Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue6.2000:

Advances in Water Resources 23 (2000) 531±543

A mass conservative 3-D numerical model for predicting solute ¯uxes
in estuarine waters
Yan Wu, Roger A. Falconer *
Cardi€ School of Engineering, Cardi€ University of Wales, P.O. Box 686, Cardi€ CE2 3TB, UK
Received 27 January 1999; received in revised form 24 July 1999; accepted 21 August 1999

Abstract
A re®ned three-dimensional layer-integrated model to predict accurately salt and cohesive sediment transport in estuarine waters
is described herein. A splitting algorithm has been used to split the three-dimensional transport equation into a horizontal twodimensional equation and a vertical one-dimensional equation due to the di€erent length scales. An additional source term associated with the layer average of the free-surface ¯ow is introduced in the conservative form of the layer-integrated pollutant
transport equation. The one-dimensional QUICKEST scheme has been extended to two dimensions and included in the layer-integrated advective±di€usion equation. A modi®ed one-dimensional ULTIMATE algorithm has also been added to avoid unphysical
numerical oscillations. Numerical tests for discontinuities have been carried out to study the performance of the ULTIMATE
QUICKEST scheme used in the present model. The model has also been used to simulate solute transport in an idealized harbor. It
has been found that the additional source term was crucial for the mass conservation of pollutant. Finally the re®ned model has been
applied to simulate salt and cohesive sediment transport in the Humber Estuary, UK. Good agreement has been obtained with the
®eld measured data. Ó 2000 Elsevier Science Ltd. All rights reserved.
Keywords: Estuarine and coastal hydrodynamics; Solute and sediment transport; Salinity; Cohesive sediment; Mass conservation;
ULTIMATE QUICKEST

1. Introduction

With a growing awareness of pollution problems in
coastal and inland waters, in recent years there has been
a considerable increased e€ort in developing and applying numerical models to predict ¯ow ®elds and solute
concentration distributions in coastal and estuarine
waters. The requirement for any model to simulate accurately the advective±di€usion processes is a vital aspect in model development.
It is widely known that the ®rst-order unwind scheme
su€ers from serious numerical di€usion and the highorder central di€erencing schemes are plagued by grid
scale oscillations (or wiggle) when they are used to solve
convection-dominated problems. A number of schemes
have been developed to overcome these diculties in the
®eld of Computational Fluid Dynamics (CFD). For
example, the second-order Lax±Wendro€ scheme, which
*

Corresponding author. Tel.: +44-01222-874280; fax: +44-01222874597.
E-mail address: falconerra@cardi€.ac.uk (R.A. Falconer).

was introduced by Lax and Wendro€ [1] for computing
¯ows with a shock wave; the MacCormark scheme [2],
which includes a predictor±corrector version of the Lax±

Wendro€ scheme; and the second-order upwind scheme
of Warming and Beam [3]. However, it was found that
these schemes also introduce numerical dispersion and
did not eliminate unphysical spurious oscillations near
discontinuities. Leonard [4] developed QUICK and
QUICKEST schemes of advective±di€usion equation
for steady and unsteady ¯ows, with these schemes being
based on the quadratic upstream interpolation. These
schemes have been widely used due to their high numerical accuracy, i.e. third-order. However, even though
the QUICKEST scheme greatly reduces the unphysical
oscillations caused by numerical dispersion, this scheme
still su€ers from numerical oscillations near discontinuities.
In recent times a number of high-order oscillationfree schemes have been constructed, with the total variation diminishing (TVD) scheme being one of the most
popular [5±9]. Leonard [10] pointed out that the secondorder TVD schemes achieved their oscillation-free

0309-1708/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 3 0 9 - 1 7 0 8 ( 9 9 ) 0 0 0 3 5 - 4

532

Y. Wu, R.A. Falconer / Advances in Water Resources 23 (2000) 531±543

results by the use of locally varying positive arti®cial
di€usion, which distorted local discontinuities to simple
smooth pro®les. He then proposed a ``monotonizing''
universal limiter (ULTIMATE) for the advection
problem to banish unphysical overshoot and nonmonotonic oscillations, without corrupting the expected
accuracy of the underlying method. In Leonard's ULTIMATE scheme the e€ective time-averaged normalized
cell face value, rather than the normalized cell face value
at time level n, is limited by a monotonicity-maintenance
criteria to eliminate nonmonotonic oscillations. Cahyono [11] carried out a detailed study of 36 of the most
popular ®nite di€erence schemes for the advection
problem, with his comparisons showing that the ULTIMATE scheme was particularly attractive, since it
was more general than the other schemes considered and
was easier to apply.
In this paper, details are given of the requirement of
a three-dimensional layer-integrated numerical model
to predict accurately solute transport in estuarine and
coastal waters. In Section 2 the three-dimensional
transport equation has been split into a horizontal

two-dimensional equation and a vertical one-dimensional equation due to the di€erence length scales in
the horizontal and vertical planes. A conservative form
of the horizontal two-dimensional layer-integrated advective±di€usion equation has been derived, in which
an additional source term associated with the layer
average of the free-surface ¯ows has been introduced.
The QUICKEST scheme originally developed by
Leonard [4] for the one-dimensional advective±di€usion problem has been extended to two-dimensions for
the horizontal layer-integrated advective±di€usion
equation in Section 3, and the modi®ed one-dimensional ULTIMATE algorithm constructed by Wu and
Falconer [12] has been used to avoid unphysical numerical oscillations. For the vertical part of the advective±di€usion equation the implicit power-law
scheme has been used to predict accurately the solute
distribution over the depth.
In Section 4 numerical tests of the one-dimensional
pure advection of a square wave and the two-dimensional advection and di€usion of a square column have
been undertaken to verify the present model. The
model has also been used to study solute transport in
an idealized square harbor, to check the conservation
behavior of the scheme. The results have shown that
the additional source term associated with the layer
average of the free-surface ¯ow is crucial for the mass

conservation of pollutant. This conclusion is consistent
with the results of a study for the case of two-dimensional depth integrated advective±di€usion equation by
the authors [12]. Finally, in Section 5 the model has
been applied to predict salt and cohesive sediment
concentration distributions in the Humber Estuary,
UK.

