3-D numerical simulation for total sediment transport and its application

Hongwei Fang1 and Wolfgang Rodi2

1.Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China

2.Institute for Hydromechanics, University of Karlsruhe, Karlsruhe 76128, Germany

The calculation of flow and sediment transport is one of the most important tasks in river engineering.

The task is particularly difficult because of the many complex and interacting physical phenomena which

need to be accounted for realistically in a model that has predictive power. 3D models was developed and

tested over the years which allow the calculation of flow, both suspended and bed-load sediment transport

and the associated bed deformation for natural rivers. The model type and their interrelation are described

with their components such as hydrodynamic model including the turbulence model, bed-load and

suspended-load model and the numerical procedures for solving the model equations. Applications of the

models to the stretch of the River Elbe and the Three Gorges Reservoir Project in the Yangtze River are

presented and compared with experiments whenever possible.

Flood protection, keeping rivers navigable, securing household and irrigation water supply and

providing the conditions for hydropower generation are the prime tasks of river engineers. In fulfilling this

task, not only economical and technical considerations are important but increasingly also ecological ones

and the impact of any human measures on the environment. For planning such measures, the engineers

need to be able to predict the consequences of field and hydraulic model studies, but these are very costly

and time consuming. With the rapid increase in computer power and the advancement of numerical

methods, more and more computer models are used, and the prime task of such models is to predict the

flow and sediment transport in rivers and the impact of human measures. This is a difficult tasks because

of the many complex flow patterns, turbulence, suspended and bed-load transport with deposition and

erosion causing bed deformation.

Different levels of idealizations and empirical input have been used in computer modeling. 1D models

are widely used in practice and can be applied to long river stretches. With these models, only cross-

sectional averaged quantities are calculated so that details cannot be resolved and many influences require

an empirical treatment. Depth-averaged 2D models resolve horizontal variations and provide many more

details like the influence of changing cross-sections and irregular side boundaries. These models are based

on the assumption of hydrostatic pressure distribution and cannot account for secondary motions which are

particularly strong in bends. Currently they are also used already for solving practical problems, albeit only

for shorter river stretches. 3D models are the most powerful ones; they need not assume hydrostatic

pressure and can calculate secondary motions, but they are also the most costly ones and are so far used,

only for calculating more local phenomena. Again several models, some fully 3D, some only partly, are

available and a brief review can be found in the literatures[1-4].

In this part, 3D models for calculating the flow and sediment transport in natural rivers with complex

geometry have been developed. Both models have much in common in terms of numerical solution

procedure as well as turbulence and sediment-transport modeling. The paper gives a description of the

main features of the model and shows their interrelation. Test calculations for the natural river situations

are presented which show the performance and the potential of the model and also its relative merits.

1. Hydrodynamic models Since the concentration of suspended sediments is generally small and the bed-load layer thin, the

assumption can be made that the flow development is not influenced directly by the presence of the

sediments, i.e. the flow equations can be solved de-coupled from the sediment-transport equations.

However, the sediments may have an indirect effect on the flow in that they can deform the bed and

change the geometry and bed roughness. This is accounted for by adjusting the flow geometry and in the

calculation of the bed friction.

With the above assumptions, the three-dimensional flow field in rivers is determined by the following

Reynolds-averaged continuity and Navier-Stokes equations, written here in Cartesian coordinates

ƒƒ

u

xj

j

= 0 , (1)

ƒƒ

ƒƒ ρ

ƒƒ ρ

ƒτƒ

u

t

u u

xF

P

x xi i j

ji

i

ij

j

+ = − +( ) 1 1

, (2)

where )3,2,1( =iui are the velocity components, iF is the gravity force per unit volume, ρ the fluid

density and p the pressure. The Einstein summation convention applies. The turbulent stresses ijτ , which

in the averaged equations represent the momentum exchange through the turbulent fluctuating motion,

have to be calculated with a turbulence model. Most models used in hydraulics employ the eddy-viscosity

relation

τ ρ탃

ƒƒ

δij ti

j

j

iij

u

x

u

xk= + −( )

