Download - Breakwaters and closure dams: chapter 5
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April 12, 2012
Vermelding onderdeel organisatie
1
Chapter 5: use of theory
ct5308 Breakwaters and Closure Dams
H.J. Verhagen
Faculty of Civil Engineering and GeosciencesSection Hydraulic Engineering
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Theoretical background needed• waterlevels (tides)• flow trough gaps• stability of floating objects• waves
• basics• refraction, shoaling, breaking, diffraction, reflection• wave statistics
• short term statistics (Rayleigh)• long term statistics
• Geotechnics• sliding• squeeze• liquefaction
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Initial tidal wave by the moon and the sun
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Adding semi-diurnal constants resulting in spring and neap tide
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Adding diurnal to semi-diurnal constant
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Amphidromy in the North Sea
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typical tides
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adding the fortnightly constant
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flow pattern in a gap
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Flow over a sill
subcritical flow
critical flow
2 ( )Q mBh g H h= −
2 ( )Q hu m g H hB a a
= = −
1Q = m B a 2 g H3
( ) ( )2 1Q m B H 2 g H3 3
=
( )1u m 2 g H3
=
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modelling
xQ H+ = 0Bx t
δ δδ δ
0x2
g Q QQ ( Q u) Hg A Wt x x A RC
δ δ α δδ δ δ
+ + + − =
Solving these equation by:•physical model•mathematical model
•2 d model•1 d model•storage area approach
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Physical model
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two dimensional model
Korea, Gaduk port, Mike21, DHIOosterscheldewerken, Waqua, Rijkswaterstaat/WL
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one-dimensional model
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storage/area approach
xQ H+ = 0Bx t
δ δδ δ
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validity of storage/area approach
length of tidal wave: L= c*T = √gh * T = √10*10 *12*3600= 432 km
basin < 0.05 L = 20 km
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equations for storage/area approach
31 2
2 3 3 1
2 1 3 1
2 ( ) ( )
23
2 23 3
g Rdh
A g H h B Q tdt
h h for h H
h H for h H
μ − = −
= >
= <
Ag and B can be combined to one input parameter
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parameters needed• water level in the sea• river discharge• ratio between storage area and width of closure gap• sill height• discharge coefficient of the gap
Assume for the time being that the river discharge is zero and that the tide is always semi-diurnal
Set the discharge coefficient of the gap to 1
Remaining parameters: • tidal difference• ratio storage area/gap width• sill height
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design graph for the velocity
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example of the use of a design graph
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velocity as a function of the closure
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Stability of a submerged object
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Stability of a floating object
.5 2 3.5
112
b
b
IMCV
I yx dx LB
GVgρ
+
−
=
= =
=
∫
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Definition of a regular wave
H
H wave heightT wave periodL wave length
2 2cos2x tH
L Tπ πη ⎛ ⎞= −⎜ ⎟
⎝ ⎠
2tanh2gL hc
Lπ
π⎛ ⎞= ⎜ ⎟⎝ ⎠
22
0 1.562gTL Tπ
= = c gh=
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validity for wave theories
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breaking
by steepness H/L< 0.14by depth H/h < 0.78 but…………….
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Irregular wave
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Rayleigh graph paper
2
2
( ) s
HH
P H H e
⎡ ⎤⎛ ⎞⎢ ⎥− ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦> =
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characteristic wave heights
Name Notation H/√m0 H/Hs
Standard deviation free surface ση=√m0 1 0.250 RMS height Hrms 2√2 0.706 Mean Height H = H1 2√ln 2 0.588
Significant Height Hs= H1/3 4.005 1 Average of 1/10 highest waves H1/10 5.091 1.271 Average of 1/100 highest waves H1/100 6.672 1.666 Wave height exceeded by 2% H2% 1.4
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characteristic wave periods
Name Notation Relation to spectral moment
T/Tp
Peak period Tp 1/fp 1 Mean period Tm √(m0/m2) 0.75 to 0.85 Significant period Ts 0.9 to 0.95
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typical types of wave statistics patterns
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H/T-diagram
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waves in shallow water
shoalingrefractionbreakingdiffractionreflection
( ) ( )( )
0
1 1 4 /tanh 2 / 1
sinh 4 /
shH k
h LH h Lh L
πππ
= + =+
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the iribarren number(surf similarity parameter)
