Description of TRANSPOR2004 and
Implementation in Delft3D-ONLINE
INTERIM REPORT May 2004 ReportDG Rijkswaterstaat,
Rijksinstituut voor Kust en Zee | RIKZ
Prepared for:
DG Rijkswaterstaat,
Rijksinstituut voor Kust en Zee | RIKZ
INTERIM REPORT
L.C. van Rijn and D.J.R. Walstra
Report Z3748
Description of TRANSPOR2004 and
Implementation in Delft3D-ONLINE
Contents
1
Introduction...1—1
2
UPDATED TRANSPOR2004-MODEL ...2—1
2.1 Introduction... 2—1 2.2 Updated sand transport model TRANSPOR2004 (TR2004) ... 2—1 2.2.1 Bed roughness predictor... 2—1
2.2.2 Predictor for suspended sediment size... 2—4
2.2.3 Thickness of wave-boundary layer, fluid mixing and sediment mixing layer ... 2—4
2.2.4 Wave-induced bed-shear stress ... 2—5
2.2.5 Wave-induced streaming... 2—6
2.2.6 Shields criterion for initiation of motion ... 2—6 2.2.7 Bed-load transport ... 2—7
2.2.8 Wave-related suspended transport ... 2—8
2.2.9 Near-bed sediment mixing coefficient... 2—8
2.2.10 Reference concentration and reference level... 2—8
2.2.11 Recalibration ... 2—9
2.3 Intercomparison of transport rates based on TR2004 with TR2000 and
TR1993... 2—15 2.4 Application of TR2004-model for graded sediment ... 2—17 2.4.1 Experiments... 2—17
2.4.2 Model results ... 2—20
3
Sand transport formulations in DELFT3D model...3—1
3.1 Introduction... 3—1 3.2 Model description ... 3—2 3.2.1 Hydrodynamics ... 3—2
3.2.2 Waves... 3—8
3.2.3 Sediment dynamics and bed level evolution ... 3—9
3.2.4 Bed load transport ... 3—20
3.2.5 Wave-related suspended transport ... 3—22
4
Conclusions...4—1
4.1 Updated sand transport model TRANSPOR2004 (TR2004) ... 4—1 4.2 Sand transport formulations in DELFT3D model ... 4—1
1
Introduction
RIKZ of Rijkswaterstaat and Delft Hydraulics are working together on the development/improvement, verification/validation and evaluation of morphodynamic models within the framework K2005 of Rijkswaterstaat (see Report Z2478 of Delft Hydraulics and Website http://vop.wldelft.nl) and within the SANDPIT-project (website: http://sandpit.wldelft.nl).
In 2003 much effort has been spent in the improvement of the DELFT3D-ONLINE model based on the engineering sand transport formulations of the TRANSPOR2000 model (TR2000). This work has been described in Delft Hydraulics Report Z3624 by Van Rijn and Walstra (2003). However, the engineering sand transport model TR2000 has recently been updated into the TR2004 model within the EU-SANDPIT project. The most important improvements involve the refinement of the predictors for the bed roughness and the suspended sediment size. Up to now these parameters had to be specified by the user of the models. As a consequence of the use of predictors for bed roughness and suspended sediment size, it was necessary to recalibrate the reference concentration of the suspended sediment concentration profile. Given the updated TR2004 model, an effort is necessary to further improve the DELFT3D-ONLINE model using the formulations of the updated TR2004 sand transport model (see Chapter 2). This latter work has been reported in Chapter 3.
Chapter 2 addresses the description of the updated TR2004 model and the recalibration of the reference concentration using field and laboratory data sets. Furthermore, the results of the TR2004 model have been compared with results from older versions (TR1993 and TR2000) of the sand transport model
Chapter 3 addresses the central focus point of the study: the DELFT3D-ONLINE model. The formulations (including the newly derived formulations of the TR2004) implemented in this 3D-model are described in detail. The implementation of TR2004 in Delft3D-ONLINE is part of an update of Delft3D which involves among others: the extension of the model to be run in profile mode, an update of the SWAN wave model and the synchronisation of the roughness formulations. The present report only describes the implementation of TR2004 formulations in Delft3D-ONLINE. At the end of the project this description will be updated to completely describe the modifications and improvements in the final updated version of Delft3D-ONLINE.
2
UPDATED TRANSPOR2004-MODEL
2.1
Introduction
A new version of the TRANSPOR model has been made (TR2004) based on the results of former studies, particularly those of 2003 (Van Rijn and Walstra, 2003). The basic formulations of the TR1993-model are described in Appendix A of Van Rijn 1993. Detailed information on the Multi-fraction method can be found in Van Rijn (2000).
The modifications concern the following points:
• Predictor of bed roughness;
• Predictor of suspended sediment size • Grain roughness and friction factor;
• Wave-induced orbital velocities and streaming near the bed; • Wave-induced bed-shear stress;
• Wave-induced sand transport; • Shields criterion for fine sand; • Bed load transport model • Mixing near the bed; • Reference concentration.
In 2003 new bed roughness predictors to simulate the effective roughness of various types of bed forms were developed and implemented in the latest version of the TRANSPOR-model and in the DELFT3D-TRANSPOR-model. Experiences so far showed an unrealistic behaviour of the roughness predictors of ripples and dunes. Therefore, the predictors of mega-ripple roughness and dune roughness were adjusted slightly resulting in the updated TR2004-model. The roughness predictor of small-scale ripples in current, waves and combined current-wave conditions was not changed. In line with this the predictor of the suspended sediment size was slightly modified.
2.2
Updated sand transport model TRANSPOR2004
(TR2004)
2.2.1
Bed roughness predictor
The TR2004 model includes a bed-roughness predictor for the current-related and wave-related bed roughness parameters. In TR1993 and TR2000 both parameters have to be specified as user-related input data.
Physical current-related bed roughness
It is assumed that the physical bed roughness of movable small-scale ripples in natural conditions is approximately equal to the ripple height: k ≅∆. Furthermore, it is assumed
that the small-scale ripples are fully developed with a height equal to∆r=150d50forψ≤50 in
the lower wave-current regime and that the ripples disappear with ∆r=0 forψ≥250 in the
upper wave-current regime (sheet flow conditions).
The expressions implemented for small-scale ripples are given by:
(
)
, , 50 , , 50 , , 50 150 0 50 ( , ) 182.5 0.65 50 250 ( , ) 20 250 ( ) s c r s c r s c rk d and lower wave current regime SWR ripples
k d and upper wave current regime sheet flow
k d and linear approach in transitional regime
ψ ψ ψ ψ = ≤ ≤ − = − < < − = ≥ (2.2.1)
with:ψ= mobility parameter=Uwc2/((s-1)gd50)), (Uwc)2= (Uδ)2+ vR2+2(Uw) (vR)|cos ϕ|
Uδ= peak orbital velocity near bed= πHs/(Trsinh(2kh)), vR = depth-averaged current
velocity, ϕ= angle between wave and current motion, Hs= significant wave height,
k=2π/L, L= wave length derived from (L/Tp± vR)2=gL tanh(2πh/L)/(2π),
Tr= Tp/((1-( vRTp/L)cosϕ)= relative wave period, Tp= peak wave period, h= water
depth.
