View
217
Download
0
Category
Preview:
Citation preview
Smooth slopes: dikes up to vertical walls
New formulae including zero freeboard andvery steep slopes
Chapter 5 in EurOtop
1.E-07
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
Rel
ativ
e ov
erto
ppin
g ra
te
q/(g
Hm
03 )0.
5(H
m0/(
L m-1
,0ta
n))0
.5/
b
Relative freeboard Rc/(Hm0m-1,0bfv)
straight, smooth, deepbermrough slopesoblique long crestedoblique short crestedshallow/bi-modalverticale wall on slopesteep foreshoresteep foreshore, bi-modalLWI-1:6; 2DLWI-3DEquation 4 Battjes/TAW
+5%
-5%
Ch. 5 Coastal dikes and embankment seawalls5.3 Wave run-up
gentle slopesvery shallow watervery steep slopes
5.4 Wave overtopping dischargeslow freeboardsvery steep upto vertical
5.5 Influence factorsb, f, βeffect of currents
5.6 Effect of wave walls5.7 Overtopping wave characteristics
overtopping wave volumesflow velocities and thickness
distinction with very shallow water
new
modifiednew
similarnewProf. De Rouck
modifiedmodified
Wave run-up – gentle slopes
0,10
%2 65.1 mfbm
u
HR with a maximum of
0,10
%2 5.140.1mb
fm
u
HR
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Rel
ativ
e w
ave
run-
up R
u2%
/(f
H
m0)
[-]
Breaker parameter bm-1,0 [-]
breaking waves non breaking waves
5%
5%
Eq. 5.1
Design approach, Eq. 5.3
Wave run-up – very shallow water
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 2 4 6 8 10 12 14 16
Rel
ativ
e w
ave
run-
up R
u2%
/Hm
0[-]
Breaker parameter m-1,0 [-]
5%
5% Eq. 5.1
Slopes 1:2.5 and 1:4 with very large breaker parameters.Threshold to very shallow water: sm-1,0 < 0.005.
Wave run-up - steep slopes up to vertical walls
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 2 4 6 8 10 12 14 16
Relativ
e wave run‐up
Ru2
%/H
m0[‐]
Breaker parameter m‐1,0 [‐]
gentle slopes
very shallow
steep to vertical slopes,see Fig. 5‐8
sm‐1,0=0.06
sm‐1,0=0.01slope 1:2
slope 1:1.5
slope 1:1slope 1.5:1
slope 2:1
limit: vertical wall
Steep to vertical: based on Victor et al (2012)
Wave run-up - steep slopes up to vertical walls
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
00.511.52
Relativ
e run‐up
Ru2
%/H
m0
Slope angle cot
slope 1:2 slope 1:1 vertical wall
5%
5%
Eq. 5.4
Eq. 5.5
6.1cot8.00
%2 m
u
HR
with a minimum of 1.8 and a maximum of 3.0
Wave overtopping discharges
)/exp( 030
mc
m
HbRagHq
)1HR(-4.70.06=
gH
q
vfbops
copb3
s
exptan
)1HR(-2.30.=
gH
q
fs
c
3s
exp2
with as maximum:
breaking waves
non-breaking waves
Principal formula (Owen, 1980):A straight line on log-linear paper
1.E-07
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
Rel
ativ
e ov
erto
ppin
g ra
te
q/(g
Hm
03 )0.
5(H
m0/(
L m-1
,0ta
n))0
.5/
b
Relative freeboard Rc/(Hm0m-1,0b fv)
straight, smooth, deep
berm
rough slopes
oblique long crested
oblique short crested
shallow/bi-modal
verticale wall on slope
steep foreshore
steep foreshore, bi-modal
LWI-1:6; 2D
LWI-3D
+5%
-5%Problem: zero orlow freeboards
Wave overtopping: breaking waves (gentle slopes)
1.E-07
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
Rel
ativ
e ov
erto
ppin
g ra
te
q/(g
Hm
03 )0.
5(H
m0/(
L m-1
,0ta
n))0
.5/
b
Relative freeboard Rc/(Hm0m-1,0b fv)
straight, smooth, deep
berm
rough slopes
oblique long crested
oblique short crested
shallow/bi-modal
verticale wall on slope
steep foreshore
steep foreshore, bi-modal
LWI-1:6; 2D
LWI-3D
Equation 6
+5%
-5%
Wave overtopping: non-breaking waves (steep slopes)
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Rel
ativ
e ov
erto
ppin
g ra
te q
/(gH
m03 )
0.5
Relative freeboard Rc/(Hm0fb)
straight, smooth, deep
rough slopes
oblique long crested
oblique short crested
shallow/bi-modal; xi<6
vertical wall on slope
steep foreshore
steep foreshore
LWI-1:6 2D
108 Rc=0
Equation 7
+5%
-5%
From Exponential to Weibull
03
0
expm
c
mHR
bagHqEurOtop, Owen (1980):
exponential function
Weibull: shapeparameter c:c = 1: exponentialc = 2: Rayleigh
Fitting on EurOtop data gives c = 1.3Then a and b have to be re-fitted
c
m
c
mHRba
gHq
03
0
exp
New formulae in EurOtop Update
)1HR(-4.70.06=
gH
q
vfbops
copb3
s
exptan
)1HR(-2.30.=
gH
q
fs
c
3s
exp2with as maximum:
breaking waves
non-breaking waves
with as maximum:
EurOtop (2007)
EurOtop Update
3.1
00,10,13
0
7.2exptan023.0
vfbmm
cmb
mH
RHgq
3.1
03
0
5.1exp09.0 fm
c
mH
RHgq
Wave overtopping: vertical and steep slopes
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Rel
ativ
e ov
erto
ppin
g ra
te q
/(gH
m03 )0.
