Consideration of flood propagation model performance in relation to laboratory physical model data

JAVIER MURILLO Universidad de Zaragoza, Spain jmurillo@mafalda.cps.unizar.es JAVIER BURGUETE Universidad de Zaragoza, Spain jburguete@able.es PILAR GARCÍA-NAVARRO Universidad de Zaragoza, Spain pigar@posta.unizar.es PILAR BRUFAU Universidad de Zaragoza, Spain brufau@posta.unizar.es SUMMARY Within the research project’s deliverable D.3.2.3 of exploring the different possibilities in flood modelling over irregular valleys, the present work contributes with the application of different numerical techniques and methods to previously obtained physical model data: the propagation of a flood along the physical model of Toce river valley and the dam break flow over a sill. In the first case, the 2D method applied is designed to deal with all kind of grids, structured or unstructured, and admits local grid refinement. In the second case, the emphasis is put on the design of implicit schemes for unsteady flow with shocks. The two test cases are based on laboratory setups of rather different characteristics, geometries, and scales. 1. INTRODUCTION New schemes have been reported successful for flow in channels. However, their application to river flow and complex geometry is not so common in the literature. The presence of extreme slopes, high roughness and strong changes in the irregular geometry represent a great difficulty that can lead to important numerical errors presumably arising from the source terms of the equations. The importance of the numerical discretization of these terms makes sometimes doubtful the interest of increasing the order of accuracy of the basic advective scheme. This is even more when the problem involves propagation over irregular dry beds. In two dimensional hydraulic free surface problems, when a mesh is required to be representative of some topography, as the number of cells used to create the mesh increases, the discrete representation of the real problem improves, so the accuracy of the results is enhanced, but the computing time grows and this can be a cumbersome difficulty. On the other hand, the experience says that, when the topography is smooth, larger cells can be used to simulate a flooding event but, if the topography is highly irregular, the number of cells must be increased in order to allow for a correct flow representation. One solution is to generate a mesh with local refinement, in order to reduce the calculation effort and, at the same time, be able to achieve the accuracy in the flow simulation results that a globally fine mesh would provide. When cuadrilateral structured meshes are used, the refinement must be done in all cells contained in the rows and columns connected with the area whose cell-

density we want to increase. Unstructured triangular meshes with local mesh refinement can lead to directional cell deformation depending on how they are created, and the grid refinement procedure is more complicated. The solution proposed here consists of using a structured triangular mesh of variable density following the variation of the bed slope. This produces a grid where the local refinement is introduced in the irregular topography zones, with a refinement depending on how irregular the topography is. Furthermore, the new cells are generated following a simple algorithm, without distortions and not affecting the size of the cells in the same row or column far from the area of interest. The performance and efficiency of a finite volume upwind method on different grids will be presented and compared for a river flow simulation problem over dry bed. The influence of the grid used on a 2D finite volume numerical model is important. This is of particular interest when the flow pattern is complicated, but also when the bottom bathymetry is irregular and highly variable. In this work, the generation of new meshes is developed following the bathymetry characteristics in order to obtain a better approximation of the actual singularities of the valley geometry and to improve the accuracy of the results in the flow for those regions in which a more complex flow is associated to a more abrupt geometry. Using the proposed mesh refinement technique it is possible to generate locally refined areas without requiring a large time simulation in a simple way. When flooding events are simulated over areas associated to small slope variation in space, large cells can be used to represent the bathymetry conforming a coarse grid. However, abrupt slope changes in space mean abrupt geometries and a higher influence of the terrain on the flow behaviour, so smaller cells are needed to represent both geometry and flooding. The criterion used in this work to decide the degree of refinement required is given by the variation of the slope in distance, that is, the gradient of bed slope. Implicit schemes are well known for the property of allowing numerical stability in the resolution of partial differential equations in presence of time steps not restricted by the Courant-Friedrichs-Lewy condition. Therefore they have traditionally been the most attractive methods in CFD for steady or gradually unsteady flows. Computational Hydraulics has always been a field of frequent search for accurate and robust implicit schemes (Preissmann, Abbott). Some research has been recently oriented to the development of new implicit upwind techniques able to deal with transcritical flows in order to overcome deficiencies found in previous implicit methods. However these results are characterized by a limitation in the allowable time step size in cases of unsteady and highly discontinuous flow. It is usually found that CFL values bigger than 3 lead to instabilities in the resolution of moving front waves. This is due to poor treatment of the linearization of the implicit flux terms, which are often locally evaluated. In this work, driven by the interest in river flow modelling, new upwind implicit schemes are presented. In the line of previous upwind implicit schemes, a new way of linearizing the implicit upwind flux terms is proposed. It consists essentially of selecting the extreme positive or negative eigenvalues of the linearizing Jacobians. This modification removes the above mentioned stability restriction and produces a robust and accurate upwind implicit scheme suitable for discontinuous flows in complex topography. The performance of the proposed technique will be shown in a dambreak test case.