Equation (2.2.1) is assumed to be valid for relatively fine sand with d50in the range of 0.1 to
0.5 mm. An estimate of the bed roughness for coarse particles (d50>0.5 mm) can be obtained
by using Equation (2.2.1) for d50=0.5 mm. Thus, d50=0.5 mm for d50≥0.5 mm resulting in a
maximum bed roughness height of 0.075 m (upper limit). The lower limit will be ks,c=20d50= 0.002 m for sand with d50≤0.1 mm.
When mega-ripples and/or dunes are present on the seabed (if h=water depth>1 m and uc=depth-averaged velocity>0.3 m/s), the physical form roughness (ks,c,mr) of the
mega-ripples and dunes should also be taken into account (grain roughness is negligibly small; only form roughness). Compared with the bed roughness predictor implemented earlier (Van Rijn and Walstra, 2003), the expressions of the current-related bed roughness due to mega-ripples and dunes have been refined into:
Mega ripples:
(
)
, , , , , , , , , 0.01 0 50 1 0.3 0.011 0.00002 50 550 1 0.3 0 550 1 0.3 0.2 s c mr r s c mr r s c mr r s c mr MAXk h and and h and v
k h and and h and v
k and and h and v
k
ψ
ψ
ψ
ψ
= ≤ ≤ > > = − < < > > = ≥ > > = (2.2.2)Dunes (only applicable in rivers, .i.e. no waves):
(
)
, , , , , , , , , 0.0004 0 100 1 0.3 0.048 0.0008 100 600 1 0.3 0 600 1 0.3 1.0 s c d r s c d r s c d r s c d MAXk h and and h and v
k h and and h and v
k and and h and v
k
ψ
ψ
ψ
ψ
ψ
= ≤ ≤ > > = − < < > > = ≥ > > = (2.2.3)Equation (2.2.2) yields: ks,c,mr=0.01h forψ=50 and ks,c,mr=0 forψ=550. Hence, the maximum
value is ks,c,mr=0.01h. The absolute maximum value of the mega-ripple roughness is assumed
to be 0.2 m
Equation (2.2.3) yields: ks,c,d=0 for ψ=0, ks,c,d=0.04h for ψ=100 and ks,c,d=0 for ψ=600.
Hence, the maximum value is ks,c,d=0.04 h. The absolute maximum value of the dune
roughness is assumed to be 1.0 m.
It is remarked that Equations (2.2.2) and (2.2.3) are slightly different from those presented in 2003 (see Equations 3.1.10 and 3.1.11 of Van Rijn and Walstra, 2003), because these latter expressions showed a less realistic behaviour at larger bed-shear stresses. When mega-ripples and/or dunes are present, these values are added to the physical current-related bed roughness of the small-scale ripples by quadratic summation, as follows:
(
2 2 2)
0.5, , , , , , , s c s c r s c mr s c d
k = k +k +k (2.2.4)
The current-related friction coefficient (based on the Darcy-Weisbach approach: f=8g/C2) can be computed as:
2 2 , ,
8
0.24
12
12
18 log
log
c s c s cg
f
h
h
k
k
=
=
(2.2.5)Physical wave-related roughness of movable bed ks,w
As regards the physical wave-related bed roughness, only bed forms (ripples) with a length scale of the order of the wave orbital diameter near the bed are relevant. Bed forms (mega-ripples, ridges, sand waves) with a length scale much larger than the orbital diameter do not contribute to the wave-related roughness.
The physical wave-related roughness of small-scale ripples is given by:
(
)
, , 50
, , 50
, , 50
150 50 (lower wave-current regime, SWR ripples) 20 250 (upper wave-current regime, sheet flow)
182.5 0.65 50 250 (linear approach in transitional regime) s w r s w r s w r k d for k d for k d for
ψ
ψ
ψ
ψ
= ≤ = ≥ = − < < (2.2.6)with:ψ= mobility parameter=Uwc2/((s-1)gd50)), (Uwc)2= (Uδ)2+ vR2+2(Uw) (vR)|cos ϕ|
Uδ= peak orbital velocity near bed= πHs/(Trsinh(2kh)), vR = depth-averaged current
velocity, ϕ= angle between wave and current motion, Hs= significant wave height,
k=2π/L, L= wave length derived from (L/Tp± vR)2=gL tanh(2πh/L)/(2π),
Tr= Tp/((1-( vRTp/L)cosϕ)= relative wave period, Tp= peak wave period, h= water
depth.
Equation (2.2.6) includes grain roughness and is assumed to be valid for relatively fine sand with d50in the range of 0.1 to 0.5 mm.
0.19 , ,
exp 5.2
6
w s w rA
f
k
δ −
=
−
(2.2.7)Apparent bed roughness for flow over a movable bed
It is proposed to use the existing expression:
, ,
exp
10
a a s c R s c MAXk
U
k
and
k
v
k
δγ
=
=
(2.2.8)with: Uδ=peak orbital velocity near the bed (see Equation (3.2.15)), vR= depth-averaged
current velocity,γ=0.8+ϕ-0.3ϕ2andϕ= angle between wave direction and current direction (in radians between 0 and π; 0.5π= 90o, π= 180o). Characteristic γ-values are γ=0.8 for 0,
γ=1 for π= 180o
and γ=1.63 for 0.5π= 90o. The γ-value is maximum γ=1.63 for ϕ= 0.5π= 90o.
Equation (2.2.8) should only be applied to the bed roughness of the small-scale ripples and mega-ripples.
The current-related apparent friction coefficient (based on the Darcy-Weisbach approach:
f=8g/C) can be computed as:
, 2 2
8
0.24
12
12
18 log
log
c a a ag
f
h
h
k
k
=
=
(2.2.9)2.2.2
Predictor for suspended sediment size
Compared with the suspended sediment size predictor implemented earlier (Van Rijn and Walstra, 2003), this latter predictor has been refined into:
(
)
50 50, 50 10 50 min 0.5 1 0.0006 1 550 250 250 s s d d d d for d d d forψ
ψ
ψ
= + − − < = ≥ (2.2.10)2.2.3 Thickness of wave-boundary layer, fluid mixing and sediment mixing
layer
In TR2004 the wave boundary layer thickness according to (Davies and Villaret, 1999) is used:
0.25 , , , 0.36 w w s w r A A k δ δ
δ
− = (2.2.11)Aδ= peak orbital excursion at edge of wave boundary layer
Which replaces the wave boundary layer thickness formulation based on that of Jonsson and Carlsen (1976) used in TR1993 and TR2000.
