Transcript ENGR 693-73 Research Topics in Engineering Science I

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: [email protected]
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
Near-bed Orbital Velocities
Applying linear wave theory, the
peak value of the orbital excursion
(Aδ) and velocity (U δ) at the edge
of the wave boundary layer can be
expressed as
A 
H
2sinh(kh)
U    A 
H
T sinh(kh)
H = wave height
h = water depth
ω = angular frequency = 2π/T
k = wave number
Wave Boundary layer (1)
Video: Laboratory Wave Flume
Wave Boundary Layer (2)
z
The wave boundary layer is a thin layer forming the
transition layer between the bed and the upper layer of
irrotational oscillatory flow (Fig.). The thickness of this layer
remains thin (0.01 to 0.1 m) in short period wave (<=12s)
because the flow reverses before the layer can grow in
vertical direction. The boundary layer thickness δw can be
defined as the minimum distance between the wall and a
level where the velocity equals the peak value of the free
stream velocity Uδ.
Jonson (1980):
Manohar (1955):
w 
Uδ
2

4.6
w 



T
In case of turbulent boundary layer: δw=f(T, ks)
δw
u
ν = kinematic viscosity coefficient (m2/s)
T = wave period (s)
ks = effective bed roughness (m)
Wave Boundary layer (3)
- Bed shear stress and bed friction
Wave exerts friction forces at the bed during oscillations. The bed shear stress, which
is important for wave damping and sediment entrainment, is related to the friction
coefficient by :
 b , w (t ) 
1
 f wu 2 (t )
2
fw = friction coefficient (non-dimension), which is assumed to be constant over the
wave cycle
uδ = instantaneous fluid velocity of the free stream (just outside boundary layer
The time-averaged over half a wave cycle bed shear stress is
ˆb , w
1
  f wU 2
4
The calculations of the friction coefficient fw depend on the flow regimes of the wave
boundary layer, i.e. laminar flow, smooth turbulent flow, and rough turbulent flow. See
van Rijin (1993) for details.
Wave friction coefficient
Initiation of Motion in Waves
Critical Velocity
In oscillatory flow there is no generally accepted relationship for initiation of motion
on a plane bed. Many equations have been proposed. One of the more popular
equations is proposed by Komar and Miller (1975)
(U  ,cr )
2
( s  1) gd50
0.5

 2 A ,cr 
 0.21

d

 50 

0.25
2
A
  ,cr 

1.45



d
 50 

for d50  500 m
for d50  500 m
Uδ, cr= critical peak value of orbital velocity near the bed
A δ, cr = critical peak value of orbital excursion near bed
s = specific gravity (=ρs/ρ)
Initiation of Motion in Waves
- Critical Velocity
Initiation of Motion in Waves
- Critical Bed-Shear Stress
The experimental data for initiation of motion in waves can also be expressed in terms
of the Shield parameter using the time-averaged bed-shear stress, i.e.
 b,w,cr
 f ( D* )
( s   ) gd50
where
 b ,w,cr
1
  f w (U ,cr ) 2
4
=time-averaged over half a wave period) wave-related bed-shear stress
D*=particle parameter
 ( s  1) g 
D*  
 d 50
2
 

1/3
Initiation of motion for waves over a plane bed
based on critical bed-shear stress
motion
No motion
• The Shields curve which is valid for unidirectional flow data only.
• The variation between the results of different investigators is mainly caused by the
definition problem for initiation of motion .
• The Shields curve can also be applied as a criterion for initiation of motion for oscillatory
flow over a plane bed. It represents a critical stage at which only a minor part (1% to 10%)
of the bed surface is moving.
Shields Curve
 0.24 D*1 for 1  D*  4

