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Coastal Sediment Transport
CEPD03
Lecture 3
Prof. Dano Roelvink
Dr. Jan van de Graaff
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Contents
• Overview of problems
• Bed shear stress by waves and current
• Sediment transport by waves and current
• Longshore sediment transport• Cross-shore sediment transport
• Coastline changes
• Dune erosion• Sedimentation in channels and trenches
• Stability of channels and inlets
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Exercise about lectures 1 and 2
• V=0.7 m/s
• H=1 m
• h=2 m• T=7 s
• D50=0.2 mm; D90=0.3 mm
• r=0.05 m• Compute
0, , , , , ,c w w cw b su f S S
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Longshore transport
• Simple method to estimate longshore
velocity
• Use estimated wave height in surf zone
• Derive radiation stress gradient from this
• Balance radiation stress gradient with
bottom shear stress
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Radiation stress
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Longshore wave forces
cos sin
yy xy
y
S
F x
n x
S
y
E
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Longshore wave forces
cos sin
sincos
sin
cos
sincos
xy g
y
g
g
g
S C F E x x C
EC x C
EC x C
EC
C x
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Longshore wave forces
sincos
sincos
sin( )
y g
g
w y
F EC x C
EC C x
D F
C
=0
(Snel’s
Law)
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Assignment
• Hs0=3m, T=8s,
• Angle of incidence:
• 75,60,50,45,40,30,20,10 deg.• Compute conditions at breaker line
• Compute longshore sediment transport
using coefficient B=0.04
0.7
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Contents
• Wave energy balance
• Dispersion relation
• Wave celerity and group velocity
• Snel’s Law • Shoaling and refraction
• Wave breaking
• Dissipation
• Solving 1D energy balance
• Radiation stresses and wave forces
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Wave energy balance
2 2 2
cos sin
1 1
2 8
g g
rms
E EC EC D
t x y
E g ga gH
Wave energy Group
velocity
Wave angle Wave
dissipation
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Dispersion relation
• Relation between wave period T and wave
length L for given water depth
2
tanh( )2
2
1
2 sinh(2 ) g
gk kh
T
Lk C L T k
d khC nC C
dk kh
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Deep water
22
tanh( ) 1 sinh( )
2
2
2
1
2 g
kh kh
gk k g
T
L g gT
k C L T k
d C nC C
dk
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Shallow water
2
tanh( ) sinh( )
( ) /
2
2
1
2 sinh(2 ) g
kh kh kh kh
gk kh k gh
T
Lk C g h
L T k d kh
C nC C ghdk kh
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Wave refraction
05102030
Depth
contours
Wave rays
Wave crests
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Snel’s Law
• Valid for straight contour lines
• Relates local wave angle to deep water
wave angle
0
0
sin sin
c c
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Wave breaking
• Complex process
• Various approximations, e.g.
– Battjes and Janssen, 1978
– Thornton and Guza, 1983
– Roelvink, 1993
– Baldock, 1998
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Baldock model
2
max
2 2
max
1
4ms
H
H
w p rms D g f e H H
max
0.88tanh( )
0.88
kh H
k
Coefficient
~1
Coefficient
~0.6-0.8
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1D Wave energy balance
cos sin g g w
E EC
t y EC D
x
Wave energy Group
velocity
Wave angle Wave
dissipation
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Solving 1D wave energy balance
• Outside surf zone:
– Wave energy flux is constant
– Group velocity follows from dispersion relation
– Wave angle follows from Snel’s Law
0 0 0
cos 0
cos cos
g
g g
E C
x EC E C
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Shoaling and refraction
• Shoaling is change in wave height due to
change in group velocity
• Mostly increasing towards shore
• Refraction is bending of wave rays
towards shore, leads to decrease of wave
height because energy is spread over
wider area
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Wave refraction
05102030
Depth
contours
Wave rays
Wave crests
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Solving 1D wave energy balance
• Inside surf zone:
– Group velocity follows from dispersion relation
– Wave angle follows from Snel’s Law
– Dissipation follows from e.g. Baldock, relating wavedissipation to wave energy and water depth
– Solve E numerically, starting from known value
outside breaker zone
cos ( , ) g w EC D E h x
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Solving 1D wave energy balance
• Very simple numerical scheme:
1 , 1 1 ,
,
1
, 1 1
cos cos( , )
cos ( , )
cos
i g i i i g i i
w i i
i g i i w i i
i
g i i
E C E C
D E h x
E C D E h x
E C
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Radiation stresses
xxS
xxS
yxS yxS
yyS
yyS
xyS
xyS
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Wave forces
xy xx x
xy yy
y
S S F x y
S S F x y
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Radiation stresses
2
2
1cos
2
cos sin
1sin
2
xx
xy yx
yy
g
S n n E
S S n E
S n n E
C n
C
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Wave forces
• Follow from radiation stress gradients
• Radiation stresses are function of wave
energy, wave direction and ratio wave
celerity to group velocity
• In 1D case we can compute all these
easily
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