Coastal Modelling System Non-Equilibrium Sediment Transport Model (NET) Alex Sánchez, CHL Weiming Wu, NCCHE Coastal Inlets Research Program No. 1 of 18 Slides ERDC Coastal and Hydraulics Laboratory
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 No. 2 of 18 Slides ERDC Coastal and Hydraulics Laboratory
Introduction: CMS Overview • Sediment Module • Overview 1. Total load Coastal Modeling System • Sediment balance Eq. CMS-Wave CMS-Flow 2. Equilibrium Transport (ET) • Advection-Diffusion Eq. Salinity Sediment – Suspended load Module Module • Bed change Eq. – Morphology change • Coupling – Bed-slope effect – Wave-Flow – Bed load transport • Radiation stress 3. Non-Equil. Transport (NET) • Wave height, period and direction • Advection-Diffusion Eq. – Flow-Wave – Total load • Water elevation • Bed change Eq. • Currents – Morphology change • Updated bathymetry – Bed-slope effect Coastal Inlets Research Program No. 3 of 18 Slides ERDC Coastal and Hydraulics Laboratory
Introduction: NET Overview • 2D depth-averaged • Features (processes) – Advection • Definition of variables – Diffusion – Erosion and deposition – Bed-slope effects – Avalanching Coastal Inlets Research Program No. 4 of 18 Slides ERDC Coastal and Hydraulics Laboratory
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 No. 5 of 18 Slides ERDC Coastal and Hydraulics Laboratory
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 No. 6 of 18 Slides ERDC Coastal and Hydraulics Laboratory
Model Equations: Quick Overview • Advection-diffusion ⎛ ⎞ ∂ ∂ ∂ ∂ ∂ ∂ ⎡ ∂ ⎤ ⎡ ⎤ dC ( dUC ) ( dVC ) ( r C ) ( r C ) + + = + + + α − ⎜ t ⎟ t t s t s t Kd ⎢ Kd ⎥ w ( C C ) ⎢ ⎥ ∂ β ∂ ∂ ∂ ∂ ∂ ∂ t f t * t ⎣ ⎦ ⎣ ⎦ ⎝ ⎠ t x y x x y y t = C Concentration Capacity t * • Bed change equation α = Total Load Adapt. Coeff. t = w Sediment Fall Velocity f ∂ = h r Fraction of suspended load − = α ω − + s ' (1 p ) ( C C ) S β = ∂ m t f t * t b Total Load Correction Factor t t = K Diffusion Coefficient = S Bed-slope term • Bed-slope term b = D Bed-slopecoefficien t s ∂ ∂ ∂ ∂ ∂ ⎡ ∂ ⎤ ⎡ ⎤ G G h h = + = − + − y x S D (1 r d U C ) ⎢ D (1 r d U C ) ⎥ ⎢ ⎥ ∂ ∂ ∂ ∂ ∂ ∂ b s s t s s t ⎣ ⎦ ⎣ ⎦ x y x x y y Coastal Inlets Research Program No. 7 of 18 Slides ERDC Coastal and Hydraulics Laboratory
Concentration Capacity • The concentration that would be achieved under steady- state and equilibrium conditions = • Formula Q Suspended load transport s Q ∗ = = t C Q Bed load transport t b U d = c Q Total load transport t = f Suspended scaling factor = + s Q f Q f Q = t s s b b f Bed scaling factor b • 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 Coastal Inlets Research Program No. 8 of 18 Slides ERDC Coastal and Hydraulics Laboratory
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 No. 9 of 18 Slides ERDC Coastal and Hydraulics Laboratory
