1. ENGR 691, Fall Semester 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:

2. 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

3. Introduction
The transport of bed material particles by a flow of water can be in the form of bed-load 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:
Fluid velocity
Bed shear stress
Probabilistic particle movement
Bed form celerity
Energetics (stream power)
m = 3 ~ 5
n ≈ 1.5
:depth-averaged velocity
ucr: critical shear velocity
τ: bed shear stress
τcr: critical bed shear stress

4. Three Modes of Particle Motions
Three particle motions:
Rolling and sliding motion or both:
Saltation:
Suspended particle motion:
Bed shear velocity just exceeds the critical value
Bed shear velocity further increases
Bed shear velocity exceeds the particle fall velocity

5. 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 suspended-load 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).

6. Sediment Model Definitions
Volumetric Bed-load transport:
Cb = volumetric concentration in the bed-load layer
ub = particle velocity (m/s)
δb = thickness of bed-load layer (m)
Suspended-load transport:

7. Issues for Calculating Sediment Transport Rate
Bed-load Layer Thickness: Saltation height
Particle Velocity
Particle pick-up rate from the bed

8. Deterministic Bed-Load Transport Formulae
Meyer-Peter Mueller (1948)
Dimensionless bed-load transport rate:
Particle mobility parameter:
ks,c = effective bed roughness
dm = mean particle diameter
θcr = 0.047
Bagnold (1966)
Van Rijn(1984a)
Fig

9. Suspended Load Transport
u(z)
c(z)
uc(z) ca
z=0
z=a
z=h
Ca = reference concentration or
Van Rijn (1984)
a = reference level: bed form height

10. 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

11. 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

12. Nearshore Sediment Transport
On-off shore
Longshore transport

13. Longshore Sediment Transport
Wave
Sediment
Current &
Breaking Line
Ocean City Beach looking north, Maryland
Downloaded from:
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.

14. 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:
and
The net total time-averaged total transport rate can be expressed as:
Not a ! 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:
w= fall velocity; ε= mixing coefficient

15. 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.
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,
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

16. Total Sediment Transport Model (1)
Instantaneous Total Sediment Transport Rate
　　（Grant & Madsen, 1976）
(m2/s)
Instantaneous velocity vector near the bed
Particle mobility parameter
WS = particle fall velocity
d50 = median particle diameter of bed material
uc = near-bed current velocity
UW = near-bed orbital velocity
fw = friction factor
f = angle between current direction and wave propagation direction

17. Total Sediment Transport Model (2)
Time-averaged (over half a period) Total Sediment Transport Rate
(Grant & Madsen, 1976）
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

18. 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:
= 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

19. 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
Wave direction
current direction
shoreline
Return flow

20. Wave-Related Sediment Mixing for Non-Breaking Waves

21. Wave-Related Sediment Mixing for Breaking Waves

22. Bed-load Transport (1:only current)
van Rijn’s model (Rijn, 1984) {
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

23. 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 : 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)

24. Bed-load Transport due to Combined Current and Wave
Instantaneous bed-load transport rate
Satoh & Kabiling (1994)
Shields Number
yc = Critical Shield number
ab = empirical coefficient (=1.0)
Ub = near-bed current velocity

25. 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

26. 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

27. 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 ] =
VR = MEAN VEL. IN CURRENT DIR. [ M/S ] =
UR = MEAN VEL. IN WAVE DIR(-BACK) [ M/S ] =
UB = NEAR-BED VEL IN WAVE DIR(-BACK) [ M/S ]=
HS = SIGNIFICANT WAVE HEIGHT [ M ] =
0.5000
TP = PEAK WAVE PERIOD [ S ] =
PHI = ANGLE CURRENT AND WAVES [ DEG ]
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 ] =
RC = CURRENT-RELATED ROUGHNESS [ M ] =
RW = WAVE-RELATED ROUGHNESS [ M ] =
TE = WATER TEMPERATURE [ CELSIUS ] =
SA = SALINITY OF FLUID [ PROMILLE ] =
(2) Input different wave heights, e.g. 0.5, 0.8, and 1.5 m, investigate the variations of sediment transport rate.

28. Longshore Sediment Transport by Wave Breaking
Significant Wave Direction
Supply of Sediment:small
Sediment Rate:
Eroded shoreline
propagation of erosion
1 year
10 year
5 year
Significant Wave Direction
Sediment Rate: decrease
Stabilization of Shoreline
Beach

29. Shoreline Erosion Protection
Protection of shoreline erosion by wave dissipating breakwaters and sand nourishments near the Fuji river mouth, Japan
Protected by Coastal Structure: Artificial Headland

30. Estimation of Longshore Sediment Rate - Energy Flux Method (CERC formula)
(m3/s)
K=empirical coefficient in the CERC Formula
l = in-place sediment porosity
Energy Flux = (ECg)b Ql
Hb : Wave Height at the breaker line
(Cg)b : Wave group speed at the breaker line

31. Empirical Coefficient K - (Shore Protection Manual, 1984)
SPM
0.92
Komar &
Inman
(1970)
0.77
Takagi &
Ding
(2000)
0.06*
*Nourished sand (gravel)
+ in-place sediment
K=K(d)

32. 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= location of shoreline
dc= offshore closure depth
db=berm crest elevation
q=line source or sink of sediment
Fig. Elemental volume on equilibrium beach profile

33. Sketch of Seasonal Change between Two Headlands
Incident wave in winter
Incident wave in summer
Maximum erosion
1km
Shoreline in winter
Shoreline in summer
Variation of shoreline: seasonal wave height, wave direction,
even daily data