A Staggered Method for Simulating Shallow Water Flows along Channels with Irregular Geometry and Friction

We consider the shallow water equations along channels with non-uniform rectangular cross sections with source terms due to bottom topography, channel width, and friction factor. The system of equations consist of the mass and momentum conservation equations. We have two main goals in this paper. The first is to develop a numerical method for solving the model of shallow water equations involving those source terms. The second is to investigate effects of friction in water flows governed by the model. We limit our research to the flows of one-dimensional problems. The friction uses the Manning's formula. The mathematical model is solved numerically using a finite volume numerical method on staggered grids. We propose the use of this method, because the computation is cheap due to that no Riemann solver is needed in the flux calculation. Along with a detailed description of the scheme, in this paper, we show a strategy to include the discretization of the friction term in the staggered-grid finite volume method. Simulation results indicate that our strategy is successful in solving the problems. Furthermore, an obvious effect of friction is that it slows down water flows. We obtain that great friction values lead to slow motion of water, and at the same time, large water depth. Small friction values result in fast motion of water and small water depth. Keywords— friction term; irregular channel width; irregular geometry; irregular topography; shallow water equations.


I. INTRODUCTION
Water flow appears in everyday life. Flows can be in either a closed channel or free surface channel. An example of closed-channel flow is pipe flow. Examples of free surface flows are those in rivers, lakes, seas, and reservoirs. They can be in the forms of floods, tsunamis, and others. This paper focuses on free surface water flows.
A large number of studies have been done by researchers relating to water flows. One of the most famous results is the Saint-Venant model [1]. This model can be applied to simulate shallow water waves or flows. The original Saint-Venant model has been developed by a number of researchers. The model has currently a number of improvements and variations. Some improvements and variations involve topography, some other involve irregular channel width, and the rest involve friction factor [2], [3], [4], [5], [6]. Based on its form, the Saint-Venant model is a partial differential equation system. Based on physics, the form of the equation can be either conservative or nonconservative [7], [8], [9].
One of the Saint-Venant model variations is the system of one-dimensional shallow water equations. This system or model can be used for various types of free surface flows with irregular topography and irregular channel width. Mungkasi et al. [8] solved this model using a staggered-grid finite volume method. This method has a very simple flux calculation, which leads to cheap computation, as developed by Stelling and Duinmeijer [10]. However, Mungkasi et al. [8] did not involve any friction in their shallow water model. The work of Mungkasi et al. [8] is extended in the present paper by including the friction factor in our water flow problems.
In this paper, we have two goals. First, we develop a numerical scheme of the finite volume method on staggered grids to approximate solutions to the shallow water equations with source terms due to bottom topography, channel width, and Manning friction. Second, we investigate the influences or effects of the friction to water flows.
A finite volume method is chosen, as our problems admit continuous and discontinuous solutions. Some research of finite volume methods has been reported in the literature, such as [11], [12]. Based on the positions of grids for evaluation of the unknown variables, finite volume methods are categorized into two types, namely, the collocated and staggered methods. Some works on collocated-grid finite volume methods are [13], [14], [15], [16]. Some works on staggered-grid finite volume methods are [8], [10]. An application of these works is, for example, in simulation of the flood routing problem [17], [18], [19], [20].
The rest of this paper follows. Section II recalls the mathematical model and writes our proposed numerical method. Simulation results are presented and discussed in section III. Conclusion is given in section IV.

A. Mathematical Model
We consider one-dimensional shallow water flows with irregular bottom topography, irregular channel width, and friction. Schematic illustrations of this flow problem are shown in Fig.1 including the geometries of the transversal cross section, the longitudinal flow, and the flow space viewed from the top. Water flows in open channels such as rivers and canals can be modeled by a partial differential equation system. This system can be expressed in the forms of mass conservation equation and momentum balance equation, respectively, as follows [3]: Here, h is the surface level of water in the channel, Q represents the discharge, A denotes the wet cross-sectional area of the flow, S 0 is the slope of bottom of the channel, S f is the friction slope, g is the gravitational acceleration, x represents the longitudinal distance along the channel, and t is the time variable.
The friction slope S f can be obtained from the Manning friction formula as: where n represents the Manning's friction coefficient and R is the hydraulic radius. By substituting the following expressions, that is, (1) and (2), then we obtain: denotes the velocity, and and ( ) ( ) In this paper, (6) and (7) are the mathematical model which shall be solved numerically using the method with staggered grids. The system of (6) and (7) is the shallow water equations in one dimension that model flows along channels with irregular rectangular cross section of channel width and irregular bottom topography involving friction.

