Hydrosedimentary processes in the nearshore domain Elements for the physical approach

Hydrosedimentary processes in the nearshore domain

Elements for the physical approach

Physical and Mathematical Tools for the Study of Marine Processes of Coastal Areas

26 May – 6 June 2008, Cienfuegos, CUBA

Jean-Pierre Lefebvre, IRD (FRANCE)

Permanent stress (current)

Oscillatory stress (wave)

Non cohesive sediment (sand)

Cohesive sediment (mud)

Erosion, suspension, fluidization

Turbulence, energy dissipation, shoaling

FORCINGS

I. Permanent flow

II. Oscillatory flow

SEDIMENTSIII. Cohesive sediments

Permanent flow (laminar)

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.

µ : (absolute) dynamic viscosity (Pa.s) (1.08 10−3 Pa.s for seawater at T = 20°C and S = 35 g.kg-1)

Newton’s law of viscosity

: kinematic fluid viscosity (m².s-1)

w: density of water (kg.m-3) (≈ 1.025 for seawater for T =20°C, S = 35 g.kg-1)

z

u(z)

u’(t,z)

t0T

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

u(t,z)

t0 T

Instantaneous local velocitysteady component_ fluctuating component

(perturbation)

Permanent flow (turbulent)

Reynolds stress tensor(covariance of vertical and horizontal velocities)

Turbulent shear stress

e: kinematic eddy viscosity (m².s-1)

Permanent flow (turbulent)

Turbulent outer region influenced by the outer boundary condition of the layer, consists of about 80-90 % of the total region, velocity relatively constant due to the strong mixing of the flow.

Intermediate region (log layer) logarithmic profile of the horizontal velocityInnermost region (viscous sub layer) dominated by viscosity, linear velocity profile , very small.

Turbulent boundary layer

h

∿ 0.1hδv

OUTER REGION

LOG LAYERVISCOUS SUB-LAYER

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

Friction velocity

z

u

∿ 0.1h

δv

LOG LAYER

VISCOUS SUB LAYER

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

von Kármán constant (κ = 0.408 )

von Kármán assumption states that the correlation scale is proportional to the distance from the boundary

the kinematic eddy viscosity must also be proportional to the height above the bed.

z

u

∿ 0.1h

δv

LOG LAYER

VISCOUS SUB LAYER

-

z

u

∿ 0.1h

δv

LOG LAYER

VISCOUS SUB LAYER

Permanent flow (turbulent)

Prandtl-Kármán law of wall

z0 : hydraulic roughness of the flow

depends on viscous sub-layer, grain roughness,ripples and other bedforms, stratification,…

Permanent flow (turbulent)

Nikuradse sand roughness (physical roughness) can be approximated by the median diameter of grains of sandy bed

d50 : mean particles diameter

z

u

∿ 0.1h

δv

LOG LAYER

VISCOUS SUB LAYER

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

Thickness of the viscous sub-layer

Ratio of inertial force to viscous force

z

u

∿ 0.1h

δv

LOG LAYER

VISCOUS SUB LAYER

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* < 5the viscous sub-layer lubricates the roughness elements so they do not interact with turbulence.

Rough regimeSmoothregime

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

Rough regimeSmoothregime

Transitionalregime

+

Permanent flow (turbulent)

Bottom shear stress

friction factor

Friction factor for current (rough turbulent regime)

turbulent outer layer

log layer

transition layer

viscous sub-layer

Permanent flow (turbulent)

FORCING SEABEDS

Velocities at some elevations near the bed Sediment granular distribution

MeasurementsQuantification

Friction velocity and hydraulic roughness Physical roughness

Description

Turbulent shear stress at the bed Hydraulic turbulent regime

Prandtl-Kármán law of wallNikuradse 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 period (s)angular velocity (rad.s-1)

wavelength (m)wave number (rad.m-1)

wave amplitude (m)wave height (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 = 0For 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

BOUNDARY CONDITIONS

Dynamic boundary condition : the pressure along an iso-potential line is constant (Bernoulli )

Kinematic boundary condition : a parcel of fluid at the surface remains at the surface

Boundary condition : the bottom is not permeable to water

Equation of Laplace for the inviscid, uncompressible flow

Oscillatory flow

For small amplitude gravity wave (wave amplitude a << wavelength λ)

Oscillatory flow

SIMPLIFIED BOUNDARY CONDITIONS

Simplified dynamic boundary condition

Simplified Kinematic boundary condition

Simplified boundary condition

Linearization (only the first order terms of the Taylor series)

Laplacian equation

Oscillatory flow

General form of ∅ for a sinusoidal wave

Oscillatory flow

VELOCITY FIELD from ∅ = VSURFACE ELEVATION From = -gη at z = 0

∂∅∂t___

PRESSURE 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

WAVE CELERITYDEEP WATER DOMAIN 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

