Comparison of depth-averaged concentration and bed load flux sediment transport models of dam-break flow [original]

Comparison of depth-averaged concentration and bed load flux sediment

transport models of dam-break flow

Jia-heng Zhao

*, Ilhan

€

Ozgen

, Dong-fang Liang

, Reinhard Hinkelmann

Department of Civil Engineering, Technische Universit€

at Berlin, Berlin 13355, Germany

Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK

Received 26 April 2017; accepted 31 August 2017

Available online 21 December 2017

Abstract

This paper presents numerical simulations of dam-break flow over a movable bed. Two different mathematical models were compared: a fully

coupled formulation of shallow water equations with erosion and deposition terms (a depth-averaged concentration flux model), and shallow

water equations with a fully coupled Exner equation (a bed load flux model). Both models were discretized using the cell-centered finite volume

method, and a second-order Godunov-type scheme was used to solve the equations. The numerical flux was calculated using a Harten, Lax, and

van Leer approximate Riemann solver with the contact wave restored (HLLC). A novel slope source term treatment that considers the density

change was introduced to the depth-averaged concentration flux model to obtain higher-order accuracy. A source term that accounts for the

sediment flux was added to the bed load flux model to reflect the influence of sediment movement on the momentum of the water. In a one-

dimensional test case, a sensitivity study on different model parameters was carried out. For the depth-averaged concentration flux model,

Manning's coefficient and sediment porosity values showed an almost linear relationship with the bottom change, and for the bed load flux

model, the sediment porosity was identified as the most sensitive parameter. The capabilities and limitations of both model concepts are

demonstrated in a benchmark experimental test case dealing with dam-break flow over variable bed topography.

creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords: Shallow water; Sediment transport; Bed load flux model; Depth-averaged concentration flux model; Dam break

1. Introduction

Sediment transport in flowing water is one of the main

factors in erosion and deposition processes. The mathematical

and numerical modeling of these processes is challenging,

because the erosion and deposition processes lead to a time-

variable bottom elevation, which in return influences the

flow. In addition, the sediment concentration is often consid-

ered to influence the momentum of the water. Furthermore, the

erosion and deposition processes are usually described by

empirical laws that depend on several parameters.

Guan et al. (2015) state that there are currently four types of

sediment transport models. Two of them are so-called coupled

models that solve the hydrodynamic and morphodynamic

equations together. Based on the transport mode, coupled

models can be categorized into the depth-averaged concen-

tration flux model (CF model) and the bed load flux model (BF

model). The other two types of models are so-called decoupled

models, namely the two-layer transport model and the two-

phase flow model.

In this paper, coupled models are considered. The BF

model solves the depth-averaged shallow water equations

together with the Exner equation, which describes the sedi-

ment transport based on bed load movement through a power

law of flow velocity. The interaction between flow and sedi-

ment is accounted for by a variable parameter (Murillo and

García-Navarro, 2010). Existing literature about the Exner

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This work was supported by the China Scholarship Council.

*Corresponding author.

E-mail address: [email protected]erlin.de (Jia-heng Zhao).

Peer review under responsibility of Hohai University.

https://doi.org/10.1016/j.wse.2017.12.006

creativecommons.org/licenses/by-nc-nd/4.0/).

Water Science and Engineering 2017, 10(4): 287e294

equation treats the hydrodynamic and sediment mass conser-

vation separately, without considering the influence of sedi-

ment movement on hydrodynamics (Soares-Fraz~

ao and Zech,

2011; Hudson and Sweby, 2003; Liu et al., 2008; Liang,

2011). This model assumes that the movement of the sedi-

ment is much slower than the flow velocity. The CF model

describes the sediment transport as a fully mixed suspended

load, while the erosion and deposition processes are calculated

with empirical equations. The sediment is modeled as a con-

centration in the water column, and its fluxes are calculated

based on this concentration. Several additional parameters are

introduced to calculate mass exchange between the dissolved

sediment and the bed. In this study, source terms were intro-

duced accounting for the interaction between the sediment and

flow (Cao et al., 2004; Simpson and Castelltort, 2006; Wu

et al., 2012; Wu and Wang, 2008).