2. Governing equations
2.1. Hydrodynamic equations
The governing hydrodynamic equations describing
¯ows in coastal and estuarine waters are generally based
on the three-dimensional Reynolds equations for incompressible and unsteady turbulent ¯ows [13]. The
hydrodynamic layer-integrated equations can be derived
by integrating the three-dimensional continuity and
momentum equations over layers. A sketch of the layers
and the relative variable locations in the x±z plane are
illustrated in Fig. 1. The three-dimensional layer-integrated equations can be written as:
(i) Continuity equation for layer k

wkÿ1=2 ˆ ÿ


8 
K < o Dz
u
X
kˆk

:

ox

‡


9
o Dz
v =
oy

…1†

;

;

where w is the vertical velocity; Dz the layer thickness; K
the total number of layers; u and v are layer-integrated
velocities and are de®ned as
1
Dz

Z

kÿ1=2

u ˆ

1
Dz

Z

kÿ1=2

v ˆ

k‡1=2

k‡1=2

u… x; y; z; t† dz;

…2a†

v… x; y; z; t† dz:

…2b†

For the ®rst layer, which describes the free-surface,

the continuity Eq. (1) becomes


9
8 
K < o Dz
u
o Dz
v =
of X
‡
ˆ 0;
‡
oy ;
ot kˆ1 : ox

…3†

where f is water elevation above (or below) datum.

Fig. 1. Sketch of layers and the relative variable locations in the
vertical plane.

Y. Wu, R.A. Falconer / Advances in Water Resources 23 (2000) 531±543

(ii) Momentum equations for layer k
"
#

oqx 
o
uqx o
vqx
‡
‡
ot k
ox
oy

o/ o

o
o
‡ …u/† ‡ …v/† ‡ ‰…w ÿ ws †/Š
ot ox
oy
oz






o
o/
o
o/
o
o/
Dx
Dy
Dz
ÿ
ÿ
ˆ S;
ÿ
ox
ox
oy
oy
oz
oz

k




of 
ˆ fqy  ÿ gDz 
ox
k
‡

(

#

"

o
o
u o
u
o
o
u o
v
eh Dz
‡
‡ eh Dz
‡
ox
ox ox
oy
oy ox

 
‡ w
u

k‡1=2

ÿ

k

"

( "
ev

 
ÿ w
u

o
u ow
‡
oz ox

kÿ1=2

#)

‡

( "
ev

o
u ow
‡
oz ox

#)

#)

k

kÿ1=2

…4†

;
k‡1=2

"
#

oqy 
o
uqy o
vqy
‡
‡
ot k
ox
oy
k



of 
ˆ ÿfqx  ÿ Dzg 
oy k
k
(
"
#
"
#)
o
o
v o
u
o
o
v o
v
‡
eh Dz
‡
‡ eh Dz
‡
ox
ox oy
oy
oy oy
k
( "
#)
 
 
o
v ow
‡
‡ ev
ÿ w
v
‡ w
v
kÿ1=2
k‡1=2
oz oy
ÿ

ev

o
v ow
‡
oz oy

#)

;
k‡1=2

…5†

where q is the ¯uid density; f the Coriolis parameter;
qx ˆ uDz and qy ˆ vDz are layer-integrated velocities per
unit width in x and y directions, respectively; g the acceleration due to gravity; eh the horizontal eddy viscosity, which is assumed to be constant in the vertical
and equated to the depth-integrated eddy viscosity [14]
in the current study; and ev is the vertical eddy viscosity,
which is represented by a two-layer mixing length model
[15].
At the free surface (where k ˆ 1), the terms …w
u†kÿ1=2
and …w
v†kÿ1=2 can be eliminated using the kinematic free
surface condition
and Leibnitz rule, and the shear

stresses sxz kÿ1=2 and syz kÿ1=2 are equated to the wind
v†k‡1=2
stresses. At the bed, the terms …w
u†k‡1=2 and …w
become zero due to the no-slip boundary
condition,
and

the shear stresses sxz k‡1=2 and syz k‡1=2 are equated to the
bed shear stresses [13].

…6†

where / is the sediment concentration, salinity or other
solute constituent concentration; u; v and w are the ¯uid
velocity components in the x, y and z directions respectively; ws the apparent sediment settling velocity
which vanishes for salinity and other solute constituents;
Dx , Dy and Dz are the turbulent di€usion coecients in
the x, y and z directions, respectively; and S is the source
or sink term.
As a result of ¯oc aggregation due to inter-particle
collision and the surface electro-chemical forces, cohesive sediments settle by ¯ocs rather than by individual
particles. It was found that the settling velocity of the
¯ocs depended strongly on the suspended cohesive sediment concentration [16]. The dependence of the settling
velocity on the local concentration generally falls within
one of the three following ranges.
(i) Free settling (/ < /1 ˆ 0:1 ÿ 0:3 g/l)
ws ˆ

kÿ1=2

( "

533

…s ÿ 1†gD2s
18m

…7†

where m is the kinematic viscosity for clear water, s the
speci®c density of suspended sediment, and Ds is the ¯oc
diameter.
(ii) Flocculation settling (/1 < / < /2 ˆ 0:3 ÿ 10 g/l)
ws ˆ k1 /4=3

…8†

where k1 is an empirical coecient.
(iii) Hindered settling (/ > /2 )
ws ˆ ws0 ‰1 ÿ k2 …/ ÿ /2 †Š4:66

…9†

where ws0 is the value of ws at the concentration /2 , and
k2 is the inverse of the concentration at which ws ˆ 0.
In estuarine and coastal waters, the horizontal length
scale is generally much larger than the vertical one, thus
an operator splitting algorithm is used to split the threedimensional advective±di€usion equation (6) into a
horizontal two-dimensional equation and a vertical onedimensional equation.
The one-dimensional advective±di€usion equation is
gives as


o/ o
o
o/
‡ ‰…w ÿ ws †/Š ÿ
Dz
ˆ0
…10†
ot oz
oz
oz
with the following boundary conditions for sediment.

