An unstructured finite volume algorithm for predicting

An unstructured finite volume algorithm for predicting man-made flood routing in the Lower Yellow River

An unstructured finite volume algorithm for predicting man-made flood routing in the Lower Yellow River Junqiang Xia 1,2, Binliang Lin 1, Roger A. Falconer 1 and Baosheng Wu 2 School of Engineering, Cardiff University; 2State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing

1

Abstract To improve the flood carrying capacity in the Lower Yellow River, large scale prototype experiments in regulating water and sediment in the Yellow River have been conducted annually since 2002. The aim of this study is to predict the routing processes of these manmade floods, especially in the braided reach of the river with its complex geometry and bed topography. A two-dimensional hydrodynamic model is applied to simulate the flood routing in a wide and shallow braided reach. The model uses a finite volume algorithm based on unstructured mesh, and employs a Roes approximate Riemann solver with the MUSCL scheme and a predictor-corrector time stepping procedure which can give the second-order accuracy, both in time and space. In this model, some specific improvements have been made for simulating complex flood flows. A new method of spatial reconstruction of state variables on the inner and outer boundary interfaces is developed, which predicts more accurately the normal flow fluxes in the vicinity of wetting/drying interfaces. In addition, a reconstruction method with the second-order accuracy in space is used along the boundary interface, which can enhance the models numerical stability. The model is firstly calibrated using hydrological data from the man-made 2004 flood and then verified further by data from the man-made 2006 flood. Favorable agreements are obtained between model predictions and observed data. Simulated results also indicate that the predicted discharges at the outlet section by the enhanced method agree better with the observed values than those by the common method. Introduction Due to severe sedimentation, the shrinking of the main channel in the Lower Yellow River (LYR) has become obvious and the bankfull discharge in the braided reach decreased sharply from 7000 m3 s-1 in the 1970s to less than 20003000 m3 s-1 before the flood season in 2002. Therefore, large scale prototype experiments in regulating water and sediment is an important measure to prevent the riverbed from rising further in the LYR (Li, 2004). The Yellow River Conservancy Commission (YRCC) has experimented annually since 2002 with regulating water and sediment in the Yellow River. The released man-made floods are firstly routed along the braided reach and then propagated along the transitional and meandering reaches of the LYR. The Mengjin-Gaocun reach, with a length of about 286 km, is a famous braided reach in the LYR. Although channel deformation in the braided reach is a major component of the total LYRs channel adjustment, the bed deformation in the braided reach between Jiahetan and Gaocun is not so prominent during these experiments BHS 10th National Hydrology Symposium, Exeter, 2008

because it is located in the lower part of the braided reach. Furthermore, the released sediment concentrations at Jiahetan were relatively low, with a mean value of less than 10.0 kg/m3, and the mean scouring thickness was less than 0.06 m during the 2004 or 2006 experimental period. Therefore, the impact of morphological changes on the hydrodynamic process in this reach can be neglected during these man-made floods, and only the hydrodynamic process need be considered in a numerical model for flood forecasting in the reach between Jiahetan and Gaocun. In the literature, there are many numerical methods for the simulation of flood routing in the LYR (Wan et al., 2002; Wang et al., 2004), and the finite volume method may be a more successful approach to modelling the flood routing in this natural river with its complex geometry and bed topography. In this paper a finite-volume method model for 2D shallow water flows based on unstructured triangular grids is developed to simulate flood routing in natural rivers with complex geometry and topography. This model employs the Roe-MUSCL scheme to evaluate the normal

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J. Xia, B. Lin, R.A. Falconer and B. Wu

fluxes across the interface and a procedure of predictorcorrector time stepping, which can give the second-order accuracy both in time and space. Furthermore, this model employs a new approach to treating the wetting and drying fronts and another new method to reconstruct the state variables on the sides of an interface. The model is tested fully with the hydrological data obtained during the manmade floods and favorable agreements are obtained between the model predictions and observed data. In addition, the errors causing the predicted discharge and the sensitivity test of roughness coefficient are investigated.

