Comparison between TVD-MacCormack and ADI-Type Solvers of the shallow water equations simple

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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: [email protected]

Prof. Roger A. Falconer

Tel: +44 (0)29 2087 4280; Email: [email protected]

Dr. Binliang Lin

Tel: +44(0)29 2087 4696; Email: [email protected]

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

qq

q

H

qq

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

qq

q

h

qq

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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No. 84-20, 1984.

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[9] Leendertse JJ, Gritton EC. A water quality simulation model for well-mixed

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[10] Liang Q, Borthwick AGL, Stelling G. Simulation of dam- and dyke-break

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[19] Tan WY. Shallow water hydrodynamics. New York: Elsevier, 1992.

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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


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


Ele

vati

on

(m)

0 500 1000 15000

2

4

6

8

10

12

14

16



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


Ele

vatio

n(m

)

0 500 1000 15000

2

4

6

8

10

12

14

16



Flo

wra

te(m

2/s

)

0 500 1000 15000.7

0.72

0.74

0.76

0.78

0.8

ComputedExact



Ele

vati

on

(m)

0 500 1000 15000

2

4

6

8

10

12

14

16



Flo

wra

te(m

2/s

)

0 500 1000 1500-0.5

0

0.5

1

1.5

2

2.5

ComputedExact


Figure 2. Gradually varying flow over an irregular channel bed

30


Ele

vatio

n(m

)

0 400 800 1200 1600-2

0

2

4

6

Computed Water SurfaceBed Profile


Flo

wra

te(m

2/s

)

0 400 800 1200 16000.5

0.55

0.6

0.65

0.7 ComputedExact



Ele

vati

on

(m)

0 400 800 1200 1600-2

0

2

4

6

Computed Water SurfaceBed Profile


Flo

wra

te(m

2/s

)

0 400 800 1200 16000.5

0.55

0.6

0.65

0.7 ComputedExact


Figure 3. Rapidly varying flow over an irregular channel bed

31


De

pth

(m)

0 5 10 150

0.05

0.1

0.15

0.2

MeasuredComputed


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


De

pth

(m)

0 250 500 750 10000

2

4

6

8

10


Ve

loci

ty(m

/s)

0 250 500 750 10000

2

4

6

(a) h0 = 2 m


De

pth

(m)

0 250 500 750 10000

2

4

6

8

10


Ve

loci

ty(m

/s)

0 250 500 750 10000

2

4

6

8

10

12

(b) h0 = 0.1 m


De

pth

(m)

0 200 400 600 800 1000 12000

2

4

6

8

10


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


De

pth

(m)

0 250 500 750 10000

2

4

6

8

10


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


De

pth

(m)

0 250 500 750 10000

2

4

6

8

10


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


De

pth

(m)

0 200 400 600 800 1000 12000

2

4

6

8

10


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


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


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

Comparison between TVD-MacCormack and ADI-Type Solvers of the shallow water equations simple

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