2.2. Solute transport equation

At the free surface

The partial di€erential equation describing the
advective±di€usion in three dimensions can be written as

…w ÿ ws †/ ÿ Dz

o/
ˆ 0:
oz

…11†

534

Y. Wu, R.A. Falconer / Advances in Water Resources 23 (2000) 531±543

At the bed
ÿws / ÿ Dz

o/
ˆ qdep
oz

when sb 6 sd

…deposition†;
…12a†

ÿws / ÿ Dz

o/
ˆ qero
oz

when sb P se

…erosion†;
…12b†

ÿws / ÿ Dz

o/
ˆ0
oz

when sd < sb < se …equilibrium†;
…12c†

where sb is the bed shear stress, sd the critical shear stress
for deposition, se the critical shear stress for erosion, qdep
and qero represent deposition and erosion rates, respectively at the bed.
The deposition rate proposed by Krone [17] was used
in this study given as


sb
;
…13†
qdep ˆ ÿws /b 1 ÿ
sd
where /b is the near-bed cohesive sediment concentration, with typical values of the critical shear stress sb for
deposition being 0.04±0.15 N=m2 [17,18]. Likewise, the
erosion rate for soft natural mud can be represented by
the following empirical expression [19]:
h
i
qero
ˆ exp a…sb ÿ se †1=2 ;
…14†
qf
where a is an empirical coecient, qf is ¯oc erosion rate
when sb ÿ se ˆ 0. Typical values of the critical shear
stress se for the erosion of soft mud are 0.07±0.17 N=m2 .
For salinity or other solute constituents, the above
boundary conditions become
w/ ÿ Dz

o/
ˆ 0 at the free surface;
oz

…15†

o/
ˆ 0 at the bed:
…16†
oz
The horizontal two-dimensional advective±di€usion
equation can be written as


o/ o
o
o
o/
‡ …u/† ‡ …v/† ÿ
Dx
ot ox
oy
ox
ox


o
o/
Dy
ÿ
ˆ S:
…17†
oy
oy
At the open boundaries, the concentration data from
®eld measurements were used for in¯ow conditions, with
the concentration pro®les being obtained by extrapolation using a ®rst-order upwind di€erence scheme for
out¯ow conditions. At the bank boundaries, the normal
derivatives of the concentration were set to zero.
To be consistent with the layer-integrated hydrodynamic model, the above horizontal two-dimensional
advective±di€usion equation was integrated over the

layers to give the following two-dimensional layer-integrated equation

 o 
 o
o/Dz o 
o/
‡
DzDx
u/Dz ‡
v/Dz ÿ
ot
ox
oy
ox
ox


o
o/
DzDy
ˆ DzS:
…18†
ÿ
oy
oy
In applying the ULTIMATE algorithm, a criteria
was used to check that the solute concentration /
maintained monotonicity. In coastal and estuarine ¯ows
the water depth H may vary rapidly, with the result that
the thickness of the top and bottom layers may vary
suddenly, thus the monotonicity of /Dz may be di€erent
from the monotonicity of the solute concentration /.
Therefore the advective±di€usion equation (18) needed
to be rearranged. Partially di€erentiating the ®rst three
terms in the left-hand side of Eq. (18), and using the
continuity equations (1) and (3), we obtain the following
conservative form of the layer-integrated equation:


o/ o   o   1 o
o/
‡
DzDx
u/ ‡
v/ ÿ
ot ox
oy
Dz ox
ox


1 o
o/
…19†
DzDy
ˆ S ‡ Sa ;
ÿ
Dz oy
oy
where
8 

/
o
u
o
v
>
/
‡
wk‡1=2
ÿ Dz
>
>
ox
oy
>
>
>
>
> for the top layer;
>
>
>
>


<  ou ov  / ÿ
/ ox ‡ oy ‡ Dz wkÿ1=2 ÿ wk‡1=2
Sa ˆ
>
for the middle layers;
>
>
>
>
>


>
>
> / ou ‡ ov ‡ / wkÿ1=2
>
>
Dz
ox
oy
>
:
for the bottom layer:

…20†

Comparing with the original equation (17), an additional source term Sa now appears in the layer-integrated equation. It will be shown in the next section that
this additional source term is vital for the mass conservation of pollutant.

3. Numerical methods
The governing hydrodynamic equations were solved
using a combined explicit and implicit ®nite di€erence
scheme. Firstly, an alternating direction implicit scheme,
based on FalconerÕs model [20] was used to solve the
depth-integrated hydrodynamic equations to give the
water elevation ®eld. The layer-integrated equations (1)
and (4) were then solved to obtain the layer-integrated
velocities, using the water elevation ®eld predicted by
the depth-integrated equations. In solving the depthintegrated hydrodynamic equations, the layer-integrated
velocities were integrated to obtain the depth mean

535

Y. Wu, R.A. Falconer / Advances in Water Resources 23 (2000) 531±543

velocity. The Crank±Nicolson scheme was used to solve
the layer-integrated hydrodynamic equations, with the
vertical di€usion terms being treated implicitly and the
remaining terms being treated explicitly. Two iterations
were performed to solve the couple problem between the
depth-integrated and layer-integrated equations. The
¯ooding and drying processes were modeled using a
robust scheme developed by Falconer and Chen [21].
For details of the hydrodynamic model, see Lin and
Falconer [13].
For the solute transport processes, ®rst the layer-integrated two-dimensional advective±di€usion equation
(19) was solved horizontally, and then the one-dimensional vertical advective±di€usion equation (10) was
solved vertically.
Firstly, the two-dimensional layer-integrated equation (19) was discretized in time space by integrating
equation (19) from tn to tn ‡ Dt
Z tn ‡Dt
o/
dt ˆ /n‡1
ÿ /nP
P
ot
tn

