(1)
Masayuki Fujihara and Shinako Kinoshita
Faculty of Agriculture, Ehime University
3-5-7 Tarumi, Matsuyama, 790-8566 Japan
Tel:+81-89-946-9890,
Fax:+81-89-946-9890, E-mail:fujihara@agr.ehime-u.ac.jp
Abstract:
This paper deals with applications of a second order accurate Godunov-type
numerical model of the two-dimensional shallow water equations discretized using
finite volumes to the flow in vertical slot fishways. Roe's flux function is
used for the convection terms, and a non-linear limiter is applied to prevent
non-physical oscillations. The mathematical formulation of shallow water
equation, used in this paper, is suitable for cases where the bathymetry is
non-uniform. The model is based on hierarchical quadtree grids automatically
generated to fit the boundary configuration and moreover the quadtree grids are
also generated to adapt to inherent flow parameter, such as vorticity. The flows
in two types of vertical slot fishway are simulated and compared to the results
of hydraulic model experiments.
Keywords: vertical slot fishway, Roe’s scheme, solution adaptive quadtree grids, shallow water equation, computational fluid dynamics
As the concern for conservation or improvement of fish-friendly rivers has been widespread, fishways have attracted major interests. Fishways are hydraulic structures that enable fish to go through obstructions to their spawning and other migrations. Most of the earlier studies of fishways have been focused on the flow structure by means of hydraulic models. Since the flow structure in a fishway is complicated with free surface and sub- and super-critical conditions usually coexist, only a few papers have been published which had applied numerical methods to the flow in fishways. Tsujimoto and Shimizu (1996) applied a numerical analysis based on k-ε turbulence model to flows in stream-type fishways: Denil type, superactive-type, bottom baffles and Alaska steeppass. Gotoh et al.(1999) investigated the characteristics of the pool-and-weir fishway, using MPS (Moving Particle Semi-implicit method).
In this paper, flows in vertical slot fishways are numerically investigated. The vertical slot fishway consists of a rectangular channel that is divided into a number of pools by vertical walls with a slot. Water runs downstream through a vertical slot from one pool to the next. The water forms a jet as it goes through the slot and the energy is dissipated effectively by jet mixing in the pool.
The work presented consists of a grid generation based on solution adaptive quadtree grids, and a flow solver based on the MUSCL (Monotonic Upstream-Centred Scheme for Conservation Laws) concept with slope limiter and Roe’s approximate Riemann solver (Toro, 1999). The shallow water equations are discretized using finite volumes collocated with the grid cells, and integrated in time using a 4th order Runge-Kutta scheme. Flows in two types of vertical slot fishway are demonstrated to show the numerical model works properly.
The shallow water equations, which describe flow in
shallow water bodies where the vertical acceleration within the fluid is
negligible, can be written as
(1)
(2)
where z
is the free surface elevation above the still water level
, h ( = z
+
) the total water depth, u and v the depth-averaged velocities in the x-
and y-directions respectively, ux,
uy and vx,
vy the derivatives of the
depth-averaged velocity components in the x-
and y-directions respectively, g the
acceleration due to gravity, r
the water density, twx
and twy
the surface stresses, tbx
and tby
the bed friction stresses and n
the kinematic eddy viscosity coefficient.
The
term is usually split to give the
hyperbolic system used by many researchers (e.g., Toro, 1999). However, when
applied to non-uniformly varying bathymetries, this approach leads to problems
non-conserving mass and momentum for individual cells when using Roe’s
approximate Riemann solver. Thus, in this paper the
term and similarly the
term are split differently so that
the shallow water equations are written as (Rogers et
al., in press)
(3)
(4)
where Sox and Soy are the bed slopes in the x- and y-directions, respectively. This formulation allows Roe’s approximate Riemann solver to be used to evaluate the inviscid flux between adjacent cells for all bathymetric conditions.
The two-dimensional shallow water equations are discretized spatially using finite volumes on the quadtree grid, with Roe's flux function used to approximate the non-linear convection terms. Time integration is implemented by means of the 4th order Runge-Kutta integration scheme. Unwanted oscillations are avoided using a slope limiter.
The shallow water equations can be written in integral form as
(5)
where W is the domain of interest, q the vector of conserved variables, f and g the flux vectors, and h the vector of forcing functions. The vectors q, f, g, and h are given by
,
,
and
Equation (5) can be written as
(6)
where S
is the boundary of W,
and
the vector of flux
functions through S. The term
may also be written in
terms of inviscid and viscous fluxes as
(7)
with
and
where
and
are the Cartesian components of n, the unit normal vector to S.
The inviscid fluxes can be expressed by adopting Roe’s approximate Riemann solver (Roe, 1981) at the cell interface as follows
(8)
with
(9)
where
and
are reconstructed right and left
Riemann states at the cell interface, respectively, located between adjacent
cells i and j,
R and L
the right and left eigenvector matrices of flux Jacobian A
respectively, and |L|
is a diagonal matrix of the absolute values of the eigenvalues of A.