23

, (3)

introducing an artificial, turbulent or eddy viscosity tv . For this, often a constant, suitably adjusted value

is used. However, this does not do justice to complex flow patterns that evolve in rivers with irregular

boundaries and hence the more universal ε−k turbulence model is used here which allows to calculate

the distribution of tv over the flow field. This is achieved by relating tv to two local turbulence

parameters, namely the turbulent kinetic energy k and its dissipation rate ε , through εµ /2kcvt = and

by then determining the distribution of k and ε from the following model equations:

ƒƒ

ƒƒ

ƒƒ

νσ

ƒƒ

εk

t

u k

x x

k

xG

j

j j

t

k j

+ = + −( )

( ) , (4)

ƒεƒ

ƒ εƒ

ƒƒ

νσ

ƒεƒ

εε

εε εt

u

x x xc G c

kj

j j

t

j

+ = + −( )

( ) ( )1 2 , (5)

here Gu

x

u

x

u

xti

j

j

i

i

j

= +탃

ƒƒ

ƒƒ

( ) is the production of k through velocity gradients. The standard values

of the model coefficients[5] are used.

The water level sz can be calculated from the kinematic free surface condition or the two-dimensional

depth-integrated continuity equation. Here it is determined from a two-dimensional Poisson equation

derived from the depth-averaged two-dimensional momentum equations for shallow open channel flow[4]:

ƒƒ

ƒƒ

2

2

2

2

z

x

z

y

Q

gs s+ = , (6a)

where

Qt

U

x

V

y

U

x

U

y

V

x

V

yU

U

x

V

x y= − + − − − − +

ƒƒ

ƒƒ

ƒƒ

ƒƒ

ƒƒ

ƒƒ

ƒƒ

ƒƒ

ƒƒ ƒ

( ) ( ) ( ) ( )2 22

2

2

2

− + + + + − −VV

y

U

x y x x y y x h y hxx xy yy xb yb

( ) ( ) ( ) ( )ƒƒ

ƒƒ ƒ ρ

ƒ τƒ

ƒ τƒ ƒ

ƒ τƒ ρ

ƒƒ

τρ

ƒƒ

τ2

2

2 2

2

2 2

2

12

1 1, (6b)

where g is acceleration due to gravity, x and y are horizontal coordinate axes, U and V are depth-

averaged velocities in x and y directions, τ xx, τ xy , τ yx and τ yy are depth-averaged shear stresses, τ xb

and τ yb are bed shear stresses. Equations (6a) and (6b) are used to calculate the water level. All the depth-

averaged velocities and stresses in Q are calculated from the three-dimensional values.

On a natural river bed there are usually sand g rains and even sand waves, and it is necessary to account

for the influence of these through the bed boundary condition. In this study the wall function approach

τ ρ κ νµw c k Eu z= − 1 421 2

2/ /

*/ ln( / ) is still used, but the roughness parameter E is allowed to vary with

the roughness Reynolds number k u ks s+ = * / ν via the relation:

E B B= −exp[ ( )]κ ∆ , (7)

where κ is the von Kaman constant, ks is the equivalent roughness of the river bed, B is an additive

constant. ∆B is called the roughness function and is determined as follows[6]

∆B

k

B k k k

B k k

s

s s s

s s

=

<

− + − ≤ <

− + ≥

+

+ + +

+ +

0 2 25

8 5 1 0 4285 0 811 2 25 90

8 5 1 90

.

[ . ( / ) ln ]sin[ . (ln . )] .

. ( / ) ln

κ

κ

, (8)

where B = 52. and κ = 0 41. . The equivalent roughness height ks is usually set to d50 when there is

only sand grain roughness on the bed. However in practice, at the natural river bed there is generally also

sand wave roughness (including ripples, dunes and anti-dunes) and other kinds of roughness. For the sand

wave roughness, d50 should be replaced by the height of the sand wave, such as 3d90 by van Rijn.