Description of a 2D hydrodynamic model Governing equations for 2D shallow water flows Many practical flows in natural rivers, open channels and floodplain systems can be adequately described by a set of shallow water equations for 2D flows over a horizontal plane (Tan, 1992). The governing equations used in this model are such a set of depth-averaged 2D shallow water equations, which can be written in a general conservative form: wU wE wG wE wG S (1) wt wx wy wx wy in which U is the vector of conserved variables; E and G are the convective flux vectors of flow in the x and y and G are the diffusive directions, respectively; E vectors related to turbulent stresses in the x and y directions, respectively; and S is the source term including bed friction and slope, and Coriolis action. The above terms can be expressed in detail as:

U

ªhº «hu » E « », «¬ hv »¼

E

ª0º « » G «W xx » «W yx » , ¬ ¼

S

hu ª º « hu 2 1 gh 2 » G 2 « », «¬ »¼ huv ª0º « » «W xy » «W yy » ¬ ¼

hv ª º « huv » « », 2 2 «¬ hv 12 gh »¼

wU 'Ai ³ Fn (U)d * wt *

³* T (U)d* S(U)'A n

i

(3)

& ) n& ; 'Ai and G In which, Fn (U) (E,G ) n ; Tn ( U) (E, * denote the &area and interface of the control volume Ai , respectively; n is the outward unit vector normal to the boundary * . If the line integral can be approximated by summing the flux vector over each interface of the triangular control volume, the terms of line integral in Equation (3) can be discretized in the following way: 3

¦ F 'l

³ Fn (U)d*

ij

*

ij

and

3

³* T (U)d* ¦ T 'l n

ij

ij

(4)

j 1

j 1

In which, 'lij is the length of the interface; Fij and Tij are the numerical convective and diffusive fluxes across the interface * from cell Ai to cell A j . Tij is often estimated n ( n and ny are the n G simply by Tij &E x y x components of n in the x and y directions, respectively ). Thus, Equation (3) can be rewritten as:

wU i wt

1 'Ai

3

¦ F 'l ij

ij

1 'Ai

3

¦ (E n

x

n ) 'l S ( U ) G y ij i

(5) Therefore, the key problem in the above discretization procedure is how to evaluate Fij . At the interface * between cells Ai and A j , the problem could be taken as a locally one-dimensional Riemann problem in the direction normal to the interface according to Godunov (1959), so the normal flux can be obtained by an approximate Riemann solver. In this study, the Roe-MUSCL scheme is employed to evaluate the normal flux across an interface, which can be expressed as:

, and

ª º qs « » « hfv gh( Sbx S fx ) » « hfu gh( Sby S fy ) » ¬ ¼

volume Ai , the area-integral form of Equation (1) can be yielded easily. Assuming U to be the averaged value of each cell stored at the centroid of the cell, the area integral can be approximately used to calculate the unsteady term representing local time variation and the source term in Equation (1). Using the divergence theorem, the convective and diffusive terms in Equation (1) can be replaced by a line integral over the interface of the control volume. Thus, Equation (1) can be rewritten as:

Fij

(2)

Where u , v = depth-averaged velocities (m s-1) in the x and y directions, respectively; h = total water depth (m); qs = source (or sink) discharge per area (m2 s-1); g = gravitational acceleration (9.81m2 s-1); f = Coriolis acceleration due to the Earths rotation; Sbx and Sby = bed slopes in the x and y directions, respectively; S fx and S fy = friction slopes in the x and y directions, respectively; and W xx , W xy , W yx and W yy = turbulent shear stresses in different directions. Discretization method In this model, a kind of unstructured triangular mesh with a cell-centered finite volume method is adopted. Integrating the governing equation (1) over a control

j 1

j 1

& & & 0.5 [F ( U R )ij F( U L )ij ] n A [( U R ) ij ( U L )ij ]