The thickness of the effective fluid mixing layer in TR2004 is modelled as (in metres):
, ,
2
0.05
0.2
m w
with
m MINand
m MAXδ
=
δ
δ
=
δ
=
(2.2.12)The thickness of the effective sediment mixing layer in TR2004 is modelled as:
{
}
min 0.5, max 0.05, 2 s br wδ
= γ δ
(2.2.13) with: 0.51
s0.4
1
s0.4
br brH
H
and
for
h
h
γ
= +
−
γ
=
≤
(2.2.14)2.2.4
Wave-induced bed-shear stress
The time-averaged bed-shear stress is computed as:
( )
2 , , 1 4 b w wfw Uδ rτ
=ρ
(2.2.15) with: ρ = fluid densityfw = wave-related friction factor, Eq. (2.2.7)
In TR2004 the peak orbital velocity is refined into:
(
)
(
)
(
)
13 3 3
,r 0.5 ,for 0.5 ,back
Uδ = Uδ + Uδ (2.2.16)
Uδ,r = representative peak orbital velocity near the bed
Uδ,for = peak orbital velocity in forward direction (method of Isobe and Horikawa)
Uδ,back= peak orbital velocity in backward direction (method of Isobe and Horikawa)
2.2.5
Wave-induced streaming
Based on the results of Van Rijn and Walstra (2003), the wave-induced streaming near the bed can be represented as:
2 , , , , , , 2 , , , , , 2 , , , , ,
1 0.875 log
1
100
0.75
100
1
m m s w r s w r m w m s w r m w m s w rU
A
A
u
for
k
c
k
U
A
u
for
c
k
U
A
u
for
c
k
δ δ δ δ δ δ δ δ δ δ
= − +
<
<
=
≥
= −
≤
(2.2.17) with:uδ,m= streaming velocity at edge of wave boundary layer,
Uδ,m=0.5(Uδ,for+Uδ,back)= peak orbital velocity at edge of wave boundary layer,
c= wave propagation velocity,
Aδ= peak orbital excursion at edge of wave boundary layer=TpUδ/(2π),
Tp= peak wave period,
ks,w,r= wave-related bed roughness
In TR2004 the streaming velocity vector is added to the current-related velocity vector at level z=δ.
2.2.6
Shields criterion for initiation of motion
In TR2004 the critical bed-shear stress for initiation of motion is modelled as:
(
)
3,
1
, ,b cr pmud b cr o
τ
= +
τ
(2.2.18)τb,cr,o= critical bed-shear stress for pure sand (no mud)
pmud= fraction (0 to 0.3) of mud (Van Ledden, 2003)
In TR1993 and TR2000 the dimensionless Shields criterion for initiation of motion of very fine sediments is represented as:
* *
0.24
4
crfor D
D
Θ =
≤
(2.2.19)with Θcr=τb,cr,o/((s-1)gd50and D*=d50[(s-1)g/ν2]1/3, s=ρs/ρ= relative density, ν=kinematic
viscosity coefficient.
A better representation based on experimental data is given by (See Van Rijn, 1993):
0.5 * *
0.115
4
crD
for D
−Θ =
≤
(2.2.20)which is implemented in TR2004.
2.2.7
Bed-load transport
Bed load transport model
The net bed-load transport rate in conditions with uniform bed material is obtained by time-averaging (over the wave period T) of the instantaneous transport rate using the bed-load transport model (quasi-steady approach), as follows:
,
1
b b tq
q dt
T
=
∫
(2.2.21)with qb,t= F(instantaneous hydrodynamic and sediment transport parameters).
The formula applied, reads as:
0.5 ' ' , , , , , 0.3 50 * ,
0.5
b cw t b cw t b cr b s b crq
ρ
d D
τ
τ
τ
ρ
τ
−
−
=
(2.2.22) in which: τ/b,cw,t = instantaneous grain-related bed-shear stress due to both current and wave motion =
0.5ρ f/cw(Uδ,cw,t)2,
Uδ,cw,t= instantaneous velocity due to current and wave motion at edge of wave boundary
layer,
f/c = current-related grain friction coefficient =0.24(log(12h/ks,grain))-2,
f/w = wave-related grain friction coefficient=Exp[-6+5.2(Aδ,w/ks,grain)-0.19],
α = coefficient related to relative strength of wave and current motion:
ˆ
RU
v
δα =
,ˆ
U
δ = the peak orbital velocity, vRis the depth averaged current,βf = coefficient related to vertical structure of velocity profile,
Aδ = peak orbital excursion,
τb,cr = critical bed-shear stress according to Shields,
ρs = sediment density,
ρ = fluid density, d50 = particle size,
D* = dimensionless particle size.
The two most influential parameters of Eq. (2.2.22) are:
f
cw' and ks,grain.Various field data sets from the literature and new data sets (laboratory and field) collected within the SANDPIT project have been used to verify/improve these parameters of the bed-load transport formulations (see Van Rijn and Walstra, 2003).
(
)
' ' '1
cw f c wf
=
αβ
f
+ −
α
f
(2.2.23) , 901
3
s grain grain grain
k
=
α
d
with
α
between
and
(2.2.24)Based on the findings of Van Rijn and Walstra (2003), the following expressions have been implemented in TR2004:
(
)
' 0.5 ' 0.5 '1
cw f c wf
=
α β
f
+ −
α
f
(2.2.25) , 90 s graink
=
d
(2.2.26)2.2.8
Wave-related suspended transport
The wave-related suspended transport component is modelled as follows:
4 4 , , , 3 3 , , , for back s w m for back
U
U
q
u
cdz
U
U
δ δ δ δ δγ
−
=
+
+
∫
(2.2.27)with: Uδ,for= near-bed peak orbital velocity in onshore direction (in wave direction) and
Uδ,back= near-bed peak orbital velocity in offshore direction (against wave direction),
uδ,m= wave-induced streaming velocity near the bed, c= time-averaged concentration
and γ= phase lag function.
In TR2004 (based on the findings of Van Rijn and Walstra, 2003), the phase lag function is:γ= 0.1 in stead of γ= 0.2 as was used in TR2000.
2.2.9
Near-bed sediment mixing coefficient
The mixing coefficient near the bed is modelled as:
,
0.018
, w bed w sU
δrε
=
β δ
(2.2.28)with Uδ,raccording to Equation (2.2.16) and δsaccording to Equation (2.2.13).
2.2.10
Reference concentration and reference level
The reference level in TR2004 is described by:
(
, , , ,)
max 0.5
s c r, 0.5
s w r, 0.01
a
=
k
k
(2.2.29)with ks,c,r= current-related bed roughness height due to small-scale ripples and ks,w,r=
wave-related bed roughness height due to small-scale ripples.
Similarly as in TR1993 and TR2000, the reference concentration (single fraction approach) in TR2004 is described by:
( )
( )
1.5 50 s 0.3 ,0.015
a0.05
a a MAX sd
T
c
with c
a D
ρ
ρ
∗=
=
(2.2.30)2.2.11
Recalibration
The T-parameter of Equation (2.2.30) involves the computation of the wave-related bed-shear stress and a wave-related efficiency factor µw. This latter parameter has been
recalibrated using a dataset of 53 cases (see Table 3.2.1) from combined quasi-steady and oscillatory flow cases, resulting in:
( ) * ( ) , * ( ) , * 0.7 0.35 2 0.14 5 w w MAX w MIN D for D for D
µ
µ
µ
= = < = > ! ! ! (2.2.31)with D*= particle size parameter,
The measured concentration in the lowest measuring point above the bed (in the range of 0.015 m for laboratory cases to 0.5 m for field cases) has been used as measured reference concentration. To better understand the variability within the available dataset, some concentration profiles measured under similar conditions are presented in Figures 2.2.1A and 2.2.1B, showing differences in the range of a factor 5 to 10.
Figure 2.2.2 shows measured and computed reference concentrations for 53 datasets. Variation ranges of a factor of 2 are also indicated. About 75% of the computed reference concentrations are within a factor of 2 of the measured concentrations.
