Post on 03-Feb-2023
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Comparison Between TVD-MacCormack and ADI-Type Solvers
of the Shallow Water Equations Simple
Dr. Dongfang Liang
Tel: +44(0)29 2087 6557; Email: liangd@cf.ac.uk
Prof. Roger A. Falconer
Tel: +44 (0)29 2087 4280; Email: FalconerRA@cf.ac.uk
Dr. Binliang Lin
Tel: +44(0)29 2087 4696; Email: linbl@cf.ac.uk
Hydroenvironmental Research Centre, Cardiff School of Engineering,
Cardiff University, The Parade, CARDIFF, CF24 3AA, U.K.
Abstract
A simple TVD (Total Variation Diminishing) step is appended to the standard
MacCormack scheme for simulating shallow water dynamics. The governing equations
are solved on a uniform Cartesian grid system. Two forms of the conservative SWEs
(Shallow Water Equations) were compared for inclusion in the computations. The
comparisons showed that the numerical solutions gave more accurate predictions when
the water level was chosen as one unknown parameter, rather than the water depth. The
TVD-MacCormack model was then compared with a widely used numerical model
which employs the traditional ADI (Alternating Direction Implicit) method to predict
shallow water flows. This traditional model was unable to predict trans-critical flows
under normal conditions and artificial viscosity had to be introduced to remove the
spurious oscillations occurring near sharp gradients, while the TVD-MacCormack
Corresponding author
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model was able to recognize and reproduce all flow regimes accurately with no
adjustable parameters. Finally, a dyke-break experiment was used to verify the TVD-
MacCormack model. The computed results compared favourably with the experimental
results.
Key words: Shallow water, Dam break, Flood routing, River modelling, TVD scheme,
MacCormack scheme, Shock capturing
1. Introduction
Using the assumption of the hydrostatic pressure distribution and the kinematic
boundary condition of the free surface, the 3-D Reynolds-averaged continuity and
Navier-Stokes equations can be integrated over the water column. The resulting depth-
integrated equations are called the SWEs, which are broadly used to describe coastal,
estuarine and inland water flows. In recent years there have been a number of reviews
on the mathematical characteristics of and the numerical methods used for solving the
SWEs, e.g. Tan (1992) and Vreugdenhil (1994).
Although these equations have been derived for incompressible flows, the SWEs are
mathematically similar to the Euler equations for compressible flows if the viscosity
terms are disregarded. Since this set of differential equations is hyperbolic in structure,
then discontinuities are admitted through the weak form of these equations. These
discontinuities can take the form of hydraulic jumps and flood waves that are analogous
to shock waves in aerodynamics. The conservation of energy does not hold across these
discontinuities, where the flow conditions change abruptly. However, the physical
principle of the conservation of mass and momentum remains valid. It has been
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demonstrated by Lax and Wendroff (1960) that the finite difference method, when
satisfying certain conditions, has the ability to produce the correct weak solutions to
these equations and thereby capture these discontinuities.
Traditionally the ADI method has been used extensively to solve the SWEs, due to its
large stability domain and attractive balance between computational cost and accuracy
(e.g. Leendertse and Gritton 1971, Falconer 1980, Stelling et al. 1986). However,
Meselhe and Holly (1997) pointed out the invalidity of the Preissmann scheme for
solving the 1-D SWEs for trans-critical flows. The ADI method for 2-D SWEs is
somewhat based on the same principle as that of the Preissmann method for 1-D SWEs.
Hence, it is generally accepted that the ADI method is not suitable for calculating trans-
critical flows, with the authors’ experience confirming this finding. The switch from
one flow regime to the other can readily lead to the collapse of the numerical simulation.
This switch can be caused by either the physical nature of the flow or the temporary
numerical deviation from the convergence solution.
The existence of bore waves occurs for flows such as embankment breaks, storm
surges and flash floods. It is essential for numerical schemes to have the shock-
capturing capability in order to be used to model these types of flows. Two major
categories of shock-capturing methods have been regularly applied to shallow water
flows. The first category is the Godunov-type scheme, where a discontinuity in the
variables is assumed at the interface of every two grid cells and a Riemann solver is
used to calculate the variable flux across the interface (e.g. Mingham and Causon 1998,
Zoppou and Roberts 1999, Rogers et al. 2003, Liang et al. 2004). The second category
consists of the algebraic combination of the first-order and second-order upwind
schemes, so as to maintain the high accuracy and satisfy the TVD criterion at the same
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time. Based on this concept, the proportion of the contribution from each scheme is
adjusted within the solution such that the high-order scheme is used when the solution is
smooth and sub-critical, while a lower-order scheme is deployed at sharp gradients in
the solution where the flow is trans- or super-critical (e.g. Wang et al. 2000, Tseng and
Chu 2000, Vincent et al. 2000, Lin et al. 2003).