0.64
0.14
D
for 4  D*  10
*

 cr   0.04 D*0.1 for 10  D*  20
0.013D 0.29 for 20  D  150
*
*


0.055 for D*  150
I
II
III
IV
V
I
II
III
IV
V
Example
The water depth in a coastal sea with a plane bed is h = 5m. The wave period is T = 7s.
The bed material parameters are d50=200μm, d90=300μm, ρs=2650kg/m3. The water
temperature Te = 20oC. The kinematic viscosity coefficient ν=1.0x10-6m2/s, fluid
density ρ=1025kg/m3.
What is the wave height at initiation of motion?
Method 1
Using the figure for initiation of wave motion,
the critical peak orbital velocity can be
obtained: Uδ,cr = 0.23
Calculate the wave length by the dispersion
relation
gT
2 h
L
2
tanh(
L
)
Yielding L = 45.7m.
Online wave calculator: http://www.coastal.udel.edu/faculty/rad/wavetheory.html
Using the definition of the critical
velocity, the wave height can be
calculated
2 h
H cr  U  ,crT sinh(
) = 0.38m

L
1
Initiation of motion for combined current and wave
Longshore Currents
Ocean
Ebb
Flood
Longshore Currents
Wave
Estuary
Alluvial Sediments
River Inflow
Initiation of motion for combined current and wave
The resulting time-averaged total bed-shear stress :  b,cw   b,c   b,w
Wave Breaking
Wave Breaking (1)
• Wave breaking limit: assume that wave crest particle velocity
equals the wave celerity at the breaking point, i.e.
Fr 
u
In general
u z 0
C
C
1
H cosh k (d  z )
T
| u | z 0 
Then,
u z 0
sinh kd
H
1

T tanhkd
(
cos( kx  t )
H
)
1
L
C tanhkd
(
H
1
)
1
L tanhkd
Therefore, the wave breaking limit:
(
H
1
)b  tanh kd  0.3183 tanh kd
L

Wave Breaking (2)
The wave breaking limit (wave steepness) based on small amplitude wave theory:
H
1
is not accurate!
)b  tanh kd  0.3183 tanh kd
L

H
tanhkd  1,
( )b  0.3183
In deep water,
L
H
( ) b  0.14
According to observations
L
H
(
)b  2.0
In shallow water, tanhkd  kd ,
too late
d
H
(
) b  0.78
As a rule of thumb,
d
(
Saturated wave breaking in shallow water (McCowan 1894)
Breaking Wave Criteria in Shallow Water
Goda (1985) proposed a very useful breaking wave criteria (BWC) based on a
large amount of observation data (laboratory and field data), i.e.


Hb
h
 A 1  exp(1.5
(1  15 4 / 3 )  C ( d )
L0
L0


Hb = Breaking Wave Height
L0 = Wave Length
A = Empirical Coefficient (0.12 – 0.18)
= Sea Bed Slope
C(εd) = Coefficient (if = 1.0, BWC = Goda’s formula (Goda 1975);
if not equal to 1.0, BWC=extended Goda’s formula (Sakai et al. 1988)
Breaking Wave Criteria (3)
In deep water, wave celerity can be written:
C0 
gT
2
and the particle velocity amplitude at surface
| u | z 0 
Then,
Or,
H
T
2 2 H

1
2
C
gT
u z 0
gT 2
2
H 

0
.
0507
gT
2 2
0
b
Actually, Ramberg and Griffin (1987) found that the deep water breaking height is
best represented by
Hb0  0.021gT 2
e.g.
• T = 10s in deep water, if breaking wave, the wave height could be 21 meter.
That’s a huge wave.
• In Hurricane Katrina, offshore maximum wave height record in offshore of
Mississippi Gulf Coast is 36 ft = 10.8m. It might not be breaking yet.
• Tsunami wave in deep water, in general, is not breaking wave, because
tsunami wave in deep water is long wave.
• 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
Bed forms
Sand ripples
http://www.virtualbay.co.nz/nature/index.htm
Bed Forms
Sand Bars formed by wave breaking
Classification of Bed Forms in Unidirectional Currents
Classification of Bed
Forms in Unidirectional
Currents
(van Rijn 1984, 1989)
Regimes: Lower, transition, and upper
Bed Form = F(T, D)
Particle parameter:
 ( s  1) g 
D*  
 d 50
2
 

1/3
Excess bed-shear stress parameter:
  