Furthermore van Rijn proposed a more accurate method to calculate the roughness height ks on a movable

bed[7],

k d es = + − −3 11 19025. ( )∆ Ψ , (9)

where Ψ ∆= / λ , ∆ and λ are height and length of sand waves respectively; they can be calculated by

∆ / . ( / ) ( )( )λ = − −−0 015 1 25500.3 0.5d h e TT and λ = 7 3. h a s we l l a s Ψ ∆= / λ ,

T u u ucr cr= −[( ) ( ) ] / ( )*’

*, *,2 2 2

, u*’

is the bed-shear velocity related to sediment grains above river bed,

u cr*, is the critical bed-shear velocity based on Shield s curve.

2. Sediment transport model

The overall sediment transport in rivers in governed by the following equation, which is the sediment

mass-balance equation integrated over the water depth h:

( )1 0− ′ + − =ρƒƒz

tD Eb

b b , (10)

where bz is the local bed level above datum and sz the level of the free surface, ρ′ is the porosity of the

bed material, C is the depth-averaged sediment concentration and Txq and Tyq are the components of the

total-load sediment transport in the horizontal x- and y-directions, respectively. The second term, which is

the storage term, can be neglected for the quasi-steady flow conditions considered here. The above

equation relates the bed deformation tzb ƒƒ / to the spatial changes in the sediment-transport load. This

load is subdivided into suspended load and bed load and hence the flow domain is subdivided into a bed-

load layer with thickness bδ and the suspended-load region above it with thickness bh δ− . The exchange

of sediment between the two layers is through deposition (downward sediment flux) at rate bD and

entrainment from the bed-load layer (upward flux) at rate bE . In situations without bed-load transport,

these quantities express the exchange suspended-sediment layer and the bed is hence bb ED − . In a 3D

model the vertical variation of the sediment concentration c in the suspended-load layer is resolved with a

3D convection-diffusion equation and the sediment flux bb ED − at the lower boundary of this layer

needs to be specified as boundary condition.

2.1 Bed-load transport

The mass-balance equation for sediment transport within the bed-load layer can be written as (neglecting

again the storage term):

0)1( =ƒ

ƒ+

ƒƒ

+−+′−y

q

x

qED

t

z bbybbxbb

bαα

ƒƒρ , (11)

where bbx qα and bby qα are the components of the bed-load transport bq in the x- and y-directions

respectively. bxα and byα are the direction cosines; under the assumption that the bed load is in the

direction of the bed shear stress these cosines are known from a three-dimensional flow calculation.

Gravity-induced bed-load transport on transverse bed slopes can change these cosines and this can be

accounted for through empirical formulae[8]. In general, and especially in cases with significant secondary

flow, the direction of the bed shear stress does not correspond with the direction of the depth-averaged

velocity.

An equation for local bed-load transport bq can be obtained from (11) when a model assumption is

introduced for the bed deformation term. Following Philips and Sutherland[9] this is related to the

difference between the actual bed-load transport and bed-load transport *bq that would prevail under

equilibrium conditions:

( ) ( )*11

− ′ = −ρƒƒz

t Lq qb

sb b , (12)

where Ls is a non-equilibrium adaptation length for bed-load transport which has to be prescribed

empirically. Unfortunately, considerable uncertainty exists about this prescription, and a wide variety of

different values have been adopted. Some discussion on this can be found in Wu et al.[4] where it is

concluded that the value of sL is not so important in near-equilibrium situations but that the development

towards this state may be markedly influenced by this value. Further, sL has also an influence on the

numerical stability of the calculation which may be a determining factor when coarse grids are used. In

most of the calculations presented in this paper, the empirical relation of van Rijn[7] given below has been

used. In these calculations the bed-load transport bq is determined from the governing equation which

results from combining (11) and (12). The equilibrium bed-load transport *bq appearing in this equation

also needs to be determined from an empirical relation. Here one of the many transport formulae available

in the literature can be used; again in the calculations presented below, a formula due to van Rijn was

employed.

2.2 Suspended-load transport

In the model, the distribution of the sediment concentration in the suspended-load layer is governed by

the following convection-diffusion equation:

)(])[( 3jc

s

jjsj

j x

cv

xcu

xt

c

ƒƒ

ƒƒ=−

ƒƒ+

ƒƒ

σδϖ , (13)

where c is the local sediment concentration, sϖ the settling velocity of the sediment, 3jδ the Kronecker

delta with j=3 indicating the vertical direction and cσ the turbulent Schmidt number relating the turbulent

diffusivity of the sediment to the eddy viscosity tv . In the case of non-uniform particle size distribution,

several concentration equations for different classes or particle sizes could be solved, but in the

calculations presented below only one concentration equation for a representative particle size was used.