One reason that prevents shock-capturing methods from being widely used in
practical studies is their higher computational cost. Therefore a very simple and
efficient TVD-MacCormack algorithm has been deployed in this study for constructing
shock-capturing models. It has been found in the study herein that the choice of the
formulation of the SWEs is important for obtaining accurate numerical solutions of
these equations. Although various types of the combined TVD-MacCormack scheme
for shallow water flows have been reported in the literature (e.g. Tseng and Chu 2000,
Vincent et al. 2000, Mingham et al. 2001), the water depth and discharge (or velocity)
are usually chosen as the unknown variables in the previous studies. The results from
this study have shown that choosing the water level and the discharge as the unknown
variables can produce more stable and accurate solutions. To the authors’ knowledge,
most of the commercial software packages widely used for shallow-water flow
problems are still based on ADI-type methods, despite the growing emergence and
application of shock-capturing schemes in the fields of aerodynamic and hydrodynamic
model applications. This paper presents some comparisons between the TVD-
MacCormack model and a 2-D ADI-based model (namely DIVAST model) to show the
advantages, and sometimes necessity, in utilizing shock-capturing schemes for
numerical simulations of rapidly varying flows. The TVD-MacCormack model has also
been applied to a dyke-break problem, for which experimental data are available. The
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complex water surface variance has been reproduced by the present TVD-MacCormack
model and the computational results agree well with the experimental data.
2. Governing equations
Neglecting the Coriolis and wind forces, the SWEs may be written in the following
general form:
0
y
q
x
q
tyx (1)
yx
q
y
q
x
q
CH
qqgq
xgH
y
Hqq
x
Hq
t
q yxxyxxyxxx
2
2
2
2
2
22
222
2 (2)
yx
q
y
q
x
q
CH
qqgq
ygH
y
Hq
x
Hqq
t
qxyyyxyyyxy
2
2
2
2
2
22
222
2 (3)
where t is time; is the water surface elevation above datum; xq and yq are the
discharges per unit width in the x and y directions respectively; is the correction
factor for the non-uniform vertical velocity profile, which equates to 1.0 for a uniform
velocity distribution and 1.016 for a seventh power law velocity distribution; H (= h + η)
is the total water column depth, with h being the depth below datum; g is the
acceleration due to gravity; C is the Chezy bed roughness coefficient, which is
determined from the Manning formula in this study; and is the kinematic eddy
viscosity. It should be noted that Equations (2) and (3) are not in their conservative form
because the water depth H is outside the derivative in the surface-slope terms.
For the finite volume or finite difference method to preserve the correct movement of
the shock, it is usually necessary to deploy the conservative form of the governing
equations. In formulating these equations into their conservative form, the discrete
conservation of mass and momentum is then insured in the difference equations. In the
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current TVD-MacCormack model, the diffusion terms in Equations (2) and (3) have
been disregarded, based on the knowledge that the inertial, gravity and bottom friction
forces play a much more significant role in most practical riverine flow studies.
Equations (1)-(3) can then be rearranged into the following conservative form:
TSGFX
yxt (4)
where X, F, G, S, T are the column vectors. They can be expressed in the following
forms, where H, xq and yq are taken as unknowns:
)cb,5a,(
2
,2
,
22
22
gH
H
qH
q
H
gH
H
q
q
q
q
H
y
yx
y
yx
x
x
y
x
GFX
e)(5d,0
0
,
0
0
22
22
22
22
CH
qqgq
y
hgH
CH
qqgq
x
hgH
yxy
yxxTS
Equations (1) to (3) can also be transformed into Equation (4) in such a way that
(instead of H), xq and yq are taken as the independent functions, giving:
)cb,6a,(
2
,2
,
22
22
ghg
h
qh
q
h
ghg
h
q
q
q
q
y
yx
y
yx
x
x
y
x GFX
7
e)(6d,
)(
0
0
,
0
)(
0
22
22
22
22
Ch
qqgq
y
hg
Ch
qqgq
x
hg
yxy
yxx
TS
Most of the shock-capturing models reported in the literature choose the formulation
given by Equations (5a-e) in their solution strategy. This formulation can induce a
numerical imbalance problem associated with the treatment of the bed-slope term (e.g.
Garcia-Navarro and Vazquez-Cendon 2000, Zhou et al. 2001, Rogers et al. 2003, Tseng
2003, Tseng 2004), i.e. the quiescent flow state cannot be maintained under stationary
conditions for non-uniform bathymetries. One objective of the present study is to show
that this imbalance can be greatly alleviated by using Equations (6a-e) instead. The
reason for this imbalance alleviation has been explained by Rogers et al. (2003), in that
the latter formulation describes the balance of the forces that deviate away from the
equilibrium (still-water) state. In other words, the formulation in Equations (6a-e) can
be thought of as being derived by subtracting the still-water state equations from the
formulation in Equations (5a-e). The still-water state, i.e. (η, qx, qy)T = (0, 0, 0)T, is
satisfied explicitly in the later formulation.
3. Numerical method
Equations (1)-(3) are solved directly by using the ADI method in a 2-D generic
model. Details of the numerical schemes generally used in ADI models can be found in
Falconer (1986a) and Falconer (1986b). Only the numerical scheme for the TVD-
MacCormack model is outlined below, where Equation (4) is solved numerically.
