Hydrosedimentary processes in the nearshore domain
Elements for the physical approach
Jean-Pierre Lefebvre, IRD (FRANCE)
Physical and Mathematical Tools for the Study
of Marine Processes of Coastal Areas
26 May – 6 June 2008, Cienfuegos, CUBA
Erosion, suspension, fluidization
Permanent stress (current)
Oscillatory stress (wave)
Non cohesive sediment (sand)
Cohesive sediment (mud)
Turbulence, energy dissipation, shoaling
FORCINGS
I. Permanent flow
II. Oscillatory flow
SEDIMENTS
III. Cohesive sediments
Permanent flow (laminar)
z
u(z)
boundary layer : for a viscous flow, layer defined from the bed (non slip
condition) up to the height where the flow is no longer perturbed by the wall.
Newton’s law of viscosity
-1)
µn:: kinematic
(absolute)fluid
dynamic
viscosity
viscosity
(m².s(Pa.s)
(1.08 10−3 Pa.s for seawater
at T = 20°C and S = 35 g.kg-1)
-3
rw: density of water (kg.m )
(≈ 1.025 for seawater for T =20°C, S = 35 g.kg-1)
Permanent flow (turbulent)
turbulent flow : fluid regime characterized by
chaotic property changes. This includes high
frequenty variation of velocity in space and
time.
da Vinci sketch of a turbulent flow
Reynolds decomposition of a parameter
_
fluctuating component
(perturbation)
local velocity
u(t,z) Instantaneous
u’(t,z)
steady component
0
0
T
T
t
t
Permanent flow (turbulent)
Turbulent shear stress
Reynolds eddy
stressviscosity
tensor (m².s-1)
ne: kinematic
(covariance of vertical and horizontal velocities)
Permanent flow (turbulent)
Turbulent boundary layer
h
OUTER REGION
∿ 0.1h
δv
LOG LAYER
VISCOUS SUB-LAYER
Innermostouter
Turbulent
Intermediate
region
region
region
(viscous
(log layer)
sub layer)
 dominated
logarithmic
influenced by
profile
viscosity,
the outer
of theboundary
horizontalcondition
velocity of the layer,
 linear
consists
velocity
of about
profile
80-90
, % of the total region,
 very
velocity
small.
relatively constant due to the strong mixing of the flow.
Permanent flow (turbulent)
The characteristic velocity scale u* is a parameter of the
order of magnitude of the turbulent velocity
often called friction velocity since it is used as the actual
turbulent velocity action on the bed
z
Friction velocity
∿ 0.1h
LOG LAYER
δv
VISCOUS SUB LAYER
u
Permanent flow (turbulent)
Prandtl’s model of mixing-length in the turbulent
boundary layer, states that the turbulence is linearly
related to the averaged velocity gradient by a term Lm
called, mixing length
z
-
∿ 0.1h
von Kármán assumption states that the correlation
scale is proportional to the distance from the boundary
LOG LAYER
δv
VISCOUS SUB LAYER
u
von Kármán constant (κ = 0.408 )
the kinematic eddy viscosity must also be proportional
to the height above the bed.
Permanent flow (turbulent)
z
Prandtl-Kármán law of wall
∿ 0.1h
LOG LAYER
δv
VISCOUS SUB LAYER
u
z0 : hydraulic roughness of the flow
depends on viscous sub-layer, grain roughness,
ripples and other bedforms, stratification,…
Permanent flow (turbulent)
z
Nikuradse sand roughness (physical roughness)
can be approximated by the median diameter of
grains of sandy bed
∿ 0.1h
LOG LAYER
δv
VISCOUS SUB LAYER
u
d50 : mean particles diameter
Permanent flow (turbulent)
the viscous sub-layer is a narrow layer close to the
wall where roughness of the wall and molecular
viscosity dominate transport of momentum
z
∿ 0.1h
LOG LAYER
Ratio of inertial force to viscous force
δv
Thickness of the viscous sub-layer
VISCOUS SUB LAYER
u
Permanent flow (turbulent)
The relative roughness (ratio of hydraulic roughness z0 on the physical roughness ks)
depends on the relative length scales for the viscous sub-layer and the physical
roughness
roughness Reynolds or grain Reynolds number
Permanent flow (turbulent)
Hydraulically rough regime : Re* > 70
 the viscous sub-layer is interrupted by the bed roughness,
 roughness elements interact directly with the turbulence.
Rough regime
Permanent flow (turbulent)
Hydraulically smooth regime : Re* < 5
the viscous sub-layer lubricates the roughness elements
so they do not interact with turbulence.
