Coastal Modelling System Non-Equilibrium Sediment Transport Model - - PowerPoint PPT Presentation

Coastal Inlets Research Program ERDC Coastal and Hydraulics Laboratory

• No. 1 of 18 Slides

Coastal Modelling System Non-Equilibrium Sediment Transport Model (NET)

Alex Sánchez, CHL Weiming Wu, NCCHE

Coastal Inlets Research Program ERDC Coastal and Hydraulics Laboratory

• No. 2 of 18 Slides

Outline

• Introduction

– CMS and NET Overview – Equilibrium vs. Non-equilibrium sediment transport – Advantages of Non-equilibrium

• Model Equations: An overview

– Concentration capacity – Adaptation length – Bed-slope coefficient – Sediment diffusivity – Total load correction factor

• Avalanching
• Numerical implementation
• Questions

Coastal Inlets Research Program ERDC Coastal and Hydraulics Laboratory

• No. 3 of 18 Slides

Introduction: CMS Overview

• Overview
• Coupling

– Wave-Flow

• Radiation stress
• Wave height, period and

direction

– Flow-Wave

• Water elevation
• Currents
• Updated bathymetry

CMS-Wave Coastal Modeling System Salinity Module CMS-Flow Sediment Module

• Sediment Module

1. Total load

• Sediment balance Eq.

2. Equilibrium Transport (ET)

• Advection-Diffusion Eq.

– Suspended load

• Bed change Eq.

– Morphology change – Bed-slope effect – Bed load transport

3. Non-Equil. Transport (NET)

• Advection-Diffusion Eq.

– Total load

• Bed change Eq.

– Morphology change – Bed-slope effect

Coastal Inlets Research Program ERDC Coastal and Hydraulics Laboratory

• No. 4 of 18 Slides

Introduction: NET Overview

• 2D depth-averaged
• Features (processes)

– Advection – Diffusion – Erosion and deposition – Bed-slope effects – Avalanching

• Definition of variables

Coastal Inlets Research Program ERDC Coastal and Hydraulics Laboratory

• No. 5 of 18 Slides

Introduction: Equilibrium vs. Non-Equilibrium

• Equilibrium sediment transport

– Assume local instantaneous equilibrium for bed-load transport or total-load – Bed change is determined by mass balance equation (Exner equation)

• Non-equilibrium sediment transport models

– Do not assume any sediment transport load to be in equilibrium – Bed change is proportional to difference between local and equilibrium transport rates

Coastal Inlets Research Program ERDC Coastal and Hydraulics Laboratory

• No. 6 of 18 Slides

Advantages of NET

• Considers temporal and spatial lags between flow

and sediment transport

• Can easily handle constrained sediment loading

(over- or under-loading)

• Hard-bottom problem is no problem
• Can model suspended and bed load separately or

combined as bed-material or total load

• More stable than equilibrium sediment transport

Coastal Inlets Research Program ERDC Coastal and Hydraulics Laboratory

• No. 7 of 18 Slides
• Advection-diffusion
• Bed change equation
• Bed-slope term

Model Equations: Quick Overview

* s

Concentration Capacity Total Load Adapt. Coeff. Sediment Fall Velocity Fraction of suspended load Total Load Correction Factor Diffusion Coefficient Bed-slope term D Bed-slopecoefficien

t t f s t b

C w r K S α β = = = = = = = = t

*

( ) ( ) ( ) ( ) ( )

t t t s t s t t f t t t

dC dUC dVC r C r C Kd Kd w C C t x y x x y y α β ⎛ ⎞ ⎡ ⎤ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ⎡ ⎤ + + = + + + − ⎜ ⎟ ⎢ ⎥ ⎢ ⎥ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ⎣ ⎦ ⎣ ⎦ ⎝ ⎠ (1 ) (1 )

y x b s s t s s t

G G h h S D r d U C D r d U C x y x x y y ∂ ⎡ ⎤ ∂ ∂ ∂ ∂ ∂ ⎡ ⎤ = + = − + − ⎢ ⎥ ⎢ ⎥ ∂ ∂ ∂ ∂ ∂ ∂ ⎣ ⎦ ⎣ ⎦