5
Relative freeboard Rc/Hm0
Vertical,no foreshore5% exceedance
Vertical, deep water: caissons; harbour flood walls, lock gates
Steep slope:
3.1
03
0
)5.1(exp09.0 fm
C
mH
RHg
q
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Rel
ativ
e ov
erto
ppin
g ra
te q
/(gH
m03 )0.
5
Relative freeboard Rc/Hm0
Smooth slope, non-breaking
Vertical, no foreshore
3.1
03
0
35.2exp047.0 fm
c
mH
RHgq
Data Ghent University (Victor et al. 2012)
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Rel
ativ
e ov
erto
ppin
g ra
te q
/(gH
m03 )0.
5
Relative freeboard Rc/Hm0
cota=2.75cota=2.14cota=1.73cota=1.43cota=1.19cota=1.0cota=0.84cota=0.58cota=0.36Smooth slope non-breakingVertical cota=0
Very steep slopes and large overtopping
Determinea and b
for each cotα
with as maximum:
a = 0.09 - 0.01 (2 – cot α)2.1 with a = 0.09 for cot α > 2
b = 1.5 + 0.42 (2 – cot α)1.5 with a maximum of b = 2.35 and b = 1.5 for cot α > 2
breaking waves
non-breaking waves
Applicable for Rc ≥ 0; f = 1 for very steep slopes
One set of formulae for gentle slopesup to vertical walls
3.1
00,10,13
0
7.2exptan023.0
vfbmm
cmb
mH
RHgq
3.1
03
0
exp fm
C
mH
RbaHg
q
Application; various cotα; sm-1,0=0.04
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Rel
ativ
e ov
erto
ppin
g ra
te q
/(gH
m03 )0.
5
Relative freeboard Rc/Hm0
Smooth slope non-breakingVertical cota=0
cotα = 10; 6; 4; 3; 2; 1.5; 1.0; 0.5; 0.33; 0.25
Application; various cotα; sm-1,0=0.04
cotα = 10; 6; 4; 3; 2; 1.5; 1.0; 0.5; 0.33; 0.25
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Rel
ativ
e ov
erto
ppin
g ra
te q
/(gH
m03 )0.
5
Relative freeboard Rc/Hm0
Smooth slope non-breakingVertical cota=0
Application; various cotα; sm-1,0=0.04
cotα = 10; 6; 4; 3; 2; 1.5; 1.0; 0.5; 0.33; 0.25
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Rel
ativ
e ov
erto
ppin
g ra
te q
/(gH
m03 )0.
5
Relative freeboard Rc/Hm0
Smooth slope non-breakingVertical cota=0
Application; various cotα; sm-1,0=0.04
cotα = 10; 6; 4; 3; 2; 1.5; 1.0; 0.5; 0.33; 0.25
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Rel
ativ
e ov
erto
ppin
g ra
te q
/(gH
m03 )0.
5
Relative freeboard Rc/Hm0
Smooth slope non-breakingVertical cota=0
Application; various cotα; sm-1,0=0.04
cotα = 10; 6; 4; 3; 2; 1.5; 1.0; 0.5; 0.33; 0.25
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Rel
ativ
e ov
erto
ppin
g ra
te q
/(gH
m03 )0.
5
Relative freeboard Rc/Hm0
Smooth slope non-breakingVertical cota=0
Application; various cotα; sm-1,0=0.04
cotα = 10; 6; 4; 3; 2; 1.5; 1.0; 0.5; 0.33; 0.25
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Rel
ativ
e ov
erto
ppin
g ra
te q
/(gH
m03 )0.
5
Relative freeboard Rc/Hm0
Smooth slope non-breakingVertical cota=0
Application; various cotα; sm-1,0=0.04
cotα = 10; 6; 4; 3; 2; 1.5; 1.0; 0.5; 0.33; 0.25
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Rel
ativ
e ov
erto
ppin
g ra
te q
/(gH
m03 )0.
