Comparison between a coupled 1D-2D model and a ...

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Comparison between a coupled 1D-2D model and a fully 2D model for supercritical flow simulation in crossroads a

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Rabih Ghostine , Ibrahim Hoteit , Jose Vazquez , Abdelali Terfous , Abdellah Ghenaim & Robert Mose

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Postdoctoral Researcher, Department of Applied Mathematics, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia b

Professor, Department of Applied Mathematics, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia Email: c

Professor, Department of Fluid Mechanics, Institut de Mécanique des Fluides et des Solides, Strasbourg, France Email:

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Professor, Department of Fluid Mechanics, Institut National des Sciences Appliquées, Strasbourg, France Email: e

Professor, Department of Fluid Mechanics, Institut National des Sciences Appliquées, Strasbourg, France Email: f

Professor, Department of Fluid Mechanics, Institut de Mécanique des Fluides et des Solides, Strasbourg, France Email: Published online: 01 Dec 2014.

To cite this article: Rabih Ghostine, Ibrahim Hoteit, Jose Vazquez, Abdelali Terfous, Abdellah Ghenaim & Robert Mose (2015) Comparison between a coupled 1D-2D model and a fully 2D model for supercritical flow simulation in crossroads, Journal of Hydraulic Research, 53:2, 274-281, DOI: 10.1080/00221686.2014.974081 To link to this article: http://dx.doi.org/10.1080/00221686.2014.974081

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Journal of Hydraulic Research Vol. 53, No. 2 (2015), pp. 274–281 http://dx.doi.org/10.1080/00221686.2014.974081 © 2014 International Association for Hydro-Environment Engineering and Research

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Comparison between a coupled 1D-2D model and a fully 2D model for supercritical flow simulation in crossroads RABIH GHOSTINE, Postdoctoral Researcher, Department of Applied Mathematics, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia Email: [email protected] (author for correspondence) IBRAHIM HOTEIT, Professor, Department of Applied Mathematics, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia Email: [email protected] JOSE VAZQUEZ, Professor, Department of Fluid Mechanics, Institut de Mécanique des Fluides et des Solides, Strasbourg, France Email: [email protected] ABDELALI TERFOUS, Professor, Department of Fluid Mechanics, Institut National des Sciences Appliquées, Strasbourg, France Email: [email protected] ABDELLAH GHENAIM, Professor, Department of Fluid Mechanics, Institut National des Sciences Appliquées, Strasbourg, France Email: [email protected] ROBERT MOSE, Professor, Department of Fluid Mechanics, Institut de Mécanique des Fluides et des Solides, Strasbourg, France Email: [email protected] ABSTRACT In open channel networks, flow is usually approximated by the one-dimensional (1D) Saint-Venant equations coupled with an empirical junction model. In this work, a comparison in terms of accuracy and computational cost between a coupled 1D-2D shallow water model and a fully twodimensional (2D) model is presented. The paper explores the ability of a coupled model to simulate the flow processes during supercritical flows in crossroads. This combination leads to a significant reduction in the computational time, as a 1D approach is used in branches and a 2D approach is employed in selected areas only where detailed flow information is essential. Overall, the numerical results suggest that the coupled model is able to accurately simulate the main flow processes. In particular, hydraulic jumps, recirculation zones, and discharge distribution are reasonably well reproduced and clearly identified. Overall, the proposed model leads to a 30% reduction in run times.

Keywords: Coupled 1D-2D model; finite volume; fully 2D model; shallow water equations; supercritical flow Ghenaim, 2012; Ghostine et al., 2013; Hager, 1989; Hsu, Lee, Shieh, & Tang, 2002; Kesserwani et al., 2008; Shabayek, Steffler, & Hicks, 2002), (ii) a full two-dimensional approach (Chong, 2006; Ghostine et al. 2009; Mignot, Paquier, & Rivière, 2008; Rong & Mao, 2003; Shamloo & Pirzadeh, 2007; Shettar & Murthy, 1996), and (iii) a three-dimensional approach based on solving the Navier-Stokes equations (Ghostine et al., 2009; Goudarzizadeh, Hedayat, & Jahromi, 2010; Huang, Weber, & Lai, 2002; Neary, Sotiropoulos, & Odgaard, 1999).

