Floodplain Modelling Using Unstructured Finite Volume Technique

Floodplain Modelling Using Unstructured Finite Volume Technique

DHI

Technical Note January 2004

Floodplain Modelling Using Unstructured Finite Volume Technique January 2004

91071-01TechNote_.doc/ORS/MSD0401.lsm

Agern Allé 5 DK-2970 Hørsholm, Denmark Tel: Fax: Dept. fax: e-mail: Web:

+45 4516 9200 +45 4516 9292 [email protected] www.dhi.dk

CONTENTS 1

AIM ............................................................................................................................1-1

2 2.1 2.1.1 2.1.2 2.1.3 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.3 2.3.1 2.3.2 2.3.3 2.4 2.4.1 2.4.2 2.4.3

VERIFICATION..........................................................................................................2-1 Moving Hydraulic Jump ..............................................................................................2-1 Analytical solution ......................................................................................................2-1 Set-up ........................................................................................................................2-1 Results.......................................................................................................................2-2 Dam-break Flow through Sharp Bend ........................................................................2-3 Aims...........................................................................................................................2-3 Physical Experiment...................................................................................................2-3 Numerical Experiments ..............................................................................................2-4 Results.......................................................................................................................2-4 Skjern AA ...................................................................................................................2-8 Aim.............................................................................................................................2-8 Set-up ........................................................................................................................2-8 Results.......................................................................................................................2-9 Mariager Fjord..........................................................................................................2-11 Aim...........................................................................................................................2-11 Set-up ......................................................................................................................2-11 Results.....................................................................................................................2-13

3

REFERENCES...........................................................................................................3-1


i

DHI Water & Environment

1

AIM The aim of the work is to develop a two-dimensional floodplain model using unstructured grid approach.


1-1


2

VERIFICATION

2.1

Moving Hydraulic Jump The domain is a 200 m long channel with constant water depth of 1 m. At the east (left) boundary the incoming discharge of 10 m3/s is prescribed and the west (right) boundary is a fully reflecting wall.

2.1.1

Analytical solution H is the total water depth, U is depth-integrated velocity and C is the velocity of the front of the hydraulic jump. Subscript a indicates upstream values and subscript b downstream values. Before reflection from the east boundary: Ha = 2.51738 m, Ua = 3.97238 m/s Hb = 1.0 m, Ub = 0.0 m/s C = 6.59029 m/s After reflection from the east boundary: Ha = 4.82426 m, Ua = 0.0 m/s Hb= 2.51738 m, Ub= 3.97238 m/s C = -4.33487 m/s After 20 s the front of the jump will be located at x = 131.8058 m and after 50 s at x = 114.8097 m.

2.1.2

Set-up The computational domain is a channel with length of 200 m, width of 1 m and constant depth of 1 m. The west boundary is an open boundary and the east boundary is a fully reflecting wall. The finite volume mesh consists of quadrilateral elements with element size of 1 m. The time step is 0.05 s. At the open boundary the incoming unit discharge of 10 m*m*m/s is specified. No bed friction is included.


2-1


2.1.3

Results In Figure 2.1 the line series of the water depth is shown at T = 20 s (before the bore reaches the wall) and at T = 50 s (after the bore have been reflected from the wall).

Figure 2.1

Line series of the water level at T=20 s (top) and T=50 s (bottom). Red line: Analytical solution, Black line: New model.


2-2


2.2

Dam-break Flow through Sharp Bend The physical model to be studied combines a square-shaped upstream reservoir and an L-shaped channel. The flow will be essentially two-dimensional in the reservoir and at the angle between the two reaches of the L-shaped channel. However, there are numerical and experimental evidences that the flow will be mostly unidimensional in both rectilinear reaches.

2.2.1

Aims Two characteristics or the dam-break-resulting flow are of special interest, namely • the "damping effect" of the corner • the upstream-moving hydraulic jump which forms at the corner. The multiple reflections of the expansion wave in the reservoir will also offer an opportunity to test the 2D capabilities of the numerical models. As the flow in the reservoir will remain subcritical with relatively small-amplitude waves, computations could be checked for excessive numerical dissipation.

2.2.2

Physical Experiment Domain The channel is made of a 3.92 and a 2.92 m-long and 0.495 m-wide rectilinear reaches connected at right angle by a 0.495 x 0.495 m square element. The channel slope is equal to zero. A guillotine-type gate connects this L-shaped channel to a 2.44 x 2.39 m (nearly) square reservoir. Please note that the reservoir bottom level is 33 cm lower than the channel bed level. See Figure 2.2 for details. Initial and boundary conditions • Upstream reservoir: water initially at rest, with the free surface 20 cm above the channel bed level, i.e. the water depth in the reservoir is 53 cm. • Initial water depth in the channel: null for the dry-bed test, 1 cm for the wet-bed test. • End of the channel (downstream boundary): chute Physical parameters The Manning coefficients evaluated through steady-state flow experimentation are 0.0095 and 0.0195 s/m1/3, respectively, for the bed and the walls of the channel. Measurements The data acquisition will furnish water level at 0.1 s intervals for six gauging points, during 40 seconds. The locations of the gauges are shown in Figure 2.3.