2. TOCE RIVER TEST CASE 2D Numerical model The two-dimensional shallow water equations, which represent mass and momentum conservation in plane, can be obtained by depth averaging the Navier-Stokes equations. Neglecting diffusion of momentum due to viscosity and turbulence, wind effects and the Coriolis term, they form the following system of equations:

H=∂∂

+∂∂

+∂∂

yxtGFU

(1)

in which,

))(),(,0(,),2

,(

),2

,(,),,(

00

22

22

fyyfxxx

x

SSghSSghhqqgh

hq

q

hqqgh

hq

qqqh

yxx

yxxyx

−−=+=

+==

HG

FU (2)

where qx=uh and qy=vh. The variable h represents the water depth, g is the acceleration of the gravity and (u,v) are the depth averaged components of the velocity along the x and y coordinates respectively. The source terms in the momentum equation are the bed slopes and the friction losses along the two coordinate directions. It is useful to rewrite (1) as

HEU=∇+

∂∂

t (3)

in which E=(F,G), since this displays the conservative character of the system in the absence of source terms, and also in order to introduce the integral form of the equations over a fixed volume Ω,

∫ ∫∫∫∫ Ω ΩΩΩΩΩ=+Ω

∂∂

⇒Ω=Ω∇+Ω∂∂

∫ ddSdt

dddt S

HEnUHEU (4)

where S denotes the surface surrounding the volume Ω , n is the unit outward normal vector and Gauss theorem has been used. In order to formulate cell centered finite volume methods over a control volume where the dependent variables of the system are represented as piecewise constants, conservation equations can be written for every cell i and Riemann solvers are oriented perpendicularly to the edges of the grid cells in much a one-dimensional way. For that purpose, the normal flux En and its Jacobian are of interest. The Jacobian matrix, of the normal flux is evaluated as

( ) ( ) ( )

UG

UF

UEnJ

∂

∂+

∂∂

=∂

∂= yx nn

n (5)

and can be diagonalized, Λn = P-1 Jn P, making the schemes rely on its eigenvalues and eigenvectors as in one dimension. Using the finite volume formulation, for the updating of a single cell only the in-going contributions are taken into account when evaluating the contour flux integral.

( )ik

kkkni

ni

ni dlt

⎟⎟⎠

⎞⎜⎜⎝

⎛Ω−

Ω∆

−= ∑ −−−+ HUPPΛUU δ11

(6) were the sum in k is over the cell sides and the quantities with – superscript are calculated at every cell edge from

( ) ( ) HPΛΛ1PHΛΛΛ 11nnnnn ,

21 −−−− −=−= (7)

The procedure described in the previous section is applied for the ordinary cells, that is, those representing points at the interior of the wetted domain. The boundaries of the wetted domain are defined by the cells not completely surrounded by other cells. All these cells actually require the definition of suitable boundary conditions in order to reach the solution of a problem. However, for transient flows a distinction can be made considering either wetted domains fixed in extension, that is, limited by vertical walls, or those whose size changes as time progresses, that is, those involving moving boundaries. In this work, boundary conditions are, strictly speaking, applied only at fixed boundaries. The moving boundaries are considered as wetting fronts and hence included in the ordinary cell procedure in a through calculation that assumes zero water depth for the dry cells. This approach provides satisfactory results when dealing with wetting fronts over flat or downward sloping surfaces but can lead to difficulties in advances over adverse slopes. This is what is called in this work the wetting/drying front. 2D Mesh Generation Quadrangular structured meshes have been used for many numerical purposes, and their refinement limitations are well known. In this paper, an unstructured triangular mesh is basically used, and, taking into account its connectivity properties, a local refinement is suggested in an easy way. The basic squared cell is divided by the diagonal in two cells, which are subdivided leading to four, eight, and sixteen cells. Four refinement degrees can be used as shown in Figure 1.