Figure 2.2.3 shows measured and computed suspended sand transport rates between the lowest and highest measurement points for 34 datasets. Measured transport rates were not available for the Delta flume cases (wave-alone cases) and the Noordwijk Spring 2003 field cases. Variation ranges of a factor of 2 are also indicated. About 65% of the computed suspended transport rates (34 cases) are within a factor of 2 of the measured values.
Figures 2.2.4 to 2.2.21 show various computed and measured concentration profiles based on the recalibrated TR2004 model.
Site Sediment size d50 (mm) Water depth range (m) Wave height range (m) Flow velocity range (m/s) Reference Boscombe 1977-1978 0.25 4.8-5.3 0.45-1.05 0.2-0.4 Whitehouse et al., 1997 Maplin sands 1973-1975 0.14 2.8-3.2 0.4-0.9 0.07-0.34 Whitehouse et al., 1996 Egmond 1989-1990 0.3-0.35 1-1.6 0.2-0.9 0.06-0.55 Kroon, 1994 Wolf, 1997 Egmond 1998 0.25 2.5-3.1 0.45-1.1 0.1-0.3 Grasmeijer, 2002 Noordwijk spring 2003
0.22 13-15 2.2-2.8 0.1-0.5 Grasmeijer and Tonnon,
2003
Duck 1991 0.15 13 3.75 0.4-0.6 Madsen et al., 1993
Deltaflume 1987
0.21 1.1-2.1 0.3-1.1 0 SEDMOC sand transport
database, 2001 Deltaflume
1997
0.16-0.33 4.5 1-1.5 0 SEDMOC sand transport
database, 2001 DH Vinje lab.
basin
0.1 0.4 0.1-0.14 0.13-0.32 SEDMOC sand transport
database, 2001
TUD flume 0.2 0.5 0.12-0.15 0.1-0.45 SEDMOC sand transport
database, 2001 Table 2.2.1 Summary of field and laboratory datasets used for calibration of reference
concentration of TR2004 sand transport model
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.01 0.1 1 10 Concentration (kg/m3) H e ight a bove b e d (m EGMOND BEACH, h=2.1 m, Hs=1.1 m, Tp=7.2 s, V=0.3 m/s, d50=0.25 mm DELTAFLUME, h=2.0 m, Hs=1.1 m, Tp=5.8 s, V= 0 m/s, d50=0.21 mm
Figure 2.2.1A Comparison of concentration profiles measured under similar conditions in water depth of about 2 m (d50in range of 0.2 to 0.25 mm)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.001 0.01 0.1 1 Concentration (kg/m3) H e ight a b ov e b e d (m Maplin Sands M24-03; d50=0.14 mm, h=3.2 m, Hs=0.73 m, v=0.083 m/s Maplin Sands M22-01; d50=0.14 mm, h=3.2 m, Hs=0.68 m, v=0.1 m/s
Figure 2.2.1B Comparison of concentration profiles measured under similar conditions in water depth of about 3 m (d50of about 0.14 mm)
0.01 0.1 1 10 100 0.01 0.1 1 10 100 Ca,computed (kg/m3) C a ,m eas u red (k g /m 3
Line of perfect agreement Variation range of factor 2 Egmond 1998, d50=0.25 mm
Boscombe Pier 1977-1978, d50=0.25 mm Deltaflume 1997, d50=0.16-0.33 mm Deltaflume 1987, d50=0.21 mm Egmond 1989-1990, d50=0.3-0.35 mm Vinje Lab. basin, d50=0.1 mm TUD Lab. basin, d50=0.2 mm Maplin Sands 1973-1975, d50=0.14 mm Noordwijk 2003, d50=0.22 mm Duck 1991, d50=0.15 mm
0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 1 qs,computed (kg/s/m) qs, m easur e d (kg/ s /m )
Line of perfect agreement Egmond 89-90
Vinje Basin TUD flume
variation range of factor 2 Egmond 98
Boscombe 77-78 Maplin 73-75
Figure 2.2.3 Measured and computed suspended sand transport rates
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.00001 0.0001 0.001 0.01 0.1 1 10 Concentration (kg/m3) R e la ti v e he ight a bov e b e d (z /h Hs=1 m, v=0.3 m/s Computed Group 4
Figure 2.2.4 Boscombe Pier 1977-1978
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.00001 0.0001 0.001 0.01 0.1 1 Concentration (kg/m3) R e la ti v e he ig h t a bov e b e d (z /h Hs=0.5 m, v=0.2 m/s Computed group 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0001 0.001 0.01 0.1 1 10 Concentration (kg/m3) R e l. h e ig h t a b o v e b e d (z /h Measured 3C Measured 3C Computed 3C Figure 2.2.6 Egmond 1989-1990 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.001 0.01 0.1 1 10 Concentration (kg/m3) R e l. he ig ht a bov e b e d (z /h Measured 4A Measured 4A Measured 4A Computed 4A Figure 2.2.7 Egmond 1989-1990 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0001 0.001 0.01 0.1 1 10 Concentration (kg/m3) R e l. he ight a bov e b ed (z /h Measured Class4 Computed Figure 2.2.8 Egmond 1998 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0001 0.001 0.01 0.1 1 10 Concentration (kg/m3) R e l. h e ight above b ed (z /h Measured Class6 Computed Figure 2.2.9 Egmond 1998 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0001 0.001 0.01 0.1 1 Concentration (kg/m3) R e l. h e ig ht a b ov e b e d (z /h Measured 2206-2207Computed
Figure 2.2.10 Noordwijk Spring 2003
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0001 0.001 0.01 0.1 1 10 Concentration (kg/m3) R e l. he ig h t ab ov e b ed (z /h Measured 2209 Computed
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 1 10 Concentration (kg/m3) R e l. h e ight a bov e b e d (z /h Computed 2H Computed 2I Measured 2H Measured 2I Figure 2.2.12 Deltaflume 1987 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.001 0.01 0.1 1 10 Concentration (kg/m3) R e l. he ight a bov e b e d (z /h measured Hs/h=0.19 (2C) measured Hs/h=0.55 (2F) Computed 2C Computed 2F Figure 2.2.13 Deltaflume 1987 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.01 0.1 1 10 Concentration (kg/m3) R e l. he ight a bov e b e d (z /h measured Hs= 1 m (Hs/h=0.22), case 1A measured Hs= 1.25 m (Hs/h=0.27), case 1B Computed 1A Computed 1B Figure 2.2.14 Deltaflume 1997 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.01 0.1 1 10 Concentration (kg/m3) R e l. he ig ht a bov e b e d (z /h measured Hs= 1 m (Hs/h=0.22; 1C) measured Hs= 1.25 m (Hs/h=0.27; 1D) measured Hs= 1.5 m (Hs/h=0.33; 1E) Computed 1C Computed 1D Computed 1E Figure 2.2.15 Deltaflume 1997 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.01 0.1 1 10 Concentration (kg/m3) R e l. he ight a bov e b e d (z /h Measured Hs=0.105 m, v=0.245 m/s Computed
Figure 2.2.16 DH Vinje Laboratory basin
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.01 0.1 1 10 Concentration (kg/m3) R e l. he ight a bov e b e d (z /h Measured Hs=0.137 m, v=0.317 m/s Computed
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.01 0.1 1 10 Concentration (kg/m3) R e l. he ig ht a b o v e b e d (z /h Measured Hs=0.133 m, v=0.13 m/s Computed
Figure 2.2.18 DH Vinje Laboratory Basin
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.00001 0.0001 0.001 0.01 0.1 1 Concentration (kg/m3) R e l. h e ig ht a b ov e b e d (z /h Measured Hs=0.123 m, v=0.22 m/s Computed
Figure 2.2.19 TUD Flume
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0001 0.001 0.01 0.1 1 10 Concentration (kg/m3) R e l. he ight a bov e b e d (z /h Measured Hs=0.119 m/s, v=0.44 m/s Computed
Figure 2.2.20 TUD Flume
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.01 0.1 1 10 Concentration (kg/m3) Rel a ti ve h e ig h t ab o v e b ed (z /h
Measured Duck Shelf 1991 (h=13 m) Computed
Figure 2.2.21 DUCK 1991
2.3
Intercomparison of transport rates based on TR2004
with TR2000 and TR1993
Figures 2.3.1 and 2.3.2 show intercomparison-results of the TR2004-model with TR2000-and TR1993-models based on reference case computations for a water depth of h=5 m TR2000-and a median particle size of d50= 0.25 mm (see Appendix A of Van Rijn, 1993).