T  b,c b ,cr
 b,cr
SAND
WAVES
PLANE BED
ANTI-DUNES
DUNES
Shape and Dimension of Bed Forms at Lower Regime in
Unidirectional Flows (1)
1. Flat bed, lower region: Before onset of particle motion
2. Ribbons and ridges, lower regime: small scale, parallel to the main flow
direction, esp. in case of fine sediments (d50<100μm), probably generated by
secondary flows and near-bed turbulence burst-sweep effect, vertical scales ≈
10d50, the width scale = 100ν/u*
3. Ripples, lower regime: Mini ripples, 3-D ripples, lunate ripples (concave shape),
linguoid ripples (convex shape), mega-ripples
Mini ripples: ripple height Δr = 50 ~ 200 d50
ripple length λr = 500 ~ 1000d50
Mega-ripple:
r
 0.02(1  e 0.1T )(10  T )
h
r
h
 0.5
for 1.0≤ D*≤ 10.0 and 3.0 ≤ T ≤ 10.0
 ( s  1) g 
D

*

 d 50
2
Particle parameter
 

 b,c   b,cr
T
Excess bed-shear stress parameter
1/3
 b,cr
Dune Characteristics after van Rijn (1982)
Nondimensional dune height vs Transport Parameter T
Dune Characteristics after van Rijn (1982)
Dune steepness vs Transport Parameter T
Shape and Dimension of Bed Forms at Lower Regime in
Unidirectional Flows (2)
4. Dune, lower regime
The dune-type bed form is a typical bed form at the lower regime. Dunes
have an asymmetrical profile with a rather steep leeside and a gentle stoss side.
A general feature of dune-type bed forms is a leeside flow separation, which
results in strong eddy motions downstream of the dune crest
The length of dunes is strongly related to the water depth with values in
the range of 3 to 15 h.
Dunes can migrate to downstream, or to upstream (anti-dune)
Estimations of dune shapes by van Rijn’s formulations:
d
d
 0.11( 50 )0.3 (1  e 0.5T )(25  T )
h
h
d
h
 7.3
Where h = water depth from water surface to mean bed level (at the half the bed
form height)
Note that the dunes are assumed to be washed out for T ≥25.0
Fig. 5.2.13
5. Bars, lower regime: the largest bed forms in the lower regime
Alternate bars, side bars (point bars & scroll bars), braid bars, and
transverse bars
Shape and Dimension of Bed Forms at transition and
upper Regime in Unidirectional Flows
Washed-out dunes and sand waves in the transition regime
It is a well-known phenomenon that the bed forms generated at low regime are
washed out at high velocities. Ultimately, relative long and smooth sand waves with a
roughness equal to the grain roughness were generated
Based on van Rijn’s result in Fig. 5.2.7, the transition regime will occur for T ≥ 15 .
The bed forms in the transition regime which will most likely occur are washed-out
dune and (symmetrical) sand waves. The dimensions of the sand waves are described
by:
 s,w
h
s , w
h
 0.15(1  e 0.5(T 15) )(1  Fr 2 )
 10.0
for T ≥ 15.0 and subcritical flow regime
The bed forms will fully disappear for T ≥ 25.
Plane bed and sand waves, upper regime
Two sub-regimes:
Subcritical upper transport regime: T≥25.0 and Fr< 0.8, symmetrical sand waves
Supercritical upper transport regime: T≥25.0 and Fr ≥ 0.8, plane bed and/or anti-dunes.
When the flow velocity further increases, finally a stage with chute and pools may be
generated
Examples and Problems (1)
A wide open channel has a mean water depth h = 3m, a mean velocity u = 1 m/s, the bed
material characteristics are d50 = 0.35mm, d90=1.0mm, sediment density ρs=2650kg/m3,
ρ=1000kg/m3, kinematic viscosity ν = 1x10-6.
Given that the bottom boundary layer flow regime is hydraulic rough flow, what types of
bed forms are generated? What are the dimensions of the bed forms?
Bed forms by van Rijn’s approach (Fig. 5.2.7)
1. Calculate the bed shear stress
Chezy roughness coefficient C = 18log(12h/3d90) = 73.4 m1/2/s
Bed shear stress τb,c = ρg(u/C)2 = 1.82 N/m2
2. Calculate the critical bed shear stress
Specific density = ρs /ρ
Particle parameter D* = ((s-1)g/ν2)1/3d50 = 8.79
According to the Shields’ curve, the critical mobility parameter:
θcr = 0.14D*-0.64=0.0348
The critical shear stress τb,cr = (ρs-ρ)gd50 θcr =0.197N/m2
3. Calculate the excess bed-shear stress parameter
T=(τb,c - τb,cr )/ τb,cr =8.23
4. Find the bed forms from Fig. 5.2.7 using the values of D* and T
Examples and Problems (2)
A wide open channel has a mean water depth h = 3m, a mean velocity u = 1 m/s, the bed
material characteristics are d50 = 0.35mm, d90=1.0mm, sediment density ρs=2650kg/m3,
ρ=1000kg/m3, kinematic viscosity ν = 1x10-6.
Given that the bottom boundary layer flow regime is hydraulic rough flow, what types of
bed forms are generated? What are the dimensions of the bed forms?
4. Find the bed forms from Fig. 5.2.7 using the values of D* and T
Bed forms: mega-ripples and dunes
Bed form dimensions by van Rijn’s approach
1. mega-ripples
Ripple height: Δmr= 0.02h(1-exp(-0.1T))(10-T) = 0.074m
Ripple length: λmr = 0.5h = 1.5m
2. Dunes
Dune height: Δd= 0.11h (d50/h)0.3(1-exp(-0.5T))(25-T) = 0.37m
Ripple length: λd = 7.3h = 21.9m
Homework (1)
A wide open channel has a mean water depth h = 2.0m, a mean velocity u = 1.2 m/s, the
bed material characteristics are d50 = 0.35mm, d90=1.0mm, sediment density
ρs=2650kg/m3, ρ=1000kg/m3, kinematic viscosity ν = 1x10-6.
Given that the bottom boundary layer flow regime is hydraulic rough flow, using van Rijn’s
method, find the types of bed forms generated by the flow, and determine the dimensions
of the bed forms.
Hint: please refer to pages 5.22-5.24 in van Rijn’s book (Principle s of Sediment Transport
in Rivers, Estuaries and Coastal Seas) or my notes for solving the problems
Bed Forms in Waves
Classification
Two typical regimes can be observed in nature:
• Lower regime with flat immobile bed, ripples and bars,
• Upper regime with flat mobile bed (i.e. sheet flow)
•A typical transition regime does not occur
Two parameters for classifying the regimes:
(U )2