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Using the operator-splitting technique, the solution to Equation (4) is obtained by
solving two 1-D problems in sequence giving:
SFX
xt, T
GX
yt (7a,b)
On a uniform rectangular grid system, the explicit discretization of Equations (7a,b) can
be written as:
njix
nji ,1
, L XX , njiy
nji ,1
, L XX (8a,b)
where Lx and Ly are the finite-difference operators, and the subscript and superscript of
X represent the spatial and temporal grid levels respectively. The finite difference
solution to Equation (4) can thus be approximated by:
njixyyx
nji ,2
, LLLL XX (9)
The TVD-MacCormack scheme is utilized to solve consecutively the two 1-D
hyperbolic equations in each time step. Taking Equation (7a) as an example, the
discretization scheme (or the finite-difference operator Lx) is given by:
txt nni
ni
ni
pi SFFXX 1
(10a)
txt ppi
pi
ni
ci SFFXX 1
(10b)
niii
niii
ci
pi
ni rrrr 2/112/11
1 )G()G()G()G(2
XXXXX (10c)
where the superscripts p and c denote the predictor and corrector steps respectively, x
and t are the spatial and time steps, G( ) is a function that will be discussed further
below and:
ni
ni
ni XXX 12/1
, ni
ni
ni 12/1 XXX (11a,b)
ni
ni
ni
ni
ir2/12/1
2/12/1
,
,
XX
XX , ni
ni
ni
ni
ir2/12/1
2/12/1
,
,
XX
XX (12a,b)
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The point brackets in the numerator and denominator of Equations (12a,b) denote the
scalar product of the two vectors within the point brackets. The function G( ) is defined
as:
)(15.0)G( xCx (13)
in which the flux limiter function is given as:
)1,2min(,0max)( xx (14)
and the variable C as:
5.0,25.0
5.0),1(
Cr
CrCrCrC (15)
with Cr being the local Courant number, defined as:
x
tgHHqCr x
(16)
It can be seen that the present TVD-MacCormack scheme modifies the standard
MacCormack scheme by appending an extra term after the corrector step. In this way,
the numerical oscillations are removed near the sharp-gradient regions and the
numerical solution is equipped with the TVD property. The present TVD scheme was
proposed by Davis (1984) and deployed in solving SWEs by Louaked and Hanich (1998)
and Mingham et al. (2001). Although Equation (10c) looks complicated, there is no
need to solve the eigenvalues of the equations. This suggests that the computational
cost is low in comparison with other TVD-MacCormack methods (e.g. Tseng and Chu
2000, Vincent et al. 2000, Tseng 2003 and 2004). At the limited expense of accuracy,
the simplicity and efficiency of the numerical method are both improved greatly. It has
also been noted that a number of researchers (e.g. Louaked and Hanich 1998, Wang et
al. 2000, Tseng 2004) use the single-step scheme, such as the explicit Euler or Lax-
Wendroff schemes, as the base solution scheme. The key benefit of using a two-stage
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scheme is that the source term can be easily treated to get second-order accuracy, both
in space and time, as can be seen in Equations (10a,b).
4. Comparison between two formulations of SWEs
The two formulations of the SWEs outlined in Equations (4-6) are the same
mathematically. However, they may behave differently with regard to the numerical
solution. In this section, several steady flows over uneven bed topographies are
simulated to compare the performance of the two formulations when implemented in
the TVD-MacCormack model. Based on the expressions in Equations (5) and (6), the
two corresponding TVD-MacCormack models are referred to as TVD-H and TVD-η
respectively in the following sub-sections.
4.1 Smooth flow or no-flow in an irregular bed channel
Figure 1 shows a situation where the water should always be quiescent. The channel
is 1500 m long, with the grid size of 5 m and a Manning roughness coefficient of 0.1.
The bed topography is shown in Figures 1(a) and (c), with further details being given
elsewhere (e.g. Zhou et al. 2001). The initial water level is 15 m across the channel. At
the two boundaries, the water level and discharge are fixed at 15 m and zero
respectively. Since the water surface is horizontal and there is no flux across the
boundaries, then the flow should remain still all the times. This result is reproduced by
the TVD-η model, as seen Figures 1(a) and (b). While for the TVD-H model the water
is not maintained still, as shown in Figures 1(c) and (d). The computed flow rates are
shown to be quite large at the sharp corners of the bed topography. The occurrence of
the fictitious flow is caused by the TVD terms that used in the solution procedure. The
significant variance in the water depth due to the irregular bathymetry is misjudged to
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be due to the presence of shocks and thereby anti-dispersive dissipation is wrongly
added at these locations.
Figure 2 shows the steady smooth flow over the same channel. In this computation,
the downstream water depth is fixed at 15 m and at the upstream boundary the specified
flow rate is 0.75 m2/s. The TVD-η model follows the exact solution closely, as shown
in Figures 2(a) and (b). It was found in the simulations that the TVD-H model could
not produce steady results. Figures 2(c) and (d) show the results for one snapshot. As
can be seen, large oscillations occur for the flow rate along the channel when the TVD-
H model is used.
4.2 Trans-critical flow over an irregular channel bed
The flow condition simulated in this case is similar to that which might arise in a
mountainous river, which is a severe test for numerical schemes since the flow varies
rapidly with several transitions between sub- and super-critical regimes. The length of
the channel is 1600 m, with a grid size of 8 m and a Manning roughness coefficient of
0.033. The topography of the channel bottom can be seen in Figures 3(a) and (c), while
the exact bed coordinates can be found in Tseng (2004). A constant unit-width-
discharge of 0.59 m2/s is imposed at the upstream boundary and the water depth is fixed
at the downstream end at 0.42 m.
In this test case the water level and water depth both change sharply, but again better
results are obtained with the TVD-η model, as shown in Figures 3(a) and (b). The
performance of TVD-H model is less accurate in two respects. Firstly, there are some
small-scale spikes in the predicted water surface, as seen in Figure 3(c). Secondly,
when the TVD-η model is used then mass conservation is less well satisfied, as can be
seen in Figure 3(d) when compared with Figure 3(b).
