Paper 3 – “A solution method for one-dimensional shallow water equations using flux limiter centered scheme for open

Venturi channels”

The conventional shallow water equations are modified to accurately capture the wall-reflection pressure-force effect in open Venturi channels. The conventional shallow water equations produce an artificial flux due to bottom width variation in the contraction and expansion regions. The artificial flux is due to the weak integration of total wall pressure acting on control volumes at contraction and expansion regions. The modified shallow water equations can be used to model both prismatic and non-prismatic channels. The total variation diminishing (TVD) scheme and the explicit Runge–Kutta fourth-order method were used to solve the modified shallow water equations. The simulated results were validated by experimental results and three-dimensional computational fluid dynamics results.

 The 1D shallow water equations need to be modified to include wall contraction and expansion effects. The pressure force from the sidewalls for a control volume might be challenging to add to the conventional shallow water method.

The wall-reflection pressure-force acts in the opposite direction to the flow direction in the contraction region, so leading to a hydraulic jump in some cases.

In the expansion region, the wall-reflection pressure-force acts in the flow direction, directing flow states such that they become supercritical. Compared to the conventional shallow water momentum-balance equation, Equation-(6.2), the expression 𝑘_𝑔ℎ²𝑔^𝜕𝑏

𝜕𝑥 is added to the new equation. Figure 6-4 shows a result comparison between the modified and conventional shallow water equations with experimental result.

𝜕(𝐴𝑢)

𝜕𝑡 = −𝜕(𝐴𝑢²)

𝜕𝑥 − 𝑘_𝑔𝜕(𝐴ℎ)

𝜕𝑥 𝑔 + 𝑘_𝑔ℎ²𝑔𝜕𝑏

𝜕𝑥+ 𝐴𝑔(sin 𝛼 − 𝑆_𝑓). (6.2)

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Figure 6-4. Quasi-steady state results, water flow rate at 400 kg/min, a comparison between the modified and conventional shallow water equations with experimental results.

 Poor treatment of the source term (due to irregular geometry, changes of width) produces significant oscillation in the flow depth. The conservativeness of the scheme can be severely damaged by this (Garcia-Navarro and Vazquez-Cendon, 2000). Studies propose pointwise and upwind approaches to the discretization of the source term (Garcia-Navarro and Vazquez-Cendon, 2000; Vázquez-Cendón, 1999). One main reason for this is poor capture of wall pressure. In this study, a higher order discretization method is suggested for the channel irregularity source term. The wall reflection pressure-force effect 𝐒_𝑹(𝐔)_𝑗^𝑚 is taken into consideration using the centered-TVD discretization method. The cell center value takes into account 𝐒_𝑹(𝐔)_𝑗^𝑚. The term has the first-order accuracy in space. Even though other researchers suggest wall-reflection pressure-force as a source term, we leave all pressure force in the advection term. This helps to remove artificial acceleration within the TVD scheme and to avoid this propagating into the ODE solver.

 Low flow depth initial conditions can reduce the stability of the numerical scheme at the high inclination angles. Due to high gravitational force in the flow direction, dry beds and discontinuities are generated downstream of the channel during start up when initial flow depth is low. This behavior is very common in long channels, where the channel length is greater than 2 m and inlet velocity is low. Flow depth furthermore becomes negative and the numerical scheme can

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break down. As suggested in Paper 1, a threshold value of flow depth helps to maintain water depth non-negative.

 A dry bed condition is one of the main difficulties in shallow water flow modeling.

Friction slopes can become very large, giving unphysical results when water depth approaches zero near the wet/dry interface (Tseng, 2004). A non-zero threshold flow depth can be used as a solution for this. In this study, the cell flux is forced to become zero where dry bed occurs, ℎ < 10⁻⁸m.

 Heat transfer from the liquid to the atmosphere is very small. This is because the fluid is at room temperature, and the temperature rise due friction is very small.

The energy equation is, therefore, not solved as a conservation equation. The entropy of the system is a considerable factor in hyperbolic equation solving using TVD schemes in gas dynamics. Entropy is produced with the admissible shock, but would be reduced across an expansion shock (LeVeque, 2002).

However, the entropy concept is not much popular in shallow water flow. The hyperbolic equation is an imperfect model for real open channel flow with friction due to the non-zero friction terms. Tseng (2004) used an entropy fix function for the approximate Riemann solvers method for the shallow water equations. Gassner et al. (2016) proved that the total energy based interface flux function precisely preserves entropy in the shallow water equations. The superbee flux limiter function is defined, in this study, in terms of the total energy of control volume.

 Abdo et al. (2018) noticed that shallow water equations cannot predict steady supercritical flow in a straight wall contraction. A possible reason for this may be the inaccurate estimation of the source term or the neglect of turbulence resistance (Hsu et al., 1998). The Manning’s formula was used in this study as the turbulent friction model. A constant roughness value (𝑘_𝑚) was also used throughout all flow regimes. However, some researchers argue that Manning’s friction might be a function of flow depth and Froude number (Hsu et al., 1998;

Thomas and Hanif, 1995). Turbulent viscosity was set to zero in the 1D

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simulation. According to the Thomas and Hanif (1995) study, varying the turbulent viscosity does not affect the converged solution.

 The 1D shallow water equations are well suited to open channel flow modeling using the high-resolution numerical schemes described in Chapter 4. The developed high-resolution scheme has strong stability at hydraulic jumps. The modified shallow water equations well matched with the experimental results in both unsteady and steady state.

6.4 Paper 4 – “Computational Fluid Dynamics Study of Shear