Figure 1. Refinement levels

Level 1 Level 2 Level 3 Level 4

Levels 1 and 2 can connect perfectly their nodes to each other, and the same happens between Levels 3 and 4. However, connection cells must be supplied for any pair of Levels 1 or 2 with 3 or 4. Using connection cells a local refinement can be easily achieved, as in Figure 2. The first step of the refinement technique consists of dividing the whole domain in squared cells, of side length equal to the maximum length edge desired in the final triangular mesh, and so that each of the cells contains several topography data points. For each pair of data points within the cell i of the domain, the slope, is calculated in both coordinate directions, x and y. Once all the and cell values are calculated, their maximum and minimum in both directions is searched, comparing the slopes calculated inside the cell. With this information the maximum slope gradient is stored for each cell. This value is calculated as:

Figure 2. Connection cells. For those cells in which the local value of the maximum slope gradient exceeds the absolute maximum, it is replaced by the absolute maximum. There is an important reason to limit the size of the initial cells used for slope searching to the same size as the largest edge of the final triangular mesh. If larger cells are adopted and if maximum refinement is required, the amount of refinement will be performed in a zone greater than necessary, increasing the final number of cells without an actual improvement of the bathymetry representation. Now, making use of the relative value of the maximum gradient in the cell, a refinement degree will be set. Although it seems natural to use a direct rule using the relative value of the maximum gradient, this criterion is not adequate. If the number of cells close to the absolute maximum is an important part of all the cells, an excessive refinement will be done; on the other hand, if the number of cells close to the absolute minimum slope gradient represents most of the cells, a poor refinement can be achieved. In order to avoid such tendencies the following technique is used in which the amount of cells with a maximum slope gradient less or equal to the average value is estimated. This shows how irregular is the distribution of the maximum gradient slopes in the full mesh. As four different refinement levels can be achieved whit this technique, in order to avoid an underestimation or an overestimation of the final number of cells involved, three new parameters are defined, which represent the upper limit value of the maximum gradient slope for the 25%, 50% and, 75% of the cells respectively. Therefore, some cells will be refined using refinement Level 1, some cells with refinement Level 2, other cells with refinement Level 3, and finally, cells with maximum gradient slope greater than a certain condition, with refinement Level 4. Once reached to this point the refinement that must be applied to each initial cell is known. In order to maintain the correct connectivity between cells it is necessary to generate the adequate connecting cells. The resulting grids

always have a bigger number of elements than the original one, which results in a smooth transition from non refined cells to maximum refinement cells. Test Case In order to illustrate the performance of this meshing technique an example is presented. The test case consists of generating a suitable mesh for a scaled model of the Toce river valley, a watercourse of the occidental Alps, in Italy. The model scale factor is 1:100 and the approximate model dimensions are 50x11 m. The model reproduces the details of the real geometry. The bathymetry is supplied by means of a grid of points, with a typical distance of 5 cm. Figure 3 shows a view of the topography of the valley.