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It appears that the flow rate is not well predicted, even with the TVD-η model, as
shown in Figure 3(b), with small oscillations being observed around the accurate values,
especially close to the sharp water level drops at x = 500 m and x = 1100 m. Similar
findings have been reported by other researchers using the high-order accurate Godunov
method near hydraulic jumps (e.g. Zhou et al. 2001, Rogers et al. 2003). This can be
explained by the fact that the second and third order terms in the Taylor series
expansion become significant near sudden changes because of their discontinuous
nature. This does not necessarily have anything to do with the treatment of the bed-
slope term. To illustrate this point, the next example shows the case for flow over a flat
bed.
4.3 Hydraulic jump in a flat channel
A hydraulic jump induced by bottom friction was next simulated over a flat channel.
The discharge per unit width at the inlet was specified as 0.1178 m2/s. The water depths
at the inlet and outlet were fixed at 0.064 m and 0.17 m respectively. For this flow
scenario the roughness of the bed had a significant influence on the position of the
hydraulic jump, which needed to be tuned to obtain a favourable comparison with the
experimental result. The Chezy value was set to 95.0 m1/2/s and a spatial step of 0.3 m
was used in the present computations. Since the bed was flat, the computations were
the same for both the TVD-η and TVD-H models. Figure 4 shows that there is an
apparent disturbance in the flow rate near the hydraulic jump, although the computed
water depth agrees very well with the experimental data (Gharangik 1988).
It has been noticed that the TVD-MacCormack scheme reported by Tseng (2003,
2004) generates better mass conservation near strong gradients than the present TVD-
MacCormack scheme. However, that scheme involves the local characteristic
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transformation to determine the TVD dissipation term, which is much more complicated
and time-consuming than the present method. No matter which kind of numerical
method is deployed, it appears that discharge disturbances near the very sharp water
surface variations are inevitable for steady flows. However, mass conservation is
regained when the variation of the flow becomes gradual again. For unsteady flows,
such as the dam-break problem (discussed later in this paper), the flow rate agrees well
with the analytical solution. Due to this concern, no recourse to the sophisticated
algorithm was made in order to retain the efficiency of the model. It was also noted that
the basic hypothesis for the shallow-flow model was not valid in high-gradient regions,
so strictly speaking the SWEs do not make sense physically at these locations.
Therefore, the modelling in these regions is ad hoc and some degree of inaccuracy can
be accepted in practical circumstances.
5. Comparison with the ADI scheme
The TVD-MacCormack method was then tested against the traditional ADI method.
For this purpose, the DIVAST model was chosen to be a representative numerical
model which deploys the traditional ADI scheme to solve the SWEs. Although the
numerical background between the ADI and the present schemes is very different, the
ADI method appears similar to the operator-splitting method in the sense that the
scheme breaks up the 2-D problem into two 1-D problems. In the present TVD-
MacCormack model all of the variables are stored and computed at the same location,
while in most ADI models reported in the literature a staggered computational grid is
used, with the water elevations and velocities (or discharges) being located at different
grid locations. Nevertheless, the computational grids for both models are uniformly
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distributed in two orthogonal directions. The following examples highlight the different
behaviours of the two model types in simulating flows with strong water level gradients
and regime transitions.
5.1 Introduction to dam-break problem
Figure 5 illustrates the dam-break problem, which was used in the comparisons
reported later. This can be considered as a profile perpendicular to the dam, with h1 and
h0 being the water depths at the upstream headwater and downstream tailwater
respectively. The tailwater depth h0 can be set to zero, which represents the dry bed
scenario. In the numerical model a thin layer of water, with a depth of 10-4 m, is
imposed on top of the dry bed for the convenience in computations. If the dam
collapses all together, then the problem is 1-D. However, if the dam fails only partially,
then the problem is 2-D in nature. It has been found that the bed roughness has
practically little influence on this type of flow (e.g. Fraccarollo and Toro 1995, Tseng
and Chu 2000), since the acceleration force dominates over the bed friction force in this
situation, except for the dry bed case (e.g. Stelling and Zijlema 2003, Liang et al. 2004).
5.2 1-D dam-break
The scenario considered here was the total and instantaneous dam-failure on a flat
and frictionless bed. This provides an ideal benchmark test case for shock-capturing
schemes since analytical solution has been known. The present configuration is the
same as that used by Wang et al. (2000), where the initial upstream water depth was 10
m, and the downstream water depth was either 2 m, 0.1 m or zero. The dam was located
at x = 500 m, and the grid spacing was 10 m. The time step was 0.5 s, unless otherwise
specified. Figure 6 shows the water surface positions 30 s after dam failure using the
TVD-MacCormack model, where the solid line represents the analytical solution and
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the circles illustrate the predicted results. It can be seen that the shallower the
downstream water depth, then the faster the flood wave travels. Large differences exist
between the results for h0 = 0.1 m and those obtained for h0 = 10-4 m.
The agreement between the analytical and numerical solutions is good for the first
two downstream water depths. In contrast, seemingly large errors exist in the predicted
velocities for the case of h0 = 0, although the accuracy is quite acceptable with respect
to depth predictions. This can be attributed to the singularity of the problem when the
water depth tends to zero. It might be of theoretical interest to capture the large velocity
near the wet and dry interface, but for practical applications this result is not particularly
meaningful. The water depth is very small near the dry region, so the actual momentum
of the flow is insignificant, despite the seemingly large velocities. For practical
scenarios bed friction always exists, which will significantly reduce the flow velocity
when the water depth is very shallow.
