NUMERICAL SIMULATION OF EFFLUENT FLOW AT THE NATORI RIVER MOUTH USING BOUNADARY-FITTED COORDINATES HITOSHI TANAKA Department of Civil Engineering, Tohoku University, 06 Aoba, Sendai 980-8579, Japan e-mail:
[email protected]
NOBUAKI ITONAGA Hiroshima Branch Office, West Japan Railway Company, 1-25 Kubo, Kaita, Aki, Hiroshima 736-0046, Japan e-mail:
[email protected] A detailed comparison has been made between rectangular grid and BFC ( Boundary Fitted Coordinates ) grid computations for effluent current through a curved channel formed by the jetties at the Natori River mouth, Japan. In the computation of the former grid system, it is found that the discretization of the curved jetties causes roughness effect, resulting in rise of water level and reduction of the velocity in the region between the jetties. It is observed the reduction of the velocity becomes more considerable with the increase of grid size. In contrast to this result, much smoother distribution of water surface profile can be obtained using BFC grid system.
1
Introduction
The change in coastal morphology in an estuary is affected by the sediment supply from a river to the coastal area due to flood. There have been a number of studies related to morphological change during a flood at the Natori river mouth in the past ( Tanaka et al. [7], Tanaka and Samad [8] ). Actual flood events were considered in these studies and a good agreement was found between the model prediction and the field data of water level and river mouth topography. This model is equipped to handle the spiral flow occurring in the curved channel at the river outlet portion. According to the latest computation of Itonaga et al. [3], however, the reduction of the velocity becomes more considerable in the jettied region with the increase of grid size, implying that the descretization error of the curved jetties becomes more predominant with the increase of grid spacing in the computation of hydrodynamic model. In the present study, a comparison has been made between rectangular and BFC grid predictions for hydrodynamic behavior through a curved channel formed by the jetties in the outlet region of Natori River. This comparison enables us to further validate the numerical model and observe the detailed flow pattern between the jetties.
2
Field Site
Natori river mouth is located in the Miyagi Prefecture, Tohoku District, Japan. The river has a length of 55km and catchment area of 939km2 (Figure 1). There are two jetties at the river mouth and the sand spit located near the left jetty normally remain attached to it. The usual height of this sand spit is 2.5m. In order to avoid very high water levels in the upstream areas the flushing of this sand spit is required so that the flood discharge may be passed to ocean without causing severe backwater effect. Te izan Ca na l
N
0
1km Idou ra La goo n
Natori River
N
SE NDAI Hiro ura La go on
JAPAN
Figure 1. Map of the Natori River mouth
3
Methodology of Numerical Computation
3.1 Governing Equations In the present computation, two kinds of coordinate systems are employed; (1) ordinary Cartesian coordinates and (2) boundary-fitted coordinates (BFC). The governing equations of the hydrodynamic model for both grid systems consist of continuity and the momentum conservation equations, expressed in terms of depth-averaged quantities. Since detailed explanation for the former computation method applied to the Natori River mouth has already been reported elsewhere ( Tanaka et al. [7] and Itonaga et al. [3] ), only the governing equations in BFC coordinates will be shown, expressed in terms of transformed variables (ξ,η). ∂ h ∂ Uh ∂ Vh + + =0 (1) ∂t J ∂ξ J ∂η J ξ ∂z η ∂z τ ∂ M ∂ UM ∂ VM (2) + + = −gh x s + z s − b x ∂t J ∂ξ J ∂η J J ∂η ρJ J ∂ξ ξ y ∂z s η y ∂z s τ b y ∂ N ∂ UN ∂ VN + − (3) + + = − gh ∂t J ∂ξ J ∂η J J ∂η ρJ J ∂ξ where (M,N): the discharge flux in (x,y)-direction, zs : the water level, h: the water