Equation (13) is solved with the following boundary condition: at the free surface the vertical sediment

flux is set to zero while at the lower boundary of the suspended-sediment layer, the net flux bb ED −

across this boundary has to be specified as mentioned above. The deposition rate at the boundary, which is

located at bz δ=′ , is bsb cD ϖ= while for the entrainment rate bE a model has to be introduced.

Following van Rijn[7] and Celik and Rodi[10] it is assumed that the entrainment is equal to the one under

equilibrium conditions (i.e. when bb DE = ) so that *bsb cE ϖ= where *bc is the equilibrium

concentration at bz δ=′ . The sediment flux at the lower boundary of the suspended layer is therefore

prescribed as

D E c cb b b b b− = −ω ( )* , (14)

For determining the equilibrium concentration *bc again an empirical relation of van Rijn given below is

used. The reference level bz δ=′ at which this concentration is to be determined is taken as 502d for flat

bed and 13/2 − of the height of roughness elements in the case of beds with such elements.

2.3 Empirical input

In the sediment transport models described above, the near-bed equilibrium concentration at reference

level, *bc , the equilibrium bed-load transport *bq and the non-equilibrium adaptation length sL have to

be provided through empirical relations. Here the following relations proposed by van Rijn[7] are used:

cd T

aDb*

.

*..= 0 015 501 5

0 3 , q Rgd T

Db*

. .

*..= 0 053 50

1 5 2 1

0 3 , L d D Ts = 3 500 6 0 9*

. . , (15)

where the particle-size diameter 3/1250* ]/)[( vgdD s ρρρ −= and the non-dimensional excess bed

shear stress T u u ucr cr= −[( ) ( ) ] / ( )*’

*, *,2 2 2

. In these relations, =50d median diameter of the bed

material, sρ = density of this material, ρ = density of water, u′ = effective bed shear velocity related to

the grain and cru* = critical bed shear velocity for sediment motion given by the Shields diagram.

2.4 Calculation of bed deformation

Once the bed load and the suspended sediment concentration have been determined with the above

models, the resulting change of the bed level bz is calculated from equation (10), with the storage term

(second term) neglected. The total-load components Txq and Tyq are composed of the bed-load

components bxq and byq and the suspended-load components sxq and syq . Here bbxbx qq α= and

bbyby qq α= and the suspend-load components are determined from

′=s

b

z

isj zcduqδ

(16)

3. Numerical solution procedures

The partial differential equations for the mean flow, for the turbulence model and for the suspended-

sediment concentration are solved with extended versions of the finite-volume codes which is described in

detail in Majumdar et al.[11]. Finite-volume codes use non-staggered curvilinear numerical grids. In the 3D

model the grid is adjusted to the changing free surface and bed boundaries as the solution proceeds in time;

The connective fluxes are discretized with the standard hybrid upwind/central differencing scheme and the

pressure velocity coupling is achieved by using the SIMPLE pressure correction algorithm. The discretized

algebraic equations are solved with strongly implicit line relaxation procedure of Stone. The non-

equilibrium bed-load transport equation (11) with (12) is solved by a 2D finite-volume procedure using

second-order upwind differencing. Finally the bed-load changes are computed either with a Lax-Wendroff

scheme or a predictor-corrector scheme.

4. Calculation examples

In this section, two calculation results obtained with the 3D models introduced above are presented.