The significant wave height varies between 0 and 3 m; the depth-averaged current velocity varies between 0.1 and 2 m/s. The wave-current angle is 90 degrees. Other parameters are: d90= 0.5 mm, water temperature= 15oCelsius and salinity= 30 promille.
The TR2004-model results (total sand transport rates) are based on predicted bed roughness and suspended sediment size values, whereas the TR-2000 and TR1993-model results are based on prescribed values in the range of ks=0.02 to 0.1 m and ds= 0.17 to 0.25 mm (see
Van Rijn, 1993). Measured transport rates (mainly suspended sand transport; see Van Rijn, 2000) for the current-alone cases (no waves) are also shown in Figures 2.3.1 and 2.3.2.
Figure 2.3.1 shows that the TR2004 results are considerably smaller than those of the TR2000-model for wave heights of Hs=0.5 and 1 m. This effect is caused by a less
pronounced effect of the bed roughness on the sand transport rate in the TR2004-model. The results of the TR2004 and TR2000 models are in reasonably good agreement for wave heights of Hs= 2 and 3 m. 0.001 0.01 0.1 1 10 100 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
Depth-averaged current velocity (m/s)
To ta l c urre nt -r e lat e d s a n d tra nsport (k g /s /m TRANSPOR 2000 TRANSPOR 2004
Eastern and Western Scheldt data (Netherlands) Nile river data (Egypt)
Mississippi river data (USA) Hs=0 Hs=0.5 h = 5 m d50=0.25 mm d90=0.50 mm Hs=1 m Hs=2 m Hs=3 m
Figure 2.3.1 Intercomparison of TR2004 and TR2000 model results for constant water depth of 5 m and particle size of 0.25 mm
The TR2004-model yields smaller transport rates for current-alone cases (no waves), particularly for current-velocities larger than 1.4 m/s. This latter effect is also caused by the modelling of the bed roughness; the TR2004-model yields smaller values in the upper regime. The TR2004 results are in good agreement with the measured data points (current-alone cases), whereas the TR2004-model seems to over predict the measured transport rates (bed-load transport is assumed to negligibly small).
Figure 2.3.2 shows that the TR2004 results are quite close to the TR1993 results for wave heights of Hs= 1, 2 and 3 m. The TR2004 model yields smaller transport rates for the wave
heights of Hs=0.5 and Hs=0 m (current-alone case), particularly for current velocities larger
than 1.4 m/s. This latter effect is caused by smaller bed roughness values in the upper regime using the TR2004-model.. The TR2004 results are in good agreement with the measured data points (current-alone cases), whereas the TR1993-model seems to over predict the measured transport rates (bed-load transport is assumed to negligibly small).
0.001 0.01 0.1 1 10 100 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
Depth-averaged current velocity (m/s)
T o ta l c ur re nt -r e lat e d s a n d tra ns po rt (k g/ s /m TRANSPOR 1993 TRANSPOR 2004
Eastern and Western Scheldt data (Netherlands) Nile river data (Egypt)
Mississippi river data (USA) Hs=0 Hs=0.5 h = 5 m d50=0.25 mm d90=0.50 mm Hs=1 m Hs=2 m Hs=3 m
Figure 2.3.2 Intercomparison of TR2004 and TR1993 model results for constant water depth of 5 m and particle size of 0.25 mm
2.4
Application of TR2004-model for graded sediment
2.4.1
Experiments
Experiments over a horizontal sand bed have been carried out in a small-scale wave-current flume of the Fluids Mechanics Laboratory of the Delft University of Technology (Jacobs and Dekker, 2000 and Sistermans, 2000). Two types of sand have been used in the experimental program: uniform sand with d50of about 0.16 mm and graded sand with d50 of
about 0.25 mm. The water depth was about 0.5 m in all tests. The hydrodynamic conditions are: irregular waves superimposed on a following current. The significant wave heights are in the range of 0.12 to 0.2 m. The depth-averaged current velocities are in the range of 0.1 to 0.3 m/s (following current). Time-averaged suspended sand concentrations and suspended transport rates have been measured. Instantaneous velocities and sand concentrations at various elevations above the bed have been measured by use of an acoustic instrument. Instantaneous fluid velocities have also been measured by use of an electro-magnetic velocity meter. Time-averaged sand concentration profiles have been obtained by using a pump sampling instrument consisting of 10 intake tubes (internal opening of 3 mm; sampling time of about 20 min).