depth, J : Jacobian, (U,V): the contravariant component of the velocity vector，g: the gravity acceleration, ρ: the density of fluid, and (τbx,τby ): the bottom shear stress. The governing equations have been solved by using a leap-frog scheme, the detail of which is provided by Tanaka and Qin [6]. Computation of fluid motion using BFC has already been carried out for various kinds of configuration in the field of hydraulic engineering ( e.g. [1], [5] ), and it is reported that use of BFC can improve the accuracy of boundary presentation in hydrodynamic simulation. 3.2 Computation Conditions In most of the previous computations around a river mouth, prediction of sediment movement and resulting morphological change has been performed. For this purpose, the conservation equation of sediment mass should solved along with the hydro-dynamic equation explained in the previous section. In this study, however, sediment movement will be ignored, with the objective to make straightforward investigation on the effect of grid system on computed result of hydro-dynamic properties, such as velocity and water depth. The grid systems employed in this study are shown in Table 1. The computation for Type1 and 2 are based on rectangular grid system with different grid spacing, whereas Type 3 computation is carried out using BFC for the grid system illustrated in Figure 2. 3500
Table 1 Computational cases Type 1 Squared Grid (Δx=Δy=15m) Type 2 Squared Grid (Δx=Δy=7.5m) Type 3 BFC
3000
2500
2000
3000
y(m)
Case 2
Q(m /s)
Case 1 2000
3
1500
1000
tidal le vel(T.P.m)
1000
500
0
0
500
1000 x(m)
Figure 2. BFC grid
1500
0 1 0 -1 0
12 Aug.5
24
12 Aug.6
Figure 3. Hydrograph and tidal level variation During the flood in August, 1986
24
The boundary conditions at the upstream and downstream ends are determined based on the actual flood event occurred in August, 1986. Numerical computation, including morphological change, for this particular flood has already been carried out by Tanaka et al. [7] using a rectangular grid system. The hydrograph and tidal variation are given in Figure 3. The first case, Case 1 is taken from the very early stage of the flood, whereas the second one is for the flood stage at the peak of the hydrograph. The detail of the computation conditions is given in Table 3.
Discharge Tidal level
4
Table 3 Case 1 924.1m3 /s +0.41
Case 2 2356.2m3 /s -0.47m
Results and Discussions
4.1
Case 1
The longitudinal profile of water surface along the line A − A ′ in the jetties ( see Figure 2 ) is plotted in Figure 4 for three computational cases. Although all of them show the same water level at the downstream end, it is observed that the water level rise in the mouth is considerable in Type 1 and Type 2 for squared grid system, while the BFC computation shows smaller water level rise in the mouth. Figure 5 shows longitudinal profile of the magnitude of velocity along the section A − A ′ . It is seen that computation using Type 3 grid shows highest value among them, in accordance with the result for the water level profile shown in Figure 4. A bird's-eye view of the water surface in the jetties is shown for each case in Figure 6. In the computations using a rectangular grid system, the step-like approximation of jettied boundary at the river entrance causes “roughness” effect to the fluid motion between the structure, resulting in water level rise like a spike at the corner of the grid approximating curved structure. In contrast to these results for 2
3
2.5
1.5 Type2
1 Type3
Vel ocit y (m /s)
Water level (m)
Type1
Type3 2 Type2 1.5
0.5
800
Type1 900 1000 1100 1200 1300 1400 1500 1600 1700 Distance (m)
Figure 4. Water surface elevation along A − A ′ line (Case 1)
1 800
900 1000 1100 1200 1300 1400 1500 1600 1700 Distance (m)
Figure 5. Velocity profile along A − A ′ line (Case 1)
(a) Type 1
(b) Type 2
(c) Type 3
Figure 6. Water surface profile in jettied area (Case 1)
Type 1 and Type 2 using rectangular grid system, unrealistic “spike-shaped” water level rise can not be observed in the computation of Type 3. In this respect, BFC computation is remarkably effective for computation of effluent flow at jettied river entrance. 4.2
Case 2 4 Type1 1 3.5 Velocity (m/s)
Water level (m)
Type2 0.5
0
3
2.5
Type3
Type3
Type2
Type1
-0.5 800