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 − = α − E D w ( C C ) f t * t • Formulas – Suspended Load ω ⎛ ⎞ U d α = + = • Lin (1984) f ⎜ ⎟ c 3.25 0.55ln L κ α s ⎝ ⎠ U w * c f • Amanini and Silvio (1986) ⎡ − ⎤ 1/6 ⎛ ⎞ ⎛ ⎞ w 1 b b b = = + − − ⎢ f ⎥ ⎜ ⎟ ⎜ ⎟ b 33 o z 1 exp 1.5 α ⎝ ⎠ ⎝ ⎠ ⎢ ⎥ d d d U ⎣ ⎦ * ⎧ ⎪ A d =⎨ given – Bed Load L b ⎪ L ⎩ b given , ⎧ − + (1 r ) L r L s b s s – Total Load ⎪ = ⎨ U d L max( L L , ) α = Most Stable c t b s ω ⎪ L L t f ⎩ , t given Coastal Inlets Research Program No. 10 of 18 Slides ERDC Coastal and Hydraulics Laboratory
Bed-slope Coefficient D s • Bed change equation ⎡ ⎤ ∂ ∂ ∂ ∂ ∂ ⎡ ⎤ h h h − = αω − + − + − ' ⎢ ⎥ (1 p ) ( C C ) D (1 r d U C ) D (1 r d U C ) ⎢ ⎥ ∂ ∂ ∂ ∂ ∂ m f t * t ⎣ s s t ⎦ s s t ⎣ ⎦ t x x y y • Method first adapted by Watanabe (1985) • Smoothes bathymetry • Improves stability • Related to sediment properties and flow characteristics • Reported values – Watanabe (1985) used D s = 10 – Larson et al. (2003) and Karambas (2003) used D s = 2 • Recommended values: 1-10 Coastal Inlets Research Program No. 11 of 18 Slides ERDC Coastal and Hydraulics Laboratory
Sediment Diffusivity Coefficient • Coefficient is related to the strength of horizontal mixing in a depth-averaged sense • Directly related to eddy viscosity = K Diffusion coefficient ν = ν = K Eddy viscosity σ σ = Smidth number s s ν = − θ ν + θ ν • Eddy viscosity (1 ) m t m w ⎡ ⎤ U 1 1.15 – Falconer = c ⎢ ⎥ v gd C t 2 2 ⎣ ⎦ – Subgrid model ( ) ( ) 2 ν = + α 2 + 2 v u d l S t 0 0 * c h – Waves Λ = empiricalcoeff. 3 ⎛ ⎞ H ν = Λ θ = ⎜ = u H ⎟ u Wave orbital velocity w m m m ⎝ ⎠ d = H Wave height Coastal Inlets Research Program No. 12 of 18 Slides ERDC Coastal and Hydraulics Laboratory
Total-load Correction Factor • Accounts for the lag between the depth-averaged sediment and flow velocities • For total load transport = ∫ z s u cdz 1 U s β = + z a c U b + − t sed z ∫ r U (1 r ) u s cdz s sed s b + z a b Velocity profile Concentration profile ⎧ ⎫ w ⎛ ⎞ = − − f ⎨ ⎬ u z c z ( ) c exp ( z a ) = * c ⎜ ⎟ ε u z ( ) ln a ⎩ ⎭ κ s ⎝ ⎠ z 0 ⎛ ⎞ w d z β ≈ β = f 0.7 o ⎟ ⎜ f , For sands t ε t ⎝ ⎠ d Coastal Inlets Research Program No. 13 of 18 Slides ERDC Coastal and Hydraulics Laboratory
Hard-Bottom Problem • Concentration capacity at hard-bottom cells ( ) = ← Only allows deposition C min C , C ∗ ∗ t , hb t t • Bed change equation ∂ h [ ] ( ) ′ − = α ω − + 1 p min( C , C ) C S ∂ m t f t t * t hb t • Advection-diffusion equation ( ) ( ) ∂ ⎛ ⎞ ⎡ ⎤ ⎡ ⎤ ∂ ∂ ∂ ∂ ∂ ∂ dU C ( ) ( ) dC dU C r C r C ⎜ ⎟ + + = + y t t x t s t s t ⎢ ⎥ ⎢ ⎥ K d K d ⎜ ⎟ ∂ β ∂ ∂ ∂ ∂ ∂ ∂ x y ⎣ ⎦ ⎣ ⎦ ⎝ ⎠ t x y x x y y t [ ] + α ω − min( C , C ) C t f t t * t Coastal Inlets Research Program No. 14 of 18 Slides ERDC Coastal and Hydraulics Laboratory
Avalanching • Avalanching process is simulated as = h depth Δ = + Δ − + Δ l distancebetween cellnodes ( h h ) ( h h ) = γ φ i i p p tan φ = p i Δ Reposeangle r l r < ⎧ ⎪ 1 for h h (upslope) i p i γ = ⎨ -1 for > ⎪ h h (downslope) ⎩ p i • 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 Coastal Inlets Research Program No. 15 of 18 Slides ERDC Coastal and Hydraulics Laboratory
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