Z tn ‡Dt   
o
o 
u/ ‡
v/ dt
ˆ
ox
oy
tn

Z tn ‡Dt  
1
o
o/
DzDx
dt
‡
Dz tn
ox
ox

Z tn ‡Dt  
1
o
o/
DzDy
dt
‡
Dz tn
oy
oy
Z tn ‡Dt
…S ‡ Sa † dt
‡
tn

ˆ

tn ‡Dt




o  o 
u/ ‡
v/ dt
ox
oy
tn
n 
 
Dt o
o/
DzDx
‡
Dz ox
ox
 
n 
Dt o
o/
DzDy
‡ DtS n
‡
Dz oy
oy
Z

‡ DtSan ;

…21†

where the explicit of forward Euler method were used
for di€usion and source terms.
Integrating Eq. (21) over the ®nite volume shown in
Fig. 2, we have the following ®nite di€erence equation:

 

/Pn‡1 ˆ /nP ÿ cxe /e ÿ cxw /w ÿ cyn /n ÿ cys /s

n 
n 
Dt
o/
o/
DzDx
ÿ DzDx
‡
DxDz
ox e
ox w
n 
n 

Dt
o/
o/
DzDy
ÿ DzDy
‡
DyDz
oy n
oy s
n
n


…22†
‡ Dt S ‡ DtS ;

Fig. 2. Sketch of the control voluem for the two-dimensional
horizontal layer-integrated transport equation.

cxe ˆ

ue Dt
;
Dx

cxw ˆ

uw Dt
;
Dx

cyn ˆ

vn Dt
;
Dy

cys ˆ

vs Dt
Dy

are Courant numbers at east, west, north and south cell
faces; Sn represents the average of S n over the control
volume, and San represents the average of San over the
control volume. The superscript n is an index for time,
the subscript n denotes the northern cell face.
The gradient terms at cell faces in above ®nite difference equation are approximated by central di€erencing and the average cell face values /ne , /nw , /nn and /ns
are estimated using the two-dimensional QUICKEST
scheme, extended from the one-dimensional QUICKEST scheme of Leonard [4].
In a similar manner to the one-dimensional case
treated by Leonard [4], consideration of the following
exact integral formulation for two-dimensional purely
advective ¯ows
Z Dx=2 Z Dy=2
Z Dx=2 Z Dy=2
n‡1
/ dx dy ÿ
/n dx dy
ÿDx=2

ˆ

Z

ÿDy=2

Dy=2

ÿDy=2

‡

Z

Z

Dx=2

ÿDx=2

ÿDx=2

Dt

Uw /w ds dy ÿ

0

Z

Dt
0

Vs /s ds dx ÿ

ÿDy=2

Dy=2

Z

ÿDy=2

Z

Dx=2

ÿDx=2

Dt

Z

Ue /e ds dy

0

Z

Dt

Vn /n ds dx:
0

…23†

For pure advection of a scalar / at a constant vector
velocity …U ; V †, the exact transient solution over a time
interval s can be shown to be

a

where / represents the respective average cell face values
over the time increment Dt

/…x; y; s† ˆ /…x ÿ U s; y ÿ V s; 0†
ˆ /n …x ÿ U s; y ÿ V s†:

…24†

536

Y. Wu, R.A. Falconer / Advances in Water Resources 23 (2000) 531±543

Using the above concept and a Taylor series expansion, the ®rst integral on the right-hand side of Eq. (23)
can be approximated as
Z Dy=2 Z Dt
Uw /w ds dy
ÿDy=2

0

ˆ Uw

Z

 Uw

Z

Dy=2

ÿDy=2
Dy=2
ÿDy=2

Z

Dt

…25†

…26†

Dt

Vs /s ds dx
"
cxs
o/n
ˆ DxDycys /ns ÿ Dx s
2
ox

ÿDx=2

0

cys
o/n
1
o2 /n c2
o2 /n
Dy s ‡
Dx2 2s ‡ xs Dx2 2s
oy
ox
ox
2
6
24
#
2
n
n
2
2
cys
o/
cxs cys
o /s
DxDy
‡ Dy 2 2s ‡
;
oy
3
oxoy
6

cyn
o/n
1
o2 /n c2
o2 /n
Dy n ‡
Dx2 2n ‡ xn Dx2 2n
2
oy
ox
ox
6
24
#
2
n
n
2
2
cyn 2 o /n cxn cyn
o /n
Dy
DxDy
‡
‡
:
2
6
oy
3
oxoy

…28†

ÿDy=2

ÿDx=2

ÿDy=2



Dx o2 /Pn‡1 o2 /nP
ÿ n‡1
n
ˆ DxDy /P ÿ /P ‡
ÿ
ox2
ox2
24
 2 n‡1

2
2 n
Dy o /P
o /P
ÿ
‡
2
oy
oy 2
24



cxe 2 o2 /ne cxw 2 o2 /nw
Dx
Dx
ÿ
 DxDy /Pn‡1 ÿ /nP † ÿ
ox2
ox2
24
24


cyn 2 o2 /nn cys 2 o2 /ns
Dx
ÿ Dx
ÿ
24
24
ox2
ox2


2 n
cxe 2 o /e cxw 2 o2 /nws
Dy
Dy
ÿ
ÿ
24
24
oy 2
oy 2


n
cyn 2 o2 /n cys 2 o2 /ns
Dy
Dy
ÿ
:
…29†
ÿ
oy 2
oy 2
24
24
2



Substituting Eqs. (25)±(29) into Eq. (23) and using
the QUICK formulae for /nw , /ne , /ns and /nn , we can get
the average cell face values /w , /e , /s and /n over the
time interval Dt. Here only the average value at west cell
face is given for convenience as
"
1 n
cxw o/nw cyw
o/n

Dx
Dy w
ÿ
…/W ‡ /nP † ÿ
/w ˆ
2
2
ox
oy
2
c2xw ÿ 1 2 o2 /nw c2yw 2 o2 /nw
Dy
Dx
‡
ox2
oy 2
6
6
#
cxw cyw
o2 /nw
DxDy
‡
:
oxoy
3