( s  1) gd50
 b,w

( s   ) gd50
Ripple regime for
Sheet flow regime for
  250 or   0.8
  250 or   0.8
Bed form classification diagram for waves after
Allen (1982)
Bed Forms in the Coastal Zone (1)
- Clifton (1976) and Shipp (1984)
Sand Waves
Bed Forms in the Coastal Zone (2)
- Clifton (1976) and Shipp (1984)
Upper shore face: linear ripples, asymmetric ripples, flat bed (sheet flow)
Longshore trough: Linear ripples (λr = 0.7m, Δr = 0.15m)
Landward slope of bar: cross ripple, irregular ripples and linear ripples (from top to
bottom)
Longshore bar crest: irregular and cross ripples for low energy conditions lunate megaripples (λr = 0.7m, Δr = 0.15m) for higher energy conditions
Seaward slope of bar: Cross-ripples and linear ripples
Transitional zone: linear ripples of fine sand (0.2mm), locally coarse grain deposits
(0.6mm) forming linear mega-ripples
Offshore: linear ripples of fine sand (0.15 – 0.20 mm)
Dimensions of Bed Forms (1)
- Observed ripple height
(a) For regular waves
(b) For irregular waves
Dimensions of Bed Forms (1)
- Observed ripple length
(a) For regular waves
(b) For irregular waves
Dimensions of Bed Forms (2)
For non-breaking irregular waves - van Rijn’s Formulations
Dimensionless ripple height:
Ripple steepness:
0.22 for   10