5
Relative freeboard Rc/Hm0
Smooth slope non-breakingVertical cota=0
Application; various cotα; sm-1,0=0.04
cotα = 10; 6; 4; 3; 2; 1.5; 1.0; 0.5; 0.33; 0.25
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Rel
ativ
e ov
erto
ppin
g ra
te q
/(gH
m03 )0.
5
Relative freeboard Rc/Hm0
Smooth slope non-breakingVertical cota=0
Application; various cotα; sm-1,0=0.04
cotα = 10; 6; 4; 3; 2; 1.5; 1.0; 0.5; 0.33; 0.25
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Rel
ativ
e ov
erto
ppin
g ra
te q
/(gH
m03 )0.
5
Relative freeboard Rc/Hm0
Smooth slope non-breakingVertical cota=0
Application; various cotα; sm-1,0=0.04
cotα = 10; 6; 4; 3; 2; 1.5; 1.0; 0.5; 0.33; 0.25
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Rel
ativ
e ov
erto
ppin
g ra
te q
/(gH
m03 )0.
5
Relative freeboard Rc/Hm0
Smooth slope non-breakingVertical cota=0
Application; various cotα; sm-1,0=0.04
cotα = 10; 6; 4; 3; 2; 1.5; 1.0; 0.5; 0.33; 0.25
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Rel
ativ
e ov
erto
ppin
g ra
te q
/(gH
m03 )0.
5
Relative freeboard Rc/Hm0
Smooth slope non-breakingVertical cota=0
Wave overtopping – influence of cot
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
0 1 2 3 4 5 6 7 8
Dim
ensi
onle
ss o
vert
oppi
ng q
/(gH
m03 )0
.5
Slope angle cot
Wave steepness sm-1,0=0.04
Wave steepness sm-1,0=0.01
Rc/Hm0 = 2.5
Influence factors – effect of currentsb, f, β: no real changes
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 10 20 30 40 50 60 70 80 90
Influ
ence factor
Wave direction [°]
Wave run‐up; short‐crested waves; recommended
Wave overtopping; short‐crested waves; recommended
Wave overtoppinglong‐crested waves; special application
Influence factors – effect of currents
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-60 -40 -20 0 20 40 60
Influ
ence
fact
or
Combined angle of attack 0.5(+e) (degr)
Slope 1:3Slope 1:6
u
u
U
Un = Usinβ
Dike
U
cg relative
e
Dike
No significant effect
Best if in is replaced by: + e)/2
Overtopping wave characteristics
Random in time
‐0.20
‐0.15
‐0.10
‐0.05
0.00
0.05
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0 10 20 30 40 50 60 70 80 90 100 110 120
Water su
rface elevation (m
)
Flow
thickn
ess a
t the
crest (m
)
Time (s)
Flow thicknesswater surface elevation
Overtopping in reality
Overtopping wave volume
0.00
0.05
0.10
0.15
0.20
0
1
2
3
4
5
6
7
8
105 106 107 108 109 110 111 112
Flow
thickn
ess (m)
Velocity (m
/s)
Time (s)
Velocity
Flow thickness Overtopping wave volume V = 0.9 m3 per m
hvtovt
Overtopping wave volume, VFlow velocity, vFlow thickness, h
Overtopping wave volumes
P % P V V expVa ∙ 100%
a1
Γ 1 1b
qTP EurOtop (2007): b = 0.75
EurOtop Update:
b=0.73+55q
gHm0Tm−1,0
0.8
0.0
1.0
2.0
3.0
4.0
5.0
6.0
1.E‐06 1.E‐05 1.E‐04 1.E‐03 1.E‐02 1.E‐01 1.E+00
Weibu
ll b
Relative discharge q/(gHm0Tm‐1,0)
Smooth
Fit for b
Emergedstructures
Submergedstructures
Rc = 0
Rubble mound:
b = 0.85+1500q
gHm0Tm−1,0
1.3
0.0
1.0
2.0
3.0
4.0
5.0
6.0
1.E‐06 1.E‐05 1.E‐04 1.E‐03 1.E‐02 1.E‐01 1.E+00
Weibu
ll b
Relative discharge q/(gHm0Tm‐1,0)
SmoothClash rubble moundLow crested rubble moundEquation 5.38 smoothEquation 6.18 rubble mound
Pow < 5% with b > 1.4
Rc = 0
Overtopping wave volumes
0
500
1000
1500
2000
2500
3000
3500
4000
0 500 1000 1500 2000 2500
Ove
rtop
ping
wav
e vo
lum
e (m
3 /m)
Number of overtopping wave, in ascending order
0.1 l/s per m
1 l/s per m
10 l/s per m
50 l/s per m
100 l/s per m
200 l/s per m
Hs = 1 m; Tm-1,0 = 3.6sec
Overtopping wave volumes
Modifications to flow velocity and flow depth
Front velocity in wave run-up is quite constant during run-up. Breaking gives acceleration.
Velocities of overtopping wave volumes accelerate on the landward slope. Theory in EurOtop (2007) was validated by the Wave Overtopping Simulator with f = 0.01.
Recommended