' *

(1 ) ( )

m t f t t b

h p C C S t α ω ∂ − = − + ∂

Coastal Inlets Research Program ERDC Coastal and Hydraulics Laboratory

• No. 8 of 18 Slides

Concentration Capacity

• The concentration that would be achieved under steady-

state and equilibrium conditions

• Formula
• Larger scaling factors produce larger sediment loads and

therefore larger morphology changes

• It is one of the most important parameters (driving force)

– Controls largely the magnitude and distribution of the sediment concentration field

t t c

Q C U d

∗ = t s s b b

Q f Q f Q = +

Suspended load transport Bed load transport Total load transport Suspended scaling factor Bed scaling factor

s b t s b

Q Q Q f f = = = = =

Coastal Inlets Research Program ERDC Coastal and Hydraulics Laboratory

• No. 9 of 18 Slides

Concentration Capacity - Continued

• Options for sediment transport capacity:

– Lund-CIRP (ERDC/CHL CR-07-01)

• Separate equations for suspended and bed loads

– Van Rijn (J. Hydraulic Eng. 2007)

• Separate equations for suspended and bed loads

– Watanabe (Proc. Coastal Sediments 1987)

• One equation for total load
• These equations represent the sediment transport under

equilibrium conditions

Coastal Inlets Research Program ERDC Coastal and Hydraulics Laboratory

• No. 10 of 18 Slides

Adaptation Coefficient

• Related to how much time/distance it takes to reach equilibrium
• The larger the coefficient the more rapid the system goes into

equilibrium and the larger the erosion and deposition

• Formulas

– Suspended Load

• Lin (1984)
• Amanini and Silvio (1986)

– Bed Load – Total Load

c t f

U d L α ω =

*

3.25 0.55ln

f c

U ω α κ ⎛ ⎞ = + ⎜ ⎟ ⎝ ⎠

c s f

U d L w α =

,

(1 ) max( , )

s b s s t b s t given

r L r L L L L L ⎧ − + ⎪ = ⎨ ⎪ ⎩

, given b b given

A d L L ⎧ ⎪ =⎨ ⎪ ⎩ 33 o b z =

1/6 *

1 1 exp 1.5

f

w b b b d d d U α

−

⎡ ⎤ ⎛ ⎞ ⎛ ⎞ = + − − ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎢ ⎥ ⎣ ⎦

Most Stable

*

( )

f t t

E D w C C α − = −

Coastal Inlets Research Program ERDC Coastal and Hydraulics Laboratory

• No. 11 of 18 Slides

Bed-slope Coefficient Ds

• Bed change equation
• Method first adapted by Watanabe (1985)
• Smoothes bathymetry
• Improves stability
• Related to sediment properties and flow characteristics
• Reported values

– Watanabe (1985) used Ds = 10 – Larson et al. (2003) and Karambas (2003) used Ds = 2

• Recommended values: 1-10

' *

(1 ) ( ) (1 ) (1 )

m f t t s s t s s t

h h h p C C D r d U C D r d U C t x x y y αω ⎡ ⎤ ∂ ∂ ∂ ∂ ∂ ⎡ ⎤ − = − + − + − ⎢ ⎥ ⎢ ⎥ ∂ ∂ ∂ ∂ ∂ ⎣ ⎦ ⎣ ⎦

Coastal Inlets Research Program ERDC Coastal and Hydraulics Laboratory

• No. 12 of 18 Slides

Sediment Diffusivity Coefficient

• Coefficient is related to the strength of horizontal mixing

in a depth-averaged sense

• Directly related to eddy viscosity
• Eddy viscosity

– Falconer – Subgrid model – Waves

s

K ν σ =

s

Diffusion coefficient Eddy viscosity Smidth number K ν σ = = =

( )