r 
 2.8 1013 (250  )5 for 10    250
A 
0 for   250

0.18 for   10

r 
 2.0 107 (250  ) 2.5 for 10    250
r 
0 for   250

Sheet flow regime
The upper regime with sheet flow conditions is assumed to be present for Ψ ≥ 250.0
Surf –zone bars or longshore bars
These type of bars have their orientation (crests) parallel to the coastline and are
formed in the surf zone near the breakline. It may be generated by net offshoredirected current in the surf zone (undertow flow).
Examples and Questions
A coastal sea has a water depth of h = 5m. Irregular waves with a peak period of
Tp=7s are present. The bed material characteristics are d50 = 0.3mm,
d90=0.5mm. Other coastal water parameters: ρ = 1025kg/m3, ρs=2650kg/m3,
ν=1x10-6m2/s
What is the significant wave height at the initiation of sheet flow?
Significant wave height (Hs) is the average wave height (trough to crest) of the one-third largest waves
Solution:
Initiation of sheet flow: Ψ ≥ 250.0
(U )2

( s  1) gd50
and s = ρs/ρ = 2.585
U   (s 1) gd50  250.0*(2.585  1)*9.8*0.0003  1.08m / s
H
U 
T sinh(2 h / L)
The wave length of the wave (Tp=7s): LS = 45.7m
HS  U Tp sinh(2 h / LS ) /   1.08*7.0*sinh(2.0*3.14*5/ 45.7) / 3.14  1.79m
Answer: A significant wave height of 1.79m (or higher) will initiate sheet flow on the sea bed.
Homework (2)
A coastal sea has a water depth of h = 5m. Irregular waves with a significant wave
height Hs = 1 m and a peak period of Tp=7s are present. The bed material
characteristics are d50 = 0.3mm, d90=0.5mm. Other coastal water parameters: ρ =
1025kg/m3, ρs=2650kg/m3, ν=1x10-6m2/s
What are the ripple height and the length according to van Rijn’s method?
What is the flow regime if the wave height Hs = 2m and Tp=7.0s?
Bed Forms in Currents and waves (1)
Currents in the nearshore and surf zone:
Cross-shore return flow, longshore current, and tidal current when weather is clam.
The cross-shore return flows refer to a general seaward flow or to a channelized narrow
seaward rip current due to wave breaking. Seaward-facing mega-ripples have been
commonly observed.
Longshore currents refer to the current in the zone between the longshore bar and the
shoreline.Complex ripples patterns are found in the areas.
In the offshore zone, tidal currents may become dominant, which may be following,
opposing, or oblique to the wave direction. Bed forms in tidal seas are related to the peak
current velocities, water depth, sediment diameter and the availability of sediment.
Tidal inlet
Ripple Patterns in Combined Current and Wave
Conditions
uc current
wave
uw
Symmetrical
wave-induced
ripples (2.5D)
Asymmetrical
current-induced
ripples (3D)
wave-current
induced ripples
(honeycomb
pattern, 3D)
Ripples created by ebb-tide
http://www.geograph.org.uk/reuse.php?id=652440
Honeycomb ripples
Others patterns
Patterns on Borth Sands
As the tide ebbs, it has left a scallop-shaped
pattern on the sandbank.
http://www.geograph.org.uk/reuse.php?id=652440
Bed Forms in Currents and waves (2)
Four types of bed forms in combined current and wave conditions:
• Symmetrical wave-induced ripples with their crest almost perpendicular to the wave
direction in case of weak tidal current velocities
• Asymmetrical current-induced ripples and large symmetrical sand waves with their crest
perpendicular to the tidal current direction in case of a strong tidal current and weak orbital
velocities
• Wave-current ripples in a honeycomb pattern in case of equal strength of the current and
peak orbital velocities.
• Longitudinal furrows, ribbons, ridges and banks with crests and troughs almost parallel to
the peak tidal current direction
The bed forms generated by combined currents and waves bear some features of both
hydraulic effects. Where the wave component dominates, the bed forms are similar to fully
