theory on steady state flow considerations. Their friction factors are different by nature and therefore determined differently with regard to those derived from regression analysis in the present study. More importantly, these values have obviously no physical meaning.
3.4.3
Relationship between flow velocity and water-layer thickness
Owning to the assumption of a steady flow condition along the landward slope, previous authors usually make use of a relationship between flow velocity and water-layer thickness, (hyC,2%× uyC,2%).
Resulted from the continuity equation, this term is considered constant and equal to (hB,2%× uB,2%)
at the crest edge. Experimental data are plotted against the dimensionless distance to the crest edge in Figure 3.14(a) showing that this product fluctuates widely on the landward slope. Obviously, there is hardly any trend supporting the continuity of a steady flow. Besides, the computed values of hyC,2%
and uyC,2% fit relatively well the measured results as in panel (b).
0 0.5 1 0 0.05 0.1 0.15 0.2 0.25 yC/tanβ/Lm−1,0 (hy C ,2% *uy C ,2% ) measurement
(a) Along the landward slope
0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25 (hy C,2% *uy C,2% ) computation (hy C ,2% *uy C ,2% ) measurement series A series B series C series D series Dr
(b) Water-layer thickness and flow velocity
Figure 3.14: Product of water-layer thickness and flow velocity on the landward slope,
mea-surement of Van Gent [2002] vs. computation with Equation (3.19) using f
Ch= 0.1 and
Equation (3.20) utilising f
Cu= 0.1.
In Chapter 2, we have seen how the simulator generates water flows which are expected to be similar to real overttopping flows on dike slopes. The set of equations developed above will be verified with measurements of flows released from the simulator in Chapter 4. In addition to checking the velocity u and depth h separately, the product of these two parameters, i.e. (h×u), should be examined along the slopes under investigation. Furthermore, comparing this term with what derived from real overtopping flows such as in a wave flume might probably help to assess the simulator performance.
3.5
Summary
The set of equations quantifying flow parameters, velocity u and water-layer thickness h, on a sea dike developed in the previous sections are summarised here. Note that these equations are based on the measurements of 2% exceedance probability so that to represent parameters of 2% of exceedance probability. In Chapter 4, we will see how they are addapted to the case of an arbitrary overtopping volume or an arbitrary value of exceedance probability.
At the seaward crest edge
When the effect due to the surface roughness of a landward dike slope is neglected, water-layer thickness and flow velocity can be expressed as
hA,2%= chA V2%sin α Ru,2%− Rc when 0 ≤ hA,2%/Hm0≤ 0.18 and uA,2% = cuAcos α r gV2% sin αHm0 when 0 ≤ uA,2%/ p gHm0≤ 1.49 with ch A = 1.1 and cuA = 0.88. On the crest
The variation of flow parameters along a horizontal dike crest are given by exponential functions. Water-layer thickness is written as
hxB,2%= exp bhB xB Lm−1,0 (1 + fBh)chA V2%sin α Ru,2%− Rc or in short form hxB,2%= exp bhB xB Lm−1,0 (1 + fBh)hA,2%
Flow velocity can be calculated with
uxB,2%= exp buB xB Lm−1,0 (1 − fBu)cuAcos α r gV2% sin αHm0 or in short form uxB,2%= exp buB xB Lm−1,0 (1 − fBu)uA,2% where bh B = −2.24, b u B = −1.82 and f u B= 3f h B.
On the landward slope Water-layer thickness hyC,2%= exp bhC yC tan βLm−1,0 hB,2%(1 + fCh) Flow velocity uyC,2% = 1 + buC yC tan βLm−1,0 uB,2%(1 − fCu) With bh C = -5.4, b u C = 2.82 and f u C= f h
C. In case of the same material applied for both the crest and
landward slope one might first estimate fh C = f
h
B. Values of all coefficients are obtained by performing
Applications of the new formulas
4.1
Equation adaptations
The operation of the simulator is principally based on the distribution of wave volumes during a storm characterised with a mean overtopping discharge, a wave height and a wave period. During some hydraulic measurements, different water volumes were released from the simulator creating flows running on dike crests and then grass covered slopes. The associated velocity and water-layer thickness (depth) of these flows were observed and measured at various points by using digital camcorders, paddle wheels and surfboards. The equations proposed in Chapter 3 are now adapted for any arbitrary wave volume instead of V2%that is exceeded by 2% of the coming waves.
For a wave condition including wave height (Hm0), wave periods (Tp, Tmand Tm−1,0) and a given
value of mean overtopping discharge q, the wave run-up level Ru,2% on a tan α-slope and the required
crest freeboard Rc are calculated with Equations(1.3) and (1.2).