( )

2 2 2 * t c h

v u d l S ν α = + +

2

1 1.15 2

c t

U v gd C ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦

w m

u H ν = Λ (1 )

m t m w

ν θ ν θ ν = − +

empiricalcoeff. Wave orbital velocity Wave height

m

u H Λ = = =

3 m

H d θ ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠

Coastal Inlets Research Program ERDC Coastal and Hydraulics Laboratory

• No. 13 of 18 Slides
• Accounts for the lag between the depth-averaged

sediment and flow velocities

• For total load transport

Velocity profile Concentration profile

Total-load Correction Factor

,

f

• t

w d z f d β ε ⎞ ⎛ = ⎟ ⎜ ⎝ ⎠

1 (1 )

c t s sed s b

U r U r u β = + −

s b s b

z s z a sed z z a

u cdz U cdz

+ +

= ∫

∫

0.7

t

β ≈

For sands

*

( ) ln

c s

u z u z z κ ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ ( ) exp ( )

f a

w c z c z a ε ⎧ ⎫ = − − ⎨ ⎬ ⎩ ⎭

Coastal Inlets Research Program ERDC Coastal and Hydraulics Laboratory

• No. 14 of 18 Slides

Hard-Bottom Problem

• Concentration capacity at hard-bottom cells
• Bed change equation
• Advection-diffusion equation

( )

∗ ∗

=

t t hb t

C C C , min

,

( ) [ ]

*

1 min( , )

m t f t t t hb

h p C C C S t α ω ∂ ′ − = − + ∂

( )

( )

[ ]

t t t f t t s y t s x t y t x t t

C C C y C r d K y x C r d K x y C dU x C dU dC t − + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ = ∂ ∂ + ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ) , min( ) ( ) (

*

ω α β ← Only allows deposition

Coastal Inlets Research Program ERDC Coastal and Hydraulics Laboratory

• No. 15 of 18 Slides

Avalanching

• Avalanching process is simulated as
• Two approaches available

1. 9-point mass balance approach (Wu 2007)

• May be used at large time intervals
• Iterates until convergence

2. Relaxation method (5- and 9-point)

• Requires smaller time steps (morphologic time step)
• No iterations
• Very simple and stable

( ) ( ) tan

i i p p r

h h h h l γ φ + Δ − + Δ = Δ

depth distancebetween cellnodes Reposeangle

r

h l φ = Δ = = 1 for (upslope)

• 1 for

(downslope)

p i p i

h h h h γ < ⎧ ⎪ = ⎨ > ⎪ ⎩

p i i

Coastal Inlets Research Program ERDC Coastal and Hydraulics Laboratory

• No. 16 of 18 Slides

Avalanching: Relaxation Method

• Mass balance between neighboring cells
• Repose slope condition
• Depth change due to avalanching

i i p p

A h A h Δ + Δ =

( )

tan tan

i r p p i

A l h R A A φ γ φ Δ − Δ = + tan

i p

h h l φ − = Δ

p p i i

h A h A Δ Δ = − ( ) ( ) tan

i i p p r

h h h h l γ φ + Δ − + Δ = Δ Relaxation factor 0.1-0.2 R = ≈

Coastal Inlets Research Program ERDC Coastal and Hydraulics Laboratory

• No. 17 of 18 Slides

Numerical Methods

• Finite volume method
• Time integration

– Explicit Euler scheme

• Advection

– Upwind – Hybrid Linear/Parabolic Approximation (HLPA)

• Diffusion

– Central difference

• Bed-slope term (conc.)

– Central difference

• Boundary conditions

– Ocean boundaries

• Inflow: “Zero-gradient” BC
• Outflow: Open BC

– Land boundary

• Zero flux BC

– Future versions will have

• User specified concentration

(river)

• Equilibrium BC

Coastal Inlets Research Program ERDC Coastal and Hydraulics Laboratory

• No. 18 of 18 Slides

Questions?