B. Numerical Method
In this subsection, we continue the work of Mungkasi et al. [8] as well as Stelling and Duinmeijer [10] in developing the staggered-grid finite volume method, or the staggered method, in short. There was no variation of channel width in the work of Stelling and Duinmeijer [10] for onedimensional problems. There was no friction in the model solved by Mungkasi et al. [8]. The problems in the present paper involve irregular topography, irregular channel width, and friction.
To discretize the governing equations, without loss of generality, in this subsection we consider (6) and (7) with the spatial domain L x ≤ ≤ 0 using the partition: . ..., , , , ..., , 0 As we work on the numerical method with staggered-grids, values of depth h are approximated at full grid points i x for N i ..., , 1 = and the values of velocity u are approximated at half grid points We consider (6) and (7) on different control volumes, as shown in Fig. 2. We calculate the depth h using the mass conservation (6) at the and the velocity u using the momentum balance (7) at the control volume [ ].
, 1 We denote for all involved integers i and .
k The spatial discretization of (6) and (7) are: The values of h and u with asterisks must be approximated, because their values are not known on the corresponding grid points. We calculate the values 2 / 1 * + i h and i u * using the upwind approximation of the first-order, as follows: (11) and The next step is converting (9) into a semidiscrete scheme to obtain the rate of change of u with respect to t. Based on (8), we have the formulation: Substituting (13) into (9), we obtain: Hence, our mass conservative and momentum balance equations (6) and (7) become semidiscrete schemes (8) and (14).
To discretize (8) and (14) we can apply a discrete time integration using a standard solver of ordinary differential equations, such as, the first-order explicit Euler method. Therefore, the fully discrete schemes of our staggered-grid finite volume method are: and With these formulations, we have extended the work of Mungkasi et al. [8] as well as Stelling and Duinmeijer [10]. The strategy of involving the friction term due to Manning has been shown in the above derivations.
Up to here, we have achieved the first goal of this paper. Note that our first goal is to develop a numerical scheme (a finite volume method on staggered grids) to approximate the solutions to the shallow water equations involving source terms due to bottom topography, channel width, and Manning friction. The numerical schemes (15) and (16) are our proposed staggered method for solving the system of (6) and (7).

III. RESULTS AND DISCUSSION
In this section, we conduct five numerical tests to achieve our second goal of this paper. That is, we want to analyze the influences of the friction factor with respect to water flows in the considered Saint-Venant model. The problems for the tests in this paper are taken from the paper of Mungkasi et al. [8].
For the discussion in this section, we define two additional variables. The first is the stage ), , that is, the vertical position of water surface measured from the horizontal x-axis reference. The second is the discharge ) , = which is actually the momentum in our one-dimensional problems when the channel has a constant width. Here, ) , All measured quantities are assumed to have MKS units in the SI system. With this assumption, we omit the writing of units in all values of variables. In addition, the gravitational acceleration is . 81 . 9 = g

A. Case for constant topography and constant width
In this case, we take a dam break having width 1 ) as studied by Stoker [20]. The numerical schemes (15) and (16)   . 0 = n respectively. We observe that the motion of water becomes slower when friction factor is greater. Slow motion of water has consequences in the increase of depth. This means that the greater the friction factor leads to the slower the water flow and the greater the depth.

B. Case for constant topography and irregular width
In this second test, the numerical method is applied to simulate flow generated by the a radial dam which collapses instantaneously on a horizontal bottom topography. In this case, the channel width is The numerical schemes (15) and (16)  .
Before the vertical dam is removed, the depth has the initial condition: (18) and is the initial velocity. At the boundary, we have: , Results of this numerical simulations are shown in Fig. 4 for time .
We vary the Manning friction coefficients to be .
Similar results to the those of the first test are obtained. Water motion becomes slower when the friction factor is larger. Slow motion of water has increased the depth. This means that the greater the friction factor results in the slower the water flow and the greater the depth. Fig. 4. Results of simulation for the radial dam break problem at t = 3 using 400 cells.

C. Case for irregular topography and constant width
In this third test, we have a parabolic obstacle on the topography. Let us take the steady state case of water flowswith constant width .
Due to the obstacle, we have irregular bottom topography. The bottom topography is assumed to be defined by: The numerical schemes (15) and (16)  We observe that the velocity increases when water approaches the peak of the obstacle and decreases when water has passed the peak of the obstacle. Our inference is the same as in the two previous tests. That is, the water motion becomes slower when friction factor is involved in modeling. Slow motion of water has consequences to the increase of depth. The greater the friction factor leads to the slower the water flow and the greater the depth.

D. Case for irregular topography and irregular width
In this fourth test, we take the steady state of "a lake at rest" with irregular channel width and irregular bottom topography. The numerical schemes (15) and (16) as the initial conditions. At the boundary we have: is the channel width. The channel profile viewed from the top is shown in Fig. 6, which is given in between the curves Table 1 provides the bottom topography function, and together with the initial stage, it is plotted in Fig. 7. Results of this simulation is shown in Fig. 8  We observe that our proposed numerical method is able to simulate the steady state problem of "a lake at rest" involving irregular bottom topography, irregular channel width, and friction. For time , 0 > t where in Fig. 8 7. Initial stage (water surface) and topography of "a lake at rest." Fig. 8. Results for the steady state of "a lake at rest" at t = 10.

E. Simulation of irregular topography and irregular width
Here, we simulate an unsteady state case due to the collapse of a dam involving irregular channel width and irregular bottom topography. The numerical schemes in (15) and (16)  The bottom topography in this case is: . 100 Here, the channel width is the same as in the previous test, that is, (20); and the channel width is shown in Fig. 6. The stage is initially given by:  We observe once again that water motion becomes slower when friction is involved in the problem.

IV. CONCLUSIONS
We have achieved our two goals in this paper. First, we have provided a numerical method for solving the shallow water equations in one dimension involving irregular bottom topography, irregular channel width, and friction. Second, using our proposed numerical method we have investigated some effects of friction to water flows along channels having irregular bottom topography, irregular channel width, and friction.
Our remarks are as follows. Our numerical method is able to solve problems of steady state and unsteady one, with horizontal topography and irregular one, with constant channel width and irregular one, without friction and with friction, as well as of combination of these all. In addition, larger friction factor leads to slower water motion, and consequently due to the momentum balance, larger friction factor also leads to greater depth when water is in motion.
This research is limited to one-dimensional problems with the friction due to Manning. Future research could be directed to work on higher dimensional problems and/or other friction formulations.