Figure 7 shows the computational results using the ADI model for the same dam-
break problem. Using the normal viscosity coefficient, the presence of spurious
oscillations is observed in the numerical predictions, especially at the front of the bore
as the wave propagates downstream. The water level drop due to the propagation of a
depression wave upstream is well predicted, since the flow in this region is relatively
slow. Negative water depths may occur due to numerical oscillations and may cause
computational difficulties. In the present ADI model, the water depth is reset to zero if
it becomes negative. In order to remove non-physical oscillations, an exceptionally
large viscosity coefficient can be specified, with the exact value needing to be
determined by trial and error. For h0 = 10-4 m, the ADI model predictions diverge when
using a time step of 0.5 s. Hence a time step of 0.2 s is used for h0 = 10-4 m and the
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results are shown in Figures 7 (e) and (f). It should be noted that the analytical
solutions assume zero viscosity of the fluid, and the same is true for the TVD-
MacCormack model. Consequently the viscosity introduced in the ADI model can only
be understood as artificial viscosity.
5.2 2-D dam-break
This is another widely used example for dam-break simulations. In this test case the
computational domain was set to be 200 m wide and 200 m long. A dam with a
thickness of 15 m divides the domain into two parts at x = 100 m. Initially, the
upstream water depth was 10 m, while the downstream water depth was set to either 5
m, 0.1 m or 10-4 m in the current study. The breach in the dam started from y = 95 m
and ended at y = 170 m. The bed was again flat and frictionless. The water surface was
studied in the following analysis at 7.2 s after the breach. Although there are no
theoretical solutions available to this problem, there have been a number of numerical
simulations reported in the literature (e.g. Mingham and Causon 1998, Louaked and
Hanich 1998, Wang et al. 2000, Tseng and Chu 2000). To be consistent with most of
these studies, the grid size was set to be 5 m and the time step was set at 0.2 s.
Figure 8 shows the results for the TVD-MacCormack model. They agree well with
the results of the other simulations reported in the literature. As for the 1-D dam-break
case, the bore wave travels faster when the downstream depth becomes smaller. A
smaller water depth means a smaller momentum flux, if the water columns have the
same speed. Moreover, the smaller downstream water depth means a larger water level
difference and therefore a larger driving force within the flow, which in turn leads to the
water being released faster from the reservoir to the tailwater. Although the flood wave
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travels more slowly for the case with a larger downstream water depth, the wave front is
steeper with a larger momentum flux and thus having a more damaging capability.
For the 2-D ADI model, the computational results are shown in Figure 9. Violent
non-physical oscillations affect the predicted water surface levels without the use of a
very large viscosity coefficient. The computation was not stable for h0 = 0.1 m. After
several attempts it was found that the numerical simulations could be performed only by
specifying a minimum water depth of 0.02m over the domain, below which the water
depth was adjusted back to 0.02 m and the velocities were set to zero. Several failed
attempts were experienced for the case of h0 = 10-4 m, therefore no results are shown
herein. It is anticipated that the numerical instability might be overcome by decreasing
the time step. However, this has not been tested since the ADI scheme implies an
implicit approach and it is not practical to use very small time steps for real flood
predictions. It was also found that the spurious oscillations could not be effectively
removed by simply increasing the viscosity coefficient when h0 was set to 0.1 m (Figure
9d). Although the spurious oscillations still existed near the bore wave front for a
viscosity of 2.0 m2/s, extra diffusion was apparent in the vicinity of the depression wave
front when compared with the corresponding results for the TVD-MacCormack model,
as shown in Figure 8(b).
For the dam-break problem with h0 = 0.1 m and 10-4 m, the test was more severe for
numerical models, since part of the flows were super-critical. In contrast, the flow was
sub-critical across the domain for the case of h0 = 5.0 m. This condition may explain
why the 2-D ADI model was found to be less stable when simulating flows with smaller
downstream water depths. Apart from this defect, the other problem with the 2-D ADI
model was that the right amount of artificial viscosity needed was hard to determine
18
without some numerical experiments. It was found that the results were quite sensitive
to the value of the viscosity coefficient. As an example, Figure 10 shows the results
obtained in using two values of the viscosity coefficient, with the difference between
both sets of results shown to be quite large.
5.3 Trans-critical flow in a frictionless channel with a bump
A 1-D open channel flow was simulated, with the aim being to study the ability of the
model to predict flow regime transitions. The channel was 25 m long, with the bottom
elevation being described as follows:
otherwise0
128when1005.02.0)(
2 xxxzb
(17)
The upstream flow rate was set to 0.18 m2/s, and the downstream water level was set to
either 0.43 m or 0.33 m. The initial water level was specified to be the same as that at
the downstream boundary and the spatial step was 0.1 m. When the downstream depth
was 0.43 m, the flow was sub-critical throughout the channel, whereas when the
downstream depth was 0.33 m, a hydraulic jump occurred towards the rear side of the
bump. The analytical solution can be acquired from the Bernoulli equation, except in
the vicinity of the hydraulic jump, where the jump relationship applies.
The results are shown in Figure 11, where it can be seen that the present TVD-
MacCormack model predicts the water level positions very accurately for both cases.