From Rc one can calculate overtopping probability Pov (how many percentage of waves that
overtops) and scale factor a of the volume probability distribution with Equation (1.8). With a given value of overtopping volume V (x% of exceedance), the probability that a wave volume (chosen from waves that overtop) is less than V is given by (4.1)
PV = P (V ≤ V ) = 1 − exp " − V a 0.75# (4.1) With regard to the total number of incoming waves, the exceedance probability of a wave with volume V is
Pov,x%= (1 − PV) · Pov (4.2)
As aforementioned in Chapter 1, in case of V2% we have (1 − PV2%) = 0.02/Pov. Run-up level
Ru,x% associated with Pov,x% is calculated with Equation (4.3) as follow
Ru,x= Ru,2%·
r lnPov
ln0.02 (4.3)
Replace Ru,2%with Ru,x%in Equations (3.3) and (3.9) giving formulas estimating flow parameters
of any arbitrary value of exceedance probability x% as follows hA,x%= chA Vx%· sin α Ru,x%− Rc (4.4) uA,x%= cuA· cos α r gVx% sin αHm0 (4.5) The equations of hA,x% and uA,x% are used in examining the velocity and water-layer thickness
4.2
Measurements in the Netherlands and Belgium
In this section, the equations adapted in the previous section are applied in calculating flow velocity and water-layer thickness at various positions on sea dikes which will be then compared with mea-surements in the Netherlands and Belgium. The following is the wave conditions for designing the Dutch simulator, more details can be found in report of Van der Meer [2007].
• significant wave height Hm0= 2.0 m;
• wave period Tp = 5.7 s, Tm= Tp/1.2 = 4.75 s and Tm−1,0 = Tp/1.1 = 5.18 s;
• seaward-side slope tan α = 1/4.
With regard to the natural sequence, characteristics of overtopping flow are examined on the dike crest and then along the landward slope.
4.2.1
In front of the simulator
Equations (4.4) and (4.5) are applied to calculating flow velocity and water-layer thickness at the seaward crest edge of the 1/4-inclination slope for various values of discharge. The experimental conditions of Van Gent [2002] is recalled here
• high crest level 0 < (Ru,2%− Rc)/Hs< 1.0;
thus resulting in a flow rate not larger than 35 l/s per m.
The computed values are compared with measurement results achieved during the simulator pro-totype calibration test K-series [van der Meer, 2007] and at Belgium [Steendam et al., 2012] (given in Chapter 2) as graphically depicted in Figure 4.1.
0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 hA,x% /H m0 V/Hm02
Seaward crest edge, xB = 0 m
5 l/s per m 20 l/s per m 35 l/s per m 50 l/s per m calibration K Belgium SB1 van Gent
(a) water-layer thickness
0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 V/Hm02 uA,x% /(gH m0 ) 0.5
Seaward crest edge, xB = 0 m
5 l/s per m 20 l/s per m 35 l/s per m 50 l/s per m calibration K Belgium PW1 Belgium PW2 van Gent (b) flow velocity
Figure 4.1: Water-layer thickness and flow velocity measured during the simulator prototype
calibration test K and at Belgium vs. computed with Equations (4.4) and (4.5) at the seaward
crest edge. Wave height H
m0= 2.0 m, wave period T
p= 5.7 s and seaward slope tan α =
1/4.
Solid lines are drawn indicating the application range of the two Equations as • wave volume 0 ≤ V2%/Hm02 ≤ 0.89;
• water-layer thickness 0 ≤ hA,2%/Hm0≤ 0.18;
• flow velocity 0 ≤ uA,2%/
√
gHm0≤ 1.49.
Different values of discharge give different curves presenting the variation of water-layer thickness with increasing wave volume but only one common tendency of flow velocity. Larger flow rates generate higher values of water-layer thickness, e.g. curve of 35 l/s per m lies above curve of 5 l/s per m. In general, the experimental results of hA,2% fall well within the computed lines. While Equation (4.5)
4.2.2
Vechtdike slope
Measurement positions on Vechtdijk slope are given in Figure 2.2 and Table 2.1, Chapter 2. Here and in later calculations, a discharge of 35 l/s per m is chosen to illustrate the relationship between volume V and the associated flow characteristics u and h.
At the landward crest edge
Using Equation (3.14) to estimate the water-layer thickness at the landward crest edge, xB is
about 3 m from the simulator. Computed values are compared with measurement of surfboard SB5 as depicted in Figures 4.2 with fh
B= 0.1. At this position, there is no experimental data of flow velocity.
0 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2 0.25 V/Hm02 hx B ,2% /H m0
Landward crest edge xB = 3.0 m, fBh = 0.1
35 l/s per m 100 l/s per m SB5
Figure 4.2: Water-layer thickness measured at the Vechtdijk crest edge x
B= 3 m away from
the simulator vs. values computed with Equation (3.14). Wave conditions H
m0= 2.0 m, T
p= 5.7 s; seaward slope tan α = 1/4; crest width B = 3 m and f
Bh= 0.1.
