Wen-Dar Guo

Finite-volume component-wise TVD schemes for 2D shallow water equations

Abstract Four finite-volume component-wise total variation diminishing (TVD) schemes are proposed for solving the two-dimensional shallow water equations. In the framework of the finite volume method, a proposed algorithm using the flux-splitting technique is established by modifying the MacCormack scheme to preserve second-order accuracy in both space and time. Based on this algorithm, four component-wise TVD schemes, including the Liou–Steffen splitting (LSS), van Leer splitting, Steger–Warming splitting and local Lax–Friedrichs splitting schemes, are developed. These schemes are verified through the simulations of the 1D dam-break, the oblique hydraulic jump, the partial dam-break and circular dam-break problems. It is demonstrated that the proposed schemes are accurate, efficient and robust to capture the discontinuous shock waves without any spurious oscillations in the complex flow domains with dry-bed situation, bottom slope or friction. The simulated results also show that the LSS scheme has the best numerical accuracy among the schemes tested.

Hybrid Flux-Splitting Finite-Volume Scheme for the Shallow Water Flow Simulations with Source Terms

The extension of the hybrid flux-splitting finite-volume (HFF) scheme to the shallow water equations with source terms is presented. Based on the monotonic upstream schemes for conservation laws (MUSCL) method, the scheme is second-order-accurate both in space and time. An accurate and efficient surface gradient method (SGM), in conjunction with the HFF scheme, is adopted for the discretization of source terms, including the bed slopes and friction slopes. The resulting scheme has several desirable properties: ease of implementation, satisfaction of entropy condition, sharp shock resolution and preservation of well-balancing. The HFF scheme with SGM is verified through the simulations of steady transcritical flow over a hump and steady flow over an irregular bed. Besides, the effects of the limiter functions, the grid sizes and the Manning roughness coefficients on the simulated results are investigated for the steady transcritical flow problems. Using the laboratory measurements, the scheme is also applied to the dam-break flows: with an adverse slope, with a triangular hump, and with a constriction. Furthermore, the HFF scheme is employed in the simulation of typhoon flood flow with natural-irregular river topography to demonstrate the practical engineering application. The results show good agreements compared with the exact solutions, the experimental data and the field measurements.

High-resolution TVD schemes in finite volume method for hydraulic shock wave modeling

High-resolution total variation diminishing (TVD) schemes in the framework of the finite volume method are presented and evaluated for hydraulic shock wave modeling. Three approximate Riemann solvers, namely the FVS, Roe and Osher schemes, are extended to high-resolution TVD schemes based on the direct MUSCL-Hancock (DMH) slope limiter approach. The TVD schemes are then used to develop numerical models to compute water depth and velocity. The numerical models developed are then verified through simulations of the dam-break flows, the oblique hydraulic jump, and the shock-on-shock interaction. The numerical models with TVD schemes are capable of capturing discontinuous shock waves without any spurious oscillation. A comparison of numerical efficiency shows that the Osher-DMH scheme coupled with van Leer limiter performs the best among the proposed TVD schemes. Applications of the Osher-DMH scheme to flows of partial dam-break experiments have shown that the simulated water depths agree well with the measured data.

Simulation of Hydraulic Shock Waves by Hybrid Flux-Splitting Schemes in Finite Volume Method

In the framework of the finite volume method, a robust and easily implemented hybrid flux-splitting finite-volume (HFF) scheme is proposed for simulating hydraulic shock waves in shallow water flows. The hybrid flux-splitting algorithm without Jacobian matrix operation is established by applying the advection upstream splitting method to estimate the cell-interface fluxes. The scheme is extended to be second-order accurate in space and time using the predictor-corrector approach with monotonic upstream scheme for conservation laws. The proposed HFF scheme and its second-order extension are verified through simulations of the 1D idealized dam-break problem, the 2D oblique hydraulic shock-wave problem, and the 2D dam-break experiments with channel contraction as well as wet/dry beds. Comparisons of the HFF and several well-known first-order upwind schemes are made to evaluate numerical performances. It is demonstrated that the HFF scheme captures the discontinuities accurately and produces no entropy-violating solutions. The HFF scheme and its second-order extension are proven to achieve the numerical benefits combining the efficiency of flux-vector splitting scheme and the accuracy of flux-difference splitting scheme for the simulation of hydraulic shock waves.

Performance of high‐resolution TVD schemes for 1D dam‐break simulations

Abstract The performance of high‐resolution total variation diminishing (TVD) schemes for simulating dam‐break problems are presented and evaluated. Three robust and reliable first‐order upwind schemes, namely FVS, Roe and HLLE schemes, are extended to six second‐order TVD schemes using two different approaches, the Sweby flux limiter approach and the direct MUSCL‐Hancock slope limiter. For idealized dam‐break flows, comparisons of the simulated results with the exact solutions show that the flux vector splitting (FVS) scheme coupled with the direct MUSCL‐Hancock (DMH) slope limiter approach has the best numerical performance among the presented schemes. Application of the FVS‐DMH scheme to a dam‐break experiment with sloping dry bed shows that the simulated water depths agree well with the measured.

CHARACTERISTICS OF HYDRAULIC SHOCK WAVES IN AN INCLINED CHUTE CONTRACTION - EXPERIMENTS

This paper presents the experimental results of the characteristics of hydraulic shock waves in an inclined chute contraction with consideration of the effects of sidewall deflection angle φ, bottom inclination angle θ and approach Froude number Fr 0 . Seventeen runs of laboratory experiments were conducted in the range of 27.45° ≤φ ≤ 40.17°, 6.22° ≤ θ ≤ 25.38° and 1.04 ≤ Fr 0 ≤ 3.51. Based on the experimental data, three empirical dimensionless relations for the shock angle, maximum shockwave height, and corresponding position of maximum shockwave were obtained by regression analyses, respectively. These empirical relations would be useful for hydraulic engineers in designing chute contraction structures.

Flood threshold value for bridge scour prediction and warning

In Taiwan, owing to deep slope, frequency of extreme rail fall, typhoon attacking and earthquake impacts, huge amount of sediment would generate from mountain area and flow with flood toward downstream river. Then, the bridge safety issue is serious discussed during flood event. Therefore, bridge-scour problems have attracted considerable attention in Taiwan, spatially in Zhuo-shui River. In this study, the effects of bend and contraction scours could be neglected because of the river reach near the bridge is roughly straight and the channel width are substantially larger than the pier diameters, respectively. In addition, according to the river bathymetry survey, the bed elevation is generally steady around the bridge area. Therefore, the total scour depth of monitoring bridge is dominated by the local scour. It indicates that a two-dimensional numerical model is adopted to simulate flow field and water depth without sediment transport calculation for collected local scour depth formulas. An appropriate turbulence model, K-epsilon (k-e) turbulence model, is the most common model used in Computational Fluid Dynamics (CFD) to simulate mean flow characteristics for turbulent flow conditions, spatially near bridge piers. Herein, the adapted empirical equations have been validated experimentally; using return-period hydrograph events, and they can give satisfactory simulation results. Then, flood threshold value for bridge scour prediction and warning can be established in the future. In the present study, based on the experimental results, those empirical equations will be suggested to compute the local scour depth evolution under unsteady flow caused by rapid changes of flow depth and velocity in field.