A fifth order WENO scheme for numerical simulation of shallow granular two-phase flow model

Omar Rabbani,Saqib Zia,Asad Rehman

doi:10.1177/16878140211045240

Abstract

In this article, a weighted essentially non-oscillatory (WENO) scheme is implemented to simulate two-phase shallow granular flow (TPSF) model. The flow is assumed to be incompressible and it is regarded as shallow layer of granular and liquid material. The mathematical model consists of two phases, that is, solid and liquid. Each phase has its continuity and momentum equation. The presence of the equations are coupled together involving the derivatives of unknowns which make it more challenging to solve. An efficient numerical technique is needed to tackle the numerical complexities. Our main intrigue is the numerical approximation of the above-mentioned solid-liquid model. The weighted essentially non-oscillatory (WENO) scheme of order 5 is utilized to handle the shock waves and contact discontinuities appear in the solution. The results are compared with the results already available in the literature by conservation element and solution element (CESE) scheme. It is observed the WENO scheme produces less errors as compared to CESE scheme and also effectively handle the shocks.

Highlights

We investigate depth-averaged two-phase shallow granular flow (TPSF) model for gravity-driven mixtures of solid and liquid
The development of a finite volume weighted essentially non-oscillatory (WENO) numerical technique for numerical approximation of two-phase flow model is presented
An incompressible two-phase shallow granular flow model is numerically investigated by fifth order WENO scheme

Summary

Introduction

We investigate depth-averaged two-phase shallow granular flow (TPSF) model for gravity-driven mixtures of solid and liquid. The proposed numerical scheme is more simple and fundamentally distinctive than the computational methods that were devised to solve the mentioned model in the articles.[14,15] Initially in 1987, Harten and Osher[30] introduced the finite volume essentially non-oscillatory (ENO) numerical scheme. This scheme obtained arbitrarily high order accuracy in smooth regions, resolved the steep gradients efficiently and did not produce spurious oscillations in the vicinity of sharp gradients. In compact form the system of equations are expressed as[21]

Introduction to fifth order WENO Scheme

Dxi ð xi

Conclusions