M218g graded M220g graded M418g graded h=0.5 m Hs=0.155 m Tp=2.7 s v=0.2 m/s d10=0.09 mm d50=0.26 mm d90=0.42 mm ds=0.1-0.09 mm ∆r=0.022 m λr=0.15 m Te=24oC Fractions (mm), (%) 0.075 10 0.105 10 0.130 10 0.175 10 0.230 10 0.285 10 0.325 10 0.365 10 0.400 10 0.450 10 h=0.525 m Hs=0.2 m Tp=2.7 s v=0.17 m/s d10=0.09 mm d50=0.26 mm d90=0.42 mm ds=0.11-0.1 mm ∆r=0.022 m λr=0.18 m Te=24oC Fractions (mm), (%) 0.075 10 0.105 10 0.130 10 0.175 10 0.230 10 0.285 10 0.325 10 0.365 10 0.400 10 0.450 10 h=0.52 m Hs=0.15 m Tp=2.6 s v=0.29 m/s d10=0.09 mm d50=0.27 mm d90=0.41 mm ds=0.12-0.1 mm ∆r=0.022 m λr=0.2 m Te=24oC Fractions (mm), (%) 0.075 10 0.105 10 0.130 10 0.175 10 0.230 10 0.285 10 0.325 10 0.365 10 0.400 10 0.450 10 z (m) c (kg/m3) z (m) c (kg/m3) z (m) c (kg/m3) 0.032 0.042 0.052 0.067 0.092 0.122 0.157 0.192 0.232 0.282 1.08 0.86 0.71 0.57 0.42 0.31 0.19 0.15 0.12 0.09 0.02 0.03 0.04 0.055 0.08 0.11 0.145 0.18 0.22 0.27 6.6 1.86 1.41 1.11 0.72 0.5 0.33 0.24 0.17 0.14 0.031 0.041 0.051 0.066 0.091 0.121 0.156 0.191 0.231 0.281 1.86 1.53 1.2 0.97 0.75 0.54 0.37 0.21 0.17 0.13
Table 2.4.1 Basic data of experiments with graded sand bed in small-scale wave-current flume (Tests M218g, M220g, M418g; Jacobs and Dekker, 2000)
M015u uniform h=0.545 m Hs=0.155 m Tp=2.5 s v=0 m/s d10=0.12 d50=0.155 d90=0.23 ds=0.13-0.1 (mm) ∆r=0.008 m λr=0.1 m Te=24oC z (m) c (kg/m3) z (m) c (kg/m3) z (m) c (kg/m3) 0.016 0.026 0.036 0.051 0.076 0.106 0.141 1.15 0.77 0.52 0.26 0.085 0.022 0.0037 0.016 0.026 0.036 0.051 0.076 0.106 0.141 1.42 0.89 0.55 0.28 0.079 0.0184 0.0037 0.011 0.021 0.031 0.046 0.071 0.101 0.136 1.37 0.87 0.60 0.32 0.11 0.028 0.0037 M015g graded h=0.5 m Hs=0.15 m Tp=2.5 s v=0 m/s d10=0.08 d50=0.23 d90=0.42 ds=0.08 (mm) ∆r=0.012 m λr=0.09 m Te=24oC Fractions (mm), (%) 0.07 10 0.10 10 0.12 10 0.15 10 0.20 10 0.25 10 0.30 10 0.34 10 0.40 10 0.45 10 z (m) c (kg/m3) z (m) c (kg/m3) z (m) c (kg/m3) 0.008 0.018 0.028 0.043 0.068 0.098 0.133 0.168 0.208 2 1.35 0.89 0.7 0.38 0.14 0.023 0.009 0.0018 0.008 0.018 0.028 0.043 0.068 0.098 0.133 0.168 0.208 2.3 1.45 1.05 0.79 0.43 0.146 0.0251 0.0072 0.0018 0.005 0.015 0.025 0.04 0.065 0.095 0.13 0.165 0.203 2.47 1.5 1.15 0.93 0.53 0.185 0.031 0.009 0.0018 M018g graded h=0.5 m Hs=0.18 m Tp=2.7 s v=0 m/s d10=0.08 d50=0.24 d90=0.42 ds=0.12-0.09 (mm) ∆r=0.012 m λr=0.09 m Te=24oC Fractions (mm), (%) 0.07 10 0.10 10 0.12 10 0.15 10 0.20 10 0.25 10 0.30 10 0.34 10 0.40 10 0.45 10 z (m) c (kg/m3) z (m) c (kg/m3) z (m) c (kg/m3) 0.021 0.031 0.041 0.056 0.081 0.111 0.146 0.181 0.221 0.271 2.6 1.51 1.03 0.64 0.3 0.11 0.022 0.014 0.0036 0.0018 0.024 0.034 0.044 0.059 0.084 0.114 0.149 0.184 0.224 0.274 4.35 1.41 1.09 0.71 0.36 0.16 0.036 0.0144 0.0036 0.0018 0.025 0.035 0.045 0.06 0.085 0.115 0.15 0.185 0.225 0.275 1.72 1.28 0.99 0.66 0.39 0.18 0.049 0.02 0.0072 0.0018
Table 2.4.2 Basic data of experiments with uniform sand bed and graded sand bed in small-scale wave-current flume (Tests M015u, M015g, M018g; Jacobs and Dekker, 2000)
The suspended sand sizes based on analysis in a settling tube, are also given in Tables 2.4.1 and 2.4.2. The measured suspended sand size is about ds= 0.7 to 0.9 d50,bedfor the uniform bed
materials and about ds= 0.35 to 0.45 d50,bedfor the graded bed material. Ripple dimensions have
been determined by use of a bed profile follower.
Figure 2.4.1 shows measured sand concentration profiles (based on the pumped concentrations) for waves with Hs= 0.15 m and 0.18 m over uniform and graded bed
material. The experimental conditions are given in each plot. As can be observed by comparing the results of Figure 2.4.1Top and Middle (Hs=0.15 m for both cases), the
near-bed concentrations are significantly larger (factor 2) for the graded sediment near-bed (Middle) and the sand concentrations higher up in the water column are somewhat larger for the graded sediment bed, which is caused by the winnowing of the fine sediments from the bed. Figure 2.4.2 shows measured concentration profiles for combined wave and current conditions (3 tests). As can be observed, the concentrations are more uniformly distributed over the depth due to the mixing capacity of the current.
2.4.2
Model results
Both the Single-fraction method and the Multi-fraction method have been applied to compute the sand concentration profiles for the 6 experimental cases. The Multi-fraction method has not been used for the uniform sediment case M015U.
The results are shown in Figures 2.4.1 and 2.4.2 for 6 cases. The results are:
Waves alone (Figure 2.4.1)
• the computed sand concentrations based on the SF-method are considerably too small
compared with the measured concentrations in the near-bed region for the uniform sand (Figure 2.4.1Top) due to under-prediction of the reference concentration; the computed concentrations in the upper layers are slightly too large;
• the computed sand concentrations based on the MF-method show reasonably good
agreement with the measured concentrations in the near-bed region for the graded sand bed (Figure 2.4.1Middle and Bottom), but the computed concentrations higher up in the water column are much too large compared with the measured values; the winnowing effect of the fine fractions is overestimated by the model; the wave-related mixing coefficient is too large for z>0.1 m.
• the computed reference concentration based on the MF-method is larger than that based on
the SF-method, which is in agreement with the physics involved (larger near-bed concentrations for graded sediment than for uniform sediment).