developed wave-related bed forms. As the current component gains in strength, the bed
forms become more asymmetrical and larger in height and length, especially in case of an
opposing current. The influence of the waves is that the bed form crest will become more
rounded.
Bed Forms in Currents and waves (3)
- Classification in terms of van Rijn’s approach
Based on the available data, van Rijn proposed a classification diagram for bed forms under
combined waves and currents conditions. Two key nondimensional parameters are defined
as follows:
The current-related mobility parameter:
c 
The wave-related mobility parameter:
w 
where
(u*,c )2
( s  1) gd2 50
(u*,w )
( s  1) gd50
Current-related effective bed-shear velocity:
u*,c  0.125 f c u
Current-related friction factor:
f c  0.24(log(12h / 3d90 ) 2
Wave-related effective bed-shear velocity:
u*,w  0.125 f w U
Wave-related friction factor:
f w  exp 6  5.2( A / 3d90 )0.19 
u = depth-averaged velocity
Uδ = peak orbital velocity at bed based on relative wave period
Aδ = peak orbital excursion at bed based on relative period, U δ =ωA δ
Bed form
classification for
currents and waves
Example and Problem
A coastal sea has a water depth of h = 20m, the peak flood-current velocity is umax.flood =
0.6m/s; the peak ebb-current velocity is umax.ebb = 0.5m/s. Irregular waves perpendicular
to the flood and ebb current directions are present. The significant wave height Hs=1.5m,
the peak period Tp = 8s. The bed material characteristics are d50 = 0.3mm, d90 = 0.6mm.
Other data are ρ = 1025kg/m3, ρs=2650kg/m3, ν=1x10-6m2/s.
What type of bed forms are present in combined wave and current (flood/ebb)
conditions?
Wave length: L = 88.84 m http://www.coastal.udel.edu/faculty/rad/wavetheory.html
 HS
Peak orbital velocity:
U 
 0.304m / s
Tp sinh(2 h / L)
2
Peak orbital excursion:
A  U
 0.387m
Tp
Wave-related friction factor:
f w  exp  6  5.2( A / 3d90 ) 0.19   0.0161
Current-related friction factor:
fc  0.24(log(12h / 3d90 )2  0.0913
Bed-shear velocity by wave:
By flood current
By ebb current
u*,w  0.25 f w U  0.0193m / s
u*, flood  0.125 fc umax. flood  0.0203m / s
u*,ebb  0.125 fc umax.ebb  0.0169m / s
Example and Problem (cont.)
Mobility parameter:
By wave
By flood current
By ebb current
w 
(u*,w )2
(s  1) gd50
 flood 
ebb 
 0.0801
(u*, flood )2
(s  1) gd50
(u*,ebb )2
(s  1) gd50
 0.0881
 0.0612
Answer: Using Fig.5.5.1, van Rijn’s method, bed forms are 2D waves-ripples
superimposed on 3D current ripples in honeycomb pattern. Both types may be
superimposed on large-scale sand waves.
Homework (3)
A coastal sea has a water depth of h = 10m, the peak flood-current velocity is umax.flood =
1.0 m/s; the peak ebb-current velocity is umax.ebb = 1.2 m/s. Irregular waves perpendicular
to the flood and ebb current directions are present. The significant wave height Hs=1.5m,
the peak period Tp = 8s. The bed material characteristics are d50 = 0.3mm, d90 = 0.6mm.
Other data are ρ = 1025kg/m3, ρs=2650kg/m3, ν=1x10-6m2/s.
What type of bed forms are present in combined wave and current (flood/ebb)
conditions?
Hint: wave length will be changed to 70.93m at the 10m water depth
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
Bed Shear Stress and Bed Friction in
Unidirectional Flows
u
C
1
8
 b ,c   ghI   g ( )2   f cu 2
h = water depth
I = energy slope
u = depth-averaged velocity
C = Chézy coefficient (m^0.5/s)
fc = friction factor of Darcy-Weisbach
ρ = fluid density
Hydraulic rough flow regime:

12h 
f c  0.24 log(
)
ks 

2
ks = effective bed roughness (m)
Manning’s n
n  0.045(ks )1/6
Wave-Related Bed Shear Stress and Bed Friction
Time-averaged over half a wave cycle bed shear stress is
1
4
 b ,w   f w (U )2
fw = wave-related friction coefficient
Uδ = peak orbital velocity near the bed
Rough Turbulent Flow Regime