0
tanH Lαξ =
tan α slope of the shoreline/structureH wave heightL0 wave length at deep water
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breaker types (2)
spilling ξ < 0.5
plunging 0.5 < ξ < 3
collapsing ξ = 3
surging ξ > 3
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breaking waves
20.142 tanhbH L hLπ⎛ ⎞= ⎜ ⎟
⎝ ⎠
0.78 ( )bH solitarywaveh≈
0.4 0.5sHh≈ −
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change of distribution in shallow water
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Battjes Jansen method
{ }( )
2
11
3.6
22
( ) 1 exp
Pr
1 exp
tr
tr
HF H H HH
H HHF H H HH
⎧ ⎡ ⎤⎛ ⎞⎪ ⎢ ⎥= − − ≤⎜ ⎟⎪ ⎢ ⎥⎝ ⎠⎪ ⎣ ⎦≤ = ⎨
⎡ ⎤⎪ ⎛ ⎞⎢ ⎥= − − >⎪ ⎜ ⎟⎢ ⎥⎝ ⎠⎪ ⎣ ⎦⎩
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Influence of shallow water on the wave height
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Wave refraction
22 1
1
sin sincc
α α⎛ ⎞
= ⎜ ⎟⎝ ⎠
2 1
1 2
H bH b
=
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Diffraction behind a detached breakwater
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reflection
20.1Rr
I
HKH
ξ= ≈
tot i rη η η= + = ( ) ( ) ( ) ( ) ( ) ( )2 2 2 21 cos *cos 1 sin *sin2 2
i iH Hx t x tr rL T L Tπ π π π+ + −
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Example with Cress
run demo Cress
refractionshoaling, etcdiffraction
x(50-200;4)
y (-200,200)
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The effect of shoaling on wave parameters
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Typical wave record of the North Sea
( ) 212 iS a
ω
ω ωΔ
= Δ∑
( ) ( )cos 2i i it a f tη π ϕ= +∑
0 13.5%4sH m H= =
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Vermelding onderdeel organisatie
46
Spectral wave periodsThe use of different wave parameters to obtain better results for wave structure interaction
ct5308 Breakwaters and closure dams
H.J. Verhagen
Faculty of Civil Engineering and GeosciencesSection Hydraulic Engineering
( )0
nnm f S f df
∞
= ∫
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Example wave record
28 waves, Hs = "13% wave", Hs= wave nr 4, Hs ≈ 3.828 waves in 150 seconds, so Tm = 5.3 s
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composition of the record
H1 = 0.63 m T1= 4 sec
H2 = 1.80 m T2 = 5 sec
H3 = 1.55 m T3 = 6.67 sec
H4 = 0.90 m T4 = 10 sec Tm = 5.3 sec
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Spectrumdiscretised spectrum
0
1
2
3
4
5
6
7
0,1 0,15 0,2 0,25
frequency (Hz)
ener
gy d
ensi
ty (m
2 s)
energy density spectrum
0
1
2
3
4
5
6
7
0 0,1 0,2 0,3 0,4
frequency (Hz)
ener
gy d
ensi
ty (m
2 s)
212
a S f= ⋅Δ
2 221.558 6 [ ]
8 8 0.05HH S f S m s
f= ⋅Δ = = =
Δ ⋅
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Calculation of m0
0.60
0.050.05*1
0.150.05*3
0.300.05*6
0.100.05*2
04 3.1m m=
discretised spectrum
0
1
2
3
4
5
6
7
0,1 0,15 0,2 0,25
frequency (Hz)
ener
gy d
ensi
ty (m
2 s)
( )0
nnm f S f df
∞
= ∫
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Calculation of m1
dist * SΔf
0.098
0.0130.25*0.05
0.0300.20*0.15
0.0450.15*0.30
0.0100.10*0.10
discretised spectrum
0
1
2
3
4
5
6
7
0,1 0,15 0,2 0,25
frequency (Hz)
ener
gy d
ensi
ty (m
2 s)
( )0
nnm f S f df
∞
= ∫
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Calculation of m2 discretised spectrum
0
1
2
3
4
5
6
7
0,1 0,15 0,2 0,25
frequency (Hz)
ener
gy d
ensi
ty (m
2 s)
dist2 * SΔf
1.69 10-3
3.12 10-30.252*0.05
6.00 10-30.202*0.15
6.75 10-30.152*0.30
1.00 10-30.102*0.10
0
2
0.6010 5.69sec1.69
mTm
= = =
( )0
nnm f S f df
∞
= ∫
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Calculation of m-1 discretised spectrum
0
1
2
3
4
5
6
7
0,1 0,15 0,2 0,25
frequency (Hz)
ener
gy d
ensi
ty (m
2 s)
1/dist * SΔf
3.95
0.201/0.25*0.05
0.751/0.20*0.15
2.01/0.15*0.30
1.01/0.10*0.10
11,0
0
3.95 6.58 sec0.60m
mTm−
− = = =
( )0
nnm f S f df
∞
= ∫
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Overview
•Hm0 = 3.1 m (1.55+1.10+0.90+0.63=4.18)•Tm0 = 5.69 sec•Tm-1,0 = 6.58 sec•Tpeak = 6.67 sec
•
•Tm = 5.35 sec
•
− = =1,0
0
6.58 1.165.69
m
m
TT
For standard spectra:
Goda: Tp=1.1 T1/3
PM: Tp=1.15 T1/3
Jonswap: Tp=1.07 T1/3
TAW (vdMeer): Tp=1.1Tm-1,0
Old Test (vdMeer): Tp=1.04 Tm-1,0
Also: Tm-1,0=1.064T1/3
= =0 5.69 1.065.35
m
m
TT
Usual assumptions:Tm0 = TpT1/3 = Tm
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Overview to determine shallow water wave condition
• Determine deep water wave condition, this gives wave height, peak period and spectrum shape type (e.g. Jonswap)
• Calculate shallow water condition using spectral model (e.g. with SWAN), this gives Hm0, Tm0 and Tm-1,0
• Use Battjes-Jansen method to determine H2%
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Why these parameters ?
( )0.2
0.250.182%1,0
50
cotpl mn
H Sc P s for plunging wavesd N
α−⎛ ⎞= ⎜ ⎟Δ ⎝ ⎠
( ) ( )0.2
0.25 0.50.132%1,0 1,0
50
P
s m sn
H Sc P s for surging wavesd N
ξ− −−
− −⎛ ⎞= ⎜ ⎟Δ ⎝ ⎠
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stress relations determined by soil testing
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Dam profile after the slide
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Squeeze
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Liquefied sand