^

`

(6)

where subscripts of L and R denote the left and right sides of * ; (U L )ij and (U R )ij are the reconstructions of U & on the right& and left sides of the interface, respectively. F ( U L )ij and F ( U R )ij are the normal fluxes on the left and right sides of * , respectively; and A is the flux Jacobian matrix evaluated by Roes average (Roe 1981). Further expression of Equation (6) can be found in the related publications (Sleigh et al., 1998; Wang and Liu 2000). Spatial reconstruction of state variables The spatial construction of state variables is a very important process in solving the water shallow equations by the finite volume method, which determines the spatial accuracy and resolution for a specific solver. In simulating the process of flood routing, the numerical approximation of the solution as a piecewise constant is equivalent to a

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first-order spatial accuracy (Zhao et al. 1996; Brufau et al. 2004), which is often inadequate to achieve a desired accuracy. For this reason, several higher-order implementations have been developed, which involve either a reconstruction of piecewise linear interpolation or a gradient reconstruction (Sleigh et al., 1998; Hubbard, 1999; Wang and Liu ,2000). Extensions of onedimensional reconstruction techniques are usually used for their simplicity and computational efficiency. The Roe-MUSCL scheme is used in the present model to reconstruct the state variables ( U R and U L ), which are piecewise linear functions of the state variables inside the control volume. In this scheme, the values of state variables at the centroids of two cells Ai and A j are U i and U j , respectively. In order to spatially reconstruct the state variables, their values at nodes of m and k are also introduced by using the method of inverse-power interpolation. Thus, U R and U L can be expressed as:

U j 12 M (rj )G U j and U L

UR

U i 12 M (ri )G U i

(7)

In which, M is a slope limiter function, and the purpose of this function is to limit the slope of linear variations and to avoid the non-physical oscillations in the numerical solution. G U i and G U j are the normal gradients of the state variables on the left and right hand sides of the common interface * . In order to maintain the accuracy in highly stretched triangles, M (ri ) and G U i can be expressed respectively as: § 2 a (U j U i ) · ¸ G U i U i U m and M ( ri ) M ¨ (8) © a b Ui U m ¹ In which, a or b is the distance between the centre of cell Ai or A j and the midpoint of interface * (Tan, 1992). M (rj ) and G U j can be obtained using the similar method. In this study, the minmod flux limiter is used to preserve the positivity of solution, which was extensively used in TVD-type numerical schemes. The above scheme of RoeMUSCL can obtain second-order spatial accuracy in space In the current study, the bed topography is very complex in the braided reach, and the influence of wet or dry interface on the calculation of numerical flux is accounted for as follows. Firstly, the water depth hij at the midpoint of interface can be determined by:

a (9) ( Z j Z i ) Z bij ab In which, Z i and Z j are the water levels at the centroids of two cells Ai and A j , respectively; and Z bij is the bed elevation at the midpoint of interface. If the value of hij is hij

Zi

less than the specified minimum water depth, this interface will be considered to be dry. Therefore, no numerical flux & can & transit this interface, which means that Fn ( U L )ij and Fn ( U R )ij should be equal to zero. Time integration The Runge-Kutta method is usually used to increase the time-wise accuracy of the scheme. In this model, to maintain the stability of solution and obtain the secondorder accuracy in time, a classic predictor-corrector approach is applied (Tan, 1992). Since this approach is an explicit scheme, the time step is also restricted by the Courant-Friendrichs Lewy (CFL) condition to ensure numerical stability in the model.