For the sub-critical flow situation shown in Figure 11(a), the ADI model produces
almost the same results as the TVD-MacCormack model, with both agreeing well with
the analytical solution. However, the ADI model is unable to cope with the hydraulic
jump. When the downstream water level was set to 0.33 m, the solution obtained using
the ADI model could not reach steady state, even when a large viscosity coefficient was
used, hence an instantaneous water surface profile results as shown in Figure 11(b).
19
6. Verification against the dyke-break experiment in a flume
Figure 12 demonstrates schematically the configuration of an experiment carried out
in the Fluid Mechanics Laboratory, at Delft University of Technology. It is more
appropriate for this experiment to be called a dyke-break experiment, rather than a dam-
break experiment, due to the small breach width in comparison with the length of the
whole separating wall. The experiment was carried out in a flume, hence all boundaries
could be treated as reflective boundaries in the model. The flume had dimensions of
31.4 m 8 m. A wall extending the whole width of the flume separated the headwater
whose depth was 0.6 m and the tailwater depth was 0.05 m. A 0.4-m-wide sluice gate
was located at the centre of the separating wall, which was raised at a speed of 0.16 m/s.
Since the sluice gate was opened slowly, special treatment of this process was required
in the simulations. The water surface position at the gate was examined throughout the
computations. When it was found that the water level was higher than the opening
height, the gate was treated as an inner solid boundary as effectively occurred in the
physical model. On the left, the water level was fixed to the original value and the
velocity remained as zero. For the computational domain on the right of the gate, the
boundary condition was given as:
Gh hCH (18)
GhRGhx hChghCq 2 (19)
0yq (20)
where hC is the contraction coefficient taken to be 0.6, Gh is the height of the gate
opening, and Rh is the water level in the reservoir ( = 0.6 m). The Manning roughness
20
coefficient was set to be 0.012 across the computational domain, which was the same as
that used by other researches (Stelling and Zijlema 2003, Liang et al. 2004). The grid
size was 0.1 m and the time step was 0.02 s.
Two snapshots of the water surface are shown in Figure 13. As seen in Figure 13 (a),
the wave front was almost semi-circular at the initial stage. Close to the breach, the
flow was super-critical with a high speed, high energy and a small depth. This super-
critical flow was surrounded by hydraulic jumps. The initial flood wave was reflected
when it hit the side walls. The reflected waves interacted with the initial waves and
with one another. The computed water surface 18 s after the breach shows a complex
pattern as the result of the reflection and diffraction among the various surface waves,
as shown in Figure 13 (b). These interactions form a straight wave front perpendicular
to the side walls at the later stage. Between the straight flood front and the high-speed
jet flow near the breach, there is a plateau with a relatively small variation in the water
surface elevation. The water depth in the reservoir changes by less than 0.2 m in 18 s,
due to the large reservoir size and the small breach width.
Figure 14 shows the computed water depth histories at four gauging locations,
together with the corresponding measured values. These gauge points were all located
along the centreline of the flumes, but were 6 m, 9 m, 13 m and 17 m downstream of the
gate centre. The measured water depth histories were digitized from the figures in
Stelling and Zijlema (2003). It can be seen that the predicted time at which the water
level rises suddenly agrees closely with the experimental data, indicating that the flood
wave speed was calculated very accurately. The smaller variations in the water levels
thereafter are due to the diffraction of the waves and are more difficult to be captured
very accurately. Overall, the agreement is again good.
21
7. Conclusions
A simple TVD-MacCormack model has been developed for shallow water flows,
which does not require local characteristic transformation of the differential equations,
and thereby leads to high efficiency in the computation. Two formulations of the SWEs
have been compared. The difference between the two formulations can be significant in
the numerical solutions for flows over uneven bottom topographies. The formulation
with the water depth as an unknown variable was found to be less accurate than that
with the water level as the unknown variable. However, local disturbances in the
discharge could not be totally eradicated using the present TVD-MacCormack model
where the flow conditions change abruptly in steady flows.
The performance of the TVD-MacCormack model was also compared with a typical
2-D ADI model. The results showed that the 2-D ADI model did not have the shock-
capturing capability, which is particularly appropriate for modelling dam-break, dyke-
break and steep riverine flows. Spurious oscillations arose in the solutions near sharp
gradients in the flow and the steady hydraulic jump could not be predicted using the 2-D
ADI model. These numerical difficulties could not be overcome by simply increasing
the artificial viscosity.
Through the comparisons between different formulations of SWEs and with the ADI
model, the present TVD-MacCormack model has been validated for a range of flow
conditions, including 1-D and 2-D, steady and unsteady, hypothetical and realistic, sub-,
super- and trans-critical flows. Besides, a further validation was undertaken for a
complicated dyke-break experiment. The predicted hydrographs compared well with
the measured data. It should also be noted that the computational grids used in this
22
paper were relatively coarse. Since the computational results were encouraging, no finer
grid resolutions were considered necessary. The simulations were all completed within
1 minute on a Pentium 4 personal computer (3.2GHz CPU, 2G RAM) even for the
dyke-break case with 315×81 grid cells. The high efficiency and robustness of the
present model makes it a powerful tool for real time flood predictions.
Acknowledgements
The study is funded by the Engineering and Physical Sciences Research Council as
part of the Flood Risk Management Research Consortium.