4.1 Stretch of River Elbe

A stretch of river Elbe (km 506.0 to 513.0) in the area of the former East/West German border was

simulated by 3D models; also, both field measurements and laboratory experiments have been carried out

by German BAW. An impression of the geometry of the river in this region can be obtained from Fig. 1a,

which also shows the calculation domain (full lines) used in the model. For the discretization of this

domain a curvilinear grid with 62 points in the lateral, 701 points in the streamwise and 25 points in the

depth direction was used. In the main channel, the grid was fairly fine and conforms to the geometry of

more than 50 groynes on either bank of the river. The bed elevation at the initial state of the simulation was

interpolated from the river. In a first step, the model coefficients n was calibrated by simulating steady-

state medium-discharge Q=795m3/s. Fig. 1b and 1c show the blown-up picture at the part of the calculation

domain at the medium-discharge case from Fig. 1a. In this case, the groynes are only partly submerged and

a complex flow field develops in their vicinity with recirculation regions. The figure demonstrates that this

is resolved fairly well in the calculations. Fig. 2 compares calculated and measured water surface elevation

and velocity profiles for the medium situations at km 508.5 and km 510. Both field and laboratory

measurements are included. It can be seen that the calculated velocities agree well with the field and

laboratory measurements in the main river.

The BAW also carried out a laboratory experiment for the same Elbe stretch starting with a flat movable

bed. The horizontal length scale was 1:110 and the vertical scale 1:40 and the channel cross-section was

trapezoidal with vertical walls. At the beginning of the experiment, the channel was filled with a 10cm

thick layer of sand with a flat surface. The experiment was carried out for a low-water discharge of 12.54

l/s which corresponds to Q=350 m3/s in the natural river. At the inlet, 1070 cm3/min of model sediment

with median diameter 50d =2.1mm was fed in permanently and this was also the bed-load transport rate in

the channel. The experiment was run for 6 hours and the bed level was measured at that time. However, a

state close to equilibrium was established after 2-3 hours and the surface velocity was measured then at a

number of cross-sections. The model simulated the experiment described with the 3D model, switching on

only the bed-load sediment transport model because this was the only important mode in the experiment.

The calculations started also with a flat sand bed and this was then allowed to deform according to the

sediment transport calculated. The experimental sediment input was used as inflow condition for the bed

load. In initial calculations the non-equilibrium adaptation length sL was determined from van Rijn s

formula which yielded values considerably smaller than the grid size. These small values led to rather

erratic behavior and a decoupling of the results from the boundary condition of bed-load input. As was

found also in other sediment-transport calculations using sL , this parameter should be closer to the mesh

size and hence it was adopted to be equal to the average grid size. However, further work is necessary to

study the influence of this parameter on the calculations.

Fig. 3 allows a more quantitative comparison as here the measured and calculated bed profiles are

compared at 3 cross-sections. The model reproduces very well the development of a scour channel near the

outer left bank in the initial bend and a change-over to a profile with a scour channel on the other side in

the downstream region where the bend is in the other direction. The calculated surface velocities also agree

fairly well with the measured ones at the various cross-sections.

4.2 Three Gorges Project reservoir in the Yangtze River

The 3D model was applied to calculate the flow and sediment transport that will occur in the reservoir

which will be generated by the dam of the Three Gorges project in the Yangtze River from the year 2003.

A period of 76 years after the dam starts operating was simulated and this period was also investigated in a

laboratory experiment at Tsinghua University. As the sediment transport is almost entirely due to

suspended load, only the suspended-load model was used in these calculations. These were carried out for

a 16.7km reach upstream of the dam and the calculation domain can be seen from Fig. 6. This domain was

discretized with a numerical grid that had 234 points in the streamwise, 42 points in the lateral and 22

points in the vertical direction. The initial bed geometry was that of the natural river provided by the

National Yangtze River Committee. The time variation of flow and sediment input at the inflow section

was basically taken the same as in the experiment, but it was somewhat smoothed by averaging over

certain periods. Also, the experimental water depth at the downstream boundary (dam) was prescribed.

Various time steps were used in the calculation. The shortest one for the calculation of the flow field with

fixed bed geometry, an intermediate time step for the calculation of the sediment transport and the bed

deformation and the longest time interval for the periods of constant discharge and suspended-load input.

