ESTIMATION OF FLOW PAST A SIDE CAVITY USING A FINITE VOLUME METHOD WITH AN UNSTRUCTURED GRID Ryota Tsubaki1) and Ichiro Fujita2) 1) Department of Civil Engineering, Kobe University 1-1,Rokkodai-cho, Nada-Ku, Kobe 657-8501 Japan e-mail: [email protected] 2) Department of Civil Engineering, Kobe University 1-1,Rokkodai-cho, Nada-Ku, Kobe 657-8501 Japan e-mail: [email protected] INTRODUCTION Under natural conditions, rivers meander and rarely take a rectilinear path. Japanese river control policy was weighed heavily towards preventing disasters and ensuring enough water for industrial and domestic use, so rivers were straightened. But in recent years, that government policy was changed, and the biological and environmental condition of rivers became important. Now, controlled rivers also have a complex shape in plane and bed topography. Therefore, predicting the flow of rivers will become difficult because of increasing complexity of boundary shape. In this study, we develop a new numerical model with a Finite Volume Method using an unstructured mesh for flexibility of the boundary shape, and the MUSCL scheme is used for stability and accuracy in the numerical calculation. NUMERICAL MODEL We used a 2-dimensional depth averaged conservation equation are as follows: ∂U ∂E ∂F + + + S +ν ∂t ∂x ∂y unsteady

⎛h ⎜ U = ⎜ uh ⎜ vh ⎝

advection

model. The mass and two components of the momentum ⎛ ∂ ∂G ∂ ∂G ⎜⎜ h h + x x y ∂ ∂ ∂ ∂y ⎝

source

⎞ ∂H ∂I ⎟⎟ + + = 0 x y ∂ ∂ ⎠

viscosity

(1)

eddy viscosity

⎛ ⎞ ⎛ uh ⎞ ⎜ vh ⎟ ⎛0 ⎞ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ 1 ⎜ 2 2 ⎟ ⎟ , S = ⎜ − gh ( S 0 x − S ⎟ , E = ⎜ u h + gh ⎟ , F = ⎜ uvh 2 ⎜ ⎜ ⎟ ⎟ ⎜ − gh ( S − S ⎜ ⎟ ⎠ 0y ⎜⎜ v 2 h + 1 gh 2 ⎟⎟ ⎝ uvh ⎝ ⎠ 2 ⎝ ⎠

⎞ ⎛0⎞ ⎟ ⎜ ⎟ ⎟ G ) , = ⎜u ⎟ fx ⎟ ⎜v ⎟ ⎟ ⎝ ⎠ fy ) ⎠

(2)

where h is the depth of water， u and v are the x and y components of velocity and g is the gravitational acceleration. S fx and S fy are the friction forces of the bed. We use a two-dimensional form of Manning’s equation for this friction force S fx =

n 2u u 2 + v 2 n 2v u 2 + v 2 = , S fy h4/3 h4/3

(3)

and S 0 x and S 0 y are the bed slopes ∂z b ∂z , S0y = − b ∂x ∂y A bousinessq type eddy viscosity is used for the transverse shear. ⎛0 ⎞ ⎛0 ⎞ ⎜ ⎟ ⎜ ⎟ 2 H = h ⎜ − u ′ ⎟ , I = h ⎜ − u ′v ′ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ − v′2 ⎟ ⎜ − u ′v ′ ⎟ ⎝ ⎠ ⎝ ⎠ S0x = −

(4)

(5)

⎛ ∂v ⎞ 2 ⎛ ∂u ∂v ⎞ ⎛ ∂u ⎞ 2 (6) − u ′ 2 = 2 Dh ⎜ ⎟ − k , − u ′v ′ = Dh ⎜⎜ + ⎟⎟, − v ′ 2 = 2 Dh ⎜⎜ ⎟⎟ − k ⎝ ∂x ⎠ 3 ⎝ ∂y ⎠ 3 ⎝ ∂y ∂x ⎠ where Dh = αhu* is the depth averaged eddy viscosity coefficient and u * is bed friction velocity. The transverse averaged turbulent energy k is evaluated as k = 2.07u *2 ． Eq. (1) is integrated in the ‘Finite Volume’, and surface integrals are converted to line integral using the Gauss-Green theorem as below. ∂ Ω Ud Ω + ∫ ∂ Ω ( Edy − Fdx ) d Ω + ∫ Ω Sd Ω + ∂t ∫ (7) ⎞ ⎛ ∂G ∂G ∫ ∂ Ων ⎜⎜⎝ h ∂ x dy − h ∂ y dx ⎟⎟⎠ d Ω + ∫ ∂ Ω (Hdy − Idx )d Ω = 0

where

∫

Ω

dΩ are surface integrals on the finite Volume Ω , and

∫

∂Ω

dΩ are line integrals on the

outline of Ω . Treatment of the advection term The advection term is calculated using the MUSCL(Monotonic Upstream Scheme for Conservation Laws) scheme. The MUSCL scheme is a 2nd order accurate upwind method. Most established numerical models use a 1st order accurate upwind scheme for numerical stability, but 1st order schemes have strong numerical viscosity, so accurate estimation of flow dominated by shearing forces is difficult. Fig. 1 shows numerical viscosity in the advection equation. To estimate numerical viscosity, the velocity field is rotated (8) u = 50 − x, v = y − 50 200 triangles are used for the calculation mesh. An initial condition, the round area in Fig. 1 is set to a value of 1 and the rest of the field is 0. To rotate this field, 2nd order and 1st order schemes are used.