The aim of this study was to compare the bed load flux and

depth-averaged concentration flux sediment transport models

for dam-break flow over movable beds with regard to accuracy

and suitable conditions for each model.

2. Governing equations

The coupled governing equations, consisting of the two-

dimensional shallow water equations and sediment transport,

can be expressed as

vtþvf

vxþvg

vy¼sð1Þ

where tis time; xand yare the two-dimensional Cartesian

coordinates; qis the vector of conserved variables; fand gare

the flux vectors of the conserved variables in the x- and y-

directions, respectively; and the vector srepresents the source

terms. These vectors are expressed as

q¼

f¼

hþgh2

qyqx

qxc

qsx

g¼

qyqx

hþgh2

qyc

qsy

s¼

ð2Þ

where his the water depth; qxand qyare the unit-width dis-

charges in the x- and y-directions, respectively, and qx¼uh

and qy¼vh, with uand vbeing the depth-averaged velocity

components in the x- and y-directions, respectively; zis the

bottom elevation above the datum; cis the flux-averaged

volumetric sediment concentration; gis the gravitational ac-

celeration (9.81 m/s

); qsxand qsyare the sediment fluxes in

the x- and y-directions for the BF model, respectively, with the

corresponding fluxes of the CF model described as qxcand

qyc;hmis the mass source term of flow; Bxand Byare the

momentum source terms in the x- and y-directions,

respectively; Bzis the source term of the sediment concen-

tration; and zmis the source term for the bottom elevation. By

defining the flux and source terms associated with sediment

transport, the BF or CF model can be obtained from Eq. (2).

For the CF model,

hm¼ED

1pð3Þ

Bx¼ghvz

vxþSfxþðrsrwÞh

vxðr0rÞðEDÞu

rð1pÞð4Þ

By¼ghvz

vyþSfyþðrsrwÞh

vyðr0rÞðEDÞv

rð1pÞð5Þ

Bz¼EDð6Þ

qsx¼qsy¼0ð7Þ

zm¼DE

1pð8Þ

where pis the bed sediment porosity; Sfxand Sfyare the

friction slopes in the x- and y-directions, respectively; Eand D

are substrate entrainment and deposition fluxes across the

bottom boundary of flow, respectively; rwand rsare the

density of water and sediment, respectively; ris the density of

the sediment-water mixture, and r¼rwð1cÞþrsc; and r0

is the density of the saturated bed, and r0¼rwpþrsð1pÞ.

These equations have been presented in Simpson and

Castelltort (2006).

For the entrainment and deposition of sediment, this study

used the following equations from Cao (1999):

D¼u0ð1acÞmacð9Þ

E¼4maxðqqc;0Þu

hd0:2ð10Þ

where u0is the settling velocity of a single particle in

still water, and u0¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ð13:95n=dÞ2þ1:09ðrs=rw1Þgd

q

13:95n=d;nis the kinematic viscosity; dis the sediment

diameter; the coefficient ais calculated by

a¼min½2:0;ð1:04Þ=c;m¼2; 4is a calibration param-

eter; qis the Shields parameter calculated by q¼u2=sgd,

where s¼rs=rw1; and qcis the critical value of the Shields

parameter for the initiation of sediment motion.

For the BF model, sediment fluxes are calculated with

Grass'model (Grass, 1981) as follows:

hm¼0ð11Þ

Bx¼gh vz

vxghsfxð12Þ

By¼gh vz

vyghsfyð13Þ

288 Jia-heng Zhao et al. / Water Science and Engineering 2017, 10(4): 287e294

c¼0ð14Þ

qsx¼zAgqx

xþq2

h3ð15Þ

qsy¼zAgqy

xþq2

h3ð16Þ

zm¼0ð17Þ

where z¼1:0=ð1:0pÞ; and Agis an empirical coefficient

that indicates the intensity of the interaction between flow and

sediment, which is experimentally determined as a constant

with a value between 0 and 1 s

/m. Details of the parameters

can be found in Liang (2011). The BF model with the Exner

equation is known to overestimate the flow velocity because

sediment particles do not influence the momentum of the fluid.