900 1000 1100 1200 1300 1400 1500 1600 1700 Distance (m)
Figure 7. Water surface elevation along A − A ′ line (Case 2)
2 800 900 1000 1100 1200 1300 1400 1500 1600 1700 Distance (m)
Figure 8. Velocity profile along A − A ′ line (Case 2)
The longitudinal profile of water surface elevation and the resultant velocity along the line A − A ′ in the jetties are shown in Figures 7 and 8, respectively. It is noted that the water level rise in the mouth is considerable in Type 1 and Type 2 for squared grid system, while the BFC computation shows lower water level rise in the mouth, being similar to the results observed in Case 1. 3-D view of the water surface in the jetties is shown for each grid system in Figure 9. The difference between rectangular and BFC grid is more considerable as compared with corresponding profile for Case 1 depicted in Figure 5, due to bigger flood discharge in Case 2. In the previous studies on topography change at the Natori River mouth by Tanaka et al. [7], the bedload formula proposed by Einstein-Brown (EB) ( Brown [2] ) was used, though, it was found that direct use of EB formulation resulted in overestimation of sediment transport rate. Thus, a multiplying coefficient has been introduced to the bedload formula, which should be calibrated through a comparison with field data. In general, sediment transport rate in a river is proportional to from the 3rd to 6th power of the velocity, such as Meyer-Peter and Müller [4] and EB [2]. Therefore, the difference of the calculated velocity observed in Figures 6 and 8 leads to different calibration result for the coefficient in the sediment transport rate formula. In that sense, an accurate estimation of the velocity, especially in the area with higher magnitude of the velocity, is essential for numerical computation of river mouth morphological change. According to Figures 6 and 8, it can be concluded that
(a) Type 1
(b) Type 2
(c) Type 3
Figure 9. Water surface profile in jettied area (Case 2)
numerical simulation using BFC is very suitable for this purpose. 5
Conclusions
In the present paper, detailed comparisons have been made between rectangular and boundary–fitted grid computations for the river mouth of Natori River. It was observed that the approximation in the former grid system causes “spike-shaped” water level rise due to discretization of curved jetties at the river entrance, meanwhile BFC grid system yields much smoother profile of the water surface elevation. 6
Acknowledgements
The authors are indebted to the Sendai Construction Office, Ministry of Land, Infrastructure and Transport Construction, Japan for their kind supply of field data used in the present study. References 1. Borthwick, A.G.L. and Barber, R.W., River and reservoir flow modeling using
2. 3.
4. 5.
6.
7.
8.
the transformed shallow water equations, Int. J. Num. Methods in Fluids, 14 (1992) pp.1193-1217. Brown, C.B., Sediment Transportation, in Engineering Hydraulics ( ed. H. Rouse, John Wiley and Sons, Inc., New York , 1950) Itonaga, N., Sana, A., Tanaka, H. and Samad, M.A., Effect of grid spacing on the prediction of sediment movement in a curved channel, Proc. 4th Int. Conf. on Hydrodyn., 1 (2000) pp.653-658. Meyer-Peter, E. and Müller, R., Formulas for bed-load transport, Proc. of 2nd IAHR Congress (1948) pp.39-64. Pearson, R.V. and Barber, R.W., Mathematical simulation of the Humber Estuary using a dept-averaged boundary–fitted tidal model, Proc. 8th Int. Conf. On Num. Method in Laminar and Turbulent Flow, 2 (1993) pp.1244-1255. Tanaka, H. and Qin, H., Numerical simulation of sand terrace formation in front of a river mouth, Computational Modelling of Free and Moving Boundary Problems II (1993) pp.241-248. Tanaka, H., Shuto, N., Kuwahara, N. and Sato, K. , Numerical modeling of 2-D flow and sediment movement in the vicinity of Natori River mouth, Flow Modeling and Turbulence Measurements VI (1996) pp.813-820. Tanaka, H. and Samad, M.A.: Influence of river mouth sand spit level on the time variation of river stage and bed topography, Coastal Engineering and Marine Developments (1999) pp.3-12.