Equation (3.14) seems to overestimate the water-layer thickness of waves with VA,2%/Hm02 < 0.5.
The curve of 100 l/s per m is also plotted showing that the Equation gives higher values of flow depth for larger discharges.
On the landward slope
Water-layer thickness and flow velocity at 4 m and 12 m from the crest edge are computed with Equations (3.19) and (3.20), respectively. The obtained results are then plotted together with the data measured by surfboards and paddle wheels in Figures 4.3 and 4.4. Different values of friction coefficients are used on the crest and dike slope, fBh = fBu/3 = 0.3/3 and fCh = fCu = 0.1. The assumption of these friction coefficients are inspired by the regression analysis in Chapter 2 where roughness exerts more considerable effect on velocity than water-layer thickness along a dike crest. For the sake of simplicity, two slope parts with sin β1 = 0.25 and sin β2 = 0.19 are assumed to have
similar friction coefficients fh C and f u C. 0 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2 0.25 V/Hm02 hy C ,2% /H m0
Landward slope sc = 4 m and fCh = 0.1
35 l/s per m 100 l/s per m SB3
(a) distance to crest 4 m
0 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2 0.25 V/Hm02 hy C ,2% /H m0
Landward slope sc = 12 m and fCh = 0.1
35 l/s per m 100 l/s per m SB1
(b) distance to crest 12 m
Figure 4.3: Water-layer thickness measured at 4 and 12 m from the Vechtdijk crest edge vs.
values computed with Equation (3.19). Wave conditions H
m0= 2.0 m, T
p= 5.7 s; seaward
slope tan α = 1/4; crest width B = 3 m and f
Bh= 0.1; landward slope sin β
1= 0.25, sin β
20 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 V/Hm02 uy C ,2% /(gH m0 ) 0.5
Landward slope sC = 4 m and fCu = 0.1
35 l/s per m 100 l/s per m PW3b PW3a
(a) distance to crest 4 m
0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 V/Hm02 uy C ,2% /(gH m0 ) 0.5
Landward slope sC = 12 m and fCu = 0.1
35 l/s per m 100 l/s per m PW1
(b) distance to crest 12 m
Figure 4.4: Flow velocity measured at 4 m and 12 m from the Vechtdijk crest edge vs. values
computed with Equation (3.20). Wave conditions H
m0=2 m, T
p= 5.7 s; seaward slope tan α
= 1/4; crest width B = 3 m and f
Bu= 0.3; landward slope sin β
1= 0.25, sin β
2= 0.19 and
f
Cu= 0.1.
At sC = 4 m, the flow can be quantified relatively well with Equations (3.19) and (3.20). Further
away from the crest edge at sC = 12 m, the measured values of flow velocity are slightly higher
than the computed results while the estimated water-layer thickness curve is much lower than the experimental data.
4.2.3
Tholen slope
At the landward crest edgeMeasurements at Tholen crest edge are compared with calculation using Equations (3.14) and (3.15). Two values of friction coefficient fBh = 0.03 and fBu = 0.1 are applied presenting the effects on water-layer thickness and flow velocity due to the roughness of a 2-m-wide crest covered with grass. Note that only the curves of 35 and 100 l/s per m are shown for the sake of simplicity and clarity.
0 0.5 1 1.5 0 0.05 0.1 0.15 0.2 0.25 V/Hm02 hx B ,x% /H m0
Landward crest edge xB = 2.04 m and fBh = 0.03
35 l/s per m 100 l/s per m SB1 van Gent
(a) water-layer thickness
0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 V/Hm02 ux B ,x% /(gH m0 ) 0.5
Landward crest edge xB = 2.04 m and fBu = 0.1
35 l/s per m 100 l/s per m PW2 van Gent
(b) flow velocity
Figure 4.5: Water-layer thickness and flow velocity measured at Tholen landward crest edge
x
B= 2.04 m vs. computed with Equations (3.14) and (3.15). Wave conditions H
m0= 2 m,
T
p= 5.7 s; seaward slope tan α = 1/4, f
Bh= 0.03 and f
Bu= 0.1.
On the landward slope
Equations (3.19) and (3.20) are applied in estimating flow parameters at several points with increasing distance to the crest edge sC = 3, 6, 9 and 12 m. The obtained results are plotted together
with measurements performed on Tholen slope which is assumed to have fh
C = 0.03 and f u
C = 0.03.
Figures 4.6 and 4.7 reveal that the further away from the crest edge, the closer might Equa-tion (3.20) estimate the flow velocity, while EquaEqua-tion (3.19), in contrast, predicts the water-layer thickness worse and worse.