‡

ÿ

Z

0



Similarly, the remaining integrals on the right-hand
side of Eq. (23) can be respectively approximated as
Z Dy=2 Z Dt
Ue /e ds dy
ÿDy=2 0
"
cxe o/n
ˆ DxDycxe /ne ÿ Dx e
2
ox

Dx=2

Dt

Vn /n ds dx
"
cxn o/nn
Dx
ˆ DxDycyn /nn ÿ
2
ox

ÿDx=2

ÿDx=2

‡

Z

Z

The integrals on the left-hand side of Eq. (23) can be
similarly approximated by
Z Dx=2 Z Dy=2
Z Dx=2 Z Dy=2
/n‡1 dx dy ÿ
/n dx dy

…Uw s†2 o2 /nw …y ÿ Vw s†2 o2 /nw
‡
ox2
oy 2
2
2
#
o2 /nw
‡ Uw …y ÿ Vw s†s
ds dy
oxoy

Uw Dt2 o/nw Vw Dt2 o/nw
ÿ
ˆ DyUw Dt/nw ÿ
ox
oy
2
2
Uw2 Dt3 o2 /nw Dy 2 Dt o2 /nw Vw2 Dt3 o2 /nw
‡
‡
‡
ox2
oy 2
6
6
24 oy 2

n
Uw Vw Dt3 o2 /w
‡
3
oxoy
"
cxw
o/n cyw
o/n
Dx w ÿ
Dy w
ˆ DxDycxw /nw ÿ
2
ox
2
oy
‡

cye
o/n c2
o2 / n
1
o2 /n
Dy e ‡ xe Dx2 2e ‡
Dy 2 2e
2
oy
6
ox
oy
24
#
2
n
n
2
2
cye
o/
cxe cye
o /e
DxDy
‡ Dy 2 2e ‡
;
6
oy
oxoy
3

Dx=2

ÿ

/nw …ÿUw s; y ÿ Vw s† ds dy
0
Z Dt "
o/n
o/n
/nw ÿ Uw s w ‡ …y ÿ vw s† w
ox
oy
0

2 n
c2yw 2 o2 /nw
c2xw 2 o2 /nw
1
2 o /w
Dy
Dx
Dy
‡
‡
6
ox2
oy 2
oy 2
6
24
#
cxw cyw
o2 /nw
DxDy
:
‡
3
oxoy

Z

ÿ

…27†

…30†

Even though the QUICKEST scheme is third-order with
upwind bias, it still produces some unphysical overshoot
and undershoot when sharp changes of concentration
exist. Thus, the universal limiter designed for one-dimensional problems by Leonard [10] was used at each
control volume face to eliminate unphysical overshoot
and undershoot.
However, when using LeonardÕs universal limiter
directly at each control volume face for a two-dimensional solute transport problem, a variation in the
concentration distribution occurs when discontinuities

Y. Wu, R.A. Falconer / Advances in Water Resources 23 (2000) 531±543

exist. This variation results from the fact that the node
values in the normal direction only are used to limit the
control volume face values, which estimated using the
information in both the normal and lateral directions.
Wu and Falconer [12] divided the estimated face value
into three parts. Taking time averaged face value /w as
an example
1
1
o2 /nw cxw
o/nw
Dx
ÿ
/w ˆ …/W ‡ /P † ÿ Dx2
2
ox2
ox
2
8


n
2 n
1 2
1
o
/
c
o/
yw
w
w
cxw ÿ
Dy
Dx2
ÿ
‡
6
4
ox2
oy
2
‡

…31†

1
1
o2 /nw
uw0 ˆ …/W ‡ /P † ÿ Dx2
ox2
2
8


cxw
o/n 1 2
1
o2 /nw
Dx w ‡
cxw ÿ
uwt ˆ ÿ
;
Dx2
2
ox
ox2
6
4
cyw
o/n c2yw 2 o2 /nw cxw cyw
o2 /nw
Dy w ‡
DxDy
:
Dy
‡
2
oy
oy 2
3
oxoy
6

Note that the average value has been split into three
parts, i.e. the main part uw0 ; the time correction part,
contributed from the ¯ow normal to the cell face only
i.e. uwt ; and the time correction part, introduced by the
¯ow parallel to the cell face i.e. uwl :
By limiting the main part of the average cell face
value, and the time correction term brought in by the
normal ¯ow only, a modi®ed one-dimensional ULTIMATE algorithm for the two-dimensional problem has
been constructed to avoid the variation of local discontinuities. The modi®ed one-dimensional ULTIMATE algorithm can be expressed as
/w ˆ UL‰UL…uw0 † ‡ uwt Š ‡ uwl ;

…32†

where UL… † denotes LeonardÕs one-dimensional universal limiter, see [10] for details.
The one-dimensional advective±di€usion equation
(10) was discretized for a non-uniform grid spacing in
the vertical direction (see Fig. 1). Since the di€usion
term was very important for the vertical concentration
distribution and some of the gird sizes were very small
near the sea bed and the water surface, the backward or
implicit Euler method was used to avoid very small time
steps. The power-law scheme [22] was adapted to calculate the total mass ¯ux. The resultant discretization
equation for the vertical one-dimensional advective±
di€usion equation (10) is therefore
n‡1
n‡1
‡ aP /n‡1
ÿ aN /k‡1
ˆ b;
ÿaS /kÿ1
k



ÿ
aS ˆ DTz A  PeT  ‡ ‰F T ; 0Š; aN


ÿ
Dz
ˆ DBz A  PeB  ‡ ‰ÿF B ; 0Š; aP ˆ aS ‡ aN ‡ ;
Dt
h
i
Dz n
/ ; A…j Pej† ˆ 0; …1 ÿ 0:1j Pej†5
bˆ
Dt k

4. Numerical tests

where

uwl ˆ ÿ

where

where the symbol ‰a; bŠ is used to denote the greater of a
and b, superscripts T and B denote the control volume
face, Pe the grid Peclet number, F is the mass ¯ow rate.

c2yw 2 o2 /nw cxw cyw
o2 /nw
‡
DxDy
Dy
2
oy
oxoy
3
6

ˆ uw0 ‡ uwt ‡ uwl ;