1 Introduction In the design of flood control channels, one of the most important hydraulic problems is the analysis of the flow conditions at channel junctions. The numerical simulation of flow in sewerage and urban networks can be treated following different approaches: (i) a one-dimensional approach in branches coupled to analytical semi-empirical formulations for treating the junctions (García-Navarro & Savirón, 1992; Gurram, Karki, & Hager, 1997; Ghostine, Vazquez, Terfous, Mose, &

Received 19 July 2013; accepted 2 October 2014/Open for discussion until 29 Oct 2015. ISSN 0022-1686 print/ISSN 1814-2079 online http://www.tandfonline.com 274


Journal of Hydraulic Research Vol. 53, No. 2 (2015)

One-dimensional simulation provides a reasonable framework to model the flow in the streets, but it is not suitable to model the flow in street intersections where the flow is strongly two-dimensional or even three-dimensional. The onedimensional approach cannot be used to represent the flow characteristics within junctions such as oblique jumps, separation zones, recirculation zones, etc. Two-dimensional models can be efficiently applied to a large set of problems as they represent a good compromise between implementation efforts, computation time, and simulation accuracy. Two-dimensional models require more computational time and data than the one-dimensional models, but their results are more accurate and more informative, and allow establishing more detailed risk maps. Therefore, the combination of these two approaches can be beneficial. Coupled 1D-2D models have been developed in recent years and successfully applied to large and complex river systems (Bladé et al., 2012; Gejadze & Monnier, 2007; Marin & Monnier, 2009; Verwey, 2001). By means of a mixed approach, it is possible to use the more accurate flow description of a 2D scheme where needed, and use the 1D approach elsewhere. In this way, computer time and memory are reduced, as the number of calculation points in a 1D scheme is radically lower than in a 2D scheme. The first integrated 1D-2D models were developed as extensions of already existing 1D models (Cunge, 1975), where a one-dimensional model of looped channel flow, solving the Saint-Venant equations with the Preissmann Scheme, was coupled with a storage cell algorithm using the mass conservation equation to link domains. This method was subsequently referred to as a 1D-quasi 2D model, and was soon adopted in the first versions of Mike-11. It was also later used by others with slight variations, as in Bladé, Gómez, & Dolz (1994), and in the present version of Hec-Ras. Quasi-2D schemes can be used in an uncoupled way (Di Baldassarre, Castellarin, Montanari, & Brath, 2009) if the riverbed elevation is higher than the floodplain, or if there is a dike or embankment between them. In this case, backwater effects are not included (Hunter, Bates, Horritt, & Wilson, 2007). Quasi-2D schemes are always limited in determining the front wave advance and recession over the floodplain. Several other approaches are possible for coupling a 1D model with a 2D model. The very first research integrating fully 1D with fully 2D schemes was developed to study the hydrodynamics of the Venice Lagoon (D’Alpaos & Defina 1993) using finite element schemes. This was based on a new approach in which 1D channels would work as open channels for low water elevations, and as pressure conduits below 2D elements for high tide flows. Numerical results were compared with field data (Defina, D’Alpaos, & Matticchio, 1995) and also with a fully two-dimensional model (Carniello, D’Alpaos, Defina, & Martini, 2003; D’Alpaos & Defina 2007), and satisfactory results were obtained. Integrated 1D and 2D numerical schemes were also used for flood modelling in the Netherlands, using implicit schemes with

