ENGR 691, Fall Semester 2010-2011
Special Topic on Sedimentation Engineering
Section 73
Coastal Sedimentation
Yan Ding, Ph.D.
Research Assistant Professor, National Center for
Computational Hydroscience and Engineering (NCCHE),
The University of Mississippi, Old Chemistry 335,
University, MS 38677
Phone: 915-8969
Email: ding@ncche.olemiss.edu
Outline
• Introduction of morphodynamic processes driven by waves
and currents in coasts, estuaries, and lakes
• Initiation of motion for combined waves and currents
• Bed forms in waves and in combined waves and currents
• Bed roughness in combined waves and currents
• Sediment transport in waves
• Sediment transport in combined waves and currents
• Transport of cohesive materials in coasts and estuaries
• Mathematical models of morphodynamic processes driven
by waves and currents
• Introduction of a process-integrated modeling system
(CCHE2D-Coast) in application to coastal sedimentation
problems
Introduction
The transport of bed material particles by a flow of water can be in the form of bedload and suspended load, depending on the size of the bed material particles and the
flow conditions. The suspended load may also contain some wash load (usually,
d<0.05mm), which is generally defined as that portion of the suspended load which is
governed by the upstream supply rate and not by the composition and properties of
the bed material.
The sediment transport in a steady uniform current is assumed to be equal to the
transport capacity defined as the quantity of sediment that can be carried by the flow
without net erosion or deposition, given sufficient availability of bed material (no
amour layer).
Various types of formulae are available for predicting the bed-load and the
suspended load transport (q). The formulae can be divided in five main groups as
defined by the relevant hydraulic parameters:
m = 3 ~ 5 u :depth-averaged velocity
q  (u  ucr ) m
• Fluid velocity
n ≈ 1.5
ucr: critical shear velocity
q  (   cr ) n
• Bed shear stress
τ: bed shear stress
• Probabilistic particle movement
τcr: critical bed shear stress
• Bed form celerity
• Energetics (stream power) q   u
Three Modes of Particle Motions
Three particle motions:
• Rolling and sliding motion or both:
• Saltation:
• Suspended particle motion:
u*  ucr Bed shear velocity just exceeds the
critical value
u*  ucr Bed shear velocity further increases
u*  ws Bed shear velocity exceeds the
particle fall velocity
Bed-Load Transport Definition
Bagnold’s definition (1956): In the bed-load transport, the successive contacts of the
particles with the bed are strictly limited by the effect of gravity, while the suspendedload transport is defined as that in which the excess weight of the particles is supported
by random successions of upward impulses imported by turbulent eddies.
Einstein’s definition (1950): The bed-load transport is the transport if sediment particles
in a thin layer of 2 particle diameters thick just above the bed by sliding, rolling, and
sometimes by making jumps with a longitudinal distance of a few particle diameters.
Bed-load transport = the transport of particles by rolling, sliding, and saltating
Brigadier Ralph Alger Bagnold, OBE, FRS (3 April 1896 – 28 May 1990) was the founder and first commander of the British Army's Long Range Desert
Group during World War II. Hans Albert Einstein (May 14, 1904 – July 26, 1973) was a professor at University of California, Berkeley. He was the second
child, and the first son, of renowned physicist Albert Einstein (1879–1955) and his first wife, Mileva Marić (1875–1948).
Sediment Model Definitions
Volumetric Bed-load transport:
qb  Cbub b
Cb = volumetric concentration in the bed-load layer
ub = particle velocity (m/s)
δb = thickness of bed-load layer (m)
h
Suspended-load transport:
qs   ucdz
b
Issues for Calculating Sediment Transport Rate
Bed-load Layer Thickness: Saltation height
Particle Velocity
Particle pick-up rate from the bed
Deterministic Bed-Load Transport Formulae
Meyer-Peter Mueller (1948)
b  8(    cr )1.5
Dimensionless bedload transport rate:
b 