Treatment of boundary conditions The boundary conditions used in this model include two types of land boundary and open boundary. Existing FVM models often solve a boundary Riemann problem with the first-order accuracy in space to determine the normal flux across the boundary interface (Zhao et al., 1996; Yoon and Kang, 2004). However, the solution with the first-order accuracy can often cause numerical instability in cells including the open boundary with complex bed topography. Therefore, the developed model solves a boundary Riemann problem with the second-order accuracy in space, and treats two types of boundary conditions respectively. In the section, the state variables ( U L ) can be expressed respectively as (hL , uL , vL )T on the left side of a boundary interface, and these values can be calculated by using Equation (7). At the midpoint of a boundary interface, the state variables ( Ub ) on the T interface can be expressed as ( hb , ub , vb ) . Land boundary For a land boundary, the following condition is specified ubn 0 , ubW z 0 , and hb hL , in which the subscripts of n and W denote the outward normal and tangential directions of the interface; ubn and ubW are the velocity components in the local coordinates of n and W , and they can be expresses as ubn ub nx vb n y and ubW ub ny vb nx , respectively; hL =water depth on the left of this interface. Open boundary For the real flooding problem in natural rivers, three kinds of boundary conditions can be specified respectively. (1) As a water level hydrograph for an open boundary, hb at each node for this boundary can be obtained directly for each time level. ubn can be solved according to the theory of characteristics by ubn u Ln 2( ghL ghb ) , in which u Ln u L nx vL n y . (2) As a discharge hydrograph for an open boundary, qbn at each node for this open boundary can be estimated by distributing the total discharge according to the water depth and roughness along the inflowing profile. Thus hb and ubn can be solved iteratively from the relations of qbn hb ubn and hb (uLn 2 ghL ubn ) 2 / 4 g (Sleigh, 1998). (3) As a rating curve between water level and discharge for an open boundary, the corresponding water level and water depth at each node can be obtained by using the known discharge across this section, and the known relationship between water level and discharge. Then ubn can be obtained from step (1). For the above types of boundaries, an assumption of ubW u LW is usually adopted in a FVM model. According to the values of ubW and ubn in the local coordinates, the velocity components on the boundary interface in the x and y coordinates can be obtained from the following relations: ub ubn nx ubW n y and vb ubn ny ubW nx . Treatment of wetting and drying fronts For predicting flooding in a natural river, it is necessary to to simulate the evolution of wetting and drying fronts. The presence of extreme bed slopes and big changes in the irregular geometry often results in a great difficulty for numerical models, since an inaccurate treatment of wetting and drying fronts may lead to significant numerical errors.

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In using FVM for solving shallow water equations, a lot of improvements have been made for the treatment of wetting and drying fronts (Zhao et al., 1994; Sleigh et al., 1998). In the current study, a wetting and drying method developed for a regular grid in the finite difference model (Falconer and Chen, 1991) has been refined for the unstructured triangular grids employed herein. For detailed treatment of wetting and drying fronts in this model, see Xia et al., 2008).

reach consisted of 12 consecutive river bends with different curvatures, so it was also characterized by a meandering reach. To save computer time, only the potential submerged domain in the study reach was used in the computational domain. The total computational domain was composed of 18618 unstructured triangular grids, and the mesh in the main channel was refined locally, as shown in Figure 1. By using the known elevation of observed cross-sections, the bed elevation at each computational node can be obtained by the technique of smooth interpolation in the local region of main channels and floodplains. In this study, the simulation period was from June 16 to July 16 in 2004. The discharge hydrograph at the Jiahetan station was used as the upstream boundary condition, and a rating curve between water level and discharge at the Gaocun station was used as the downstream boundary condition. In this case study, the time step was 2 seconds, and the value of minimum water depth was 0.05 m for treating the evolution of wetting and drying fronts. The manning roughness was taken as a constant, and a value of 0.011 was used in all computational cells. Figure 2(a) shows the comparison between predicted and observed discharge hydrographs at Gaocun, and the overall agreement between them is generally good. The predicted discharge process experienced a similar change as that occurred at the inlet section; however, the maximum discharges of the second and third flood peaks were reduced to some extent. The calculated discharge of the flood peak is underestimated by about 9%, compared with the observed value. The calculated minimum discharge is 100200 m3 s-1 greater than the observed data. Figure 2(b) shows the progress in the water level with time at Jiahetan. In the initial 72 hours, the predicted water level is 0.20.4 m higher than the observed value, and the calculated water level is in good agreement with the observed data after 72 hours. Figure 3 shows the comparisons between predicted and observed water levels at different sites in the reach. It can be seen that the