A coupled 1D-2D model

275

Sobek software (Verwey 2001). Here, Sobek used two separate computational layers linked to each other via water level compatibility. Other combinations of 1D and 2D approaches have also been employed. Using the LISFLOOD-FD software (Bates & De Roo, 2000; Horritt & Bates, 2001; Huang, Rauberg, Apel, & Lindenschmidt, 2007), the flow in the river main channel is solved in 1D and the overbank inundated areas are solved in 2D by means of the diffusive wave equation (Hunter et al., 2007). In addition, finite volumes and Riemann solvers in the river have been combined with storage cells in the floodplain (Villanueva & Wright, 2006). 1D-2D integration was achieved by means of the mass conservation equation, using the Parabolic Shallow Water Equations (a simplified version of the Shallow water equations). A superposition approach was also proposed by a number of studies (Fernández-Nieto, Marin, & Monnier, 2010; Gejadze & Monnier, 2007; Marin & Monnier, 2009). In such an approach, instead of decomposing the original 1D (network) model, one superposes the 2D model (so-called “local zoom model”). The superposition approach presents some advantages. The original 1D model remains intact and the 2D local models can be performed with their own dynamics (typically, time steps and mesh grids are much smaller for 2D solvers than for 1D solvers). Morales-Hernandez, Garcia-Navarro, Burguete, & Brufau (2013) presented a conservative strategy to couple 1D and 2D models. 1D and 2D models are formulated using a conservative upwind cell-centred finite volume scheme. The discretization is based on cross-sections for the 1D model and with triangular unstructured grid for the 2D model. In this paper, we aim to study the ability of a coupled 1D-2D model to represent the set of structures developed in crossroads during supercritical flows, such as water depths, the location of the hydraulic jumps, the recirculation zones, and the flow discharge distribution in the downstream branches of the crossroad. The model is based on the numerical fluxes of the numerical scheme and should take into account the outgoing and incoming fluxes. Data from cells at the 1D-2D interfaces are used to determine the left (or right) state of the Riemann problem to be solved before calculating the fluxes through the interfaces. A fully 2D model is also applied in order to compare the two approaches. Finally, the results are evaluated in terms of accuracy and computational cost. 2 Two-dimensional Saint-Venant equations The two-dimensional shallow water equations, which represent mass and momentum conservation in a 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: ∂E ∂G ∂U2D + + = S2D ∂t ∂x ∂y

(1)

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R. Ghostine et al.

in which ⎛

U2D

⎛

⎞

⎞

hu 2 ⎝ E = hu + gh2 /2⎠ , huv ⎞

h = ⎝hu⎠ , hv ⎛

hv ⎠, huv 2 2 hv + gh /2

G=⎝


and S2D = S0 + Sf =

0 ghS0x ghS0y

+

(2)

0 −ghSfx −ghSfy

∂U2D,i Ai + ∂t

√

Sfx =

u2

+

h4/3

v2

is the source term.

,

Sfy =

n2M v

√

u2

+

v2

h4/3

The one-dimensional shallow water equations in conservative form for prismatic channel are: ∂F1D ∂U1D + = S1D ∂t ∂x

(4)

in which U1D

S1D =

F1D = ⎝ Q2

A

Q +

→ n)ds= (F2D −

S2D d i

(6)

i

4

→ n k )lk (F2D,k −

(7)

k=1

∂i

⎡ 1 → → n k = ⎣(F2D (U2D,L ) + F2D (U2D,R ))− nk F2D,k − 2 ⎛ ⎞⎤ 3 −⎝ α˜ j | lj | ej ⎠⎦

⎞ A2 ⎠ , 2B

where (U2D,L )k and (U2D,R )k are the values of U2D on the left ek are the and right sides of the edge k, respectively, lk and eigenvalues and eigenvectors of the Jacobian matrix of the Roe approximate flux vector:

0 , gA(S0 − Sf )

t n n n + tS1D,i F1D,i+1/2 − F1D,i−1/2 x

(8)

j =1

˜ + c l1 = u˜ nx + vnx

where U1D is the vector of conserved variables, F 1D is the flux vector and S 1D is the source term. A represents the wetted crosssectional area, Q is the flow discharge, S 0 is the channel slope, S f is the friction slope and B is the channel width. A finite volume numerical scheme for these equations can be written as: n Un+1 1D,i = U1D,i −

where k represents the edges index of the cell i . The vector − → nk is the unit outward normal to edge k, lk is the length of the side, and F k is the numerical flux approximated by Roe Riemann solver (Roe, 1981):

3 One-dimensional Saint-Venant equations

⎛

→ n)ds= (F2D −

→ where Ai is the area of the cell i , − n is the outward normal unit vector, F 2D = (E, G) and ds is the contour differential. The contour integral is approached by a sum over the cell edges. The normal flux is evaluated via an upwind flux difference splitting technique:

(3)

where nM is Manning’s roughness coefficient.