Extending the momentum source terms of the BF model with

jsqsxand jsqsyin the x- and y-directions, respectively, is

proposed, in order to account for momentum losses due to the

sediment movement. As the influence of these source terms on

the eigen-structure of the governing equations has been

neglected, the calibration coefficient jis introduced.

3. Numerical methods

For both models, a cell-centered Godunov-type scheme is

used for solving the equations, and a second-order monotonic

upstream-centered scheme for conservation laws (MUSCL)

(van Leer, 1979) is employed to reconstruct the value at the

middle point of the edge. To preserve the non-negative water

depth and the C-property (Gallardo et al., 2007), the hydro-

static reconstruction method (Audusse et al., 2004) is used. A

total variation diminishing scheme presented in Hou et al.

(2013b) is applied to avoid spurious oscillations.

In the CF model, the numerical fluxes are calculated by the

Harten, Lax, and van Leer approximate Riemann solver with

the contact wave restored (HLLC) (Toro et al., 1994). In the

BF model, the additional fluxes qsxand qsychange the eigen-

structure of the governing equations. Thus, the modified

HLLC flux calculation method by Liang (2011) is used in this

model.

A two-stage Runge-Kutta scheme is applied for the time

discretization.

3.1. Treatment of source terms

In sediment transport models, the influence of sediment

movement is reflected by source terms. For shallow water

equations, the friction source terms are evaluated with a

splitting point-implicit method (Bussing and Murman, 1988).

Valiani and Begnudelli (2006) presented an efficient

divergence form for the slope source term calculation and

Hou et al. (2013a) modified this slope source term treatment

to obtain higher-order accuracy for the bed slope source term

in the classical shallow water model. Based on this method, a

modified source term treatment was derived in this study,

induced by the water flow density change. It is noted that this

treatment requires flow with density change, which means

that the volumetric concentration is a necessary condition, or,

in other words, this method cannot be applied to the BF

model.

In a one-dimensional case, the source terms in the CF

model related to gravity force can be written in the integral

form as follows:

Bgx¼!Ughvz

vxþðrsrwÞh

vxdUð18Þ

where Bgxis the gravity source term, and Uis the control

volume. Source terms related to the gravity force are divided

into the bed slope source term of the flow and the gradient

source term of the sediment concentration. This assumes that

sediment concentration and flow depth are independent of

each other. The source terms are calculated separately with

one of the two values remaining constant.

The derivation process will not be shown in detail here. The

overall derivation for a bed slope can be found in Hou et al.

(2013a), and the application of the method to the source

terms in Eq. (18) is

FsðqÞ$nf¼2

nfxghMþhfrfrMzbM zbf 2

nfyghMþhfrfrMzbM zbf 2

5ð19Þ

where FsðqÞdenotes the flux of the slope source term; nfxand

nfyare the components of the outward unit normal vector nf

along the x- and y-directions, respectively, of the considered

face; hM,zbM,rM¼rwð1cMÞþrscM, and cMrepresent the

water depth, bottom elevation, density, and concentration at

the centroid of the considered cell, respectively; and hf,zbf,

rf¼rwð1cfÞþrscf, and cfrepresent the corresponding

values for the face. The integral of slope source sbover a cell

can be rewritten as

!UsbdU¼∮GFskðqÞ$nfdG¼X

k¼1

½FskðqÞ$nklkð20Þ

where kand neare the index and the number of the edges in

the considered cell, respectively.