Model tests The man-made 2004 and 2006 floods occurred during the third and fifth experiments with water and sediment regulation, respectively. During these experiments, a considerable amount of hydrological data was obtained. These observations laid the foundation to simulate the flood routing in the braided reach using the proposed hydrodynamic model. The data from the 2004 flood were used to calibrate the model and the 2006 data were used to verify the model further. The application feasibility of this model in the braided reach with complex geometry and bed topography was tested with these two case studies. In addition, the errors in the predicted discharge at the outlet section were analyzed and the sensitivity test of Manning roughness was conducted. Simulation of the man-made 2004 flood During the third prototype experiment in 2004, the manmade flood passed within the main channel only and the phenomenon of large scale overland flow did not happen. The discharge released from the Xiaolangdi Reservoir was controlled by an exact reservoir operation, and the maximum discharge released was less than 3000 m3 s-1, which was slightly less than the bankfull discharge in the study reach. The length of the braided reach from Jiahetan to Gaocun was about 80 km, with 46 observed crosssections. It can be seen from Figure 1 that this braided 18000

an het Jia WLST1

Water Level Station(WLST) 1=Jiahetan 6=Daliusi 2=Sanyizhai 7=Zhouying 3=Dongbatou 8=Qingzhuang 4=Chanfang 9=Gaocun 5=Caiji

15000

CS05

Bed Elevation (m)

___ Observed sections

60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76

WLST7 9000 CS30

WLST6

WLST8 CS35

CS15

3000

CS40 CS20

WLST2 CS10

CS45

WLST3

CS25

WLST5

W G LS ao cu T9 n

WLST4 6000

0 0

4000

8000

12000

16000

20000

24000

28000

32000

36000

40000

44000

48000

52000

56000

X (m)

CS39

CS04

Figure 1

Computational mesh and bed topography in 2004

292

60000

64000

68000

72000 CS40

CS05

Y (m)

12000

Hydrological Station 1=Jiahetan 2=Gaocun

76000


Discharge at Gaocun (m /s)

3000 3

2500 2000

(a)

1500

Observed Calculated

1000 500

76.6

0

72

144

216

(b)

76.2

288

360 432 Time (hours)

504

576

648

720

288


504

576

648

720

Observed Calculated

75.8 75.4 75.0 0

72

Figure 2

144

216

Comparison between predicted and observed data during the 2004 flood

agreements between them are satisfactory for this simulation in the complex braided reach. It can also indicate that the Manning roughness value of 0.011 is reasonable in this simulation. Simulation of the man-made 2006 flood During the 2006 experiment, the man-made flood also stayed within the main channel without the presence of overland flow. A new set of mesh with new bed topography was used in this simulation. The new domain was divided into 19636 unstructured triangular grids. The mean width of main channel in the reach was about 900 m, with a minimum value of 400 m. In this case study, the simulation period was from June 11 to July 2 in 2006. The time step, minimum water depth and roughness coefficient used in the case were identical to the values of the 2004 flood. Figure 4 shows the comparison between predicted and observed discharge hydrographs at Gaocun. The general

74.5

73.5 (b)Sanyizhai 73.0

Water level(m)

Observed Calculated

72.0

73.5

71.5

73.0 72.5

(c)Dongbatou

72.0

72.5

Observed Calculated

60 120 180 240 300 360 420 480 540 600 660 720 Time (hours)

70.5

67.5

70.0 69.5 (f)Daliusi 69.0

Observed Calculated

Water level(m)

68.0

60 120 180 240 300 360 420 480 540 600 660 720 Time (hours)

0

60 120 180 240 300 360 420 480 540 600 660 720 Time (hours)

Figure 3

(e)Caiji Observed Calculated 0

60 120 180 240 300 360 420 480 540 600 660 720 Time (hours)

64.0

63.5 67.0 66.5 (g)Zhouying 66.0

Observed Calculated

65.5

68.5

70.5

69.5 0

71.0

71.0

70.0

71.5 0

Water level(m)

74.0

Water level(m)

Water level(m)