A = , Q

∂i

In the above equations, u and v denote the velocity components in the x and y directions, respectively; h is the water depth; g is the acceleration due to gravity; (S 0x , S 0y ) are the bed slopes in the x and y directions, and (S fx , S fy ) are the friction slopes in the x and y directions, respectively. The friction slopes are estimated by using Manning’s formula: n2M u

A discrete approximation of Eq. (1) is applied in every cell i at a given time so that the volume integrals represent integrals over the area of the cell and the surface integrals represent the total flux through the cell boundaries. Denoting by U2D,i the average value of the conservative variables over the volume i at a given time, from Eq. (1) the following conservation equation can be written for every cell as:

˜ l2 = u˜ nx + vnx l3 = u˜ nx + vnx ˜ − c ⎛ ⎞ ⎛ ⎞ 1 0 ⎜ ⎟ ⎜ ⎟ e2 = ⎜ e1,3 = ⎜ u ± cnx ⎟ cny ⎟ ⎝ ⎠, ⎝− ⎠ v ± cny cnx

(5)

4 Numerical method A cell-centred finite volume method is formulated where all the dependent variables of the system are represented as piecewise constants (first order). In the two-dimensional approach presented in this work, the spatial domain of integration is covered by a set of rectangular cells.

(9)

(10)

and α˜ k are the wave strengths: h 1 ± (hu)nx + (hv)ny − (u.nx + vny)h 2 2 c 1 α˜ 2 = ((hv) − vh)nx − ((hu) − uh)ny c (11)

α˜ 1,3 =



where = ( )R − ( )L and uk , vk and ck are the averaged states of the conserved variable: √ √ √ √ hL uL + hR uR hL vL + hR vR uk = √ , vk = √ , √ √ hL + hR hL + hR hL + hR ck = g 2

t n − n n k lk + tS2D,i F → Ai k=1 2D,k 4

(13)

(15)

where UL = (h, hu, hv) comes from the two-dimensional cell and UR = (A, Q, 0) is from the one-dimensional cell. We added a third component to UR since the calculation of the twodimensional flux via Roe’s solver requires three components. For the same reason as above, the third component of UR is set to zero. The same technique is used for the lateral coupling zones. Finally, a common element in both 1D and 2D models is the evaluation of the time step. When dynamically computed from the CFL condition, t can be different in both models. The global t taken is the minimum value of the two models, that is:

5 Coupled model

t = min(t1D , t2D )

1D and 2D Saint-Venant equations are solved in a single coupled model. In areas where the flow is mainly 1D (streamlines, perpendicular to cross-sections, similar water elevation across the river), the 1D approach has the advantage of computationally being much less demanding, while in other areas where this 1D approach is not appropriate (such as recirculation areas and oblique hydraulic jumps), the 2D models are adopted. The strategy presented here is based on the flux exchange between 1D and 2D cells. It takes into account the outgoing and incoming fluxes. The information exchange between the two models is done through the flux terms. Data from cells at the 1D-2D interfaces are used to determine the left (or right) state of the Riemann problem to be solved before calculating the fluxes through the interfaces via Roe’s solver. For the frontal coupling zones, the one-dimensional flux in Eq. (5) is calculated as follows: ROE (UL , UR ) F1D = F1D

(16)

Once t is calculated, each model computes separately its own conserved variables according to Eqs. (5) and (13). 6 Numerical results and discussion In this section, the coupled 1D-2D model and the fully 2D model are compared when simulating supercritical flows in four identical rectangular channels that were joined at 90°, as indicated in Fig. 1. The four branches have the same width B = 0.3 m and length L = 2 m (two entries and two exits) as shown in Fig. 2. Qux , hux , Quy and huy are the flow discharges and the water depths at the upstream branches, respectively. Qdx , hdx , Qdy and hdy are the flow discharges and the water depths at the downstream branches, respectively. The 1D Saint-Venant equations are applied in the two upstream branches up to a distance of 2B