b) FDS(1st order upwind) a) MUSCL(2nd order upwind) Fig. 1 Numerical viscosity in the advection term The round profile is well conserved in MUSCL, compared to the result with the FDS scheme. STRAIGHT CHANNEL WITH A SIDE CAVITY

To evaluate the performance of the model, we compare numerical and experimental results of flow through a straight open-channel with a long side cavity having an aspect ratio of ten (Fig. 2). To maintain accessibility to the water surface, side cavities and stairs are installed in Japanese rivers in urban areas. These structures could be at risk of water levels rising in flood times, so predicting flow around this shape is important. The bed slope is 1/250, and the inlet flow rate is 10 l/s. The calculated and experimental results are shown in Figs. 3,4 and 5. The experimental value of the surface velocity field is measured using PIV techniques. Both velocity distributions show a similar pattern in the upstream area and shear region between the main flow and the cavity. A shock wave is seen in the measurement but this wave affects mainly only surface flow, so the induced velocity pattern is weak in depth averaged calculations. The circulatory flow on the inside of the cavity is stronger in the calculated result. A 3-dimensional complex flow occurs in the cavity and this causes strong dissipation of kinetic energy, but the 2-dimensional model has limitations in estimating such behavior.

7.5m 流れ Flow

0.2m 0.1m

2.0m 1.0m

Fig. 2 Plain view of a channel 0.2 0.1 0 0.5m/s -0.1 -1.3 -1.2 -1.1

-1

-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 m

0.1 0.2

0

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0.5

0.2 0.1 0 -0.1

m

Fig. 3

-1.4

-1.2

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Experimental result (Upper shows surface velocity (m/s). Lower shows water depth (m). ) 0.2 0.1 0 0.5m/s -0.1 -1.3 -1.2 -1.1

-1

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0.1 0.2

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m

-1.4

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Fig. 4 Calculated result (Upper shows depth averaged velocity (m/s), and lower shows water depth (m) ) 0.2

0.2

0.1

0.1

0

0

-0.1

m

-0.4

-0.2

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a) Experimental result

0.4

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b) Calculated result

Fig. 5 Contours of Froude number The locations and values of maximum water depth have good agreement. There is a local minimal value around the center of the cavity in the calculated result, at the center of the circulatory flow. This circulation is weak in the experimental result, so the depth distribution around the center of the cavity is flatter in the measurements. After contraction of the flow, a hydraulic jump stretches from the corner of the cavity to the left side bank and is reflected by the wall. Fig. 5 shows the distribution of Froude number. The calculated result shows a similar pattern to the measured one. FLOW WITH SPUR DIKES ON THE UJI RIVER

The estimation of effects of spur dikes in normal and flood periods is an important challenge. Fig. 5 a) shows bed elevation distribution of a section of the Uji river that runs through Uji, Kyoto, Japan. Figs. 5 b)

260 240 200 180 160 140

0

0

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20

120 60

80 m

a) Bed elevation (m)

100

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200 m 180 160

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Flow

0

260

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60

80 m

100

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260 240

and c) show measured and calculated velocity distortions. An ADCP (Acoustic Doppler current profiler) was used to measure bed elevation. The LSPIV (Large scale PIV) technique was applied to measure velocity distribution. An unstructured mesh was generated to represent the detailed bed form. The minimum size of a mesh triangle is about 5m and maximum is about 10m. The total number of triangles is about 5000.

b) Measured velocity intensity (m/s) Fig. 6 Flow past spur dikes on the Uji river

20

60

80

100

m

c) Calculated velocity intensity (m/s)

Here, the measured and calculated velocity profiles show good agreement especially the area around (x=80, y=140). In this area, a mound is shown in the bed elevation. The flow is accelerated while passing over this mound in both the measured and calculated results. However, the numerical results show more energy in the recirculation region than the measurements do, because of limitations on dissipation of energy in 2-D models. Though, a combination of numerical estimation and detailed measurements using ADCP or/and LSPIV is a powerful tool in measuring and recognizing flow in the field. CONCLUSIONS

An unstructured depth averaged flow model is developed in this study. In this model, an accurate advection scheme is used. First, the flow of a channel with a side-cavity is used to verify accuracy and applicability. Second, river flow past a spur dikes is calculated, and calculated results shows good agreement with measured values. The authors are grateful to Dr. Muto of Kyoto University for his cooperation. REFERENCES

Fujita I., Muste M. and Kruger A. Large-scale particle image velocimetry for flow analysis in hydraulic engineering, Journal of Hydraulic Research, IAHR, 1998, 36, pp.397-414.