74.0

trend of the flood is reasonably well predicted, although the discharge of flood peak is underestimated by about 10% compared with the observed value. The predicted discharge at Gaocun with the improved spatial reconstruction method can agree better with the observed value than those with the common reconstruction method. The predicted discharge with the common reconstruction method is greatly overestimated as the observed value is less than 1000 m3 s-1. Therefore, it is necessary to account for the effect of the wetting and drying interface on the numerical flux as the inflowing discharge is relatively small and the water level is relatively low in the channel. Figure 5 indicates the changing water level with time at Jiahetan; the predicted water level agrees better with the observed data and the maximum error between them is less than 10 cm. Figure 6 shows the comparisons between predicted and observed water levels at different sites during the 2006 flood. At Sanyizhai, the predicted water level is about 0.20.5 m higher than the observed value. At

Water level(m)

Water level at Jiahetan (m)

77.0 0

63.0 (h)Qingzhuang

62.5

Observed Calculated 62.0

0

60 120 180 240 300 360 420 480 540 600 660 720 Time (hours)

0

60 120 180 240 300 360 420 480 540 600 660 720 Time (hours)

Comparisons between predicted and observed water levels at different sites during the 2004 Flood

293


3

Discharge at Gaocun (m /s)

4000 3500 3000 Observed

2500

Calculated n=0.011 Calculated n=0.013

2000

Calculated n=0.015 Common Reconstruction method

1500 1000 0

60

Water level at Jiahetan (m)

Figure 4

120

180


360

420

480

Comparison between predicted and observed discharges at Gaocun during the 2006 flood

76.8 76.4 76.0 Observed

75.6

Calculated n=0.011 Calculated n=0.013

75.2

Calculated n=0.015

74.8 0

Figure 5

60

120

180


74.5

73.5

74.0

73.0

Water level(m)

Water level(m)

480

although the general trend in the predicted discharge in each flood was very close to reality. The factors causing this error are analyzed as follows. The inaccuracy of initial topography and the assumption of fixed bed may be the first factor to cause the error of discharge. The initial topographies used in the models were obtained by the interpolation from the

Error analysis of the predicted discharge at Gaocun From these two case studies, there was about 10% error between the predicted and observed discharge at Gaocun

73.5 (b)Sanyizhai Observed Calculated

73.0

72.5 (c)Dongbatou Observed Calculated

72.0

72.5

71.5 0

60

120 180 240 300 360 420 480 Time (hours)

0

73.0

60

120 180 240 300 360 420 480 Time (hours)

68.5

72.5

68.0

Water level(m)

Water level(m)

420

Comparison between predicted and observed water levels at Jiahetan during the 2006 flood

the observation sites of Dongbatou, Chanfang and zhouyin, agreements between the predicted and observed data are satisfactory for this simulation.

72.0 (d)Chanfang

71.5

Observed Calculated

71.0

67.5 (g)Zhouying Observed Calculated

67.0

70.5

66.5 0

Figure 6

360

60

120 180 240 300 360 420 480 Time (hours)

0

60

120 180 240 300 360 420 480 Time (hours)

Comparisons between predicted and observed water levels at different sites during the 2006 Flood

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available 46 cross sections. The average distance between two consecutive cross-sections was about 2 km, which was too large a spacing for the complex braided reach of the Lower Yellow River. Therefore, the generated topography may have represented the real one with low precision. In each simulation, the channel bed was assumed to be fixed, which could also cause error in the discharge. The treatment of Riemannn invariants was used at the boundaries neglecting the contribution of source terms, which may be the second factor to cause this error. The solution of Riemannn invariants at the boundaries is only applicable to a condition without friction and bed slope. However, bed friction and slope does exist in the cells, including the open boundary. Sensitivity test of roughness coefficient In any hydrodynamic model, the parameter of roughness coefficient is very important. A decreasing bed roughness coefficient produces higher flow velocities and therefore lower water levels and an increasing propagation velocity of the flood wave. In this section, the effect of different roughness coefficients on the discharge at Gaocun and the water level at Jiahetan was analysed for the case of the 2006 flood. Figure 4 also shows the water levels at Jiahetan for different roughness coefficients of 0.011, 0.013 and 0.015. From this figure, it can be seen that the water level will rise by 0.20.4 m as the roughness value increases from 0.011 to 0.015. The predicted water level for n=0.011 can obtain better agreement with the observed data. Figure 5 also indicates the predicted discharges at Gaocun for different roughness values. It can be found that different values of bed roughness coefficient have little effect on the predicted discharge at the outlet section, and a lower value of bed roughness coefficient can produce a better simulated result.

