# Parallel Accelerated Fifth-Order WENO Scheme-Based Pipeline Transient Flow Solution Model
## 5 Model Validation and Case Analysis
### Parameter Sensitivity Analysis  
By calculating and evaluating the numerical dissipation and surge intermittent capture performance of the pressure wave curve at the pipe's valve at different Courant numbers, the sensitivity of different simulation methods to the Courant number is verified.  
Assuming that the two calculation conditions are recorded as Case 1 and Case 2, the pipeline length is 1960 m and 39,200 m, the calculation time is 40 s and 800 s, the grid number is 20 and 40, and the Courant number is 0.1 and 0.01, respectively. Other parameters are the same. The pipe diameter is 1.0 m, the initial flow velocity is 0.5 m/s, the upstream constant water level is 10 m, the wave speed is 980 m/s, and the gravitational acceleration is 9.806 m/s\({}^{2}\). The viscous term is ignored, so the roughness of the pipe is 0, and the end valve is suddenly closed.  
Figure 5 shows the pressure fluctuation characteristics at the end valve and the 540 m upstream of the pipeline when the model Courant number is 0.1 and the model grid is 20 (case 1). According to the Joukowsky theorem, the change value of water hammer pressure:  
\[\Delta l=\frac{c\Delta v}{g}=\frac{980\times 0.5}{9.806}=49.97\text{ m} \tag{26}\]  
The maximum pressure is 59.97 m, the minimum pressure is \(-39.97\) m, and the phase length (the time for traveling one wavelength) of the pressure wave is \(T=2l/c=2\times 1960\div 980=4\) s. As shown in Figure 5, the calculation results of the MOC and the first-order Godunov scheme indicate significant numerical dissipation with increasing time, while the peak pressure waves captured by both sc  
Figure 5: Comparison of results of different schemes for Cr = 0.1. (**a**) Comparison of pressure wave curve at value. (**b**) Comparison of pressure wave curves at 540 m.  
extreme value of the pressure wave at the end valve can be captured accurately during the calculation period using the second-order Godunov scheme. The calculation results at 540 m upstream exhibit a significant error, and the scheme's numerical dissipation is still more severe. Simultaneously, the second-order Godunov scheme cannot capture the abrupt interruptions. In contrast, the fifth-order WENO scheme accurately captures the actual pressure wave peak and provides performance superior to the other three schemes in capturing the abrupt interruptions of the excitation wave. Figure 5a shows that the second-order Godunov scheme captures a complete excitation interruption with a response time of about 4 s, while the fifth-order WENO scheme captures a complete interruption within 1 s. Besides, the non-physical oscillations generated by the fifth-order WENO scheme are so minimal that their impact on the actual results can be neglected.  
Figure 6 shows the pressure fluctuation characteristics when the Courant number is taken as 0.01 and the model grid is 40 (Case 2). Similarly, the pressure change value is \(\Delta h=c\Delta v/g=49.97\) m, and the phase length of the pressure wave is \(T=21/c=2\times 39200\div 980=80\) s. Compared with Figure 5, the numerical dissipation of the MOC and the first-order Godunov scheme decreases slightly by increasing the number of computational grids. However, the scheme cannot capture the pressure wave extremes after 100 s, and the scheme calculation results are almost distorted. The second-order Godunov scheme can better capture the intermittent excitation waves and slightly reduce the dissipation of the scheme values. The fifth-order WENO scheme has no significant nonphysical oscillations and numerical dissipation while maintaining high performance in capturing intermittent excitation waves. In summary, the fifth-order WENO scheme has lower numerical dissipation and better intermittent capture capability than the traditional MOC and Godunov scheme for calculating water hammer waves in pipes with low Courant number conditions.