Combined current and waves (Figure 2.4.2)
• the computed sand concentrations based on the MF-method show reasonably good
agreement with the measured concentrations for the graded sand; the vertical distribution is predicted rather good, but the reference concentration is somewhat under predicted;
• the computed sand concentrations based on the SF-method are considerably too small if the
suspended sediment size is based on the standard prediction method (ds=0.13 mm≅
0.5d50,bed); the computed sand concentrations show reasonably good agreement with the
measured values, if the suspended sediment size is taken (calibrated) as ds= 0.4d50,bed≅0.1
Figure 2.4.1 Measured and computed sand concentration profiles for waves (no current) over uniform sand bed (Top) and graded sand bed (Middle and Bottom); 3 tests M015uniform, M015graded and M018graded
0 0.1 0.2 0.3 0.4 0.5 0.6 0.001 0.01 0.1 1 10 Concentration (kg/m 3) H e ight a bov e b e d (m )
m easured uniform sand TR2004 MF M015Uniform h= 0.54 m Hs=0.155 m Tp= 2.5 s v=0 m /s d50=0.155 m m d90=0.23 m m r=0.008 m 0 0.1 0.2 0.3 0.4 0.5 0.6 0.001 0.01 0.1 1 10 Concentration (kg/m 3) H e ig ht a bov e b e d (m )
m eas ure d graded sand TR2004 MF TR2004 SF (standard; ds=0.115 m m ) M015Graded h= 0.5 m Hs=0.15 m Tp= 2.5 s v=0 m /s d50=0.23 m m d90=0.42 m m r=0.012 m 0 0.1 0.2 0.3 0.4 0.5 0.6 0.001 0.01 0.1 1 10 Concentration (kg/m 3) H e ig h t ab o v e b ed (m )
m easured graded s and TR2004 MF TR2004 SF (standard; ds=0.12 m m ) M018Graded h= 0.5 m Hs=0.18 m Tp= 2.7 s v=0 m /s d50=0.24 m m d90=0.42 m m r=0.012 m
Figure 2.4.2 Measured and computed sand concentration profiles for combined current and waves over graded sand bed; 3 tests M218graded, M220graded and M418graded 0 0.1 0.2 0.3 0.4 0.5 0.6 0.001 0.01 0.1 1 10 Concentration (kg/m 3) H e ig h t ab o v e b ed (m )
m easured graded s and
TR2004 SF (ds=0.13 m m ; s tandard) TR2004 MF (10 fractions) TR2004 SF (ds=0.1 m m ; calibrated) M218Graded h= 0.5 m Hs=0.155 m Tp= 2.7 s v=0.2 m /s d50=0.26 m m d90=0.42 m m 0 022 0 0.1 0.2 0.3 0.4 0.5 0.6 0.001 0.01 0.1 1 10 Concentration (kg/m 3) H e ight a bov e b e d (m )
m easured graded s and TR2004 MF (10 fractions ) TR2004 SF (ds=0.13 m m ; standard) TR2004 SF (ds=0.1 m m ; calibrated) M220Graded h= 0.52 m Hs=0.2 m Tp= 2.7 s v=0.17 m /s d50=0.26 m m d90=0.42 m m r=0.022 m 0 0.1 0.2 0.3 0.4 0.5 0.6 0.001 0.01 0.1 1 10 Conce ntration (kg/m 3) H e ig h t ab o v e b ed (m )
m easured graded s and TR2004 MF (10 fractions) TR2004 SF (ds=0.135 m m ; s tandard) TR2004 SF (ds=0.1 m m ; calibrated) M418Graded h= 0.52 m Hs=0.15 m Tp= 2.6 s v=0.29 m /s d50=0.27 m m d90=0.41 m m r=0.022 m
3
Sand transport formulations in DELFT3D
model
3.1
Introduction
Section 3.2 of this chapter gives a detailed description of the implemented processes in DELFT3D-ONLINE. Sub-sections 3.2.1 and 3.2.2 present overviews of the hydrodynamics of currents and waves (largely taken from Lesser et al., 2003). Sub-Section 3.2.3 describes the sediment transport formulations based on TR2004 for non-cohesive sediment following Van Rijn (1993, 2000 and 2002)which have been implemented in DELFT3D-ONLINE as part of the present study. Besides the TR2004 approach, DELFT3D-ONLINE offers a number of extra sediment transport relations for non-cohesive sediment, see Table 3.1 below (a detailed overview of Delft3D-Online sand transport approaches is given in Table 3.2).
Formula Transport modes Waves IFORM
Engelund-Hansen (1967) Total transport No 1 Meyer-Peter-Muller (1948) Bed load transport No 2 Swanby (Ackers-White, 1973) Total transport No 3
General formula Total transport No 4
Bijker (1971) Bed load + suspended Yes 5 Van Rijn (1984) Bed load + suspended No 7 Soulsby / Van Rijn Bed load + suspended Yes 11
Soulsby Bed load + suspended Yes 12
Van Rijn (TR2004) Bed load + suspended Yes -1 Van Rijn (TR1993) Bed load + suspended Yes 0
Remarks:Application of a total transport formulation implies that total load transport is treated as bed-load transport; suspended load transport is assumed to be zero.
Table 3.1 Available sand transport formulations in DELFT3D-ONLINE.
It is emphasized that the implementation as it is reported in this chapter is focussed on the implementation of the TR2004 formulations regarding suspended sediment size, variable roughness, etc. (see Section 3.2.3). In the present version of Delft3D-ONLINE, the approximation formulas are for the bed load transport are still used. An extension to include the complete TR2004 formulations will be done during the course of the project (intra-wave approach to determine wave-related bed load transport). This upgrade will require a redesign of some parts of the code which is also influenced by upgrades of other parts of the code. The description given here should be seen as a report of the present status of the model. At the end of the project a complete overview will be given of the improved Delft3D-ONLINE model.
Type of model Spatial dimension
Transport approach
DELFT-ONLINE
2DH Bed load transport
a) Equilibrium transport based on approximation function of TR2000
b) Other equilibrium formulations (See Table 2.1.2)
Wave-related suspended transport
Equilibrium transport based on approximation method of TR2000
Current-related suspended transport
1) Depth-averaged sand concentration derived from equilibrium sand transport formulation plus adjustment factor based on method of Galappatti
2) Equilibrium suspended transport formulations (no adjustment): a)TR2000 (detailed formulations)
b)TR2000 (approximation functions) c) Other formulations; see Table 2.1.2
Bed roughness a) specified by user b) roughness predictor DELFT-ONLINE 3D and 2DV
Bed load transport
a) Equilibrium transport based on approximation function of TR2000
b) Other equilibrium formulations (See Table 2.1.2)
Wave-related suspended transport
Equilibrium transport based on approximation method of TR2000
Current-related suspended transport
1) Concentration derived from advection-diffusion equation 2) Reference concentration derived from
a) TR2000
b) Other formulations (Table 2.1.2); ref concentration is calculated backwards from equilibrium suspended transport using computed velocity profiles and mixing coefficient
Bed roughness
a) specified by user b) roughness predictor
Table 3.2 Sand transport approaches in DELFT-MOR and DELFT3D-ONLINE model.
3.2
Model description
3.2.1
Hydrodynamics
The DELFT3D-FLOW module solves the unsteady shallow-water equations in two (depth-averaged) or three dimensions. The system of equations consists of the horizontal momentum equations, the continuity equation, the transport equation, and a turbulence closure model. The vertical momentum equation is reduced to the hydrostatic pressure relation as vertical accelerations are assumed to be small compared to gravitational acceleration and are not taken into account. This makes the DELFT3D-FLOW model suitable for predicting the flow in shallow seas, coastal areas, estuaries, lagoons, rivers, and lakes. It aims to model flow phenomena of which the horizontal length and time scales are significantly larger than the vertical scales.
The user may choose whether to solve the hydrodynamic equations on a Cartesian rectangular, orthogonal curvilinear (boundary fitted), or spherical grid. In three-dimensional simulations a boundary fitted (σ-coordinate) approach is used for the vertical grid direction. For the sake of clarity the equations are presented in their Cartesian rectangular form only.
Vertical σ-coordinate system
The verticalσ-coordinate is scaled as (− ≤ ≤1
σ
0) z dζ
σ
ζ
− = + (3.2.1)The flow domain of a 3D shallow water model consists of a number of layers. In a σ-coordinate system, the layer interfaces are chosen following planes of constantσ. Thus, the number of layers is constant over the horizontal computational area. For each layer a set of coupled conservation equations is solved. The partial derivatives in the original Cartesian coordinate system are expressed in σ-coordinates by use of the chain rule. This introduces additional terms (Stelling and Van Kester, 1994).