A
f w  exp  6  5.2(  ) 0.19 
ks


With fw, max = 0.03 for Aδ/ks ≤ 1.57
An estimate of roughness ks = 3 d90
Aδ = peak value of orbital excursion near bed
Effective Bed Roughness
Nikuradse (1932) introduced the concept of an equivalent or effective sand
roughness height (ks) to simulate the roughness of arbitrary roughness
elements of the bottom boundary. In case of a movable bed consisting of
sediments, the effective bed roughness (ks) mainly consists of grain roughness
generated by skin friction forces and of form roughness generated by
pressure forces acting on the bed forms
ks  ks  ks
Fluid Pressure and Shear-stress Distribution
along a Dune
Available Methods for Determining ks
Basically, two approaches can be found in the literature to estimate the bed roughness.
• Methods based on bed-form and grain-related parameters such as bed-form length,
height, steepness and bed-material size
• Methods based on integral parameters such as mean depth, mean-velocity and bedmaterial size.
The first method is more universal and can also be used to determine the roughness of a
movable bed in non-steady conditions, provided that the bed-form characteristics are
known.
Based on the bed-form parameters, the bed-shear stress (τb )in an alluvial channel can be
divided into:
• Grain-related bed-shear stress (τ’b )
• Form-related bed shear stress (τ”b )
Grain Roughness
Grain roughness is the roughness of individual moving or non-moving sediment particles
as present in the top layer of a natural plane movable or non-movable bed.
Van Rijn (1982) analyzed about 120 sets of flume and field data with and without a
mobile bed to determine the grain roughness. The grain roughness k’s was calucated by
using the Chézy coefficient, which is derived from the measured water depth (h), depthaveraged velocity (u), and energy slope (I), i.e.
 12h 
C  18log 
 k  
 s ,c 
Hydraulic rough flow regime
u  C hI
Van Rijn’s results: Grain roughness in the lower regime is mainly related to the largest
particles of the top layer of the bed
k’s,c = 2 ~ 3 d90 for non-movable plane bed
k’s,c = 3 ~ 5 d90 for movable plane bed
Based on the data in the upper regime (mobility parameter θ > 1, van Rijn proposes to use
 3d90
ks,c  
3 d90
for   1 (lower regime)
for   1 (upper regime)
u*2

(s  1) gd50
u*  gu / C
Bed-form
Roughness
ks,c  f (,


, )
Δ = bed-form height
Δ/λ = bed-form steepness
γ = bed-form shape
ks,c  ks,r  ks,d  ks,sw
Current-related form roughness
= ripple-related roughness
+ dune-related roughness
+ sandwave-related roughness
Bed-form Roughness
Ripple-related roughness
ks,r  20 r r (
r
)
r
γs = ripple presence factor(=1.0 for ripples alone, = 0.7
for ripples superimposed on dunes or sand wave
Dune-related roughness
Symmetrical Sand Wave:
ks,sw  0
The leeside slopes of symmetrical sand waves are relatively mild. Hence, flow separation will not
occur. Therefore, the form roughness of symmetrical sand waves is assumed to be zero.
Example and Problems
A wide channel with a depth h = 8m has a bed covered with dunes. Ripples are
superimposed on the dunes. The dune dimensions are Δd = 1.0m, λd = 50.0m. The ripple
dimensions are Δr = 0.2m, λr = 3.0m. The bed material characteristics are d50 = 0.3mm,
d90 = 0.5mm.
What is the effective bed roughness, the Chézy-coefficient, and the Manning’s n?
Solution:
Grain roughness (lower regime) :
Ripple form roughness (γs = 0.7):
ks  3d90 =0.0015m

ks,r  20 r  r r =0.187m
r
d
Dune form roughness
ks,d  1.1 d  d (1  exp(25
Effective bed roughness:
ks  ks  ks  ks  ks,r  ks,d =0.492m
Chézy-coefficient:
C  18log(
Manning’s n:
h1/6
n
= 0.0342
C
d
12h
) =41.3 m1/2/s
ks
) =0.303m
Wave-related Bed Roughness
The effective wave-related bed roughness also consists of two components:
ks,w  ks,w  ks,w
In which k’s,w = wave-related grain roughness height (m)
k”s,w = wave-related bed-form roughness height (m)
The wave-related friction factor (fw) for rough oscillatory flow is