DEEP WATERINTERMEDIATE

ZONESHALLOW

WATER

Limit of lower orbital motions

Slight erosion of the seabedNo erosion of the seabed

shoaling

Wavebreaking

Strong erosion

SWL

∿ λ__2

h ∿ λ__20

h

Oscillatory flow

Orbital velocity at the bed

Stokes’ drift

Oscillatory flow (turbulent)

Wave boundary layer thickness

Turbulent wave shear stress

maximum shear velocity

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)

Nikuradse approximation

FORCING SEABEDS

Oscillatory flow (turbulent)

Sediment granular distribution

Maximum shear velocity and hydraulic roughness Physical roughness

Measurements

Maximum shear stress at the bed Hydraulic turbulent regime

Surface wave parameters and wave height

Grant-Madsen Law of wall

DescriptionQuantification

Potential Energy

Oscillatory flow

Kinetic Energy

Energy density (J.m-2)

Oscillatory flow

Flux of energy (J)

Group Velocity (m.s-1)

In deep water domain (kh→∞) and Cg = C/2In shallow water domain (kh→0) and Cg = C

Oscillatory flow

Shoaling

01020304050607080901000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Wav

e he

ight

(m)

Water depth (m)

Wave period : 8 s

hdw= 49.6 m hsw= 0.8

2.8

2.3∿

h__ 0.8H

many empirical expressions exist for coupling permanent and oscillatory stresses

Combined current and wave stresses

(Soulsby, 1995)

seabed

Bed-load transport The rolling, sliding and jumping grains in almost continuous contact with the bed. Intergranular collision forces play an important role

Suspended-load transport Grains are almost continuously suspended in the water column The turbulence mixing processes are dominant Sheet flow a layer with a thickness of several grain layers (10–100) and large sediment concentrations is transported along the bed.

Transport mode for marine sediments

2µ 4 8 16 32 64 125 250 500 1mm 2 4 8 16 32 64

seabed

Sediment cohesion : domination of interparticle forces or the gravitational force in the behavior of sediment.

CLAY SILT SAND GRAVEL COBBLES

veryfine

veryfine

fine finemedium medium medium coarsecoarsecoarse verycoarse

verycoarse

peagravel

gravel cobblesclay

Cohesive sediments : material with strong interparticle forces due to their surface ionic charges

Non cohesive sediments : granular material dominated by the gravitational force

seabed

Erosion

In situ sampling of unperturbed seabedextraction of the unperturbed interfaceMeasurements of the erosion (erodimeter, IFREMER)

fully controled flow (flow, chenal dimensions, fixed bottom roughness)Erosion and transport ( bedload and suspension)Non cohesive sediment trapping (gravitation)Suspended fine sediments measured with OBS

Grain size spectrum of defloculated material

critical shear stress

Flocculation

Mud flocs are characterized by four main physical properties: size (diameter) Df

density ρfloc

settling velocity Ws

floc strength Fc

turbulent motions will cause particles, carried by the eddies to collide and form flocs

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 shear may disrupt the flocs again, causing floc breakup

Flocculation

clay < 4µm fine silt (4 ∿ 10µm)

flocculus

flocculus

microfloc

Microfloc (< 100µm)

Macrofloc ( ∿O(2) µm up to ∿ O(1) mm)

strong interparticle forces due to surface ionic chargesstrong bound by sticky material produced by biological organismsloosely bound and very fragile

Self similarity

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

floc diameter

Sediment density ρs for clay 1390 kg.m∿ -3

defloculated particles diameter

Flocculation

Floc limitation by turbulence

the cut-off floc diameter is determined by the local balance of floc growth and rupture within a turbulent fluid regime.

Rate of turbulent shear

volume average value of ε (J)

the energy dissipation rate per unit mass ε expresses the process of energy transfer

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

velocity of a spherical object settling through a fluid when the flow around the object is laminar

Stokes settling velocity

Settling

gravity flocculation hindered settling

The expression of the settling velocity for flocs must combine three effects:

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 settling velocity of estuarine mud flocs is largely affected by some physical parameters:

Settling

Hindered settling velocity

At high concentrations, the return flow of water around a particle may create an upward drag on neighboring particles.

Volume concentration

depends on grain Reynolds number

CURRENT

SEABED

Turbulent boundary layer

WAVE

Water height

Airy model

Bottom shear stress

Bed roughness

Turbulence within the boundary layer

COHESIVE SEDIMENT

Bottom shear stress

Turbulence within the boundary layer

Turbulent boundary layer

Erosion

Flo

ccul

ation

Sett

lings