(14)

Qdy, hdy

where UL = (A, Q) comes from the last cell of the onedimensional mesh and UR = (hm , hm um ) from the average of the two-dimensional cells located at the 1D-2D interface, as shown in Fig. 2. The third component of UR is set to zero since the flow is in the x-direction. The two-dimensional flux in Eq. (13)

Colour online, B/W in print


Finally, the solution at each cell i at the time increment n + 1 is written as follows: n Un+1 2D,i = U2D,i −

is calculated as follows: ROE (UL , UR ) F2D = F2D

(12)

277

2m

0.3 m Qdx, hdx

Qux, hux

y x

0.3 m

Quy, huy

Figure 1

Coupled 1D-2D strategy for the crossroad

Figure 2 Schematic layout of the crossroad

2m

278

Figure 3



R. Ghostine et al.

Description of the main supercritical flow types in a crossroad

from the junction to ensure that the flow is one-dimensional in this area. Because the flow is supercritical and the branches are small, the flow is two-dimensional in the other areas where the 2D Saint-Venant equations are applied. The comparison is carried out in terms of accuracy and computational cost. The main objective of this application is to demonstrate the capability of the coupled model to represent the set of structures developed in crossroads during supercritical flows. Due to the slopes and the experimental fixed flow discharges, the uniform flows are in a supercritical regime and should remain so until the downstream ends of the branches if no singularity was introduced. However, when reaching the junction, the two flows meet and suddenly deviate from each other compared with their initial direction, creating two hydraulic jumps that can be oblique and confined in the junction, or normal and located in the upstream branches. The zones of recirculation appear within each of the two downstream branches. In addition, gravity waves appear in the downstream branches and reflect on the walls of the output branches until the end of these branches. Several forms of flow can appear according to the slopes and the used flow discharges. These flows can be classified into three types corresponding to the supercritical/subcritical transitions already observed by Gisonni & Hager (2002) for free-surface flow junctions in pipes. Concerning junctions at open channels, two types were observed by Nania, Gomez, & Dolz (2004) and a third one by Mignot et al. (2008). They are indicated in Fig. 3 by Type I, Type II and Type III, respectively. Type I corresponds to a flow with a normal hydraulic jump within each upstream branch. Type II corresponds to a flow for which a normal hydraulic jump takes place within the upstream minor branch and an oblique hydraulic jump takes place within the junction. Type III is a type of flow for which two oblique hydraulic jumps appear in the junction. Four supercritical flows are calculated numerically using the coupled 1D-2D model on 16,500 cells and the fully 2D model on 24,900 cells (x = y = 1 cm). The characteristics of these flows are presented in Table 1. The initial conditions are zero flow discharges in the four branches and equal water depth for the four cases, respectively. The Manning roughness coefficient used is 0.0087 sm–1/3 . Since the flow is supercritical, water depths and flow discharges are imposed at the inlets and free outfall conditions are imposed at the outlets. Figure 4 presents the contour of the water depth calculated by the coupled model and the full 2D model for the four flows

Table 1 Characteristics of the four studied flows Case C1 C2 C3 C4

Qux (l s–1 )

hux (cm)

Quy (l s–1 )

huy (cm)