Figure 3. Toce river valley bathymetry.

p 13

p 12

Figure 4: Location of measuring probes near the lateral reservoir. Using these data to generate a uniform structured mesh, this results in a mesh with a total of 279530 cells, which is excessive if a small simulation time is required. Analysing the valley it can be seen that the bathymetry can be accurately represented with larger cells in some parts. A second attempt to use a uniform grid with cells 4 times the initial grid distance, 20cm, leads to a dramatic reduction in the number of cells involved in the new mesh, but also some bathymetry details, like the river bed or the reservoir limits, clearly loose accuracy. In order to obtain a better result, the mesh algorithm proposed here is applied. Using a basic grid distance of 20 cm, a new final mesh with 56351 cells is generated. Figure 5 shows a detail of the reservoir using the initial density mesh of 5cm. Figure 6 shows the same detail using a mesh

with a constant density of 20 cm. On the other hand, Figure 7 shows the result when the meshing technique proposed here is used with an initial grid density of 20 cm, and Figure 8 the result using an initial grid density of 40 cm. In Figure 8, a wider region is displayed for the latter density. Using local refinement, a good representation of the bathymetry is achieved in those places presenting a more complex geometry. For the Toce model good results are obtained when using local refinement over meshes with original cell edge of 20 and 40 cm. In order to compare the results produced in a flooding simulation, a discharge hydrograph characterised by a peak discharge value leading to overtopping of the reservoir is used. In this case several probes were located to measure the depth of water at different locations. For the sake of clarity, only the results at two probes are shown here.

Y

Z

X

Figure 5. Detail of the reservoir. Constant density mesh, edge 5 cm.

Y

Z

X

Figure 6. Detail of the reservoir. Constant density mesh, edge 20 cm.

Y

Z

X

Figure 7. Detail of the reservoir. Variable density mesh, initial edge 20 cm.

Y

Z

X

Figure 8. Detail of the reservoir. Variable density mesh, initial edge 40 cm.

These probes are located in places were the flow conditions are clearly conditioned by the surrounding geometry. Figure 3 shows their location. Probe number 12 is located inside the reservoir, which will be flooded by the wave, and probe 13 is located in the river bed, close to the reservoir. Figure 9 shows the results at those probes depending on the mesh size. As can be clearly seen from the figures, both meshes lead to similar results. The main differences are found in probe 12. For the mesh with cell density given by 40 cm, higher depths are obtained, since the discrete bathymetry is a smoother representation of reality and offers less resistance than the mesh with a greater mesh density. 2D Numerical Stability and Time Step In all the cases the time step is calculated depending on the flow conditions using a constant CFL parameter equal to 0.9. The average time step is about 0.005 seconds and all the simulations are performed using times lower than 10 minutes.

Figure 9: Detail of the reservoir. Variable density mesh, initial density 40 cm.

probe 12

7,3657,37

7,3757,38

7,3857,39

7,3957,4

7,4057,41

7,4157,42

20 70 120 170

depth (m)

time

(s)

measured40cm20cm

probe 13

7,3

7,35

7,4

7,45

7,5

7,55

20 40 60 80 100 120 140 160 180

depth (m)

time

(s)

measured40cm20cm

Figure 10: Measured and computed water levels at probes 12 and 13.

2. DAM BREAK WAVE OVER A SILL 1D Numerical Model We are interested in solving as efficiently as possible 1D hyperbolic systems with source terms. In a general conservative form

HFU=+

∂∂

dxd

t (8)

where U is the vector of conserved variables, F the vector of fluxes and H that of source terms. Our interest is led by the numerical modelling of one-dimensional shallow water flows of practical application in Hydraulics such as open channel and river flows. In that case

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

fSSgAgIA

QQ

QA

01

2(

0,, HFU (9)

where Q is the discharge, A is the wetted cross section, g is the acceleration of gravity and S0 is the bed slope. The rest of the terms account for pressure forces and for friction forces, with Sf associated to wall friction and represented by the empirical Manning (Cunge et al., 1980). It is very important to remark that in the conservative form the total derivative dF/dx is used to represent both the increments due to the pure spatial variations in x and those due to the variations of the conserved variable U whereas the partial derivative is reserved to represent only the variation due to the x with U constant. The difference is significant. From the equations in conservative form (8), it is possible to pass to an associated non-conservative form

'HUJU=

∂∂

+∂∂

xt (10)

where J is the Jacobian matrix of the original system and

x∂∂

−=FHH' (11)

The characteristic form of the equations is important for the correct formulation of upwind schemes and boundary conditions. This form is obtained from a diagonalization of the Jacobian in. Calling P and P-1 the matrices that make diagonal J,