Fig. 4 shows the calculated surface velocities at the beginning of the dam operation period and after 76

years of operation. At the beginning, when the natural bed still exists, the flow occupies the entire

calculation domain and obviously is faster in the narrower upstream reach and has a lower velocity and a

more complex wider. After 76 years, much sediment has deposited on the sides and the river flows only in

fairly narrow cross-section similar to its behaviour before the dam was erected. The change in the bed and

hence the flow cross-sections can be seen from Fig. 5 where the bed profiles are given for the year zero

and for the 54th year, showing clearly the rise of the bed elevation due to the sediment deposition over the

years. The figure compares the calculated bed profiles with those measured in the laboratory experiment

and the agreement can be seen to be quite good. For the same cross-sections and times, the figure also

compares the profiles of calculated and measured surface velocity, and again the agreement is good, even

in the region near the left bank at cross-section 8 where negative velocities occur. Altogether, these results

demonstrate that the model is able of simulating the flow and sediment transport processes in this

practically important application example realistically and hence can be used as a tool in river engineering.

5. Discussions

3D models developed for calculating the flow and sediment transport in rivers have been presented. The

models consist of a hydrodynamic module, which includes the ε−k model, and bed-load and suspended-

load sediment transport modules with a model for calculating the interchange between them. The bed-load

transport and the suspended load is calculated by solving the convection-diffusion equation for sediment

concentration. Various empirical relations due to van Rijn appear in the sediment-transport models. The

bed deformation is finally calculated from the depth-integrated sediment mass balance equation.

Applications of the models have been presented for the various natural river situations with complex

geometry. The capabilities of the models for calculating flow with complex geometry and bed-load and

suspended-load transport were thereby tested. In general, the main flow behaviour was simulated well by

both models, including complex patterns in the vicinity of structures such as groynes and a proper response

to changes in cross-section. A laboratory experiment in curved channels were simulated in which the

mobile bed was initially flat and then deformed mainly due to bed-load transport. This process could be

reproduced quite well by the models, which predicted realistically the development of a scour channel near

the outer bank and of a point bar near the inner bank. There is still some uncertainty about the specification

of the non-equilibrium adaptation length sL appearing in the bed-load model; so far the comparison with

data was for near-equilibrium situations where the value of this parameter is of little influence. Further

tests should be carried out for really non-equilibrium situations and also for large-scale natural conditions.

The suspended-load model involves fewer uncertainties, especially when it is run for cases where

deposition is predominant. Application to the Three Gorges Project Reservoir has shown that the long-term

development of the bed and the associated flow can be predicted quite well for realistic situations so that

the model can be used as a predictive tool.

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construction period, Science in China, Ser.A, 1995, 38(5): 618.

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phase flow, ACTA Mechanic Sinica, 1996, 12(1): 85.

3. Fang Hongwei, Wang Guangqian, Three-dimensional mathematical model for suspended sediment

transport, ASCE J. of Hydraulic Eng., 2000, 126(8): 578.

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ASCE J. Hydraulic Eng., 2000, 126(1): 4.

5. Rodi, W., Turbulence Models and their Application in Hydraulics, Balkema Rotterdam: 3rd. ed., IAHR

Monograph, 1993.

6. Cebeci, T., Bradshaw, P., Momentum Transfer in Boundary Layers, New York: Hemisphere Publishing

Corporation, 1977.

7. van Rijn, L.C., Sediment transport, part _: bed forms and alluvial roughness, ASCE J. of Hydraulic

Eng., 1984, 110(12):1733.

8. Struiksma, N., Olesen, K.W., Flokstra, C. et al., Bed deformation in curved alluvial channels, J.

Hydraulic Research, 1985, 23(2): 57.

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Research, 1989, 27(1): 115.

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Hydraulic Eng., 1988, 114(11): 1157.

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with complex boundaries, J. Fluids Eng., 1992, 114(4): 496.

Fig.1a River Elbe: calculation domain

Fig.1b, c River Elbe: blown-up picture at the part of the calculation domain

Fig.2a Calibration of the water level along the river streamwise direction

Fig.2b Comparison of the surface velocity in two cross sections

Fig.3 River Elbe: comparison of bed profiles calculated by 3D model and measured in the laboratory

Fig.4 Three Gorges project reservoir in Yangtze River: surface velocity vectors calculated by 3D model

Fig.5 Three Gorges project reservoir in Yangtze River: comparison of surface velocity and bed profiles

calculated by 3D model and measured in the laboratory