A
f w  exp  6  5.2(  ) 0.19 
ks


Time-averaged over half a wave cycle bed shear stress is
1
4
 b ,w   f w (U )2
Wave-related Grain Roughness
A number of empirical formulations based on experimental and field data on nonmovable and movable bed. Van Rijn’s approach is introduced as follows:
According to van Rijn, the effective grain roughness of a sheet flow bed is of the order
of the sheet flow layer thickness or the boundary layer thickness (k’s,w ≈δw). The sheet
flow layer is a high-concentration layer of bed material particles. Van Rijn (1989)
proposed the following values to calculate the grain roughness:
(in the ripple regime)
 3d90 for   1
ks,w  
3 d90 for   1 (in the sheet flow regime)
inwhich

uˆ*,2 w
(s  1) gd50
u*  0.25 f wU2


A
f w  exp  6  5.2(
) 0.19  for friction factor in transition regime
ks, w  3.3 m / uˆ*, w


νm = kinematic viscosity of fluid-sediment mixture in near-bed region (νm ≈ 10ν)
The grain roughness equations have to be solved iteratively. Typically, this approach yields a
value in the range of 3 ~30 d for θ = 1 ~ 10.
Wave-related Form Roughness
Ripples are the dominant bed forms generated by oscillatory flows. Ripples may be
present on a horizontal bed or superimposed on large sand waves. Large-scale sand
waves have no friction effect on the water waves, because the water waves experience
the sand waves as a gradual bottom topography. When the nesr-bed orbital excursion is
larger than the ripple length, the ripples are the dominant roughness elements for the
wave motion in the sea waters.
Apparently, bed-form roughness depends on the bed form height and length. There are a
number of empirical formulations for estimating the ripple roughness. They can be
described as
ks,w  f (r , r / r ,...)
Van Rijn (1989) proposed
ks,w  20 r r (
r
r
)
γs = ripple presence factor(=1.0 for a ripple covered
bed, = 0.7 for ripples superimposed on sand wave s
Raudkivi (1988)
ks, w  16 r (
r
r
)
Bed Roughness in Combined Currents and Waves
The most important bed form regime created by currents and waves:
• Ripples in case of weak (tidal) currents and low waves
• Sand waves with ripples in case of (tidal) current and low waves
• Plane bed with sheet flow in case of strong (tidal) currents and high waves (surf zone)
• Sand waves with sheet flow in case of strong (tidal) currents and high waves (outside surf zone
ks  ks  ks
More complicated!
No universal solutions
Grain Roughness (k’s)in Combined Currents and Waves
Grain roughness is dominant for both the wave-related and current-related friction when
the bed is plane.
When bed forms are present and the peak orbital excursion at the bed is smaller than the
bed form length (i.e. Aδ < λ), the grain roughness is also dominant for the wave-related
friction. In that case the bed forms act as topographic features for the waves.
For wave motion:
(in the ripple regime)
 3d90 for   1

ks , w  
3 d90 for   1 (in the sheet flow regime)
For current motion:
 3d90

k s ,c  
3 d90
for   1 (lower regime)
for   1 (upper regime)
Note that the calculation of the mobility parameter θ for current are different from that
for wave motion
Form Roughness (k”s)in Combined Currents and Waves
When the bed is covered with ripples, the ripple roughness is dominant for the
current-related friction. Ripple roughness is also dominant for the wave-related
friction when the peak value of the orbital excursion at the bed is larger than the
ripple length (i.e. Aδ < λr). The ripple roughness is calculated by
ks,c  20 r r (
r
r
)
When sand waves with or without (mega or mini) ripples are present, the largescale sand waves act as topographic features for the waves motion because the
sand waves have a length much larger than the orbital excursion at the bed. Thus,
the wave-related friction factor is not determined by the large-scale sand wave
dimensions, but by the small-scale ripples (if present) on the back of the sand
waves.
Dune on Mars ?
Three pairs of before and after images from the High Resolution Imaging Science Experiment (HiRISE) camera on NASA's Mars
Reconnaissance Orbiter illustrate movement of ripples on dark sand dunes in the Nili Patera region of Mars. Image Credit:
NASA/JPL-Caltech/University of Arizona/International Research School of Planetary Sciences