JPPΛPPΛJ 11 , −− == (12)

Conservative Schemes The conservation law contains an important physical meaning. By spatial integration it expresses that the time variation in the conserved variable in a given volume is equal to the difference between the incoming and the outgoing fluxes plus the contribution of the source term. When discretizing a conservation law of this kind, bad numerical approximations can lead to bad behaviour in the solution and unacceptable error. Schemes properly approximating the physical conservation equation are called conservative schemes (Hirsch 1989, Toro, 2000). Schemes so defined will produce a good approximation of the physical equation cancelling the contributions of the flux at the grid interfaces, being the global variation of the conserved variable due only to the source terms and to the flux at the boundaries The most common definition of a conservative scheme is based on a numerical flux structure and a pointwise evaluation of the source terms, derived from the conservative form of the equations. Conservative schemes can also be derived from the non-conservative form of the equations (10). The advantage is that the latter form tends to be simpler to deal with than the conservative form and, in the shallow water case, avoids the use of the pressure integrals. Two equivalent forms of conservative schemes with non-centred source terms are possible. In any case

Ri

Li

it 2/12/1 +− +=

∆∆

GGU

(13)

where

2/12/12/1 '

+++ ⎟

⎠⎞

⎜⎝⎛

δδ

−=⎟⎠⎞

⎜⎝⎛

δδ

−=ii

i xxUJHFHG (14)

the decomposition in left and right parts is to be defined in every particular numerical scheme, ∆ will be used for time increments, and δ represents spatial increment Finally, conservative schemes based in the characteristic form of the equations are the basis for the wave decomposition of upwind schemes (Burguete and Garcia-Navarro 2002). From (13) it is possible to rewrite a discrete wave decomposition into left and right moving contributions

( ) ( ) 2/11

2/11

+−

−− +=

∆∆

iR

iLi

tGPPΩGPPΩ

U (15)

Note that this discretization requires again a non-centred formulation of the source terms. First Order Implicit Upwind Schemes Tridiagonal scheme Upwind schemes are based on the idea of approximating the spatial derivatives by non-centred differences biased in the sense of propagation of information in the physical problem.

In order to construct a first order scheme, suitable for left and right moving propagation velocities, the following is usually written

( ) ( ) ⎥⎦⎤

⎢⎣⎡ +∆=∆

θ++

−θ+−

+ ni

nii

n t 2/12/1 GGU (16)

where G- is associated to negative velocities and G+ to positive velocities and the notation fn+θ = θ fn+1 + (1- θ)fn is used. For this scheme the following wave decomposition is assumed in order to select the appropriate influence region in every case.

( )[ ] ( )[ ]ΛIΩΩΛIΩΩ signsign RL −==+== −+

21,

21 (17)

( ) GPPΩGΛΛΛ 11 , −±±− ==sign A linear analysis of the homogeneous equations shows that the stability condition is

(1-2θ)CFL ≤ 1 (18) with

xtaCFL k δ

∆= max (19)

the Courant-Friedrichs-Lewy number and ak being the eigenvalues of the Jacobian. It is unconditionally stable if θ ≥ 1/2. This analysis shows also that the scheme is TVD if

θ−≤

11CFL (20)

being this condition more restrictive. The scheme is unconditionally TVD if θ =1. With complex non-linear equations this scheme requires to include F n+1 and H n+1, which represents a difficulty. In order to avoid this problem, the following linearisation can be made

Fn+1 ≈ Fn + Jn ∆U n (21)

Hn+1 ≈ Hn + Kn ∆U n with K= ∂H/∂ U the Jacobian of the source term. This leads to the conservative implicit tri-diagonal scheme in characteristic form. It is worth noting that, in this scheme, it is simpler to make the upwind treatment of the source terms in the explicit operator keeping a central discretization for K in the linearized implicit operator. This linearized implicit scheme was developed in the context of Gas Dynamics and successfully used for steady problems without source terms. Applications of this kind of method in Hydraulics can be found in Bermúdez and Vázquez, (1995), Garcia-Navarro et al. (1994) and Delis (2000). The performance for steady problems is highly sensitive to the CFL used during the transient phase. When shocks are present in the solution of a non-linear equation and a high time step is used, the propagation velocities (eigenvalues in the system

