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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)


berm

rough slopes



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)


rough slopes



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


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


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


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


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


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


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.

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