23
References
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[20] Tseng MH. The improved surface gradient method for flows simulation in
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26
Figure captions
Figure 1. Still water over an irregular channel bed. (a) TVD-η, water level; (b) TVD-η,
flowrate; (c) TVD-H, water level; (d) TVD-H, flowrate.
Figure 2. Gradually varying flow over an irregular channel bed. (a) TVD-η, water level;
(b) TVD-η, flowrate; (c) TVD-H, water level; (d) TVD-H, flowrate.
Figure 3. Rapidly varying flow over an irregular channel bed. (a) TVD-η, water level;
(b) TVD-η, flowrate; (c) TVD-H, water level; (d) TVD-H, flowrate.
Figure 4. Hydraulic jump in a flat channel. (a) Water level; (b) Flowrate.
Figure 5. Illustration of the dam-break problem.
Figure 6. 1-D dam-break simulation using the TVD-MacCormack method. Left: Depth;
Right: Velocity. (a) h0 = 2 m; (b) h0 = 0.1 m; (c) h0 = 10-4 m.
Figure 7. 1-D dam-break simulation using the ADI model. (a) h0 = 2 m, υ = 10-6 m2/s;
(b) h0 = 2 m, υ = 15.0 m2/s; (c) h0 = 0.1 m, υ = 10-6 m2/s; (d) h0 = 0.1 m, υ = 7.0 m2/s; (e)
h0 = 10-4 m, υ = 10-6 m2/s; (f) h0 = 10-4 m, υ = 2.0 m2/s.
Figure 8. 2-D dam-break simulation using the TVD-MacCormack model. (a) h0 = 5 m;
(b) h0 = 0.1 m; (c) h0 = 10-4 m.
Figure 9. 2-D dam-break simulation using a 2-D ADI model. (a) h0 = 5 m, υ = 10-6 m2/s;
(b) h0 = 5 m, υ = 5.0 m2/s; (c) h0 = 0.1 m, υ = 10-6 m2/s; (d) h0 = 0.1 m, υ = 2.0 m2/s.
Figure 10. Influence of viscosity for a 2-D ADI model, h0 = 5 m. (a) υ = 3.0 m2/s; (b) υ
= 10.0 m2/s.
Figure 11. Steady flow over a bump. (a) Downstream depth of 0.43 m; (b) Downstream
depth of 0.33 m.
Figure 12. Layout of dyke-break experiment.
27
Figure 13. Snapshots of the predicted water surface for the dyke-break experiment. (a) t
= 4 s; (b) t =18 s.
Figure 14. Water depth history at four gauge points for dyke-break experiment. (a) 6 m
downstream the gate; (b) 9 m downstream the gate; (c) 13 m downstream the gate; (d)
17 m downstream the gate.
28
Streamwise distance (m)
Ele
vatio
n(m
)
0 500 1000 15000
2
4
6
8
10
12
14
16
ComputedExactBed Profile
Streamwise distance (m)
Flo
wra
te(m
2/s
)
0 500 1000 1500-0.04
-0.02
0
0.02
0.04
ComputedExact
(a) TVD-η, water level (b) TVD-η, flowrate
Streamwise distance (m)
Ele
vati
on
(m)
0 500 1000 15000
2
4
6
8
10
12
14
16
ComputedExactBed Profile
Streamwise distance (m)
Flo
wra
te(m
2/s
)
0 500 1000 1500-2
-1
0
1
2
ComputedExact
(c) TVD-H, water level (d) TVD-H, flowrate
Figure 1. Still water over an irregular channel bed
29
Streamwise distance (m)
Ele
vatio
n(m
)
0 500 1000 15000
2
4
6
8
10
12
14
16
ComputedExactBed Profile
Streamwise distance (m)
Flo
wra
te(m
2/s
)
0 500 1000 15000.7
0.72
0.74
0.76
0.78
0.8
ComputedExact
(a) TVD-η, water level (b) TVD-η, flowrate
Streamwise distance (m)
Ele
vati
on
(m)
0 500 1000 15000
2
4
6
8
10
12
14
16
ComputedExactBed Profile
Streamwise distance (m)
Flo
wra
te(m
2/s
)
0 500 1000 1500-0.5
0
0.5
1
1.5
2
2.5
ComputedExact
(c) TVD-H, water level (d) TVD-H, flowrate
Figure 2. Gradually varying flow over an irregular channel bed
30
Streamwise distance (m)
Ele
vatio
n(m
)
0 400 800 1200 1600-2
0
2
4
6
Computed Water SurfaceBed Profile
Streamwise distance (m)
Flo
wra
te(m
2/s
)
0 400 800 1200 16000.5
0.55
0.6
0.65
0.7 ComputedExact
(a) TVD-η, water level (b) TVD-η, flowrate
Streamwise distance (m)
Ele
vati
on
(m)
0 400 800 1200 1600-2
0
2
4
6
Computed Water SurfaceBed Profile
Streamwise distance (m)
Flo
wra
te(m
2/s
)
0 400 800 1200 16000.5
0.55
0.6
0.65
0.7 ComputedExact
(c) TVD-H, water level (d) TVD-H, flowrate
Figure 3. Rapidly varying flow over an irregular channel bed
31
Streamwise distance (m)
De
pth
(m)
0 5 10 150
0.05
0.1
0.15
0.2
MeasuredComputed
Streamwise distance (m)
Flo
wra
te(m
2/s
)
0 5 10 150.1
0.11
0.12
0.13
0.14
ComputedExact
(a) Water level (b) Flowrate
Figure 4. Hydraulic jump in a flat channel
h1
h0
Figure 5. Illustration of the dam-break problem
32
Streamwise distance (m)
De
pth
(m)
0 250 500 750 10000
2
4
6
8
10
Streamwise distance (m)
Ve
loci
ty(m
/s)
0 250 500 750 10000
2
4
6
(a) h0 = 2 m
Streamwise distance (m)
De
pth
(m)
0 250 500 750 10000
2
4
6
8
10
Streamwise distance (m)
Ve
loci
ty(m
/s)
0 250 500 750 10000
2
4
6
8
10
12
(b) h0 = 0.1 m
Streamwise distance (m)
De
pth
(m)
0 200 400 600 800 1000 12000
2
4
6
8
10
Streamwise distance (m)
Ve
loci
ty(m
/s)
0 200 400 600 800 1000 12000
5
10
15
20
(c) h0 = 10-4 m
Figure 6. 1-D dam-break simulation using the TVD-MacCormack method.