Slope

3.50 5.00 5.11 5.02

1.10 1.15 1.15 1.15

3.55 2.00 1.01 3.99

1.11 0.66 0.44 1.00

3% 5% 5% 5%

quoted above. It appears from Fig. 4 that the set of structures observed in the results of the fully 2D model are reproduced by the coupled model. In fact, the prediction and the location of the right hydraulic jumps in the upstream branches, the oblique hydraulic jumps in the junction, the recirculation zone, and the beads near the downstream corner of the junction are clearly identified by the coupled model. However, differences between predicted flows by the two models appear. First, we note that the deflection angles (oblique jumps) and the locations of the right jumps predicted by the coupled model differ from those predicted by the full model. The source of these errors can be due to the fact that the flow at the 1D-2D interfaces is not completely one-dimensional. The velocity predicted in the coupled model at which the flow reaches the junction is slightly greater than that predicted by the full model. Therefore, the deflection angles are underestimated by the coupled model and the right jumps are closer to the junction (see Fig. 4). Consequently, one can see differences in (i) water levels next to the hydraulic jumps; (ii) the size and shape of beads predicted near the corners; and (iii) the size and shape of the recirculation zones. These differences are particularly related to the differences in the prediction of the hydraulic jumps by the two models. Concerning the flow discharges, the differences in the distribution of the flow discharges in the downstream branches between measured (when available) and predicted values are presented in Table 2. For a more accurate study of the distribution of the flow discharges in the downstream branches, we introduce an estimator of simulation quality E QT defined as: EQT = max

|Qdxc − Qdxm | |Qdyc − Qdym | , QT QT

(17)

where QT is the total upstream flow discharge (QT = Qux + Quy ). The values of the indicator E QT are presented in Table 3.



Figure 4


279

Water depth fields (in mm) of computed flow configurations from Table 1: (left) coupled 1D-2D model, (right) full 2D model Table 2 Values of the downstream flow discharges measured and calculated Measured

C2 C3 C4

Coupled model

Fully 2D model

Qdx (l s–1 )

Qdy (l s–1 )

Qdx (l s–1 )

Qdy (l s–1 )

Qdx (l s–1 )

Qdy (l s–1 )

6.26 5.73 5.42

0.71 0.39 3.59

6.42 5.83 5.77

0.58 0.29 3.24

6.36 5.80 5.75

0.64 0.32 3.26

The results suggest that the coupled 1D-2D is able to satisfactorily predict the distribution of the flow discharges in the downstream branches. It also appears that the major downstream flow rate is overestimated for all the cases. Nevertheless, the indicator E QT is limited to about 1.5% for cases C2 and

C3, and 3.9% for the case C4. In view of these results, the coupled 1D-2D model is believed to be useful for reducing running time while preserving the solution accuracy. Overall, in our specific application the achieved CPU time reduction is around 30%.

280


R. Ghostine et al. Table 3 Values of the indicator E QT Case C2 C3 C4

Coupled model

Fully model

1.57% 1.63% 3.88%

0.71% 1.14% 3.66%

Subscripts d u

= downstream branch = upstream branch

References


7 Conclusion The present study analysed the ability of a coupled 1D-2D model to simulate the flow processes during supercritical flows in crossroads. The one-dimensional Saint-Venant equations are used in branches where the flow is one-dimensional and was coupled to the two-dimensional Saint-Venant equations to give detailed description of the flow in the junction domain. A fully 2D model is also applied in order to compare the two approaches in terms of accuracy and computational cost. The two approaches were implemented using an explicit finite volume method based on Roe’s Riemann solver. The coupled model proved to be robust, providing stable results in terms of both water depth and flow discharge. The results showed that the model was able to simulate the overall flow processes occurring in supercritical flows. Hydraulic jumps, recirculation zones, beads, and discharges’ distribution are therefore reasonably well reproduced and clearly identified. Moreover, the CPU time reduction is around 30% in our setting. In view of these results, the coupled 1D-2D model is believed to be useful for reducing running time while preserving the solution accuracy and level of detail.

Notation A B E, F, G g h J l n nM Q S S0 Sf t U u, v x, y t

= = = = = = = = = = = = = = = = = =

area (m2 ) channel width (m) flux vectors (–) gravity acceleration (ms−2 ) water depth (m) Jacobian matrix (–) length of an edge (m) normal vector (–) Manning’s roughness coefficient (sm–1/3 ) flow discharge (m3 s–1 ) source term (–) bed slope (m2 s–2 ) friction slope (m2 s–2 ) time (s) conservative variables (–) velocities in the x and y directions (m s–1 ) Cartesian coordinates (m) time step (s)

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