Particle mobility
parameter:
q
b
0.5 0.5
( s  1) g d m1.5
 b ,c
( s   ) gd m
  (C / C )
C  18log(12h / ks ,c )
C   18log(12h / d90 )
ks,c = effective bed roughness
dm = mean particle diameter
θcr = 0.047
Bagnold (1966)
Van Rijn(1984a)
Fig. 7.2.12
Suspended Load Transport
z=h
u(z)
c(z)
uc(z)
ca
z=a
z=0
h
qs   ucdz
a
or
qs  ca uhF
F 
h
a
uc z
d( )
uca h
Ca = reference concentration
Van Rijn (1984)
d50 T 1.5
ca  0.015
a D*0.3
a = reference level: bed form height
Bed Material Suspension and Transport in Waves
Wave motion over an erodible sand bed can
generate a sediment suspension with relatively large
sediment concentration in the near-bed region in
the case of non-breaking waves.
The key role of the breaking waves on the sediment
concentration field in the coastal zone is obvious.
The concentrations are maximum near the plunging
point and decrease sharply on the both sides of the
plunging area.
The bed material is usually coarsest in the most
energetic area and finer inside the surf zone and
outside the surf zone.
Wave-induced transport processes are related to the
currents generated by waves. Net onshore transport
is dominant in non-breaking wave conditions.
Net offshore transport is dominant in breaking wave
conditions.
Fig. 8.1.1
Transport Processes in Breaking Waves (Surf Zone)
Mechanisms:
• net backward (offshore) transport due to the generation of a net return flow
(undertow) in the near-bed region in spilling and plunging breaking waves.
• net forward (onshore) transport by asymmetrical wave motion in weakly spilling
breaking waves.
• longshore and offshore-directed transport due to the generation of large-scale
circulation cells with longshore currents and offshore rip current.
• gravity-induced transport (bed load) in downsloping direction
Longshore Sediment Transport
Ocean City Beach looking north, Maryland
Downloaded from: http://images.usace.army.mil/main.html
Observations on natural beaches as well as in laboratory wave basins
have confirmed that the longshore current is largely confined to the surf
zone. This longshore current drives the shoreward movement of
longshore sediment transport.
Computation of Sediment Transport Rates Induced by Waves
Two Approaches:
• Sediment transport models representing both the instantaneous fluid velocity and
concentration profile
• Sediment transport formulae similar to the current-related bed-load formulae.
1. Sediment Transport Models (Local 1-D)
This approach is useful when the phase difference between the instantaneous velocities
and sediment concentrations at different elevations above the bed can not be neglected.
Dividing the instantaneous velocity (U) and concentration (C) into two parts:
U (t )  u  U (t )
and
C (t )  c  C (t )
The net total time-averaged total transport rate can be expressed as:
Not a !
h
h
0
0
qt   ucdz   UCdz
qc
+
qw
= the current-related part + the wave-related part
Neglecting the horizontal convection, the horizontal diffusion and the vertical
convection, the simplified concentration equation reads as:
C
C  
C 
 ws ,m
   s ,w
0
t
z z 
z 
w= fall velocity; ε= mixing coefficient
Sediment Transport Formulae
Since the major part of the sediment suspension in wave conditions is confined to a region
close the bed (within 3 to 5 the ripple height or the sheet flow layer thickness), it seems
reasonable to compute the wave-related sediment transport by a simple formula in
analogy with the bed-load transport formulae applied in steady currents. A division
between bed load and suspended load is only academic interest.
The existing formulae are generally based on empirical concepts, as used in steady uniform
flow, using experimental data of oscillating flow in wave tanks, wave tunnels, etc.
Assumptions:
• No phase differences between the instantaneous bed shear stress (τb,w) and the
velocity outside the boundary layer (U)
• No phase differences between instantaneous bed shear stresses and instantaneous
transport rates
Measures of sediment transport rates in waves:
qw,half = the sediment transport rate in half a period of an oscillatory flow
qw,net = the net sediment transport rate per wave cycle (period)
In general,
qw  f (U , f w , d , )
Total Sediment Transport Model (1)
• Instantaneous Total Sediment Transport Rate
（Grant & Madsen, 1976）
U
q(t )  40ws d 50
|U |
3
(m2/s)
Instantaneous velocity vector near the bed
U  U w2  uc2  2U wuc cos
Particle mobility parameter
0.5 f wU 2

