Weighted Average Flux Research Articles

Exact Riemann Solution and Weighted Average Flux Method for Power-law Channel Section

The aim of this work is to extend the Weighted Averaged Flux (WAF) method by E.F. Toro from rectangular to generic regular cross sections, in order to represent the geometric effects of the cross-section on wave propagation along channels. Assuming a power-law variation of the channel width, the 1D conservative shallow water equations, their characteristic form and the shock propagation equations are presented. The exact Riemann solver is derived and is applied to the dam-break problem in valleys with different shape in order to test its efficiency and to check the accuracy order of solutions obtained by approximating the real cross-section with an equivalent rectangle. The WAF method is extended to a power-law channel section and is used to solve the 1D proposed equations taking into account all the source terms that are incorporated into the local Riemann problem. A code based on this method has been developed and results of numerical applications to a Venturi channel and to the attenuation of waves are presented, in order to verify, for well known situations, how accurately the source terms are represented. The code is applied to reproduce one of Brock's experiments (1969; 1970) on roll waves generation in a rectangular channel and results are compared with those obtained with a Godunov-type code developed at CEMAGREF, which is based on a Roe scheme with a later average evaluation of source terms.

An efficient solver for nearshore flows based on the WAF method

We describe an efficient and robust flow solver for the integration of the classic Nonlinear Shallow Water Equations (NSWE) on a beach of arbitrary topography. On the basis of the ‘shock capturing’ Weighted Average Flux (WAF) method, a numerical code is implemented which employs a space-splitting technique to integrate the NSWE over a horizontally two-dimensional domain (2DH). Special care is put in handling the moving shoreline and a new, efficient formulation of the shoreline boundary conditions is presented. This is based on the hydrodynamic equivalent of the cavitation condition used in gas dynamics. Model capabilities are illustrated by means of a number of tests (both 1DH and 2DH). These reveal that, though primarily conceived for modelling nearshore hydrodynamics, the solver can be successfully adopted even for studying the run-up of large tsunami waves.

Three-dimensional shock wave reflection over a corner of two intersecting wedges

This paper presents an experimental and numerical investigation of three-dimensional shock wave reflections over a corner of two wedges intersecting perpendicularly in a shock tube. Experiments were conducted in a \(100\,{\rm mm} \times 180\,{\rm mm}\) diaphragmless shock tube equipped with double-exposure diffuse holographic interferometry in which the time interval between the first and second exposure was set to be \(1 \,{\rm \mu s}\). This arrangement clearly visualized complex configurations of three-dimensional shock wave reflections. A numerical study was also carried out for interpreting these holographic interferometric observations by using the Weighted Average Flux (WAF) method to solve the three-dimensional unsteady compressible Euler equations. It was found that along the line of the intersection of these two wedges, two Mach stems intersected each other resulting in the formation of a Mach stem which leaned forward.

A unified Riemann-problem-based extension of the Warming-Beam and Lax-Wendroff schemes

A unified, Riemann-problem-based extension of the Warming-Beam and Lax-Wendroff schemes for nonlinear systems of hyperbolic conservation laws is presented. This is achieved by employing the weighted average flux (WAF) approach with the numerical flux given by a space-time integral average of the solution of local Riemann problems. The local wave structure dictates switching between schemes automatically with no need for special conservative switching operators. The resulting Godunov-type scheme has CFL-number 2 stability restriction and when used in conjunction with Riemann-solver adaption procedures is significantly more efficient than Riemann-problem-based methods currently in use.

Direct Riemann solvers for the time-dependent Euler equations

Approaches for finding direct, approximate solutions to the Riemann problem are presented. These result in three approximate Riemann solvers. Here we discuss the time-dependent Euler equations but the ideas are applicable to other systems. The approximate solvers are (i) assessed on local Riemann problems with exact solutions and (ii) used in conjunction with the Weighted Average Flux (WAF) method to solve the two-dimensional Euler equations numerically. The resulting numerical technique is assessed on a shock reflection problem. Comparison with experimental observation is carried out.

Three-Dimensional Shock Wave Reflection over a Corner of Two Intersecting Wedges.

This paper describes the three-dimensional shock wave reflection over a corner of two intersecting wedges. A numerical study was carried out to investigate shock wave reflection in three-dimensional flow fields. The Weighted Average Flux (WAF) method was applied to solve three-dimensional unsteady compressible Euler equations. The results were compared with a numerical simulation using the Harten-Yee TVD finite difference scheme. The WAF method gave high resolution and was found to be useful in simulating this three-dimensional shock wave reflection without use of fine computational grids. The three-dimensional shock wave reflection was also analyzed with the shock polar method valid for two-dimensional obligue shock waves. The results obtained from the shock polar analysis agreed with the numerical simulations. A shock tube experiment was also conducted using double-exposure holographic interferometry. In order to obtain three-dimensional images of shock waves, the time interval between the first and the second exposures was set to be two or three micro seconds. This enabled clear observation of the structures of the three-dimensional shock wave reflection. Good agreement between this flow visualization and the results of the numerical simulations was observed.