2-3


2.2.3

Numerical Experiments Computational mesh Two unstructured meshes are used. A coarse mesh containing 18311 triangular elements and a fine mesh containing 36830 elements. For the coarse mesh the minimum edge length is 0.01906 m and the maximum edge length is 0.06125 m and for the fine grid the minimum edge length is 0.0099 m and the maximum edge length is 0.04343 m. The time step is 0.002 s for simulations using the coarse mesh and 0,001 s using the fine mesh. Initial and boundary conditions • Upstream reservoir: water initially at rest, with the free surface 20 cm above the channel bed level, i.e. the water depth in the reservoir is 53 cm. • At the downstream boundary, a free outfall (absorbing) boundary condition is applied. Physical and numerical parameters A constant Manning coefficient of 105.26 m1/3/s is applied. The flooding depth and drying depth are 0.0001 m and 0.00005 m, respectively.

2.2.4

Results In Figure 2.3 time series of calculated surface elevations at the locations of the six gauges are compared to the measurements. It is seen that the differences between the results obtained using the coarse and the fine mesh are very small. Only at location five small differences can be identified. In Figure 2.4 contour plots of the surface elevations are shown at T = 1.6, 3.2 and 4.8 s.


2-4


Figure 2.2

Set-up of the experiment by Franzao and Zech (2002)


2-5


Figure 2.3

Time evolution of the water level at the six gauge locations. Blue: Numerical calculation coarse grid, Black: Numerical calculation – fine grid and Red: Measurements by Franzao and Zech (2002)


2-6


Figure 2.4

Contour plots of the surface elevation at T = 1.6 s (top), T = 3.2 s (middle) and T = 4.8 s (bottom)


2-7


2.3

Skjern AA

2.3.1

Aim Use the finite volume ((FV) model to predict flood behaviour on the Skjern AA flood plain. The results with the FV model are compared with the results obtained using a finite difference (FD) model (Standard MIKE 21). Simulations are performed with the same structured mesh. In addition, a simulation is performed with the FD model using an unstructured mesh.

2.3.2

Set-up For the structured mesh a grid size of 25 m was chosen. For the FV simulations quadrilateral elements is used. The structured mesh contains 29506 elements. The unstructured mesh contains 6936 triangular elements. A fine resolution is applied to resolve the main rivers in the domain (the rivers are introduced as internal boundaries in the mesh with a resolution of 25 m), while at coarse resolution is applied to resolve the flood plains. The minimum edge length is 17.37 m and the maximum edge length is 207.13 m. The model topography is shown in Figure 2.5 and the unstructured mesh is shown in Figure 2.6. For all three simulations (FV and FD model) the time step is 1 s and the duration time is 30 hours (108000 time steps). At (x,y) = (-316723.06, 170739.14) (the upper right corner of domain) a source is specified with constant discharge of 20 m3/s. All boundaries are land boundaries (fully reflecting wall).

Figure 2.5

Computational domain and topography


2-8


Figure 2.6

Computational mesh

The bed friction is modelled using the Manning formulation and a constant Manning number of 20 is used. In the FD simulation eddy viscosity is applied (Smagorinsky coefficient of 0.5). For the flooding and drying depth is set to 0.01 m and 0.005 m, respectively. 2.3.3

Results Contour plot of the surface elevation for the three simulations are shown in Figure 2.7, Figure 2.8 and Figure 2.9.

Figure 2.7

Contour plot of the surface elevation for the FD model


2-9


Figure 2.8

Contour plot of the surface elevation for the FV model with structured mesh.

Figure 2.9

Contour plot of the surface elevation for the FV model with structured mesh.


2-10


2.4

Mariager Fjord Mariager Fjord is a narrow estuary with strong curvature. The topography of the domain is shown in Figure 2.10. The bed topography in the lower estuary is characterised by at narrow channel and tidal flats along the embankments. The depth of the channel is approximately 6 m. In the upper estuary the depth is up to 28 m. A fine resolution of the channel is essential for accurate simulations of the tidal dynamics.

2.4.1

Aim To demonstrate the application of the FD model to complicated domains.

2.4.2

Set-up The computational mesh is shown in Figure 2.11 and a close-up at the entrance is shown in Figure 2.12. The mesh contains 20428 elements. The minimum edge length is 23.31 m and the maximum edge length is 951.32 m. There are three open boundaries. At the east boundary a fully reflection boundary condition (wall) is applied and at the north and south boundary a level boundary condition is applied. The level variation at the north and south boundary is obtained by a time shift of the measured surface elevations at the entrance to Mariager Fjord. For the bed friction the Manning formulation is applied with a constant coefficient of 50 and a constant wind friction coefficient of 0.0008 is used. For the flooding and drying depth is set to 0.01 m and 0.005 m, respectively.

Figure 2.10

Computational domain and topography


2-11


Figure 2.11

Computational mesh

Figure 2.12

Close-up of the computational mesh


2-12


2.4.3

Results In Figure 2.13 the time series of the surface elevation at three locations are compared to measurements. The comparison is generally very good.

Figure 2.13

Time series of surface elevation at three locations. Red: Measurements and Black: New model.


2-13


3

REFERENCES /1/

Soares Franzäo S. and Zech Y., 2002, Effects of an sharp bend on dam-break flow


3-1