537

…33†

A series of numerical tests were carried out to verify
the performance of the present model. First the convection of a square wave in a uniform ¯ow was studied,
then the advective±di€usion of an initial square column
concentration distribution was investigated to show the
performance of the present model, and ®nally the model
was used to simulate the solute transport in an idealized
square harbor to illustrate the importance of the additional source term associated with the layer average of
free surface ¯ows.
4.1. Pure advection of a square wave and advective±
di€usion of a square column
Firstly, a square wave advecting in a one-dimensional
uniform stream was studied to test the performance of
the present model when discontinuities existed. The
problem had the following initial condition

1; x1 6 x 6 x2 ;
…34†
/…x; t ˆ 0† ˆ
0; 0 6 x < x1 ; or x2 < x 6 1;
where 0 6 x 6 1, x1 ˆ 0:05, x2 ˆ 0:15.
In the computation the advective velocity was unity,
the grid size was 0.01, the time step was 0.002, and the
total simulating time was 0.6. Fig. 3 shows the results
using the ULTIMATE QUICKEST scheme in the
present model, together with results for the ®rst-order
upwind, second-order upwind, Lax±Wendro€, MacCormark, and QUICKEST schemes.
The results in Fig. 3(a) show that the ®rst-order
upwind scheme introduced excessive di€usion. Not
surprisingly, the results of the second-order upwind,
Lax±Wendro€ and MacCormark schemes presented in
Fig. 3(b)±(d) show serious unphysical oscillations
around discontinuities due to the numerical dispersion
term. Even though the QUICKEST scheme exhibits less
numerical oscillations as shown in Fig. 3(e), it still
largely overestimates the peak concentration by about
10%. Fig. 3(f) shows the results computed using the
ULTIMATE QUICKEST scheme where, as expected,
the results given by ULTIMATE QUICKEST scheme

538

Y. Wu, R.A. Falconer / Advances in Water Resources 23 (2000) 531±543

show no oscillations and predict the discontinuities with
very high resolution.
Secondly, the advection and di€usion of a square
column concentration distribution in a uniform twodimensional ¯ow has been simulated. The computational domain, boundary and initial conditions of the
problem studied were described using the following
initial and boundary value partial di€erential equations
o/
o/
o/
o2 /
o2 /
‡u
‡v
ˆ mx 2 ‡ my 2 ;
ot
ox
oy
ox
oy

0 < x; y < 1;
…35a†

/…x; y; 0† ˆ



/…x; y; t† ˆ 0;

1;
0;

x1 6 x 6 x2 ; y1 6 y 6 y2 ;
other x; y;
x ˆ 0; 1; or y ˆ 0; 1

with the analytical solution being given as

…35b†
…35c†

/…x; y; z; t†
 



1
x2 ÿ x ‡ ut
ÿx1 ‡ x ÿ ut
p
p
ˆ
erf
‡ erf
4
2 mx t
2 mx t
 



y2 ÿ y ‡ vt
ÿy1 ‡ y ÿ vt
‡ erf
; …36†
 erf
p
p
2 my t
2 my t

where erf… † is the error function.
In this test, the advective velocity components u and v
were set to unity, the di€usion coecients mx and my were
both equated to 0.0005, a mesh of 100  100 with grid
sizes Dx and Dy of 0.01 was used, and the time step Dt
was 0.003, with a total simulating time t of 0.6. Fig. 4
shows the above exact solution and the model results. A
comparison of the numerical results and the exact solution at the section y ˆ 0:7 are given in Fig. 5. The
results illustrate the excellent performance of the ULTIMATE QUICKEST scheme as used in present model.

Fig. 3. Pure advection of square wave: (a) ®rst-order upwind; (b) second-order upwind; (c) Lax±Wendro€; (d) MacCormark (e) QUICKEST; (f)
ULTIMATE QUICKEST.

Y. Wu, R.A. Falconer / Advances in Water Resources 23 (2000) 531±543

539

Fig. 6. Sketch of the idealized square harbor.

Fig. 4. Exact solution and numerical results of advective±di€usion of a
square column: (a) exact solution; (b) numberical results.

Fig. 5. Comparison of numerical results and exact sollution at section
y ˆ 0:7.

4.2. Solute transport in a square harbor
To investigate the contribution of the additional
source term introduced by the layer average of the free
surface ¯ow, the transport of a solute driven by tidal
currents in an idealized square harbor, was also
simulated. The harbor had plan-form dimensions of
2.5 km2.5 km, with a symmetric entrance of width
250 m, and an open sea area of 1.0 km2.5 km sited just
beyond the harbor entrance, see Fig. 6.
A horizontal bed with the mean water depth of 6 m
was assumed. The tidal ¯ow was generated by a sinusoidal water elevation variation at the open boundary,

with a period of 12.4 h and an amplitude of 1 m. The
initial concentrations inside and outside the harbor were
50 and 0 respectively. A mesh of 70  50 grid squares,
with a uniform grid size of 50 m was used. Five layers
were used in the vertical, with the thickness of the top
layer being 2 m at mean water level and with the other
layers being 1 m thick. The time step was set at 15 s. In
order to check the conservativeness of the model, all
computations were stopped after 8.0 h of simulation
since some solute was started to advect out of the
computational domain beyond this time.
Fig. 7 gives the contours of the ®rst layer solute
concentration distribution in harbor, calculated both
with and without the additional source term. It can be
seen that: the results obtained with the additional source
term, as given in Fig. 7(a), appear to be reasonable and
are within the initial range of 0±50, whilst the results
without the additional source term, as given in Fig. 7(b),
exceed the initial range with the maximum value being
larger than 60. Since there was no mass e‚ux from
within the computational domain during the simulation
period, then the total amount of solute mass within the
domain should remain unchanged up to this time. The
model with the additional source term was found to give
perfectly conservative results, whereas the model without the additional source term proved to be mass productive. Fig. 8 shows the percentage mass produce by
the model missing the additional source term as a
function of time, where it can be seen that the model
without the additional source term has led to a large
production of solute mass during the simulation time.