3.2. Main procedure of models'application

The models use a ghost cell treatment to impose boundary

conditions (LeVeque, 2002). The updating procedure of the

models can be summarized in the following steps:

Step 1: Update the ghost cells.

Step 2: Interpolate the edge values by means of the

MUSCL reconstruction from Hou et al. (2013b).

Step 3: Carry out the hydrostatic reconstruction (Audusse

et al., 2004).

Step 4: Use the novel slope source treatment to calculate

the divergence form of the bed and density slope source terms

289Jia-heng Zhao et al. / Water Science and Engineering 2017, 10(4): 287e294

for the CF model and use the divergence form to calculate the

bed slope source terms while neglecting the density change

with regard to rf=rM¼1 for the BF model, then calculate

interface fluxes of mass and momentum in the same loop with

the slope source fluxes, with different HLLC approximate

solvers for the CF and BF models.

Step 5: Calculate source terms that are evaluated based on

cell values.

Step 6: Calculate the intermediate values of cells.

Step 7: Carry out the Runge-Kutta time integration by

repeating steps 1 through 6 using the intermediate values.

4. Test cases

In this section, two test cases are presented. The first case is

a one-dimensional dam-break flow over a movable bed. The

second case is dam-break flow over variable bed topography.

4.1. Error calculation

The volume weighted L

-norm (Sun and Takayama, 2003)

was used to quantify the difference between results:

L1ðqÞ¼P

i¼1

Aijqiq0j

i¼1

ð21Þ

where NCis the total number of the cells, iis the cell index, Ai

is the area of the computational cell i,qiis the exact solution

of the computational cell i, and q0is the numerical solution.

4.2. One-dimensional dam-break flow over movable bed

This test case was initially presented by Cao et al. (2004),

who described a numerical solution for reference. The domain

was 4000 m long. A dam was set up at a position of 2000 m.

The water surface elevation on the left side of the dam was

40 m, and the water surface elevation on the right side of the

dam was 2 m. The sensitivity of Manning's coefficient, sedi-

ment diameters, and sediment porosity were investigated.

This case used the same parameters as Cao et al. (2004) to

verify the model. This case was also used as the reference case

for the sensitivity study. The parameters were as follows: the

diameter of the bed sediment was set to 8 mm, the porosity of

the bed sediment was 0.4, and the density of the sediment was

2650 kg/m

. The calibration parameter was set to 4¼0.015 in

the CF model.The modified source term in the BF model was

omitted, i.e., j¼0, to demonstrate the overestimation of

velocities by the BF model. Results are shown in Fig. 1. While

the CF model shows strong agreement with the results re-

ported in Cao et al. (2004), the BF model results do not agree.

This is expected, as the CF model uses similar model concepts

to the model by Cao et al. (2004), i.e., suspended load trans-

port that influences the momentum balance, while in the BF

model the sediment is transported solely as bed load, i.e., the

sediment is not transported as suspended load. Thus, the im-

mediate change in the bottom elevation in the BF model is

described by the sediment concentration in the CF model.

Therefore, the BF model overestimates the erosion upstream

and the deposition downstream. It can be argued that the CF

model captures the physical process better, as the results seem

to agree with the experimental results of Spinewine and Zech

(2007), who replicated this test case in a laboratory. It is noted

that, in reality, distinguishing suspended load from bed load is

not trivial.

In addition, as discussed in Audusse et al. (2012),using

¼0.2 s

/m for unsteady flow will lead to almost constant

water level downstream, which might also explain the greater

deviation of the BF model, and because the influence from

the sediment movement has been neglected, the water level

wave front of the BF model is significantly faster than that of

the CF model. Fig. 2 shows the unit-width discharge distri-

bution in the domain. It is noted that the BF model tends to

overestimate flow velocities, because the sediment movement

does not influence the fluid dynamics.

Fig. 1. Comparison of water surface elevations and bottom elevations

of BF model and CF model after 60 s.

Fig. 2. Unit-width discharge after 60 s.