( s  1) gd 50
WS = particle fall velocity
d50 = median particle diameter of bed material
uc = near-bed current velocity
UW = near-bed orbital velocity
fw = friction factor
  angle between current direction and wave propagation direction
Total Sediment Transport Model (2)
Time-averaged (over half a period) Total Sediment Transport Rate
(Grant & Madsen, 1976）
qw,half  12.5ws d50 ( )3
0.5 f wU2
 
( s  1) gd50
Other formulae
Bailard-Bagnold (1981) : the instantaneous total transport rate (bed-load+suspended load)
Sato-Horikawa (1986): the net transport rate based on wave tunnel experiments with
regular asymmetric wave motion over a ripple sand bed
Van Rijn (1989): asymmetric regular swell waves, asymmetric irregular wind waves, mean
(weak) current in presence of waves, and bound long waves
Bed Material Suspension and Transport in
Combined Waves and Currents
Currents in Coasts and Oceans:
• tidal current
• wind-induced currents
•wave-induced currents (especially wave breaking)
• vertical mixing due to bottom boundary turbulence
These current will result in additional upward transport of particles yielding larger
concentrations in the upper layer. The basic mechanism is the entrainment of particles
by the stirring wave action and the transport of the particles by the current motion.
The total sediment transport in combined waves and currents can be divided into:
qt (t )  qt ,c (t )  qt , w (t ) = current-related transport rate + wave-related transport rate
qt,c: the transport of particles by the time-averaged current velocities
qt,w: the transport of particles by the oscillating fluid motions (orbital velocities
Issues in Sediment Transport in Combined
Waves and Currents
• Non-breaking Waves:
• Breaking Waves
• Angle between wave and current:
wave-current interaction: following current and oppose current
Return
flow
Wave
direction
shoreline
current
direction
Wave-Related Sediment Mixing for
Non-Breaking Waves
Wave-Related Sediment Mixing for
Breaking Waves
Bed-load Transport (1:only current)
• van Rijn’s model (Rijn, 1984)
qb  Cb bub
Cb  0.18C0T / D*
 b  0.3d 50 D*0.5T 0.5
ub  7.0u*/  7.0( b/ ,c /  )0.5
qb = Volumetric bed-load transport rate
Cb = Volumetric concentration
δb = Thickness of bed load layer
ub = Bed load particle velocity
D* : Dimensional Particle paramter
d50 : median particle diameter of bed material (m)
/b,c :effective bed-shear stress due to combined current (N/m2)
T: dimensionless bed-shear parameter
Bed-load Transport (2: wave+current)
• Van Rijn’s model (Rijn, 1993)
Time-averaged bed-load transport rate can be obtained by
averaging the instantaneous values qb(t)over the whole wave
period
qb (t )  0.25d50D
0.3
*
[
/
b,cw
/  ] [(
0.5
/
b,cw
  b,cr ) /  b,cr ]
qb : instantaneous Volumetric bed-load transport rate (m2/s)
D* : Dimensional Particle parameter
d50 : median particle diameter of bed material (m)
b,cr :critical bed-shear stress according to Shields (N/m2)
/b,cw :grain-related instantaneous bed-shear stress due to combined
current (N/m2)
: fluid density (kg/m3)
:calibration factor = 1-(Hs/h)0.5 , Hs: significant wave height (m)
1.5
Bed-load Transport due to Combined Current and Wave
Instantaneous bed-load transport rate
Satoh & Kabiling (1994)
Ub
qb (t )  ( s  1) gd  b  (t ) max{ (t )   c , }
| Ub |
3
Shields Number
b