Smooth
regime
Rough regime
Permanent flow (turbulent)
hydraulically transitional regime : 5 ≤ Re* ≤ 70
For 0.26 < ks/v < 8.62 the near-wall flow is transitional
between the hydraulically smooth and hydraulically rough regimes
+
Smooth
regime
Transitional
regime
Rough regime
Permanent flow (turbulent)
Bottom shear stress
turbulent outer layer
friction factor
log layer
Friction factor for current
(rough turbulent regime)
transition layer
viscous sub-layer
Permanent flow (turbulent)
Quantification
Measurements
Description
FORCING
SEABEDS
Velocities at some elevations near the bed
Sediment granular distribution
Friction velocity and hydraulic roughness
Physical roughness
Turbulent shear stress at the bed
Hydraulic turbulent regime
Prandtl-Kármán
law of wall
Nikuradse approximation
Oscillatory flow
Waves can be defined by their superficial properties
 wave height (distance between its trough and crest)
 wave length (distance between two crests)
 wave period (duration for the propagation of two successive extrema at a given location)
wave amplitude (m)
angular velocity (rad.s-1)
wave number (rad.m-1)
wave height (m)
wave period (s)
wavelength (m)
Oscillatory flow
Airy wave : model for monochromatic progressive sinusoidal waves
Wave with multispectral components
Oscillatory flow
velocity potential ∅
Assuming an oscillatory flow V of an inviscid , incompressible fluid,
with no other motions interfering (i.e. no currents)
 irrotational flow (i.e. no curl between the water particles trajectories) :  V = 0
 satisfying the continuity equation : . V = 0
For a sinusoidal wave field, it exists an ideal potential flow solution: ∅ = V
from which we can derivate the expressions of all the pressure and flow fields.
Oscillatory flow
Equation of Laplace for the inviscid, uncompressible flow
BOUNDARY CONDITIONS
Kinematic boundary condition : a parcel of fluid at the surface remains at the surface
Dynamic boundary condition : the pressure along an iso-potential line is constant (Bernoulli )
Boundary condition : the bottom is not permeable to water
Oscillatory flow
For small amplitude gravity wave (wave amplitude a << wavelength λ)
Oscillatory flow
Linearization (only the first order terms of the Taylor series)
Laplacian equation
SIMPLIFIED BOUNDARY CONDITIONS
Simplified Kinematic boundary condition
Simplified dynamic boundary condition
Simplified boundary condition
Oscillatory flow
General form of ∅ for a sinusoidal wave
Oscillatory flow
SURFACE ELEVATION
VELOCITY FIELD
PRESSURE
∂∅
From ___= -gη at z = 0
∂t
from ∅ = V
∂∅
From p = -ρw ___
∂t
Oscillatory flow
DISPERSION EQUATION
The relation between the angular velocity ω and the wave number
(from the simplified Laplace equation)
WAVE CELERITY
velocity of the wave crest ( m.s-1)
Oscillatory flow
DISPERSION EQUATION
DEEP WATER DOMAIN
WAVE CELERITY
The water height is much greater than the wavelength (h >> λ)
Oscillatory flow
DISPERSION EQUATION
WAVE CELERITY
SHALLOW WATER DOMAIN
The wavelength is much greater than the water height (λ >>h)
Oscillatory flow
INTERMEDIATE DOMAIN
Oscillatory flow
shoaling
SWL
Wave
breaking
Limit of lower orbital motions
No erosion of the seabed
Strong
erosion
Slight erosion of the seabed
INTERMEDIATE
ZONE
DEEP WATER
λ
h ∿ __
2
SHALLOW
WATER
λ
h ∿ __
20
Oscillatory flow
Orbital velocity at the bed
Stokes’ drift
Oscillatory flow (turbulent)
Wave boundary layer thickness
maximum shear velocity
Turbulent wave shear stress
Oscillatory flow (turbulent)
Law of wall (Grant and Madsen)
Phase lead
Oscillatory flow (turbulent)
Shear stress generated by the oscillatory flow
where
Oscillatory flow (turbulent)
Maximum shear stress
friction factor for wave
Friction factor for wave (rough turbulent regime)
Oscillatory flow (turbulent)
Measurements
Quantification
Description
FORCING
SEABEDS
Surface wave parameters and wave height
Sediment granular distribution
Maximum shear velocity and hydraulic roughness
Maximum shear stress at the bed
Physical roughness
Hydraulic turbulent regime
Grant-Madsen
Law of wall
Nikuradse approximation
Oscillatory flow
Energy density (J.m-2)
Potential Energy
Kinetic Energy
Oscillatory flow
Flux of energy (J)
Group Velocity (m.s-1)
In deep water domain (kh→∞) and Cg = C/2
In shallow water domain (kh→0) and Cg = C
Oscillatory flow
Shoaling
5
4.5
Wave height (m)
4
Wave period : 8 s
3.5
H
__
∿ 0.8
h
3
2.5
2.3
2
1.5
hdw= 49.6 m
1
hsw= 0.8
0.5
0
100
90
80
70
60
50
40
Water depth (m)
30
20
10
0
2.8
Combined current and wave stresses
many empirical expressions exist for coupling permanent and oscillatory stresses
(Soulsby, 1995)
seabed
Transport mode for marine sediments
Bed-load
Thea rolling,
sliding
and
jumping
grains(10–100)
insuspended
almostand
continuous
contact
with the bed.