case) can change strongly between the two time levels. The implicit linearized methods are not unconditionally stable in these cases because the linearization involves the evaluation of the Jacobian at time level n. In the references the test cases presented confirm this behaviour. Bidiagonal scheme The scheme built as in the previous section leads, in general, to a linear block tridiagonal system of equations. In order to work with a simpler block bi-diagonal system and to avoid spurious errors in the transcritical points, the original scheme is transformed in a two-step block bi-diagonal system by splitting in increments due to positive and negative propagations and defining the final updating at a point as the sum of both (Burguete and Garcia-Navarro 2004). This discretization also requires a wave decomposition of the source term. For similitude with the wave decomposition of the flux, the following is assumed:

K± = P Ω± P-1 K (22) Despite the improvement achieved by the splitting philosophy, this scheme is still not able to cope with high CFL numbers in presence of shocks. Semi-explicit scheme In the search for stability and simplicity, a semi-explicit version of the bidiagonal scheme can be built (Burguete and Garcia-Navarro 2004). This version replaces the Jacobians in the implicit operators by the maximum eigenvalue stabilizing the scheme in all cases. A linear analysis of the homogeneous equations shows that the stability condition is

(1-2θ)CFL ≤ 1 (23) being unconditionally stable if θ ≥ 1/2. This analysis shows also that the scheme is TVD if

θ−≤

11CFL (24)

being unconditionally TVD if θ =1. Corrected scheme The semi-explicit scheme is very diffusive and cannot reproduce sharp gradients in moving fronts. In a different approach the bidiagonal scheme can be improved by means of a spatial weighting parameter locally defined. From a non-linear analysis based on a scalar equation some conditions on the parameters can be imposed to ensure monotonicity (Burguete and Garcia-Navarro 2004). For the extension to a nonlinear system of conservation laws, the question of the optimal value for the correcting parameter remains open.

Numerical Results The second example deals with a dambreak flow simulation. The test case has been taken from an experimental setup described in (Alcrudo and Soares-Frazao, 2000) where the details can be found. In summary, it is a rectangular channel in which a gate separates a reservoir containing still water from a dry part in which a bump of triangular shape is placed. The removal of the gate generates a wave that advances over the triangular obstacle and gets reflected downstream.

Fig.11 Experimental setup In the original experiment, the wave underwent several reflections until steady state was achieved. In the numerical results presented here we will focus on intermediate states to check the shock capturing ability of the schemes during the transient phase. For this purpose, the initial conditions and the profiles at $t=8s$ and $t=12s$ are shown in Figs12-14. Fig. 12 shows the results obtained with a second order TVD explicit scheme, already validated with the experimental data and used here as reference solution. Figs. 13 and 14 correspond to the results from the semi-explicit and corrected schemes both using $CFL=10$ and $CFL=100$.

Fig. 11

Fig. 12

Fig.13

Fig.14

Fig. 15

3. CONCLUSIONS In the work presented, a grid generation technique where the local refinement is introduced in the irregular topography zones, with a refinement depending on the irregularity of the topography has been described. The new cells are generated following a simple algorithm, without grid distortions and not affecting the size of the cells in the same row or column far from the area of interest. The performance and efficiency of a finite volume upwind method on different grids has been analysed for a river flow test case simulation problem over dry bed. Conservative implicit methods belonging to the family of linearized upwind schemes have been presented. They are in general suitable for the numerical treatment of a variety of hydraulic problems. Among them, the basic linearized scheme is restricted to low CFL values for the simulation of unsteady transcritical flow with shocks. Two modifications of this scheme have been derived introducing the idea of splitting the scheme in a superposition of negative and positive wave influences: A semi-explicit and a corrected scheme. The performance of the implicit schemes has been tested in a dam-break problem arising from the shallow water equations. Both the semi-explicit and the corrected scheme are able to handle unsteady problems involving transcritical shocks with no stability restriction on the CFL. The best results are obtained with the corrected scheme. Future work is oriented towards a better understanding of the correcting parameter.

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