The inviscid flux Jacobian is given by
(10)
which has eigenvalues given by
(11)
The associated right and left eigenvector matrices are
(12)
and the left eigenvector matrix is given by
(13)
The variables u,
v, and c in (9) through (13) are evaluated by Roe’s
average state
and the minmod limiter (Toro, 1999) is employed as a
slope limiter. Riemann invariants have been used to implement inflow
open boundary.
Two types of vertical slot fishway are used to compare the results of hydraulic experiments with those of numerical simulations. One is called C-type, in this paper, which has a slot at the center of baffles (Fig.1) and the other is P-type, which is a popular type shown in Fig. 5.
Figure 1 shows the fishway configuration used in this case. Hydraulic model experiments were carried out by Izumi et al.(1999). The experimental channel is 8.8 m long and 0.8 m wide. The 0.8 m long horizontal channel followed by the 6.0 m long sloping fishway was installed in the experimental channel. Bottom gradient at the fishway was 1/15. The length of a pool was 0.76 m and the width of a slot was 0.08 m. Guide walls were installed at the slots to prevent the flow from being unstable due to meandering. The channel was divided into a large number of square grids 0.02 m for computations. Time increment was set as 0.001 s and the kinematic eddy viscosity was 0.002 m2/s. The bottom friction was ignored, but non-slip condition was applied along the wall. As a boundary condition, the values of water depths at the up- and downstream were given in order to supply the required discharge (Q).
Figure 2 shows the computed and measured water depths along the center line in the forth pool, when discharges are 0.01 and 0.005 m3/s. In Fig. 2, PL denotes the pool length and X means the longitudinal distance from a slot. Computed and measured water depths once decrease at the downstream of slot and after that those water depths gradually increase as the water flows. Other pools have same pattern. Overall the computed results show good agreement with the experimental results, except the location of minimum depth occurred.
The computed velocity field in a pool is depicted in Fig. 3, when the discharge is 0.01 m3/s. A jet through a slot forms vortex-pair. In other words, the jet generates a clockwise eddy along the right bank side of the pool and a counter clockwise eddy along the left bank side. Therefore, upstream currents are seen along the both banks. While main stream current flows downstream around the center of the channel. Maximum velocity is 1.252 m/s. The computed flow structure is very similar to the experimental one (not shown here). The Froude number distribution is shown in Fig. 4. Supercritical flow is seen at the center of jet, but most of the flow region is subcritical flow.
The plan view of P-type fishway is depicted in Fig. 5. The bottom gradient is set as 1/15. Figure 6 shows a corresponding initial non-uniform quadtree grids used for the present simulation where the smallest cells are space square grids 0.0195 m. Both the upper and lower stream depths were set to be constant as 0.5 m and the resultant discharge was 0.191 m3/s. The time step was set as 0.001 s and the kinematic eddy viscosity was 0.005 m2/s. The bottom friction was also ignored in this case.
During the time-steping procedure, the quadtree grid was adapted according the criteria based on the magnitude of the depth-averaged vorticity,
(14)
Quadtree grid was subdivided into four squares
where
was greater than 1.0. If
was greater than 2.0, then the grid
was subdivided again. The number of initial quadtree cells was 8065 and finally
it increased up to 12640 at the end of computaiton. Final quadtree grids are
depicted in Fig. 7.
The computed results in second pool are visualised in Figs. 8 through 10. The velocity vector field is shown in Figure 8. The maximum velocity is 2.3 m/s. The water flowing through a slot gradually turns left as the water flows and heads for the next slot, making an anticlockwise eddy at the left side of the main stream flow. On the other hand, the water flows upstream along the right-hand wall of a pool. These flow properties agree well with the measured one (Rajaratnam et al., 1986). Figure 9 shows equi-velocity lines. The main stream from one slot to next can more easily be seen in this figure. Since the average velocity in the slot is about 1.5 m/s in this discharge, the target fish should have enough ability to swim with the over 1.5 m/s speed. In other words, the fish, which could not swim at the speed of over 1.5 m/s, can not reach the upper pools. However, the velocity decreases near walls due to friction of wall, some clever fish could swim up in this region at the speed of under 1.5 m/s. In the main stream, the water flows with the speed of over 1 m/s. The equi-Froude number lines with an interval of 0.1 are shown in Fig. 10. Supercritical flow is occurring in slots with the maximum Froude number of 2.93. In most part of the fishway, the Froude number is less than 0.5 and thus the flow is subcritical.
A second-order accurate Godunov-type solver of
the shallow water equations in finite volume form on solutio-adaptive quadtree
grids was described. The numerical model was applied to the flows in two types
of vertical slot fishway and validated by the experimental data. The computed
results were in good agreements with the measured ones. Taking into
consideration of the applicability of quadtree grids to complicated natural
region, it is possible to calculate efficiently the flow in riverine reach
including fishway with this presented model.
Acknowledgements
The authors wishes to express their sincere gratitude to Dr. A. G. L. Borthwick and Mr. B. Rogers of University of Oxford for helping computer program coding of quadtree grids.