Sheet
flowtransport
a layertransport
with
thickness
of
grain
layers
large
sediment
concentrations
Suspended-load
Grains
areseveral
almost
continuously
in
the water
column
Intergranular
collision
forces
an important
role
Thethe
turbulence
mixingplay
processes
are dominant
is transported
along
bed.
seabed
Sediment cohesion : domination of interparticle forces or the gravitational force
in the behavior of sediment.
Cohesive sediments : material with strong interparticle forces due to their surface ionic charges
Non cohesive sediments : granular material dominated by the gravitational force
CLAY
very
fine
clay
2µ
SILT
4
fine
8
medium coarse
16
32
GRAVEL
SAND
very
coarse
64
125
very
fine
fine
medium coarse
250 500
1mm
very
coarse
2
4
COBBLES
pea
medium gravel coarse cobbles
gravel
8
16
32
64
seabed
Erosion
critical shear stress
fully
Suspended
controled
flow
sediments
(flow, (chenal
measured
dimensions,
with
OBS
fixed bottom roughness)
Erosion
andfine
transport
bedload
and
suspension)
Non
cohesive
sediment
trapping
(gravitation)
Measurements
the
erosion
(erodimeter,
IFREMER)
In
extraction
situ
sampling
ofofthe
ofunperturbed
unperturbed
interface
seabed
Grain
size
spectrum
of defloculated
material
Flocculation
Mud flocs are characterized by four main physical properties:
 size (diameter) Df
 density ρfloc
 settling velocity Ws
 floc strength Fc
Mud floc properties are governed by four mechanisms:
 Brownian motions cause the particles to collide to form aggregates

particles with a large settling velocity will overtake particles
with a low settling velocity and aggregate
 turbulent motions will cause particles, carried by the eddies
to collide and form flocs
 turbulent shear may disrupt the flocs again, causing floc breakup
Flocculation
Self similarity
clay < 4µm
fine silt (4 ∿
flocculus
Microfloc
(<10µm)
100µm)
Macrofloc
microfloc
flocculus
( ∿O(2) µm up to ∿ O(1) mm)
strong
bound
by sticky loosely
material
produced
biological
organisms
strong
interparticle
forces due
to surface
ionic
charges
bound
and by
very
fragile
Flocculation
Floc size
The fractal dimension nf is obtained from the description of a growing object
with linear size αL and volume V(αL)
α ( linear size of the primary object (seed) (arbitrary = 1)
number of seeds
in estuarine and coastal environments 1.7 < nfloc < 2.2
Flocculation
Floc excess density
defloculated particles diameter
floc diameter
Sediment density ρs for clay ∿ 1390 kg.m-3
Flocculation
Floc limitation by turbulence
the energy
dissipation
rate perisunit
mass ε expresses
the balance
process of
of floc
energy
transfer
the cut-off
floc diameter
determined
by the local
growth
and rupture within a turbulent fluid regime.
volume average value of ε (J)
Rate of turbulent shear
Flocculation
Taylor microscale
The Taylor microscale λ is representative of the energy transfer from large to small scales.
For large Reynolds numbers, the structure of turbulence tends to be approximately isotropic
Normalized Taylor microscale
Flocculation
Kolmogorov microscale
At very small length scales, viscosity becomes effective in smoothing out velocity fluctuations
preventing the generation of infinitely small scales by dissipating small-scale energy into heat.
The smallest scale of motion automatically adjusts itself to the value of the viscosity.
The Kolmogorov length defines the smallest length scale of turbulent motion
and is location dependent thru λ(z)
Flocculation
Kolmogorov microscale
Turbulent mixing induces aggregation and, at the same time, subjects aggregates to
higher shear stresses causing breakup for flocs of diameter greater than dmax
Settling
Stokes settling velocity
velocity of a spherical object settling through a fluid
when the flow around the object is laminar
Settling
The settling velocity of estuarine mud flocs is largely affected by some physical parameters:
 turbulence, shear or bottom shear stress
 salinities
 floc strength
 fractal structure
 concentration
 sediment composition
 time spent in an equilibrium state (residence time of flocs )
The expression of the settling velocity for flocs must combine three effects:
 gravity
 flocculation
 hindered settling
Settling
Hindered settling velocity
depends on grain Reynolds number
Volume concentration
At high concentrations, the return flow of water around a particle
may create an upward drag on neighboring particles.
CURRENT
WAVE
Airy model
Turbulent boundary layer
Water height
Turbulence within the boundary layer
Bottom shear stress
Bed roughness
SEABED
Settlings
Flocculation
Turbulent boundary layer
Turbulence within the boundary layer
BottomErosion
shear stress
COHESIVE SEDIMENT