5. Model application to the Humber Estuary
The Humber Estuary is one of the main estuaries in
UK, situated along the coast of northern England, and
strategically important in terms of its input to the North
Sea and shipping links to mainland Europe. The main
part of the estuary is about 62 km long, from Trent Falls
to Spurn Head, and the tidal in¯uence extends for

540

Y. Wu, R.A. Falconer / Advances in Water Resources 23 (2000) 531±543

Fig. 7. Contours of solute concentration distribution in the harbor: (a) with the additonal source term; (b) without the additional source term.

Fig. 8. Percentage mass produced by the model missing the aditonal
source term.

another 62 km up the River Ouse and about 72 km up the
River Trent. A six-year project LOIS (Land-Ocean Interaction Study) was set up in 1992. It aims to quantify
and simulate the ¯uxes and transformation of solute
(including: sediments, nutrients, and contaminants) into
and out of the coastal zone. The main study area is the
UK East Coast from Berwick upon Tweed to Great
Yarmouth, concentrating on the Humber and its catchment, and to a lesser extent the River Tweed. As part of
this study within the LOIS project, the present model has
been set up and applied to simulate the salt and cohesive
sediment transport in the Humber Estuary from Trent
Falls to just beyond Spurn Head as shown in Fig. 9.
Field measured data of water elevations, velocities,
salinity and suspended sediment concentrations were
collected by Plymouth Marine Laboratory during the
LOIS project at ®ve locations, including: Station SG13
(53° 35.190 N, 0° 13.900 E), SG24 (53° 32.060 N, 0°
15.290 E), SG10 (53° 36.820 N, 0° 11.350 E), which were
positioned in Humber plume; Station SG23 (53°
35.710 N, 0° 2.170 E) was positioned within the estuary at
Hawke; and Station at the Trent Falls region of the
upper Humber (53° 42.80 N, 0° 37.80 W).

The model area was represented horizontally using a
mesh of 118  56 uniform grid squares, with a length of
500 m. Vertically, eight layers were used with the
thickness of the top layer being 4 m at mean water level,
and with the other layers each being 3 m thick. The
streamline boundary was located such that the boundary was as parallel as possible to the dividing streamline
separating the north±south alongshore current and the
estuarine in¯ow±out¯ow. The water elevation recorded
at Station G13 was chosen as the seaward boundary
condition to drive the tidal current. Due to the lack of
data regarding the ¯uid volume ¯ux data at Trent falls,
the water elevation at Trent falls was used as the landward boundary condition. A bed roughness of 10 mm
was used, as suggested by model calibration for a previous study [13]. Since measurements taken at the survey
sites mentioned above were not simultaneous, i.e. the
measurements at Station G13 and Hawke were taken for
neap tide, whereas conditions at Trent Falls were taken
during the mid-tide, the water elevations at Trent Falls
therefore had to be adjusted to neap tide conditions.
Calibration was carried out during the adjustment according to the water elevations and mean velocities
measured at Hawke. A comparison of the velocities for
the di€erent layers is given in Fig. 10. As can be seen a
reasonable level of agreement was obtained between the
model predicted and ®eld measured results.
Since the Humber Estuary is generally a well-mixed
estuary with salinity varying by typically less than 5 ppt
over the depth, only one layer was used to compute the
salinity distribution within the estuary, which was
therefore equivalent to using a two-dimensional depthintegrated model. For the sediment transport predictions, as mentioned previously eight layers were used to
predict the vertical distribution of cohesive sediments.
The comparison of the ®eld measured and model predicted salinity at Hawke is shown in Fig. 11, with good

Y. Wu, R.A. Falconer / Advances in Water Resources 23 (2000) 531±543

541

Fig. 9. Map of the Humber estuary and the locations of ®eld observation stations.

Fig. 10. Comparison of predicted and ®eld measured velocities of
Hawke, taken on 8 June 1995.

critical shear stress for erosion assumed to be:
se ˆ 0:15 N=m2 . These values were in the ranges proposed respectively by Krone [17] and Thorn and Parsons
[23], and were re®ned by trial and error. The ¯oc erosion
parameters of a ˆ 8:3 and qf ˆ 0:0000042 were used as
suggested by Thorn and Parsons [23]. Fig. 12 shows a
comparison of the ®eld measured data and the model
predicted suspended cohesive sediment concentration
variation with time. As the concentration variation with
depth from the model predictions was small, only the
depth mean concentration was plotted in Fig. 12. The
results show good agreement between the model predictions and the ®eld measured data in the regime of
deposition. However in the regime of erosion, the ®eld
data show a large variation with depth, which cannot be
represented in the model predictions. This was thought
to be due to the exclusion of non-cohesive sediment
transport in this study, wherein a large amount of noncohesive sediment may be suspended in the water column during the erosion period, whilst due to its larger
settling speed than the cohesive sediments, most of the

Fig. 11. Comparison of predicted and ®eld measured salinity at
Hawke, taken on 8 June 1995.

agreement being obtained between the predicted and
measured data.
Due to a lack of information on the cohesive sediment size measured at Hawke Anchorage, the sediment
(¯oc) size of 20 lm was assumed, with this being a size at
which aggregation becomes negligible. The critical shear
stress for deposition was set to: sd ˆ 0:07 N=m2 , and the

Fig. 12. Comparison of predicted and ®eld measured cohesive sediment concentratons at Hawke, taken on 8 June 1995.

542

Y. Wu, R.A. Falconer / Advances in Water Resources 23 (2000) 531±543

non-cohesive sediments may settle out. The mixture of
cohesive and non-cohesive sediments should therefore
be studied further in the future.

UK. The authors are grateful to NERC for supporting this study and to Dr. Uncles for the provision of
data.