Conclusions A 2D hydrodynamic model based on the unstructured finite volume algorithm was developed to predict the manmade floods routing in a braided reach of the Lower Yellow River. This model adopted an approximate Riemann solver with the MUSCL scheme and a procedure of predictor-corrector time stepping. Furthermore, an effective treatment method of wetting and drying fronts was used to solve the problem of a moving boundary, and a new method of spatial reconstruction of state variables on inner and boundary interfaces was proposed to account for the influence of wetting or drying interfaces on the calculation of numerical flow flux. Therefore, the model has the capability to simulate the flood routing process in natural rivers with complex geometry and bed topography. The model was tested in detail with the hydrological data from the man-made 2004 and 2006 floods. Comparisons between predicted results and measurements indicate the applicability of this model to predict the flood routing in the braided reach of the LYR. In addition, the errors causing the predicted discharge at the outlet section were analyzed, and the effect of different roughness values on the flood routing processes was investigated by a sensitivity test.

Acknowledgements The research in this paper was supported by the Flood Risk Management Research Consortium (phase II) in UK, and also by the National Key Technologies R&D Program of China (Grant No. 2006BAB06B04) and the Science Fund for Creative Research Groups of the Natural Science Foundation of China (Grant No. 50221903).

References Brufau, P., Garcia-Navarro, P. and Vazquez-Cendon, M.E., 2004. Zero mass error using unsteady wetting drying conditions in shallow flows over dry irregular topography. Int. J. Numerical Methods in Fluids, 45, 10471082. Falconer, R.A. and Chen, Y., 1991. An improved representation of flooding and drying and wind stress effects in a 2D tidal numerical model. Proc. Inst. Civ. Eng., Part 2, 2, 659672. Godunov, S.K.,1959. A difference method for the numerical calculation of discontinuous solutions of hydrodynamic equations. Hubbard, M.E., 1999. Multidimensional slope limiters for MUSCL-type finite volume schemes on unstructured grids. J. Comput. Phys., 155, 5474. Li, G.Y., 2004. Ponderation and practice of the Yellow River Control. Yellow River Conservancy Press. Roe, P.L., 1981. Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys., 43, 357372. Sleigh, P.A., Gaskell, P.H. et al., 1998. An unstructured finite-volume algorithm for predicting flow in rivers and estuaries. Computers and Fluids, 27, 479508. Tan, W.Y., 1992. Shallow water hydrodynamics. New York: Elsevier. Wan, Q., Wan, H.T., Zhou, C.H. and Wu, Y.X., 2002. Simulating the hydraulic characteristics of the Lower Yellow River by the finite-volume technique. Hydrol. Process., 16, 27672779. Wang, G.Q., Xia, J.Q. and Wu, B.S., 2004. Twodimensional composite mathematical alluvial model for the braided reach in the Lower Yellow River. Water Internat., 29,455466. Wang, J.W. and Liu, R.X., 2000. A comparative study of finite volume methods on unstructured meshes for simulation of 2D shallow water wave problems. Mathe. Comput. Simulation, 53,171184. Xia, J.Q., Falconer, R A., Lin, B.L. and Wang, G.Q., 2008. Modelling floods routing on initially dry beds by a TVD finite volume method. Hydrol.Process, (under review). Yoon, T.H. and Kang, S.K., 2004. Finite volume model for two-dimensional shallow water flows on unstructured grids. ASCE J. Hydrau. Eng., 130, 678688. Zhao, D.H., Shen, H.W. et al., 1994. Finite-volume twodimensional unsteady flow model for river basins. ASCE J. Hydrau. Eng., 120, 863883. Zhao, D.H., Shen, H.W. et al., 1996. Approximate Riemann solvers in FVM for 2D hydraulic shock wave modeling. AASCE J. Hydrau. Eng, 122, 692 702.

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