290 Jia-heng Zhao et al. / Water Science and Engineering 2017, 10(4): 287e294

To capture the shock wave position downstream, the co-

efficient jof the BF model was set to 0.05. A comparison

between the CF model and the modified BF model is shown in

Fig. 3. After both models were verified, a sensitivity analysis

was carried out. Manning's coefficient, the sediment diameter,

and the sediment porosity were varied and their respective

influences on the results were quantified by means of the L

norm of the bottom elevation with regard to the reference case.

In each simulation run, one parameter was increased or

decreased by 50% of itself, while the values of all other pa-

rameters were the same as those in the reference case. The

results are summarized in Table 1. The results for varying

Manning's coefficient are shown in Fig. 4. Increasing friction

causes different types of erosion. In the BF model, Manning's

coefficient is not considered in the sediment movement pro-

cess, so the friction only decreases the flow velocity, which

leads to less erosion. In contrast, with an increasing friction

parameter in the CF model, qwill grow larger, causing more

erosion.

For the CF model, Manning's coefficient and the sediment

porosity show an almost linear relationship with the change of

bottom elevation. For the BF model, the sediment porosity is

identified as the most sensitive parameter. This is partly

because the BF model in this study used a constant A

calculate the sediment flux and Manning's coefficient was not

used.

4.3. Simulation of laboratory dam-break experiment over

movable discontinuous bottom

Dam-break experiments over movable beds were carried

out at the laboratory of the Department of Civil and Envi-

ronmental Engineering at Universit

e Catholique de Louvain,

Belgium, by Spinewine (2005). In comparison to the previous

example, the bottom topography in these cases was more

complex, as initial discontinuities were introduced in the

initial domain.

The domain was a rectangular glass flume, which was 6 m

long, 0.25 m wide, and 0.70 m high. At the middle of the

flume there was a thin gate that simulated the dam. Sand bed

material had a uniform diameter, d¼1.82 mm, sediment

density was r

¼2683 kg/m

, and porosity was p¼0.47.

Different initial conditions were imposed by adjusting the

initial water depth and thickness of the bed material upstream

and downstream of the gate. All four scenarios had the same

Fig. 3. Comparison of water surface elevations and bottom elevations

of modified BF model and CF model after 60 s.

Table 1

-norm summary of bottom elevation in dependency of model parameters.

Parameter L

-norm of bottom elevation

CF model BF model

n¼0.015 s$m

1/3

0.729 0.168

n¼0.045 s$m

1/3

0.666 0.227

d¼4 mm 1.130 0.125

d¼12 mm 0.424 0.134

p¼0.2 0.610 0.376

p¼0.6 0.730 0.483

Fig. 4. Comparison of results with varying Manning's coefficient for

modified BF model and CF model.

Table 2

Upstream and downstream elevations for four scenarios.

Scenario Water surface elevation (m) Bottom elevation (m)

Upstream Downstream Upstream Downstream

1 0.35 0 0 0

2 0.35 0 0.1 0

3 0.35 0 0.1 0

4 0.35 0.1 0.1 0

291Jia-heng Zhao et al. / Water Science and Engineering 2017, 10(4): 287e294

upstream water surface elevation of 0.35 m and downstream

bottom elevation of 0 m. The upstream bed elevation and the

downstream water surface elevation for the four scenarios are

shown in Table 2. A detailed discussion of the experiments can

be found in El Kadi Abderrezzak and Paquier (2011).

The experimental results were replicated with both models,

in order to study the applicability of both models in different

scenarios with complex topography.

For the BF model, A

was set to 0.0006. This is because the

water depth here was comparably low, implying less interac-

tion between flow and sediment. The parameter jwas set to

0.1 for the flow momentum source term dealing with the

sediment movement. For the CF model, 4was set to 0.002.

The results for scenarios 1, 2, 3, and 4 are plotted in Fig. 5,in

which the experimental data are from Spinewine (2005).