Left: Depth; Right: Velocity
33
Streamwise distance (m)
De
pth
(m)
0 250 500 750 10000
2
4
6
8
10
Streamwise distance (m)
De
pth
(m)
0 250 500 750 10000
2
4
6
8
10
(a) h0 = 2 m, υ = 10-6 m2/s (b) h0 = 2 m, υ = 15.0 m2/s
Streamwise distance (m)
De
pth
(m)
0 250 500 750 10000
2
4
6
8
10
Streamwise distance (m)
De
pth
(m)
0 250 500 750 10000
2
4
6
8
10
(c) h0 = 0.1 m, υ = 10-6 m2/s (d) h0 = 0.1 m, υ = 7.0 m2/s
Streamwise distance (m)
De
pth
(m)
0 200 400 600 800 1000 12000
2
4
6
8
10
Streamwise distance (m)
De
pth
(m)
0 200 400 600 800 1000 12000
2
4
6
8
10
(e) h0 = 10-4 m, υ = 10-6 m2/s (f) h0 = 10-4 m, υ = 2.0 m2/s
Figure 7. 1-D dam-break simulation using the ADI model
34
6
8
10
De
pth
(m)
0
50100
150200
x (m)
0
50
100150
200
y (m)
0
2
4
6
8
10
De
pth
(m)
0
50100
150200
x (m)
0
50
100150
200
y (m)
(a) h0 = 5 m (b) h0 = 0.1 m
0
2
4
6
8
10D
ep
th(m
)
0
50100
150200
x (m)
0
50
100150
200
y (m)
(c) h0 = 10-4 m
Figure 8. 2-D dam-break simulation using the TVD-MacCormack model
35
6
8
10
De
pth
(m)
0
50100
150200
x (m)
0
50
100150
200
y (m)
6
8
10
De
pth
(m)
0
50100
150200
x (m)
0
50
100150
200
y (m)
(a) h0 = 5 m, υ = 10-6 m2/s (b) h0 = 5 m, υ = 5.0 m2/s
0
2
4
6
8
10
De
pth
(m)
0
50100
150200
x (m)
0
50
100150
200
y (m)
0
2
4
6
8
10
De
pth
(m)
0
50100
150200
x (m)
0
50
100150
200
y (m)
(c) h0 = 0.1 m, υ = 10-6 m2/s (d) h0 = 0.1 m, υ = 2.0 m2/s
Figure 9. 2-D dam-break simulation using a 2-D ADI model
36
6
8
10
De
pth
(m)
0
50100
150200
x (m)
0
50
100150
200
y (m)
6
8
10
De
pth
(m)
0
50100
150200
x (m)
0
50
100150
200
y (m)
(a) υ = 3.0 m2/s (b) υ = 10.0 m2/s
Figure 10. Influence of viscosity for a 2-D ADI model, h0 = 5 m
Streamwise distance (m)
De
pth
(m)
0 5 10 15 20 250
0.1
0.2
0.3
0.4
0.5
TVD-Mac ResultDIVAST ResultAnalytical ResultBed Profile
Streamwise distance (m)
De
pth
(m)
0 5 10 15 20 250
0.1
0.2
0.3
0.4
0.5
TVD-Mac ResultDIVAST ResultAnalytical ResultBed Profile
(a) Downstream depth of 0.43 m (b) Downstream depth of 0.33 m
Figure 11. Steady flow over a bump
37
0.4 m
2.5 m 28.9 m
8.0 m
0.6 m
0.26 m
Res
ervo
ir
Dyke
Basin
0.05
m
Figure 12. Layout of dyke-break experiment
(a) t = 4 s (b) t =18 s
Figure 13. Snapshots of the predicted water surface for the dyke-break experiment
38
Time (s)
De
pth
(m)
0 5 10 15 200
0.05
0.1
0.15
MeasuredComputed
Time (s)
De
pth
(m)
0 5 10 15 200
0.05
0.1
0.15
MeasuredComputed
(a) 6 m downstream the gate (b) 9 m downstream the gate
Time (s)
De
pth
(m)
0 5 10 15 200
0.05
0.1
0.15
MeasuredComputed
Time (s)
De
pth
(m)
0 5 10 15 200
0.05
0.1
0.15
MeasuredComputed
(c) 13 m downstream the gate (d) 17 m downstream the gate
Figure 14. Water depth history at four gauge points for dyke-break experiment