Generalised Lagrangian mean (GLM) reference frame
In simulations including waves the hydrodynamic equations are written and solved in a GLM reference frame (Andrews and McIntyre, 1978; Groeneweg and Klopman, 1998; and Groeneweg 1999). In GLM formulation the 2DH and 3D flow equations are very similar to the standard Eulerian equations, however, the wave-induced driving forces averaged over the wave period are more accurately expressed. The relationship between the GLM velocity and the Eulerian velocity is given by:
s s U u u V v v = + = + (3.2.2)
where U and V are GLM velocity components, u and v are Eulerian velocity components, and us and vsare the Stokes’ drift components. For details and verification results we refer to Walstra et al. (2000).
Hydrostatic pressure assumption
Under the so-called “shallow water assumption” the vertical momentum equation reduces to the hydrostatic pressure equation. Under this assumption vertical acceleration due to buoyancy effects or sudden variations in the bottom topography is not taken into account. The resulting expression is:
P
g h
∂
ρ
∂σ
= − (3.2.3)Horizontal momentum equations The horizontal momentum equations are
2 0 2 0 1 1 1 1 x x x V y y y V U U U U u U v fV P F M t x y h h V V V V v U V fU P F M t x y h h ∂ ∂ ∂ ω ∂ ∂ ν ∂ ∂ ∂ ∂ ∂σ ρ ∂σ ∂σ ∂ ∂ ∂ ω ∂ ∂ ν ∂ ∂ ∂ ∂ ∂σ ρ ∂σ ∂σ + + + − = − + + + + + + − = − + + + (3.2.4)
in which the horizontal pressure terms, Px and Py, are given by (Boussinesq approximations) 0 0 0 0 0 0 1 1 x y h P g g d x x x h P g g d y y y σ σ
∂ζ
∂ρ ∂σ ∂ρ
σ
ρ
∂
ρ
∂
∂ ∂σ
∂ζ
∂ρ ∂σ ∂ρ
σ
ρ
∂
ρ
∂
∂ ∂σ
′ ′ = + + ′ ′ ′ = + + ′ ∫
∫
(3.2.5)The horizontal Reynold’s stresses, Fx and Fy, are determined using the eddy viscosity
concept (e.g. Rodi, 1984). For large scale simulations (when shear stresses along closed boundaries may be neglected) the forces Fx and Fy reduce to the simplified formulations
2 2 2 2 2 2 2 2 x H y H U U V V F F x y x y
∂
∂
∂
∂
ν
ν
∂
∂
∂
∂
= + = + (3.2.6)in which the gradients are taken along σ-planes. In Eq. (3.2.4) Mx and Myrepresent the
contributions due to external sources or sinks of momentum (external forces by hydraulic structures, discharge or withdrawal of water, wave stresses, etc.).
Continuity equation
The depth-averaged continuity equation is given by
hU hV S t x y
∂
∂
∂ζ
∂
∂
∂
+ + = (3.2.7)in which
S
represents the contributions per unit area due to the discharge or withdrawal of water, evaporation, and precipitation.Transport equation
The advection-diffusion equation reads
[ ]
[
]
[
]
( )
1 H H V hc hUc hVc c t x y c c c h D D D hS x x y y h∂
∂
∂
∂ ω
∂
∂
∂
∂σ
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂σ
∂σ
+ + + = + + + (3.2.8)in which S represents source and sink terms per unit area.
In order to solve these equations the horizontal and vertical viscosity (
ν
H andν
V ) anddiffusivity (DH and DV) need to be prescribed. In DELFT3D-FLOW the horizontal
viscosity and diffusivity are assumed to be a superposition of three parts: 1) molecular viscosity, 2) “3D turbulence”, and 3) “2D turbulence”. The molecular viscosity of the fluid (water) is a constant value O(10-6). In a 3D simulation “3D turbulence” is computed by the selected turbulence closure model (see the turbulence closure model section below). “2D turbulence” is a measure of the horizontal mixing that is not resolved by advection on the horizontal computational grid. 2D turbulence values may either be specified by the user as a constant or space-varying parameter, or can be computed using a sub-grid model for horizontal large eddy simulation (HLES). The HLES model available in DELFT3D-FLOW is based on theoretical considerations presented by Uittenbogaard (1998) and is fully discussed by Van Vossen (2000).
For use in the transport equation, the vertical eddy diffusivity is scaled from the vertical eddy viscosity according to
DV V
c =
ν
σ
(3.2.9)in which
σ
c is the Prandtl-Schmidt number given byσ
c =σ
c0F Riσb g
(3.2.10)where
σ
c0 is purely a function of the substance being transported. In the case of the algebraic turbulence model, F Riσb g
is a damping function that depends on the amount of density stratification present via the gradient Richardson’s number (Simonin et al., 1989). The damping function, F Riσb g
, is set equal to 1.0 if thek
−
ε
turbulence model is used, as the buoyancy term in thek
−
ε
model automatically accounts for turbulence-damping effects caused by vertical density gradients.We note that the vertical eddy diffusivity used for calculating the transport of “sand” sediment constituents may, under some circumstances, vary somewhat from that given by Eq. (3.2.9) above. The diffusion coefficient used for sand sediment is described in more detail in Section 3.2.3.
Turbulence closure models
Several turbulence closure models are implemented in DELFT3D-FLOW. All models are based on the so-called “eddy viscosity” concept (Kolmogorov, 1942; Prandtl, 1945). The eddy viscosity in the models has the following form
in which
c
µ′
is a constant determined by calibration,L
is the mixing length, and k is the turbulent kinetic energy.Two types of turbulence closure models are available in DELFT3D-FLOW. The first is the “algebraic” turbulence closure model that uses algebraic/analytical formulas to determine k and L and therefore the vertical eddy viscosity. The second is the k
−
ε
turbulence closure model in which both the turbulent energy k and the dissipationε
are produced by production terms representing shear stresses at the bed, surface, and in the flow. The “concentrations” of k andε
in every grid cell are then calculated by transport equations. The mixing length L is determined fromε
and k according toL
=
cDk kε
(3.2.12)in which cD is another calibration constant.
3.2.1.1 Boundary Conditions
In order to solve the systems of equations, the following boundary conditions are required:
Bed and free surface boundary conditions
In theσ-coordinate system the bed and the free surface correspond with σ-planes. Therefore the vertical velocities at these boundaries are simply
ω
b g
− =
1 0 andω
b g
0=
0 (3.2.13)Friction is applied at the bed as follows:
1 1 by V u bx V v h σ h σ
τ
ν ∂
τ
ν ∂
∂σ
=−=
ρ
∂σ
=−=
ρ
(3.2.14)where
τ
bx andτ
by are bed shear stress components that include the effects of wave-current interaction.Friction due to wind stress at the water surface may be included in a similar manner. For the transport boundary conditions the vertical diffusive fluxes through the free surface and bed are set to zero.
Lateral boundary conditions
Along closed boundaries the velocity component perpendicular to the closed boundary is set to zero (a free-slip condition). At open boundaries one of the following types of boundary conditions must be specified: water level, velocity (in the direction normal to the boundary), discharge, or Riemann (weakly reflective boundary condition, Verboom and Slob, 1984). Additionally, in the case of 3D models, the user must prescribe the use of either a uniform or logarithmic velocity profile at inflow boundaries.