As shown in Fig. 5, for scenarios 1 and 2, both the modified

BF model and the CF model show strong agreement with the

experimental data for the bottom elevation. The agreement of

the water surface elevation is poor for both models, but the

location of the shock is captured accurately. The deviation

between the models and experiments might be due to the

mathematical model limitations of the shallow water equa-

tions, which cannot account for the non-hydrostatic pressure

distribution that is expected at the beginning of a dam break.

For scenario 1, the CF model provides better results than the

modified BF model. The reason for this is the same as in the

first test case, i.e., the sediment in the CF model is still dis-

solved in the fluid while the modified BF model causes im-

mediate deposition. The CF model has stiff source terms, i.e.,

for cases with dry beds, the erosion rate goes to infinity. As a

numerical treatment, if the water depth is less than the sedi-

ment diameter, the erosion rate is set to zero. This enables a

robust simulation of sediment transport on a dry bed. For

scenario 2, both models provide about the same agreement for

the bottom elevation. However, the modified BF model shows

better agreement with the experimental data for water surface

elevation.

The results for scenario 3 show that the modified BF model

provides better agreement for the bottom elevation. Both

models fail to capture the shock properly. For the water sur-

face elevation, the modified BF model results may be

considered better. For scenario 4, the modified BF model

Fig. 5. Water surface elevation and bottom elevation at t¼1.247 s for different scenarios.

292 Jia-heng Zhao et al. / Water Science and Engineering 2017, 10(4): 287e294

provides strong agreement for the bottom elevation. The water

surface elevation could not be reproduced by either of the

models, but the modified BF model provides better agreement.

The shock is again not captured properly. It can be concluded

that the sharply descending bottom is numerically more

challenging for both models and leads to errors in the hy-

drodynamics and morphodynamics. The shock cannot be

captured accurately in these cases. For the flat bottom and the

sharply ascending bottom, these errors are not observed; the

shock is captured accurately.

Overall, the results are comparable to the ones reported in

El Kadi Abderrezzak and Paquier (2011).

5. Conclusions

A bed load flux (BF) model and a depth-averaged con-

centration flux (CF) model were used to simulate sediment

transport in two different ways: as bed load and as suspended

load. The bottom change in the BF model is calculated using

the sediment flux of the bed load. Therefore, the morphody-

namics are instantaneous. In contrast, in the CF model, the

sediment transport is a process that depends on the entrain-

ment and deposition that are calculated separately using

empirical formulas. The water carries the sediment and the

sediment particle movement influences the water momentum.

Thus, the CF model concept is more complex than the BF

model concept.

A simple heuristic modification is proposed that introduces

additional source terms to the momentum balance. A modified

bed slope source term treatment that considers the density

change of the flow in the CF model meant to obtain higher-

order accuracy, is presented in this paper.

Two test cases were described that compare the BF model

with the CF model. In the first test case, both models were

verified by comparison with results from Cao et al. (2004). After

the verification, a sensitivity test was carried out for different

parameters. For the CF model, Manning's coefficient and the

sediment porosity show an almost linear relationship with the

bottom change. For the BF model, the sediment porosity is

considered the most sensitive parameter. The influence of the

friction in both models shows the opposite impact.

In the second test case, the modified BF model and the CF

model were compared with experimental data. The BF and CF

models both provide good results for the morphodynamics, but

for the water surface, the numerical models show greater water

depth in the wave front, which might result from non-

hydrostatic flow conditions in the experiment. However, in

the investigated cases, the sediment movement and its influ-

ence on the water surface elevation were relatively small, so

that both models lead to similar surface water levels. For real-

world applications such as real flood events, the difference

between the morphodynamics may lead to larger differences in

the water surface levels.

The empirical coefficient jin the BF model will be

investigated in the future. It can be concluded that the

modified BF model can be used for sediment transport

modeling for relatively steady flow, and the CF model can be

used for unsteady flow. However, it is acknowledged that in

practice it is difficult to separate these cases.

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