(  s   ) gd
c = Critical Shield number
b = empirical coefficient (=1.0)
Ub = near-bed current velocity
Van Rijn’s Sediment Transport Model:
TRANSPOR
Input Parameters:
h = water depth
VR= depth-averaged velocity vector of
current
uR = time-averaged and depth-averaged
return flow velocity (- in offshore
direction)
ub = time-averaged near-bed velocity due
to waves
Hs = significant wave height
Tp = peak wave period
φ = angle between wave and current (deg)
d50= median diameter of bed materials
d90= 90% diameter of bed materials
ds= representative diameter of suspended
material
ks,c=current-related bed roughness height
ks,w=wave-related bed roughness height
Te = water temperature (oC)
SA = salinity
Fig. Schematic illustration of current and
wave directions
Van Rijn’s Sediment Transport Model: TRANSPOR
TRANSPOR.for
TRANSPOR.exe
Appendix A in Van Rijn’s Book
My version: cf_tanaka.for, cf_tanaka.exe
Code: in my folder /TRANSPOR
Homework 4 (optional)
(1) Using TRANSPOR sediment transport model and the following parameters, calculate
sediment transport rates by wave and current.
HD = WATER DEPTH [ M ] =
1.0000000
VR = MEAN VEL. IN CURRENT DIR. [ M/S ] =
0.200000
UR = MEAN VEL. IN WAVE DIR(-BACK) [ M/S ] =
0.4000000
UB = NEAR-BED VEL IN WAVE DIR(-BACK) [ M/S ]=
0.3000000
HS = SIGNIFICANT WAVE HEIGHT [ M ] =
0.5000
TP = PEAK WAVE PERIOD
[S]=
4.50000
PHI = ANGLE CURRENT AND WAVES 0-360 [ DEG ]
60.0000000
D50 = MEDIAN PARTICLE SIZE OF BED [ M ] =
0.0002
D90 = 90 0/0 PARTICLE SIZE OF BED [ M ] =
0.0004
DSS = SUSPENDED SEDIMENT SIZE [ M ] =
0.00015
RC = CURRENT-RELATED ROUGHNESS [ M ] =
0.0200000
RW = WAVE-RELATED ROUGHNESS [ M ] =
0.0200000
TE = WATER TEMPERATURE [ CELSIUS ] =
20.0000000
SA = SALINITY OF FLUID [ PROMILLE ] =
35.0000000
(2) Input different wave heights, e.g. 0.5, 0.8, and 1.5 m, investigate the variations of
sediment transport rate.
Longshore Sediment Transport by Wave Breaking
Significant Wave Direction
Supply of Sediment:small
Sediment Rate:
Eroded shoreline
10 year
1 year
5 year
Significant Wave Direction
Sediment Rate: decrease
Stabilization of Shoreline
Beach
Shoreline Erosion Protection
Protected by Coastal Structure: Artificial
Headland
Protection of shoreline erosion by wave
dissipating breakwaters and sand nourishments
near the Fuji river mouth, Japan
Estimation of Longshore Sediment Rate
- Energy Flux Method (CERC formula)
Ql 
K
(EC g )b sin αb cos αb
(ρs  ρ)g( 1 λ)
K=empirical coefficient in the CERC Formula
l = in-place sediment porosity
Energy Flux = (ECg)b
Eb 
gH b2
8
Hb : Wave Height at the
breaker line
(Cg)b : Wave group speed at
the breaker line
(m3/s)
Empirical Coefficient K
- (Shore Protection Manual, 1984)
K
SPM
0.92
Komar &
Inman
0.77
(1970)
Takagi &
0.06*
Ding
(2000)
*Nourished sand (gravel)
+ in-place sediment
K=K(d)
Shoreline Evolution Model (Long-term)
- One Line Model (Hanson & Kraus, 1989)
• Assume the beach
profile is displaced
parallel to itself in the
cross-shore direction
- Parallel Contour Lines
y
1  Ql


 q  0

t d b  d c  x

y= location of shoreline
dc= offshore closure depth
db=berm crest elevation
Fig. Elemental volume on equilibrium beach profile
q=line source or sink of sediment
Sketch of Seasonal Change between Two
Headlands
Incident wave in winter
Incident wave in summer
Maximum erosion
Shoreline in winter
1km
Shoreline in summer
Variation of shoreline: seasonal wave height, wave direction,
even daily data