6. Conclusion

References

A three-dimensional layer-integrated numerical
model has been re®ned to simulate salinity and cohesive
sediment transport ¯uxes in estuarine and coastal waters. A combined layer-integrated and depth-integrated
scheme was used to solve the hydrodynamic equations.
The original three-dimensional advective±di€usion
equation was split into a two-dimensional horizontal
and a one-dimensional vertical equation to implement
the variation between the horizontal and vertical scales.
To be consistent with the three-dimensional layer-integrated hydrodynamic model, the two-dimensional horizontal advective±di€usion equation was integrated over
the layers to give the layer-integrated advective±di€usion equation. An additional source term associated
with the layer average of free surface ¯ows was introduced into the conservative form of the layer-integrated
equation. It has been shown by numerical tests that the
additional source term is vital for mass conservation of
pollutant. A two-dimensional modi®ed QUICKEST
scheme has been obtained and used in this study to
achieve high accuracy. To avoid the numerical oscillations when a large concentration gradient exists, a
modi®ed one-dimensional universal limiter ULTIMATE has also been used. Two test cases having exact
analytical solutions have been used to test the mode,
with the results illustrating that the model gave accurate
results.
Finally, the model was applied to predict the salt and
cohesive sediment distribution in the Humber Estuary,
UK, where good agreement has generally been obtained
between the predicted and measured results. The model
was found to predict non-cohesive suspended sediment
accurately in a previous study and to predict cohesive
sediment concentration levels accurately during the deposition regime. However, the predictions were not so
accurate during the erosion regime and further laboratory and ®eld studies are required to attain a better
understanding of the governing rheological processes
involved.

[1] Lax PD, Wendro€ B. Systems of conservation laws. Commun
Pure Appl Math 1960;13:217±37.
[2] MacCormark RW. The e€ect of viscosity in hypervelocity impact
cratering. AIAA Paper 69-354, Ohio: Cinicinnati, 1969.
[3] Warming RF, Beam RM. Upwind second-order di€erence
schemes and applications in unsteady aerodynamic ¯ows. AIAA
J 1976;24:1241±9.
[4] Leonard BP. A stable and accurate convective modeling procedure based on quadratic upstream interpolation. Comput Meth
Appl Mech Eng 1979;19:59±98.
[5] Harten A. High-resolution schemes for hyperbolic conservation
laws. J Compt Phys 1983;49:357±85.
[6] Chakravarthy SR, Osher S. High-resolution application of the
Osher upwind scheme for the Euler equations. AIAA Paper 831943, MA: Danvers, 1983.
[7] Sweby PK. High resolution schemes using ¯ux limiter for
hyperbolic conservation laws. SIAM J Num Anal 1984;21:995±
1011.
[8] Roe PL. Some contributions to the modeling of discontinuous
¯ows. In: Proceedings of the 1983 AMS-SIAM Summer Seminar
on Large Scale Computing in Fluid Mechanics; Lect Appl Math
1985;22:163±93.
[9] Hirsch C. Numerical computation of internal and external ¯ows,
vol. 2, computational methods for inviscid and viscous ¯ows.
[10] Leonard BP. The ULTIMATE conservative di€erence scheme
applied to unsteady one-dimensional advection. Comp Meth Appl
Mech Eng 1991;88:17±74.
[11] Cahyono M. Three-dimensional numerical modeling of sediment
transport processes in non-strati®ed estuarine and coastal waters.
PhD thesis, Department of Civil Engineering, University of
Bradford, UK, 1993.
[12] Wu Y, Falconer RA. Two-dimensional ULTIMATE QUICKEST
scheme for pollutant transport in free-surface ¯ows. In: Third
International Conference on Hydroscience and Engineering,
Cottbus/Berlin, Germany, 1998.
[13] Lin B, Falconer RA. Three-dimensional layer-integrated modeling
of estuarine ¯ows with ¯ooding and drying. Estuarine, Coast.
Shelf Sci. 1997;44:737±51.
[14] Falconer RA. An introduction to nearly horizontal ¯ows. In:
Abbott MB, Price WA, editors. Coastal estuarial and harbor
engineersÕ reference book. London: E & FN Spon, 1993, p. 27±36.
[15] Rodi W. Turbulence models and their application in hydraulics.
2nd ed. Delft: IAHR Publication, 1984.
[16] Mehta A.J. Hydraulic behavior of ®ne sediment. In: Abbott MB,
Price WA, editors. Coastal, Estuarial and Harbor EngineersÕ
Reference Book. London: E & FN Spon, 1993:577±84.
[17] Krone RB. Flume studies of the transport of sediment in estuarial
processes. Final Report, Hydraulic Engineering Laboratory and
Sanitary Engineering Research Laboratory, University of California Berkeley, 1962.
[18] Mehta AJ. Deposition behavior of cohesive sediment. PhD thesis,
University of Florida, Gainesville, 1973.
[19] Raudkivi AJ. Loose boundary hydraulics. 3rd ed. Oxford:
Pergamon Press, 1990.
[20] Falconer RA. A two-dimensional mathematical model study of
the nitrate levels in an inland natural basin. In: Proceedings of the
International Conference on Water Quality Modeling in the
Inland Natural Environment, BHRA Fluid Engineering, Paper
J1. Bournemouth, UK, 1986, p. 325±44.

Acknowledgements
This is a LOIS publication number 357 of the LOIS
Community Research Programme, carried out under a
Special Topic Award from the Natural Environment
Research Council. The data collected for the
Humber Estuary model application were provided by
Dr. R.J. Uncles of the Plymouth Marine Laboratory,

Y. Wu, R.A. Falconer / Advances in Water Resources 23 (2000) 531±543
[21] Falconer RA, Chen Y. An improved representation of ¯ooding
and drying and wind stress e€ects in a two-dimensional numerical
model. In: Proceedings of the Institution of Civil Engineers, Part
2, Research and Theory. 1991;91:659±87.
[22] Patanker SV. In: Minkowycz WJ. et al., editor. Elliptic systems:
®nite di€erence method I, handbook of numerical heat transfer.
New York: Wiley, 1988.

543

[23] Thorn MFC, Parsons JG. Properties of Grangemouth mud. Rep.
No. EX 781, Hydraulic Research Station, Wallingford, UK,
1977.